i34 pre ib geometry mathematics curriculum … school curriculum/i34 pre ib...i34 pre ib geometry...

I34

Pre IB Geometry

Mathematics

Curriculum Essentials

Document

Boulder Valley School District

Department of Curriculum and Instruction

January 2012

6/21/2012 BVSD Curriculum Essentials 2

Boulder Valley School District Mathematics – An Introduction to The Curriculum Essentials

Document

Background

The 2009 Common Core State Standards (CCSS) have brought about a much needed move towards consistency in mathematics throughout the state and nation. In December 2010, the Colorado Academic Standards revisions for Mathematics were adopted by the State Board of Education. These standards aligned the previous state standards to the Common Core State Standards to form the Colorado Academic Standards (CAS). The CAS include additions or changes to the CCSS needed to meet state legislative requirements around Personal Financial Literacy. The Colorado Academic Standards Grade Level Expectations (GLE) for math are being adopted in their entirety and without change in the PK-8 curriculum. This decision was made based on the thorough adherence by the state to the CCSS. These new standards are specific, robust and comprehensive. Additionally, the essential linkage between the standards and the proposed 2014 state assessment system, which may include interim, formative and summative assessments, is based specifically on these standards. The overwhelming opinion amongst the mathematics teachers, school and district level administration and district level mathematics coaches clearly indicated a desire to move to the CAS without creating a BVSD version through additions or changes. The High School standards provided to us by the state did not delineate how courses should be created. Based on information regarding the upcoming assessment system, the expertise of our teachers and the writers of the CCSS, the decision was made to follow the recommendations in the Common Core State Standards for Mathematics- Appendix A: Designing High School Math Courses Based on the Common Core State Standards. The writing teams took the High School CAS and carefully and thoughtfully divided them into courses for the creation of the 2012 BVSD Curriculum Essentials Documents (CED).

The Critical Foundations of the 2011 Standards The expectations in these documents are based on mastery of the topics at specific grade levels with the understanding that the standards, themes and big ideas reoccur throughout PK-12 at varying degrees of difficulty, requiring different levels of mastery. The Standards are: 1) Number Sense, Properties, and Operations; 2) Patterns, Functions, and Algebraic Structures; 3) Data Analysis, Statistics, and Probability; 4) Shape, Dimension, and Geometric Relationships. The information in the standards progresses from large to fine grain, detailing specific skills and outcomes students must master: Standards to Prepared Graduate Competencies to Grade Level/Course Expectation to Concepts and Skills Students Master to Evidence Outcomes. The specific indicators of these different levels of mastery are defined in the Evidence Outcomes. It is important not to think of these standards in terms of ―introduction, mastery, reinforcement.‖ All of the evidence outcomes in a certain grade level must be

mastered in order for the next higher level of mastery to occur. Again, to maintain consistency and coherence throughout the district, across all levels, adherence to this idea of mastery is vital. In creating the documents for the 2012 Boulder Valley Curriculum Essentials Documents in mathematics, the writing teams focused on clarity, focus and understanding essential changes from the BVSD 2009 standards to the new 2011 CAS. To maintain the integrity of these documents, it is important that teachers throughout the district follow the standards precisely so that each child in every classroom can be guaranteed a viable education, regardless of the school they attend or if they move from another school, another district or another state. Consistency, clarity and coherence are essential to excellence in mathematics instruction district wide.

Components of the Curriculum Essentials Document

The CED for each grade level and course include the following: -A-Glance page containing:

o approximately ten key skills or topics that students will master during the year

o the general big ideas of the grade/course

o the Standards of Mathematical Practices

o assessment tools allow teachers to continuously monitor student progress for planning and pacing needs

o description of mathematics at that level

with additional topics or more in-depth coverage of topics included in bold text.

teachers should be familiar and comfortable using during instruction. It is not a comprehensive list of vocabulary for student use.

-12 Prepared Graduate Competencies

-12 At-A-Glance Guide from the CAS with notes from the CCSS

-12

Explanation of Coding In these documents you will find various abbreviations and coding used by the Colorado Department of Education. MP – Mathematical Practices Standard PFL – Personal Financial Literacy CCSS – Common Core State Standards Example: (CCSS: 1.NBT.1) – taken directly from the Common Core State Standards with an reference to the specific CCSS domain, standard and cluster of evidence outcomes. NBT – Number Operations in Base Ten OA – Operations and Algebraic Thinking MD – Measurement and Data G – Geometry


Standards for Mathematical Practice from

The Common Core State Standards for Mathematics

The Standards for Mathematical Practice have been included in the Nature of Mathematics section in

each Grade Level Expectation of the Colorado Academic Standards. The following definitions and

explanation of the Standards for Mathematical Practice from the Common Core State Standards can be

found on pages 6, 7, and 8 in the Common Core State Standards for Mathematics. Each Mathematical

Practices statement has been notated with (MP) at the end of the statement.

Mathematics | Standards for Mathematical Practice

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at

all levels should seek to develop in their students. These practices rest on important ―processes and

proficiencies‖ with longstanding importance in mathematics education. The first of these are the NCTM

process standards of problem solving, reasoning and proof, communication, representation, and

connections. The second are the strands of mathematical proficiency specified in the National Research

Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding

(comprehension of mathematical concepts, operations and relations), procedural fluency (skill in

carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition

(habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in

diligence and one’s own efficacy).

1. Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and

looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They

make conjectures about the form and meaning of the solution and plan a solution pathway rather than

simply jumping into a solution attempt. They consider analogous problems, and try special cases and

simpler forms of the original problem in order to gain insight into its solution. They monitor and

evaluate their progress and change course if necessary. Older students might, depending on the

context of the problem, transform algebraic expressions or change the viewing window on their

graphing calculator to get the information they need. Mathematically proficient students can explain

correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of

important features and relationships, graph data, and search for regularity or trends. Younger students

might rely on using concrete objects or pictures to help conceptualize and solve a problem.

Mathematically proficient students check their answers to problems using a different method, and they

continually ask themselves, ―Does this make sense?‖ They can understand the approaches of others to

solving complex problems and identify correspondences between different approaches.

2. Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem

situations. They bring two complementary abilities to bear on problems involving quantitative

relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically

and manipulate the representing symbols as if they have a life of their own, without necessarily

attending to their referents—and the ability to contextualize, to pause as needed during the

manipulation process in order to probe into the referents for the symbols involved. Quantitative

reasoning entails habits of creating a coherent representation of the problem at hand; considering the

units involved; attending to the meaning of quantities, not just how to compute them; and knowing

and flexibly using different properties of operations and objects.

3. Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously

established results in constructing arguments. They make conjectures and build a logical progression

of statements to explore the truth of their conjectures. They are able to analyze situations by breaking

them into cases, and can recognize and use counterexamples. They justify their conclusions,

communicate them to others, and respond to the arguments of others. They reason inductively about

data, making plausible arguments that take into account the context from which the data arose.

Mathematically proficient students are also able to compare the effectiveness of two plausible

arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in


an argument—explain what it is. Elementary students can construct arguments using concrete

referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be

correct, even though they are not generalized or made formal until later grades. Later, students learn

to determine domains to which an argument applies. Students at all grades can listen or read the

arguments of others, decide whether they make sense, and ask useful questions to clarify or improve

the arguments.

4. Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in

everyday life, society, and the workplace. In early grades, this might be as simple as writing an

addition equation to describe a situation. In middle grades, a student might apply proportional

reasoning to plan a school event or analyze a problem in the community. By high school, a student

might use geometry to solve a design problem or use a function to describe how one quantity of

interest depends on another. Mathematically proficient students who can apply what they know are

comfortable making assumptions and approximations to simplify a complicated situation, realizing that

these may need revision later. They are able to identify important quantities in a practical situation

and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and

formulas. They can analyze those relationships mathematically to draw conclusions. They routinely

interpret their mathematical results in the context of the situation and reflect on whether the results

make sense, possibly improving the model if it has not served its purpose.

5. Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem.

These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a

spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software.

Proficient students are sufficiently familiar with tools appropriate for their grade or course to make

sound decisions about when each of these tools might be helpful, recognizing both the insight to be

gained and their limitations. For example, mathematically proficient high school students analyze

graphs of functions and solutions generated using a graphing calculator. They detect possible errors by

strategically using estimation and other mathematical knowledge. When making mathematical models,

they know that technology can enable them to visualize the results of varying assumptions,

explore consequences, and compare predictions with data. Mathematically proficient students at

various grade levels are able to identify relevant external mathematical resources, such as digital

content located on a website, and use them to pose or solve problems. They are able to use

technological tools to explore and deepen their understanding of concepts.

6. Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear

definitions in discussion with others and in their own reasoning. They state the meaning of the symbols

they choose, including using the equal sign consistently and appropriately. They are careful about

specifying units of measure, and labeling axes to clarify the correspondence with quantities in a

problem. They calculate accurately and efficiently, express numerical answers with a degree of

precision appropriate for the problem context. In the elementary grades, students give carefully

formulated explanations to each other. By the time they reach high school they have learned to

examine claims and make explicit use of definitions.

7. Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for

example, might notice that three and seven more is the same amount as seven and three more, or

they may sort a collection of shapes according to how many sides the shapes have. Later, students will

see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive

property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7.

They recognize the significance of an existing line in a geometric figure and can use the strategy of

drawing an auxiliary line for solving problems. They also can step back for an overview and shift

perspective. They can see complicated things, such as some algebraic expressions, as single objects or

as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive

number times a square and use that to realize that its value cannot be more than 5 for any real

numbers x and y.


8. Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general

methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they

are repeating the same calculations over and over again, and conclude they have a repeating decimal.

By paying attention to the calculation of slope as they repeatedly check whether points are on the line

through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3.

Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1),

and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series.

As they work to solve a problem, mathematically proficient students maintain oversight of the process,

while attending to the details. They continually evaluate the reasonableness of their intermediate

results.

Connecting the Standards for Mathematical Practice to the Standards for Mathematical

Content

The Standards for Mathematical Practice describe ways in which developing student practitioners of the

discipline of mathematics increasingly ought to engage with the subject matter as they grow in

mathematical maturity and expertise throughout the elementary, middle and high school years.

Designers of curricula, assessments, and professional development should all attend to the need to

connect the mathematical practices to mathematical content in mathematics instruction. The

Standards for Mathematical Content are a balanced combination of procedure and understanding.

Expectations that begin with the word ―understand‖ are often especially good opportunities to connect

the practices to the content. Students who lack understanding of a topic may rely on procedures too

heavily. Without a flexible base from which to work, they may be less likely to consider analogous

problems, represent problems coherently, justify conclusions, apply the mathematics to practical

situations, use technology mindfully to work with the mathematics, explain the mathematics accurately

to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In

short, a lack of understanding effectively prevents a student from engaging in the mathematical

practices. In this respect, those content standards which set an expectation of understanding are

potential ―points of intersection‖ between the Standards for Mathematical Content and the Standards

for Mathematical Practice. These points of intersection are intended to be weighted toward central and

generative concepts in the school mathematics curriculum that most merit the time, resources,

innovative energies, and focus necessary to qualitatively improve the curriculum, instruction,

assessment, professional development, and student achievement in mathematics.


21st Century Skills and Readiness Competencies in Mathematics

Mathematics in Colorado’s description of 21st century skills is a synthesis of the essential abilities

students must apply in our rapidly changing world. Today’s mathematics students need a repertoire of

knowledge and skills that are more diverse, complex, and integrated than any previous generation.

Mathematics is inherently demonstrated in each of Colorado 21st century skills, as follows:

Critical Thinking and Reasoning

Mathematics is a discipline grounded in critical thinking and reasoning. Doing mathematics involves

recognizing problematic aspects of situations, devising and carrying out strategies, evaluating the

reasonableness of solutions, and justifying methods, strategies, and solutions. Mathematics provides

the grammar and structure that make it possible to describe patterns that exist in nature and society.

Information Literacy

The discipline of mathematics equips students with tools and habits of mind to organize and interpret

quantitative data. Informationally literate mathematics students effectively use learning tools,

including technology, and clearly communicate using mathematical language.

Collaboration

Mathematics is a social discipline involving the exchange of ideas. In the course of doing mathematics,

students offer ideas, strategies, solutions, justifications, and proofs for others to evaluate. In turn, the

mathematics student interprets and evaluates the ideas, strategies, solutions, justifications and proofs

of others.

Self-Direction

Doing mathematics requires a productive disposition and self-direction. It involves monitoring and

assessing one’s mathematical thinking and persistence in searching for patterns, relationships, and

sensible solutions.

Invention

Mathematics is a dynamic discipline, ever expanding as new ideas are contributed. Invention is the key

element as students make and test conjectures, create mathematical models of real-world

phenomena, generalize results, and make connections among ideas, strategies and solutions.


Colorado Academic Standards Mathematics

The Colorado academic standards in mathematics are the topical organization of the concepts and

skills every Colorado student should know and be able to do throughout their preschool through

twelfth-grade experience.

1. Number Sense, Properties, and Operations

Number sense provides students with a firm foundation in mathematics. Students build a deep

understanding of quantity, ways of representing numbers, relationships among numbers, and

number systems. Students learn that numbers are governed by properties and understanding

these properties leads to fluency with operations.

2. Patterns, Functions, and Algebraic Structures

Pattern sense gives students a lens with which to understand trends and commonalities.

Students recognize and represent mathematical relationships and analyze change. Students

learn that the structures of algebra allow complex ideas to be expressed succinctly.

3. Data Analysis, Statistics, and Probability

Data and probability sense provides students with tools to understand information and

uncertainty. Students ask questions and gather and use data to answer them. Students use a

variety of data analysis and statistics strategies to analyze, develop and evaluate inferences

based on data. Probability provides the foundation for collecting, describing, and interpreting

data.

4. Shape, Dimension, and Geometric Relationships

Geometric sense allows students to comprehend space and shape. Students analyze the

characteristics and relationships of shapes and structures, engage in logical reasoning, and use

tools and techniques to determine measurement. Students learn that geometry and

measurement are useful in representing and solving problems in the real world as well as in

mathematics.

Modeling Across the Standards

Modeling links classroom mathematics and statistics to everyday life, work, and decision-making.

Modeling is the process of choosing and using appropriate mathematics and statistics to analyze

empirical situations, to understand them better, and to improve decisions. When making mathematical

models, technology is valuable for varying assumptions, exploring consequences, and comparing

predictions with data. Modeling is best interpreted not as a collection of isolated topics but rather in

relation to other standards, specific modeling standards appear throughout the high school standards

indicated by a star symbol (*).


I34 Pre-IB Geometry Course Overview

Course Description

PIB Geometry will cover the concepts of Geometry with more emphasis on problem solving strategies. PIB Geometry

presents a thorough study of the structure of the postulate system and development of formal two-column proof. It considers the topics of congruence, parallelism, perpendicularity, properties of polygons, similarity, and the relationships of circles, spheres, lines, and planes with respect to space as well as the plane. Basic principles of probability will be introduced. The use of algebraic skills is

expected. As an advanced course, this course goes beyond the curriculum expectations of a standard course and addresses specific prerequisite skills needed for the study of further International Baccalaureate mathematics by increasing the depth and complexity. Students are engaged in dynamic, high‐level learning.

Topics at a Glance

Congruence

Similarity

Angles and Triangles

Coordinate Geometry

Areas and Volumes

Introduction to Trigonometry

Parallel and Perpendicular Lines

Geometric Inequalities

Circles & Spheres

Lines, Planes, Separation and Space

Logic & The study of reasoning

Probability

Assessments

Teacher created assessments

Common Assessments and Finals

Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the

reasoning of others. 4. Model with mathematics.

5. Use appropriate tools strategically. 6. Attend to precision. 7. 7. Look for and make use of structure. 8. Look for and express regularity in repeated

reasoning.

Notes on this document:

Plain text indicates standards which are

part of the Colorado Academic Standards

as defined by the Common Core State

Standards for the general (M41) Geometry

course.

Evidence Outcomes in italics are items

that have been added or changed from the

Colorado Academic Standards as needed

to fit the specific needs of BVSD students.

Evidence outcomes in bold indicate

items that were added to (M41)

Geometry standards for additional

depth and understanding in Advanced

Geometry.

Items in bold blue italic font indicate

specific topics added to meet the

prerequisites for the International

Baccalaureate program in

mathematics.

Grade Level Expectations

Standard Big Ideas for Pre IB Geometry

1. Number Sense, properties, and operations

1. Number sets and their properties form a basis for algebraic number sense.

2. Patterns, Functions, & Algebraic Structures

1. Properties of the real number system can be applied algebraically.

2. The coordinate plane allows us to apply algebraic understandings to Geometric concepts.

3. Data Analysis, Statistics, & Probability

1. Probability models outcomes for situations in which there is inherent randomness.

4. Shape, Dimension, & Geometric Relationships

1. The use of Geometric definitions and deductive logic form the foundation for expanding understanding to applied Geometry.

2. Objects in the plane can be transformed, and those transformations can be described and analyzed mathematically.

3. Concepts of similarity are foundational to geometry and its applications.

4. Logic and the study of reason provide the processes for students to formulate a strategic plan for problem-solving.

5. Attributes of two- and three-dimensional objects are measurable and can be quantified.

6. Objects in the real world can be modeled using geometric concepts.


Course Information: Students enrolling in PIB Geometry should have successfully completed an advanced Algebra I course.

Students wishing to take this course should be highly motivated and hard working. PIB Geometry is

designed to prepare students for success in both IB and AP classes later on in high school.

The Grade Level Expectations and Evidence Outcomes listed in this course for Standard 3: Data

Analysis, Statistics and Probability are a part of the Geometry standards for the Colorado Academic

Standards and should be addressed as a part of this course because students will not receive exposure to these concepts elsewhere.

