k 12 mathematics standards

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Washington State K-12 Mathematics Learning Standards

Prepared by Charles A. Dana Center for Science and Mathematics Education The University of Texas at Austin Mathematics Teaching and Learning Office of Superintendent of Public Instruction

Dr. Terry Bergeson Superintendent of Public Instruction

Catherine Davidson, Ed. D.

Chief of Staff

Lexie Domaradzki Assistant Superintendent, Teaching and Learning

July 2008

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SUPERINTENDENT OF PUBLIC INSTRUCTION

DR. TERRY BERGESON OLD CAPITOL BUILDING • PO BOX 47200 • OLYMPIA WA 98504-7200 • http://www.k12.wa.us

It is with great pride that I, Dr. Terry Bergeson, State Superintendent of Public Instruction officially adopt the revised K-12 Mathematics Standards as the new essential academic learning requirements for the state of Washington.

Teams of national experts and talented Washington state mathematics educators, curriculum directors, and mathematicians have worked tirelessly since October 2007 to develop the best set of K-12 mathematics standards for our state. Since the announcement of the first draft in December 2007, these standards have received input from thousands of educators and stakeholders throughout the state, including in-depth input from the State Board of Education’s Math Panel.

As per 2008 Senate Bill 6534, the State Board of Education (SBE) contracted with an independent contractor, Strategic Teaching to conduct a final review and analysis of the K-12 standards. On April 28, 2008, the SBE voted to approve the K-8 standards for adoption by the Office of Superintendent of Public Instruction (OSPI). The 9-12 standards were approved for OSPI adoption on July 30, 2008.

These standards set more challenging and rigorous expectations at each grade level. In addition, they provide more clarity to support all students in developing and sharpening their mathematical skills, deepening their understanding of concepts and processes, and utilizing their problem-solving, reasoning and communication abilities. For students to develop this deeper level of understanding, their knowledge must be connected not only to a variety of ideas and skills across topic areas and grade levels in mathematics, but also to other subjects taught in school and to situations outside the classroom.

The revised K-12 Mathematics Standards are the first step in improving the mathematics learning of all students in Washington and are now at the vanguard of the nation’s mathematics education improvement movement. The standards will strongly support teachers as they prepare the state’s young people for graduation, college and the workforce.

K-12 Mathematics Standards adopted on this 1st day of August, 2008 by

Sincerely,

Dr. Terry Bergeson, Superintendent Office of Superintendent of Public Instruction

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July 2008Washington State K–12 Mathematics Standards

Table of Contents

Introduction .................................................................................................................................i

Kindergarten ..............................................................................................................................1K.1. Core Content: Whole numbers ............................................................................................................3K.2. Core Content: Patterns and operations ...............................................................................................5K.3. Core Content: Objects and their locations ..........................................................................................6K.4. Additional Key Content ........................................................................................................................7K.5. Core Processes: Reasoning, problem solving, and communication ...................................................8

Grade 1 .......................................................................................................................................91.1. Core Content: Whole number relationships ...................................................................................... 111.2. Core Content: Addition and subtraction .............................................................................................141.3. Core Content: Geometric attributes ...................................................................................................171.4. Core Content: Concepts of measurement .........................................................................................181.5. Additional Key Content ......................................................................................................................191.6. Core Processes: Reasoning, problem solving, and communication .................................................20

Grade 2 .....................................................................................................................................212.1. Core Content: Place value and the base ten system .........................................................................232.2. Core Content: Addition and subtraction .............................................................................................242.3. Core Content: Measurement ..............................................................................................................262.4. Additional Key Content ......................................................................................................................272.5. Core Processes: Reasoning, problem solving, and communication .................................................29

Grade 3 .....................................................................................................................................313.1. Core Content: Addition, subtraction, and place value ........................................................................333.2. Core Content: Concepts of multiplication and division .....................................................................343.3. Core Content: Fraction concepts .......................................................................................................383.4. Core Content: Geometry ...................................................................................................................403.5. Additional Key Content ......................................................................................................................413.6. Core Processes: Reasoning, problem solving, and communication .................................................42

Grade 4 .....................................................................................................................................434.1. Core Content: Multi-digit multiplication ...............................................................................................454.2. Core Content: Fractions, decimals, and mixed numbers ..................................................................484.3. Core Content: Concept of area .........................................................................................................514.4. Additional Key Content .......................................................................................................................534.5. Core Processes: Reasoning, problem solving, and communication .................................................55

Grade 5 .....................................................................................................................................575.1. Core Content: Multi-digit division ........................................................................................................595.2. Core Content: Addition and subtraction of fractions and decimals ....................................................615.3. Core Content: Triangles and quadrilaterals .......................................................................................635.4. Core Content: Representations of algebraic relationships ................................................................655.5. Additional Key Content ......................................................................................................................675.6. Core Processes: Reasoning, problem solving, and communication .................................................68


Grade 6 .....................................................................................................................................696.1. Core Content: Multiplication and division of fractions and decimals ..................................................716.2. Core Content: Mathematical expressions and equations .................................................................746.3. Core Content: Ratios, rates, and percents .........................................................................................766.4. Core Content: Two- and three-dimensional figures ...........................................................................786.5. Additional Key Content .......................................................................................................................806.6. Core Processes: Reasoning, problem solving, and communication .................................................81

Grade 7 .....................................................................................................................................837.1. Core Content: Rational numbers and linear equations ......................................................................857.2. Core Content: Proportionality and similarity .......................................................................................887.3. Core Content: Surface area and volume ...........................................................................................927.4. Core Content: Probability and data ....................................................................................................937.5. Additional Key Content ......................................................................................................................957.6. Core Processes: Reasoning, problem solving, and communication .................................................96

Grade 8 .....................................................................................................................................978.1. Core Content: Linear functions and equations ..................................................................................998.2. Core Content: Properties of geometric figures ................................................................................1018.3. Core Content: Summary and analysis of data sets ........................................................................1038.4. Additional Key Content .....................................................................................................................1078.5. Core Processes: Reasoning, problem solving, and communication ...............................................109

Algebra 1 ................................................................................................................................ 111A1.1. Core Content: Solving problems ................................................................................................... 113A1.2. Core Content: Numbers, expressions, and operations ................................................................ 116A1.3. Core Content: Characteristics and behaviors of functions ............................................................120A1.4. Core Content: Linear functions, equations, and inequalities .........................................................123A1.5. Core Content: Quadratic functions and equations ........................................................................126A1.6. Core Content: Data and distributions.............................................................................................128A1.7. Additional Key Content ..................................................................................................................131A1.8. Core Processes: Reasoning, problem solving, and communication .............................................132

Geometry ................................................................................................................................133G.1. Core Content: Logical arguments and proofs ................................................................................135G.2. Core Content: Lines and angles .....................................................................................................137G.3. Core Content: Two- and three-dimensional figures ........................................................................138G.4. Core Content: Geometry in the coordinate plane ...........................................................................143G.5. Core Content: Geometric transformations .....................................................................................145G.6. Additional Key Content ...................................................................................................................146G.7. Core Processes: Reasoning, problem solving, and communication ..............................................149

Algebra 2 ................................................................................................................................151A2.1. Core Content: Solving problems ....................................................................................................153A2.2. Core Content: Numbers, expressions, and operations .................................................................157A2.3. Core Content: Quadratic functions and equations ........................................................................159A2.4. Core Content: Exponential and logarithmic functions and equations ...........................................161A2.5. Core Content: Additional functions and equations ........................................................................163A2.6. Core Content: Probability, data, and distributions ........................................................................ 165A2.7. Additional Key Content ...................................................................................................................167A2.8. Core Processes: Reasoning, problem solving, and communication .............................................168


Mathematics 1 ........................................................................................................................171M1.1. Core Content: Solving problems ..................................................................................................173M1.2. Core Content: Characteristics and behaviors of functions ...........................................................176M1.3. Core Content: Linear functions, equations, and relationships ..................................................... 179M1.4. Core Content: Proportionality, similarity, and geometric reasoning ..............................................183M1.5. Core Content: Data and distributions ............................................................................................185M1.6. Core Content: Numbers, expressions, and operations .................................................................187M1.7. Additional Key Content ..................................................................................................................190M1.8. Core Processes: Reasoning, problem solving, and communication ............................................192

Mathematics 2 ........................................................................................................................193M2.1. Core Content: Modeling situations and solving problems ............................................................195M2.2. Core Content: Quadratic functions, equations, and relationships ................................................198M2.3. Core Content: Conjectures and proofs ........................................................................................202M2.4. Core Content: Probability ............................................................................................................ 208M2.5. Additional Key Content ..................................................................................................................209M2.6. Core Processes: Reasoning, problem solving, and communication ............................................ 211

Mathematics 3 ........................................................................................................................213M3.1. Core Content: Solving problems ...................................................................................................215M3.2. Core Content: Transformations and functions ..............................................................................218M3.3. Core Content: Functions and modeling ........................................................................................220M3.4. Core Content: Quantifying variability .............................................................................................223M3.5. Core Content: Three-dimensional geometry ................................................................................225M3.6. Core Content: Algebraic properties ...............................................................................................227M3.7. Additional Key Content ..................................................................................................................229M3.8. Core Processes: Reasoning, problem solving, and communication ............................................231

July 2008Washington State K–12 Mathematics Standards i

Introduction

OverviewThe Washington State K–12 Mathematics Standards outline the mathematics learning expectations for all students in Washington. These standards describe the mathematics content, procedures, applications, and processes that students are expected to learn. The topics and mathematical strands represented across grades K–12 constitute a mathematically complete program that includes the study of numbers, operations, geometry, measurement, algebra, data analysis, and important mathematical processes.

Organization of the standardsThe Washington State K–12 Mathematics Standards are organized by grade level for grades K–8 and by course for grades 9–12, with each grade/course consisting of three elements: Core Content, Additional Key Content, and Core Processes. Each of these elements contains Performance Expectations and Explanatory Comments and Examples.

Core Content areas describe the major mathematical focuses of each grade level or course. A limited number of priorities for each grade level in grades K–8 and for each high school course are identified, so teachers know which topics call for the most time and emphasis. Each priority area includes a descriptive paragraph that highlights the mathematics addressed and its role in a student’s overall mathematics learning.

Additional Key Content contains important expectations that do not warrant the same amount of instructional time as the Core Content areas. These are expectations that might extend a previously learned skill, plant a seed for future development, or address a focused topic, such as scientific notation. Although they need less classroom time, these expectations are important, are expected to be taught, and may be assessed as part of Washington State’s assessment system. The content in this section allows students to build a coherent knowledge of mathematics from year to year.

Core Processes include expectations that address reasoning, problem solving, and communication. While these processes are incorporated throughout other content expectations, they are presented in this section to clearly describe the breadth and scope of what is expected in each grade or course. In Core Processes, at least two rich problems that cut across Core or Key Content areas are included as examples for each grade or course. These problems illustrate the types and breadth of problems that could be used in the classroom.

Performance Expectations, in keeping with the accepted definition of standards, describe what students should know and be able to do at each grade level. These statements are the core of the document. They are designed to provide clear guidance to teachers about the mathematics that is to be taught and learned.

Explanatory Comments and Examples accompany most of the expectations. These are not technically performance expectations. However, taken together with the Performance Expectations, they provide a full context and clear understanding of the expectation.

The comments expand upon the meaning of the expectations. Explanatory text might clarify the parameters regarding the type or size of numbers, provide more information about student expectations regarding mathematical understanding, or give expanded detail to mathematical definitions, laws, principles, and forms included in the expectation.

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The example problems include those that are typical of the problems students should do, those that illustrate various types of problems associated with a particular performance expectation, and those that illustrate the expected limits of difficulty for problems related to a performance expectation. Teachers are not expected to teach these particular examples or to limit what they teach to these examples. Teachers and quality instructional materials will incorporate many different types of examples that support the teaching of the content described in any expectation.

In some instances, comments related to pedagogy are included in the standards as familiar illustrations to the teacher. Teachers are not expected to use these particular teaching methods or to limit the methods they use to the methods included in the document. These, too, are illustrative, showing one way an expectation might be taught.

Although, technically, the performance expectations set the requirements for Washington students, people will consider the entire document as the Washington mathematics standards. Thus, the term standards, as used here, refers to the complete set of Performance Expectations, Explanatory Comments and Examples, Core Content, Additional Key Content, and Core Processes. Making sense of the standards from any grade level or course calls for understanding the interplay of Core Content, Additional Key Content, and Core Processes for that grade or course.

What standards are not Performance expectations do not describe how the mathematics will be taught. Decisions about instructional methods and materials are left to professional teachers who are knowledgeable about the mathematics being taught and about the needs of their students.

The standards are not comprehensive. They do not describe everything that could be taught in a classroom. Teachers may choose to go beyond what is included in this document to provide related or supporting content. They should teach beyond the standards to those students ready for additional challenges. Standards related to number skills, in particular, should be viewed as a floor—minimum expectations—and not a ceiling. A student who can order and compare numbers to 120 should be given every opportunity to apply these concepts to larger numbers.

The standards are not test specifications. Excessive detail, such as the size of numbers that can be tested and the conditions for assessment, clouds the clarity and usability of a standards document, generally, and a performance expectation, specifically. For example, it is sufficient to say “Identify, describe, and classify triangles by angle measure and number of congruent sides,” without specifying that acute, right, and obtuse are types of triangles classified by their angle size and that scalene, isosceles, and equilateral are types of triangles classified by their side length. Sometimes this type of information is included in the comments section, but generally this level of detail is left to other documents.

What about strands?Many states’ standards are organized around mathematical content strands—generally some combination of numbers, operations, geometry, measurement, algebra, and data/statistics. However, the Washington State K–12 Mathematics Standards are organized according to the priorities described as Core Content rather than being organized in strands. Nevertheless, it is still useful to know what content strands are addressed in particular Core Content and Additional Key Content areas. Thus, mathematics content strands are identified in parentheses at the beginning of each Core Content or Additional Key Content area. Five content strands have been identified for this purpose: Numbers, Operations, Geometry/Measurement, Algebra, and Data/Statistics/Probability. For each of these strands, a separate K–12 strand

July 2008Washington State K–12 Mathematics Standards iii

document allows teachers and other readers to track the development of knowledge and skills across grades and courses. An additional strand document on the Core Processes tracks the development of reasoning, problem solving, and communication across grades K–12.

A well-balanced mathematics program for all studentsAn effective mathematics program balances three important components of mathematics—conceptual understanding (making sense of mathematics), procedural proficiency (skills, facts, and procedures), and problem solving and mathematical processes (using mathematics to reason, think, and apply mathematical knowledge). These standards make clear the importance of all three of these components, purposefully interwoven to support students’ development as increasingly sophisticated mathematical thinkers. The standards are written to support the development of students so that they know and understand mathematics.

Conceptual understanding (making sense of mathematics)Students who understand a concept are able to identify examples as well as non-examples, describe the concept (for example, with words, symbols, drawings, tables, or models), provide a definition of the concept, and use the concept in different ways. Conceptual understanding is woven throughout these standards. Expectations with verbs like demonstrate, describe, represent, connect, and justify, for example, ask students to show their understanding. Furthermore, expectations addressing both procedures and applications often ask students to connect their conceptual understanding to the procedures being learned or problems being solved.

Procedural proficiency (skills, facts, and procedures)Learning basic facts is important for developing mathematical understanding. In these standards, clear expectations address students’ knowledge of basic facts. The use of the term basic facts typically encompasses addition and multiplication facts up to and including 10 + 10 and 10 x 10 and their related subtraction and division facts. In these standards, students are expected to “quickly recall” basic facts. “Quickly recall” means that the student has ready and effective access to facts without having to go through a development process or strategy, such as counting up or drawing a picture, every time he or she needs to know a fact. Simply put, students need to know their basic facts.

Building on a sound conceptual understanding of addition, subtraction, multiplication, and division, Washington’s standards include a specific discussion of students’ need to understand and use the standard algorithms generally seen in the United States to add, subtract, multiply, and divide whole numbers. There are other possible algorithms students might also use to perform these operations and some teachers may find value in students learning multiple algorithms to enhance understanding.

Algorithms are step-by-step mathematical procedures that, if followed correctly, always produce a correct solution or answer. Generalized procedures are used throughout mathematics, such as in drawing geometric constructions or going through the steps involved in solving an algebraic equation. Students should come to understand that mathematical procedures are a useful and important part of mathematics.

The term fluency is used in these standards to describe the expected level and depth of a student’s knowledge of a computational procedure. For the purposes of these standards, a student is considered fluent when the procedure can be performed immediately and accurately. Also, when fluent, the student knows when it is appropriate to use a particular procedure in a problem or situation. A student who is fluent in a procedure has a tool that can be applied reflexively and doesn’t distract from the task of solving the problem at hand. The procedure is stored in long-term memory, leaving working memory available to focus on the problem.

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Problem solving and mathematical processes (reasoning and thinking to apply mathematical content)Mathematical processes, including reasoning, problem solving, and communication, are essential in a well-balanced mathematics program. Students must be able to reason, solve problems, and communicate their understanding in effective ways. While it is impossible to completely separate processes and content, the standards’ explicit description of processes at each grade level calls attention to their importance within a well-balanced mathematics program. Some common language is used to describe the Core Processes across the grades and within grade bands (K–2, 3–5, 6–8, and 9–12). The problems students will address, as well as the language and symbolism they will use to communicate their mathematical understanding, become more sophisticated from grade to grade. These shifts across the grades reflect the increasing complexity of content and the increasing rigor as students deal with more challenging problems, much in the same way that reading skills develop from grade to grade with increasingly complex reading material.

TechnologyThe role of technology in learning mathematics is a complex issue, because of the ever-changing capabilities of technological tools, differing beliefs in the contributions of technology to a student’s education, and equitable student access to tools. However, one principle remains constant: The focus of mathematics instruction should always be on the mathematics to be learned and on helping students learn that mathematics.

Technology should be used when it supports the mathematics to be learned, and technology should not be used when it might interfere with learning.

Calculators and other technological tools, such as computer algebra systems, dynamic geometry software, applets, spreadsheets, and interactive presentation devices are an important part of today’s classroom. But the use of technology cannot replace conceptual understanding, computational fluency, or problem-solving skills.

Washington’s standards make clear that some performance expectations are to be done without the aid of technology. Elementary students are expected to know facts and basic computational procedures without using a calculator. At the secondary level, students should compute with polynomials, solve equations, sketch simple graphs, and perform some constructions without the use of technology. Students should continue to use previously learned facts and skills in subsequent grade levels to maintain their fluency without the assistance of a calculator.

At the elementary level, calculators are less useful than they will be in later grades. The core of elementary school—number sense and computational fluency—does not require a calculator. However, this is not to say that students couldn’t use calculators to investigate mathematical situations and to solve problems involving complicated numbers, lots of numbers, or data sets.

As middle school students deal with increasingly complex statistical data and represent proportional relationships with graphs and tables, a calculator or technological tool with these functions can be useful for representing relationships in multiple ways. At the high school level, graphing calculators become valuable tools as all students tackle the challenges of algebra and geometry to prepare for a range of postsecondary options in a technological world. Graphing calculators and spreadsheets allow students to explore and solve problems with classes of functions in ways that were previously impossible.

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While the majority of performance expectations describe skills and knowledge that a student could demonstrate without technology, learning when it is helpful to use these tools and when it is cumbersome is part of becoming mathematically literate. When students become dependent upon technology to solve basic math problems, the focus of mathematics instruction to help students learn mathematics has failed.

Connecting to the Washington Essential Academic Learning Requirements (EALRs) and Grade Level Expectations (GLEs) The new Washington State K–12 Mathematics Standards continue Washington’s longstanding commitment to teaching mathematics content and mathematical thinking. The new standards replace the former Essential Academic Learning Requirements (EALRs) and Grade Level Expectations (GLEs). The former mathematics EALRs, listed below, represent threads in the mathematical content, reasoning, problem solving, and communication that are reflected in these new standards.

EALR 1: The student understands and applies the concepts and procedures of mathematics.

EALR 2: The student uses mathematics to define and solve problems.

EALR 3: The student uses mathematical reasoning.

EALR 4: The student communicates knowledge and understanding in both everyday and mathematical language.

EALR 5: The student understands how mathematical ideas connect within mathematics, to other subjects.

System-wide standards implementation activitiesThese mathematics standards represent an important step in ramping up mathematics teaching and learning in the state. The standards provide a critical foundation, but are only the first step. Their success will depend on the implementation efforts that match many of the activities outlined in Washington’s Joint Mathematics Action Plan. This includes attention to:

Aligning the Washington Assessment for Student Learning to these standards;•

Identifying mathematics curriculum and instructional support materials;•

Providing systematic professional development so that instruction aligns with the standards;•

Developing online availability of the standards in various forms and formats, with additional • example problems, classroom activities, and possible lessons embedded.

As with any comprehensive initiative, fully implementing these standards will not occur overnight. This implementation process will take time, as teachers become more familiar with the standards and as students enter each grade having learned more of the standards from previous grades. There is always a tension of balancing the need to raise the bar with the reality of how much change is possible, and how quickly this change can be implemented in real schools with real teachers and real students.

Change is hard. These standards expect more of students and more of their teachers. Still, if Washington’s students are to be prepared to be competitive and to reach their highest potential, implementing these standards will pay off for years to come.

Kindergarten

July 2008Washington State K–12 Mathematics Standards 3

KindergartenK.1. Core Content: Whole numbers (Numbers, Operations)

Students begin to develop basic notions of numbers and use numbers to think about objects and the world around them. They practice counting objects in sets, and they think about how numbers

are ordered by showing the numbers on the number line. As they put together and take apart simple numbers, students lay the groundwork for learning how to add and subtract. Understanding numbers is perhaps the most central idea in all of mathematics, and if students build and maintain a strong foundation of number sense and number skills, they will be able to succeed with increasingly sophisticated numerical knowledge and skills from year to year.

Performance Expectations Explanatory Comments and Examples

Students are expected to:

K.1.A Rote count by ones forward from 1 to 100 and backward from any number in the range of 10 to 1.

K.1.B Read aloud numerals from 0 to 31. Shown numeral cards in random order from 0 to 31, students respond with the correct name of the numerals. Students also demonstrate that they can distinguish 12 from 21 and 13 from 31—a common challenge for kindergartners.

The choice of 31 corresponds to the common use of calendar activities in kindergarten.

K.1.C Fluently compose and decompose numbers to 5.

Students should be able to state that 5 is made up of 4 and 1, 3 and 2, 2 and 3, or 1 and 4. They should understand that if I have 3, I need 2 more to make 5, or that if I have 4, I need only 1 more to make 5. Students should also be able to recognize the number of missing objects without counting.

The words compose and decompose are used to describe actions that young students learn as they acquire knowledge of small numbers by putting them together and taking them apart. This understanding is a bridge between counting and knowing number combinations. It is how instant recognition of small numbers develops and leads naturally to later understanding of fact families.

Example:

• Here are 5 counters. I will hide some. If you see 2, how many am I hiding?

K.1.D Order numerals from 1 to 10. The student takes numeral cards (1 to 10) that have been shuffled and puts them in correct ascending order.

Kindergarten

July 2008Washington State K–12 Mathematics Standards4



K.1.E Count objects in a set of up to 20, and count out a specific number of up to 20 objects from a larger set.

K.1.F Compare two sets of up to 10 objects each and say whether the number of objects in one set is equal to, greater than, or less than the number of objects in the other set.

K.1.G Locate numbers from 1 to 31 on the number line. Students should be able to do this without having to start counting at 1.

K.1.H Describe a number from 1 to 9 using 5 as a benchmark number.

Students should make observations such as “7 is 2 more than 5” or “4 is 1 less than 5.” This is helpful for mental math and lays the groundwork for using 10 as a benchmark number in later work with base-ten numbers and operations.

Kindergarten


Kindergarten K.2. Core Content: Patterns and operations (Operations, Algebra)

Students learn what it means to add and subtract by joining and separating sets of objects. Working with patterns helps them strengthen this understanding of addition and subtraction and moves them

toward the important development of algebraic thinking. Students study simple repetitive patterns in preparation for increasingly sophisticated patterns that can be represented with algebraic expressions in later grades.



K.2.A Copy, extend, describe, and create simple repetitive patterns.

Students can complete these activities with specified patterns of the type AB, AAB, AABB, ABC, etc.

Examples:

Make a type AB pattern of squares and circles with • one square, one circle, one square, one circle, etc.

Here is a type AAB pattern using colored cubes: red, • red, blue, red, red, blue, red, red. What comes next?

A shape is missing in the type AB pattern below. • What is it?

K.2.B Translate a pattern among sounds, symbols, movements, and physical objects.

Red, red, yellow, red, red, yellow could translate to clap, clap, snap, clap, clap, snap.

Students should be able to translate patterns among all of these representations. However, when they have demonstrated they can do this, they need not use all representations every time.

K.2.C Model addition by joining sets of objects that have 10 or fewer total objects when joined and model subtraction by separating a set of 10 or fewer objects.

Seeing two sets of counters or other objects, the student determines the correct combined total. The student may count the total number of objects in the set or use some other strategy in order to arrive at the sum. The student establishes the total number of counters or objects in a set; then, after some have been removed, the student figures out how many are left.

Examples:

• Get 4 counting chips. Now get 3 counting chips. How many counting chips are there altogether?

• Get 8 counting chips. Take 2 away. How many are left?

K.2.D Describe a situation that involves the actions of joining (addition) or separating (subtraction) using words, pictures, objects, or numbers.

Students can be asked to tell an addition story or a subtraction story.

Kindergarten


KindergartenK.3. Core Content: Objects and their locations (Geometry/Measurement)

Students develop basic ideas related to geometry as they name simple two- and three-dimensional figures and find these shapes around them. They expand their understanding of space and location

by describing where people and objects are. Students sort and match shapes as they begin to develop classification skills that serve them well in both mathematics and reading—matching numbers to sets, shapes to names, patterns to rules, letters to sounds, and so on.



K.3.A Identify, name, and describe circles, triangles, rectangles, squares (as special rectangles), cubes, and spheres.

Students should be encouraged to talk about the characteristics (e.g., round, four-cornered) of the various shapes and to identify these shapes in a variety of contexts regardless of their location, size, and orientation. Having students identify these shapes on the playground, in the classroom, and on clothing develops their ability to generalize the characteristics of each shape.

K.3.B Sort shapes using a sorting rule and explain the sorting rule.

Students could sort shapes according to attributes such as the shape, size, or the number of sides, and explain the sorting rule. Given a selection of shapes, students may be asked to sort them into two piles and then describe the sorting rule. After sorting, a student could say, “I put all the round ones here and all the others there.”

K.3.C Describe the location of one object relative to another object using words such as in, out, over, under, above, below, between, next to, behind, and in front of.

Examples:

Put this pencil under the paper. •

I am between José and Katy. •

Kindergarten


Kindergarten K.4. Additional Key Content (Geometry/Measurement)

Students informally develop early measurement concepts. This is an important precursor to Core Content on measurement in later grades, when students measure objects with tools. Solving

measurement problems connects directly to the student’s world and is a basic component of learning mathematics.



K.4.A Make direct comparisons using measurable attributes such as length, weight, and capacity.

Students should use language such as longer than, shorter than, taller than, heavier than, lighter than, holds more than, or holds less than.

Kindergarten


KindergartenK.5. Core Processes: Reasoning, problem solving, and communication

Students begin to build the understanding that doing mathematics involves solving problems and discussing how they solved them. Problems at this level emphasize counting and activities that

lead to emerging ideas about addition and subtraction. Students begin to develop their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?”



K.5.A Identify the question(s) asked in a problem.

K.5.B Identify the given information that can be used to solve a problem.

K.5.C Recognize when additional information is required to solve a problem.

K.5.D Select from a variety of problem-solving strategies and use one or more strategies to solve a problem.

K.5.E Answer the question(s) asked in a problem.

K.5.F Describe how a problem was solved.

K.5.G Determine whether a solution to a problem is reasonable.

Descriptions of solution processes and explanations can include numbers, words (including mathematical language), pictures, or physical objects. Students should be able to use all of these representations as needed. For a particular solution, students should be able to explain or show their work using at least one of these representations and verify that their answer is reasonable.

Examples:

Grandma went to visit her two grandchildren and • discovered that the gloves they were each wearing had holes in every finger, even the thumbs. She will fix their gloves. How many glove fingers (including thumbs) need to be fixed?

Students are given drinking straws or coffee • stirrers cut to a variety of different lengths: 6″, 5″, 4″, 3″, and 2″. They are to figure out which sets of three lengths, when joined at their ends, will form triangles and which sets of three will not.

Kindergarten


Grade 1


Grade 1 1.1. Core Content: Whole number relationships (Numbers, Operations)

Students continue to work with whole numbers to quantify objects. They consider how numbers relate to one another. As they expand the set of numbers they work with, students start to develop

critical concepts of ones and tens that introduce them to place value in our base ten number system. An understanding of how ones and tens relate to each other allows students to begin adding and subtracting two-digit numbers, where thinking of ten ones as one ten and vice versa is routine. Some students will be ready to work with numbers larger than those identified in the Expectations and should be given every opportunity to do so.



1.1.A Count by ones forward and backward from 1 to 120, starting at any number, and count by twos, fives, and tens to 100.

Research suggests that when students count past 100, they often make errors such as “99, 100, 200” and “109, 110, 120.” However, once a student counts to 120 consistently, it is highly improbable that additional counting errors will be made.

Example:

Start at 113. Count backward. I’ll tell you when • to stop. [Stop when the student has counted backward ten numbers.]

1.1.B Name the number that is one less or one more than any number given verbally up to 120.

1.1.C Read aloud numerals from 0 to 1,000. The patterns in the base ten number system become clearer to students when they count in the hundreds. Therefore, learning the names of three-digit numbers supports the learning of more difficult two-digit numbers (such as numbers in the teens and numbers ending in 0 or 1).

1.1.D Order objects or events using ordinal numbers. Students use ordinal numbers to describe positions through the twentieth.

Example:

John is fourth in line.•

1.1.E Write, compare, and order numbers to 120. Students arrange numbers in lists or talk about the relationships among numbers using the words equal to, greater than, less than, greatest, and least.

Example:

Write the numbers 27, 2, 111, and 35 from least • to greatest.

Students might also describe which of two numbers is closer to a given number. This is part of developing an understanding of the relative value of numbers.

Grade 1




1.1.F Fluently compose and decompose numbers to 10.

Students put together and take apart whole numbers as a precursor to addition and subtraction.

Examples:

Ten is 2 + 5 + 1 + 1 + 1.•

Eight is five and three.•

Here are twelve coins. I will hide some. If you • see three, how many am I hiding? [This example demonstrates how students might be encouraged to go beyond the expectation.]

1.1.G Group numbers into tens and ones in more than one way.

Students demonstrate that the value of a number remains the same regardless of how it is grouped. Grouping of numbers lays a foundation for future work with addition and subtraction of two-digit numbers, where renaming may be necessary.

For example, twenty-seven objects can be grouped as 2 tens and 7 ones, regrouped as 1 ten and 17 ones, and regrouped again as 27 ones. The total (27) remains constant.

27

27 = 10 + 10 + 7

27

27 = 10 + 17

can beshown as

can beshown as

27 can beshown as

1.1.H Group and count objects by tens, fives, and twos.

Given 23 objects, the student will count them by tens as 10, 20, 21, 22, 23; by fives as 5, 10, 15, 20, 21, 22, 23; and by twos as 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 23.

Grade 1




1.1.I Classify a number as odd or even and demonstrate that it is odd or even.

Students use words, objects, or pictures to demonstrate that a given number is odd or even.

Examples:

13 is odd because 13 counters cannot be • regrouped into two equal piles.

20 is even because every counter in this set of • 20 counters can be paired with another counter in the set.

Grade 1


Grade 1 1.2. Core Content: Addition and subtraction (Operations, Algebra)

Students learn how to add and subtract, when to add and subtract, and how addition and subtraction relate to each other. Understanding that addition and subtraction undo each other is an

important part of learning to use these operations efficiently and accurately. Students notice patterns involving addition and subtraction, and they work with other types of patterns as they learn to make generalizations about what they observe.



1.2.A Connect physical and pictorial representations to addition and subtraction equations.

The intention of the standard is for students to understand that mathematical equations represent situations. Simple student responses are adequate.

Combining a set of 3 objects and a set of 5 objects to get a set of 8 objects can be represented by the equation 3 + 5 = 8. The equation 2 + 6 = 8 could be represented by drawing a set of 2 cats and a set of 6 cats making a set of 8 cats. The equation 9 – 5 = 4 could be represented by taking 5 objects away from a set of 9 objects.

1.2.B Use the equal sign (=) and the word equals to indicate that two expressions are equivalent.

Students need to understand that equality means is the same as. This idea is critical if students are to avoid common pitfalls in later work with numbers and operations, where they may otherwise fall into habits of thinking that the answer always follows the equal sign.

Examples:

7 = 8 – 1•

5 + 3 equals 10 – 2•

1.2.C Represent addition and subtraction on the number line.

Examples:

4 + 3 = 7 •

0 1 2 3 4 5 6 7 8

7 – 4 = 3•

0 1 2 3 4 5 6 7 8

Grade 1




1.2.D Demonstrate the inverse relationship between addition and subtraction by undoing an addition problem with subtraction and vice versa.

The relationship between addition and subtraction is an important part of developing algebraic thinking. Students can demonstrate this relationship using physical models, diagrams, numbers, or acting- out situations.

Examples:

3 + 5 = 8, so 8 – 3 = 5•

Annie had ten marbles, but she lost three. How • many marbles does she have? Joe found her marbles and gave them back to her. Now how many does she have?

1.2.E Add three or more one-digit numbers using the commutative and associative properties of addition.

Examples:

3 + 5 + 5 = 3 + 10 • (Associativity allows us to add the last two addends first.)

(5 + 3) + 5 = 5 + (5 + 3) = (5 +5) + 3 = 13 • (Commutativity and associativity allow us to reorder addends.)

This concept can be extended to address a problem like 3 + + 2 = 9, which can be rewritten as 5 + = 9.

1.2.F Apply and explain strategies to compute addition facts and related subtraction facts for sums to 18.

Strategies for addition include counting on, but students should be able to move beyond counting on to use other strategies, such as making 10, using doubles or near doubles, etc.

Subtraction strategies include counting back, relating the problem to addition, etc.

1.2.G Quickly recall addition facts and related subtraction facts for sums equal to 10.

Adding and subtracting zero are included.

1.2.H Solve and create word problems that match addition or subtraction equations.

Students should be able to represent addition and subtraction sentences with an appropriate situation, using objects, pictures, or words. This standard is about helping students connect symbolic representations to situations. While some students may create word problems that are detailed or lengthy, this is not necessary to meet the expectation. Just as we want students to be able to translate 5 boys and 3 girls sitting at a table into 5 + 3 = 8, we want students to look at an expression like 7 – 4 = 3 and connect it to a situation or problem using objects, pictures, or words.

Grade 1




1.2.H Cont. Example:

For the equation 7 + ? = 10, a possible story • might be:

Jeff had 7 marbles in his pocket and some marbles in his drawer. He had 10 marbles altogether. How many marbles did he have in his drawer? Use pictures, words, or objects to show your answer.

1.2.I Recognize, extend, and create number patterns. Example:

Extend the simple addition patterns below and tell • how you decided what numbers come next:

1, 3, 5, 7, . . .

2, 4, 6, 8, 10, . . .

50, 45, 40, 35, 30, . . .

Grade 1


Grade 1 1.3. Core Content: Geometric attributes (Geometry/Measurement)

Students expand their knowledge of two- and three-dimensional geometric figures by sorting, comparing, and contrasting them according to their characteristics. They learn important

mathematical vocabulary used to name the figures. Students work with composite shapes made out of basic two-dimensional figures as they continue to develop their spatial sense of shapes, objects, and the world around them.

