5 grade math unit guide core/pacing guides... · jackson county school district year at a glance...

5th Grade

Math Unit Guide

2014-2015

Jackson County School District Year At A Glance 5th Grade Math

Unit 1 Understanding volume 8 Days Unit 2 Developing multiplication and division strategies 12 Days Unit 3 Using equivalency to add and subtract fractions with unlike denominators 12 Days Unit 4 Expanding understanding of place value to decimals 10 Days Unit 5 Comparing and rounding decimals 10 Days Unit 6 Understanding the concept of multiplying fractions by fractions 12 Days Unit 7 Interpreting multiplying fractions as scaling 12 Days Unit 8 Developing the concept of dividing unit fractions 10 Days Unit 9 Solving problems involving volume 10 Days Unit 10 Performing operations with decimals 12 Days Unit 11 Classifying two-dimensional geometric figures 10 Days Unit 12 Solving problems with fractional quantities 12 Days Unit 13 Representing algebraic thinking 10 Days Unit 14 Exploring the coordinate plane 10 Days Unit 15 Finalizing multiplication and division with whole numbers 10 Days

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5th Grade Math

In previous grades, students learned various strategies for multiplication and division and demonstrated fluency in addition and subtraction. They developed understanding of structure of the place value system, and applied understanding of operations with whole numbers to begin developing computational strategies with fractions, paying special attention to unit fractions as the building blocks of all fractions. Students gained understanding that geometric figures can be analyzed and classified based on their properties.

The Grade 5 year, as sequenced here, begins with developing conceptual understanding of volume as an attribute of solid figures. This is a new concept for Grade 5 and provides an engaging context that supports problem solving throughout the year. Students practice and refine their multiplication and division strategies, attaining fluency in multiplication with whole numbers by the end of the year. The domain of Number and Operations in Base Ten is finalized this year as students generalize their understanding of the base-‐ten system to include decimals.

This document reflects the Dana Institute’s current thinking related to the intent of the Common Core State Standards for Mathematics (CCSSM) and assumes 160 days for instruction, divided among 15 units. The number of days suggested for each unit assumes 45-‐minute class periods and is included to convey how instructional time should be balanced across the year. The units are sequenced in a way that we believe best develops and connects the mathematical content described in the CCSSM; however, the order of the standards included in any unit does not imply a sequence of content within that unit. Some standards may be revisited several times during the course; others may be only partially addressed in different units, depending on the focus of the unit. Strikethroughs in the text of the standards are used in some cases in an attempt to convey that focus, and comments are included throughout the document to clarify and provide additional background for each unit.

Throughout Grade 5, students should continue to develop proficiency with the Common Core's eight Standards for Mathematical Practice:

1. Make sense of problems and persevere in solving them. S. Use appropriate tools strategically. 2. Reason abstractly and quantitatively. 6. Attend to precision. 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure. 4. Model with mathematics. 8. Look for and express regularity in repeated reasoning.

These practices should become the natural way in which students come to understand and do mathematics. While, depending on the content to be understood or on the problem to be solved, any practice might be brought to bear, some practices may prove more useful than others. Opportunities for highlighting certain practices are indicated in different units in this document, but this highlighting should not be interpreted to mean that other practices should be neglected in those units.

When using this document to help in planning your district's instructional program, you will also need to refer to the CCSSM document, relevant progressions documents for the CCSSM, and the appropriate assessment consortium framework.

Unit 1: Understanding volume Suggested number of days: 8

Learning Targets Notes/Comments Unit Materials and Resources

Unit Overview:

Students expand their understanding of geometric measurement and spatial structuring to include volume as an attribute of three-‐dimensional space. In this unit, students develop this understanding using concrete models to discover strategies for finding volume, whereas in unit 9 students generalize this understanding in real-‐world problems and apply strategies and formulas. Volume is addressed in two units (unit 1 and unit 9) because it is a major emphasis in Grade 5. The connection to multiplication and addition provides an opportunity for students to start the year off by applying the multiplication and addition strategies they learned in previous grades in a new, interesting context. Common Core State Standards for Mathematical Content

Measurement and Data – 5.MD C. Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. 3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

a. A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume.

b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

Common Core State Standards for Mathematical Practice

3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure.

5.MD.3a.1 Understand that unit cubes are used

to measure volume of solid figures.

5.MD.3b.1 Understand that unit cubes cannot have gaps or overlap.

5.MD.4.1 Use a visual model to measure volume by counting unit cubes.

5.MD.4.2 Measure volume by counting unit cubes.

Students decompose and recompose geometric figures to make sense of the spatial structure of volume (MP.7). In particular, students explain their thinking and analyze others' reasoning as they practice partitioning figures into layers and each layer into rows and each row into cubes (MP.3).