Pre-International Baccalaureate Geometry Curriculum Map: Semester 1 Semester 2

http://www.bvsd.org/curriculum/math/_layouts/WordViewer.aspx?id=/curriculum/math/High%20School/Pre-IB%20Geometry%20Curriculum%20Map%20Sem%201.docx&Source=http%3A%2F%2Fwww%2Ebvsd%2Eorg%2Fcurriculum%2Fmath%2FPages%2FHigh-School-Curriculum%2Easpx&DefaultItemO

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1. Number Sense, Properties, and Operations

Number sense provides students with a firm foundation in mathematics. Students build a deep understanding of quantity, ways of

representing numbers, relationships among numbers, and number systems. Students learn that numbers are governed by properties, and

understanding these properties leads to fluency with operations.

Prepared Graduates

The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado

education system must master to ensure their success in a postsecondary and workforce setting.

Prepared Graduate Competencies in the Number Sense, Properties, and Operations

Standard are:

Understand the structure and properties of our number system. At their most basic level

numbers are abstract symbols that represent real-world quantities

Understand quantity through estimation, precision, order of magnitude, and comparison.

The reasonableness of answers relies on the ability to judge appropriateness, compare,

estimate, and analyze error

Are fluent with basic numerical and symbolic facts and algorithms, and are able to select

and use appropriate (mental math, paper and pencil, and technology) methods based on

an understanding of their efficiency, precision, and transparency

Make both relative (multiplicative) and absolute (arithmetic) comparisons between

quantities. Multiplicative thinking underlies proportional reasoning

Understand that equivalence is a foundation of mathematics represented in numbers,

shapes, measures, expressions, and equations

Apply transformation to numbers, shapes, functional representations, and data


Content Area: Mathematics - Pre-IB Geometry

Standard: 1. Number Sense, Properties, and Operations

Prepared Graduates:

Understand the structure and properties of our number system. At their most basic level numbers are abstract symbols that represent

real-world quantities.

GRADE LEVEL EXPECTATION:

Concepts and skills students master:

1. Number sets and their properties form a basis for algebraic number sense.

Evidence Outcomes 21st Century Skills and Readiness Competencies

Students can:

a. Learn the meaning and correct notations for universal

set, null set, subset, member or element of a set.

b. Be able to explain the union, intersection, and

complement of sets using proper notation.

c. Use Venn Diagrams to solve set problems, specifically

classification in 2 and 3 set problems and finding the

number of members in a set.

Inquiry Questions:

1. When you extend to a new number systems (e.g., from

integers to rational numbers and from rational numbers

to real numbers), what properties apply to the

extended number system?

Relevance and Application:

1. The understanding of the closed system of real, rational

and irrational numbers is the foundation for the truth of

properties and operations throughout mathematical

development.

2. Venn Diagrams are a useful means of modeling set

theory and problem solving complex situations with

overlapping characteristics.

Nature of the Discipline:

1. Mathematicians build a deep understanding of quantity, ways

of representing numbers, and relationships among numbers

and number systems.

2. Mathematics involves making and testing conjectures,

generalizing results, and making connections among ideas,

strategies, and solutions.

3. Mathematicians look for and make use of structure. (MP)

4. Mathematicians look for and express regularity in repeated

reasoning. (MP)


2. Patterns, Functions, and Algebraic Structures

Pattern sense gives students a lens with which to understand trends and commonalities. Being a student of mathematics

involves recognizing and representing mathematical relationships and analyzing change. Students learn that the structures

of algebra allow complex ideas to be expressed succinctly.

Prepared Graduates

The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who

complete the Colorado education system must have to ensure success in a postsecondary and workforce setting.

Prepared Graduate Competencies in the 2. Patterns, Functions, and Algebraic Structures Standard are:

Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate

(mental math, paper and pencil, and technology) methods based on an understanding of their efficiency,

precision, and transparency

Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures,

expressions, and equations

Make sound predictions and generalizations based on patterns and relationships that arise from numbers,

shapes, symbols, and data

Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by

relying on the properties that are the structure of mathematics

Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present

and defend solutions



Standard: 2: Patterns, Functions, and Algebraic Structures

Prepared Graduates:

Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and

pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency



1. Properties of the real number system can be applied algebraically.


Students Can:

a. Review Algebra I Concepts to be Used in Geometry

i. Real number system

ii. Number line

iii. Commutative, associative, distributive properties

iv. Properties of equality

v. Properties of inequality

vi. Trichotomy property

vii. Absolute value

Inquiry Questions:

1. When you extend to a new number systems (e.g., from

integers to rational numbers and from rational numbers to real

numbers), what properties apply to the extended number

system?

2. How is the number system extended and applied to geometric

figures and properties?

3. Why don’t we have a different number system for shapes and

angles?


1. The understanding of the closed system of real, rational and

irrational numbers is the foundation for the truth of properties

and operations throughout mathematical development.


Standards for Mathematical Practice.









Standard: 2: Patterns, Functions, and Algebraic Structures

Prepared Graduates:

Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data

GRADE LEVEL EXPECTATION


2. The coordinate plane allows us to apply algebraic understandings to Geometric concepts.


Students Can:

a. Explore algebra and geometry relationships concerning

coordinate geometry

b. Understand the history behind coordinate geometry and

its usefulness today

c. Cartesian coordinate system (parts and notation)

i. Label points on a graph

ii. Find projection of a point to the x and y axis

d. Understand a three dimensional coordinate system

i. Ordered triple, octants, graphing, projections, to

XY, YZ, XZ planes

e. Be able to discuss fallacies in graphing and improper

graphing techniques

f. Be able to graph inequalities and absolute value

relationships on a coordinate plane to parallel and no

parallel lines

g. Express and understand of slope; verbally, graphically

and computationally

h. Know the difference between positive, negative, zero,

and undefined slope

i. Prove the theorems for the slope of two parallel and

two perpendicular lines and apply in problems with

quadrilaterals and triangles.

j. Prove the distance and midpoint formula and apply

them in two and three dimensional problems.

k. Prove previously proven theorems in geometry by

coordinate geometry using slope, distance and midpoint

formulas

l. Describe a line by an equation

i. Be familiar with slope intercept form, point slope

form, and standard form for a linear equation

ii. Write the equation of a line given:

1. Slope, Y-intercept

Inquiry Questions:

1. How can the 2 and 3-dimensional coordinate planes be used to

systematize applications of Geometric properties?


1. Visualization and the use of coordinate Geometry is used in

professions such as architecture, robotics, animation, film and

computer graphics, navigation, manufacturing, engineering,

urban planning, interior design, construction management and

military design.




2. Construct viable arguments and critique the reasoning of

others.





* Indicates a part of the standard connected to the mathematical

practice of Modeling.


2. Slope, A point

3. 2 points 4. ║or ┴ another line through a given point

m. Know how to graph using slope and intercept the

intercept method or a table of values

n. Demonstrate graphing 3 variable equations

o. Equation of circles in the coordinate plane

p. Be able to write the equation of a circle at the origin

given the radius and center (and vice versa)

q. Be able to write the equation of a circle not at the origin

given the center and radius (vice versa)

r. Be able to change an equation of a circle in vertex form

to standard form (vice versa)

s. Identify whether the graph of the equation

X2+Y2+AX+BY+C=0 is a circle a point or the empty set.


3. Data Analysis, Statistics, and Probability

Data and probability sense provides students with tools to understand information and uncertainty. Students ask

questions and gather and use data to answer them. Students use a variety of data analysis and statistics

strategies to analyze, develop and evaluate inferences based on data. Probability provides the foundation for

collecting, describing, and interpreting data.

Prepared Graduates

The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students

who complete the Colorado education system must master to ensure their success in a postsecondary and

workforce setting.

Prepared Graduate Competencies in the 3. Data Analysis, Statistics, and Probability Standard are:

Recognize and make sense of the many ways that variability, chance, and randomness appear in a

variety of contexts

Solve problems and make decisions that depend on understanding, explaining, and quantifying the

variability in data

Communicate effective logical arguments using mathematical justification and proof. Mathematical

argumentation involves making and testing conjectures, drawing valid conclusions, and justifying

thinking

Use critical thinking to recognize problematic aspects of situations, create mathematical models, and

present and defend solutions



Standard: 3. Data Analysis, Statistics, and Probability

Prepared Graduates:

Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts.



1. Probability models outcomes for situations in which there is inherent randomness.


Students can:

a. Understand independence and conditional probability and use them

to interpret data. (CCSS: S-CP)

i. Describe events as subsets of a sample space5 using

characteristics (or categories) of the outcomes, or as

unions, intersections, or complements of other events.6

(CCSS: S-CP.1)

ii. Explain that two events A and B are independent if the

probability of A and B occurring together is the product of

their probabilities, and use this characterization to

determine if they are independent. (CCSS: S-CP.2)

iii. Using the conditional probability of A given B as P(A and

B)/P(B), interpret the independence of A and B as saying

that the conditional probability of A given B is the same as

the probability of A, and the conditional probability of B

given A is the same as the probability of B. (CCSS: S-CP.3)

iv. Construct and interpret two-way frequency tables of data

when two categories are associated with each object being

classified. Use the two-way table as a sample space to

decide if events are independent and to approximate

conditional probabilities.7 (CCSS: S-CP.4)

v. Recognize and explain the concepts of conditional

probability and independence in everyday language and

everyday situations.8 (CCSS: S-CP.5)

b. Use the rules of probability to compute probabilities of compound

events in a uniform probability model. (CCSS: S-CP)

i. Find the conditional probability of A given B as the fraction

of B’s outcomes that also belong to A, and interpret the

answer in terms of the model. (CCSS: S-CP.6)

ii. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and

B), and interpret the answer in terms of the model. (CCSS: S-CP.7)

Inquiry Questions:

1. Can probability be used to model all types of uncertain

situations? For example, can the probability that the 50th

president of the United States will be female be

determined?