Performance Expectations Explanatory Comments and ExamplesStudents are expected to:

1.3.A Compare and sort a variety of two- and three-dimensional figures according to their geometric attributes.

The student may sort a collection of two-dimensional figures into those that have a particular attribute (e.g., those that have straight sides) and those that do not.

1.3.B Identify and name two-dimensional figures, including those in real-world contexts, regardless of size or orientation.

Figures should include circles, triangles, rectangles, squares (as special rectangles), rhombi, hexagons, and trapezoids.

Contextual examples could include classroom clocks, flags, desktops, wall or ceiling tiles, etc. Triangles should appear in many positions and orientations and should not all be equilateral or isosceles.

1.3.C Combine known shapes to create shapes and divide known shapes into other shapes.

Students could be asked to trace objects or use a drawing program to show different ways that a rectangle can be divided into three triangles. They can also use pattern blocks or plastic shapes to make new shapes. The teacher can give students cutouts of shapes and ask students to combine them to make a particular shape.

Example:

What shapes can be made from a rectangle and a • triangle? Draw a picture to show your answers.

Grade 1


Grade 11.4. Core Content: Concepts of measurement (Geometry/Measurement)

Students start to learn about measurement by measuring length. They begin to understand what it means to measure something, and they develop their measuring skills using everyday objects. As

they focus on length, they come to understand that units of measure must be equal in size and learn that standard-sized units exist. They develop a sense of the approximate size of those standard units (like inches or centimeters) and begin using them to measure different objects. Students learn that when a unit is small, it takes more of the unit to measure an item than it does when the units are larger, and they relate and compare measurements of objects using units of different sizes. Over time they apply these same concepts of linear measurement to other attributes such as weight and capacity. As students practice using measurement tools to measure objects, they reinforce their numerical skills and continue to develop their sense of space and shapes.


1.4.A Recognize that objects used to measure an attribute (length, weight, capacity) must be consistent in size.

Marbles can be suitable objects for young children to use to measure weight, provided that all the marbles are the same weight. Paper clips are appropriate for measuring length as long as the paper clips are all the same length.

1.4.B Use a variety of non-standard units to measure length.

Use craft sticks, toothpicks, coffee stirrers, etc., to measure length.

1.4.C Compare lengths using the transitive property. Example:

If Jon is taller than Jacob, and Jacob is taller than • Luisa, then Jon is taller than Luisa.

1.4.D Use non-standard units to compare objects according to their capacities or weights.

Examples can include using filled paper cups to measure capacity or a balance with marbles or cubes to measure weight.

1.4.E Describe the connection between the size of the measurement unit and the number of units needed to measure something.

Examples:

It takes more toothpicks than craft sticks to • measure the width of my desk. The longer the unit, the fewer I need.

It takes fewer marbles than cubes to balance my • object. The lighter the unit, the more I need.

It takes more little medicine cups filled with water • than larger paper cups filled with water to fill my jar. The less my unit holds, the more I need.

1.4.F Name the days of the week and the months of the year, and use a calendar to determine a day or month.

Examples:

Name the days of the week in order.•

Name the months of the year in order.•

How many days until your birthday?•

What month comes next?•

What day was it yesterday? •

Grade 1


Grade 1 1.5. Additional Key Content (Data/Statistics/Probability)

Students are introduced to early ideas of statistics by collecting and visually representing data. These ideas reinforce their understanding of the Core Content areas related to whole numbers and addition

and subtraction as students ask and answer questions about the data. As they move through the grades, students will continue to apply what they learn about data, making mathematics relevant and connecting numbers to applied situations.


1.5.A Represent data using tallies, tables, picture graphs, and bar-type graphs.

In a picture graph, a single picture represents a single object. Pictographs, where a symbol represents more than one unit, are introduced in grade three when multiplication is developed.

Students are expected to be familiar with all representations, but they need not use them all in every situation.

1.5.B Ask and answer comparison questions about data.

Students develop questions that can be answered using information from their graphs. For example, students could look at tallies showing the number of pockets on pants for each student today.

Andy

SaraChris

They might ask questions such as:

Who has the most pockets?—

Who has the fewest pockets?—

How many more pockets does Andy have — than Chris?

Grade 1


Grade 1 1.6. Core Processes: Reasoning, problem solving, and communication

Students further develop the concept that doing mathematics involves solving problems and discussing what they did to solve them. Problems in first grade emphasize addition, subtraction, and solidifying

number concepts, and sometimes include precursors to multiplication. Students continue to develop their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?”; “Why did you do that?”; and “How do you know that?” Students begin to build their mathematical vocabulary as they use correct mathematical language appropriate to first grade.


1.6.A Identify the question(s) asked in a problem.

1.6.B Identify the given information that can be used to solve a problem.

1.6.C Recognize when additional information is required to solve a problem.

1.6.D Select from a variety of problem-solving strategies and use one or more strategies to solve a problem.

1.6.E Answer the question(s) asked in a problem.

1.6.F Identify the answer(s) to the question(s) in a problem.

1.6.G Describe how a problem was solved.

1.6.H Determine whether a solution to a problem is reasonable.


Examples:

Think about how many feet a person has. How • many feet does a cat have? How many feet does a snail have? How about a fish or a snake?

There are ten feet living in my house. Who could be living in my house?

Come up with a variety of ways you can have a total of ten feet living in your house. Use pictures, words, or numbers to show how you got your answer.

You are in charge of setting up a dining room with • exactly twenty places for people to sit. You can use any number and combination of different-shaped tables. A hexagon-shaped table seats six people. A triangle-shaped table seats three people. A square-shaped table seats four people.

Draw a picture showing which tables and how many of each you could set up so that twenty people have a place to sit. Is there more than one way to do this? How many ways can you find?

Grade 1


Grade 2


Grade 2 2.1. Core Content: Place value and the base ten system (Numbers)

Students refine their understanding of the base ten number system and use place value concepts of ones, tens, and hundreds to understand number relationships. They become fluent in writing and

renaming numbers in a variety of ways. This fluency, combined with the understanding of place value, is a strong foundation for learning how to add and subtract two-digit numbers.


2.1.A Count by tens or hundreds forward and back-ward from 1 to 1,000, starting at any number.

Example:

Count forward by tens out loud starting at 32.•

2.1.B Connect place value models with their numerical equivalents to 1,000.

Understanding the relative value of numbers using place value is an important element of our base ten number system. Students should be familiar with representing numbers using words, pictures (including those involving grid paper), or physical objects such as base ten blocks. Money can also be an appropriate model.

2.1.C Identify the ones, tens, and hundreds place in a number and the digits occupying them.

Examples:

4 is located in what place in the number 834? •

What digit is in the hundreds place in 245?•

2.1.D Write three-digit numbers in expanded form. Examples:

573 = 500 + 70 + 3•

600 + 30 + 7 = 637•

2.1.E Group three-digit numbers into hundreds, tens, and ones in more than one way.

Students should become fluent in naming and renaming numbers based on number sense and their understanding of place value.

Examples:

In the number 647, there are 6 hundreds, there are • 64 tens, and there are 647 ones.

There are 20 tens in 200 and 10 hundreds in 1,000. •

There are 23 tens in 230.•

3 hundreds + 19 tens + 3 ones describes the same • number as 4 hundreds + 8 tens + 13 ones.

2.1.F Compare and order numbers from 0 to 1,000. Students use the words equal to, greater than, less than, greatest, or least and the symbols =, <, and >.

Grade 2


Grade 2 2.2. Core Content: Addition and subtraction (Operations, Geometry/Measurement, Algebra)

Students focus on what it means to add and subtract as they become fluent with single-digit addition and subtraction facts and develop addition and subtraction procedures for two-digit numbers.

Students make sense of these procedures by building on what they know about place value and number relationships and by putting together or taking apart sets of objects. This is students’ first time to deal formally with step-by-step procedures (algorithms)—an important component of mathematics where a generalizable technique can be used in many similar situations. Students begin to use estimation to determine if their answers are reasonable.


2.2.A Quickly recall basic addition facts and related subtraction facts for sums through 20.

2.2.B Solve addition and subtraction word problems that involve joining, separating, and comparing and verify the solution.

Problems should include those involving take-away situations, missing addends, and comparisons.

The intent of this expectation is for students to show their work, explain their thinking, and verify that the answer to the problem is reasonable in terms of the original context and the mathematics used to solve the problem. Verifications can include the use of numbers, words, pictures, or physical objects.

Example:

Hazel and Kimmy each have stamp collections. • Kimmy’s collection has 7 more stamps than Hazel’s. Kimmy has a total of 20 stamps. How many stamps are in Hazel’s collection? Explain your answer.

[Students may verify their work orally, with pictures, or in writing. For instance, students might give the equation below or might use the picture.]

20 – 7 = 13

are 20

Hazel’s

Kimmy’s

USA USA USA USA USA USA USAand

2.2.C Add and subtract two-digit numbers efficiently and accurately using a procedure that works with all two-digit numbers and explain why the procedure works.

Students should be able to connect the numerical procedures with other representations, such as words, pictures, or physical objects.

This is students’ first exposure to mathematical algorithms. It sets the stage for all future work with computational procedures.

The standard algorithms for addition and subtraction are formalized in grade three.

Grade 2



2.2.D Add and subtract two-digit numbers mentally and explain the strategies used.

Examples of strategies include

Combining tens and ones: • 68 + 37 = 90 + 15 = 105

Compensating: 68 + 37 = 65 + 40 = 105•

Incremental: 68 + 37 = 68 + 30 + 7 = 105•

2.2.E Estimate sums and differences. Example:

Students might estimate that 198 + 29 is a little • less than 230.

2.2.F Create and state a rule for patterns that can be generated by addition and extend the pattern.

Examples:

2, 5, 8, 11, 14, 17, . . .•

Look at the pattern of squares below. Draw a • picture that shows what the next set of squares might look like and explain why your answer makes sense.

A B C

2.2.G Solve equations in which the unknown number appears in a variety of positions.

Students need this kind of experience with equivalence to accompany their first work with addition and subtraction. Flexible use of equivalence and missing numbers sets the stage for later work when solving equations in which the variable is in different positions.

Examples:

8 + 3 = • + 5

10 – 7 = 2 + •

• = 9 + 4 + 2

2.2.H Name each standard U.S. coin, write its value using the $ sign and the ¢ sign, and name combinations of other coins with the same total value.

Students should be expected to express, for example, the value of a quarter as twenty-five cents, $0.25, and 25¢, and they should be able to give other combinations of coins whose value is 25¢. This is a precursor to decimal notation.

2.2.I Determine the value of a collection of coins totaling less than $1.00.

Grade 2


Grade 2 2.3. Core Content: Measurement (Geometry/Measurement)

Students understand the process of measuring length and progress from measuring length with objects such as toothpicks or craft sticks to the more practical skill of measuring length with standard

units and tools such as rulers, tape measures, or meter sticks. As students are well acquainted with two-digit numbers by this point, they tell time on different types of clocks.


2.3.A Identify objects that represent or approximate standard units and use them to measure length.

At this level, students no longer rely on non-standard units. Students find and use approximations for standard length units, like a paper clip whose length is about an inch, or the width of a particular student’s thumbnail that might be about a centimeter. They might also use commonly available classroom objects like inch tiles or centimeter cubes.

2.3.B Estimate length using metric and U.S. customary units.

Students could make observations such as, “The ceiling of the classroom is about 8 feet high.”

2.3.C Measure length to the nearest whole unit in both metric and U.S. customary units.

Standard tools may include rulers, yardsticks, meter sticks, or centimeter/inch measuring tapes. Students should measure some objects that are longer than the measurement tool being used.

2.3.D Describe the relative size among minutes, hours, days, weeks, months, and years.

Students should be able to describe relative sizes using statements like, “Since a minute is less than an hour, there are more minutes than hours in one day.”

2.3.E Use both analog and digital clocks to tell time to the minute.

Grade 2


Grade 2 2.4. Additional Key Content (Numbers, Operations, Geometry/Measurement, Data/Statistics/Probability)

Students make predictions and answer questions about data as they apply their growing understanding of numbers and the operations of addition and subtraction. They extend their spatial understanding

of Core Content in geometry developed in kindergarten and grade one by solving problems involving two- and three-dimensional geometric figures. Students are introduced to a few critical concepts that will become Core Content in grade three. Specifically, they begin to work with multiplication and division and learn what a fraction is.


2.4.A Solve problems involving properties of two- and three-dimensional figures.

A critical component in the development of students’ spatial and geometric understanding is the ability to solve problems involving the properties of figures. At the primary level, students must move from judging plane and space shapes by their appearance as whole shapes to focusing on the relationship of the sides, angles, or faces. At the same time, students must learn the language important for describing shapes according to their essential characteristics. Later, they will describe properties of shapes in more formal ways as they progress in geometry.

Examples:

How many different ways can you fill the outline of • the figure with pattern blocks? What is the greatest number of blocks you can use? The least number? Can you fill the outline with every whole number of blocks between the least number of blocks and the greatest number of blocks?

Build a figure or design out of five blocks. Describe • it clearly enough so that someone else could build it without seeing it. Blocks may represent two-dimensional figures (i.e., pattern blocks) or three-dimensional figures (i.e., wooden geometric solids).

2.4.B Collect, organize, represent, and interpret data in bar graphs and picture graphs.

In a picture graph, a single picture represents a single object. Pictographs, where a symbol represents more than one unit, are introduced in grade three when multiplication skills are developed.

2.4.C Model and describe multiplication situations in which sets of equal size are joined.

Multiplication is introduced in grade two only at a conceptual level. This is a foundation for the more systematic study of multiplication in grade three. Small numbers should be used in multiplication problems that are posed for students in grade two.

Example:

You have 4 boxes with 3 apples in each box. How • many apples do you have?

Grade 2



2.4.D Model and describe division situations in which sets are separated into equal parts.

Division is introduced in grade two only at a conceptual level. This is a foundation for the more systematic study of division in grade three. Small numbers should be used in division problems that are posed for students in grade two.

Example:

You have 15 apples to share equally among • 5 classmates. How many apples will each classmate get?

2.4.E Interpret a fraction as a number of equal parts of a whole or a set.

Examples:

Juan, Chan, and Hortense are going to share • a large cookie in the shape of a circle. Draw a picture that shows how you can cut up the cookie in three fair shares, and tell how big each piece is as a fraction of the whole cookie.

Ray has two blue crayons, one red crayon, and • one yellow crayon. What fraction of Ray’s crayons is red? What fraction of the crayons is blue?

Grade 2



Students further develop the concept that doing mathematics involves solving problems and talking about what they did to solve those problems. Second-grade problems emphasize addition and

subtraction with increasingly large numbers, measurement, and early concepts of multiplication and division. Students communicate their mathematical thinking and make increasingly more convincing mathematical arguments. Students participate in mathematical discussions involving questions like “How did you get that?”; “Why did you use that strategy?”; and “Why is that true?” Students continue to build their mathematical vocabulary as they use correct mathematical language appropriate to grade two when discussing and refining solutions to problems.


2.5.A Identify the question(s) asked in a problem and any other questions that need to be answered in order to solve the problem.

2.5.B Identify the given information that can be used to solve a problem.

2.5.C Recognize when additional information is required to solve a problem.

2.5.D Select from a variety of problem-solving strategies and use one or more strategies to solve a problem.

2.5.E Identify the answer(s) to the question(s) in a problem.

2.5.F Describe how a problem was solved.

2.5.G Determine whether a solution to a problem is reasonable.


Examples:

A bag full of jellybeans is on the table. There are • 10 black jellybeans in the bag. There are twice as many red jellybeans as black jellybeans. There are 2 fewer red jellybeans than yellow jellybeans. There are half as many pink jellybeans as yellow jellybeans. How many jellybeans are in the bag? Explain your answer.

Suzy, Ben, and Pedro have found 1 quarter, 1 • dime, and 4 pennies under the sofa. Their mother has lots of change in her purse, so they could trade any of these coins for other coins adding up to the same value. She says they can keep the money if they can tell her what coins they need so the money can be shared equally among them. How can they do this?

Grade 2

Grade 2


Grade 3


Grade 3 3.1. Core Content: Addition, subtraction, and place value (Numbers, Operations)

Students solidify and formalize important concepts and skills related to addition and subtraction. In particular, students extend critical concepts of the base ten number system to include large numbers,

they formalize procedures for adding and subtracting large numbers, and they apply these procedures in new contexts.


3.1.A Read, write, compare, order, and represent numbers to 10,000 using numbers, words, and symbols.

This expectation reinforces and extends place value concepts.

Symbols used to describe comparisons include <, >, =.

Examples:

Fill in the box with <, >, or = to make a true • sentence: 3,546 4,356.

Is 5,683 closer to 5,600 or 5,700? •

3.1.B Round whole numbers through 10,000 to the nearest ten, hundred, and thousand.

Example:

Round 3,465 to the nearest ten and then to the • nearest hundred.

3.1.C Fluently and accurately add and subtract whole numbers using the standard regrouping algorithms.

Teachers should be aware that in some countries the algorithms might be recorded differently.

3.1.D Estimate sums and differences to approximate solutions to problems and determine reasonableness of answers.

Example:

Marla has $10 and plans to spend it on items • priced at $3.72 and $6.54. Use estimation to decide whether Marla’s plan is a reasonable one, and justify your answer.

3.1.E Solve single- and multi-step word problems involving addition and subtraction of whole numbers and verify the solutions.

The intent of this expectation is for students to show their work, explain their thinking, and verify that the answer to the problem is reasonable in terms of the original context and the mathematics used to solve the problem. Verifications can include the use of numbers, words, pictures, or equations.

Grade 3


Grade 3 3.2. Core Content: Concepts of multiplication and division (Operations, Algebra)

Students learn the meaning of multiplication and division and how these operations relate to each other. They begin to learn multiplication and division facts and how to multiply larger numbers.

Students use what they are learning about multiplication and division to solve a variety of problems. With a solid understanding of these two key operations, students are prepared to formalize the procedures for multiplication and division in grades four and five.


3.2.A Represent multiplication as repeated addition, arrays, counting by multiples, and equal jumps on the number line, and connect each representation to the related equation.

Students should be familiar with using words, pictures, physical objects, and equations to represent multiplication. They should be able to connect various representations of multiplication to the related multiplication equation. Representing multiplication with arrays is a precursor to more formalized area models for multiplication developed in later grades beginning with grade four.

The equation 3 × 4 = 12 could be represented in the following ways:

Equal sets: — Equal sharing:Equal groups:

An array: —

Repeated addition: 4 + 4 + 4—

Three equal jumps forward from 0 on the — number line to 12:

0 1 2 3 4 5 6 7 8 9 10 11 12

Grade 3



3.2.B Represent division as equal sharing, repeated subtraction, equal jumps on the number line, and formation of equal groups of objects, and connect each representation to the related equation.

Students should be familiar with using words, pictures, physical objects, and equations to represent division. They should be able to connect various representations of division to the related equation.

Division can model both equal sharing (how many in each group) and equal groups (how many groups).

The equation 12 ÷ 4 = 3 could be represented in the following ways:

Equal groups: Equal sharing: — Equal sharing:Equal groups:

An array: —

Repeated subtraction: The expression — 12 – 4 – 4 – 4 involves 3 subtractions of 4.

Three equal jumps backward from 12 to 0 — on the number line:

0 1 2 3 4 5 6 7 8 9 10 11 12

3.2.C Determine products, quotients, and missing factors using the inverse relationship between multiplication and division.

Example:

To find the value of • N in 3 x N = 18, think 18 ÷ 3 = 6.

Students can use multiplication and division fact families to understand the inverse relationship between multiplication and division.

Examples:

3 • × 5 = 15 5 × 3 = 15 15 ÷ 3 = 5 15 ÷ 5 = 3

15

3 5

x or ÷

Grade 3



3.2.D Apply and explain strategies to compute multiplication facts to 10 X 10 and the related division facts.

Strategies for multiplication include skip counting (repeated addition), fact families, double-doubles (when 4 is a factor), “think ten” (when 9 is a factor, think of it as 10 – 1), and decomposition of arrays into smaller known parts.

Number properties can be used to help remember basic facts.

5 × 3 = 3 × 5 (Commutative Property)

1 × 5 = 5 × 1 = 5 (Identity Property)

0 × 5 = 5 × 0 = 0 (Zero Property)

5 × 6 = 5 × (2 × 3) = (5 × 2) × 3 = 10 × 3 = 30 (Associative Property)

4 × 6 = 4 (5 + 1) = (4 × 5) + (4 × 1) = 20 + 4 = 24 (Distributive Property)

4 x 6

4 g

rou

ps

of 5

4 gro

up

s of 1

Division strategies include using fact families and thinking of missing factors.

3.2.E Quickly recall those multiplication facts for which one factor is 1, 2, 5, or 10 and the related division facts.

Many students will learn all of the multiplication facts to 10 X 10 by the end of third grade, and all students should be given the opportunity to do so.

3.2.F Solve and create word problems that match multiplication or division equations.

The goal is for students to be able to represent multiplication and division sentences with an appropriate situation, using objects, pictures, or written or spoken words. This standard is about helping students connect symbolic representations to the situations they model. While some students may create word problems that are detailed or lengthy, this is not necessary to meet the expectation. Just as we want students to be able to translate 5 groups of 3 cats into 5 x 3 = 15; we want students to look at an equation like 12 ÷ 4 = 3 and connect it to a situation using objects, pictures, or words.

Grade 3



3.2.F cont. Example:

Equation: 3 • × 9 = ?

[Problem situation:

There are 3 trays of cookies with 9 cookies on each tray. How many cookies are there in all?]

3.2.G Multiply any number from 11 through 19 by a single-digit number using the distributive property and place value concepts.

Example:

6 • × 12 can be thought of as 6 tens and 6 twos, which equal 60 and 12, totaling 72.

6 x 10 = 60

21 = 2 x 6

10 2

6

6 groups of 10 6 groups of 2

3.2.H Solve single- and multi-step word problems involving multiplication and division and verify the solutions.

Problems include using multiplication to determine the number of possible combinations or outcomes for a situation, and division contexts that require interpretations of the remainder.

The intent of this expectation is for students to show their work, explain their thinking, and verify that the answer to the problem is reasonable in terms of the original context and the mathematics used to solve the problem. Verifications can include the use of numbers, words, pictures, physical objects, or equations.

Examples:

Determine the number of different outfits that can be • made with four shirts and three pairs of pants.

There are 14 soccer players on the boys’ team and • 13 on the girls’ team. How many vans are needed to take all players to the soccer tournament if each van can take 5 players?

Grade 3


Grade 3 3.3. Core Content: Fraction concepts (Numbers, Algebra)

Students learn about fractions and how they are used. Students deepen their understanding of fractions by comparing and ordering fractions and by representing them in different ways. With a

solid knowledge of fractions as numbers, students are prepared to be successful when they add, subtract, multiply, and divide fractions to solve problems in later grades.


3.3.A Represent fractions that have denominators of 2, 3, 4, 5, 6, 8, 9, 10, and 12 as parts of a whole, parts of a set, and points on the number line.

The focus is on numbers less than or equal to 1. Students should be familiar with using words, pictures, physical objects, and equations to represent fractions.

3.3.B Compare and order fractions that have denominators of 2, 3, 4, 5, 6, 8, 9, 10, and 12.

Fractions can be compared using benchmarks

(such as 1

2 or 1), common numerators, or common

denominators. Symbols used to describe comparisons include <, >, =.

Fractions with common denominators may be compared and ordered using the numerators as a guide.

26

36

56

< <

Fractions with common numerators may be compared and ordered using the denominators as a guide.

3

1038

34

< <

Fractions may be compared using 12

as a benchmark.

12

18

56

3.3.C Represent and identify equivalent fractions with denominators of 2, 3, 4, 5, 6, 8, 9, 10, and 12.

Students could represent fractions using the number line, physical objects, pictures, or numbers.

Grade 3



3.3.D Solve single- and multi-step word problems involving comparison of fractions and verify the solutions.

The intent of this expectation is for students to show their work, explain their thinking, and verify that the answer to the problem is reasonable in terms of the original context and the mathematics used to solve the problem. Verifications can include the use of numbers, words, pictures, physical objects, or equations.

Examples:

Emile and Jordan ordered a medium pizza. Emile •

ate 13 of it and Jordan ate 1

4 of it. Who ate more

pizza? Explain how you know.

Janie and Li bought a dozen balloons. Half of them •

were blue, 13 were white, and 1

6 were red. Were

there more blue, red, or white balloons? Justify your answer.

Grade 3


Grade 3 3.4. Core Content: Geometry (Geometry/Measurement)

Students learn about lines and use lines, line segments, and right angles as they work with quadrilaterals. Students connect this geometric work to numbers, operations, and measurement as

they determine simple perimeters in ways they will use when calculating perimeters of more complex figures in later grades.


3.4.A Identify and sketch parallel, intersecting, and perpendicular lines and line segments.

3.4.B Identify and sketch right angles.

3.4.C Identify and describe special types of quadrilaterals.

Special types of quadrilaterals include squares, rectangles, parallelograms, rhombi, trapezoids and kites.

3.4.D Measure and calculate perimeters of quadrilaterals.

Example:

Sketch a parallelogram with two sides 9 cm long • and two sides 6 cm long. What is the perimeter of the parallelogram?

3.4.E Solve single- and multi-step word problems involving perimeters of quadrilaterals and verify the solutions.

Example:

Julie and Jacob have recently created two • rectangular vegetable gardens in their backyard. One garden measures 6 ft by 8 ft, and the other garden measures 10 ft by 5 ft. They decide to place a small fence around the outside of each garden to prevent their dog from getting into their new vegetables. How many feet of fencing should Julie and Jacob buy to fence both gardens?

Grade 3


Grade 3 3.5. Additional Key Content (Algebra, Geometry/Measurement, Data/Statistics/Probability)

Students solidify and formalize a number of important concepts and skills related to Core Content studied in previous grades. In particular, students demonstrate their understanding of equivalence as

an important foundation for later work in algebra. Students also reinforce their knowledge of measurement as they use standard units for temperature, weight, and capacity. They continue to develop data organization skills as they reinforce multiplication and division concepts with a variety of types of graphs.


3.5.A Determine whether two expressions are equal and use “=” to denote equality.

Examples:

Is• 5 × 3 = 3 × 5 a true statement?

Is 24 • ÷ 3 = 2 × 4 a true statement?

A common error students make is using the mathematical equivalent of a run-on sentence to solve some problems—students carry an equivalence from a previous expression into a new expression with an additional operation. For example, when adding 3 + 6 + 7, students sometimes incorrectly write:

3 + 6 = 9 + 7 = 16

Correct sentences:

3 + 6 = 9

9 + 7 = 16

3.5.B Measure temperature in degrees Fahrenheit and degrees Celsius using a thermometer.

The scale on a thermometer is essentially a vertical number line. Students may informally deal with negative numbers in this context, although negative numbers are not formally introduced until grade six.

Measure temperature to the nearest degree.

3.5.C Estimate, measure, and compare weight and mass using appropriate-sized U.S. customary and metric units.

3.5.D Estimate, measure, and compare capacity using appropriate-sized U.S. customary and metric units.

3.5.E Construct and analyze pictographs, frequency tables, line plots, and bar graphs.

Students can write questions to be answered with information from a graph. Graphs and tables can be used to compare sets of data.

Using pictographs in which a symbol stands for multiple objects can reinforce the development of both multiplication and division skills. Determining appropriate scale and units for the axes of various types of graphs can also reinforce multiplication and division skills.

Grade 3



Students in grade three solve problems that extend their understanding of core mathematical concepts—such as geometric figures, fraction concepts, and multiplication and division of whole

numbers—as they make strategic decisions that bring them to reasonable solutions. Students use pictures, symbols, or mathematical language to explain the reasoning behind their decisions and solutions. They further develop their problem-solving skills by making generalizations about the processes used and applying these generalizations to similar problem situations. These critical reasoning, problem-solving, and communication skills represent the kind of mathematical thinking that equips students to use the mathematics they know to solve a growing range of useful and important problems and to make decisions based on quantitative information.


3.6.A Determine the question(s) to be answered given a problem situation.

3.6.B Identify information that is given in a problem and decide whether it is necessary or unnecessary to the solution of the problem.

3.6.C Identify missing information that is needed to solve a problem.

3.6.D Determine whether a problem to be solved is similar to previously solved problems, and identify possible strategies for solving the problem.

3.6.E Select and use one or more appropriate strategies to solve a problem.

3.6.F Represent a problem situation using words, numbers, pictures, physical objects, or symbols.

3.6.G Explain why a specific problem-solving strategy or procedure was used to determine a solution.

3.6.H Analyze and evaluate whether a solution is reasonable, is mathematically correct, and answers the question.

3.6.I Summarize mathematical information, draw conclusions, and explain reasoning.

3.6.J Make and test conjectures based on data (or information) collected from explorations and experiments.

Descriptions of solution processes and explanations can include numbers, words (including mathematical language), pictures, physical objects, or equations. Students should be able to use all of these representations as needed. For a particular solution, students should be able to explain or show their work using at least one of these representations and verify that their answer is reasonable.

Examples:

Whitney wants to put a fence around the perimeter • of her square garden. She plans to include a gate that is 3 ft wide. The length of one side of the garden is 19 ft. The fencing comes in two sizes: rolls that are 18 ft long and 24 ft long. Which rolls and how many of each should Whitney buy in order to have the least amount of leftover fencing? Justify your answer.

A soccer team is selling water bottles with soccer • balls painted on them to raise money for new equipment. The team bought 10 boxes of water bottles. Each box cost $27 and had 9 bottles. At what price should the team sell each bottle in order to make $180 profit to pay for new soccer balls? Justify your answer.

Grade 3


Grade 4


Grade 4 4.1. Core Content: Multi-digit multiplication (Numbers, Operations, Algebra)

Students learn basic multiplication facts and efficient procedures for multiplying two- and three-digit numbers. They explore the relationship between multiplication and division as they learn related

division and multiplication facts in the same fact family. These skills, along with mental math and estimation, allow students to solve problems that call for multiplication. Building on an understanding of how multiplication and division relate to each other, students prepare to learn efficient procedures for division, which will be developed in fifth grade. Multiplication of whole numbers is not only a basic skill, it is also closely connected to Core Content of area in this grade level, and this connection reinforces understanding of both concepts. Multiplication is also central to students’ study of many other topics in mathematics across the grades, including fractions, volume, and algebra.


4.1.A Quickly recall multiplication facts through 10 X 10 and the related division facts.

4.1.B Identify factors and multiples of a number. Examples:

The factors of 12 are 1, 2, 3, 4, 6, 12.•

The multiples of 12 are 12, 24, 36, 48, . . . •

4.1.C Represent multiplication of a two-digit number by a two-digit number with place value models.

Representations can include pictures or physical objects, or students can describe the process in words (14 times 16 is the same as 14 times 10 added to 14 times 6).

The algorithm for multiplication is addressed in expectation 4.1.F.

Example:

14 • × 16 = 224

100 4 tens

6 tens 24ones

16

14100 + 40 + 60 + 24 = 224

Grade 4



4.1.D Multiply by 10, 100, and 1,000. Multiplying by 10, 100, and 1,000 extends place value concepts to large numbers through the millions. Students can use place value and properties of operations to determine these products.

Examples:

10 • × 5,000 = 50,000 100 × 5,000 = 500,000 1,000 × 5,000 = 5,000,000

40 • × 300 = (4 × 10) × (3 × 100) = (4 × 3) × (10 × 100) = 12 × 1,000 = 12,000

4.1.E Compare the values represented by digits in whole numbers using place value.

Example:

Compare the values represented by the digit 4 in • 4,000,000 and 40,000. [The value represented by the 4 in the millions place is 100 times as much as the value represented by the 4 in the ten-thousands place.]

4.1.F Fluently and accurately multiply up to a three-digit number by one- and two-digit numbers using the standard multiplication algorithm.

Example:

• 2 4 5

7x2 4 5

7x

5171

33

Teachers should be aware that in some countries the algorithm might be recorded differently.

4.1.G Mentally multiply two-digit numbers by numbers through 10 and by multiples of 10.

Examples:

4 • × 32 = (4 × 30) + (4 × 2)

4 • × 99 = 400 – 4

25 • × 30 = 75 × 10

4.1.H Estimate products to approximate solutions to problems and determine reasonableness of answers.

Example:

28 • × 120 is approximately 30 times 100, so the product should be around 3,000.

4.1.I Solve single- and multi-step word problems involving multi-digit multiplication and verify the solutions.


Problems could include multi-step problems that use operations other than multiplication.

Grade 4



4.1.J Solve single- and multi-step word problems involving division and verify the solutions.


Division problems should reinforce connections between multiplication and division. The example below can be solved using multiplication along with some addition and subtraction.

Example:

A group of 8 students shares a box containing 187 • animal crackers. What is each student’s equal share? How many crackers are left over?

Division algorithms, including long division, are developed in fifth grade.

Grade 4


Grade 4 4.2. Core Content: Fractions, decimals, and mixed numbers (Numbers, Algebra)

Students solidify and extend their understanding of fractions (including mixed numbers) to include decimals and the relationships between fractions and decimals. Students work with common factors

and common multiples as preparation for learning procedures for fraction operations in grades five and six. When they are comfortable with and knowledgeable about fractions, students are likely to be successful with the challenging skills of learning how to add, subtract, multiply, and divide fractions.


4.2.A Represent decimals through hundredths with place value models, fraction equivalents, and the number line.

Students should know how to write decimals and show them on the number line and should understand their mathematical connections to place value models and fraction equivalents. Students should be able to represent decimals with words, pictures, or physical objects, and connect these representations to the corresponding decimal.

4.2.B Read, write, compare, and order decimals through hundredths.

Decimals may be compared using benchmarks, such as 0, 0.5, 1, or 1.5. Decimals may also be compared using place value.

Examples:

List in increasing order: 0.7, 0.2, 1.4.•

Write an inequality that compares 0.05 and 0.50.•

4.2.C Convert a mixed number to a fraction and vice versa, and visually represent the number.

Students should be able to use either the fraction or mixed-number form of a number as appropriate to a given situation, and they should be familiar with representing these numbers with words, pictures, and physical objects.

4.2.D Convert a decimal to a fraction and vice versa, and visually represent the number.

Students should be familiar with using pictures and physical objects to visually represent decimals and fractions. For this skill at this grade, fractions should be limited to those that are equivalent to fractions with denominators of 10 or 100.

Examples:

310

• = 0.3

Grade 4

310



4.2.D cont.0.42 = • 42

100

520

• = 0.25

4.2.E Compare and order decimals and fractions (including mixed numbers) on the number line, in lists, and with the symbols <, >, or =.

Examples:

Compare each pair of numbers using <, >, or =: • 6

100 8 .

112

32

0 75 1

2.

Correctly show • 35

, 0.35, 3 1

2

on the number line.

Order the following numbers from least to greatest: •

76

, 6.2, 1

12, 0.88.

4.2.F Write a fraction equivalent to a given fraction. Example:

Write at least two fractions equivalent to each • fraction given below:

1

2, 56 ,

2

3

4.2.G Simplify fractions using common factors.

4.2.H Round fractions and decimals to the nearest whole number.

Grade 4

42100

520

610

11

2

0.75 1

2

35

13

2

76

32



4.2.I Solve single- and multi-step word problems involving comparison of decimals and fractions (including mixed numbers), and verify the solutions.


Example:

Ms. Ortiz needs • 112

pounds of sliced turkey. She

picked up a package labeled “1.12 lbs.” Would she have enough turkey with this package? Explain why or why not.