Videos www.khanacademy.org www.teachingchannel.org www.youtube.com Math Fact Fluency Practice www.mathwire.com www.oswego.org/ocsd-‐web/games/ http://mathfactspro.com/mathfluencygame.html#/math-‐facts-‐addition-‐games http://jerome.northbranfordschools.org/Content/Math_Fact_Fluency_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathematics.org/5 https://learnzillion.com/ www.AECSD5thGradeMathematicsdoc http://maccss.ncdpi.wikispaces.net/Fifth+Grade www.dpi.state.nc.us

http://harcourtschool.com/search/search.html www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐five http://www.onlinemathlearning.com/common-‐core-‐grade5.html http://www.mathgoodies.com/standards/alignments/grade5.html http://www.k-‐5mathteachingresources.com/5th-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.com/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/index.php/common-‐core-‐standards http://map.mathshell.org/materials/stds.php#standard1159

Vocabulary Essential Questions

• Measurement • Unit • Unit Cubes • Volume

• What tools and units are used to measure the attributes of an object? • How are the units of measure within a standard system related? • How do you decide which unit of measurement to use? • How can I measure length, mass and capacity by using non-‐standard units? • How do I choose the appropriate tool and unit when measuring?

• How do I estimate and measure?

Formative Assessment Strategies

• Anecdotal Note Cards -‐ The teacher can create a file folder with 5" x 7" note cards for each student for helpful tips and hints to guide students to remembering a process or procedure.

• Labels or Sticky Notes -‐Teachers can carry a clipboard with a sheet of labels or a pad of sticky notes and make observations as they circulate throughout the classroom. After the class, the labels or sticky notes can be placed in the observation notebook in the appropriate student's section and use the data collected to adjust instruction to meet student needs.

• Questioning -‐ Asking questions that give students opportunity for deeper thinking and provide teachers with insight into the degree and depth of student understanding. Questions should go beyond the typical factual questions requiring recall of facts or numbers.

• Discussion -‐ Teacher presents students with an open-‐ended question that build knowledge and develop critical and creative thinking skills. The teacher can assess student understanding by listening to responses and taking anecdotal notes.

Unit 2: Developing multiplication and division strategies Suggested number of days: 12


Unit Overview: In this unit students build on their work from previous grade levels to refine their strategies for multiplication and division in order to reach fluency in multiplication by the end of the year. Students continue to develop more sophisticated strategies for division to become flexible and efficient with the standard algorithm in Grade 6. Students begin to find quotients with two-‐digit divisors early in the year to build strategies for accurate computations. Common Core State Standards for Mathematical Content

Number and Operations in Base Ten - 5.NBT B. Perform operations with multi-‐digit whole numbers and with decimals to hundredths. 5. Fluently multiply multi-‐digit whole numbers using the standard algorithm.

6. Find whole-‐number quotients of whole numbers with up to four-‐digit dividends and two-‐digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.


1. Make sense of problem and persevere in solving them. 8. Look for and express regularity in repeated reasoning.

5.NBT.5.1 Fluently multiply multi-‐digit whole

numbers.

5.NBT.6.1 5.NBT.6.2

Determine the quotient of whole numbers with up to 3 digit dividends and 1 digit divisors, which are multiples of ten. Determine the quotient of whole numbers with up to 4 digit dividends and 1 digit divisors, which are multiples of ten.

5.NBT.6.3 Determine the quotient of whole numbers with up to 4 digit dividends and 2 digit divisors.

5.NBT.6.4 Illustrate and explain division using equations, rectangular arrays, and/or area models.

In this unit 5.NBT.B.5 and 5.NBT.B.6 will focus on operations with whole numbers only. Operations with decimals will be introduced in unit 10. These standards will be finalized in unit 15, but should be practiced throughout the year to provide opportunities for students to develop proficiency with these operations.

Students look for regularity in their work with multiplication and division use their understanding of the structure (MP.8) to make sense of their solutions and understand the approaches of other students (MP.1).

Videos www.khanacademy.org www.teachingchannel.org www.youtube.com Math Fact Fluency Practice www.mathwire.com www.oswego.org/ocsd-‐web/games/ http://mathfactspro.com/mathfluencygame.html#/math-‐facts-‐addition-‐games http://jerome.northbranfordschools.org/Content/Math_Fact_Fluency_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathematics.org/5 https://learnzillion.com www.AECSD5thGradeMathematicsdoc http://maccss.ncdpi.wikispaces.net/Fifth+Grade www.dpi.state.nc.us


• Algorithims • Dividend • Equal groups • Factor • Multiples • Multiplication • Partial products • Product • Multiplication symbols

• When given a problem with “errors,” how would you demonstrate and explain the process of multiplying multi-‐digit whole numbers using the standard algorithm?

• How can the methods and strategies you used to learn whole number multiplication help you to multiply decimals?

• Why does multiplying by a decimal result in a product less than one or both of the factors?

• How can you use what you know about place value to explain the relationship between expressions such as 3 x 5, 3 x 0.5, and 0.3 x 0.5?