2. How and why are simulations used to determine

probability when the theoretical probability is unknown?

3. How does probability relate to obtaining insurance? (PFL)


1. Comprehension of probability allows informed decision-

making, such as whether the cost of insurance is less

than the expected cost of illness, when the deductible on

car insurance is optimal, whether gambling pays in the

long run, or whether an extended warranty justifies the

cost. (PFL)

2. Probability is used in a wide variety of disciplines including

physics, biology, engineering, finance, and law. For

example, employment discrimination cases often present

probability calculations to support a claim.


1. Some work in mathematics is much like a game.

Mathematicians choose an interesting set of rules and

then play according to those rules to see what can

happen.

2. Mathematicians explore randomness and chance through

probability.

3. Mathematicians construct viable arguments and critique

the reasoning of others. (MP) 4. Mathematicians model with mathematics. (MP)

5 The set of outcomes. (CCSS: S-CP.1) 6 "Or," "and," "Not". (CCSS: S-CP.1) 7 For example, collect data from a random sample of students in


your school on their favorite subject among math, science, and

English. Estimate the probability that a randomly selected

student from your school will favor science given that the

student is in tenth grade. Do the same for other subjects and

compare the results. (CCSS: S-CP.4) 8 For example, compare the chance of having lung cancer if you

are a smoker with the chance of being a smoker if you have

lung cancer. (CCSS: S-CP.5)


4. Shape, Dimension, and Geometric Relationships

Geometric sense allows students to comprehend space and shape. Students analyze the characteristics and

relationships of shapes and structures, engage in logical reasoning, and use tools and techniques to determine

measurement. Students learn that geometry and measurement are useful in representing and solving problems

in the real world as well as in mathematics.

Prepared Graduates

The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all

students who complete the Colorado education system must master to ensure their success in a postsecondary

and workforce setting.

Prepared Graduate Competencies in the 4. Shape, Dimension, and Geometric

Relationships standard are:

Understand quantity through estimation, precision, order of magnitude, and comparison.

The reasonableness of answers relies on the ability to judge appropriateness, compare,

estimate, and analyze error

Make sound predictions and generalizations based on patterns and relationships that arise

from numbers, shapes, symbols, and data

Apply transformation to numbers, shapes, functional representations, and data

Make claims about relationships among numbers, shapes, symbols, and data and defend

those claims by relying on the properties that are the structure of mathematics

Use critical thinking to recognize problematic aspects of situations, create mathematical

models, and present and defend solutions



Standard: 4. Shape, Dimension, and Geometric Relationships

Prepared Graduates:

Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are

the structure of mathematics



1. The use of Geometric definitions and deductive logic to form the foundation for expanding understanding to applied

Geometry.


Students can:

Introduction to Geometry

a. State precise definitions of angle, circle, perpendicular

line, parallel line, and line segment, based on the

undefined notions of point, line, distance along a line,

and distance around a circular arc. (CCSS: G-CO.1)

i. A Deductive System of Reasoning

ii. Learn the 3 undefined terms of geometry- point,

line, plane

iii. Definitions- be able to write accurate definitions

and recognize faulty definitions

iv. Distinguish between postulates and theorems

v. Beginning Postulates in Geometry

vi. Distance postulate

vii. Ruler postulate

viii. Line postulate

b. Betweeness Property (segment addition)

c. Formal Definitions

i. Segment, length, endpoint

ii. Ray, opposite ray, midpoint, bisector

d. Point Plotting Theorem

e. Midpoint Theorem

i. Lines, Planes, Separation and Space

f. Describe how Points, Lines and Planes are Related (2

Dimensional and 3 Dimensional)

i. Know the definition of space

ii. Be able to draw 3 dimensional figures (one point

and two point perspective)

iii. Line postulate

iv. Plane-space postulate

v. Discuss the intersection of 2 lines and line with a

plane

Inquiry Questions:

1. Does the postulate system of Geometry lead to more or less

uniformity of thought?

2. Why do we need to know the formal definitions, properties,

postulates and theorems to be able to apply logic to

Geometry?

3. How can mathematical concepts be ―undefined‖? What does

this mean for our understanding of other concepts that

depend on the undefined?

4. When is deductive reasoning a more appropriate tool than

inductive reasoning?


1. The understanding of foundational definitions and properties

allows for the development of strategic problem-solving skills.



vi. Plane postulate

vii. Intersection of 2 planes postulate

viii. Determination of a plane

1. 3 noncolinear points

2. line and point not on the line

3. two intersecting lines

g. Describe How Planes and Space are Separated by Lines

and Planes

i. Definition convex set

ii. Plane separation postulate

iii. Space separation postulate

iv. Angles and Triangles

h. Define Basic Terms using Correct Notation

i. Angle

ii. Triangle

iii. Interior

iv. Exterior

v. Perimeter

i. Use Postulates for Measuring Angles

i. Angle measurement postulate

ii. Angle addition postulate

iii. Angle construction postulate

iv. Supplement postulate

j. Discuss the Relationship Between Angles and How They

are Related

i. Right angles

ii. Obtuse angles

iii. Acute angles

iv. Perpendicular angles

v. Vertical angles

vi. Adjacent angles

vii. Linear pair

viii. Supplementary angles

ix. Complimentary angles

k. Know the Equivalence Relation for Angles

i. Symmetric property

ii. Reflexive property

iii. Transitive property

l. Use Theorems Concerning Angles in problems

i. Supplement theorem

ii. Complement theorem

iii. Vertical angles theorem

7 Copying a segment; copying an angle; bisecting a segment;

bisecting an angle; constructing perpendicular lines, including

the perpendicular bisector of a line segment; and constructing a

line parallel to a given line through a point not on the line.

(CCSS: G-CO.12)

8 Compass and straightedge, string, reflective devices, paper

folding, dynamic geometric software, etc. (CCSS: G-CO.12)


m. Proof- Using Angles

i. Hypothesis- conclusion

ii. Acceptable forms

n. Acceptable reasons used in proofs

o. Make geometric constructions. (CCSS: G-CO)

p. Make formal geometric constructions7 with a variety of tools

and methods.8 (CCSS: G-CO.12)

q. Construct an equilateral triangle, a square, and a regular

hexagon inscribed in a circle. (CCSS: G-CO.13)




Prepared Graduates:

Apply transformation to numbers, shapes, functional representations, and data.



2. Objects in the plane can be transformed, and those transformations can be described and analyzed mathematically.


Students can:

Transformations

a. Experiment with transformations in the plane. (CCSS: G-CO)

b. State precise definitions of angle, circle, perpendicular line,

parallel line, and line segment, based on the undefined notions

of point, line, distance along a line, and distance around a

circular arc. (CCSS: G-CO.1)

c. Represent transformations in the plane using1 appropriate

tools. (CCSS: G-CO.2)

d. Describe transformations as functions that take points in the

plane as inputs and give other points as outputs. (CCSS: G-

CO.2)

e. Compare transformations that preserve distance and angle to

those that do not.2 (CCSS: G-CO.2)

f. Given a rectangle, parallelogram, trapezoid, or regular

polygon, describe the rotations and reflections that carry it

onto itself. (CCSS: G-CO.3)

g. Develop definitions of rotations, reflections, and translations in

terms of angles, circles, perpendicular lines, parallel lines, and

line segments. (CCSS: G-CO.4)

h. Given a geometric figure and a rotation, reflection, or

translation, draw the transformed figure using appropriate

tools.3 (CCSS: G-CO.5)

i. Specify a sequence of transformations that will carry a given

figure onto another. (CCSS: G-CO.5)

Congruence

j. Understand congruence in terms of rigid motions. (CCSS: G-

CO)

k. Use geometric descriptions of rigid motions to transform

figures and to predict the effect of a given rigid motion on a

given figure. (CCSS: G-CO.6)

l. Given two figures, use the definition of congruence in terms of

rigid motions to decide if they are congruent. (CCSS: G-CO.6)

Inquiry Questions:

1. What happens to the coordinates of the vertices of shapes

when different transformations are applied in the plane?

2. How would the idea of congruency be used outside of

mathematics?

3. What does it mean for two things to be the same? Are there

different degrees of ―sameness?‖

4. What makes a good definition of a shape?

5. What conditions create unique polygons?

6. What does it mean for two lines to be parallel?

7. How can slope and distance be used to create and

investigate the relationship of the triangles created by

joining the mid-segments of a triangle?


1. Comprehension of transformations aids with innovation and

creation in the areas of computer graphics and animation.

2. Knowledge of right triangle trigonometry allows modeling and

application of angle and distance relationships such as

surveying land boundaries, shadow problems, angles in a

truss, and the design of structures.

3. Points of concurrency are used in fields such as

architecture, engineering and physics.