Grade 4

11

2


Grade 4 4.3. Core Content: Concept of area (Geometry/Measurement, Algebra)

Students learn how to find the area of a rectangle as a basis for later work with areas of other geometric figures. They select appropriate units, tools, and strategies, including formulas, and use

them to solve problems involving perimeter and area. Solving such problems helps students develop spatial skills, which are critical for dealing with a wide range of geometric concepts. The study of area is closely connected to Core Content on multiplication, and connections between these concepts should be emphasized whenever possible.


4.3.A Determine congruence of two-dimensional figures.

At this grade level, students determine congruence primarily by making direct comparisons (e.g., tracing or cutting). They may also use informal notions of transformations described as flips, turns, and slides. Both the language and the concepts of transformations are more formally developed in grade eight.

4.3.B Determine the approximate area of a figure using square units.

Examples:

Draw a rectangle 3.5 cm by 6 cm on centimeter • grid paper. About how many squares fit inside the rectangle?

Cover a footprint with square tiles or outline it on • grid paper. About how many squares fit inside the footprint?

4.3.C Determine the perimeter and area of a rectangle using formulas, and explain why the formulas work.

This is an opportunity to connect area to the concept of multiplication, a useful model for multiplication that extends into algebra. Students should also work with squares as special rectangles.

Example:

Outline on grid paper a rectangle that is 4 units • long and 3 units wide. Without counting the squares, how can you determine the area? Other than measuring, how could you use a shortcut to find the perimeter of the rectangle?

4.3.D Determine the areas of figures that can be broken down into rectangles.

Example:

Find the area of each figure: •

7

3

16

3

3

61

Grade 4



4.3.E Demonstrate that rectangles with the same area can have different perimeters, and that rectangles with the same perimeter can have different areas.

Example:

Draw different rectangles, each with an area of • 24 square units, and compare their perimeters. What patterns do you notice in the data? Record your observations.

4.3.F Solve single- and multi-step word problems involving perimeters and areas of rectangles and verify the solutions.


Problems include those involving U.S. customary and metric units, including square units.

Grade 4


Grade 4 4.4. Additional Key Content (Geometry/Measurement, Algebra, Data/Statistics/Probability)

Students use coordinate grids to connect numbers to basic ideas in algebra and geometry. This connection between algebra and geometry runs throughout advanced mathematics and allows

students to use tools from one branch of mathematics to solve problems related to another branch. Students also extend and reinforce their work with whole numbers and fractions to describe sets of data and find simple probabilities. Students combine measurement work with their developing ideas about multiplication and division as they do basic measurement conversions. They begin to use algebraic notation while solving problems in preparation for formalizing algebraic thinking in later grades.


4.4.A Represent an unknown quantity in simple expressions, equations, and inequalities using letters, boxes, and other symbols.

Example:

There are 5 jars. Lupe put the same number • of marbles in each jar. Write an equation or expression that shows how many marbles are in each jar if there are 40 marbles total.

[5 × = 40 or 5 × M = 40; M represents the number of marbles]

4.4.B Solve single- and multi-step problems involving familiar unit conversions, including time, within either the U.S. customary or metric system.

Examples:

Jill bought 3 meters of ribbon and cut it into pieces • 25 centimeters long. How many 25-centimeter pieces of ribbon did she have?

How many quarts of lemonade are needed to • make 25 one-cup servings?

4.4.C Estimate and determine elapsed time using a calendar, a digital clock, and an analog clock.

4.4.D Graph and identify points in the first quadrant of the coordinate plane using ordered pairs.

Example:

1

1

2

2

3

3

4

4

5

5

(0, 0)

(1, 4)

(2, 2)

(3, 1)

(5, 3)

(5, 5)

0x

y

Grade 4



4.4.E Determine the median, mode, and range of a set of data and describe what each measure indicates about the data.

Example:

What is the median number of siblings that • students in this class have? What is the mode of the data? What is the range of the number of siblings? What does each of these values tell you about the students in the class?

0 1 2 3 4 5 6

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

Freq

uenc

y

Number of Siblings of Class 4A

Siblings of Class 4A

4.4.F Describe and compare the likelihood of events. For this introduction to probability, an event can be described as certain, impossible, likely, or unlikely. Two events can be compared as equally likely, not equally likely, or as one being more likely or less likely than the other.

4.4.G Determine a simple probability from a context that includes a picture.

Probability is expressed as a number from 0 to 1.

Example:

What is the probability of a blindfolded person • choosing a black marble from the bowl?

4.4.H Display the results of probability experiments and interpret the results.

Displays include tallies, frequency tables, graphs, pictures, and fractions.

Grade 4



Students in grade four solve problems that extend their understanding of core mathematical concepts—such as multiplication of multi-digit numbers, area, probability, and the relationships

between fractions and decimals—as they make strategic decisions that bring them to reasonable solutions. Students use pictures, symbols, or mathematical language to explain the reasoning behind their decisions and solutions. They further develop their problem-solving skills by making generalizations about the processes used and applying these generalizations to similar problem situations. These critical reasoning, problem-solving, and communication skills represent the kind of mathematical thinking that equips students to use the mathematics they know to solve a growing range of useful and important problems and to make decisions based on quantitative information.



4.5.B Identify information that is given in a problem and decide whether it is essential or extraneous to the solution of the problem.

4.5.C Identify missing information that is needed to solve a problem.


4.5.E Select and use one or more appropriate strategies to solve a problem and explain why that strategy was chosen.







Examples:

Jake’s family adopted a small dog, Toto. They • have a rectangular dog pen that is 10 feet by 20 feet. Toto needs only half that area, so Jake plans to make the pen smaller by cutting each dimension in half. Jake’s mother asked him to rethink his plan or Toto won’t have the right amount of space.

Whose reasoning is correct—Jake’s or his — mother’s? Why?

According to Jake’s plan, what fractional — part of the old pen will be the area of the new pen? Give the answer in simplest form.

Make a new plan so that the area of the — new pen is half the area of the old pen.

The city is paying for a new deck around the • community pool. The rectangular pool measures 50 meters by 25 meters. The deck, which will measure 5 meters wide, will surround the pool like a picture frame. If the cost of the deck is $25 for each square meter, what will be the total cost for the new deck? Explain your solution.

Grade 4

Grade 4


Grade 5


Grade 5 5.1. Core Content: Multi-digit division (Operations, Algebra)

Students learn efficient ways to divide whole numbers. They apply what they know about division to solve problems, using estimation and mental math skills to decide whether their results are

reasonable. This emphasis on division gives students a complete set of tools for adding, subtracting, multiplying, and dividing whole numbers—basic skills for everyday life and further study of mathematics.


5.1.A Represent multi-digit division using place value models and connect the representation to the related equation.

Students use pictures or grid paper to represent division and describe how that representation connects to the related equation. They could also use physical objects such as base ten blocks to support the visual representation. Note that the algorithm for long division is addressed in expectation 5.1.C.

5.1.B Determine quotients for multiples of 10 and 100 by applying knowledge of place value and properties of operations.

Example:

Using the fact that 16 • ÷ 4 = 4, students can generate the related quotients 160 ÷ 4 = 40 and 160 ÷ 40 = 4.

5.1.C Fluently and accurately divide up to a four-digit number by one- or two-digit divisors using the standard long-division algorithm.

The use of ‘R’ or ‘r’ to indicate a remainder may be appropriate in most of the examples students encounter in grade five. However, students should also be aware that in subsequent grades, they will learn additional ways to represent remainders, such as fractional or decimal parts.

Example:

7 9 361 3 2

61 91 8

1 31 2

1

r 1

-

-

-

Teachers should be aware that in some countries the algorithm might be recorded differently.

5.1.D Estimate quotients to approximate solutions and determine reasonableness of answers in problems involving up to two-digit divisors.

Example:

The team has saved $45 to buy soccer balls. If the • balls cost $15.95 each, is it reasonable to think there is enough money for more than two balls?

Problems like 54,596 ÷ 798, which can be estimated by 56,000 ÷ 800, while technically beyond the standards, could be included when appropriate. The numbers are easily manipulated and the problems support the ongoing development of place value.

Grade 5



5.1.E Mentally divide two-digit numbers by one-digit divisors and explain the strategies used.

5.1.F Solve single- and multi-step word problems involving multi-digit division and verify the solutions.


Problems include those with and without remainders.

Grade 5


Grade 5 5.2. Core Content: Addition and subtraction of fractions and decimals (Numbers, Operations, Algebra)

Students extend their knowledge about adding and subtracting whole numbers to learning procedures for adding and subtracting fractions and decimals. Students apply these procedures, along with

mental math and estimation, to solve a wide range of problems that involve more of the types of numbers students see in other school subjects and in their lives.


5.2.A Represent addition and subtraction of fractions and mixed numbers using visual and numerical models, and connect the representation to the related equation.

This expectation includes numbers with like and unlike denominators. Students should be able to show these operations on a number line and should be familiar with the use of pictures and physical materials (like fraction pieces or fraction bars) to represent addition and subtraction of mixed numbers. They should be able to describe how a visual representation connects to the related equation.

Example:

0 1

412

34

54

32

74

1 2

34

32

– =

5.2.B Represent addition and subtraction of decimals using place value models and connect the representation to the related equation.

Students should be familiar with using pictures and physical objects to represent addition and subtraction of decimals and be able to describe how those representations connect to related equations. Representations may include base ten blocks, number lines, and grid paper.

5.2.C Given two fractions with unlike denominators, rewrite the fractions with a common denominator.

Fraction pairs include denominators with and without common factors.

When students are fluent in writing equivalent fractions, it helps them compare fractions and helps prepare them to add and subtract fractions.

Examples:

Write equivalent fractions with a common •

denominator for 2

3 and

34

.

Write equivalent fractions with a common •

denominator for 38

and 1

6.

Grade 5

34

38



5.2.D Determine the greatest common factor and the least common multiple of two or more whole numbers.

Least common multiple (LCM) can be used to determine common denominators when adding and subtracting fractions.

Greatest common factor (GCF) can be used to simplify fractions.

5.2.E Fluently and accurately add and subtract fractions, including mixed numbers.

Fractions can be in either proper or improper form. Students should also be able to work with whole numbers as part of this expectation.

5.2.F Fluently and accurately add and subtract decimals.

Students should work with decimals less than 1 and greater than 1, as well as whole numbers, as part of this expectation.

5.2.G Estimate sums and differences of fractions, mixed numbers, and decimals to approximate solutions to problems and determine reasonableness of answers.

Example:

Jared is making a frame for a picture that is•

10 34

inches wide and 15 1

8 inches tall.

He has a 4-ft length of metal framing material. Estimate whether he will have enough framing material to frame the picture.

5.2.H Solve single- and multi-step word problems involving addition and subtraction of whole numbers, fractions (including mixed numbers), and decimals, and verify the solutions.


Multi-step problems may also include previously learned computational skills like multiplication and division of whole numbers.

Grade 5

34


Grade 5 5.3. Core Content: Triangles and quadrilaterals (Geometry/Measurement, Algebra)

Students focus on triangles and quadrilaterals to formalize and extend their understanding of these geometric shapes. They classify different types of triangles and quadrilaterals and develop formulas

for their areas. In working with these formulas, students reinforce an important connection between algebra and geometry. They explore symmetry of these figures and use what they learn about triangles and quadrilaterals to solve a variety of problems in geometric contexts.


5.3.A Classify quadrilaterals. Students sort a set of quadrilaterals into their various types, including parallelograms, kites, squares, rhombi, trapezoids, and rectangles, noting that a square can also be classified as a rectangle, parallelogram, and rhombus.

5.3.B Identify, sketch, and measure acute, right, and obtuse angles.

Example:

Use a protractor to measure the following angles • and label each as acute, right, or obtuse.

5.3.C Identify, describe, and classify triangles by angle measure and number of congruent sides.

Students classify triangles by their angle size using the terms acute, right, or obtuse.

Students classify triangles by the length of their sides using the terms scalene, isosceles, or equilateral.

5.3.D Determine the formula for the area of a parallel-ogram by relating it to the area of a rectangle.

Students relate the area of a parallelogram to the area of a rectangle, as shown below.

5.3.E Determine the formula for the area of a triangle by relating it to the area of a parallelogram.

Students relate the area of a triangle to the area of a parallelogram, as shown below.

5.3.F Determine the perimeters and areas of triangles and parallelograms.

Students may be given figures showing some side measures or may be expected to measure sides of figures. If students are not given side measures, but instead are asked to make their own measurements, it is important to discuss the approximate nature of any measurement.

Grade 5



5.3.G Draw quadrilaterals and triangles from given information about sides and angles.

Examples:

Draw a triangle with one right angle and no • congruent sides.

Draw a rhombus that is not a square.•

Draw a right scalene triangle.•

5.3.H Determine the number and location of lines of symmetry in triangles and quadrilaterals.

Example:

Draw and count all the lines of symmetry in the • square and isosceles triangle below. (Lines of symmetry are shown as dotted lines.)

5.3.I Solve single- and multi-step word problems about the perimeters and areas of quadrilaterals and triangles and verify the solutions.


Grade 5


Grade 5 5.4. Core Content: Representations of algebraic relationships (Operations, Algebra)

Students continue their development of algebraic thinking as they move toward more in-depth study of algebra in middle school. They use variables to write simple algebraic expressions describing

patterns or solutions to problems. They use what they have learned about numbers and operations to evaluate simple algebraic expressions and to solve simple equations. Students make tables and graphs from linear equations to strengthen their understanding of algebraic relationships and to see the mathematical connections between algebra and geometry. These foundational algebraic skills allow students to see where mathematics, including algebra, can be used in real situations, and these skills prepare students for success in future grades.


5.4.A Describe and create a rule for numerical and geometric patterns and extend the patterns.

Example:

The picture shows a sequence of towers • constructed from cubes. The number of cubes needed to build each tower forms a numeric pattern. Determine a rule for the number of cubes in each tower and use the rule to extend this pattern.

Tower 1 Tower 2 Tower 3

5.4.B Write a rule to describe the relationship between two sets of data that are linearly related.

Rules can be written using words or algebraic expressions.

Example:

The table below shows numerators (top row) and • denominators (bottom row) of fractions equivalent to

a given fraction (1

3). Write a rule that could be used

to describe how the two rows could be related.

1

3 6 9 ?

2 3 4

Grade 5



5.4.C Write algebraic expressions that represent simple situations and evaluate the expressions, using substitution when variables are involved.

Students should evaluate expressions with and without parentheses. Evaluating expressions with parentheses is an initial step in learning the proper order of operations.

Examples:

Evaluate (4 • × n) + 5 when n = 2.

If 4 people can sit at 1 table, 8 people can sit at • 2 tables, and 12 people can sit at 3 tables, and this relationship continues, write an expression to describe the number of people who can sit at n tables and tell how many people can sit at 67 tables.

Compare the answers to A and B below. •

A: (3 x 10) + 2

B: 3 x (10 + 2)

5.4.D Graph ordered pairs in the coordinate plane for two sets of data related by a linear rule and draw the line they determine.

Example:

The table shows the total cost of purchasing • different quantities of equally priced DVDs.

0 2

$0 $10

numberpurchased

total cost

5

$25

Graph the ordered pairs (0, 0), (2, 10), and (5, 25) and the line connecting the ordered pairs. Use the line to determine the total cost when 3 DVDs are purchased.

0 1 2 3 4 5

5

10

15

20

25

Number of DVDs Purchased

Co

st o

f DV

Ds

Purc

has

ed

Cost and Number of DVDs Purchased

Grade 5


Grade 5 5.5. Additional Key Content (Numbers, Data/Statistics/Probability)

Students extend their work with common factors and common multiples as they deal with prime numbers. Students extend and reinforce their use of numbers, operations, and graphing to describe

and compare data sets for increasingly complex situations they may encounter in other school subjects and in their lives.


5.5.A Classify numbers as prime or composite. Divisibility rules can help determine whether a number has particular factors.

5.5.B Determine and interpret the mean of a small data set of whole numbers.

At this grade level, numbers for problems are selected so that the mean will be a whole number.

Examples:

Seven families report the following number of pets. • Determine the mean number of pets per family.

0, 3, 3, 3, 5, 6, and 8

[One way to interpret the mean for this data set is to say that if the pets are redistributed evenly, each family will have 4 pets.]

The heights of five trees in front of the school are • given below. What is the average height of these trees? Does this average seem to represent the ‘typical’ size of these trees? Explain your answer.

3 ft, 4 ft, 4 ft, 4 ft, 20 ft

5.5.C Construct and interpret line graphs. Line graphs are used to display changes in data over time.

Example:

Below is a line graph that shows the temperature • of a can of juice after the can has been placed in ice and salt over a period of time. Describe any conclusions you can make about the data.

Grade 5

30

35

40

45

50

55

60

31 52 4

Tem

pera

ture

inD

egre

es F

ahre

nhei

t

Time in Minutes

Temperature of Apple Juice After Cooling in Salt and Ice



Students in grade five solve problems that extend their understanding of core mathematical concepts—such as division of multi-digit numbers, perimeter, area, addition and subtraction of

fractions and decimals, and use of variables in expressions and equations—as they make strategic decisions leading to reasonable solutions. Students use pictures, symbols, or mathematical language to explain the reasoning behind their decisions and solutions. They further develop their problem-solving skills by making generalizations about the processes used and applying these generalizations to similar problem situations. These critical reasoning, problem-solving, and communication skills represent the kind of mathematical thinking that equips students to use the mathematics they know to solve a growing range of useful and important problems and to make decisions based on quantitative information.



5.6.B Identify information that is given in a problem and decide whether it is essential or extraneous to the solution of the problem.

5.6.C Determine whether additional information is needed to solve the problem.


5.6.E Select and use one or more appropriate strategies to solve a problem, and explain the choice of strategy.







Examples:

La Casa Restaurant uses rectangular tables. • One table seats 6 people, with 1 person at each end and 2 people on each long side. However, 2 tables pushed together, short end to short end, seat only 10 people. Three tables pushed together end-to-end seat only 14 people. Write a rule that describes how many can sit at n tables pushed together end-to-end. The restaurant’s long banquet hall has tables pushed together in a long row to seat 70. How many tables were pushed together to seat this many people? How do you know?

The small square in the tangram figure below is • 1

8

the area of the large square.

For each of the 7 tangram pieces that make up the large square, tell what fractional part of the large square that piece represents. How do you know?

Grade 5


Grade 6


Grade 66.1. Core Content: Multiplication and division of fractions and decimals (Numbers, Operations, Algebra)

Students have done extensive work with fractions and decimals in previous grades and are now prepared to learn how to multiply and divide fractions and decimals with understanding. They can

solve a wide variety of problems that involve the numbers they see every day—whole numbers, fractions, and decimals. By using approximations of fractions and decimals, students estimate computations and verify that their answers make sense.


6.1.A Compare and order non-negative fractions, decimals, and integers using the number line, lists, and the symbols <, >, or =.

Examples:

List the numbers•

2 13

, 45

, 0.94, 54

, 1.1, and 4350

increasing order, and then graph the numbers on the number line.

Compare each pair of numbers using <, >, or =. •

4

51 2

7

41

3

4

27

82 5

.

.

6.1.B Represent multiplication and division of non-negative fractions and decimals using area models and the number line, and connect each representation to the related equation.

This expectation addresses the conceptual meaning of multiplication and division of fractions and decimals. Students should be familiar with the use of visual representations like pictures (e.g., sketching the problem, grid paper) and physical objects (e.g., tangrams, cuisenaire rods). They should connect the visual representation to the corresponding equation.

The procedures for multiplying fractions and decimals are addressed in 6.1.D and 6.1.E.

6.1.C Estimate products and quotients of fractions and decimals.

Example:

0.28 • ÷ 0.96 ≈ 0.3 ÷ 1; 0.3 ÷ 1 = 0.3

0.24 x 12.414

x 12.4 ; 14

x 12.4 = 3.1

313

2041

x 1 4

1 2

x 1 4

1 2

1 8

x; =

Grade 6

13

45

2 0.94, in1.1, and , , 54

4350

,

45

74

72 8

31 4

2.5



6.1.D Fluently and accurately multiply and divide non-negative fractions and explain the inverse relationship between multiplication and division with fractions.

Students should understand the inverse relationship between multiplication and division, developed in grade three and now extended to fractions. Students should work with different types of rational numbers, including whole numbers and mixed numbers, as they continue to expand their understanding of the set of rational numbers.

Example:

Multiply or divide. •

4

5

2

3×

6 3

8÷

21

43

1

2×

4

1

51

2

3÷

6.1.E Multiply and divide whole numbers and decimals by 1000, 100, 10, 1, 0.1, 0.01, and 0.001.

This expectation extends what students know about the place value system and about multiplication and division and expands their set of mental math tools. As students work with multiplication by these powers of 10, they can gain an understanding of how numbers relate to each other based on their relative sizes.

Example:

Mentally compute 0.01 x 435. •

6.1.F Fluently and accurately multiply and divide non-negative decimals.

Students should understand the inverse relationship between multiplication and division, developed in grade three and now extended to decimals. Students should work with different types of decimals, including decimals greater than 1, decimals less than 1, and whole numbers, as they continue to expand their understanding of the set of rational numbers.

Example:

Multiply or divide. • 0.84 × 1.5 2.04 × 32 7.85 ÷ 0.32 17.28 ÷ 1.2

6.1.G Describe the effect of multiplying or dividing a number by one, by zero, by a number between zero and one, and by a number greater than one.

Examples:

Without doing any computation, list • 74, 0.43 × 74, and 74 ÷ 0.85 in increasing order and explain your reasoning.

Explain why • 4

0 is undefined.

Grade 6



6.1.H Solve single- and multi-step word problems involving operations with fractions and decimals and verify the solutions.


Example:

Every day has 24 hours. Ali sleeps 3/8 of the day. • Dawson sleeps 1/3 of the day. Maddie sleeps 7.2 hours in a day. Who sleeps the longest? By how much?

Grade 6


Grade 6 6.2. Core Content: Mathematical expressions and equations (Operations, Algebra)

Students continue to develop their understanding of how letters are used to represent numbers in mathematics—an important foundation for algebraic thinking. Students use tables, words, numbers,

graphs, and equations to describe simple linear relationships. They write and evaluate expressions and write and solve equations. By developing these algebraic skills at the middle school level, students will be able to make a smooth transition to high school mathematics.


6.2.A Write a mathematical expression or equation with variables to represent information in a table or given situation.

Examples:

What expression can be substituted for the • question mark?

x 1 2 3 ...4 x

y 2.5 5 7.5 ...10 ?

A t-shirt printing company charges $7 for each • t-shirt it prints. Write an equation that represents the total cost, c, for ordering a specific quantity, t, of these t-shirts.

6.2.B Draw a first-quadrant graph in the coordinate plane to represent information in a table or given situation.

Example:

Mikayla and her sister are making beaded • bracelets to sell at a school craft fair. They can make two bracelets every 30 minutes. Draw a graph that represents the number of bracelets the girls will have made at any point during the 6 hours they work.

6.2.C Evaluate mathematical expressions when the value for each variable is given.

Examples:

Evaluate 2• s + 5t when s = 3.4 and t = 1.8.

Evaluate • 2

3 x – 14 when x = 60.

6.2.D Apply the commutative, associative, and distributive properties, and use the order of operations to evaluate mathematical expressions.

Examples:

Sim• plify 6 +

1

2

1

3, with and without the use of

the distributive property.

Evaluate • b – 3(2a – 7) when a = 5.4 and b = 31.7.

Grade 6



6.2.E Solve one-step equations and verify solutions. Students solve equations using number sense, physical objects (e.g., balance scales), pictures, or properties of equality.

Example:

Solve for the variable in each equation below.•

112 = 7a

1.4y = 42

= +2 b

1

2

1

3

=

y

45

7

15

6.2.F Solve word problems using mathematical expressions and equations and verify solutions.


Example:

Zane and his friends drove across the United • States at an average speed of 55 mph. Write expressions to show how far they traveled in 12 hours, in 18 hours, and in n hours. How long did it take them to drive 1,430 miles? Verify your solution.

Grade 6


Grade 6 6.3. Core Content: Ratios, rates, and percents (Numbers, Operations, Geometry/Measurement,

Algebra, Data/Statistics/Probability)

Students extend their knowledge of fractions to develop an understanding of what a ratio is and how it relates to a rate and a percent. Fractions, ratios, rates, and percents appear daily in the media

and in everyday calculations like determining the sale price at a retail store or figuring out gas mileage. Students solve a variety of problems related to such situations. A solid understanding of ratios and rates is important for work involving proportional relationships in grade seven.


6.3.A Identify and write ratios as comparisons of part-to-part and part-to-whole relationships.

Example:

If there are 10 boys and 12 girls in a class, what • is the ratio of boys to girls? What is the ratio of the number of boys to the total number of students in the class?

6.3.B Write ratios to represent a variety of rates. Example:

Julio drove his car 579 miles and used 15 gallons • of gasoline. How many miles per gallon did his car get during the trip? Explain your answer.

6.3.C Represent percents visually and numerically, and convert between the fractional, decimal, and percent representations of a number.

In addition to general translations among these representations, this expectation includes the quick recall of equivalent forms of common fractions (with denominators like 2, 3, 4, 5, 8, and 10), decimals, and percents. It also includes the understanding that a fraction represents division, an important conceptual background for writing fractions as decimals.

Examples:

Represent • 75

100 as a percent using numbers, a

picture, and a circle graph.

Represent 40% as a fraction and as a decimal. •

Write • 13

16 as a decimal and as a percent.

Grade 6



6.3.D Solve single- and multi-step word problems involving ratios, rates, and percents, and verify the solutions.


Examples:

An item is advertised as being 25% off the regular • price. If the sale price is $42, what was the original regular price? Verify your solution.

Sally had a business meeting in a city 100 miles • away. In the morning, she drove an average speed of 60 miles per hour, but in the evening when she returned, she averaged only 40 miles per hour. How much longer did the evening trip take than the morning trip? Explain your reasoning.

6.3.E Identify the ratio of the circumference to the diameter of a circle as the constant

π, and recognize 22

7 and 3.14 as

common approximations of π.

Example:

Measure the diameter and circumference of several • circular objects. Divide each circumference by its diameter. What do you notice about the results?

6.3.F Determine the experimental probability of a simple event using data collected in an experiment.

The term experimental probability refers here to the relative frequency that was observed in an experiment.

Example:

Tim is checking the apples in his orchard for • worms. Selecting apples at random, he finds 9 apples with worms and 63 apples without worms. What is the experimental probability that a given apple from his orchard has a worm in it?

6.3.G Determine the theoretical probability of an event and its complement and represent the probability as a fraction or decimal from 0 to 1 or as a percent from 0 to 100.

Example:

A bag contains 4 green marbles, 6 red marbles, • and 10 blue marbles. If one marble is drawn randomly from the bag, what is the probability it will be red? What is the probability that it will not be red?

Grade 6


Grade 6 6.4. Core Content: Two- and three-dimensional figures (Geometry/Measurement, Algebra)

Students extend what they know about area and perimeter to more complex two-dimensional figures, including circles. They find the surface area and volume of simple three-dimensional figures. As they

learn about these important concepts, students can solve problems involving more complex figures than in earlier grades and use geometry to deal with a wider range of situations. These fundamental skills of geometry and measurement are increasingly called for in the workplace and they lead to a more formal study of geometry in high school.


6.4.A Determine the circumference and area of circles.

Examples:

Determine the area of a circle with a diameter of • 12 inches.

Determine the circumference of a circle with a • radius of 32 centimeters.

6.4.B Determine the perimeter and area of a composite figure that can be divided into triangles, rectangles, and parts of circles.

Although students have worked with various quadrilaterals in the past, this expectation includes other quadrilaterals such as trapezoids or irregular quadrilaterals, as well as any other composite figure that can be divided into figures for which students have calculated areas before.

Example:

Determine the area and perimeter of each of the • following figures, assuming that the dimensions on the figures are in feet. The curved portion of the second figure is a semi-circle.

3

7

82

6

8

6.4.C Solve single- and multi-step word problems involving the relationships among radius, diameter, circumference, and area of circles, and verify the solutions.


Example:

Captain Jenkins determined that the distance • around a circular island is 44 miles. What is the distance from the shore to the buried treasure in the center of the island? What is the area of the island?

Grade 6



6.4.D Recognize and draw two-dimensional representations of three-dimensional figures.

The net of a rectangular prism consists of six rectangles that can then be folded to make the prism. The net of a cylinder consists of two circles and a rectangle.

Example:

6.4.E Determine the surface area and volume of rectangular prisms using appropriate formulas and explain why the formulas work.

Students may determine surface area by calculating the area of the faces and adding the results.

6.4.F Determine the surface area of a pyramid.

6.4.G Describe and sort polyhedra by their attributes: parallel faces, types of faces, number of faces, edges, and vertices.

Prisms and pyramids are the focus at this level.

Examples:

How many pairs of parallel faces does each • polyhedron have? Explain your answer.

What type of polyhedron has two parallel triangular • faces and three non-parallel rectangular faces?

Grade 6


Grade 6 6.5. Additional Key Content (Numbers, Operations)

Students extend their mental math skills now that they have learned all of the operations—addition, subtraction, multiplication, and division—with whole numbers, fractions, and decimals. Students

continue to expand their understanding of our number system as they are introduced to negative numbers for describing positions or quantities below zero. These numbers are a critical foundation for algebra, and students will learn how to add, subtract, multiply, and divide positive and negative numbers in seventh grade as further preparation for algebraic study.


6.5.A Use strategies for mental computations with non-negative whole numbers, fractions, and decimals.

Examples:

John wants to find the total number of hours he • worked this week. Use his time card below to find the total.

Days Monday Tuesday Wednesday Thursday Friday

Days 4 14

3 6 12

7 12

1 12

What is the total cost for items priced at $25.99 • and $32.95? (A student may think of something like 25.99 + 32.95 = (26 + 33) – 0.06 = 58.94.)

6.5.B Locate positive and negative integers on the number line and use integers to represent quantities in various contexts.

Contexts could include elevation, temperature, or debt, among others.

6.5.C Compare and order positive and negative integers using the number line, lists, and the symbols <, >, or =.

Examples:

Compare each pair of numbers using <, >, or =. •

-11 -14

-7 4

-101 -94

Grade 6


Grade 66.6. Core Processes: Reasoning, problem solving, and communication

Students refine their reasoning and problem-solving skills as they move more fully into the symbolic world of algebra and higher-level mathematics. They move easily among representations—

numbers, words, pictures, or symbols—to understand and communicate mathematical ideas, to make generalizations, to draw logical conclusions, and to verify the reasonableness of solutions to problems. In grade six, students solve problems that involve fractions and decimals as well as rates and ratios in preparation for studying proportional relationships and algebraic reasoning in grade seven.


6.6.A Analyze a problem situation to determine the question(s) to be answered.

6.6.B Identify relevant, missing, and extraneous information related to the solution to a problem.

6.6.C Analyze and compare mathematical strategies for solving problems, and select and use one or more strategies to solve a problem.

6.6.D Represent a problem situation, describe the process used to solve the problem, and verify the reasonableness of the solution.

6.6.E Communicate the answer(s) to the question(s) in a problem using appropriate representations, including symbols and informal and formal mathematical language.

6.6.F Apply a previously used problem-solving strategy in a new context.

6.6.G Extract and organize mathematical information from symbols, diagrams, and graphs to make inferences, draw conclusions, and justify reasoning.

6.6.H Make and test conjectures based on data (or information) collected from explorations and experiments.


Examples:

As part of her exercise routine, Carmen jogs • twice around the perimeter of a square park that

measures 5

8 mile on each side. On Monday, she

started at one corner of the park and jogged 2

3

of the way around in 17 minutes before stopping at a small pond in the park to feed some ducks. How far had Carmen run when she reached the pond? What percent of her planned total distance had Carmen completed when she stopped to feed the ducks? If it took Carmen 17 minutes to jog to the point where she stopped, assuming that she continued running in the same direction at the same pace and did not stop again, how long would it have taken her to get back to her starting point? Explain your answers.

At Springhill Elementary School’s annual fair, • Vanessa is playing a game called “Find the Key.” A key is randomly placed somewhere in one of the rooms shown on the map below. (The key cannot be placed in the hallway.)

Grade 6



6.6 cont. To win the game, Vanessa must correctly guess the room where the key is placed. Use what you know about the sizes of the rooms to determine the probability that the key is placed in the gym, the office, the café, the book closet, or the library. Write each probability as a simplified fraction, a decimal, and a percent. Which room should Vanessa select in order to have the best chance of winning? Justify the solution.

GymLibrary

BookCloset

CafeO�ce

Hal

lway

Grade 6


Grade 7


Grade 7 7.1. Core Content: Rational numbers and linear equations (Numbers, Operations, Algebra)

Students add, subtract, multiply, and divide rational numbers—fractions, decimals, and integers—including both positive and negative numbers. With the inclusion of negative numbers, students can move more

deeply into algebraic content that involves the full set of rational numbers. They also approach problems that deal with a wider range of contexts than before. Using generalized algebraic skills and approaches, students can approach a wide range of problems involving any type of rational number, adapting strategies for solving one problem to different problems in different settings with underlying similarities.


7.1.A Compare and order rational numbers using the number line, lists, and the symbols <, >, or =.

Examples:

List the number• s 2

3, -

2

3, 1.2, 4

3, - 4

3, -1.2, and - 7

4

in increasing order, and graph the numbers on the number line.

Compare each pair of numbers using <, >, or =.•

11 20

13 21

7 5 -1.35

-2.75 3 4 -2

-

- -

7.1.B Represent addition, subtraction, multiplication, and division of positive and negative integers visually and numerically.

Students should be familiar with the use of the number line and physical materials, such as colored chips, to represent computation with integers. They should connect numerical and physical representations to the computation. The procedures are addressed in 7.1.C.

Examples:

Use a picture, words, or physical objects to • illustrate 3 – 7; -3 – 7; -3 – (-7); (-3)(-7); 21 ÷ (-3).

At noon on a certain day, the temperature was • 13°; at 10 p.m. the same day, the temperature was -8°. How many degrees did the temperature drop between noon and 10 p.m.?

Grade 7



7.1.C Fluently and accurately add, subtract, multiply, and divide rational numbers.

This expectation brings together what students know about the four operations with positive and negative numbers of all kinds—integers, fractions, and decimals. Some of these skills will have been recently learned and may need careful development and reinforcement.

This is an opportunity to demonstrate connections among the operations and to show similarities and differences in the performance of these operations with different types of numbers. Visual representations may be helpful as students begin this work, and they may become less necessary as students become increasingly fluent with the operations.

Examples:

• −4

3

3

4- =

• 272 8

=

(3.5)(• -6.4) =

7.1.D Define and determine the absolute value of a number.

Students define absolute value as the distance of the number from zero.

Examples:

Explain why 5 and -5 have the same absolute value.•

Evaluate |7.8 – 10.3|.•

7.1.E Solve two-step linear equations. Example:

Solve 3.5• x – 12 = 408 and show each step in the process.

7.1.F Write an equation that corresponds to a given problem situation, and describe a problem situation that corresponds to a given equation.

Students have represented various types of problems with expressions and particular types of equations in previous grades. Many students at this grade level will also be able to deal with inequalities.

Examples:

Meagan spent $56.50 on 3 blouses and a pair of • jeans. If each blouse cost the same amount and the jeans cost $25, write an algebraic equation that represents this situation and helps you determine how much one blouse cost.

Describe a problem situation that could be solved • using the equation 15 = 2x – 7.

Grade 7



7.1.G Solve single- and multi-step word problems involving rational numbers and verify the solutions.