• How can you use an area model to show the product of ___ x___? Formative Assessment Strategies

• Visual Representations/Drawings -‐ Graphic organizers can be used as visual representations of concepts in the content areas. Many of the graphic organizers contain a section where the student is expected to illustrate his/her idea of the concept.

• The Mind Map -‐ requires that students use drawings, photos or pictures from a magazine to represent a specific concept. • Think/Pair/Share for Math Problem Solving -‐ Place problem on the board. Ask students to think about the steps they would use to solve the problem, but do not let them figure out

the actual answer. Without telling the answer to the problem, have students discuss their strategies for solving the problem. Then let them work out the problem individually and then compare answers.

• Math Center Fun-‐ Practicing how to read large numbers, learning how to round numbers to various places, reviewing place value, solving word problems (as described above), recalling basic geometric terms, discussing the steps of division, discussing how to rename a fraction to lowest terms.

Unit 3: Using equivalency to add and subtract fractions with unlike denominators Suggested number of days: 12


Unit Overview:

In this unit students use what they've learned in Grades 3 and 4 about equivalency in terms of visual models and benchmarks to extend understanding of adding and subtracting fractions, including mixed numbers. They reason about size of fractions to make sense of their answers-‐e.g. they understand that the sum of 1/2 and 2/3 will be greater than 1.

It is important to note that in some cases it may not be necessary to find least common denominator to add fractions with unlike denominators (any common denominator may apply). Students should be encouraged to use their conceptual understanding of fractions rather than just using the algorithm for adding fractions. In addition, there is no mathematical reason for students to write fractions in simplest form.

Common Core State Standards for Mathematical Content

Number and Operations-‐Fractions - 5.NF A. Use equivalent fractions as a strategy to add and subtract fractions. 1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

5.NF.1.1 Rewrite fractions as equivalent fractions. 5.NF.1.2 Write my answer in simplest form. 5.NF.1.3 Add or subtract two fractions with unlike

denominators (2, 4, 5, or 10). 5.NF.1.4 Add or subtract two mixed numbers with

unlike denominators (2, 4, 5, or 10). 5.NF.1.5 Add and subtract two fractions with unlike

denominators (any denominator). 5.NF.1.6 Add and subtract two mixed numbers with

unlike denominators (any denominator). 5.NF.1.7 Add and subtract any number of fractions

with unlike denominators (any denominator).

5.NF.1.8 Add and subtract any number of mixed numbers with unlike denominators (any denominator).

5.NF.1.9 Add and subtract fractions within the same expression

In this unit, 5.NF.A.1 involves students using the same method from Grade 4 to generate equivalent fractions (4.NF.A.1). In unit 7 students will extend this understanding of equivalency to understand that multiplying by a fraction equivalent to 1 (e.g. 4/4) will result in an equivalent fraction

(5.NF.B.5b).4

Videos www.khanacademy.org www.teachingchannel.org www.youtube.com Math Fact Fluency Practice www.mathwire.com www.oswego.org/ocsd-‐web/games/ http://mathfactspro.com/mathfluencygame.html#/math-‐facts-‐addition-‐games http://jerome.northbranfordschools.org/Content/Math_Fact_Fluency_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathematics.org/5 https://learnzillion.com www.AECSD5thGradeMathematicsdoc http://maccss.ncdpi.wikispaces.net/Fifth+Grade

2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.


2. Reason abstractly and quantitatively. 4. Model with mathematics.

5.NF.2.1 Use benchmark numbers (0, ¼, ½, ¾, 1) to estimate sums and differences of fractions.

5.NF.2.2 Relate estimation to my answers to see if they make sense.

5.NF.2.3 5.NF.2.4 5.NF.2.5

Create a visual fraction model to represent the fractions in a word problem. Create an equation to represent a word problem. Create a word problem involving addition and subtraction of fractions.

Students use visual models and equations to solve problems involving the addition and subtraction of fractions, moving flexibly between the abstract and concrete representations (MP.2, MP.4).

www.dpi.state.nc.us http://harcourtschool.com/search/search.html www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐five http://www.onlinemathlearning.com/common-‐core-‐grade5.html http://www.mathgoodies.com/standards/alignments/grade5.html http://www.k-‐5mathteachingresources.com/5th-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.com/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/index.php/common-‐core-‐standards


• Benchmark Fraction • Common Denominator • Denominator • Difference • Equivalent Fraction • Fraction Bar • Improper Fraction • Like Denominator • Mixed Number • Number Lines • Numerator • Proper Fraction • Sum • Unlike Denominator

• How can you use a visual model to represent equivalent fractions? • How can you add and subtract fractions with unlike denominators? • In adding or subtracting fractions, when is it necessary to find a common

denominator? Give an example to support your answer. • How can you find a common denominator using equivalent fractions? • How can benchmark fractions be used to estimate an answer? Give an example to

support your thinking. • Given the following model/picture/visual representation (fractions of the same

whole with different denominator), what is a situation that could be represented by this model? Write and evaluate an expression to represent your situation.