1. Geometry involves the investigation of invariants. Geometers

examine how some things stay the same while other parts

change to analyze situations and solve problems.

2. Mathematicians construct viable arguments and critique the

reasoning of others. (MP)

3. Mathematicians attend to precision. (MP)

4. Mathematicians look for and make use of structure. (MP)




m. Use Congruence Postulates for Triangles in Proofs

i.Understand one to one correspondence

ii.Know the definition of congruent triangles, segments

and angles

iii.Definition- included side, included angle

iv.Equivalence relation- triangles

v.Work with the SAS, ASA, and SSS postulates in

triangle proofs

vi. Use the Angle Bisector Theorem in Proofs

vii. Use the definition of congruence in terms of rigid

motions to show that two triangles are congruent if and

only if corresponding pairs of sides and corresponding

pairs of angles are congruent. (CCSS: G-CO.7)

viii. Explain how the criteria for triangle congruence (ASA,

SAS, and SSS) follow from the definition of congruence

in terms of rigid motions. (CCSS: G-CO.8)

n. Name the parts of Isosceles and Equilateral Triangles

o. Use Quadrilaterals in Proofs

ix.Rectangle, squares

p. Use Medians in Proofs

x. Know definition- median

q. Prove geometric theorems. (CCSS: G-CO)

r. Prove theorems about lines and angles.4 (CCSS: G-CO.9)

s. Prove theorems about triangles.5 (CCSS: G-CO.10)

i. Use the Isosceles Triangle Theorem and its

Converse in Proofs

ii. Use the Properties of Equilateral and Equiangular

Triangle in Proofs

1. Classify triangles by sides and angles

2. Understand the nature of corollaries

3. Work with overlapping triangles in proofs

t. Prove theorems about parallelograms (rectangles,

rhombuses, squares).6 (CCSS: G-CO.11)

u. Use circle properties (involving chords and angles) to

prove theorems about triangles.

v. Points of concurrency

w. Segments of irrational length (e.g. using equilateral

triangles and squares)

x. Use specific properties of quadrilaterals and triangles,

to construct each. Examples: Construct a parallelogram

given lengths of diagonals. Given lengths of two sides


6. Mathematicians make sense of problems and persevere in

solving them. (MP)



*Indicates a part of the standard connected to the mathematical

practice of Modeling

1 e.g., Transparencies and geometry software. (CCSS: G-CO.2)

2 e.g., Translation versus horizontal stretch. (CCSS: G-CO.2)

3 e.g., Graph paper, tracing paper, or geometry software. (CCSS: G-

CO.5)

4 Theorems include: vertical angles are congruent; when a

transversal crosses parallel lines, alternate interior angles are

congruent and corresponding angles are congruent; points on a

perpendicular bisector of a line segment are exactly those

equidistant from the segment’s endpoints. (CCSS: G-CO.9)

5 Theorems include: measures of interior angles of a triangle sum to

180°; base angles of isosceles triangles are congruent; the

segment joining midpoints of two sides of a triangle is parallel to

the third side and half the length; the medians of a triangle meet

at a point. (CCSS:G-CO.10)

6 Theorems include: opposite sides are congruent, opposite angles are

congruent, the diagonals of a parallelogram bisect each other, and

conversely, rectangles are parallelograms with congruent

diagonals. (CCSS: G-CO.11)

11 For example, prove or disprove that a figure defined by four given

points in the coordinate plane is a rectangle; prove or disprove

that the point (1, √3) lies on the circle centered at the origin and

containing the point (0, 2). (CCSS: G-GPE.4)

12 e.g., Find the equation of a line parallel or perpendicular to a

given line that passes through a given point. (CCSS: G-GPE.5)


and a non-included angle, construct two triangles.

i. Determine uniqueness.

Coordinate Geometry

a. Express Geometric Properties with Equations. (CCSS: G-GPE)

b. Translate between the geometric description and the equation

for a conic section. (CCSS: G-GPE)

i. Derive the equation of a circle of given center and

radius using the Pythagorean Theorem. (CCSS: G-

GPE.1)

ii. Complete the square to find the center and radius of a

circle given by an equation. (CCSS: G-GPE.1)

iii. Derive the equation of a parabola given a focus and

directrix. (CCSS: G-GPE.2)

c. Use coordinates to prove simple geometric theorems

algebraically. (CCSS: G-GPE)

d. Use coordinates to prove simple geometric theorems11

algebraically. (CCSS: G-GPE.4)

e. Prove the slope criteria for parallel and perpendicular lines and

use them to solve geometric problems.12 (CCSS: G-GPE.5)

f. Find the point on a directed line segment between two given

points that partitions the segment in a given ratio. (CCSS: G-

GPE.6)

g. Use coordinates and the distance formula to compute

perimeters of polygons and areas of triangles and rectangles.★

(CCSS: G-GPE.7) Extend to solve problems with similar

figures (e.g. triangles created by connecting midpoints

of sides of a triangle)

h. Use distance and slope to further investigate and

informally prove properties of points of concurrency in

triangles (orthocenter, incenter, circumcenter, centroid)

Geometric Inequalities

a. Parts Theorem

i.Know the definition of less than for angles and

segments

b. Use the Exterior Angle Theorem in Inequality Proofs

i.Know definition of remote interior and exterior

angles

c. Be able to use AAS and HL Congruence Postulates in

Proof

d. Work With Inequalities in a Single Triangle

i.Larger angle opposite longest side and converse


e. Use the First Minimum Theorem in Proofs

i.Know the definition of distance between a line and

an external point

f. Understand the Triangle Inequality Theorem in

Problems and Planes

g. Work With the Hinge Theorem and its Converse in

Proofs

h. Know the definition of an altitude

i.Use of altitudes in proofs

Lines and Planes in Space

a. Perpendicular lines and planes in space

i. Learn definition – line and plane perpendicular in

space

ii. Understand the basic theorem on perpendiculars

and its corollary

iii. Using theorems in Unit 6, be able to discuss the

relationships between intersecting and parallel

lines to a plane in space

iv. Learn the second minimum theorem and the

definition of distance from a point to a plane

b. Parallel lines and planes in space

i. Know the definition of Parallel planes

ii. Understand the relationship between parallel

planes in space, intersecting other lines or planes

iii. Know the definition , parts and notation for a

dihedral angle

iv. Discuss the projection of a point or line onto a

plane

c. Parallel Lines in a Plane

i. Understand the facts about parallel lines that are

not dependent on the parallel postulate

ii. Know the definition of parallel and skew line

iii. Be able to name the four ways to determine a plane

(emphasize 2 lines perpendicular to a third line)

iv. Discuss the existence and uniqueness of line

through a point parallel to another line

v. Identify transversals and angles created by 2 lines

and a transversal

vi. If 3 parallel lines intercept congruent segments on

one transversal; they intersect congruent segments on

any other transversal.

vii. Know conditions which guarantee two parallel lines


and the proofs of those conditions.

1. AIP Theorem

2. CAP Theorem

3. Same side Interior angles

supplementary

4. 2 lines ┴ to 3rd line (in a plane only)

viii. Be able to solve problems finding angle measures

using the above theorems

ix. Understand the parallel postulate and the facts about

parallel lines that are dependent upon the parallel

postulate

x. Given the parallel postulate know the conditions and

the proofs of those conditions that allow you to find

angle relationships and measures

1. PAI Theorem

2. PCA Theorem 3. ║ Lines → same side interior angles

supplementary 4. Lines ┴ to one of two parallel lines is ┴ to

the other

xi. Prove triangle relationships that use the parallel

postulate and be able to use these relationships to solve

angle measurements

1. Sum of interior angles of a triangle equals 180◦

2. Two angles of a triangle congruent to corresponding parts of another →triangle 3rd

angles congruent

3. Acute angles of a right triangle are

complementary

4. Exterior angle theorem (equality)

xii. Using the knowledge of Parallel lines learn some

fact about quadrilaterals and triangles

xiii. Know the definition of a quadrilateral, diagonal,

opposite, and consecutive sides and angles.

1. Know the definition of a;

a. Parallelogram

b. Trapezoid

c. Rectangle

d. Rhombus

e. Square

f. Kite

xiv. Name the six properties of a parallelogram and be


able to prove the ones which are theorems (5!)

Proofs with Quadrilaterals

a. Given a quadrilateral, name the four ways to prove the

quadrilateral is parallelogram.

b. Know the midline theorem, its proof and be able to

solve segment length and angle measure problems

using the theorem.

c. Know all the properties of rectangles rhombus squares

trapezoids and kites using the family “tree” of

quadrilaterals and be able to justify them by proof.

d. Use the knowledge of quadrilaterals to prove facts and

right triangles and medians of triangles and trapezoids

e. Median to the hypotenuse is half as long as the

hypotenuse.

f. If the acute of a right triangle has a measure of 3o

(degrees), the opposite side is half as long as the

hypotenuse.

g. Medians of a triangle are concurrent. Point of

intersection is 2/3 the way from the vertex along the

median.

h. Median of a trapezoid is parallel to the bases and the

length is one half the sums of the bases and the length

is one half the sums of the bases

i. Line bisects one side of a triangle and is parallel to a

second side, and then it bisects the third side




Prepared Graduates:

Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions.