Example:

Tom wants to buy some candy bars and • magazines for a trip. He has decided to buy three times as many candy bars as magazines. Each candy bar costs $0.70 and each magazine costs $2.50. The sales tax rate on both types of items is

612

%. How many of each item can he buy if he has

$20.00 to spend?

Grade 7


Grade 7 7.2. Core Content: Proportionality and similarity (Operations, Geometry/Measurement, Algebra)

Students extend their work with ratios to solve problems involving a variety of proportional relationships, such as making conversions between measurement units or finding the percent

increase or decrease of an amount. They also solve problems involving the proportional relationships found in similar figures, and in so doing reinforce an important connection between numerical operations and geometric relationships. Students graph proportional relationships and identify the rate of change as the slope of the related line. The skills and concepts related to proportionality represent some of the most important connecting ideas across K–12 mathematics. With a good understanding of how things grow proportionally, students can understand the linear relationships that are the basis for much of high school mathematics. If learned well, proportionality can open the door for success in much of secondary mathematics.


7.2.A Mentally add, subtract, multiply, and divide simple fractions, decimals, and percents.

Example:

A shirt is on sale for 20% off the original price of • $15. Use mental math strategies to calculate the sale price of the shirt.

7.2.B Solve single- and multi-step problems involving proportional relationships and verify the solutions.


Problems include those that involve rate, percent increase or decrease, discount, markup, profit, interest, tax, or the conversion of money or measurement (including multiplying or dividing amounts in recipes).

More complex problems, such as dividing 100 into more than two proportional parts (e.g., 4:3:3), allow students to generalize what they know about proportional relationships to a range of situations.

Examples:

At a certain store, 48 television sets were sold • in April. The manager at the store wants to encourage the sales team to sell more TVs and is going to give all the sales team members a bonus if the number of TVs sold increases by 30% in May. How many TVs must the sales team sell in May to receive the bonus? Explain your answer.

After eating at a restaurant, you know that the bill • before tax is $52.60 and that the sales tax rate is 8%. You decide to leave a 20% tip for the waiter based on the pre-tax amount. How much should you leave for the waiter? How much will the total bill be, including tax and tip? Show work to support your answers.

Grade 7



7.2.B cont. Joe, Sam, and Jim completed different amounts • of yard work around the school. They agree to split the $200 they earned in a ratio of 5:3:2, respectively. How much did each boy receive?

7.2.C Describe proportional relationships in similar figures and solve problems involving similar figures.

Students should recognize the constant ratios in similar figures and be able to describe the role of a scale factor in situations involving similar figures. They should be able to connect this work with more general notions of proportionality.

Example:

The length of the shadow of a tree is 68 feet at the • same time that the length of the shadow of a 6-foot vertical pole is 8 feet. What is the height of the tree?

7.2.D Make scale drawings and solve problems related to scale.

Example:

On an 80:1 scale drawing of the floor plan of • a house, the dimensions of the living room are

17

82

1

2

′′×

′′. What is the actual area of the living

room in square feet?

7.2.E Represent proportional relationships using graphs, tables, and equations, and make connections among the representations.

Proportional relationships are linear relationships whose graphs pass through the origin and can be written in the form y = kx.

Example:

The relationship between the width and length • of similar rectangles is shown in the table below. Write an equation that expresses the length, l, in terms of the width, w, and graph the relationship between the two variables.

4 12 18 ...

10 30 45 ...

w

?

width

length

7.2.F Determine the slope of a line corresponding to the graph of a proportional relationship and relate slope to similar triangles.

This expectation connects the constant rate of change in a proportional relationship to the concept of slope of a line. Students should know that the slope of a line is the same everywhere on the line and realize that similar triangles can be used to demonstrate this fact. They should recognize how proportionality is reflected in slope as it is with similar triangles. A more complete discussion of slope is developed in high school.

Grade 7



7.2.G Determine the unit rate in a proportional relationship and relate it to the slope of the associated line.

The associated unit rate, constant rate of change of the function, and slope of the graph all represent the constant of proportionality in a proportional relationship.

Example:

Coffee costs $18.96 for 3 pounds. What is the • cost per pound of coffee? Draw a graph of the proportional relationship between the number of pounds of coffee and the total cost, and describe how the unit rate is represented on the graph.

7.2.H Determine whether or not a relationship is proportional and explain your reasoning.

A proportional relationship is one in which two quantities are related by a constant scale factor, k. It can be written in the form y = kx. A proportional relationship has a constant rate of change and its graph passes through the origin.

Example:

Determine whether each situation represents a • proportional relationship and explain your reasoning.

— x 1 2 3

y 4.5 9 13.5

4

18

y— = 3x + 2

One way to calculate a person’s maximum — target heart rate during exercise in beats per minute is to subtract the person’s age from 200. Is the relationship between the maximum target heart rate and age proportional? Explain your reasoning.

Grade 7



7.2.I Solve single- and multi-step problems involving conversions within or between measurement systems and verify the solutions.


Students should be given the conversion factor when converting between measurement systems.

Examples:

The lot that Dana is buying for her new one-story • house is 35 yards by 50 yards. Dana’s house plans show that her house will cover 1,600 square feet of land. What percent of Dana’s lot will not be covered by the house? Explain your work.

Joe was planning a business trip to Canada, so he • went to the bank to exchange $200 U.S. dollars for Canadian dollars (at a rate of $1.02 CDN per $1 US). On the way home from the bank, Joe’s boss called to say that the destination of the trip had changed to Mexico City. Joe went back to the bank to exchange his Canadian dollars for Mexican pesos (at a rate of 10.8 pesos per $1 CDN). How many Mexican pesos did Joe get?

Grade 7


Grade 7 7.3. Core Content: Surface area and volume (Algebra, Geometry/Measurement)

Students extend their understanding of surface area and volume to include finding surface area and volume of cylinders and volume of cones and pyramids. They apply formulas and solve a range of

problems involving three-dimensional objects, including problems people encounter in everyday life, in certain types of work, and in other school subjects. With a strong understanding of how to work with both two-dimensional and three-dimensional figures, students build an important foundation for the geometry they will study in high school.


7.3.A Determine the surface area and volume of cylinders using the appropriate formulas and explain why the formulas work.

Explanations might include the use of models such as physical objects or drawings.

A net can be used to illustrate the formula for finding the surface area of a cylinder.

7.3.B Determine the volume of pyramids and cones using formulas.

7.3.C Describe the effect that a change in scale factor on one attribute of a two- or three-dimensional figure has on other attributes of the figure, such as the side or edge length, perimeter, area, surface area, or volume of a geometric figure.

Examples:

A cube has a side length of 2 cm. If each side • length is tripled, what happens to the surface area? What happens to the volume?

What happens to the area of a circle if the • diameter is decreased by a factor of 3?

7.3.D Solve single- and multi-step word problems involving surface area or volume and verify the solutions.


Examples:

Alexis needs to paint the four exterior walls of a • large rectangular barn. The length of the barn is 80 feet, the width is 50 feet, and the height is 30 feet. The paint costs $28 per gallon, and each gallon covers 420 square feet. How much will it cost Alexis to paint the barn? Explain your work.

Tyesha has decided to build a solid concrete • pyramid on her empty lot. The base will be a square that is forty feet by forty feet and the height will be thirty feet. The concrete that she will use to construct the pyramid costs $70 per cubic yard. How much will the concrete for the pyramid cost Tyesha? Justify your answer.

Grade 7


Grade 7 7.4. Core Content: Probability and data (Data/Statistics/Probability)

Students apply their understanding of rational numbers and proportionality to concepts of probability. They begin to understand how probability is determined, and they make related predictions. Students

revisit how to interpret data, now using more sophisticated types of data graphs and thinking about the meaning of certain statistical measures. Statistics, including probability, is considered one of the most important and practical fields of study for making sense of quantitative information, and it plays an important part in secondary mathematics in the 21st century.


7.4.A Represent the sample space of probability experiments in multiple ways, including tree diagrams and organized lists.

The sample space is the set of all possible outcomes.

Example:

José flips a penny, Jane flips a nickel, and Janice • flips a dime, all at the same time. List the possible outcomes of the three simultaneous coin flips using a tree diagram or organized list.

7.4.B Determine the theoretical probability of a particular event and use theoretical probability to predict experimental outcomes.

Example:

A triangle with a base of 8 units and a height of • 7 units is drawn inside a rectangle with an area of 90 square units. What is the probability that a randomly selected point inside the rectangle will also be inside the triangle?

There are 5 blue, 4 green, 8 red, and 3 yellow • marbles in a paper bag. Rosa runs an experiment in which she draws a marble from the bag, notes the color on a sheet of paper, and puts the marble back in the bag, repeating the process 200 times. About how many times would you expect Rosa to draw a red marble?

7.4.C Describe a data set using measures of center (median, mean, and mode) and variability (maximum, minimum, and range) and evaluate the suitability and limitations of using each measure for different situations.

As a way to understand these ideas, students could construct data sets for a given mean, median, mode, or range.

Examples: Kiley keeps track of the money she spends each • week for two months and records the following amounts: $6.30, $2.25, $43.00, $2.25, $11.75, $5.25, $4.00, and $5.20. Which measure of center is most representative of Kiley’s weekly spending? Support your answer.

Construct a data set with five data points, a mean • of 24, a range of 10, and without a mode.

A group of seven adults have an average age of • 36. If the ages of three of the adults are 45, 30, and 42, determine possible ages for the remaining four adults.

Grade 7



7.4.D Construct and interpret histograms, stem-and-leaf plots, and circle graphs.

7.4.E Evaluate different displays of the same data for effectiveness and bias, and explain reasoning.

Example:

The following two bar graphs of the same data • show the number of five different types of sodas that were sold at Blake High School. Compare and contrast the two graphs. Describe a reason why you might choose to use one graph over the other.

Fizzy Cola

BakenSoda

Dr. SaltMountainDon’t

SnapcracklePop

1,541

1,542

1,543

1,544

1,545

Num

ber o

f Can

s of

Sod

a So

ld

Figure 1

Fizzy Cola

BakenSoda

Dr. SaltMountainDon’t

SnapcracklePop

1,535

1,540

1,545

1,550

1,555

Num

ber o

f Can

s of

Sod

a So

ld

Figure 2

Grade 7


Grade 7 7.5. Additional Key Content (Numbers, Algebra)

Students extend their coordinate graphing skills to plotting points with both positive and negative coordinates on the coordinate plane. Using pairs of numbers to locate points is a necessary skill for

reading maps and tables and a critical foundation for high school mathematics. Students further prepare for algebra by learning how to use exponents to write numbers in terms of their most basic (prime) factors.


7.5.A Graph ordered pairs of rational numbers and determine the coordinates of a given point in the coordinate plane.

Example:

Graph and label the points A(1, 2), B(• -1, 5), C(-3, 2), and D(-1, -5). Connect the points in the order listed and identify the figure formed by the four points.

Graph and label the points A(1, • -2), B(-4, -2), and C(-4, 3). Determine the coordinates of the fourth point (D) that will complete the figure to form a square. Graph and label point D on the coordinate plane and draw the resulting square.

7.5.B Write the prime factorization of whole numbers greater than 1, using exponents when appropriate.

Writing numbers in prime factorization is a useful tool for determining the greatest common factor and least common multiple of two or more numbers.

Example:

Write the prime factorization of 360 using exponents. •

Grade 7


Grade 77.6. Core Processes: Reasoning, problem solving, and communication

Students refine their reasoning and problem-solving skills as they move more fully into the symbolic world of algebra and higher-level mathematics. They move easily among representations—

numbers, words, pictures, or symbols—to understand and communicate mathematical ideas, to make generalizations, to draw logical conclusions, and to verify the reasonableness of solutions to problems. In grade seven, students solve problems that involve positive and negative numbers and often involve proportional relationships. As students solve these types of problems, they build a strong foundation for the study of linear functions that will come in grade eight.











Examples:

When working on a report for class, Catrina • read that a person over the age of 30 can lose approximately 0.06 centimeters of height per year. Catrina’s 80-year-old grandfather is 5 feet 7 inches tall. Assuming her grandfather’s height has decreased at this rate, about how tall was he at age 30? Catrina’s cousin, Richard, is 30 years old and is 6 feet 3 inches tall. Assuming his height also decreases approximately 0.06 centimeters per year after the age of 30, about how tall will you expect him to be at age 55? (Remember that 1 inch ≈ 2.54 centimeters.) Justify your solution.

If one man takes 1.5 hours to dig a • 5-ft × 5-ft × 3-ft hole, how long will it take three men working at the same pace to dig a 10-ft × 12-ft × 3-ft hole? Explain your solution.

Grade 7


Grade 8


Grade 8

Grade 8 8.1. Core Content: Linear functions and equations (Algebra)

Students solve a variety of linear equations and inequalities. They build on their familiarity with proportional relationships and simple linear equations to work with a broader set of linear

relationships, and they learn what functions are. They model applied problems with mathematical functions represented by graphs and other algebraic techniques. This Core Content area includes topics typically addressed in a high school algebra or a first-year integrated math course, but here this content is expected of all middle school students in preparation for a rich high school mathematics program that goes well beyond these basic algebraic ideas.


8.1.A Solve one-variable linear equations. Examples:

Solve each equation for x.

91 – 2.5• x = 26

• ( )78

2 119x− =

-• 3x + 34 = 5x

114 = • -2x – 8 + 5x

3(• x – 2) – 4x = 2(x + 22) – 5

8.1.B Solve one- and two-step linear inequalities and graph the solutions on the number line.

The emphasis at this grade level is on gaining experience with inequalities, rather than on becoming proficient at solving inequalities in which multiplying or dividing by a negative is necessary.

Example:

Graph the solution of 4• x – 21 > 57 on the number line.

8.1.C Represent a linear function with a verbal description, table, graph, or symbolic expression, and make connections among these representations.

Translating among these various representations of functions is an important way to demonstrate a conceptual understanding of functions.

8.1.D Determine the slope and y-intercept of a linear function described by a symbolic expression, table, or graph.

Examples:

Determine the slope and • y-intercept for the function described by

y x= −

23

5

The following table represents a linear function. • Determine the slope and y-intercept.

x

y

2

5

3

8

5

14

8

23

12

35


Grade 8


8.1.E Interpret the slope and y-intercept of the graph of a linear function representing a contextual situation.

Example:

A car is traveling down a long, steep hill. The • elevation, E, above sea level (in feet) of the car when it is d miles from the top of the hill is given by E = 7500 – 250d, where d can be any number from 0 to 6. Find the slope and y-intercept of the graph of this function and explain what they mean in the context of the moving car.

8.1.F Solve single- and multi-step word problems involving linear functions and verify the solutions.


Example:

Mike and Tim leave their houses at the same time • to walk to school. Mike’s walk can be represented by d1 = 4000 – 400t, and Tim’s walk can be represented by d2 = 3400 – 250t, where d is the distance from the school in feet and t is the walking time in minutes. Who arrives at school first? By how many minutes? Is there a time when Mike and Tim are the same distance away from the school? Explain your reasoning.

8.1.G Determine and justify whether a given verbal description, table, graph, or symbolic expression represents a linear relationship.

Examples:

Could the data presented in the table represent a • linear function? Explain your reasoning.

x -1 0 1 2 3 4 5y 0 -1 0 3 8 15 24

Does • = −y x 514

represent a linear function?

Explain your reasoning.


Grade 8

Grade 8 8.2. Core Content: Properties of geometric figures (Numbers, Geometry/Measurement)

Students work with lines and angles, especially as they solve problems involving triangles. They use known relationships involving sides and angles of triangles to find unknown measures, connecting

geometry and measurement in practical ways that will be useful well after high school. Since squares of numbers arise when using the Pythagorean Theorem, students work with squares and square roots, especially in problems with two- and three-dimensional figures. Using basic geometric theorems such as the Pythagorean Theorem, students get a preview of how geometric theorems are developed and applied in more formal settings, which they will further study in high school.


8.2.A Identify pairs of angles as complementary, supplementary, adjacent, or vertical, and use these relationships to determine missing angle measures.

Example:

Determine the measures of • ∠BOA, ∠EOD, ∠FOB, and ∠FOE and explain how you found each measure. As part of your explanation, identify pairs of angles as complementary, supplementary, or vertical.

O 36

A

B

C

D

F

E

8.2.B Determine missing angle measures using the relationships among the angles formed by parallel lines and transversals.

Example:

Determine the measures of the indicated angles. • ∠1: _____ ∠2: _____ ∠3: _____ ∠4: _____

40º 4 25º

1

3 2 m

n

m n



8.2.C Demonstrate that the sum of the angle measures in a triangle is 180 degrees, and apply this fact to determine the sum of the angle measures of polygons and to determine unknown angle measures.

Examples:

Determine the measure of each interior angle in a • regular pentagon.

In a certain triangle, the measure of one angle is • four times the measure of the smallest angle, and the measure of the remaining angle is the sum of the measures of the other two angles. Determine the measure of each angle.

8.2.D Represent and explain the effect of one or more translations, rotations, reflections, or dilations (centered at the origin) of a geometric figure on the coordinate plane.

Example:

Consider a trapezoid with vertices (1, 2), (1, 6), • (6, 4), and (6, 2). The trapezoid is reflected across the x-axis and then translated four units to the left. Graph the image of the trapezoid after these two transformations and give the coordinates of the new vertices.

8.2.E Quickly recall the square roots of the perfect squares from 1 through 225 and estimate the square roots of other positive numbers.

Students can use perfect squares of integers to determine squares and square roots of related numbers, such as 1.96 and 0.0049.

Examples:

Determine: • 36 0 25 144 196, , and . , .

Between which two consecutive integers does the • square root of 74 lie?

8.2.F Demonstrate the Pythagorean Theorem and its converse and apply them to solve problems.

One possible demonstration is to start with a right triangle, use each of the three triangle sides to form the side of a square, and draw the remaining three sides of each of the three squares. The areas of the three squares represent the Pythagorean relationship.

Examples:

Is a triangle with side lengths 5 cm, 12 cm, and • 13 cm a right triangle? Why or why not?

Determine the length of the diagonal of a rectangle • that is 7 ft by 10 ft.

8.2.G Apply the Pythagorean Theorem to determine the distance between two points on the coordinate plane.

Example:

Determine the distance between the points (• -2, 3) and (4, 7).

Grade 8


Grade 8 8.3. Core Content: Summary and analysis of data sets (Algebra, Data/Statistics/Probability)

Students build on their extensive experience organizing and interpreting data and apply statistical principles to analyze statistical studies or short statistical statements, such as those they might

encounter in newspapers, on television, or on the Internet. They use mean, median, and mode to summarize and describe information, even when these measures may not be whole numbers. Students use their knowledge of linear functions to analyze trends in displays of data. They create displays for two sets of data in order to compare the two sets and draw conclusions. They expand their work with probability to deal with more complex situations than they have previously seen. These concepts of statistics and probability are important not only in students’ lives, but also throughout the high school mathematics program.


8.3.A Summarize and compare data sets in terms of variability and measures of center.

Students use mean, median, mode, range, and inter-quartile range to summarize and compare data sets, and explain the influence of outliers on each measure.

Example:

Captain Bob owns two charter boats, the • Sock-Eye-To-Me and Old Gus, which take tourists on fishing trips. On Saturday, the Sock-Eye-To-Me took four people fishing and returned with eight fish weighing 18, 23, 20, 6, 20, 22, 18, and 20 pounds. On the same day, Old Gus took five people fishing and returned with ten fish weighing 38, 18, 12, 14, 17, 42, 12, 16, 12, and 14 pounds.

Using measures of center and variability, compare the weights of the fish caught by the people in the two boats.

Make a summary statement telling which boat you would charter for fishing based on these data and why.

What influence, if any, do outliers have on the particular statistics for these data?

8.3.B Select, construct, and analyze data displays, including box-and-whisker plots, to compare two sets of data.

Previously studied displays include stem-and-leaf plots, histograms, circle graphs, and line plots. Here these displays are used to compare data sets. Box-and-whisker plots are introduced here for the first time as a powerful tool for comparing two or more data sets.

Grade 8



8.3.B cont. Example:

As part of their band class, Tayla and Alyssa are • required to keep practice records that show the number of minutes they practice their instruments each day. Below are their practice records for the past fourteen days:

Tayla: 55, 45, 60, 45, 30, 30, 90, 50, 40, 75, 25, 90, 105, 60

Alyssa: 20, 120, 25, 20, 0, 15, 30, 15, 90, 0, 30, 30, 10, 30

Of stem-and-leaf plot, circle graph, or line plot, select the data display that you think will best compare the two girls’ practice records. Construct a display to show the data. Compare the amount of time the two girls practice by analyzing the data presented in the display.

8.3.C Create a scatterplot for a two-variable data set, and, when appropriate, sketch and use a trend line to make predictions.

Example:

Kera randomly selected seventeen students from • her middle school for a study comparing arm span to standing height. The students’ measurements are shown in the table below.

Comparison of Arm Span and Standing Height (in cm) at Icicle River Middle School

Height (cm)

Height (cm)

Arm Span (cm)

Arm Span (cm)

135

142

158

177

150

158

160

138

135

147

145

174

152

162

160

145

175

162

150

142

149

160

173

155

177

160

152

143

149

165

170

150

160 158

Create a scatterplot for the data shown.

If appropriate, sketch a trendline.

Use these data to estimate the arm span of a student with a height of 180 cm, and the height of a student with an arm span of 130 cm. Explain any limitations of using this process to make estimates.

Grade 8



8.3.D Describe different methods of selecting statistical samples and analyze the strengths and weaknesses of each method.

Students should work with a variety of sampling techniques and should be able to identify strengths and weaknesses of random, census, convenience, and representative sampling.

Example:

Carli, Jamar, and Amberly are conducting a • survey to determine their school’s favorite Seattle professional sports team. Carli selects her sample using a convenience method—she surveys students on her bus during the ride to school. Jamar uses a computer to randomly select 30 numbers from 1 through 600, and then surveys the corresponding students from a numbered, alphabetical listing of the student body. Amberly waits at the front entrance before school and surveys every twentieth student entering. Analyze the strengths and weaknesses of each method.

8.3.E Determine whether conclusions of statistical studies reported in the media are reasonable.

8.3.F Determine probabilities for mutually exclusive, dependent, and independent events for small sample spaces.

Examples:

Given a standard deck of 52 playing cards, what is • the probability of drawing a king or queen? [Some students may be unfamiliar with playing cards, so alternate examples may be desirable.]

Leyanne is playing a game at a birthday party. • Beneath ten paper cups, a total of five pieces of candy are hidden, with one piece hidden beneath each of five cups. Given only three guesses, Leyanne must uncover three pieces of candy to win all the hidden candy. What is the probability she will win all the candy?

A bag contains 7 red marbles, 5 blue marbles, and • 8 green marbles. If one marble is drawn at random and put back in the bag, and then a second marble is drawn at random, what is the probability of drawing first a red marble, then a blue marble?

Grade 8



8.3.G Solve single- and multi-step problems using counting techniques and Venn diagrams and verify the solutions.


Counting techniques include the fundamental counting principle, lists, tables, tree diagrams, etc.

Examples:

Jack’s Deli makes sandwiches that include a • choice of one type of bread, one type of cheese, and one type of meat. How many different sandwiches could be made given 4 different bread types, 3 different cheeses, and 5 different meats? Explain your reasoning.

A small high school has 57 tenth-graders. Of • these students, 28 are taking geometry, 34 are taking biology, and 10 are taking neither geometry nor biology. How many students are taking both geometry and biology? How many students are taking geometry but not biology? How many students are taking biology but not geometry? Draw a Venn diagram to illustrate this situation.

Grade 8


Grade 8 8.4. Additional Key Content (Numbers, Operations)

Students deal with a few key topics about numbers as they prepare to shift to higher level mathematics in high school. First, they use scientific notation to represent very large and very small numbers,

especially as these numbers are used in technological fields and in everyday tools like calculators or personal computers. Scientific notation has become especially important as “extreme units” continue to be identified to represent increasingly tiny or immense measures arising in technological fields. A second important numerical skill involves using exponents in expressions containing both numbers and variables. Developing this skill extends students’ work with order of operations to include more complicated expressions they might encounter in high school mathematics. Finally, to help students understand the full breadth of the real-number system, students are introduced to simple irrational numbers, thus preparing them to study higher level mathematics in which properties and procedures are generalized for the entire set of real numbers.


8.4.A Represent numbers in scientific notation, and translate numbers written in scientific notation into standard form.

Examples:

Represent 4.27 • x 10-3 in standard form.

Represent 18,300,000 in scientific notation.•

Throughout the year 2004, people in the city of • Cantonville sent an average of 400 million text messages a day. Using this information, about how many total text messages did Cantonville residents send in 2004? (2004 was a leap year.) Express your answer in scientific notation.

8.4.B Solve problems involving operations with numbers in scientific notation and verify solutions.

Units include those associated with technology, such as nanoseconds, gigahertz, kilobytes, teraflops, etc.

Examples:

A supercomputer used by a government agency • will be upgraded to perform 256 teraflops (that is, 256 trillion calculations per second). Before the upgrade, the supercomputer performs 1.6 x 1013 calculations per second. How many more calculations per second will the upgraded supercomputer be able to perform? Express the answer in scientific notation.

A nanosecond is one billionth of a second. How • many nanoseconds are there in five minutes? Express the answer in scientific notation.

Grade 8



8.4.C Evaluate numerical expressions involving non-negative integer exponents using the laws of exponents and the order of operations.

Example:

Simplify and write the answer in exponential form:•

( )7

7

4 2

3

Some students will be ready to solve problems involving simple negative exponents and should be given the opportunity to do so.

Example:

Simplify and write the answer in exponential form:•

(54)25-3

8.4.D Identify rational and irrational numbers. Students should know that rational numbers are numbers that can be represented as the ratio of two integers; that the decimal expansions of rational numbers have repeating patterns, or terminate; and that there are numbers that are not rational.

Example:

Identify whether each number is rational or • irrational and explain your choice.

3.14, 4.6,1

112 25 π, , ,

Grade 8



Students refine their reasoning and problem-solving skills as they move more fully into the symbolic world of algebra and higher level mathematics. They move easily among representations—

numbers, words, pictures, or symbols—to understand and communicate mathematical ideas, to make generalizations, to draw logical conclusions, and to verify the reasonableness of solutions to problems. In grade eight, students solve problems that involve proportional relationships and linear relationships, including applications found in many contexts outside of school. These problems dealing with proportionality continue to be important in many applied contexts, and they lead directly to the study of algebra. Students also begin to deal with informal proofs for theorems that will be proven more formally in high school.










Descriptions of solution processes and explanations can include numbers, words (including mathematical language), pictures, or equations. Students should be able to use all of these representations as needed. For a particular solution, students should be able to explain or show their work using at least one of these representations and verify that their answer is reasonable.

Examples:

The dimensions of a room are • 12 feet by 15 feet by 10 feet. What is the furthest distance between any two points in the room? Explain your solution.

Miranda’s phone service contract ends this • month. She is looking for ways to save money and is considering changing phone companies. Her current cell phone carrier, X-Cell, calculates the monthly bill using the equation c = $15.00 + $0.07m, where c represents the total monthly cost and m represents the number of minutes of talk time during a monthly billing cycle. Another company, Prism Cell, offers 300 free minutes of talk time each month for a base fee of $30.00 with an additional $0.15 for every minute over 300 minutes. Miranda’s last five phone bills were $34.95, $36.70, $37.82, $62.18, and $36.28. Using the data from the last five months, help Miranda decide whether she should switch companies. Justify your answer.

Grade 8

Grade 8


Algebra 1


Algebra I

Algebra 1A1.1. Core Content: Solving problems (Algebra)

Students learn to solve many new types of problems in Algebra 1, and this first core content area highlights the types of problems students will be able to solve after they master the concepts and skills

in this course. Students are introduced to several types of functions, including exponential and functions defined piecewise, and they spend considerable time with linear and quadratic functions. Each type of function included in Algebra 1 provides students a tool to solve yet another class of problems. They learn that specific functions model situations described in word problems, and so functions are used to solve various types of problems. The ability to determine functions and write equations that represent problems is an important mathematical skill in itself. Many problems that initially appear to be very different from each other can actually be represented by identical equations. Students encounter this important and unifying principle of algebra—that the same algebraic techniques can be applied to a wide variety of different situations.


A1.1.A Select and justify functions and equations to model and solve problems.

Students can analyze the rate of change of a function represented with a table or graph to determine if the function is linear. Students also analyze common ratios to determine if the function is exponential.

After selecting a function to model a situation, students describe appropriate domain restrictions. They use the function to solve the problem and interpret the solution in the context of the original situation.

Examples:A cup is 6 cm tall, including a 1.1 cm lip. Find a • function that represents the height of a stack of cups in terms of the number of cups in the stack. Find a function that represents the number of cups in a stack of a given height. For the month of July, Michelle will be dog-sitting • for her very wealthy, but eccentric, neighbor, Mrs. Buffett. Mrs. Buffett offers Michelle two different salary plans:

Plan 1: $100 per day for the 31 days of — the month.Plan 2: $1 for July 1, $2 for July 2, $4 — for July 3, and so on, with the daily rate doubling each day.

Write functions that model the amount of a. money Michelle will earn each day on Plan 1 and Plan 2. Justify the functions you wrote.State an appropriate domain for each of the b. models based on the context.Which plan should Michelle choose to c. maximize her earnings? Justify your recommendation mathematically.Extension: Write an algebraic function for the d. cumulative pay for each plan based on the number of days worked.



A1.1.B Solve problems that can be represented by linear functions, equations, and inequalities.

It is mathematically important to represent a word problem as an equation. Students must analyze the situation and find a way to represent it mathematically. After solving the equation, students think about the solution in terms of the original problem.

Examples:

The assistant pizza maker makes 6 pizzas an • hour. The master pizza maker makes 10 pizzas an hour but starts baking two hours later than his assistant. Together, they must make 92 pizzas. How many hours from when the assistant starts baking will it take?

What is a general equation, in function form, that could be used to determine the number of pizzas that can be made in two or more hours?

A swimming pool holds 375,000 liters of water. • Two large hoses are used to fill the pool. The first hose fills at the rate of 1,500 liters per hour and the second hose fills at the rate of 2,000 liters per hour. How many hours does it take to fill the pool completely?

A1.1.C Solve problems that can be represented by a system of two linear equations or inequalities.

Examples:

An airplane flies from Baltimore to Seattle (assume • a distance of 2,400 miles) in 7 hours, but the return

flight takes only 41

2 hours. The air speed of the

plane is the same in both directions. How many miles per hour does the plane fly with respect to the wind? What is the wind speed in miles per hour?

A coffee shop employee has one cup of 85% milk • (the rest is chocolate) and another cup of 60% milk (the rest is chocolate). He wants to make one cup of 70% milk. How much of the 85% milk and 60% milk should he mix together to make the 70% milk?

Two plumbing companies charge different rates • for their service. Clyde’s Plumbing Company charges a $75-per-visit fee that includes one hour of labor plus $45 dollars per hour after the first hour. We-Unclog-It Plumbers charges a $100-per-visit fee that includes one hour of labor plus $40 per hour after the first hour. For how many hours of plumbing work would Clyde’s be less expensive than We-Unclog-It?

Note: Although this context is discrete, students can model it with continuous linear functions.

Algebra I



A1.1.D Solve problems that can be represented by quadratic functions and equations.

Examples:

Find the solutions to the simultaneous equations • y = x + 2 and y = x2.

If you throw a ball straight up (with initial height of • 4 feet) at 10 feet per second, how long will it take to fall back to the starting point? The function h(t) = -16t2 + v0t + h0 describes the height, h in feet, of an object after t seconds, with initial velocity v0 and initial height h0.

Joe owns a small plot of land 20 feet by 30 feet. • He wants to double the area by increasing both the length and the width, keeping the dimensions in the same proportion as the original. What will be the new length and width?

What two consecutive numbers, when multiplied • together, give the first number plus 16? Write the equation that represents the situation.

A1.1.E Solve problems that can be represented by exponential functions and equations.

Students approximate solutions with graphs or tables, check solutions numerically, and when possible, solve problems exactly.

Examples:

E. coli bacteria reproduce by a simple process • called binary fission—each cell increases in size and divides into two cells. In the laboratory, E. coli bacteria divide approximately every 15 minutes. A new E. coli culture is started with 1 cell.

Find a function that models the E. coli a. population size at the end of each 15-minute interval. Justify the function you found.

State an appropriate domain for the model b. based on the context.

After what 15-minute interval will you have at c. least 500 bacteria?

Estimate the solution to 2• x = 16,384.

Algebra I


Algebra 1A1.2. Core Content: Numbers, expressions, and operations (Numbers, Operations, Algebra)

Students see the number system extended to the real numbers represented by the number line. They work with integer exponents, scientific notation, and radicals, and use variables and expressions to

solve problems from purely mathematical as well as applied contexts. They build on their understanding of computation using arithmetic operations and properties and expand this understanding to include the symbolic language of algebra. Students demonstrate this ability to write and manipulate a wide variety of algebraic expressions throughout high school mathematics as they apply algebraic procedures to solve problems.


A1.2.A Know the relationship between real numbers and the number line, and compare and order real numbers with and without the number line.

Although a formal definition of real numbers is beyond the scope of Algebra 1, students learn that every point on the number line represents a real number, either rational or irrational, and that every real number has its unique point on the number line. They locate, compare, and order real numbers on the number line.

Real numbers include those written in scientific notation or expressed as fractions, decimals, exponentials, or roots.

Examples:

Without using a calculator, order the following on • the number line:

82 , 3π, 8.9, 9, 374

, 9.3 × 100

A star’s• color gives an indication of its temperature and age. The chart shows four types of stars and the lowest temperature of each type.

Type Lowest Temperature(in ºF)

Color

A

B

G

P

1.35 x 104

2.08 x 104

9.0 x 103

4.5 x 104

Blue-White

Blue

Blue

Yellow

List the temperatures in order from lowest to highest.

Algebra I



A1.2.B Recognize the multiple uses of variables, determine all possible values of variables that satisfy prescribed conditions, and evaluate algebraic expressions that involve variables.

Students learn to use letters as variables and in other ways that increase in sophistication throughout high school. For example, students learn that letters can be used:

To represent fixed and temporarily unknown values • in equations, such as 3x + 2 = 5;

To express identities, such as • x + x = 2x for all x;

As attributes in formulas, such as • A = lw;

As constants such as • a, b, and c in the equation y = ax2 + bx + c;

As parameters in equations, such as the • m and b for the family of functions defined by y = mx + b;

To represent varying quantities, such as • x in f(x) = 5x;

To represent functions, such as • f in f(x) = 5x; and

To represent specific numbers, such as • π.

Expressions include those involving polynomials, radicals, absolute values, and integer exponents.

Examples:

For what values of • a and n, where n is an integer greater than 0, is an always negative?

For what values of • a is 1a

an integer?

For what values of • a is 5 − a defined? For what values of • a is -a always positive?

A1.2.C Interpret and use integer exponents and square and cube roots, and apply the laws and properties of exponents to simplify and evaluate exponential expressions.

Examples:

2• -3 = 1

23

• -22 3 5

2 3 5

2

-3 22= 3

2 54

5

• a b c

a b c

b

a c

-

-

2 2

2 3 2

5

4=

• 8 2 2 2 2 2= • • =

• a b a b• = •3 3 3

Algebra I



A1.2.D Determine whether approximations or exact values of real numbers are appropriate, depending on the context, and justify the selection.

Decimal approximations of numbers are sometimes used in applications, such as carpentry or engineering; while at other times, these applications may require exact values. Students should understand the difference and know that the appropriate approximation depends upon the necessary degree of precision needed in given situations.

For example, 1.414 is an approximation and not an

exact solution to the equation x2 – 2 = 0, but 2 is an exact solution to this equation.