• Summaries and Reflections -‐ Students stop and reflect, make sense of what they have heard or read, derive personal meaning from their learning experiences, and/or increase their metacognitive skills. These require that students use content-‐specific language.

• Lists, Charts, and Graphic Organizers -‐ Students will organize information, make connections, and note relationships through the use of various graphic organizers. • Visual Representations of Information -‐ Students will use both words and pictures to make connections and increase memory, facilitating retrieval of information later on. This “dual

coding” helps teachers address classroom diversity, preferences in learning style, and different ways of “knowing.” • Collaborative Activities -‐ Students have the opportunity to move and/or communicate with others as they develop and demonstrate their understanding of concepts. • Do’s and Don’ts -‐ List 3 Dos and 3 Don’ts when using/applying/relating to the content (e.g., 3 Dos and Don’ts for solving an equation). Example of Student Response: When adding

fractions, DO find a common denominator, DO add the numerators once you’ve found a common denominators, DON’T simply add the denominators • Three Most Common Misunderstandings -‐ List what you think might be the three most common misunderstandings of a given topic based on an audience of your peers. Example of

Student Response: In analyzing tone, most people probably confuse mood and tone, forget to look beyond the diction to the subtext as well, and to strongly consider the intended audience.

• Yes/No Chart -‐ List what you do and don’t understand about a given topic—what you do on the left, what you don’t on the right; overly-‐vague responses don’t count. Specificity matters!

Unit 4: Expanding understanding of place value to decimals Suggested number of days: 10


Unit Overview: In this unit students expand their previous understanding of place value to include decimal numbers. Grade 5 is the last grade in which the NBT domain appears in CCSSM. Later work in the base-‐ten system relies on the meanings and properties of operations. This also contributes to deepening students' understanding of computation and algorithms in the new domains that start in Grade 6. Common Core State Standards for Mathematical Content

Number and Operations in Base Ten – 5.NBT A. Understand the place value system. 1. Recognize that in a multi-‐digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-‐number exponents to denote powers of 10.

5.NBT.1.1 Represent place values of whole

numbers through 100,000,000 and decimals to the thousandths with manipulatives or visual models.

5.NBT.1.2 Recognize that in a multi-‐digit number, the digit to the left is 10x larger and the right is 1/10 smaller.

5.NBT.2.1 Show repeated multiplication of tens as an exponent.

5.NBT.2.2 Use manipulatives to explain patterns in the number of zeros of the product when multiplying a number by powers of 10.

5.NBT.2.3 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10.

5.NBT.2.4 Use manipulatives to explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.

5.NBT.2.5 Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.

Powers of 10 is a fundamental aspect of the base-‐ten system, thus 5.NBT.A.2 can help students extend their understanding of place value to incorporate decimals to hundredths.6

Videos www.khanacademy.org www.teachingchannel.org www.youtube.com Math Fact Fluency Practice www.mathwire.com www.oswego.org/ocsd-‐web/games/ http://mathfactspro.com/mathfluencygame.html#/math-‐facts-‐addition-‐games http://jerome.northbranfordschools.org/Content/Math_Fact_Fluency_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathematics.org/5 https://learnzillion.com www.AECSD5thGradeMathematicsdoc http://maccss.ncdpi.wikispaces.net/Fifth+Grade www.dpi.state.nc.us http://harcourtschool.com/searc

3. Read, write, and compare decimals to thousandths. a. Read and write decimals to

thousandths using base-‐ten numerals, number names, and expanded form, e.g., 347.392 = 3 x 100 + 4 x 10 + 7 x 1 + 3 x (1/10) + 9 x (1/100) + 2 x (1/1000).


6. Attend to precision. 7. Look for and make use of structure.

5.NBT.3a.1 Read and write decimals to the tenths place using numerals, number names, and expanded form.

5.NBT.3a.2 Read and write decimals to the hundredth place using numerals, number names, and expanded form.

5.NBT.3a.3 Read and write decimals to any place using numerals, number names, and expanded form.

5.NBT.3a.1 Read and write decimals to the tenths place using numerals, number names, and expanded form.

[5.NBT.A.3a] Students will be reading and writing decimals in this unit. Comparing decimals (5.NBT.A.3b ) will be addressed in unit 6.

Students use their understanding of structure of whole numbers to generalize this understanding to decimals (MP.7) and explain the relationship between the numerals (MP.6).

h/search.html www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐five http://www.onlinemathlearning.com/common-‐core-‐grade5.html http://www.mathgoodies.com/standards/alignments/grade5.html http://www.k-‐5mathteachingresources.com/5th-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.com/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/index.php/common-‐core-‐standards http://map.mathshell.org/materials/stds.php#standard1159


• Algorithim • Equal Group • Factor • Multiples • Multiplication • Partial Products • Product • Symbols

• What effect does multiplying or dividing by ten or a power of ten have on a number?