3. Concepts of similarity are foundational to geometry and its applications.


Students can:

a. Understand similarity in terms of similarity transformations.

(CCSS: G-SRT)

i. Verify experimentally the properties of dilations given by a

center and a scale factor. (CCSS: G-SRT.1)

1. Show that a dilation takes a line not passing through the

center of the dilation to a parallel line, and leaves a line

passing through the center unchanged. (CCSS: G-SRT.1a)

2. Show that the dilation of a line segment is longer or

shorter in the ratio given by the scale factor. (CCSS: G-

SRT.1b)

ii. Given two figures, use the definition of similarity in terms of

similarity transformations to decide if they are similar. (CCSS:

G-SRT.2)

iii. Explain using similarity transformations the meaning of

similarity for triangles as the equality of all corresponding

pairs of angles and the proportionality of all corresponding

pairs of sides. (CCSS: G-SRT.2)

a. Definition of ratio and proportion

b. Properties for ratio

c. Similar units (exception-rates)

d. Notation (fraction, semicolon)

b. Properties for proportions

1. Cross product

2. Inverting

3. Numerators over denominator

4. Adding, subtracting numerators to

denominators (vice versa)

iv. Use the properties of similarity transformations to establish

the AA criterion for two triangles to be similar. (CCSS: G-

SRT.3)

b. Prove theorems involving similarity. (CCSS: G-SRT)

c. Prove theorems about triangles.9 (CCSS: G-SRT.4)

d. Prove that all circles are similar. (CCSS: G-C.1)

Inquiry Questions:

1. What happens to the coordinates of the vertices of shapes

when different transformations are applied in the plane?

2. How would the idea of congruency be used outside of

mathematics?

3. What does it mean for two things to be the same? Are there

different degrees of ―sameness?‖


1. Comprehension of transformations aids with innovation and

creation in the areas of computer graphics and animation.

2. Special triangles are commonly used in engineering,

surveying and physics.







3. Mathematicians attend to precision. (MP) 4. Mathematicians look for and make use of structure. (MP)



9 Theorems include: a line parallel to one side of a triangle divides the

other two proportionally, and conversely; the Pythagorean Theorem

proved using triangle similarity. (CCSS: G-SRT.4)

10Include the relationship between central, inscribed, and

circumscribed angles; inscribed angles on a diameter are right

angles; the radius of

a circle is perpendicular to the tangent where the radius intersects

the circle. (CCSS: G-C.2)


e. Use congruence and similarity criteria for triangles to solve

problems and to prove relationships in geometric figures.

(CCSS: G-SRT.5)

f. Use similarity criteria to discover Pythagorean triplet

patterns.

g. Review the algebraic properties for the geometric mean

h. Definition of geometric mean, arithmetic mean

i. Similarity for triangles

a. Definition similarity for triangles

b. Find lengths of sides and angle measurements in

triangles using the definition of similar triangles

Proofs with Similarity

a. Proportionality theorem (in a triangle and parallel lines)

b. Prove and apply the “basic proportionality” theorem

and its converse triangle

c. Prove and apply the “angle bisector” proportional

theorem to a triangle

d. Prove and apply the “parallels proportional segment”

theorem for 3 or more parallel lines

e. Methods to prove triangles similar

f. AAA, similarity theorem and AA corollary

g. Prove; apply proportion properties to solve various

problems involved segment length, angle measure,

ECT…

h. Line parallel to one size of a triangle intersects the

other two sides creating a triangle similar to the

original triangle i. Similarity of triangles is and equivalence relation →

transit for triangle similarity.

j. SAS similarity and SSS similarity (prove and apply in

problems)

k. Similarity in Right Triangles

l. Altitude to the hypotenuse separates the triangle into

two triangles similar to each other and to the original

triangle

i. Altitude to the hypotenuse is the geometric mean

between the segments it separates on the

hypotenuse.

ii. leg of a right triangle is the geometric mean

between the hypotenuse and the segment of the

hypotenuse adjacent to the legs (projection!!)

m. Prove the Pythagorean Theorem using geometric mean


n. Know the relationship between are, perimeter and

lengths of side, altitudes, and medians for similar

triangles and be able to solve problems




Prepared Graduates:


the structure of mathematics.



4. Logic and the study of reason provide the processes for students to formulate a strategic plan for problem-solving.


Students can:

Logic- Study of Reasoning

a. Identify Mathematical Sentences

i.Open

ii.Closed

iii.Truth value

b. Understand the Connectives used in Logic Statements

i. And

ii. Or

iii. If-then

iv. If and only if

c. Truth Value and Truth Tables

i. Negations

ii. Conjunctions

iii. Disjunctions

iv. Conditional

v. BI- conditional

vi. Compound statements

d. Identify Tautologies

i. Conjunctive statements- disjunctive statements

e. Use the Law of Detachment (Modus Poens) in Logical

arguments

i.Hidden conditional

f. Use the Law of the Contrapositive in Logical arguments

i.Inverse, converse

ii.Logically equivalent statements

g. Use the Law of Modus Tollens in logical arguments

h. Recognize Invalid Arguments

i. Conditional and converse

ii. Conditional and inverse

i. Compare the Chain Rule used in Logic with the transitive

property in algebra

Inquiry Questions:

1. Why is the study of logic important for building a personal

life-long problem-solving schematic?


1. The study of reason forms the foundation for problem-solving

in law, science, computer programming and research.




2. Construct viable arguments and critique the reasoning of

others.




j. Negations and DE Morgan’s Laws

i. Double negation

ii. Negation of a conjunction

iii. Negation of a disjunction

k. Law of Conjunction

l. Law of Simplification

m. Law of Disjunctive Addition

n. Quantifiers Used in Logic Statements

i.Universal quantifier

ii.Existential quantifier

iii.Negation of universal and existential quantifier

o. Writing Logic Proofs

p. More Techniques in Proofs

q. Performance Standards and Objectives

r. Learn to Write Indirect Proofs

i. Review the deductive system of reasoning

ii. Understand the parts of an indirect proof

s. Statement to be proved

t. Assumption or supposition

u. Conclusion from assumption

v. Know contradictory fact to conclusion

i. Work with existence and uniqueness proofs

ii. Use the perpendicular bisector theorem and its

corollary in proofs

iii. Understand auxiliary sets and their contribution in

proofs




Prepared Graduates:


the structure of mathematics.



5. Attributes of two- and three-dimensional objects are measurable and can be quantified.


Students can:

a. Visualize relationships between two-dimensional and three-

dimensional objects. (CCSS: G-GMD)

b. Identify the shapes of two-dimensional cross-sections of

three-dimensional objects, and identify three-dimensional

objects generated by rotations of two-dimensional objects.

(CCSS: G-GMD.4)

Polygonal regions and their areas

a. Learn the postulates for areas polygonal regions

b. Learn the definition of triangular region, polygonal

region and identify polygonal regions

i. Area postulates

ii. Congruent postulate

iii. Area addition postulate

iv. Unit postulate (area of a square region)

c. Learn the formulas and be able to prove them for

rectangles, triangles, parallelograms, rhombuses, and

kites

d. Be able to solve area problems using the appropriate

the appropriate formulas

e. Be able to solve area ratio problems involving changes

in bases and height

f. Use ratios to solve problems within different dimensions

(e.g. given the ratio of the surface area of two similar

solids, find the ratio of their volumes and corresponding

lengths).

Circles and Spheres

a. Basic definitions

b. Explain volume formulas and use them to solve problems.

(CCSS: G-GMD)

a. Give an informal argument13 for the formulas for the

circumference of a circle, area of a circle, volume and

Inquiry Questions:

1. How might surface area and volume be used to explain

biological differences in animals?

2. How is the area of an irregular shape measured?

3. How can surface area be minimized while maximizing volume?


1. Understanding areas and volume enables design and building.

For example, a container that maximizes volume and

minimizes surface area will reduce costs and increase

efficiency. Understanding area helps to decorate a room, or

create a blueprint for a new building.


1. Mathematicians use geometry to model the physical world.

Studying properties and relationships of geometric objects

provides insights in to the physical world that would otherwise

be hidden.


solving them. (MP)


reasoning of others. (MP) 4. Mathematicians model with mathematics. (MP)



13Use dissection arguments, Cavalieri’s principle, and informal limit

arguments. (CCSS: G-GMD.1 10 Include the relationship between central, inscribed, and

circumscribed angles; inscribed angles on a diameter are right

angles; the radius of a circle is perpendicular to the tangent where

the radius intersects the circle. (CCSS: G-C.2)


surface area of a cylinder, pyramid, sphere and cone.

(CCSS: G-GMD.1)

b. Use volume formulas for cylinders, pyramids, cones,

and spheres to solve problems.★ (CCSS: G-GMD.3)

c. Extend prior knowledge of surface area to

cylinder, cones, and spheres.

c. Understand and apply theorems about circles. (CCSS: G-C)

d. Identify and describe relationships among inscribed angles,

radii, and chords.10 (CCSS: G-C.2)

e. Construct the inscribed and circumscribed circles of a triangle.