Example:

Using a common engineering formula, an • engineering student represented the maximum

safe load of a bridge to be 1000(99 – 70 2) tons.

He used 1.41 as the approximation for 2 in his calculations. When the bridge was built and tested in a computer simulation to verify its maximum weight-bearing load, it collapsed! The student had estimated the bridge would hold ten times the weight that was applied to it when it collapsed.

Calculate the weight that the student — thought the bridge could bear using 1.41

as the estimate for 2.

Calculate other weight values using — estimates of 2 that have more decimal places. What might be a reasonable degree of precision required to know how much weight the bridge can handle safely? Justify your answer.

A1.2.E Use algebraic properties to factor and combine like terms in polynomials.

Algebraic properties include the commutative, associative, and distributive properties.

Factoring includes:

Factoring a monomial from a polynomial, such as • 4x2 + 6x = 2x(2x + 3);

Factoring the difference of two squares, such as • 36x2 – 25y2 = (6x + 5y)(6x – 5y) and x4 – y4 = (x + y)(x – y)(x2 + y2);

Factoring perfect square trinomials, such as • x2 + 6xy + 9y2 = (x + 3y)2;

Factoring quadratic trinomials, such as • x2 + 5x + 4 = (x + 4)(x + 1); and

Factoring trinomials that can be expressed as the • product of a constant and a trinomial, such as 0.5x2 – 2.5x – 7 = 0.5(x + 2)(x – 7).

Algebra I



A1.2.F Add, subtract, multiply, and divide polynomials. Write algebraic expressions in equivalent forms using algebraic properties to perform the four arithmetic operations with polynomials.

Students should recognize that expressions are essentially sums, products, differences, or quotients. For example, the sum 2x2 + 4x can be written as a product, 2x(x + 2).

Examples:

(3• x2 – 4x + 5) + (-x2 + x – 4) + (2x2 + 2x + 1)

(2• x2 – 4) – (x2 + 3x – 3)

• 29

62

2

4

xx

•

• x – – 2 2 31x

x +

Algebra I


Algebra 1A1.3. Core Content: Characteristics and behaviors of functions (Algebra)

Students formalize and deepen their understanding of functions, the defining characteristics and uses of functions, and the mathematical language used to describe functions. They learn that functions are

often specified by an equation of the form y = f(x), where any allowable x-value yields a unique y-value. While Algebra 1 has a particular focus on linear and quadratic equations and systems of equations, students also learn about exponential functions and those that can be defined piecewise, particularly step functions and functions that contain the absolute value of an expression. Students learn about the representations and basic transformations of these functions and the practical and mathematical limitations that must be considered when working with functions and when using functions to model situations.


A1.3.A Determine whether a relationship is a function and identify the domain, range, roots, and independent and dependent variables.

Functions studied in Algebra 1 include linear, quadratic, exponential, and those defined piecewise (including step functions and those that contain the absolute value of an expression).

Given a problem situation, students should describe further restrictions on the domain of a function that are appropriate for the problem context.

Examples:

Which of the following are functions? Explain why • or why not.

The age in years of each student in your — math class and each student’s shoe size.

The number of degrees a person rotates a — spigot and the volume of water that comes out of the spigot.

A function• f(n) = 60n is used to model the distance in miles traveled by a car traveling 60 miles per hour in n hours. Identify the domain and range of this function. What restrictions on the domain of this function should be considered for the model to correctly reflect the situation?

What is the domain of • f(x) = 5 − x ?

Algebra I



A1.3.A cont. Which of the following equations, inequalities, or • graphs determine y as a function of x?

y— = 2

x— = 3

y— = |x|

— yx xx x

=+ ≤− >

3 12 1,,

x— 2 + y2 = 1

1

1

-1

-1 x

y

A1.3.B Represent a function with a symbolic expression, as a graph, in a table, and using words, and make connections among these representations.

This expectation applies each time a new class (family) of functions is encountered. In Algebra 1, students should be introduced to a variety of additional functions

that include expressions such as x3, x , 1

x, and

absolute values. They will study these functions in depth in subsequent courses.

Students should know that f(x) = a

x represents an

inverse variation. Students begin to describe the graph of a function from its symbolic expression, and use key characteristics of the graph of a function to infer properties of the related symbolic expression.

Translating among these various representations of functions is an important way to demonstrate conceptual understanding of functions.

Students learn that each representation has particular advantages and limitations. For example, a graph shows the shape of a function, but not exact values. They also learn that a table of values may not uniquely determine a single function without some specification of the nature of that function (e.g., it is quadratic).

Algebra I



A1.3.C Evaluate f(x) at a (i.e., f(a)) and solve for x in the equation f(x) = b.

Functions may be described and evaluated with symbolic expressions, tables, graphs, or verbal descriptions.

Students should distinguish between solving for f(x) and evaluating a function at x.

Example:

Roses-R-Red sells its roses for $0.75 per stem • and charges a $20 delivery fee per order.

What is the cost of having 10 roses delivered?—

How many roses can you have delivered — for $65?

Algebra I


Algebra 1A1.4. Core Content: Linear functions, equations, and inequalities (Algebra)

Students understand that linear functions can be used to model situations involving a constant rate of change. They build on the work done in middle school to solve sets of linear equations and

inequalities in two variables, learning to interpret the intersection of the lines as the solution. While the focus is on solving equations, students also learn graphical and numerical methods for approximating solutions to equations. They use linear functions to analyze relationships, represent and model problems, and answer questions. These algebraic skills are applied in other Core Content areas across high school courses.


A1.4.A Write and solve linear equations and inequalities in one variable.

This expectation includes the use of absolute values in the equations and inequalities.

Examples:Write an absolute value equation or inequality for•

all the numbers 2 units from 7, and— all the numbers that are more than — b units from a.

Solve |• x – 6| ≤ 4 and locate the solution on the number line.Write an equation or inequality that has •

no real solutions; — infinite numbers of real solutions; and — exactly one real solution.—

Solve for • x in 2(x – 3) + 4x = 15 + 2x.Solve• 8.5 < 3x + 2 ≤ 9.7 and locate the solution on the number line.

A1.4.B Write and graph an equation for a line given the slope and the y-intercept, the slope and a point on the line, or two points on the line, and translate between forms of linear equations.

Linear equations may be written in slope-intercept, point-slope, and standard form.

Examples:Find an equation for a line with • y-intercept equal to 2 and slope equal to 3.Find an equation for a line with a slope of 2 that • goes through the point (1, 1).Find an equation for a line that goes through the • points (-3, 5) and (6, -2).For each of the following, use only the equation • (without sketching the graph) to describe the graph.

y— = 2x + 3y— – 7 = 2(x – 2)

Write the equation 3• x + 2y = 5 in slope intercept form.Write the equation • y – 1 = 2(x – 2) in standard form.

Algebra I



A1.4.C Identify and interpret the slope and intercepts of a linear function, including equations for parallel and perpendicular lines.

Examples:

The graph shows the relationship between time and • distance from a gas station for a motorcycle and a scooter. What can be said about the relative speed of the motorcycle and scooter that matches the information in the graph? What can be said about the intersection of the graphs of the scooter and the motorcycle? Is it possible to tell which vehicle is further from the gas station at the initial starting point represented in the graph? At the end of the time represented in the graph? Why or why not?

scooter

motorcycle

Time

Dis

tanc

e

A 1,500-gallon tank contains 200 gallons of water. • Water begins to run into the tank at the rate of 75 gallons per hour. When will the tank be full? Find a linear function that models this situation, draw a graph, and create a table of data points. Once you have answered the question and completed the tasks, explain your reasoning. Interpret the slope and y-intercept of the function in the context of the situation.

Given that the figure below is a square, find the • slope of the perpendicular sides AB and BC. Describe the relationship between the two slopes.

p

pq

q

A

B

C

D

Algebra I



A1.4.D Write and solve systems of two linear equations and inequalities in two variables.

Students solve both symbolic and word problems, understanding that the solution to a problem is given by the coordinates of the intersection of the two lines when the lines are graphed in the same coordinate plane.

Examples:

Solve the following simultaneous linear equations • algebraically:

-2— x + y = 2

x— + y = -1

Graph the above two linear equations on the same • coordinate plane and use the graph to verify the algebraic solution.

An academic team is going to a state mathematics • competition. There are 30 people going on the trip. There are 5 people who can drive and 2 types of vehicles, vans and cars. A van seats 8 people, and a car seats 4 people, including drivers. How many vans and cars does the team need for the trip? Explain your reasoning.

Let v = number of vans and c = number of cars.

v + c ≤ 5

8v + 4c > 30

A1.4.E Describe how changes in the parameters of linear functions and functions containing an absolute value of a linear expression affect their graphs and the relationships they represent.

In the case of a linear function y = f(x), expressed in slope-intercept form (y = mx + b), m and b are parameters. Students should know that f(x) = kx represents a direct variation (proportional relationship).

Examples:

Graph a function of the form • f(x) = kx, describe the effect that changes on k have on the graph and on f(x), and answer questions that arise in proportional situations.

A gas station’s 10,000-gallon underground storage • tank contains 1,000 gallons of gasoline. Tanker trucks pump gasoline into the tank at a rate of 400 gallons per minute. How long will it take to fill the tank? Find a function that represents this situation and then graph the function. If the flow rate increases from 400 to 500 gallons per minute, how will the graph of the function change? If the initial amount of gasoline in the tank changes from 1,000 to 2,000 gallons, how will the graph of the function change?

Compare and contrast the functions • y = 3|x| and

y x=1

3- .

Algebra I


Algebra 1A1.5. Core Content: Quadratic functions and equations (Algebra)

Students study quadratic functions and their graphs, and solve quadratic equations with real roots in Algebra 1. They use quadratic functions to represent and model problems and answer questions

in situations that are modeled by these functions. Students solve quadratic equations by factoring and computing with polynomials. The important mathematical technique of completing the square is developed enough so that the quadratic formula can be derived.


A1.5.A Represent a quadratic function with a symbolic expression, as a graph, in a table, and with a description, and make connections among the representations.

Example:

Kendre and Tyra built a tennis ball cannon that • launches tennis balls straight up in the air at an initial velocity of 50 feet per second. The mouth of the cannon is 2 feet off the ground. The function h(t) = -16t2 + 50t + 2 describes the height, h, in feet, of the ball t seconds after the launch.Make a table from the function. Then use the table to sketch a graph of the height of the tennis ball as a function of time into the launch. Give a verbal description of the graph. How high was the ball after 1 second? When does it reach this height again?

A1.5.B Sketch the graph of a quadratic function, describe the effects that changes in the parameters have on the graph, and interpret the x-intercepts as solutions to a quadratic equation.

Note that in Algebra 1, the parameter b in the term bx in the quadratic form ax2 + bx + c is not often used to provide useful information about the characteristics of the graph.

Parameters considered most useful are:

a• and c in f(x) = ax2 + c

a• , h, and k in f(x) = a(x – h)2 + k, and

a• , r, and s in f(x) = a(x – r)(x – s)

Example:

A particular quadratic function can be expressed in • the following two ways:f(x) = -(x – 3)2 + 1 f(x) = -(x – 2)(x – 4)

What information about the graph can be — directly inferred from each of these forms? Explain your reasoning.

Sketch the graph of this function, showing — the roots.

Algebra I



A1.5.C Solve quadratic equations that can be factored as (ax + b)(cx + d) where a, b, c, and d are integers.

Students learn to efficiently solve quadratic equations by recognizing and using the simplest factoring methods, including recognizing special quadratics as squares and differences of squares.

Examples:

2• x2 + x – 3 = 0; (x – 1)(2x + 3) = 0; x = 1, -−32

4• x2 + 6x = 0; 2x(2x + 3) = 0; x = 0, -−32

36• x2 – 25 = 0; (6x + 5)(6x – 5) = 0; x = ±

56

x• 2 + 6x + 9 = 0; (x + 3)2 = 0; x = -3

A1.5.D Solve quadratic equations that have real roots by completing the square and by using the quadratic formula.

Students solve those equations that are not easily factored by completing the square and by using the quadratic formula. Completing the square should also be used to derive the quadratic formula.

Students learn how to determine if there are two, one, or no real solutions.

Examples:

Complete the square to solve • x2 + 4x = 13.

x2 + 4x – 13 = 0

x2 + 4x + 4 = 17

(x + 2)2 = 17

x + 2 = ± 17x = -2 ± 17x ≈ 2.12, -6.12

Use the quadratic formula to solve 4• x2 – 2x = 5.

- x b b aca

= ± 2 4

2 –

- - - ( ) 2 2x

x

x

x

=

= ±

= ±

=

–( ) ( ) 2 4

2 8 4 8

2 2 21 8

1

2

± ± 21 4

4(4 -5)

x ≈ 1.40, -0.90

Algebra I


Algebra 1A1.6. Core Content: Data and distributions (Data/Statistics/Probability)

Students select mathematical models for data sets and use those models to represent, describe, and compare data sets. They analyze data to determine the relationship between two variables and make

and defend appropriate predictions, conjectures, and generalizations. Students understand limitations of conclusions based on results of a study or experiment and recognize common misconceptions and misrepresentations in interpreting conclusions.


A1.6.A Use and evaluate the accuracy of summary statistics to describe and compare data sets.

A univariate set of data identifies data on a single variable, such as shoe size.

This expectation extends what students have learned in earlier grades to include evaluation and justification. They both compute and evaluate the appropriateness of measure of center and spread (range and interquartile range) and use these measures to accurately compare data sets. Students will draw appropriate conclusions through the use of statistical measures of center, frequency, and spread, combined with graphical displays.

Examples:

The local minor league baseball team has a salary • dispute. Players claim they are being underpaid, but managers disagree.

Bearing in mind that a few top players — earn salaries that are quite high, would it be in the managers’ best interest to use the mean or median when quoting the “average” salary of the team? Why?

What would be in the players’ best interest?—

Each box-and-whisker plot shows the prices of • used cars (in thousands of dollars) advertised for sale at three different car dealers. If you want to go to the dealer whose prices seem least expensive, which dealer would you go to? Use statistics from the displays to justify your answer.

Cars are US

Better-than-New

Yours Now

50 10

Algebra I



A1.6.B Make valid inferences and draw conclusions based on data.

Determine whether arguments based on data confuse association with causation. Evaluate the reasonableness of and make judgments about statistical claims, reports, studies, and conclusions.

Example:

Mr. Shapiro found that the amount of time his • students spent doing mathematics homework is positively correlated with test grades in his class. He concluded that doing homework makes students’ test scores higher. Is this conclusion justified? Explain any flaws in Mr. Shapiro’s reasoning.

A1.6.C Describe how linear transformations affect the center and spread of univariate data.

Examples:

A company decides to give every one of its • employees a $5,000 raise. What happens to the mean and standard deviation of the salaries as a result?

A company decides to double each of its • employee’s salaries. What happens to the mean and standard deviation of the salaries as a result?

A1.6.D Find the equation of a linear function that best fits bivariate data that are linearly related, interpret the slope and y-intercept of the line, and use the equation to make predictions.

A bivariate set of data presents data on two variables, such as shoe size and height.

In high school, the emphasis is on using a line of best fit to interpret data and on students making judgments about whether a bivariate data set can be modeled with a linear function. Students can use various methods, including technology, to obtain a line of best fit.

Making predictions involves both interpolating and extrapolating from the original data set.

Students need to be able to evaluate the quality of their predictions, recognizing that extrapolation is based on the assumption that the trend indicated continues beyond the unknown data.

Algebra I



A1.6.E Describe the correlation of data in scatterplots in terms of strong or weak and positive or negative.

Example:

Which words—• strong or weak, positive or negative—could be used to describe the correlation shown in the sample scatterplot below?

100 200 300 400 500-4

-3.5

-3

-2.5

-2

-1.5

-1

xx

xx

xxxxx

xxxx

x

xxxxx

xx

Scatterplot

X

Y

Algebra I


Algebra 1A1.7. Additional Key Content (Algebra)

Students develop a basic understanding of arithmetic and geometric sequences and of exponential functions, including their graphs and other representations. They use exponential functions to analyze

relationships, represent and model problems, and answer questions in situations that are modeled by these nonlinear functions. Students learn graphical and numerical methods for approximating solutions to exponential equations. Students interpret the meaning of problem solutions and explain limitations related to solutions.


A1.7.A Sketch the graph for an exponential function of the form y = abn where n is an integer, describe the effects that changes in the parameters a and b have on the graph, and answer questions that arise in situations modeled by exponential functions.

Examples:

Sketch the graph of • y = 2n by hand.

You have won a door prize and are given a choice • between two options:

$150 invested for 10 years at 4% compounded annually.$200 invested for 10 years at 3% compounded annually.

How much is each worth at the end of — each year of the investment periods?

Are the two investments ever equal in — value? Which will you choose?

A1.7.B Find and approximate solutions to exponential equations.

Students can approximate solutions using graphs or tables with and without technology.

A1.7.C Express arithmetic and geometric sequences in both explicit and recursive forms, translate between the two forms, explain how rate of change is represented in each form, and use the forms to find specific terms in the sequence.

Examples:

Write a recursive formula for the arithmetic • sequence 5, 9, 13, 17, . . . What is the slope of the line that contains the points associated with these values and their position in the sequence? How is the slope of the line related to the sequence?

Given that • u(0) = 3 and u(n + 1) = u(n) + 7 when n is a positive integer,

a. find u(5);

b. find n so that u(n) = 361; and

c. find a formula for u(n).

Write a recursive formula for the geometric sequence • 5, 10, 20, 40, . . . and determine the 100th term.

Given that • u(0) = 2 and u(n + 1) = 3u(n),

a. find u(4), and

b. find a formula for u(n).

A1.7.D Solve an equation involving several variables by expressing one variable in terms of the others.

Examples:

Solve • A = p + prt for p.

Solve • V = πr 2h for h or for r.

Algebra I


Algebra 1A1.8. Core Processes: Reasoning, problem solving, and communication

Students formalize the development of reasoning in Algebra 1 as they use algebra and the properties of number systems to develop valid mathematical arguments, make and prove conjectures, and find

counterexamples to refute false statements, using correct mathematical language, terms, and symbols in all situations. They extend the problem-solving practices developed in earlier grades and apply them to more challenging problems, including problems related to mathematical and applied situations. Students formalize a coherent problem-solving process in which they analyze the situation to determine the question(s) to be answered, synthesize given information, and identify implicit and explicit assumptions that have been made. They examine their solution(s) to determine reasonableness, accuracy, and meaning in the context of the original problem. The mathematical thinking, reasoning, and problem-solving processes students learn in high school mathematics can be used throughout their lives as they deal with a world in which an increasing amount of information is presented in quantitative ways and more and more occupations and fields of study rely on mathematics.


A1.8.A Analyze a problem situation and represent it mathematically.

A1.8.B Select and apply strategies to solve problems.

A1.8.C Evaluate a solution for reasonableness, verify its accuracy, and interpret the solution in the context of the original problem.

A1.8.D Generalize a solution strategy for a single problem to a class of related problems, and apply a strategy for a class of related problems to solve specific problems.

A1.8.E Read and interpret diagrams, graphs, and text containing the symbols, language, and conventions of mathematics.

A1.8.F Summarize mathematical ideas with precision and efficiency for a given audience and purpose.

A1.8.G Synthesize information to draw conclusions, and evaluate the arguments and conclusions of others.

A1.8.H Use inductive reasoning about algebra and the properties of numbers to make conjectures, and use deductive reasoning to prove or disprove conjectures.

Examples:

Three teams of students independently conducted • experiments to relate the rebound height of a ball to the rebound number. The table gives the average of the teams’ results.Construct a scatterplot of the data, and describe the function that relates the height of the ball to the rebound number. Predict the rebound height of the ball on the tenth rebound. Justify your answer.

ReboundNumber

ReboundHeight (cm)

0 200

1 155

2 116

3 88

4 66

5 50

6 44

Prove (• a + b)2 = a2 + 2ab + b2.

A student writes (• x + 3)2 = x2 + 9. Explain why this is incorrect.

Prove formally that the sum of two odd numbers is • always even.

Algebra I


Geometry


Geometry G.1. Core Content: Logical arguments and proofs (Logic)

Students formalize the reasoning skills they have developed in previous grades and solidify their understanding of what it means to prove a geometric statement mathematically. In Geometry,

students encounter the concept of formal proof built on definitions, axioms, and theorems. They use inductive reasoning to test conjectures about geometric relationships and use deductive reasoning to prove or disprove their conclusions. Students defend their reasoning using precise mathematical language and symbols.


G.1.A Distinguish between inductive and deductive reasoning.

Students generate and test conjectures inductively and then prove (or disprove) their conclusions deductively.

Example:

A student first hypothesizes that the number of • degrees in a polygon = 180 • (s – 2), where s represents the number of sides, and then proves this is true. When was the student using inductive reasoning? When was s/he using deductive reasoning? Justify your answers.

G.1.B Use inductive reasoning to make conjectures, to test the plausibility of a geometric statement, and to help find a counterexample.

Examples:

Investigate the relationship among the medians of • a triangle using paper folding. Make a conjecture about this relationship.

Using dynamic geometry software, decide if the • following is a plausible conjecture: If segment AM is a median in triangle ABC, then ray AM bisects angle BAC.

G.1.C Use deductive reasoning to prove that a valid geometric statement is true.

Valid proofs may be presented in paragraph, two-column, or flow-chart formats. Proof by contradiction is a form of deductive reasoning.

Example:

Prove that the diagonals of a rhombus are • perpendicular bisectors of each other.

G.1.D Write the converse, inverse, and contrapositive of a valid proposition and determine their validity.

Examples:

If • m and n are odd integers, then the sum of m and n is an even integer. State the converse and determine whether it is valid.

If a quadrilateral is a rectangle, the diagonals• have the same length. State the contrapositive and determine whether it is valid.

G.1.E Identify errors or gaps in a mathematical argument and develop counterexamples to refute invalid statements about geometric relationships.

Example:

Identify errors in reasoning in the following proof:• Given ∠ABC ≅ ∠PRQ, AB ≅ PQ, and BC ≅ QR, then ∆ABC ≅ ∆PQR by SAS.

Geometry



G.1.F Distinguish between definitions and undefined geometric terms and explain the role of definitions, undefined terms, postulates (axioms), and theorems.

Students sketch points and lines (undefined terms) and define and sketch representations of other common terms. They use definitions and postulates as they prove theorems throughout geometry. In their work with theorems, they identify the hypothesis and the conclusion and explain the role of each.

Students describe the consequences of changing assumptions or using different definitions for subsequent theorems and logical arguments.

Example:

There are two definitions of trapezoid that can be • found in books or on the web. A trapezoid is either

a quadrilateral with exactly one pair of — parallel sides or

a quadrilateral with at least one pair of — parallel sides.

Write some theorems that are true when applied to one definition but not the other, and explain your answer.

Geometry


GeometryG.2. Core Content: Lines and angles (Geometry/Measurement)

Students study basic properties of parallel and perpendicular lines, their respective slopes, and the properties of the angles formed when parallel lines are intersected by a transversal. They prove

related theorems and apply them to solve both mathematical and practical problems.


G.2.A Know, prove, and apply theorems about parallel and perpendicular lines.

Students should be able to summarize and explain basic theorems. They are not expected to recite lists of theorems, but they should know the conclusion of a theorem when given its hypothesis.

Examples:

Prove that a point on the perpendicular bisector of • a line segment is equidistant from the ends of the line segment.

If each of two lines is perpendicular to a given line, • what is the relationship between the two lines? How do you know?

G.2.B Know, prove, and apply theorems about angles, including angles that arise from parallel lines intersected by a transversal.

Examples:

Prove that if two parallel lines are cut by a • transversal, then alternate-interior angles are equal.

Take two parallel lines • l and m, with (distinct) points A and B on l and C and D on m.

If AC intersects BD at point E, prove that ∆ABE ≅ ∆CDE.

G.2.C Explain and perform basic compass and straightedge constructions related to parallel and perpendicular lines.

Constructions using circles and lines with dynamic geometry software (i.e., virtual compass and straightedge) are equivalent to paper and pencil constructions.

Example:

Construct and mathematically justify the steps to:•

Bisect a line segment.—

Drop a perpendicular from a point to a line.—

Construct a line through a point that is — parallel to another line.

G.2.D Describe the intersections of lines in the plane and in space, of lines and planes, and of planes in space.

Example:

Describe all the ways that three planes can • intersect in space.

Geometry


GeometryG.3. Core Content: Two- and three-dimensional figures (Geometry/Measurement)

Students know and can prove theorems about two- and three-dimensional geometric figures, both formally and informally. They identify necessary and sufficient conditions for proving congruence,

similarity, and properties of figures. Triangles are a primary focus, beginning with general properties of triangles, working with right triangles and special triangles, proving and applying the Pythagorean Theorem and its converse, and applying the basic trigonometric ratios of sine, cosine, and tangent. Students extend their learning to other polygons and the circle, and do some work with three-dimensional figures.


G.3.A Know, explain, and apply basic postulates and theorems about triangles and the special lines, line segments, and rays associated with a triangle.

Examples:

Prove that the sum of the angles of a triangle is 180• °.

Prove and explain theorems about the incenter, • circumcenter, orthocenter, and centroid.

The rural towns of Atwood, Bridgeville, and • Carnegie are building a communications tower to serve the needs of all three towns. They want to position the tower so that the distance from each town to the tower is equal. Where should they locate the tower? How far will it be from each town?

G.3.B Determine and prove triangle congruence, triangle similarity, and other properties of triangles.

Students should identify necessary and sufficient conditions for congruence and similarity in triangles, and use these conditions in proofs.

Examples:

Prove that congruent triangles are similar.•

For a given • ∆RST, prove that ∆XYZ, formed by joining the midpoints of the sides of ∆RST, is similar to ∆RST.

Show that not all SSA triangles are congruent. •

G.3.C Use the properties of special right triangles (30°–60°–90° and 45°–45°–90°) to solve problems.

Examples:

Determine the length of the altitude of an equilateral • triangle whose side lengths measure 5 units.

If one leg of a right triangle has length 5 and the • adjacent angle is 30°, what is the length of the other leg and the hypotenuse?

If one leg of a 45• °–45°–90° triangle has length 5, what is the length of the hypotenuse?

Geometry



G.3.C cont. The pitch of a symmetrical roof on a house 40 • feet wide is 30º. What is the length of the rafter, r, exactly and approximately?

30º 40

‘

r

G.3.D Know, prove, and apply the Pythagorean Theorem and its converse.

Examples:

Consider any right triangle with legs • a and b and hypotenuse c. The right triangle is used to create Figures 1 and 2. Explain how these figures constitute a visual representation of a proof of the Pythagorean Theorem.

Q

P

Figure 1 Figure 2

a

bc

A juice box is 6 cm by 8 cm by 12 cm. A straw is • inserted into a hole in the center of the top of the box. The straw must stick out 2 cm so you can drink from it. If the straw must be long enough to touch each bottom corner of the box, what is the minimum length the straw must be? (Assume the diameter of the straw is 0 for the mathematical model.)

8 cm

12 cm

6 cm

In • ∆ABC, with right angle at C, draw the altitude CD from C to AB. Name all similar triangles in the diagram. Use these similar triangles to prove the Pythagorean Theorem.

Apply the Pythagorean Theorem to derive the • distance formula in the (x, y) plane.

Geometry



G.3.E Solve problems involving the basic trigonometric ratios of sine, cosine, and tangent.

Examples:

A 12-foot ladder leans against a wall to form a 63• ° angle with the ground. How many feet above the ground is the point on the wall at which the ladder is resting?

Use the Pythagorean Theorem to establish that • sin2ø + cos2ø = 1 for ø between 0° and 90°.

G.3.F Know, prove, and apply basic theorems about parallelograms.

Properties may include those that address symmetry and properties of angles, diagonals, and angle sums. Students may use inductive and deductive reasoning and counterexamples.

Examples:

Are opposite sides of a parallelogram always • congruent? Why or why not?

Are opposite angles of a parallelogram always • congruent? Why or why not?

Prove that the diagonals of a parallelogram bisect • each other.

Explain why if the diagonals of a quadrilateral • bisect each other, then the quadrilateral is a parallelogram.

Prove that the diagonals of a rectangle are • congruent. Is this true for any parallelogram? Justify your reasoning.

G.3.G Know, prove, and apply theorems about properties of quadrilaterals and other polygons.

Examples:

What is the length of the apothem of a regular • hexagon with side length 8? What is the area of the hexagon?

If the shaded pentagon were removed, it could be • replaced by a regular n-sided polygon that would exactly fill the remaining space. Find the number of sides, n, of a replacement polygon that makes the three polygons fit perfectly.

P

Geometry



G.3.H Know, prove, and apply basic theorems relating circles to tangents, chords, radii, secants, and inscribed angles.

Examples:

Given a line tangent to a circle, know and explain • that the line is perpendicular to the radius drawn to the point of tangency.

Prove that two chords equally distant from the • center of a circle are congruent.

Prove that if one side of a triangle inscribed in a circle • is a diameter, then the triangle is a right triangle.

Prove that if a radius of a circle is perpendicular to • a chord of a circle, then the radius bisects the chord.

G.3.I Explain and perform constructions related to the circle.

Students perform constructions using straightedge and compass, paper folding, and dynamic geometry software. What is important is that students understand the mathematics and are able to justify each step in a construction.

Example:

In each case, explain why the constructions work:• a. Construct the center of a circle from two chords.b. Construct a circumscribed circle for a triangle.c. Inscribe a circle in a triangle.

G.3.J Describe prisms, pyramids, parallelepipeds, tetrahedra, and regular polyhedra in terms of their faces, edges, vertices, and properties.

Examples:

Given the number of faces of a regular polyhedron, • derive a formula for the number of vertices.

Describe symmetries of three-dimensional • polyhedra and their two-dimensional faces.

Describe the lateral faces that are required for a • pyramid to be a right pyramid with a regular base. Describe the lateral faces required for an oblique pyramid that has a regular base.

Geometry



G.3.K Analyze cross-sections of cubes, prisms, pyramids, and spheres and identify the resulting shapes.

Examples:

Start with a regular tetrahedron with edges of unit • length 1. Find the plane that divides it into two congruent pieces and whose intersection with the tetrahedron is a square. Find the area of the square. (Requires no pencil or paper.)

Start with a cube with edges of unit length 1. Find • the plane that divides it into two congruent pieces and whose intersection with the cube is a regular hexagon. Find the area of the hexagon.

Start with a cube with edges of unit length 1. • Find the plane that divides it into two congruent pieces and whose intersection with the cube is a rectangle that is not a face and contains four of the vertices. Find the area of the rectangle.

Which has the larger area, the above rectangle or • the above hexagon?

Geometry


GeometryG.4. Core Content: Geometry in the coordinate plane (Geometry/Measurement, Algebra)

Students make connections between geometry and algebra by studying geometric properties and attributes that can be represented on the coordinate plane. They use the coordinate plane to

represent situations that are both purely mathematical and that arise in applied contexts. In this way, they use the power of algebra to solve problems about shapes and space.


G.4.A Determine the equation of a line in the coordinate plane that is described geometrically, including a line through two given points, a line through a given point parallel to a given line, and a line through a given point perpendicular to a given line.

Examples:

Write an equation for the perpendicular bisector of • a given line segment.

Determine the equation of a line through the points • (5, 3) and (5, -2).

Prove that the slopes of perpendicular lines are • negative inverses of each other.

G.4.B Determine the coordinates of a point that is described geometrically.

Examples:

Determine the coordinates for the midpoint of a • given line segment.

Given the coordinates of three vertices of a • parallelogram, determine all possible coordinates for the fourth vertex.

Given the coordinates for the vertices of a • triangle, find the coordinates for the center of the circumscribed circle and the length of its radius.

G.4.C Verify and apply properties of triangles and quadrilaterals in the coordinate plane.

Examples:

Given four points in a coordinate plane that are the • vertices of a quadrilateral, determine whether the quadrilateral is a rhombus, a square, a rectangle, a parallelogram, or none of these. Name all the classifications that apply.

Given a parallelogram on a coordinate plane, • verify that the diagonals bisect each other.

Given the line with • y-intercept 4 and x-intercept 3, find the area of a square that has one corner on the origin and the opposite corner on the line described.

Geometry



G.4.C cont. Below is a diagram of a miniature golf hole as • drawn on a coordinate grid. The dimensions of the golf hole are 4 feet by 12 feet. Players must start their ball from one of the three tee positions, located at (1, 1), (1, 2), or (1, 3). The hole is located at (10, 3). A wall separates the tees from the hole. At which tee should the ball be placed to create the shortest “hole-in-one” path? Sketch the intended path of the ball, find the distance the ball will travel, and describe your reasoning. (Assume the diameters of the golf ball and the hole are 0 for the mathematical model.)

....

TeesWall

Hole

G.4.D Determine the equation of a circle that is described geometrically in the coordinate plane and, given equations for a circle and a line, determine the coordinates of their intersection(s).

Examples:

Write an equation for a circle with a radius of 2 • units and center at (1, 3).

Given the circle • x2 + y2 = 4 and the line y = x, find the points of intersection.

Write an equation for a circle given a line segment • as a diameter.

Write an equation for a circle determined by a • given center and tangent line.

Geometry


GeometryG.5. Core Content: Geometric transformations (Geometry/Measurement)

Students continue their study of geometric transformations, focusing on the effect of such transformations and the composition of transformations on the attributes of geometric figures.

They study techniques for establishing congruence and similarity by means of transformations.


G.5.A Sketch results of transformations and compositions of transformations for a given two-dimensional figure on the coordinate plane, and describe the rule(s) for performing translations or for performing reflections about the coordinate axes or the line y = x.

Transformations include translations, rotations, reflections, and dilations.

Example:

Line • m is described by the equation y = 2x + 3. Graph line m and reflect line m across the line y = x. Determine the equation of the image of the reflection. Describe the relationship between the line and its image.

G.5.B Determine and apply properties of transformations.

Students make and test conjectures about compositions of transformations and inverses of transformations, the commutativity and associativity of transformations, and the congruence and similarity of two-dimensional figures under various transformations.

Examples:

Identify transformations (alone or in composition) • that preserve congruence.

Determine whether the composition of two • reflections of a line is commutative.

Determine whether the composition of two rotations • about the same point of rotation is commutative.

Find a rotation that is equivalent to the composition • of two reflections over intersecting lines.

Find the inverse of a given transformation.•

G.5.C Given two congruent or similar figures in a coordinate plane, describe a composition of translations, reflections, rotations, and dilations that superimposes one figure on the other.

Examples:

Find a sequence of transformations that • superimposes the segment with endpoints (0, 0) and (2, 1) on the segment with endpoints (4, 2) and (3, 0).

Find a sequence of transformations that • superimposes the triangle with vertices (0, 0), (1, 1), and (2, 0) on the triangle with vertices (0, 1), (2, -1), and (0, -3).

G.5.D Describe the symmetries of two-dimensional figures and describe transformations, including reflections across a line and rotations about a point.

Although the expectation only addresses two-dimensional figures, classroom activities can easily extend to three-dimensional figures. Students can also describe the symmetries, reflections across a plane, and rotations about a line for three-dimensional figures.

Geometry


GeometryG.6. Additional Key Content (Measurement)

Students extend and formalize their work with geometric formulas for perimeter, area, surface area, and volume of two- and three-dimensional figures, focusing on mathematical derivations of these formulas

and their applications in complex problems. They use properties of geometry and measurement to solve problems in purely mathematical as well as applied contexts. Students understand the role of units in measurement and apply what they know to solve problems involving derived measures like speed or density. They understand that all measurement is approximate and specify precision in measurement problems.


G.6.A Derive and apply formulas for arc length and area of a sector of a circle.