• How do you determine which decimal is greater than, less than, or equal to another?

• How does expanded form help you understand the value of each digit in a number?

• When comparing numbers with decimals to the thousandths place, how does expanded form help you?

• What are the mathematical properties that govern addition and multiplication? How would you use them?

• How can multiples be used to solve problems? • What strategies aid in mastering multiplication and division facts? • How can numbers be broken down into its smallest factors? • How can multiples be used to solve problems? • How do you find the prime factors and multiples of a number? • How can multiples be used to solve problems?

• How can I use the array model to explain multiplication?


• Numbered Heads Together -‐ Students sit in groups and each group member is given a number. The teacher poses a problem and all four students discuss. The teacher calls a number and that student is responsible for sharing for the group.

• Gallery Walk -‐ After teams have generated ideas on a topic using a piece of chart paper, they appoint a person to stay with their work. Teams rotate around examining other team’s ideas and ask questions of the person left at the paper. Teams then meet together to discuss and add to their information so the person there also can learn from other teams.

• Graffiti – Groups receive a large piece of paper and felt pens of different colors. Students generate ideas in the form of graffiti. Groups can move to other papers and discuss/add to the ideas.

• One Question and One Comment -‐Students are assigned a chapter or passage to read and create one question and one comment generated from the reading. In class, students will meet in either small or whole class groups for discussion. Each student shares at least one comment or question. As the discussion moves student by student around the room, the next person can answer a previous question posed by another student, respond to a comment, or share their own comments and questions. As the activity builds around the room, the conversation becomes in-‐depth with opportunity for all students to learn new perspectives on the text.

Unit 5: Comparing and rounding decimals Suggested number of days: 10


Unit Overview:

In this unit students apply both their understanding of comparing fractions and their understanding of place value to compare decimals.

Common Core State Standards for Mathematical Content

Number and Operations in Base Ten - 5.NBT A. Understand the place value system. 3. Read, write, and compare decimals to thousandths. b. Compare two decimals to

thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

4. Use place value understanding to round decimals to any place.


6. Attend to precision. 7. Look for and make use of structure.

5.NBT.3b.1 Compare decimals to the tenths

place using inequality symbols (<,>,=).

5.NBT.3b.2 Compare decimals to the hundredths place using inequality symbols (<,>,=).

5.NBT.3b.3 Compare decimals to any place using inequality symbols (<,>,=).

5.NBT.3b.1 Compare decimals to the tenths place using inequality symbols (<,>,=).

5.NBT.4.1 Use place value understanding to round decimals to any place.

Students apply their understanding of the structure within the base-‐ten system and fraction-‐decimal equivalencies to precisely communicate their understanding of relative sizes of decimal numbers (MP.6, MP.7).



• Decimal • Division • Hundredths • Inequality • Place Value

• How does understanding place value help you solve double digit addition and subtraction problems?

• How are place value patterns repeated in large numbers? • What effect does multiplying or dividing by ten or a power of ten have on a

number? • How do you determine which decimal is greater than, less than, or equal to

another? • How does expanded form help you understand the value of each digit in a number? • When comparing numbers with decimals to the thousandths place, how does

expanded form help you?


• Whip Around -‐ The teacher poses a question or a task. Students then individually respond on a scrap piece of paper listing at least 3 thoughts/responses/statements. When they have done so, students stand up. The teacher then randomly calls on a student to share one of his or her ideas from the paper. Students check off any items that are said by another student and sit down when all of their ideas have been shared with the group, whether or not they were the one to share them. The teacher continues to call on students until they are all seated. As the teacher listens to the ideas or information shared by the students, he or she can determine if there is a general level of understanding or if there are gaps in students’ thinking.”

• Word Sort -‐ Given a set of vocabulary terms, students sort in to given categories or create their own categories for sorting • Triangular Prism (Red/Green/Yellow)Students give feedback to teacher by displaying the color that corresponds to their level of understanding • Take and Pass -‐ Cooperative group activity used to share or collect information from each member of the group; students write a response, then pass to the right, add their

response to next paper, continue until they get their paper back, then group debriefs. • Student Data Notebooks -‐ A tool for students to track their learning: Where am I going? Where am I now? How will I get there? • Slap It -‐ Students are divided into two teams to identify correct answers to questions given by the teacher. Students use a fly swatter to slap the correct response posted on the

wall. • Say Something -‐ Students take turns leading discussions in a cooperative group on sections of a reading or video

Unit 6: Understanding the concept of multiplying fractions by fractions Suggested number of days: 12


Unit Overview:

In this unit students extend their understanding of multiplying a fraction by a whole number to multiplying fractions by fractions. In previous grades, students have developed understanding of fractions as numbers. In this grade level, students develop an understanding of the connection between fractions and division. They will use this understanding to explore the relationship of multiplication and division when multiplying fractions as explained in 5.NF.B.4a. Common Core State Standards for Mathematical Content

Number and Operations-‐Fractions – 5.NF B. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-‐pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

5.NF.3.1 Interpret a fraction as a division

problem. Ex. ¼ = 1 ÷ 4

5.NF.3.2 Interpret a division problem as a fraction. Ex. 1 ÷ 4 = ¼

5.NF.3.3 5.NF.3.4

Solve division word problems and express the quotient as a fraction or mixed number by using visual fraction models. Solve division word problems and express the quotient as a fraction or mixed number by using equations.