(CCSS: G-C.3)

f. Prove properties of angles for a quadrilateral inscribed in a

circle. (CCSS: G-C.3)

g. Find arc lengths and areas of sectors of circles. (CCSS: G-C)

h. Derive using similarity the fact that the length of the arc

intercepted by an angle is proportional to the radius, and

define the radian measure of the angle as the constant of

proportionality. (CCSS: G-C.5)

i. Derive the formula for the area of a sector. (CCSS: G-C.5)

j. Derive the formula for the surface area of a cone.

k.

l. Circle, sphere, diameter, radius, chord, secant, tangent,

interior and exterior of a circle, great circle, externally

and internally tangent circles, the following theorems in

circles problems. a. Line ┴ to the radius at its outer endpoint is

tangent to the circle b. Every tangent to the circle is ┴ to the radius at

the point of contact

c. The perpendicular from the center of a circle to a

chord bisects the chord

i. The segment from the center of a circle to

the midpoint of a chord (not a diameter) is ┴ to the chord.

d. In the plane of a circle, the ┴ bisector of a chord

pasts through the center of the circle

e. In the same circle or congruent circles, congruent

chords are equal is that from the center

(converse is true)

f. Apply the theorems above to spheres and tangent

planes

m. Measurement of arcs and angles


n. Learn definitions:

a. Minor arc

b. Major arc

c. Semicircle

d. Central angle

e. Inscribed angle

f. Tangent -secant angles

g. 2 Chord angles

h. 2 secant, secant -tangent, 2 tangent angles

o. Prove the theorems for the above angles and arcs and

apply in problems

p. Know the difference between an angle intercepting an

arc and being inscribed in an arc

q. Understand what it means to inscribe or circumscribe

polygons and circles

a. define secant segment, tangent segment

b. learn and prove the “power” theorems and apply

in problems to find the segment length

i. 2 Tangent theorem

ii. 2 Secant theorem

iii. Tangent- secant theorem

iv. 2 chord theorem

r. Learn common tangents (external and internal) and

apply in problems in circles.

Areas of circles and sectors (using polygons)

a. Learn the definition of a polygon and the angle sum

formulas

b. Know the names for polygons with 3 to 10 sides (most

commonly used!)

c. Learn the definition of convex polygon

d. Develop the formulas for the following;

i. Number of diagonals in a polygon with n

sides

ii. Sum of the interior angles of a polygon

with n sides

iii. Sum of exterior angles (one angle at each

vertex) of a polygon with n sides

e. Work with regular polygons finding areas, lengths of

sides, and measurement of angles

f. Definition of a regular polygon and the apothem

g. Learn the formula for the area of a regular polygon

h. Develop the formulas for circumference and area of a


circle

i. Prove these formulas using inscribed regular polygons

and the concept of a limit

j. Apply in problems;

i. Find the circumference and area of inscribed and

circumscribed circles given regular polygons

(especially 3, 4, 6 sides)

ii. Find the area of an annulus

iii. Understand the relationship between area,

circumference, and radii in ratio problems

k. Develop the formulas for length of an area of a sector

and area of a segment

l. Define length of an arc and compare this definition with

the measure of an arc

m. Define sector and segment

n. Apply these formulas in problems (especially the

“continuous belt” program

Solids and their volumes

a. Study the properties of solid figures, base areas,

volumes and surface area’s

b. Learn the definition of a prism

i. Know the terms right prism, base, altitude, and

cross section




Prepared Graduates:

Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions.



6. Objects in the real world can be modeled using geometric concepts.


Students can:

a. Apply geometric concepts in modeling situations. (CCSS: G-

MG)

b. Use geometric shapes, their measures, and their properties to

describe objects.14 ★ (CCSS: G-MG.1)

c. Apply concepts of density based on area and volume in

modeling situations.15 ★ (CCSS: G-MG.2)

d. Apply geometric methods to solve design problems.16 ★ (CCSS:

G-MG.3)

Right Triangle Trigonometry

a. Define trigonometric ratios and solve problems involving right

triangles. (CCSS: G-SRT)

b. Explain that by similarity, side ratios in right triangles are

properties of the angles in the triangle, leading to definitions

of trigonometric ratios for acute angles. (CCSS: G-SRT.6)

c. Explain and use the relationship between the sine and cosine

of complementary angles. (CCSS: G-SRT.7)

d. Use trigonometric ratios to discover the ratio properties

of special right triangles.

e. Use trigonometric ratios and the Pythagorean Theorem to

solve right triangles in applied problems.★ (CCSS: G-SRT.8)

f. Prove and apply trigonometric identities. (CCSS: F-TF)

g. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1. (CCSS:

F-TF.8)

h. Use the Pythagorean identity to find sin(θ), cos(θ), or tan(θ)

given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

(CCSS: F-TF.8)

i. Prove and use the Pythagorean Theorem for right

triangles

j. Be able to give several different proofs of the

Pythagorean Theorem

Inquiry Questions:

1. How are mathematical objects different from the physical

objects they model?

2. What makes a good geometric model of a physical object or

situation?

3. How are mathematical triangles different from built triangles

in the physical world? How are they the same?


1. Geometry is used to create simplified models of complex

physical systems. Analyzing the model helps to understand the

system and is used for such applications as creating a floor

plan for a house, or creating a schematic diagram for an

electrical system.


1. Mathematicians use geometry to model the physical world.

Studying properties and relationships of geometric objects

provides insights in to the physical world that would otherwise

be hidden.


solving them. (MP)



4. Mathematicians model with mathematics. (MP)

14 e.g., Modeling a tree trunk or a human torso as a cylinder. (CCSS:

G-MG.1)

15 e.g., Persons per square mile, BTUs per cubic foot. (CCSS: G-

MG.2)

16 e.g., Designing an object or structure to satisfy physical constraints

or minimize cost; working with typographic grid systems based on

ratios. (CCSS: G-MG.3)


k. Solve problems using the Pythagorean Theorem

involving the length of sides, altitude, and area with

quadrilaterals and triangles.

l. Prove the relationship between the sides of a 30-60-90

right triangle and a 45-45-90 right triangle using the

Pythagorean Theorem.

m. Solve area problems using special right triangles

(including Isosceles and equilateral triangle problems)

n. Develop the basic trigonometric ratios

a. Sine, cosine, tangent ratios

b. Develop the above ratios for special right

triangles

o. Develop the ability to use a trigonometry table and a

calculator to evaluate trig ratios

p. Find sides and angles missing in triangle problems

q. Be able to solve angle elevation and depression

problems

r. (optional) understand the process of interpolation in

estimating trig values

s. Develop the ratios to the winding function (the unit

circle)

t. Distinguish between degree measure and radian

measure on the unit circle

u. Understand directed angles, terminal and initial sides

v. Use the definition of the sine, cosine, and tangent of

directed angles to find the value of various trigametric

functions of numbers.




Pre- IB Geometry Academic Vocabulary for Students

Standard 1: complement of a set, element of a set, intersection, null set, union, universal set, Venn

diagram

Standard 2: associative property, commutative property, distance formula, distributive property,

midpoint formula, parallel lines, perpendicular lines, Real numbers, Trichotomy property, undefined slope

Standard 3: bivariate data, box plot, compound events, dot plot, frequency table, first quartile,

Independently combined probability models, independent events, inter-quartile range, line plot, mean,

mean absolute variation, median, probability distribution, probability, probability model, sample space,

scatter plot, third quartile, uniform probability model

Standard 4: associative property, inverses, commutative property, Centroid, Circumcenter, congruent,

dilation, identity property of 0, Incenter, Median, Midsegment, multiplicative inverses, Kite, Orthocenter,

Points of Concurrency, properties of equality, properties of inequality, properties of operations, rectilinear

figure, Rhombus, rigid motion, similarity transformations, transitivity principle for indirect measurement,

theorem, transformations, trigonometric ratios, Tessellation, Unique

Reference Glossary for Teachers * These are words available for your reference. Not all words below are listed above

because these are words that you will see above are for students to know and use while the

list below includes words that you, as the teacher, may see in the standards and materials.

Word Definition

Additive inverses Two numbers whose sum is 0 are additive inverses of one another. Example:

3/4 and – 3/4 are additive inverses of one another because

3/4 + (– 3/4) = (– 3/4) + 3/4 = 0.

Associative property

of addition

The SUM is the same no matter what way you group the addends.

In general, the associative property of addition can be written as:

(a + b) + c = a + (b + c).

Associative property

of multiplication

Notice that the PRODUCT is the same no matter what way you group

the factors. In general, the associative property of multiplication can

be written as: (a × b) × c = a × (b × c).

Bivariate data Pairs of linked numerical observations. Example: a list of heights

and weights for each player on a football team.

Box plot A method of visually displaying a distribution of data values by using

the median, quartiles, and extremes of the data set. A box shows the middle

50% of the data.1

Centroid That point where the medians of a triangle intersect.

Circumcenter That point where any two perpendicular bisectors of the sides of a polygon

inscribed in the circle intersect.

Commutative property The Commutative Property of Addition states that changing the order of

addends does not change the sum, i.e. if a and b are two real numbers,

then a + b = b + a.

Complement of a set In set theory, a complement of a set A refers to things not in (that is, things outside of), A.