Example:

Find the area and perimeter of the Reuleaux • triangle below.The Reuleaux triangle is constructed with three arcs. The center of each arc is located at the vertex of an equilateral triangle. Each arc extends between the two opposite vertices of the equilateral triangle.The figure below is a Reuleaux triangle that circumscribes equilateral triangle ABC. ∆ABC has side length of 5 inches. AB has center C, BC has center A, and CA has center B, and all three arcs have the same radius equal to the length of the sides of the triangle.

A

BC

G.6.B Analyze distance and angle measures on a sphere and apply these measurements to the geometry of the earth.

Examples:

Use a piece of string to measure the distance • between two points on a ball or globe; verify that the string lies on an arc of a great circle.

On a globe, show with examples why airlines use • polar routes instead of flying due east from Seattle to Paris.

Show that the sum of the angles of a triangle on a • sphere is greater than 180 degrees.

Geometry



G.6.C Apply formulas for surface area and volume of three-dimensional figures to solve problems.

Problems include those that are purely mathematical as well as those that arise in applied contexts.

Three-dimensional figures include right and oblique prisms, pyramids, cylinders, cones, spheres, and composite three-dimensional figures.

Examples:

As Pam scooped ice cream into a cone, she began • to formulate a geometry problem in her mind. If the ice cream was perfectly spherical with diameter 2.25'' and sat on a geometric cone that also had diameter 2.25'' and was 4.5'' tall, would the cone hold all the ice cream as it melted (without her eating any of it)? She figured the melted ice cream would have the same volume as the unmelted ice cream.Find the solution to Pam’s problem and justify your reasoning.

A rectangle is 5 inches by 10 inches. Find the • volume of a cylinder that is generated by rotating the rectangle about the 10-inch side.

G.6.D Predict and verify the effect that changing one, two, or three linear dimensions has on perimeter, area, volume, or surface area of two- and three-dimensional figures.

The emphasis in high school should be on verifying the relationships between length, area, and volume and on making predictions using algebraic methods.

Examples:

What happens to the volume of a rectangular • prism if four parallel edges are doubled in length?

The ratio of a pair of corresponding sides in two • similar triangles is 5:3. The area of the smaller triangle is 108 in2. What is the area of the larger triangle?

G.6.E Use different degrees of precision in measurement, explain the reason for using a certain degree of precision, and apply estimation strategies to obtain reasonable measurements with appropriate precision for a given purpose.

Example:

The U.S. Census Bureau reported a national • population of 299,894,924 on its Population Clock in mid-October of 2006. One can say that the U.S. population is 3 hundred million (3 × 108) and be precise to one digit. Although the population had surpassed 3 hundred million by the end of that month, explain why 3 × 108 remained precise to one digit.

Geometry



G.6.F Solve problems involving measurement conversions within and between systems, including those involving derived units, and analyze solutions in terms of reasonableness of solutions and appropriate units.

This performance expectation is intended to build on students’ knowledge of proportional relationships. Students should understand the relationship between scale factors and their inverses as they relate to choices about when to multiply and when to divide in converting measurements.

Derived units include those that measure speed, density, flow rates, population density, etc.

Example:

A digital camera takes pictures that are 3.2 • megabytes in size. If the pictures are stored on a 1-gigabyte card, how many pictures can be taken before the card is full?

Geometry


GeometryG.7. Core Processes: Reasoning, problem solving, and communication

Students formalize the development of reasoning in Geometry as they become more sophisticated in their ability to reason inductively and begin to use deductive reasoning in formal proofs. They

extend the problem-solving practices developed in earlier grades and apply them to more challenging problems, including problems related to mathematical and applied situations. Students use a coherent problem-solving process in which they analyze the situation to determine the question(s) to be answered, synthesize given information, and identify implicit and explicit assumptions that have been made. They examine their solution(s) to determine reasonableness, accuracy, and meaning in the context of the original problem. They use correct mathematical language, terms, symbols, and conventions as they address problems in Geometry and provide descriptions and justifications of solution processes. The mathematical thinking, reasoning, and problem-solving processes students learn in high school mathematics can be used throughout their lives as they deal with a world in which an increasing amount of information is presented in quantitative ways, and more and more occupations and fields of study rely on mathematics.


G.7.A Analyze a problem situation and represent it mathematically.

G.7.B Select and apply strategies to solve problems.

G.7.C Evaluate a solution for reasonableness, verify its accuracy, and interpret the solution in the context of the original problem.

G.7.D Generalize a solution strategy for a single problem to a class of related problems, and apply a strategy for a class of related problems to solve specific problems.

G.7.E Read and interpret diagrams, graphs, and text containing the symbols, language, and conventions of mathematics.

G.7.F Summarize mathematical ideas with precision and efficiency for a given audience and purpose.

G.7.G Synthesize information to draw conclusions and evaluate the arguments and conclusions of others.

G.7.H Use inductive reasoning to make conjectures, and use deductive reasoning to prove or disprove conjectures.

Examples:

• AB is the diameter of the semicircle and the radius of the quarter circle shown in the figure below. BC is the perpendicular bisector of AB.

A C B

ED

F

Imagine all of the triangles formed by AB and any arbitrary point lying in the region bounded by AC , CD, and AD, seen in bold below.

A C B

ED

F

Use inductive reasoning to make conjectures about what types of triangles are formed based upon the region where the third vertex is located. Use deductive reasoning to verify your conjectures.

Geometry



G.7 cont. Rectangular cartons that are 5 feet long need to • be placed in a storeroom that is located at the end of a hallway. The walls of the hallway are parallel. The door into the hallway is 3 feet wide and the width of the hallway is 4 feet. The cartons must be carried face up. They may not be tilted. Investigate the width and carton top area that will fit through the doorway.

54

3

R

A

S

C

T

Generalize your results for a hallway opening of x feet and a hallway width of y feet if the maximum carton dimensions are c feet long and x2 + y2 = c2.

Geometry


Algebra 2


Algebra 2A2.1. Core Content: Solving problems

The first core content area highlights the type of problems students will be able to solve by the end of Algebra 2, as they extend their ability to solve problems with additional functions and equations.

When presented with a word problem, students are able to determine which function or equation models the problem and use that information to solve the problem. They build on what they learned in Algebra 1 about linear and quadratic functions and are able to solve more complex problems. Additionally, students learn to solve problems modeled by exponential and logarithmic functions, systems of equations and inequalities, inverse variations, and combinations and permutations. Turning word problems into equations that can be solved is a skill students hone throughout Algebra 2 and subsequent mathematics courses.


A2.1.A Select and justify functions and equations to model and solve problems.

Examples:

A manufacturer wants to design a cylindrical • soda can that will hold 500 milliliters (ml) of soda. The manufacturer’s research has determined that an optimal can height is between 10 and 15 centimeters. Find a function for the radius in terms of the height, and use it to find the possible range of radius measurements in centimeters. Explain your reasoning.

Dawson wants to make a horse corral by creating • a rectangle that is divided into 2 parts, similar to the following diagram. He has a 1200-foot roll of fencing to do the job.

What are the dimensions of the enclosure — with the largest total area?

What function or equation best models — this situation?

l

w

Algebra 2



A2.1.B Solve problems that can be represented by systems of equations and inequalities.

Examples:

Mr. Smith uses the following formula to calculate • students’ final grades in his Algebra 2 class: 0.4E + 0.6T = C, where E represents the score on the final exam, and T represents the average score of all tests given during the grading period. All tests and the final exam are worth a maximum of 100 points. The minimum passing score on tests, the final exam, and the course is 60. Determine the inequalities that describe the following situation and sketch a system of graphs to illustrate it. When necessary, round scores to the nearest tenth.

Is it possible for a student to have a failing — test score average (i.e., T < 60 points) and still pass the course?

If you answered “yes,” what is the — minimum test score average a student can have and still pass the course? What final exam score is needed to pass the course with a minimum test score average?

A student has a particular test score — average. How can (s)he figure out the minimum final exam score needed to pass the course?

Data derived from an experiment seems to be • parabolic when plotted on a coordinate grid. Three observed data points are (2, 10), (3, 8), and (4, 4). Write a quadratic equation that passes through the points.

Algebra 2



A2.1.C Solve problems that can be represented by quadratic functions, equations, and inequalities.

In addition to solving area and velocity problems by factoring and applying the quadratic formula to the quadratic equation, students use the vertex form of the equation to solve problems about maximums, minimums, and symmetry.

Examples:

The Gateway Arch in St. Louis has a special • shape called a catenary, which looks a lot like a parabola. It has a base width of 600 feet and is 630 feet high. Which is taller, this catenary arch or a parabolic arch that has the same base width but has a height of 450 feet at a point 150 feet from one of the pillars? What is the height of the parabolic arch?

Fireworks are launched upward from the ground • with an initial velocity of 160 feet per second. The formula for vertical motion is h(t) = 0.5at2 + vt + s, where the gravitational constant, a, is -32 feet per square second, v represents the initial velocity, and s represents the initial height. Time t is measured in seconds, and height h is measured in feet.For the ultimate effect, the fireworks must explode after they reach the maximum height. For the safety of the crowd, they must explode at least 256 ft above the ground. The fuses must be set for the appropriate time interval that allows the fireworks to reach this height. What range of times, starting from initial launch and ending with fireworks explosion, meets these conditions?

A2.1.D Solve problems that can be represented by exponential and logarithmic functions and equations.

Examples:

If you need $15,000 in 4 years to start college, • how much money would you need to invest now? Assume an annual interest rate of 4% compounded monthly for 48 months.

The half-life of a certain radioactive substance is • 65 days. If there are 4.7 grams initially present, how long will it take for there to be less than 1 gram of the substance remaining?

Algebra 2



A2.1.E Solve problems that can be represented by

inverse variations of the forms f(x) = ax

+ b,

f(x) = ax2 + b, and f(x) =

( )a

bx c+ .

Examples:

At the You’re Toast, Dude! toaster company, the • weekly cost to run the factory is $1400, and the cost of producing each toaster is an additional $4 per toaster.

Find a function to represent the weekly — cost in dollars, C(x), of producing x toasters. Assume either unlimited production is possible or set a maximum per week.

Find a function to represent the total — production cost per toaster for a week.

How many toasters must be produced — within a week to have a total production cost per toaster of $8?

A person’s weight varies inversely as the square of • his distance from the center of the earth. Assume the radius of the earth is 4000 miles. How much would a 200-pound man weigh

1000 miles above the surface of the earth?—


A2.1.F Solve problems involving combinations and permutations.

Examples:

The company Ali works for allows her to invest • in her choice of 10 different mutual funds, 6 of which grew by at least 5% over the last year. Ali randomly selected 4 of the 10 funds in which to invest. What is the probability that 3 of Ali’s funds grew by 5%?

Four points (• A, B, C, and D) lie on one straight line, n, and five points (E, F, G, H, and J) lie on another straight line, m, that is parallel to n. What is the probability that three points, selected at random, will form a triangle?

Algebra 2


Algebra 2A2.2. Core Content: Numbers, expressions, and operations (Numbers, Operations, Algebra)

Students extend their understanding of number systems to include complex numbers, which they will see as solutions for quadratic equations. They grow more proficient in their use of algebraic

techniques as they continue to use variables and expressions to solve problems. As problems become more sophisticated and the level of mathematics increases, so does the complexity of the symbolic manipulations and computations necessary to solve the problems. Students refine the foundational algebraic skills they need to be successful in subsequent mathematics courses.


A2.2.A Explain how whole, integer, rational, real, and complex numbers are related, and identify the number system(s) within which a given algebraic equation can be solved.

Example:

Within which number system(s) can each of the • following be solved? Explain how you know.

3— x + 2 = 5

x— 2 = 1

x— 2 = 1

4

x— 2 = 2

x— 2 = -2

— x

7 = π

A2.2.B Use the laws of exponents to simplify and evaluate numeric and algebraic expressions that contain rational exponents.

Examples:

Convert the following from a radical to exponential • form or vice versa.

24— 1

3

— 165

— x2 1+

— xx

2

Evaluate • x-2/3 for x = 27.

Algebra 2



A2.2.C Add, subtract, multiply, divide, and simplify rational and more general algebraic expressions.

In the same way that integers were extended to fractions, polynomials are extended to rational expressions. Students must be able to perform the four basic arithmetic operations on more general expressions that involve exponentials.

The binomial theorem is useful when raising expressions to powers, such as (x + 3)5.

Examples:

• xx

xx

++

−−−

11

3 312 2( )

• Divide ( ) x

x +

+2 1

3 / 2

by x

x

+−

2

12

Algebra 2


Algebra 2A2.3. Core Content: Quadratic functions and equations (Algebra)

As students continue to solve quadratic equations and inequalities in Algebra 2, they encounter complex roots for the first time. They learn to translate between forms of quadratic equations,

applying the vertex form to evaluate maximum and minimum values and find symmetry of the graph, and they learn to identify which form should be used in a particular situation. This opens up a whole range of new problems students can solve using quadratics. These algebraic skills are applied in subsequent high school mathematics and statistics courses.


A2.3.A Translate between the standard form of a quadratic function, the vertex form, and the factored form; graph and interpret the meaning of each form.

Students translate among forms of a quadratic function to convert to one that is appropriate—e.g., vertex form—to solve specific problems.

Students learn about the advantages of the standard form (f(x) = ax2 + bx + c), the vertex form (f(x) = a(x – h)2 + d), and the factored form (f(x) = a(x – r)(x – s)). They produce the vertex form by completing the square on the function in standard form, which allows them to see the symmetry of the graph of a quadratic function as well as the maximum or minimum. This opens up a whole range of new problems students can solve using quadratics. Students continue to find the solutions of the equation, which in Algebra 2 can be either real or complex.

Example:

Find the minimum, the line of symmetry, and • the roots for the graphs of each of the following functions: f(x) = x2 – 4x + 3f(x) = x2 – 4x + 4f(x) = x2 – 4x + 5

A2.3.B Determine the number and nature of the roots of a quadratic function.

Students should be able to recognize and interpret the discriminant.

Students should also be familiar with the Fundamental Theorem of Algebra, i.e., that all polynomials, not just quadratics, have roots over the complex numbers. This concept becomes increasingly important as students progress through mathematics.

Example:

For what values of • a does f(x) = x2 – 6x + a have 2 real roots, 1 real root, and no real roots?

Algebra 2



A2.3.C Solve quadratic equations and inequalities, including equations with complex roots.

Students solve equations that are not easily factored by completing the square and by using the quadratic formula.

Examples:

x• 2 – 10x + 34 = 0

3• x2 + 10 = 4x

Wile E. Coyote launches an anvil from 180 • feet above the ground at time t = 0. The equation that models this situation is given by h = -16t2 + 96t + 180, where t is time measured in seconds and h is height above the ground measured in feet.a. What is a reasonable domain restriction for t in

this context?b. Determine the height of the anvil two seconds

after it was launched.c. Determine the maximum height obtained by

the anvil.d. Determine the time when the anvil is more

than 100 feet above ground.

Farmer Helen wants to build a pigpen. With 100 • feet of fence, she wants a rectangular pen with one side being a side of her existing barn. What dimensions should she use for her pigpen in order to have the maximum number of square feet?

Algebra 2


Algebra 2 A2.4. Core Content: Exponential and logarithmic functions and equations (Algebra)

Students extend their understanding of exponential functions from Algebra 1 with an emphasis on inverse functions. This leads to a natural introduction of logarithms and logarithmic functions. They

learn to use the basic properties of exponential and logarithmic functions, graphing both types of function to analyze relationships, represent and model problems, and answer questions. Students employ these functions in many practical situations, such as applying exponential functions to determine compound interest and applying logarithmic functions to determine the pH of a liquid.


A2.4.A Know and use basic properties of exponential and logarithmic functions and the inverse relationship between them.

Examples:

Given • f(x) = 4x, write an equation for the inverse of this function. Graph the functions on the same coordinate grid.

Find — f(-3).

Evaluate the inverse function at 7.—

Derive the formulas:•

log— ba ⋅ logab = 1

log— aN = logbN ⋅ logab

Find the exact value of • x in:

l— ogx16 = 43

log— 381 = x

Solve for • y in terms of x:

log— a y

x = x

100 = — x ⋅ 10y

A2.4.B Graph an exponential function of the form f(x) = abx and its inverse logarithmic function.

Students expand on the work they did in Algebra 1 to functions of the form y = abx. Although the concept of inverses is not fully developed until Precalculus, there is an emphasis in Algebra 2 on students recognizing the inverse relationship between exponential and logarithmic functions and how this is reflected in the shapes of the graphs.

Example:

Find the equation for the inverse function of • y = 3x. Graph both functions. What characteristics of each of the graphs indicate they are inverse functions?

Algebra 2



A2.4.C Solve exponential and logarithmic equations. Examples:

A recommended adult dosage of the cold • medication NoMoreFlu is 16 ml. NoMoreFlu causes drowsiness when there are more than 4 ml in one’s system, making it unsafe to drive, operate machinery, etc. The manufacturer wants to print a warning label telling people how long they should wait after taking NoMoreFlu for the drowsiness to pass. If the typical metabolic rate is such that one quarter of the NoMoreFlu is lost every four hours, and a person takes the full dosage, how long should adults wait after taking NoMoreFlu to ensure that there will be

Less than 4 ml of NoMoreFlu in their — system?

Less than 1 ml in their system? —

Less than 0.1 ml in their system?—

Solve for • x in 256 22 1= −x .

Solve for • x in log5(x – 4) = 3.

Algebra 2


Algebra 2A2.5. Core Content: Additional functions and equations (Algebra)

Students learn about additional classes of functions including square root, cubic, logarithmic, and those involving inverse variation. Students plot points and sketch graphs to represent these functions and

use algebraic techniques to solve related equations. In addition to studying the defining characteristics of each of these classes of functions, students gain the ability to construct new functions algebraically and using transformations. These extended skills and techniques serve as the foundation for further study and analysis of functions in subsequent mathematics courses.


A2.5.A Construct new functions using the transformations f(x – h), f(x) + k, cf(x), and by adding and subtracting functions, and describe the effect on the original graph(s).

Students perform simple transformations on functions, including those that contain the absolute value of expressions, quadratic expressions, square root expressions, and exponential expressions, to make new functions.

Examples:

What sequence of transformations changes • f(x) = x2 to g(x) = -5(x – 3)2 + 2?

Carly decides to earn extra money by making • glass bead bracelets. She purchases tools for $40.00. Elastic bead cord for each bracelet costs $0.10. Glass beads come in packs of 10 beads, and one pack has enough beads to make one bracelet. Base price for the beads is $2.00 per pack. For each of the first 100 packs she buys, she gets $0.01 off each of the packs. (For example, if she purchases three packs, each pack costs $1.97 instead of $2.00.) Carly plans to sell each bracelet for $4.00. Assume Carly will make a maximum of 100 bracelets.

Find a function — C(b) that describes Carly’s costs.

Find a function — R(b) that describes Carly’s revenue.

Carly’s profit is described by P(b) = R(b) – C(b).

Find — P(b).

What is the minimum number of bracelets — that Carly must sell in order to make a profit?

To make a profit of $100?—

Algebra 2



A2.5.B Plot points, sketch, and describe the graphs

of functions of the form = ( ) f x a x − c + d , and solve related equations.

Students solve algebraic equations that involve the square root of a linear expression over the real numbers. Students should be able to identify extraneous solutions and explain how they arose.

Students should view the function g(x) = x as the inverse function of f(x) = x2, recognizing that the functions have different domains for x greater than or equal to 0.

Example:

Analyze the following equations and tell what • you know about the solutions. Then solve the equations.

— 2 5 7x + =

— 5 6 2x − = −

— 2 15x x+ =

— 2 5 7x x− = +

A2.5.C Plot points, sketch, and describe the graphs

of functions of the form f x( )ax

b= + ,

f xax

b( ) = +2 , and ( )f x

abx c

( )=+ , and solve

related equations.

Examples:

Sketch the graphs of the four functions •

f xax

b( ) = +2 when a = 4 and 8 and b = 0 and 1.


( )f x

4bx c

( )=+

when b = 1 and 4 and c = 2 and 3.

A2.5.D Plot points, sketch, and describe the graphs of cubic polynomial functions of the form f(x) = ax3 + d as an example of higher order polynomials and solve related equations.

Example:Solve for • x in 60 = -2x3 + 6.

Algebra 2


Algebra 2A2.6. Core Content: Probability, data, and distributions (Data/Statistics/Probability)

Students formalize their study of probability, computing both combinations and permutations to calculate the likelihood of an outcome in uncertain circumstances and applying the binominal theorem

to solve problems. They extend their use of statistics to graph bivariate data and analyze its shape to make predictions. They calculate and interpret measures of variability, confidence intervals, and margins of error for population proportions. Dual goals underlie the content in the section: students prepare for the further study of statistics and become thoughtful consumers of data.


A2.6.A Apply the fundamental counting principle and the ideas of order and replacement to calculate probabilities in situations arising from two-stage experiments (compound events).

A2.6.B Given a finite sample space consisting of equally likely outcomes and containing events A and B, determine whether A and B are independent or dependent, and find the conditional probability of A given B.

Example:

What is the probability of drawing a heart from • a standard deck of cards on a second draw, given that a heart was drawn on the first draw and not replaced?

A2.6.C Compute permutations and combinations, and use the results to calculate probabilities.

Example:

Two friends, Abby and Ben, are among five students • being considered for three student council positions. If each of the five students has an equal likelihood of being selected, what is the probability that Abby and Ben will both be selected?

A2.6.D Apply the binomial theorem to solve problems involving probability.

The binominal theorem is also applied when computing with polynomials.

Examples:

Use Pascal’s triangle and the binomial theorem • to find the number of ways six objects can be selected four at a time.

In a survey, 33% of adults reported that they • preferred to get the news from newspapers rather than television. If you survey 5 people, what is the probability of getting exactly 2 people who say they prefer news from the newspaper?

Write an equation that can be used to — solve the problem.

Create a histogram of the binomial — distribution of the probability of getting 0 through 5 responders saying they prefer the newspaper.

A2.6.E Determine if a bivariate data set can be better modeled with an exponential or a quadratic function and use the model to make predictions.

In high school, determining a formula for a curve of best fit requires a graphing calculator or similar technological tool.

Algebra 2



A2.6.F Calculate and interpret measures of variability and standard deviation and use these measures and the characteristics of the normal distribution to describe and compare data sets.

Students should be able to identify unimodality, symmetry, standard deviation, spread, and the shape of a data curve to determine whether the curve could reasonably be approximated by a normal distribution.

Given formulas, students should be able to calculate the standard deviation for a small data set, but calculators ought to be used if there are very many points in the data set. It is important that students be able to describe the characteristics of the normal distribution and identify common examples of data that are and are not reasonably modeled by it. Common examples of distributions that are approximately normal include physical performance measurements (e.g., weightlifting, timed runs), heights, and weights.

Apply the Empirical Rule (68–95–99.7 Rule) to approximate the percentage of the population meeting certain criteria in a normal distribution.

Example:

Which is more likely to be affected by an outlier • in a set of data, the interquartile range or the standard deviation?

A2.6.G Calculate and interpret margin of error and confidence intervals for population proportions.

Students will use technology based on the complexity of the situation.

Students use confidence intervals to critique various methods of statistical experimental design, data collection, and data presentation used to investigate important problems, including those reported in public studies.

Example:

In 2007, 400 of the 500 10• th graders in Local High School passed the WASL. In 2008, 375 of the 480 10th graders passed the test. The Local Gazette headline read “10th Grade WASL Scores Decline in 2008!” In response, the Superintendent of Local School District wrote a letter to the editor claiming that, in fact, WASL performance was not significantly lower in 2008 than it was in 2007. Who is correct, the Local Gazette or the Superintendent? Use mathematics to find the margin of error to justify your conclusion. (Formula for the margin of error

(E): E zc= p(1−p)

n; z95 = 1.96, where n is the sample

size, p is the proportion of the sample with the trait of interest, c is the confidence level, and zc is the multiplier for the specified confidence interval.)

Algebra 2


Algebra 2A2.7. Additional Key Content (Algebra)

Students study two important topics here. First, they extend their ability to solve systems of two equations in two variables to solving systems of three equations in three variables, which leads

to the full development of matrices in Precalculus. Second, they formalize their work with series as they learn to find the terms and partial sums of arithmetic series and the terms and partial and infinite sums of geometric series. This conceptual understanding of series lays an important foundation for understanding calculus.


A2.7.A Solve systems of three equations with three variables.

Students solve systems of equations using algebraic and numeric methods.

Examples:

Jill, Ann, and Stan are to inherit $20,000. Stan is • to get twice as much as Jill, and Ann is to get twice as much as Stan. How much does each get?

Solve the following system of equations.• 2x – y – z = 73x + 5y + z = -104x – 3y + 2z = 4

A2.7.B Find the terms and partial sums of arithmetic and geometric series and the infinite sum for geometric series.

Students build on the knowledge gained in Algebra 1 to find specific terms in a sequence and to express arithmetic and geometric sequences in both explicit and recursive forms.

Examples:

A ball is dropped from a height of 10 meters. Each •

time it hits the ground, it rebounds 34

of the distance

it has fallen. What is the total sum of the distances it falls and rebounds before coming to rest?

Show that the sum of the first 10 terms of the •

geometric series 1 + 13 + 1

9 + 1

27 + ... is twice the

sum of the first 10 terms of the geometric series

1 – 13 + 1

9 – 1

27 + ...

Algebra 2


Algebra 2A2.8. Core Processes: Reasoning, problem solving, and communication

Students formalize the development of reasoning at high school as they use algebra and the properties of number systems to develop valid mathematical arguments, make and prove conjectures, and find

counterexamples to refute false statements using correct mathematical language, terms, and symbols in all situations. They extend the problem-solving practices developed in earlier grades and apply them to more challenging problems, including problems related to mathematical and applied situations. Students formalize a coherent problem-solving process in which they analyze the situation to determine the question(s) to be answered, synthesize given information, and identify implicit and explicit assumptions that have been made. They examine their solution(s) to determine reasonableness, accuracy, and meaning in the context of the original problem. The mathematical thinking, reasoning, and problem-solving processes students learn in high school mathematics can be used throughout their lives as they deal with a world in which an increasing amount of information is presented in quantitative ways and more and more occupations and fields of study rely on mathematics.


A2.8.A Analyze a problem situation and represent it mathematically.

A2.8.B Select and apply strategies to solve problems.

A2.8.C Evaluate a solution for reasonableness, verify its accuracy, and interpret the solution in the context of the original problem.

A2.8.D Generalize a solution strategy for a single problem to a class of related problems and apply a strategy for a class of related problems to solve specific problems.

A2.8.E Read and interpret diagrams, graphs, and text containing the symbols, language, and conventions of mathematics.

A2.8.F Summarize mathematical ideas with precision and efficiency for a given audience and purpose.

Examples:

Show that • aa+b ≠ + b , for all positive real values of a and b.

Show that the product of two odd numbers is • always odd.

Leo is painting a picture on a canvas that • measures 32 inches by 20 inches. He has divided the canvas into four different rectangles, as shown in the diagram.

He would like the upper right corner to be a rectangle that has a length 1.6 times its width. Leo wants the area of the larger rectangle in the lower left to be at least half the total area of the canvas.

Algebra 2



A2.8.G Use inductive reasoning and the properties of numbers to make conjectures, and use deductive reasoning to prove or disprove conjectures.

A2.8.H Synthesize information to draw conclusions and evaluate the arguments and conclusions of others.

Describe all the possibilities for the dimensions of the upper right rectangle to the nearest hundredth, and explain why the possibilities are valid.

If Leo uses the largest possible dimensions for the smaller rectangle:

What will the dimensions of the larger — rectangle be?

Will the larger rectangle be similar to the — rectangle in the upper right corner? Why or why not?

Is the original canvas similar to the — rectangle in the upper right corner?

(A rectangle whose length and width are in the ratio 1

2+ 5 (approximately equal to 1.6) is called a “golden

rectangle” and is often used in art and architecture.)

A relationship between variables can be represented • with a table, a graph, an equation, or a description in words.

How can you decide from a table — whether a relationship is linear, quadratic, or exponential?

How can you decide from a graph — whether a relationship is linear, quadratic, or exponential?

How can you decide from an equation — whether a relationship is linear, quadratic, or exponential?

Algebra 2

Algebra 2


Mathematics 1


Mathematics 1

In Mathematics 1, students begin to formalize mathematics by exploring function concepts with emphasis on the family of linear functions and their applications. Students extend their work with graphical and numerical data analysis to include bivariate data involving linear relationships. Students identify and prove relationships about lines in the plane and similar triangles. Proportionality is a common thread in Mathematics 1 that connects linear functions, data analysis, and coordinate geometry. Throughout this course, students develop their reasoning skills by making conjectures and predictions or creating simple proofs related to algebraic, geometric, and statistical relationships.


Mathematics 1M1.1. Core Content: Solving problems (Algebra)

Students learn to solve many new types of problems in Mathematics 1, and this first core content area highlights the types of problems students will be able to solve after they master the concepts and skills

in this course. Throughout Mathematics 1, students spend considerable time with linear functions and are introduced to other types of functions, including exponential functions and functions defined piecewise. They learn that specific functions model situations described in word problems, and thus they learn the broader notion that functions are used to solve various types of problems. The ability to write an equation that represents a problem is an important mathematical skill in itself, and each new function provides students the tool to solve yet another class of problems. Many problems that initially appear to be very different from each other can actually be represented by identical equations. This is an important and unifying principle of algebra—that the same algebraic techniques can be applied to a wide variety of different situations.


M1.1.A Select and justify functions and equations to model and solve problems.

Students can analyze the rate of change of a function represented with a table or graph to determine if the function is linear. Students also analyze common ratios to determine if the function is exponential. After selecting a function to model a situation, students describe appropriate domain restrictions. They use the function to solve the problem and interpret the solution in the context of the original situation.

Examples:

A cup is 6 cm tall, including a 1.1 cm lip. Find a • function that represents the height of a stack of cups in terms of the number of cups in the stack. Find a function that represents the number of cups in a stack of a given height.

For the month of July, Michelle will be dog-sitting • for her very wealthy, but eccentric, neighbor, Mrs. Buffett. Mrs. Buffett offers Michelle two different salary plans:Plan 1: $100 per day for the 31 days of the month.Plan 2: $1 for July 1, $2 for July 2, $4 for July 3, and so on, with the daily rate doubling each day.

Write functions that model the amount of a. money Michelle will earn each day on Plan 1 and Plan 2. Justify the functions you wrote.State an appropriate domain for each of the b. models based on the context.Which plan should Michelle choose to c. maximize her earnings? Justify your recommendation mathematically.Extension: Write an algebraic function for the d. cumulative pay for each plan based on the number of days worked.

Math 1



M1.1.B Solve problems that can be represented by linear functions, equations, and inequalities.

It is mathematically important to represent a word problem as an equation. Students must analyze the situation and find a way to represent it mathematically. After solving the equation, students think about the solution in terms of the original problem.

Examples:

The assistant pizza maker makes 6 pizzas an • hour. The master pizza maker makes 10 pizzas an hour but starts baking two hours later than his assistant. Together, they must make 92 pizzas. How many hours from when the assistant starts baking will it take? What is a general equation, in function form, that could be used to determine the number of pizzas that can be made in two or more hours?

A swimming pool holds 375,000 liters of water. • Two large hoses are used to fill the pool. The first hose fills at the rate of 1,500 liters per hour and the second hose fills at the rate of 2,000 liters per hour. How many hours does it take to fill the pool completely?

M1.1.C Solve problems that can be represented by a system of two linear equations or inequalities.

Examples:

An airplane flies from Baltimore to Seattle (assume • a distance of 2,400 miles) in 7 hours, but the return

flight takes only 41

4 hours. The air speed of the

plane is the same in both directions. How many miles per hour does the plane fly with respect to the wind? What is the wind speed in miles per hour?

A coffee shop employee has one cup of 85% milk • (the rest is chocolate) and another cup of 60% milk (the rest is chocolate). He wants to make one cup of 70% milk. How much of the 85% milk and 60% milk should he mix together to make the 70% milk?

Two plumbing companies charge different rates • for their service. Clyde’s Plumbing Company charges a $75-per-visit fee that includes one hour of labor plus $45 dollars per hour after the first hour. We-Unclog-It Plumbers charges a $100-per-visit fee that includes one hour of labor plus $40 per hour after the first hour. For how many hours of plumbing work would Clyde’s be less expensive than We-Unclog-It?

Note: Although this context is discrete, students can model it with continuous linear functions.

Math 1



M1.1.D Solve problems that can be represented by exponential functions and equations.

Students recognize common examples of exponential growth or decay, such as applying exponential functions to determine compound interest, population growth, and radioactivity. They approximate solutions with graphs or tables, check solutions numerically, and when possible, solve problems exactly.

Example:

Mr. Tsu invests $1000 in a 5-year CD that pays 4% • interest compounded yearly. Present to Mr. Tsu his expected balance at the end of years 1, 3, and 5 and the process you used to arrive at each value.

Math 1


Mathematics 1M1.2. Core Content: Characteristics and behaviors of functions (Algebra)

Students formalize and deepen their understanding of functions, the defining characteristics and uses of functions, and the mathematical language used to describe functions. They learn that functions

are often specified by an equation of the form y = f(x), where any allowable x-value yields a unique y-value. Mathematics 1 has a particular focus on linear functions, equations, and systems of equations and on functions that can be defined piecewise, particularly step functions and functions that contain the absolute value of an expression. Students compare and contrast non-linear functions, such as quadratic and exponential, with linear functions. They learn about the representations and basic transformations of these functions and the practical and mathematical limitations that must be considered when working with functions and when using functions to model situations.


M1.2.A Determine whether a relationship is a function and identify the domain, range, roots, and independent and dependent variables.

Functions studied in Mathematics 1 include linear and those defined piecewise (including step functions and those that contain the absolute value of an expression). They compare and contrast non-linear functions, such as quadratic and exponential, to linear functions.

Given a problem situation, students should describe further restrictions on the domain of a function that are appropriate for the problem context.

Examples:

Which of the following are functions? Explain why • or why not.

The age in years of each student in your — math class and each student’s shoe size.The number of degrees a person rotates a — spigot and the volume of water that comes out of the spigot.

A function• f(n) = 60n is used to model the distance in miles traveled by a car traveling 60 miles per hour in n hours. Identify the domain and range of this function. What restrictions on the domain of this function should be considered for the model to correctly reflect the situation?

What is the domain of • f(x) = 5 − x?

Math 1



M1.2.A cont. Which of the following equations, inequalities, or • graphs determine y as a function of x?

y— = 2x— = 3y— = |x|

— yx xx x

=+ ≤− >

3 12 1,,

x— 2 + y2 = 1

1

1

-1

-1 x

y

M1.2.B Represent a function with a symbolic expression, as a graph, in a table, and using words, and make connections among these representations.

This expectation applies each time a new class (family) of functions is encountered. In Mathematics 1, students should be introduced to a variety of additional

functions that include expressions such as x3, x , 1

x,

and absolute values. They will study these functions in depth in subsequent courses.

Students should know that f(x) = a

x represents an

inverse variation. Students begin to describe the graph of a function from its symbolic expression, and use key characteristics of the graph of a function to infer properties of the related symbolic expression.

Translating among these various representations of functions is an important way to demonstrate conceptual understanding of functions.

Students learn that each representation has particular advantages and limitations. For example, a graph shows the shape of a function, but not exact values. They also learn that a table of values may not uniquely determine a single function without some specification of the nature of that function (e.g., it is quadratic).

Math 1



M1.2.C Evaluate f(x) at a (i.e., f(a)) and solve for x in the equation f(x) = b.

Functions may be described and evaluated with symbolic expressions, tables, graphs, or verbal descriptions.