4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (alb) x q as a parts

of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a x q ÷ b. For example, use a visual fraction model to show (2/3) x 4 = 8/3, and create a story context for this equation. Do the same with (2/3) x (4/5) = 8/15. (In general, (a/b) x (c/d) = ac/bd.)

b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.


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

4. Model with mathematics. 5. Use appropriate tools strategically.

5.NF.4a.1 Represent a whole number as a fraction.

5.NF.4a.2 5.NF.4a.3

Multiply a fraction by a fraction. Multiply a fraction by a whole number.

5.NF.4a.4 5.NF.4a.5

Use a visual fraction model to represent multiplication of fractions. Create a context for a problem involving multiplication of fractions.

5.NF.4b.1 5.NF.4b.2 5.NF.4b.3

Find the area of a rectangle with fractional side lengths by tiling it with unit squares. Relate different strategies for calculating the area of a rectangle. (tiling vs. formula) Multiply fractional side lengths to find areas of rectangles.

5.NF.4b.4 Apply my understanding of the area of rectangles to include fractional units.

Representing multiplication of fractions with visual and concrete models is fundamental to this unit in order for students to make sense of multiplying fractions by fractions (MP.1, MP.4). Students select and use a variety tools to explore these concepts (MP.5).

http://harcourtschool.com/search/search.html www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐five http://www.onlinemathlearning.com/common-‐core-‐grade5.html http://www.mathgoodies.com/standards/alignments/grade5.html http://www.k-‐5mathteachingresources.com/5th-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.com/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/index.php/common-‐core-‐standards


• Area • Division • Fraction • Mixed Numbers • Side length • Tile • Unit Square • Visual Model • Whole Number

• How is fraction notation related to division? • How can you use a visual model, such as a fraction bar, to represent division? • How would you represent a fraction as division? • How do you find the area of a rectangle using tiles? • How can I use fractions in real life? • How can decimals be rounded to the nearest whole number? • How can models be used to compute fractions with like and unlike denominators? • How many ways can we use models to determine and compare equivalent

fractions? • How are models used to show how fractional parts are combined or separated? • How do I identify and record the fraction of a whole or group? • How do I identify the whole? • How do I explain the meaning of a fraction and its numerator and denominator,

and use my understanding to represent and compare fractions? • How do I explain how changing the size of the whole affects the size or amount of a

fraction?


• Fill In Your Thoughts -‐ Written check for understanding strategy where students fill the blank. (Another term for rate of change is ____ or ____.) • Circle, Triangle, Square -‐ Something that is still going around in your head (Triangle) Something pointed that stood out in your mind (Square) Something that “Squared” or agreed

with your thinking. • ABCD Whisper -‐ Students should get in groups of four where one student is A, the next is B, etc. Each student will be asked to reflect on a concept and draw a visual of his/her

interpretation. Then they will share their answer with each other in a zigzag pattern within their group. • Onion Ring -‐ Students form an inner and outer circle facing a partner. The teacher asks a question and the students are given time to respond to their partner. Next, the inner circle

rotates one person to the left. The teacher asks another question and the cycle repeats itself.

Unit 7: Interpreting multiplying fractions as scaling Suggested number of days: 12


Unit Overview:

In this unit students build on their work with "compare" problems in Grade 4 (4.0A.A.1) to develop a foundational understanding of multiplication as scaling. They interpret, represent, and explain situations involving multiplication of fractions. Students apply their whole number work with multiplication to develop conceptual understanding of multiplying a fraction by a fraction.

Scaling is foundational for developing an understanding of ratios and proportion in future grade levels. Common Core State Standards for Mathematical Content

Number and 0perations-‐Fractions - 5.NF B. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5. Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the

size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

5.NF.5a.1 5.NF.5a.2 5.NF.5a.3 5.NF.5a.4

Use visual models or manipulatives to interpret multiplication scaling and correctly perform the indicated multiplication. Interpret multiplication scaling by performing the indicated multiplication where one factor is a fraction. Interpret multiplication scaling without performing the indicated multiplication where one factor is a mixed number. Interpret multiplication scaling without performing the indicated multiplication where both factors are fractions.