Compound events A combination of multiple simple events, can be independent or dependent.

Congruent Two plane or solid figures are congruent if one can be obtained from

the other by rigid motion (a sequence of rotations, reflections, and

translations).

Dilation A transformation that moves each point along the ray through the

point emanating from a fixed center, and multiplies distances from the

center by a common scale factor.

http://dictionary.reference.com/browse/the

http://dictionary.reference.com/browse/polygon


Distance Formula The distance between (x1) and (x2) is the length of the line segment between them:

Distributive property Multiplication of real numbers distributes over addition of real numbers. Ex: 2 × (1 + 3) = (2 × 1) + (2 × 3).

Dot plot A data display method which records frequency using a ―•‖ notation as

shown below.

First quartile For a data set with median M, the first quartile is the median of the data

values less than M. Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15,

22, 120}, the first quartile is 6.2 See also: median, third quartile, inter-

quartile range.

Frequency table A table that lists items and uses tally marks to show the number of times

they occur.

Identity property of 0 Identity property of addition states that the sum of zero and any number or

variable is the number or variable itself. For example, 4 + 0 = 4, - 11 + 0 =

- 11, y + 0 = y are few examples illustrating the identity property of

addition.

Incenter That point where the bisectors of the angles of a triangle or of a regular

polygon intersect. The point where the three angle bisectors of a

triangle meet.

Independent events Two events, A and B, are independent if the fact that A occurs does not

affect the probability of B occurring.

Independently

combined probability

models

Two probability models are said to be combined independently if the

probability of each ordered pair in the combined model equals the product of

the original probabilities of the two individual outcomes in the ordered pair.

Inter-quartile Range A measure of variation in a set of numerical data, the inter-quartile range is

the distance between the first and third quartiles of the data set. Example:

For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the inter-quartile

range is 15 – 6 = 9. See also: first quartile, third quartile.

Intersection A set that contains elements shared by two or more given sets.

http://encyclopedia.thefreedictionary.com/Line+(mathematics)

http://encyclopedia.thefreedictionary.com/Multiplication

http://encyclopedia.thefreedictionary.com/Addition

http://dictionary.reference.com/browse/the

http://dictionary.reference.com/browse/polygon

http://www.mathopenref.com/bisectorangle.html


Kite A quadrilateral that has two distinct pairs of consecutive equilateral sides.

Line plot A method of visually displaying a distribution of data values where

each data value is shown as a dot or mark above a number line. Also known

as a dot plot

Mean A measure of center in a set of numerical data, computed by adding the

values in a list and then dividing by the number of values in the list.4

Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean is

21.

Mean absolute

deviation

A measure of variation in a set of numerical data, computed by adding the

distances between each data value and the mean, then dividing by the

number of data values. Example: For the data set {2, 3, 6, 7, 10,12, 14, 15,

22, 120}, the mean absolute deviation is 20.

Median (Statistical) A measure of center in a set of numerical data. The median of a list of

values is the value appearing at the center of a sorted version of the list—or

the mean of the two central values, if the list contains an even number of

values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 90}, the

median is 11.

Median of a Triangle The segment connecting the vertex of an angle in a triangle to the midpoint

of the side opposite it.

Midpoint formula Midpoint of a line segment is the point that is halfway between the endpoints of the

line segment. In two-dimensional coordinate plane, the midpoint of a line with coordinates of its endpoints as (x1, y1) and

(x2, y2) is given by

Midsegment A line segment joining the midpoints of two sides of a triangle.

Multiplicative inverses Two numbers whose product is 1 are multiplicative inverses of one another.

Example: 3/4 and 4/3 are multiplicative inverses of one

another because 3/4 × 4/3 = 4/3 × 3/4 = 1.

Null set a set that is empty; a set with no members

Orthocenter The point where the three altitudes of a triangle intersect.

http://library.thinkquest.org/2647/geometry/glossary.htm?tql-iframe#quad

http://library.thinkquest.org/2647/geometry/glossary.htm?tql-iframe#equilateral

http://www.mathopenref.com/altitude.html

http://www.mathopenref.com/triangle.html


Parallel lines Parallel Lines are distinct lines lying in the same plane and they never

intersect each other. Parallel lines have the same slope.

In the figure below, lines PQ and RS are parallel and the lines l and m are

parallel.

Perpendicular lines Perpendicular lines are lines that intersect at right angles. If two lines are

perpendicular to each other, then the product of their slopes is equal to – 1.

In the figure shown below, the lines AB and EF are perpendicular to each other.

Points of Concurrency The place where three or more lines, rays, or segments intersect at the

same point. See point H in diagram above.

Probability A number between 0 and 1 used to quantify likelihood for processes that

have uncertain outcomes (such as tossing a coin, selecting a person at

random from a group of people, tossing a ball at a target, or testing for a

medical condition).

Probability

distribution

The set of possible values of a random variable with a probability assigned

to each.

Probability model A probability model is used to assign probabilities to outcomes of a chance

process by examining the nature of the process. The set

of all outcomes is called the sample space, and their probabilities sum to 1.

See also: uniform probability model.

Real number The real numbers are sometimes thought of as points on an infinitely long line named number line or real line.

Rectilinear figure Rectilinear figures are figures bounded by straight lines.

Rigid motion A transformation of points in space consisting of a sequence of one or more

translations, reflections, and/or rotations. Rigid motions are here

assumed to preserve distances and angle measures.

Sample space In a probability model for a random process, a list of the individual outcomes

that are to be considered

Scatter plot A graph of plotted points that show the relationship between two sets

of data (bivariate).

Similarity

transformation

A rigid motion followed by a dilation.

http://encyclopedia.thefreedictionary.com/Number+line

http://encyclopedia.thefreedictionary.com/Real+line


Tessellation A covering of a plane with congruent copies of the same region with no holes

or overlaps.

Theorem An important mathematical statement which can be proven by postulates,

definitions, and previously proved theorems.

Third quartile For a data set with median M, the third quartile is the median of the data

values greater than M. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15,

22, 120}, the third quartile is 15. See also: median, first quartile,

interquartile range.

Transformations Informally, moving a shape so that it is in a different position, but still has

the same size, area, angles and line lengths. Formally, a correspondence

between two sets of points such that each point in the pre-image has a

unique image and that each point in the image has exactly one pre-image.

Transitivity principle

for indirect

measurement

If the length of object A is greater than the length of object B, and the

length of object B is greater than the length of object C, then the length of

object A is greater than the length of object C. This principle applies to

measurement of other quantities as well.

Trichotomy property The property that for natural numbers a and b , either a is less

than b , a equals b , or a is greater than b .

Trigonometric ratios A ratio that describes the relationship between a side and an angle of a

triangle. Sine, Cosine, Tangent

Undefined slope The "slope" of a vertical line. A vertical line has undefined slope because

all points on the line have the same x-coordinate. As a result

the formula used for slope has a denominator of 0, which makes the

slope undefined.

Uniform probability

model

A probability model which assigns equal probability to all outcomes. See

also: probability model

Union A set, every member of which is an element of one or another of two or more given sets.

Unique Limited to a single outcome or result; without alternative possibilities.

Universal Set A set containing all elements of a problem under consideration.

http://library.thinkquest.org/2647/geometry/glossary.htm?tql-iframe#congruent

http://www.mathwords.com/s/slope_of_a_line.htm

http://www.mathwords.com/v/vertical_line_equation.htm

http://www.mathwords.com/p/point.htm

http://www.mathwords.com/c/coordinates.htm

http://www.mathwords.com/c/coordinates.htm

http://www.mathwords.com/f/formula.htm

http://www.mathwords.com/d/denominator.htm


Venn Diagram Diagrams that show all hypothetically possible logical relations between a

finite collection of sets (aggregation of things).

Definitions adapted from:

Boulder Valley School District Curriculum Essentials Document, 2009.

―Math Dictionary‖ www.icoachmath.com/math_dictionary/mathdictionarymain.html. Copyright © 1999

- 2011 HighPoints Learning Inc. December 30, 2011.

―The Mathematics Glossary.‖ Common Core Standards for Mathematics.

http://www.corestandards.org/the-standards/mathematics/glossary/glossary/ Copyright

2011. June 23, 2011.

―Thesaurus.‖ http://www.thefreedictionary.com/statistical+distribution . Copyright © 2012 Farlex, Inc.

January 5, 2012.

(n.d.). Retrieved from sas.com.

Statistics Glossary (n.d.). Retrieved from University of Glasgow:

http://www.stats.gla.ac.uk/steps/glossary/index.html.

―Illustrated Mathematics Dictionary.‖ http://www.mathsisfun.com/definitions/Copyright © 2011

MathsIsFun.com. January 14, 2012.

―Math Open Reference.‖ http://www.mathopenref.com/triangleincenter.html 2009 Copyright Math

Open Reference. January 15, 2012.

http://www.icoachmath.com/math_dictionary/mathdictionarymain.html

http://www.corestandards.org/the-standards/mathematics/glossary/glossary/

http://www.thefreedictionary.com/statistical+distribution

http://www.mathsisfun.com/definitions/

http://www.mathopenref.com/triangleincenter.html