Students should distinguish between solving for f(x) and evaluating a function at x.

Example:

Roses-R-Red sells its roses for $0.75 per stem • and charges a $20 delivery fee per order.

What is the cost of having 10 roses delivered?— How many roses can you have delivered — for $65?

M1.2.D Plot points, sketch, and describe the graphs of

functions of the form f(x) = ax

+ b.

Mathematics 1 addresses only rational functions

of the form f(x) = ax

+ b. Rational functions of the

form f(x) = ax2 + b and f(x) =

( )a

bx c+ are addressed in

Mathematics 3.

Example:

Sketch the graphs of the four functions • f(x) = ax

+ b

when a = 4 and 8 and b = 0 and 1.

Math 1


Mathematics 1M1.3. Core Content: Linear functions, (Algebra, Geometry/Measurement, equations, and relationships Data/Statistics/Probability)

Students understand that linear functions can be used to model situations involving a constant rate of change. They build on the work done in middle school to solve systems of linear equations and

inequalities in two variables, learning to interpret the intersection of lines as the solution. While the focus is on solving equations, students also learn graphical and numerical methods for approximating solutions to equations. They use linear functions to analyze relationships, represent and model problems, and answer questions. These algebraic skills are applied in other Core Content areas across high school courses.


M1.3.A Write and solve linear equations and inequalities in one variable.

This expectation includes the use of absolute values in the equations and inequalities.

Examples:Write an absolute value equation or inequality for • all the numbers 2 units from 7, and all the numbers that are more than b units from a.Solve |• x – 6| ≤ 4 and locate the solution on the number line.Write an equation or inequality that has no real • solutions; infinite numbers of real solutions; and exactly one real solution.Solve for • x in 2(x – 3) + 4x = 15 + 2x.Solve• 8.5 < 3x + 2 ≤ 9.7 and locate the solution on the number line.

M1.3.B Describe how changes in the parameters of linear functions and functions containing an absolute value of a linear expression affect their graphs and the relationships they represent.

In the case of a linear function y = f(x), expressed in slope-intercept form (y = mx + b), m and b are parameters. Students should know that f(x) = kx represents a direct variation (proportional relationship).

Examples: Graph a function of the form • f(x) = kx, describe the effect that changes on k have on the graph and on f(x), and answer questions that arise in proportional situations.A gas station’s 10,000-gallon underground storage • tank contains 1,000 gallons of gasoline. Tanker trucks pump gasoline into the tank at a rate of 400 gallons per minute. How long will it take to fill the tank? Find a function that represents this situation and then graph the function. If the flow rate increases from 400 to 500 gallons per minute, how will the graph of the function change? If the initial amount of gasoline in the tank changes from 1,000 to 2,000 gallons, how will the graph of the function change?

Compare and contrast the functions • y = 3|x| and

y = -

|x|.

Math 1

1

3



M1.3.C Identify and interpret the slope and intercepts of a linear function, including equations for parallel and perpendicular lines.

Examples:

The graph shows the relationship between time and • distance from a gas station for a motorcycle and a scooter. What can be said about the relative speed of the motorcycle and scooter that matches the information in the graph? What can be said about the intersection of the graphs of the scooter and the motorcycle? Is it possible to tell which vehicle is further from the gas station at the initial starting point represented in the graph? At the end of the time represented in the graph? Why or why not?

scooter

motorcycle

TimeD

ista

nce

A 1,500-gallon tank contains 200 gallons of water. • Water begins to run into the tank at the rate of 75 gallons per hour. When will the tank be full? Find a linear function that models this situation, draw a graph, and create a table of data points. Once you have answered the question and completed the tasks, explain your reasoning. Interpret the slope and y-intercept of the function in the context of the situation.

Given that the figure below is a square, find the • slope of the perpendicular sides AB and BC. Describe the relationship between the two slopes.

p

pq

q

A

B

C

D

Math 1



M1.3.D Write and graph an equation for a line given the slope and the y-intercept, the slope and a point on the line, or two points on the line, and translate between forms of linear equations.

Linear equations may be written in slope-intercept, point-slope, and standard form.

Examples:

Find an equation for a line with • y-intercept equal to 2 and slope equal to 3.

Find an equation for a line with a slope of 2 that • goes through the point (1, 1).

Find an equation for a line that goes through the • points (-3, 5) and (6, -2).

For each of the following, use only the equation • (without sketching the graph) to describe the graph.

y— = 2x + 3y— – 7 = 2(x – 2)

Write the equation 3• x + 2y = 5 in slope intercept form.

Write the equation • y – 1 = 2(x – 2) in standard form.

M1.3.E Write and solve systems of two linear equations and inequalities in two variables.

Students solve both symbolic and word problems, understanding that the solution to a problem is given by the coordinates of the intersection of the two lines when the lines are graphed in the same coordinate plane.

Examples:

Solve the following simultaneous linear equations • algebraically:-2x + y = 2x + y = -1

Graph the above two linear equations on the same • coordinate plane and use the graph to verify the algebraic solution.

An academic team is going to a state mathematics • competition. There are 30 people going on the trip. There are 5 people who can drive and 2 types of vehicles, vans and cars. A van seats 8 people, and a car seats 4 people, including drivers. How many vans and cars does the team need for the trip? Explain your reasoning.Let v = number of vans and c = number of cars.v + c ≤ 58v + 4c > 30

Math 1



M1.3.F Find the equation of a linear function that best fits bivariate data that are linearly related, interpret the slope and y-intercept of the line, and use the equation to make predictions.

A bivariate set of data presents data on two variables, such as shoe size and height.

In high school, the emphasis is on using a line of best fit to interpret data and on students making judgments about whether a bivariate data set can be modeled with a linear function. Students can use various methods, including technology, to obtain a line of best fit.

Making predictions involves both interpolating and extrapolating from the original data set.

Students need to be able to evaluate the quality of their predictions, recognizing that extrapolation is based on the assumption that the trend indicated continues beyond the unknown data.

M1.3.G Describe the correlation of data in scatterplots in terms of strong or weak and positive or negative.

Example:

Which words—• strong or weak, positive or negative—could be used to describe the correlation shown in the sample scatterplot below?

100 200 300 400 500

-4

-3.5

-3

-2.5

-2

-1.5

-1

xx

xx

xxxxx

xxxx

x

xxxxx

xx

Scatterplot

X

Y

M1.3.H Determine the equation of a line in the coordinate plane that is described geometrically, including a line through two given points, a line through a given point parallel to a given line, and a line through a given point perpendicular to a given line.

Examples:

Write an equation for the perpendicular bisector of • a given line segment.

Determine the equation of a line through the points • (5, 3) and (5, -2).

Prove that the slopes of perpendicular lines are • negative inverses of each other.

Math 1


Mathematics 1M1.4. Core Content: Proportionality, similarity, and geometric reasoning (Geometry/Measurement)

Students extend and formalize their knowledge of two-dimensional geometric figures and their properties, with a focus on properties of lines, angles, and triangles. They explain their reasoning

using precise mathematical language and symbols. Students study basic properties of parallel and perpendicular lines, their respective slopes in the coordinate plane, and the properties of the angles formed when parallel lines are intersected by a transversal. They prove related theorems and apply them to solve problems that are purely mathematical and that arise in applied contexts. Students formalize their prior work with similarity and proportionality by making and proving conjectures about triangle similarity.


M1.4.A Distinguish between inductive and deductive reasoning.

Students generate and test conjectures inductively and then prove (or disprove) their conclusions deductively.

Example:

A student first hypothesizes that the sum of the • angles of a triangle is 180 degrees and then proves this is true. When was the student using inductive reasoning? When was s/he using deductive reasoning? Justify your answers.

M1.4.B Use inductive reasoning to make conjectures, to test the plausibility of a geometric statement, and to help find a counterexample.

Example:

Using dynamic geometry software, decide if the • following is a plausible conjecture: If two parallel lines are cut by a transversal, then alternate interior angles are equal.

M1.4.C Use deductive reasoning to prove that a valid geometric statement is true.


Example:

Prove that if two parallel lines are cut by a • transversal, then alternate interior angles are equal.

M1.4.D Determine and prove triangle similarity. Similarity in Mathematics 1 builds on proportionality concepts from middle school mathematics. Determining and proving triangle congruence and other properties of triangles are included in Mathematics 2.

Students should identify necessary and sufficient conditions for similarity in triangles, and use these conditions in proofs.

Example:

For a given • ∆RST, prove that ∆XYZ, formed by joining the midpoints of the sides of ∆RST, is similar to ∆RST.

Math 1



M1.4.E Know, prove, and apply theorems about parallel and perpendicular lines.

Students should be able to summarize and explain basic theorems. They are not expected to recite lists of theorems, but they should know the conclusion of a theorem when given its hypothesis.

Examples:

Prove that a point on the perpendicular bisector of • a line segment is equidistant from the ends of the line segment.

If each of two lines is perpendicular to a given line, • what is the relationship between the two lines? How do you know?

M1.4.F Know, prove, and apply theorems about angles, including angles that arise from parallel lines intersected by a transversal.

Example:

Take two parallel lines • l and m, with (distinct) points A and B on l and C and D on m.

If AC intersects BD at point E, prove that

∆ABE ≅ ∆CDE.

M1.4.G Explain and perform basic compass and straightedge constructions related to parallel and perpendicular lines.

Constructions using circles and lines with dynamic geometry software (i.e., virtual compass and straight-edge) are equivalent to paper and pencil constructions.

Example:

Construct and mathematically justify the steps to:• Bisect a line segment.— Drop a perpendicular from a point to a line.— Construct a line through a point that is — parallel to another line.

Math 1


Mathematics 1M1.5. Core Content: Data and distributions (Data/Statistics/Probability)

Students select mathematical models for data sets and use those models to represent, describe, and compare data sets. They analyze the linear relationship between two statistical variables and make

and defend appropriate predictions, conjectures, and generalizations based on data. Students understand limitations of conclusions drawn from the results of a study or an experiment and recognize common misconceptions and misrepresentations.


M1.5.A Use and evaluate the accuracy of summary statistics to describe and compare data sets.

A univariate set of data identifies data on a single variable, such as shoe size.

This expectation extends what students have learned in earlier grades to include evaluation and justification. They both compute and evaluate the appropriateness of measure of center and spread (range and interquartile range) and use these measures to accurately compare data sets. Students will draw appropriate conclusions through the use of statistical measures of center, frequency, and spread, combined with graphical displays.

Examples:

The local minor league baseball team has a salary • dispute. Players claim they are being underpaid, but managers disagree.

Bearing in mind that a few top players — earn salaries that are quite high, would it be in the managers’ best interest to use the mean or median when quoting the “average” salary of the team? Why?What would be in the players’ best interest?—

Each box-and-whisker plot shows the prices of • used cars (in thousands of dollars) advertised for sale at three different car dealers. If you want to go to the dealer whose prices seem least expensive, which dealer would you go to? Use statistics from the displays to justify your answer.

Cars are US

Better-than-New

Yours Now

50 10

Math 1



M1.5.B Describe how linear transformations affect the center and spread of univariate data.

Examples:

A company decides to give every one of its • employees a $5,000 raise. What happens to the mean and standard deviation of the salaries as a result?

A company decides to double each of its • employee’s salaries. What happens to the mean and standard deviation of the salaries as a result?

M1.5.C Make valid inferences and draw conclusions based on data.

Determine whether arguments based on data confuse association with causation. Evaluate the reasonableness of and make judgments about statistical claims, reports, studies, and conclusions.

Example:

Mr. Shapiro found that the amount of time his • students spent doing mathematics homework is positively correlated with test grades in his class. He concluded that doing homework makes students’ test scores higher. Is this conclusion justified? Explain any flaws in Mr. Shapiro’s reasoning.

Math 1


Mathematics 1M1.6. Core Content: Numbers, expressions, and operations (Numbers, Operations, Algebra)

Students see the number system extended to the real numbers represented by the number line. They use variables and expressions to solve problems from purely mathematical as well as applied

contexts. They build on their understanding of and ability to compute with arithmetic operations and properties and expand this understanding to include the symbolic language of algebra. Students demonstrate this ability to write and manipulate a wide variety of algebraic expressions throughout high school mathematics as they apply algebraic procedures to solve problems.


M1.6.A Know the relationship between real numbers and the number line, and compare and order real numbers with and without the number line.

Although a formal definition of real numbers is beyond the scope of Mathematics 1, students learn that every point on the number line represents a real number, either rational or irrational, and that every real number has its unique point on the number line. They locate, compare, and order real numbers on the number line.

Real numbers include those written in scientific notation or expressed as fractions, decimals, exponentials, or roots.

Examples:

Without using a calculator, order the following on • the number line:

82 , 3π, 8.9, 9, 374

, 9.3 × 100

A star’s color gives an indication of its temperature • and age. The chart shows four types of stars and the lowest temperature of each type.

Type Lowest Temperature(in ºF)

Color

A

B

G

P

1.35 x 104

2.08 x 104

9.0 x 103

4.5 x 104

Blue-White

Blue

Blue

Yellow

List the temperatures in order from lowest to highest.

Math 1



M1.6.B Determine whether approximations or exact values of real numbers are appropriate, depending on the context, and justify the selection.

Decimal approximations of numbers are sometimes used in applications, such as carpentry or engineering; while at other times, these applications may require exact values. Students should understand the difference and know that the appropriate approximation depends upon the necessary degree of precision needed in given situations.

For example, 1.414 is an approximation and not an exact solution to the equation x2 – 2 = 0, but 2 is an exact solution to this equation.

Example:

Using a common engineering formula, an • engineering student represented the maximum safe load of a bridge to be 1000(99 – 70 2) tons.

He used 1.41 as the approximation for 2 in his calculations. When the bridge was built and tested in a computer simulation to verify its maximum weight-bearing load, it collapsed! The student had estimated the bridge would hold ten times the weight that was applied to it when it collapsed.

Calculate the weight that the student — thought the bridge could bear using 1.41 as the estimate for 2. Calculate other weight values using — estimates of 2 that have more decimal places. What might be a reasonable degree of precision required to know how much weight the bridge can handle safely? Justify your answer.

M1.6.C Recognize the multiple uses of variables, determine all possible values of variables that satisfy prescribed conditions, and evaluate algebraic expressions that involve variables.

Students learn to use letters as variables and in other ways that increase in sophistication throughout high school. For example, students learn that letters can be used:

To represent fixed and temporarily unknown values • in equations, such as 3x + 2 = 5;

To express identities, such as • x + x = 2x for all x;

As attributes in formulas, such as • A = lw;

As constants such as • a, b, and c in the equation y = ax2 + bx + c;

As parameters in equations, such as the • m and b for the family of functions defined by y = mx + b;

To represent varying quantities, such as • x in f(x) = 5x;

To represent functions, such as • f in f(x) = 5x; and

To represent specific numbers, such as • π.

Math 1



M1.6.C cont. Expressions include those involving polynomials, radicals, absolute values, and integer exponents.

Examples:

For what values of • a and n, where n is an integer greater than 0, is an always negative?

For what values of • a is 1a

an integer?

For what values of • a is 5 − a defined? For what values of • a is -a always positive?

M1.6.D Solve an equation involving several variables by expressing one variable in terms of the others.

Examples:

Solve • A = p + prt for p.

Solve • V = πr 2h for h or for r.

Math 1


Mathematics 1M1.7. Additional Key Content (Numbers, Algebra)

Students develop a basic understanding of arithmetic and geometric sequences and of exponential functions, including their graphs and other representations. They use exponential functions to analyze

relationships, represent and model problems, and answer questions in situations that are modeled by these nonlinear functions. Students learn graphical and numerical methods for approximating solutions to exponential equations. Students interpret the meaning of problem solutions and explain limitations related to solutions.


M1.7.A Sketch the graph for an exponential function of the form y = abn where n is an integer, describe the effects that changes in the parameters a and b have on the graph, and answer questions that arise in situations modeled by exponential functions.

Examples:

Sketch the graph of • y = 2n by hand.

You have won a door prize and are given a choice • between two options:

$150 invested for 10 years at 4% compounded annually.$200 invested for 10 years at 3% compounded annually.

How much is each worth at the end of each year of the investment periods?

Are the two investments ever equal in value? Which will you choose?

M1.7.B Find and approximate solutions to exponential equations.

Students can approximate solutions using graphs or tables with and without technology.

M1.7.C Interpret and use integer exponents and square and cube roots, and apply the laws and properties of exponents to simplify and evaluate exponential expressions.

Examples:

2• -3 = 1

23

• -22 3 5

2 3 5

2

-3 22= 3

2 54

5

• a b c

a b c

b

a c

-

-

2 2

2 3 2

5

4=

• 8 2 2 2 2 2= • • =

• a b a b• = •3 3 3

Math 1



M1.7.D Express arithmetic and geometric sequences in both explicit and recursive forms, translate between the two forms, explain how rate of change is represented in each form, and use the forms to find specific terms in the sequence.

Examples:

Write a recursive formula for the arithmetic • sequence 5, 9, 13, 17, . . . What is the slope of the line that contains the points associated with these values and their position in the sequence? How is the slope of the line related to the sequence?

Given that • u(0) = 3 and u(n + 1) = u(n) + 7 when n is a positive integer, a. find u(5);b. find n so that u(n) = 361; andc. find a formula for u(n).

Write a recursive formula for the geometric • sequence 5, 10, 20, 40, . . . and determine the 100th term.

Given that • u(0) = 2 and u(n + 1) = 3u(n),

a. find u(4), and

b. find a formula for u(n).

Math 1


Mathematics 1M1.8. Core Processes: Reasoning, problem solving, and communication

Students formalize the development of reasoning in Mathematics 1 as they use algebra, geometry, and statistics to make and defend generalizations. They justify their reasoning with accepted standards

of mathematical evidence and proof, using correct mathematical language, terms, and symbols in all situations. They extend the problem-solving practices developed in earlier grades and apply them to more challenging problems, including problems related to mathematical and applied situations. Students formalize a coherent problem-solving process in which they analyze the situation to determine the question(s) to be answered, synthesize given information, and identify implicit and explicit assumptions that have been made. They examine their solution(s) to determine reasonableness, accuracy, and meaning in the context of the original problem. The mathematical thinking, reasoning, and problem-solving processes students learn in high school mathematics can be used throughout their lives as they deal with a world in which an increasing amount of information is presented in quantitative ways and more and more occupations and fields of study rely on mathematics.


M1.8.A Analyze a problem situation and represent it mathematically.

M1.8.B Select and apply strategies to solve problems.

M1.8.C Evaluate a solution for reasonableness, verify its accuracy, and interpret the solution in the context of the original problem.

M1.8.D Generalize a solution strategy for a single problem to a class of related problems, and apply a strategy for a class of related problems to solve specific problems.

M1.8.E Read and interpret diagrams, graphs, and text containing the symbols, language, and conventions of mathematics.

M1.8.F Summarize mathematical ideas with precision and efficiency for a given audience and purpose.

M1.8.G Synthesize information to draw conclusions, and evaluate the arguments and conclusions of others.

M1.8.H Use inductive reasoning to make conjectures, and use deductive reasoning to prove or disprove conjectures.

Examples:

Three teams of students independently conducted • experiments to relate the rebound height of a ball to the rebound number. The table gives the average of the teams’ results.

ReboundNumber

ReboundHeight (cm)

0 200

1 155

2 116

3 88

4 66

5 50

6 44

Construct a scatterplot of the data, and describe the function that relates the height of the ball to the rebound number. Predict the rebound height of the ball on the tenth rebound. Justify your answer.

Prove formally that the sum of two odd numbers is • always even.

Math 1


Mathematics 2


Mathematics 2

Mathematics 2 extends the study of functions to include quadratic functions, providing tools for modeling a greater variety of real-world situations. Students develop computational and algebraic skills that support analysis of these functions and their multiple representations. Students extend their ability to reason mathematically. They distinguish between inductive and deductive thinking, make conjectures, and prove theorems. Students become skilled in writing more involved proofs through their study of triangles, lines, and quadrilaterals. Finally, the study of probability extends students’ understanding of proportional reasoning and relationships with the inclusion of counting methods and lays the groundwork for the study of data and variability in the next course.


Mathematics 2M2.1. Core Content: Modeling situations and solving problems (Algebra)

This first core content area highlights the types of problems students will be able to solve by the end of Mathematics 2. Students extend their ability to model situations and solve problems with additional

functions and equations in this course. Additionally, they deepen their understanding and proficiency with functions they encountered in Mathematics 1 and use these functions to solve more complex problems. When presented with a word problem, students determine which function or equation models the problem and then use that information to write an equation to solve the problem. Turning word problems into equations that can be solved is a skill students hone throughout the course.



Example:

Dawson wants to make a horse corral by creating • a rectangle that is divided into 2 parts, similar to the following diagram. He has a 1200-foot roll of fencing to do the job. What are the dimensions of the enclosure with the largest total area? What function or equation best models this situation?

l

w

M2.1.B Solve problems that can be represented by systems of equations and inequalities.

Example:

Data derived from an experiment seems to be • parabolic when plotted on a coordinate grid. Three observed data points are (2, 10), (3, 8), and (4, 4). Write a quadratic equation that passes through the points.

M2.1.C Solve problems that can be represented by quadratic functions, equations, and inequalities.

Students solve problems by factoring and applying the quadratic formula to the quadratic equation, and use the vertex form of the equation to solve problems about maximums, minimums, and symmetry.

Examples:

Find the solutions to the simultaneous equations • y = x + 2 and y = x2.

Math 2



M2.1.C cont. If you throw a ball straight up (with initial height of • 4 feet) at 10 feet per second, how long will it take to fall back to the starting point? The function h(t) = -16t2 + v0t + h0 describes the height, h in feet, of an object after t seconds, with initial velocity v0 and initial height h0.

Joe owns a small plot of land 20 feet by 30 feet. • He wants to double the area by increasing both the length and the width, keeping the dimensions in the same proportion as the original. What will be the new length and width?

What two consecutive numbers, when multiplied • together, give the first number plus 16? Write the equation that represents the situation.

The Gateway Arch in St. Louis has a special • shape called a catenary, which looks a lot like a parabola. It has a base width of 600 feet and is 630 feet high. Which is taller, this catenary arch or a parabolic arch that has the same base width but has a height of 450 feet at a point 150 feet from one of the pillars? What is the height of the parabolic arch?

M2.1.D Solve problems that can be represented by exponential functions and equations.

Students extend their use of exponential functions and equations to solve more complex problems. They approximate solutions with graphs or tables, check solutions numerically, and when possible, solve problems exactly.

Examples:

E. coli bacteria reproduce by a simple process • called binary fission—each cell increases in size and divides into two cells. In the laboratory, E. coli bacteria divide approximately every 15 minutes. A new E. coli culture is started with 1 cell.

Find a function that models the E. coli a. population size at the end of each 15-minute interval. Justify the function you found.

State an appropriate domain for the model b. based on the context.

After what 15-minute interval will you have at c. least 500 bacteria?

Estimate the solution to 2• x = 16,384

Math 2



M2.1.E Solve problems involving combinations and permutations.

Examples:

The company Ali works for allows her to invest • in her choice of 10 different mutual funds, 6 of which grew by at least 5% over the last year. Ali randomly selected 4 of the 10 funds in which to invest. What is the probability that 3 of Ali’s funds grew by 5%?

Four points (• A, B, C, and D) lie on one straight line, n, and five points (E, F, G, H, and J) lie on another straight line, m, that is parallel to n. What is the probability that three points, selected at random, will form a triangle?

Math 2


Mathematics 2M2.2. Core Content: Quadratic functions, equations, and relationships (Algebra)

Students learn that exponential and quadratic functions can be used to model some situations where linear functions may not be the best model. They use graphical and numerical methods with

exponential functions of the form y = abx and quadratic functions to analyze relationships, represent and model problems, and answer questions. Students extend their algebraic skills and learn various methods of solving quadratic equations over real or complex numbers, including using the quadratic formula, factoring, graphing, and completing the square. They learn to translate between forms of quadratic equations, applying the vertex form to evaluate maximum and minimum values and find symmetry of the graph, and they learn to identify which form should be used in a particular situation. They interpret the meaning of problem solutions and explain their limitations. Students recognize common examples of situations that can be modeled by quadratic functions, such as maximizing area or the height of an object moving under the force of gravity. They compare the characteristics of quadratic functions to those of linear and exponential functions. The understanding of these particular types of functions, together with students’ understanding of linear functions, provides students with a powerful set of tools to use mathematical models to deal with problems and situations in advanced mathematics courses, in the workplace, and in everyday life.


M2.2.A Represent a quadratic function with a symbolic expression, as a graph, in a table, and with a description, and make connections among the representations.

Example:

Kendre and Tyra built a tennis ball cannon that • launches tennis balls straight up in the air at an initial velocity of 50 feet per second. The mouth of the cannon is 2 feet off the ground. The function h(t) = -16t2 + 50t + 2 describes the height, h, in feet, of the ball t seconds after the launch.Make a table from the function. Then use the table to sketch a graph of the height of the tennis ball as a function of time into the launch. Give a verbal description of the graph. How high was the ball after 1 second? When does it reach this height again?

M2.2.B Sketch the graph of a quadratic function, describe the effects that changes in the parameters have on the graph, and interpret the x-intercepts as solutions to a quadratic equation.

Note that in Mathematics 2, the parameter b in the term bx in the quadratic form ax2 + bx + c is not often used to provide useful information about the characteristics of the graph.

Parameters considered most useful are:

a• and c in f(x) = ax2 + c

a• , h, and k in f(x) = a(x – h)2 + k, and

a• , r, and s in f(x) = a(x – r)(x – s)

Math 2



M2.2.B cont. Example:

A particular quadratic function can be expressed in • the following two ways:f(x) = -(x – 3)2 + 1 f(x) = -(x – 2)(x – 4)What information about the graph can be directly inferred from each of these forms? Explain your reasoning.Sketch the graph of this function, showing the roots.

M2.2.C Translate between the standard form of a quadratic function, the vertex form, and the factored form; graph and interpret the meaning of each form.

Students translate among forms of a quadratic function to convert to one that is appropriate—e.g., vertex form—to solve specific problems.

Students learn about the advantages of the standard form (f(x) = ax2 + bx + c), the vertex form (f(x) = a(x – h)2 + d), and the factored form (f(x) = a(x – r)(x – s)). They produce the vertex form by completing the square on the function in standard form, which allows them to see the symmetry of the graph of a quadratic function as well as the maximum or minimum. This opens up a whole range of new problems students can solve using quadratics. Students continue to find the solutions of the equation, which can be either real or complex.

Example:

Find the minimum, the line of symmetry, and • the roots for the graphs of each of the following functions: f(x) = x2 – 4x + 3f(x) = x2 – 4x + 4f(x) = x2 – 4x + 5

M2.2.D Solve quadratic equations that can be factored as (ax + b)(cx + d) where a, b, c, and d are integers.

Students learn to efficiently solve quadratic equations by recognizing and using the simplest factoring methods, including recognizing special quadratics as squares and differences of squares.

Examples:

2• x2 + x – 3 = 0; (x – 1)(2x + 3) = 0; x = 1, -−32

4• x2 + 6x = 0; 2x(2x + 3) = 0; x = 0, -−32

36• x2 – 25 = 0; (6x + 5)(6x – 5) = 0; x = ±

56

x• 2 + 6x + 9 = 0; (x + 3)2 = 0; x = -3

Math 2



M2.2.E Determine the number and nature of the roots of a quadratic function.

Students should be able to recognize and interpret the discriminant.

Students should also be familiar with the Fundamental Theorem of Algebra, i.e., that all polynomials, not just quadratics, have roots over the complex numbers. This concept becomes increasingly important as students progress through mathematics.

Example:

For what values of • a does f(x) = x2 – 6x + a have 2 real roots, 1 real root, and no real roots?

M2.2.F Solve quadratic equations that have real roots by completing the square and by using the quadratic formula.

Students solve those equations that are not easily factored by completing the square and by using the quadratic formula. Completing the square should also be used to derive the quadratic formula.

Students learn how to determine if there are two, one, or no real solutions.

Examples:

Complete the square to solve • x2 + 4x = 13.

x2 + 4x – 13 = 0

x2 + 4x + 4 = 17

(x + 2)2 = 17

x + 2 = ± 17x = -2 ± 17x ≈ 2.12, -6.12

Use the quadratic formula to solve 4• x2 – 2x = 5.

- x b b aca

= ± 2 4

2 –

- - - ( ) 2 2x

x

x

x

=

= ±

= ±

=

–( ) ( ) 2 4

2 8 4 8

2 2 21 8

1

2

± ± 21 4

4(4 -5)

x ≈ 1.40, -0.90

Math 2



M2.2.G Solve quadratic equations and inequalities, including equations with complex roots.

Students solve equations that are not easily factored by completing the square and by using the quadratic formula.

Examples:

x• 2 – 10x + 34 = 0

3• x2 + 10 = 4x

Wile E. Coyote launches an anvil from 180 feet • above the ground at time t = 0. The equation that models this situation is given by h = -16t2 + 96t + 180, where t is time measured in seconds and h is height above the ground measured in feet.a. What is a reasonable domain restriction for t in

this context?b. Determine the height of the anvil two seconds

after it was launched.c. Determine the maximum height obtained by

the anvil.d. Determine the time when the anvil is more

than 100 feet above ground.

Farmer Helen wants to build a pigpen. With 100 • feet of fence, she wants a rectangular pen with one side being a side of her existing barn. What dimensions should she use for her pigpen in order to have the maximum number of square feet?

M2.2.H Determine if a bivariate data set can be better modeled with an exponential or a quadratic function and use the model to make predictions.

In high school, determining a formula for a curve of best fit requires a graphing calculator or similar technological tool.

Math 2


Mathematics 2M2.3. Core Content: Conjectures and proofs (Algebra, Geometry/Measurement)

Students extend their knowledge of two-dimensional geometric figures and their properties to include quadrilaterals and other polygons, with special emphasis on necessary and sufficient conditions

for triangle congruence. They work with geometric constructions, using dynamic software as a tool for exploring geometric relationships and formulating conjectures and using compass-and-straightedge and paper-folding constructions as contexts in which students demonstrate their knowledge of geometric relationships. Students define the basic trigonometric ratios and use them to solve problems in a variety of applied situations. They formalize the reasoning skills they have developed in previous grades and solidify their understanding of what it means to mathematically prove a geometric statement. Students encounter the concept of formal proof built on definitions, axioms, and theorems. They use inductive reasoning to test conjectures about geometric relationships and use deductive reasoning to prove or disprove their conclusions. Students defend their reasoning using precise mathematical language and symbols. Finally, they apply their knowledge of linear functions to make and prove conjectures about geometric figures on the coordinate plane.


M2.3.A Use deductive reasoning to prove that a valid geometric statement is true.


Example:

Prove that the diagonals of a rhombus are • perpendicular bisectors of each other.

M2.3.B Identify errors or gaps in a mathematical argument and develop counterexamples to refute invalid statements about geometric relationships.

Example:

Identify errors in reasoning in the following proof:• Given ∠ABC ≅ ∠PRQ, AB ≅ PQ, and BC ≅ QR, then ∆ABC ≅ ∆PQR by SAS.

M2.3.C Write the converse, inverse, and contrapositive of a valid proposition and determine their validity.

Examples:

If • m and n are odd integers, then the sum of m and n is an even integer. State the converse and determine whether it is valid.

If a quadrilateral is a rectangle, the diagonals• have the same length. State the contrapositive and determine whether it is valid.

M2.3.D Distinguish between definitions and undefined geometric terms and explain the role of definitions, undefined terms, postulates (axioms), and theorems.

Students sketch points and lines (undefined terms) and define and sketch representations of other common terms. They use definitions and postulates as they prove theorems throughout geometry. In their work with theorems, they identify the hypothesis and the conclusion and explain the role of each.

Students describe the consequences of changing assumptions or using different definitions for subsequent theorems and logical arguments.

Math 2



M2.3.D cont. Example:

There are two definitions of trapezoid that can be • found in books or on the web. A trapezoid is either

a quadrilateral with exactly one pair of — parallel sides or

a quadrilateral with at least one pair of — parallel sides.

Write some theorems that are true when applied to one definition but not the other, and explain your answer.

M2.3.E Know, explain, and apply basic postulates and theorems about triangles and the special lines, line segments, and rays associated with a triangle.

Examples:

Prove that the sum of the angles of a triangle is 180• °.

Prove and explain theorems about the incenter, • circumcenter, orthocenter, and centroid.

The rural towns of Atwood, Bridgeville, and • Carnegie are building a communications tower to serve the needs of all three towns. They want to position the tower so that the distance from each town to the tower is equal. Where should they locate the tower? How far will it be from each town?

M2.3.F Determine and prove triangle congruence and other properties of triangles.

Students extend their work with similarity in Mathematics 1 to proving theorems about congruence and other properties of triangles.

Students should identify necessary and sufficient conditions for congruence in triangles, and use these conditions in proofs.

Examples:

Prove that congruent triangles are similar.•

Show that not all SSA triangles are congruent. •

M2.3.G Know, prove, and apply the Pythagorean Theorem and its converse.

Students extend their work with the Pythagorean Theorem from previous grades to include formal proof.

Examples:

Consider any right triangle with legs • a and b and hypotenuse c. The right triangle is used to create Figures 1 and 2. Explain how these figures constitute a visual representation of a proof of the Pythagorean Theorem.

Q

P

Figure 1 Figure 2

a

bc

Math 2



M2.3.G cont. A juice box is 6 cm by 8 cm by 12 cm. A straw is • inserted into a hole in the center of the top of the box. The straw must stick out 2 cm so you can drink from it. If the straw must be long enough to touch each bottom corner of the box, what is the minimum length the straw must be? (Assume the diameter of the straw is 0 for the mathematical model.)

8 cm

12 cm

6 cm

In • ∆ABC, with right angle at C, draw the altitude CD from C to AB. Name all similar triangles in the diagram. Use these similar triangles to prove the Pythagorean Theorem.

Apply the Pythagorean Theorem to derive the • distance formula in the (x, y) plane.

Determine the length of the altitude of an • equilateral triangle whose side lengths measure 5 units.

M2.3.H Solve problems involving the basic trigono-metric ratios of sine, cosine, and tangent.

Students apply their knowledge of the Pythagorean Theorem from Grade 8 to define the basic trigonometric ratios. They formally prove the Pythagorean Theorem in Mathematics 2.

Examples:

A 12-foot ladder leans against a wall to form a 63• ° angle with the ground. How many feet above the ground is the point on the wall at which the ladder is resting?

Use the Pythagorean Theorem to establish that • sin2ø + cos2 ø = 1 for ø between 0° and 90°.

Math 2



M2.3.I Use the properties of special right triangles (30°–60°–90° and 45°–45°–90°) to solve problems.

Examples:

If one leg of a right triangle has length 5 and the • adjacent angle is 30°, what is the length of the other leg and the hypotenuse?

If one leg of a 45• °–45°–90° triangle has length 5, what is the length of the hypotenuse?

The pitch of a symmetrical roof on a house 40 feet • wide is 30º. What is the length of the rafter, r, exactly and approximately?

30º 40

‘

r

M2.3.J Know, prove, and apply basic theorems about parallelograms.

Properties may include those that address symmetry and properties of angles, diagonals, and angle sums. Students may use inductive and deductive reasoning and counterexamples.

Examples:

Are opposite sides of a parallelogram always • congruent? Why or why not?

Are opposite angles of a parallelogram always • congruent? Why or why not?

Prove that the diagonals of a parallelogram bisect • each other.

Explain why if the diagonals of a quadrilateral • bisect each other, then the quadrilateral is a parallelogram.

Prove that the diagonals of a rectangle are • congruent. Is this true for any parallelogram? Justify your reasoning.