In this unit, 5.NF.B.5a and 5.NF.B.5b involve only multiplication by fractions. Division by unit fractions will be introduced in unit 8.


b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (nxa)/(nxb) to the effect of multiplying a/b by 1.

6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.


2. Reason abstractly and quantitatively.

4. Model with mathematics. 6. Attend to precision.

5.NF.5b.1 Predict the size of the product based on the size of the factors. Ex: fraction x fraction = smaller fraction, fraction x whole number = a fraction of the whole number.

5.NF.5b.2 5.NF.5b.3 5.NF.5b.4

Use visual models or manipulatives to explain when multiplying by a fraction greater than one, the number increases and when multiplying by a number less than one, the number decreases. Explain when multiplying by a fraction greater than one, the number increases and when multiplying by a number less than one, the number decreases. Explain that when multiplying the numerator and denominator by the same number is the same as multiplying by one.

5.NF.6.1 Solve real-‐world problems involving multiplication of fractions and mixed numbers.

In 5.NF.B.6 students should have opportunities to work with all problem types.

Students reason abstractly and practice communicating their thinking in real world situations (MP.2, MP.6). They use number lines and other visual models to interpret situations involving multiplication by numbers larger than one (when the result will be larger than the original quantity) and involving multiplication by a fraction smaller than 1 (when the result will be smaller than the original quantity) (MP.4).

www.dpi.state.nc.us http://harcourtschool.com/search/search.html www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐five http://www.onlinemathlearning.com/common-‐core-‐grade5.html http://www.mathgoodies.com/standards/alignments/grade5.html http://www.k-‐5mathteachingresources.com/5th-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.com/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/index.php/common-‐core-‐standards


• Area • Division • Fraction • Mixed Numbers • Side length • Tile • Unit Square • Visual Model • Whole Number

• How can I use fractions in real life? • How can decimals be rounded to the nearest whole number? • How can models be used to compute fractions with like and unlike denominators? • How many ways can we use models to determine and compare equivalent

fractions? • How are models used to show how fractional parts are combined or separated? • How do I identify and record the fraction of a whole or group? • How do I identify the whole? • How do I explain the meaning of a fraction and its numerator and denominator,

and use my understanding to represent and compare fractions? • How do I explain how changing the size of the whole affects the size or amount of

a fraction? • How is fraction notation related to division? • How can you use a visual model, such as a fraction bar, to represent division? • How would you represent a fraction as division?


• Quick Write -‐ The strategy asks learners to respond in 2–10 minutes to an open-‐ended question or prompt posed by the teacher before, during, or after reading. • Direct Paraphrasing -‐ Students summarize in well-‐chosen (own) words a key idea presented during the class period or the one just past. • RSQC2 -‐ In two minutes, students recall and list in rank order the most important ideas from a previous day's class; in two more minutes, they summarize those points in a single

sentence, then write one major question they want answered, then identify a thread or theme to connect this material to the course's major goal. • I have the Question, Who has the Answer? -‐The teacher makes two sets of cards. One set contains questions related to the unit of study. The second set contains the answers to

the questions. Distribute the answer cards to the students and either you or a student will read the question cards to the class. All students check their answer cards to see if they have the correct answer. A variation is to make cards into a chain activity: The student chosen to begin the chain will read the given card aloud and then wait for the next participant to read the only card that would correctly follow the progression. Play continues until all of the cards are read and the initial student is ready to read his card for the second time.

Unit 8: Developing the concept of dividing unit fractions Suggested number of days: 10


Unit Overview:

In this unit students will use their understanding of the relationship of multiplication and division to develop a conceptual understanding of division with fractions (division of a whole number by a unit fraction or a unit fraction by a whole number). Common Core State Standards for Mathematical Content

Number and Operations-‐Fractions - 5.NF B. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

a. Interpret division of a unit fraction by a non-‐zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) x 4 = 1/3.

5.NF.7 Understand the relationship between

multiplication and division.

5.NF.7a.1 Use a visual fraction model to divide a unit fraction by a whole number.

5.NF.7a.2 Create a context for a problem involving division of a unit fraction by a whole number.


b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 x (1/5) = 4.

NOTE: Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.


1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively.

5.NF.7b.1 Use a visual fraction model to divide a whole number by a unit fraction.

5.NF.7b.2 Create a context for a problem involving division of a whole number by a unit fraction.

In this unit it is critical for students to use concrete objects or pictures to help conceptualize, create, and solve problems (MP.1, MP.2).



• Division • Fraction • Mixed Numbers • Multiplication • Unit Fraction • Visual Model • Whole Number

• How can you use a visual model to divide a whole number by a unit fraction? • What is the relationship between multiplication and division? • How can I use fractions in real life? • How can decimals be rounded to the nearest whole number? • How can models be used to compute fractions with like and unlike denominators? • How many ways can we use models to determine and compare equivalent

fractions? • How are common and decimal fractions alike and different? • What strategies can be used to solve estimation problems with common and

decimal fractions? • How are models used to show how fractional parts are combined or separated? • How do I identify and record the fraction of a whole or group? • How do I identify the whole?