Math 2



M2.3.K Know, prove, and apply theorems about properties of quadrilaterals and other polygons.

Examples:

What is the length of the apothem of a regular • hexagon with side length 8? What is the area of the hexagon?

If the shaded pentagon were removed, it could be • replaced by a regular n-sided polygon that would exactly fill the remaining space. Find the number of sides, n, of a replacement polygon that makes the three polygons fit perfectly.

P

M2.3.L Determine the coordinates of a point that is described geometrically.

Examples:

Determine the coordinates for the midpoint of a • given line segment.

Given the coordinates of three vertices of a • parallelogram, determine all possible coordinates for the fourth vertex.

Given the coordinates for the vertices of a • triangle, find the coordinates for the center of the circumscribed circle and the length of its radius.

Math 2



M2.3.M Verify and apply properties of triangles and quadrilaterals in the coordinate plane.

Examples:

Given four points in a coordinate plane that are the • vertices of a quadrilateral, determine whether the quadrilateral is a rhombus, a square, a rectangle, a parallelogram, or none of these. Name all the classifications that apply.

Given a parallelogram on a coordinate plane, • verify that the diagonals bisect each other.

Given the line with • y-intercept 4 and x-intercept 3, find the area of a square that has one corner on the origin and the opposite corner on the line described.

Below is a diagram of a miniature golf hole as • drawn on a coordinate grid. The dimensions of the golf hole are 4 feet by 12 feet. Players must start their ball from one of the three tee positions, located at (1, 1), (1, 2), or (1, 3). The hole is located at (10, 3). A wall separates the tees from the hole. At which tee should the ball be placed to create the shortest “hole-in-one” path? Sketch the intended path of the ball, find the distance the ball will travel, and describe your reasoning. (Assume the diameters of the golf ball and the hole are 0 for the mathematical model.)

....

TeesWall

Hole

Math 2


Mathematics 2M2.4. Core Content: Probability (Data/Statistics/Probability)

Students formalize their study of probability, computing both combinations and permutations to calculate the likelihood of an outcome in uncertain circumstances and applying the binominal theorem

to solve problems. They apply their understanding of probability to a wide range of practical situations, including those involving permutations and combinations. Understanding probability helps students become knowledgeable consumers who make sound decisions about high-risk games, financial issues, etc.


M2.4.A Apply the fundamental counting principle and the ideas of order and replacement to calculate probabilities in situations arising from two-stage experiments (compound events).

M2.4.B Given a finite sample space consisting of equally likely outcomes and containing events A and B, determine whether A and B are independent or dependent, and find the conditional probability of A given B.

Example:

What is the probability of drawing a heart from • a standard deck of cards on a second draw, given that a heart was drawn on the first draw and not replaced?

M2.4.C Compute permutations and combinations, and use the results to calculate probabilities.

Example:

Two friends, Abby and Ben, are among five • students being considered for three student council positions. If each of the five students has an equal likelihood of being selected, what is the probability that Abby and Ben will both be selected?

M2.4.D Apply the binomial theorem to solve problems involving probability.

The binominal theorem is also applied when computing with polynomials.

Examples:

Use Pascal’s triangle and the binomial theorem • to find the number of ways six objects can be selected four at a time.

In a survey, 33% of adults reported that they • preferred to get the news from newspapers rather than television. If you survey 5 people, what is the probability of getting exactly 2 people who say they prefer news from the newspaper?

Write an equation that can be used to solve the problem.

Create a histogram of the binomial distribution of the probability of getting 0 through 5 responders saying they prefer the newspaper.

Math 2


Mathematics 2M2.5. Additional Key Content (Algebra, Measurement)

Students grow more proficient in their use of algebraic techniques as they use these techniques to write equivalent expressions in various forms. They build on their understanding of computation using

arithmetic operations and properties and expand this understanding to include the symbolic language of algebra. Students understand the role of units in measurement, convert among units within and between different measurement systems as needed, and apply what they know to solve problems. They use derived measures such as those used for speed (e.g., feet per second) or determining automobile gas consumption (e.g., miles per gallon).


M2.5.A Use algebraic properties to factor and combine like terms in polynomials.

Algebraic properties include the commutative, associative, and distributive properties.

Factoring includes:

Factoring a monomial from a polynomial, such as • 4x2 + 6x = 2x(2x + 3);

Factoring the difference of two squares, such as • 36x2 – 25y2 = (6x + 5y)(6x – 5y) and x4 – y4 = (x + y)(x – y)(x2 + y2);

Factoring perfect square trinomials, such as • x2 + 6xy + 9y2 = (x + 3y)2;

Factoring quadratic trinomials, such as • x2 + 5x + 4 = (x + 4)(x + 1); and

Factoring trinomials that can be expressed as the • product of a constant and a trinomial, such as 0.5x2 – 2.5x – 7 = 0.5(x + 2)(x – 7).

M2.5.B Use different degrees of precision in measurement, explain the reason for using a certain degree of precision, and apply estimation strategies to obtain reasonable measurements with appropriate precision for a given purpose.

Example:

The U.S. Census Bureau reported a national • population of 299,894,924 on its Population Clock in mid-October of 2006. One can say that the U.S. population is 3 hundred million (3 × 108) and be precise to one digit. Although the population had surpassed 3 hundred million by the end of that month, explain why 3 × 108 remained precise to one digit.

M2.5.C Solve problems involving measurement conversions within and between systems, including those involving derived units, and analyze solutions in terms of reasonableness of solutions and appropriate units.

This performance expectation is intended to build on students’ knowledge of proportional relationships. Students should understand the relationship between scale factors and their inverses as they relate to choices about when to multiply and when to divide in converting measurements.

Derived units include those that measure speed, density, flow rates, population density, etc.

Math 2



M2.5.C cont. Example:

A digital camera takes pictures that are 3.2 • megabytes in size. If the pictures are stored on a 1-gigabyte card, how many pictures can be taken before the card is full?

M2.5.D Find the terms and partial sums of arithmetic and geometric series and the infinite sum for geometric series.

Students build on the knowledge gained in Mathematics 1 to find specific terms in a sequence and to express arithmetic and geometric sequences in both explicit and recursive forms.

Examples:

A ball is dropped from a height of 10 meters. •

Each time it hits the ground, it rebounds 34

of the

distance it has fallen. What is the total sum of the distances it falls and rebounds before coming to rest?

Show that the sum of the first 10 terms of the •

geometric series 1 + 13 + 1

9 + 1

27 + ... is twice the

sum of the first 10 terms of the geometric series

1 – 13 + 1

9 – 1

27 + ...

Math 2



Students formalize the development of reasoning in Mathematics 2 as they use algebra, geometry, and probability to make and defend generalizations. They justify their reasoning with accepted standards






M2.6.D Generalize a solution strategy for a single problem to a class of related problems, and apply a strategy for a class of related problems to solve specific problems.



M2.6.G Synthesize information to draw conclusions and evaluate the arguments and conclusions of others.

M2.6.H Use inductive reasoning to make conjectures, and use deductive reasoning to prove or disprove conjectures.

Examples:

• AB is the diameter of the semicircle and the radius

of the quarter circle shown in the figure below. BC is

the perpendicular bisector of AB.

A C B

ED

F

Imagine all of the triangles formed by AB and any arbitrary point lying in the region bounded by AC , CD, and AD, seen in bold below.

A C B

ED

F

Use inductive reasoning to make conjectures about what types of triangles are formed based upon the region where the third vertex is located. Use deductive reasoning to verify your conjectures.

Math 2



M2.6 cont. Rectangular cartons that are 5 feet long need to • be placed in a storeroom that is located at the end of a hallway. The walls of the hallway are parallel. The door into the hallway is 3 feet wide and the width of the hallway is 4 feet. The cartons must be carried face up. They may not be tilted. Investigate the width and carton top area that will fit through the doorway.

54

3

R

A

S

C

T

Generalize your results for a hallway opening of x feet and a hallway width of y feet if the maximum carton dimensions are c feet long and x2 + y2 = c2.

Prove (• a + b)2 = a2 + 2ab + b2.

A student writes (• x + 3)2 = x2 + 9. Explain why this is incorrect.

Math 2


Mathematics 3


Mathematics 3

In Mathematics 3, students develop a more coherent and formal view of mathematics, going beyond specific rules and procedures to emphasize generalizations. Students extend their knowledge of number systems to include complex numbers, and they evaluate possible solutions to algebraic equations. The application and visualization of geometry extends to three-dimensional figures as students study the effects of changes in one dimension on various attributes and properties of a figure. Students study the composition of transformations on geometric figures. They generalize the relationship of changes in the symbolic form of functions to transformations of their corresponding graphs. They extend their study of functions to include logarithmic, radical, and cubic functions and are introduced to the concept of inverse functions. Students study variability of data and examine the validity of generalizing results to an entire population.


Mathematics 3M3.1. Core Content: Solving problems (Algebra)

The first core content area highlights the types of problems students will be able to solve by the end of Mathematics 3, as they extend their ability to solve problems with additional functions and equations.

Additionally, they deepen their understanding of and skills related to functions they encountered in Mathematics 1 and 2, and they use these functions to solve more complex problems. When presented with a contextual problem, students identify a function or equation that models the problem and use that information to write an equation to solve the problem. For example, in addition to using graphs to approximate solutions to problems modeled by exponential functions, they use knowledge of logarithms to solve exponential equations. Turning word problems into equations that can be solved is a skill students hone throughout the course.



Example:

A manufacturer wants to design a cylindrical soda • can that will hold 500 milliliters (mL) of soda. The manufacturer’s research has determined that an optimal can height is between 10 and 15 centimeters. Find a function for the radius in terms of the height, and use it to find the possible range of radius measurements in centimeters. Explain your reasoning.

M3.1.B Solve problems that can be represented by systems of equations and inequalities.

Examples:

Mr. Smith uses the following formula to calculate • students’ final grades in his Mathematics 3 class: 0.4E + 0.6T = C, where E represents the score on the final exam, and T represents the average score of all tests given during the grading period. All tests and the final exam are worth a maximum of 100 points. The minimum passing score on tests, the final exam, and the course is 60. Determine the inequalities that describe the following situation and sketch a system of graphs to illustrate it. When necessary, round scores to the nearest tenth.

Is it possible for a student to have a failing — test score average (i.e., T < 60 points) and still pass the course?

If you answered “yes,” what is the — minimum test score average a student can have and still pass the course? What final exam score is needed to pass the course with a minimum test score average?

A student has a particular test score — average. How can (s)he figure out the minimum final exam score needed to pass the course?

Math 3


Math 3


M3.1.C Solve problems that can be represented by quadratic functions, equations, and inequalities.

In addition to solving area and velocity problems by factoring and applying the quadratic formula to the quadratic equation, students use the vertex form of the equation to solve problems about maximums, minimums, and symmetry.

Examples:

Fireworks are launched upward from the ground • with an initial velocity of 160 feet per second. The formula for vertical motion is h(t) = 0.5at2 + vt + s, where the gravitational constant, a, is -32 feet per square second, v represents the initial velocity, and s represents the initial height. Time t is measured in seconds, and height h is measured in feet.For the ultimate effect, the fireworks must explode after they reach the maximum height. For the safety of the crowd, they must explode at least 256 ft. above the ground. The fuses must be set for the appropriate time interval that allows the fireworks to reach this height. What range of times, starting from initial launch and ending with fireworks explosion, meets these conditions?

M3.1.D Solve problems that can be represented by exponential and logarithmic functions and equations.

Examples:

If you need $15,000 in 4 years to start college, • how much money would you need to invest now? Assume an annual interest rate of 4% compounded monthly for 48 months.

The half-life of a certain radioactive substance is • 65 days. If there are 4.7 grams initially present, how long will it take for there to be less than 1 gram of the substance remaining?

M3.1.E Solve problems that can be represented by

inverse variations of the forms f(x) = ax

+ b,

f(x) = ax2

+ b, and f(x) = ( )

abx c+

.

Examples:

At the You’re Toast, Dude! toaster company, the • weekly cost to run the factory is $1400, and the cost of producing each toaster is an additional $4 per toaster.

Find a function to represent the weekly — cost in dollars, C(x), of producing x toasters. Assume either unlimited production is possible or set a maximum per week.

Find a function to represent the total — production cost per toaster for a week.

How many toasters must be produced — within a week to have a total production cost per toaster of $8?


Math 3


M3.1.E cont. A person’s weight varies inversely as the square of • his distance from the center of the earth. Assume the radius of the earth is 4000 miles. How much would a 200-pound man weigh




Math 3

Mathematics 3M3.2. Core Content: Transformations and functions (Algebra, Geometry/Measurement)

Students formalize their previous study of geometric transformations, focusing on the effect of such transformations on the attributes of geometric figures. They study techniques for using

transformations to determine congruence and similarity. Students extend their study of transformations to include transformations of many types of functions, including quadratic and exponential functions.


M3.2.A Sketch results of transformations and compositions of transformations for a given two-dimensional figure on the coordinate plane, and describe the rule(s) for performing translations or for performing reflections about the coordinate axes or the line y = x.

Transformations include translations, rotations, reflections, and dilations.

Example:

Line • m is described by the equation y = 2x + 3. Graph line m and reflect line m across the line y = x. Determine the equation of the image of the reflection. Describe the relationship between the line and its image.

M3.2.B Determine and apply properties of transformations.

Students make and test conjectures about compositions of transformations and inverses of transformations, the commutativity and associativity of transformations, and the congruence and similarity of two-dimensional figures under various transformations.

Examples:

Identify transformations (alone or in composition) • that preserve congruence.

Determine whether the composition of two • reflections of a line is commutative.

Determine whether the composition of two • rotations about the same point of rotation is commutative.

Find a rotation that is equivalent to the composition • of two reflections over intersecting lines.

Find the inverse of a given transformation.•

M3.2.C Given two congruent or similar figures in a coordinate plane, describe a composition of translations, reflections, rotations, and dilations that superimposes one figure on the other.

Examples:

Find a sequence of transformations that • superimposes the segment with endpoints (0, 0) and (2, 1) on the segment with endpoints (4, 2) and (3, 0).

Find a sequence of transformations that • superimposes the triangle with vertices (0, 0), (1, 1), and (2, 0) on the triangle with vertices (0, 1), (2, -1), and (0, -3).


Math 3


M3.2.D Describe the symmetries of two-dimensional figures and describe transformations, including reflections across a line and rotations about a point.

Although the expectation only addresses two-dimensional figures, classroom activities can easily extend to three-dimensional figures. Students can also describe the symmetries, reflections across a plane, and rotations about a line for three-dimensional figures.

M3.2.E Construct new functions using the transformations f(x – h), f(x) + k, cf(x), and by adding and subtracting functions, and describe the effect on the original graph(s).

Students perform simple transformations on functions, including those that contain the absolute value of expressions, quadratic expressions, square root expressions, and exponential expressions, to make new functions.

Examples:

What sequence of transformations changes • f(x) = x2 to g(x) = -5(x – 3)2 + 2 ?

Carly decides to earn extra money by making • glass bead bracelets. She purchases tools for $40.00. Elastic bead cord for each bracelet costs $0.10. Glass beads come in packs of 10 beads, and one pack has enough beads to make one bracelet. Base price for the beads is $2.00 per pack. For each of the first 100 packs she buys, she gets $0.01 off each of the packs. (For example, if she purchases three packs, each pack costs $1.97 instead of $2.00.) Carly plans to sell each bracelet for $4.00. Assume Carly will make a maximum of 100 bracelets.

Find a function — C(b) that describes Carly’s costs.

Find a function — R(b) that describes Carly’s revenue.

Carly’s profit is described by P(b) = R(b) – C(b).

Find — P(b).

What is the minimum number of bracelets — that Carly must sell in order to make a profit?

To make a profit of $100?—


Math 3

Mathematics 3M3.3. Core Content: Functions and modeling (Algebra)

Students extend their understanding of exponential functions from Mathematics 2 with an emphasis on inverse functions. This leads to a natural introduction of logarithms and logarithmic functions. They

learn to use the basic properties of exponential and logarithmic functions, graphing both types of functions to analyze relationships, represent and model problems, and answer questions. Students apply these functions in many practical situations, such as applying exponential functions to determine compound interest and applying logarithmic functions to determine the pH of a liquid. In addition, students extend their study of functions to include polynomials of higher degree and those containing radical expressions. They formalize and deepen their understanding of real-valued functions, their defining characteristics and uses, and the mathematical language used to describe them. They compare and contrast the types of functions they have studied and their basic transformations. Students learn the practical and mathematical limitations that must be considered when working with functions or when using functions to model situations.


M3.3.A Know and use basic properties of exponential and logarithmic functions and the inverse relationship between them.

Examples:

Given • f(x) = 4x, write an equation for the inverse of this function. Graph the functions on the same coordinate grid.

Find — f(-3).

Evaluate the inverse function at 7.—

Derive the formulas:•

log— ba ⋅ logab = 1

log— aN = logbN ⋅ logab

Find the exact value of • x in:

log— x16 = 43

log— 381 = x

Solve for • y in terms of x:

log— a y

x = x

100 = x — ⋅ 10y

M3.3.B Graph an exponential function of the form f(x) = abx and its inverse logarithmic function.

Students expand on the work they did in Mathematics 2 with functions of the form y = abx. Although the concept of inverses is not fully developed until Precalculus, there is an emphasis in Mathematics 3 on students recognizing the inverse relationship between exponential and logarithmic functions and how this is reflected in the shapes of the graphs.

Example:

Find the equation for the inverse function of • y = 3x. Graph both functions. What characteristics of each of the graphs indicate they are inverse functions?


Math 3


M3.3.C Solve exponential and logarithmic equations. Examples:

A recommended adult dosage of the cold • medication NoMoreFlu is 16 ml. NoMoreFlu causes drowsiness when there are more than 4 ml in one’s system, making it unsafe to drive, operate machinery, etc. The manufacturer wants to print a warning label telling people how long they should wait after taking NoMoreFlu for the drowsiness to pass. If the typical metabolic rate is such that one quarter of the NoMoreFlu is lost every four hours, and a person takes the full dosage, how long should adults wait after taking NoMoreFlu to ensure that there will be

Less than 4 ml of NoMoreFlu in their — system?

Less than 1 ml in their system? —

Less than 0.1 ml in their system?—

Solve for • x in 256 22 1= −x .

Solve for • x in log5(x – 4) = 3.

M3.3.D Plot points, sketch, and describe the graphs of functions of the form = ( ) f x a x − c + d, and solve related equations.

Students solve algebraic equations that involve the square root of a linear expression over the real numbers. Students should be able to identify extraneous solutions and explain how they arose.

Students should view the function g(x) = x as the inverse function of f(x) = x2, recognizing that the functions have different domains for x greater than or equal to 0.

Example:

Analyze the following equations and tell what • you know about the solutions. Then solve the equations.

— 2 5 7x + =

— 5 6 2x − = −

— 2 15x x+ =

— 2 5 7x x− = +

M3.3.E Plot points, sketch, and describe the graphs

of functions of the form f xax

b( ) = +2, and

( )f x

abx c

( )=+

, and solve related equations.

Examples:

Sketch the graphs of the four functions • f xax

b( ) = +2 when a = 4 and 8 and b = 0 and 1.


( )f x

4bx c

( )=+

when b = 1 and 4 and c = 2 and 3.


Math 3


M3.3.F Plot points, sketch, and describe the graphs of cubic polynomial functions of the form f(x) = ax3 + d as an example of higher order polynomials and solve related equations.

Example:Solve for • x in 60 = -2x3 + 6.

M3.3.G Solve systems of three equations with three variables.

Students solve systems of equations using algebraic and numeric methods.

Examples:

Jill, Ann, and Stan are to inherit $20,000. Stan is • to get twice as much as Jill, and Ann is to get twice as much as Stan. How much does each get?

Solve the following system of equations.• 2x – y – z = 73x + 5y + z = -104x – 3y + 2z = 4


Math 3

Mathematics 3M3.4. Core Content: Quantifying variability (Data/Statistics/Probability)

Students extend their use of statistics as they graph bivariate data and analyze the graph to make predictions. They calculate and interpret measures of variability, confidence intervals, and margins of

error for population proportions. Dual goals underlie the content in the section: Students prepare for the further study of statistics and also become thoughtful consumers of data.


M3.4.A Calculate and interpret measures of variability and standard deviation and use these measures and the characteristics of the normal distribution to describe and compare data sets.

Students should be able to identify unimodality, symmetry, standard deviation, spread, and the shape of a data curve to determine whether the curve could reasonably be approximated by a normal distribution.

Given formulas, students should be able to calculate the standard deviation for a small data set, but calculators ought to be used if there are very many points in the data set. It is important that students be able to describe the characteristics of the normal distribution and identify common examples of data that are and are not reasonably modeled by it. Common examples of distributions that are approximately normal include physical performance measurements (e.g., weightlifting, timed runs), heights, and weights.

Apply the Empirical Rule (68–95–99.7 Rule) to approximate the percentage of the population meeting certain criteria in a normal distribution.

Example:

Which is more likely to be affected by an outlier • in a set of data, the interquartile range or the standard deviation?

M3.4.B Calculate and interpret margin of error and confidence intervals for population proportions.

Students will use technology based on the complexity of the situation.

Students use confidence intervals to critique various methods of statistical experimental design, data collection, and data presentation used to investigate important problems, including those reported in public studies.

Example:

In 2007, 400 of the 500 10• th graders in Local High School passed the WASL. In 2008, 375 of the 480 10th graders passed the test. The Local Gazette headline read “10th Grade WASL Scores Decline in 2008!”


Math 3


M3.4.B cont. In response, the Superintendent of Local School District wrote a letter to the editor claiming that, in fact, WASL performance was not significantly lower in 2008 than it was in 2007. Who is correct, the Local Gazette or the Superintendent?

Use mathematics to find the margin of error to justify your conclusion. (Formula for the margin of error

(E): E zc= p(1−p)

n; z95 = 1.96, where n is the sample

size, p is the proportion of the sample with the trait of interest, c is the confidence level, and zc is the multiplier for the specified confidence interval.)


Math 3

Mathematics 3M3.5. Core Content: Three-dimensional geometry (Geometry/Measurement)

Students formulate conjectures about three-dimensional figures. They use deductive reasoning to establish the truth of conjectures or to reject them on the basis of counterexamples. They extend

and formalize their work with perimeter, area, surface area, and volume of two- and three-dimensional figures, focusing on mathematical derivations of these formulas and their applications in complex problems. They use properties of geometry and measurement to solve both purely mathematical and applied problems. They also extend their knowledge of distance and angle measurements in a plane to measurements on a sphere.


M3.5.A Describe the intersections of lines in the plane and in space, of lines and planes, and of planes in space.

Example:

Describe all the ways that three planes can • intersect in space.

M3.5.B Describe prisms, pyramids, parallelepipeds, tetrahedra, and regular polyhedra in terms of their faces, edges, vertices, and properties.

Examples:

Given the number of faces of a regular polyhedron, • derive a formula for the number of vertices.

Describe symmetries of three-dimensional • polyhedra and their two-dimensional faces.

Describe the lateral faces that are required for a • pyramid to be a right pyramid with a regular base. Describe the lateral faces required for an oblique pyramid that has a regular base.

M3.5.C Analyze cross-sections of cubes, prisms, pyramids, and spheres and identify the resulting shapes.

Examples:

Start with a regular tetrahedron with edges of unit • length 1. Find the plane that divides it into two congruent pieces and whose intersection with the tetrahedron is a square. Find the area of the square. (Requires no pencil or paper.)

Start with a cube with edges of unit length 1. Find • the plane that divides it into two congruent pieces and whose intersection with the cube is a regular hexagon. Find the area of the hexagon.

Start with a cube with edges of unit length 1. • Find the plane that divides it into two congruent pieces and whose intersection with the cube is a rectangle that is not a face and contains four of the vertices. Find the area of the rectangle.

Which has the larger area, the above rectangle or • the above hexagon?


Math 3


M3.5.D Apply formulas for surface area and volume of three-dimensional figures to solve problems.

Problems include those that are purely mathematical as well as those that arise in applied contexts.

Three-dimensional figures include right and oblique prisms, pyramids, cylinders, cones, spheres, and composite three-dimensional figures.

Examples:

As Pam scooped ice cream into a cone, she began • to formulate a geometry problem in her mind. If the ice cream was perfectly spherical with diameter 2.25'' and sat on a geometric cone that also had diameter 2.25'' and was 4.5'' tall, would the cone hold all the ice cream as it melted (without her eating any of it)? She figured the melted ice cream would have the same volume as the unmelted ice cream.Find the solution to Pam’s problem and justify your reasoning.

A rectangle is 5 inches by 10 inches. Find the • volume of a cylinder that is generated by rotating the rectangle about the 10-inch side.

M3.5.E Predict and verify the effect that changing one, two, or three linear dimensions has on perimeter, area, volume, or surface area of two- and three-dimensional figures.

The emphasis in high school should be on verifying the relationships between length, area, and volume and on making predictions using algebraic methods.

Examples:

What happens to the volume of a rectangular • prism if four parallel edges are doubled in length?

The ratio of a pair of corresponding sides in • two similar triangles is 5:3. The area of the smaller triangle is 108 in2. What is the area of the larger triangle?

M3.5.F Analyze distance and angle measures on a sphere and apply these measurements to the geometry of the earth.

Examples:

Use a piece of string to measure the distance • between two points on a ball or globe; verify that the string lies on an arc of a great circle.

On a globe, show with examples why airlines use • polar routes instead of flying due east from Seattle to Paris.

Show that the sum of the angles of a triangle on a • sphere is greater than 180 degrees.


Math 3

Mathematics 3M3.6. Core Content: Algebraic properties (Numbers, Algebra)

Students continue to use variables and expressions to solve both purely mathematical and applied problems, and they broaden their understanding of the real number system to include complex

numbers. Students extend their use of algebraic techniques to include manipulations of expressions with rational exponents, operations on polynomials and rational expressions, and solving equations involving rational and radical expressions.


M3.6.A Explain how whole, integer, rational, real, and complex numbers are related, and identify the number system(s) within which a given algebraic equation can be solved.

Example:

Within which number system(s) can each of the • following be solved? Explain how you know.

3— x + 2 = 5x— 2 = 1

x— 2 = 1

4

x— 2 = 2x— 2 = -2

— x

7 = π

M3.6.B Use the laws of exponents to simplify and evaluate numeric and algebraic expressions that contain rational exponents.

Examples:

Convert the following from a radical to exponential • form or vice versa.

— 241

3

— 165

— x2 1+

— xx

2

Evaluate • x-2/3 for x = 27.

M3.6.C Add, subtract, multiply, and divide polynomials. Write algebraic expressions in equivalent forms using algebraic properties to perform the four arithmetic operations with polynomials.

Students should recognize that expressions are essentially sums, products, differences, or quotients. For example, the sum 2x2 + 4x can be written as a product, 2x(x + 2).


Math 3


M3.6.C cont. Examples:

(3• x2 – 4x + 5) + (-x2 + x – 4) + (2x2 + 2x + 1)

(2• x2 – 4) – (x2 + 3x – 3)

• 29

62

2

4

xx

•

• x – – 2 2 31x

x + M3.6.D Add, subtract, multiply, divide, and simplify

rational and more general algebraic expressions. In the same way that integers were extended to fractions, polynomials are extended to rational expressions. Students must be able to perform the four basic arithmetic operations on more general expressions that involve exponentials.

The binomial theorem is useful when raising expressions to powers, such as (x + 3)5.

Examples:

• xx

xx

++

−−−

11

3 312 2( )

Divide • ( ) x

x +

+2 1

3 / 2

by x

x

+−

2

12


Math 3

Mathematics 3M3.7. Additional Key Content (Geometry/Measurement)

Students formulate conjectures about circles. They use deductive reasoning to establish the truth of conjectures or to reject them on the basis of counterexamples. Students explain their reasoning using

precise mathematical language and symbols. They apply their knowledge of geometric figures and their properties to solve a variety of both purely mathematical and applied problems.


M3.7.A Know, prove, and apply basic theorems relating circles to tangents, chords, radii, secants, and inscribed angles.

Examples:

Given a line tangent to a circle, know and explain • that the line is perpendicular to the radius drawn to the point of tangency.

Prove that two chords equally distant from the • center of a circle are congruent.

Prove that if one side of a triangle inscribed in a circle • is a diameter, then the triangle is a right triangle.

Prove that if a radius of a circle is perpendicular to a • chord of a circle, then the radius bisects the chord.

M3.7.B Determine the equation of a circle that is described geometrically in the coordinate plane and, given equations for a circle and a line, determine the coordinates of their intersection(s).

Examples:

Write an equation for a circle with a radius of 2 • units and center at (1, 3).

Given the circle • x2 + y2 = 4 and the line y = x, find the points of intersection.

Write an equation for a circle given a line segment • as a diameter.

Write an equation for a circle determined by a • given center and tangent line.

M3.7.C Explain and perform constructions related to the circle.

Students perform constructions using straightedge and compass, paper folding, and dynamic geometry software. What is important is that students understand the mathematics and are able to justify each step in a construction.

Example:

In each case, explain why the constructions work:• a. Construct the center of a circle from two chords.b. Construct a circumscribed circle for a triangle.c. Inscribe a circle in a triangle.


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M3.7.D Derive and apply formulas for arc length and area of a sector of a circle.

Example:

Find the area and perimeter of the Reuleaux • triangle below.The Reuleaux triangle is constructed with three arcs. The center of each arc is located at the vertex of an equilateral triangle. Each arc extends between the two opposite vertices of the equilateral triangle.The figure below is a Reuleaux triangle that circumscribes equilateral triangle ABC. ∆ABC has side length of 5 inches. AB has center C, BC has center A, and CA has center B, and all three arcs have the same radius equal to the length of the sides of the triangle.

A

BC


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Students formalize the development of reasoning in Mathematics 3 as they use algebra, geometry, and statistics to make and defend generalizations. They justify their reasoning with accepted standards






M3.8.D Generalize a solution strategy for a single problem to a class of related problems and apply a strategy for a class of related problems to solve specific problems.



M3.8.G Synthesize information to draw conclusions and evaluate the arguments and conclusions of others.

M3.8.H Use inductive reasoning and the properties of numbers to make conjectures, and use deductive reasoning to prove or disprove conjectures.

Examples:

Show that • aa+b ≠ + b , for all positive real values of a and b.

Show that the product of two odd numbers is • always odd.

Leo is painting a picture on a canvas that • measures 32 inches by 20 inches. He has divided the canvas into four different rectangles, as shown in the diagram.

He would like the upper right corner to be a rectangle that has a length 1.6 times its width. Leo wants the area of the larger rectangle in the lower left to be at least half the total area of the canvas. Describe all the possibilities for the dimensions of the upper right rectangle to the nearest hundredth, and explain why the possibilities are valid.


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M3.8 cont. If Leo uses the largest possible dimensions for the smaller rectangle:

What will the dimensions of the larger — rectangle be?

Will the larger rectangle be similar to the — rectangle in the upper right corner? Why or why not?

Is the original canvas similar to the — rectangle in the upper right corner?

(A rectangle whose length and width are in the ratio 1

2+ 5 (approximately equal to 1.6) is called a “golden

rectangle” and is often used in art and architecture.)

A relationship between variables can be • represented with a table, a graph, an equation, or a description in words.

How can you decide from a table whether — a relationship is linear, quadratic, or exponential?How can you decide from a graph whether — a relationship is linear, quadratic, or exponential?How can you decide from an equation — whether a relationship is linear, quadratic, or exponential?


AcknowledgmentsThese K–12 mathematics standards have been developed by a team of Washington educators, mathematics faculty, and citizens with support from staff of the Office of the Superintendent of Public Instruction and invited national consultants, and facilitated by staff of the Charles A. Dana Center at The University of Texas at Austin. In addition we would like to acknowledge Strategic Teaching, who was contracted by the State Board of Education to conduct a final review and analysis of the draft K–12 Standards, as per 2008 Senate Bill 6534. The individuals who have played key roles in this project are listed below.

Washington Educators and Community Leaders

Dana Anderson, Stanwood-Camano School DistrictTim Bartlett, Granite Falls School DistrictMillie Brezinski, Nine Mile Falls School DistrictJane Broom, MicrosoftJewel Brumley, Yakima School DistrictBob Dean, Evergreen SchoolsJohn Burke, Gonzaga UniversityShannon Edwards, Chief Leschi SchoolAndrea English, Arlington School DistrictJohn Firkins, Gonzaga University (retired)Gary Gillespie, Spokane Public SchoolsRuss Gordon, Whitman CollegeKatherine Hansen, Bethel School DistrictTricia Hukee, Sumner School DistrictMichael Janski, Cascade School DistrictRuss Killingsworth, Seattle Pacific UniversityJames King, University of WashingtonArt Mabbott, Seattle SchoolsKristen Maxwell, Educational Service District 105Rosalyn O’Donnell, Ellensburg School DistrictM. Cary Painter, Chehalis School DistrictPatrick Paris, Tacoma School DistrictTom Robinson, Lake Chelan School DistrictTerry Rose, Everett School DistrictAllen Senear, Seattle SchoolsLorna Spear, Spokane SchoolsDavid Thielk, Central Kitsap School DistrictJohnnie Tucker, retired teacherKimberly Vincent, Washington State UniversityVirginia Warfield, University of WashingtonSharon Young, Seattle Pacific University

Dana Center Facilitators

P. Uri TreismanCathy SeeleySusan Hudson Hull

National Consultants

Mary Altieri, Consultant (retired teacher)Angela Andrews, National-Louis UniversityDiane Briars, Pittsburgh Schools (retired)Cathy Brown, Oregon Department of Education (retired)Dinah Chancellor, ConsultantPhilip Daro, ConsultantBill Hopkins, Dana CenterBarbara King, Dana Center Kurt Krieth, University of California at DavisBonnie McNemar, ConsultantDavid D. Molina, ConsultantSusan Eddins, Illinois Math and Science Academy (retired)Wade Ellis, West Valley College, CA (retired)Margaret Myers, The University of Texas at AustinLynn Raith, Pittsburgh Schools (retired)Jane Schielack, Texas A&M UniversityCarmen Whitman, Consultant

OSPI Project Support

George W. Bright, Special Assistant to the SuperintendentGreta Bornemann, Teaching and LearningBarbara Chamberlain, Interim DirectorLarry Davison, Math Helping Corps AdministratorLexie Domaradzki, Assistant Superintendent, Teaching . and LearningRon Donovan, Teaching & LearningDorian “Boo” Drury, Teaching & LearningLynda Eich, AssessmentKaren Hall, AssessmentRobert Hodgman, AssessmentMary Holmberg, AssessmentAnton Jackson, AssessmentYoonsun Lee, AssessmentKarrin Lewis, Teaching & LearningJessica Vavrus, Teaching and LearningJoe Willhoft, Assistant Superintendent, Assessment

Strategic Teaching Reviewers

Linda PlattnerAndrew ClarkW. Stephen Wilson

Math 3

Office of Superintendent of Public Instruction Old Capitol Building P.O. Box 47200

Olympia, WA 98504‐7200

This document is available for purchase from the Department of Printing at: www.prt.wa.gov or by calling (360) 586‐6360.

This document may also be downloaded, duplicated, and distributed as

needed from our Web site at: www.k12.wa.us/CurriculumInstruct

For more information about the contents of the document, please contact: Greta Bornemann,

OSPI e‐mail: [email protected]

Phone: 360‐725‐0437

OSPI Document Number: 09‐0013

This material is available in alternative format upon request. Contact the Resource Center at 1‐888‐59 LEARN (1‐888‐595‐3276),

TTY (360) 664‐3631

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Olympia, WA 98504-7200 2009

k 12 mathematics standards

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