• Journal Entry -‐ Students record in a journal their understanding of the topic, concept or lesson taught. The teacher reviews the entry to see if the student has gained an understanding of the topic, lesson or concept that was taught.

• Choral Response -‐ In response t o a cue, all students respond verbally at the same time. The response can be either to answer a question or to repeat something the teacher has said.

• A-‐B-‐C Summaries -‐ Each student in the class is assigned a different letter of the alphabet and they must select a word starting with that letter that is related to the topic being studied.

• Debriefing -‐ A form of reflection immediately following an activity. • Idea Spinner -‐ The teacher creates a spinner marked into 4 quadrants and labeled “Predict, Explain, Summarize, Evaluate.” After new material is presented, the teacher spins the

spinner and asks the students to answer a questions based on the location of the spinner. For example, if the spinner lands in the “Summarize” quadrant, the teacher might say, “List the key concepts just presented.”

Unit 9: Solving problems involving volume Suggested number of days: 10


Unit Overview:

This unit calls for students to apply their understanding of volume to real-‐world problems. They develop efficient strategies, including the use of formulas, to compute volumes of right rectangular prisms or other three-‐dimensional figures that can be broken down into non-‐overlapping right rectangular prisms. Common Core State Standards for Mathematical Content

Measurement and Data -‐ 5.MD C. Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. 5. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

a. Find the volume of a right rectangular prism with whole-‐number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-‐number products as volumes, e.g., to represent the associative property of multiplication.

5.MD.5a.1 Find the volume of a right rectangular

prism with whole-‐number side lengths by packing it with unit cubes.

5.MD.5a.2 Relate volume to the operation of multiplication and addition.

5.MD.5a.3 Measure volume with unit cubes and show that it is the same as: multiplying the side lengths, multiplying the edge lengths, and multiplying the height by the area of the base.

Videos www.khanacademy.org www.teachingchannel.org www.youtube.com Math Fact Fluency Practice www.mathwire.com www.oswego.org/ocsd-‐web/games/ http://mathfactspro.com/mathfluencygame.html#/math-‐facts-‐addition-‐games http://jerome.northbranfordschools.org/Content/Math_Fact_Fluency_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathematics.org/5 https://learnzillion.com www.AECSD5thGradeMathematicsdoc http://maccss.ncdpi.wikispaces.net/Fifth+Grade www.dpi.state.nc.us http://harcourtschool.com/search/search.html

b. Apply the formulas V=l x w x h and V=b x h for rectangular prisms to find volumes of right rectangular prisms with whole-‐ number edge lengths in the context of solving real world and mathematical problems.

c. Recognize volume as additive. Find volumes of solid figures composed of two non-‐overlapping right rectangular prisms by adding the volumes of the non-‐overlapping parts, applying this technique to solve real world problems.


5. Use appropriate tools strategically.

7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

5.MD.5b.1 Apply the formulas (V= l x w x h) & (V = B x h) to solve real world and mathematical problems.

5.MD.5b.2 Create real-‐world and mathematical problems that would be solved by finding volume.

5.MD.5c.1 Recognize volume is additive by finding the volume of solid figures of two non-‐overlapping parts.

5.MD.5c.2 Recognize volume is additive by finding the volume of solid figures of two or more non-‐overlapping parts.

Students pack the figures with unit cubes (MP.5) and connect this structure to multiplicative reasoning (MP.7). They solve problems by applying the generalized formulas (MP.8).

www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐five http://www.onlinemathlearning.com/common-‐core-‐grade5.html http://www.mathgoodies.com/standards/alignments/grade5.html http://www.k-‐5mathteachingresources.com/5th-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.com/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/index.php/common-‐core-‐standards http://map.mathshell.org/materials/stds.php#standard1159


• Formula • Rectangular Prism • Solid Figures • Unit Cube • Volume

• What is volume and how does it relate to the attribute of an individual figure? • What tools and units of measurement can be reasonably used to determine

length, area and volume? • How can volume help us to solve problems in everyday life?


• Index Card Summaries/Questions -‐ Periodically, distribute index cards and ask students to write on both sides, with these instructions: (Side 1) Based on our study of (unit topic), list a big idea that you understand and word it as a summary statement. (Side 2) Identify something about (unit topic) that you do not yet fully understand and word it as a statement or question.

• Hand Signals -‐ Ask students to display a designated hand signal to indicate their understanding of a specific concept, principal, or process: -‐ I understand____________ and can explain it (e.g., thumbs up). -‐ I do not yet understand ____________ (e.g., thumbs down). -‐ I’m not completely sure about ____________ (e.g., wave hand).

• One Minute Essay -‐ A one-‐minute essay question (or one-‐minute question) is a focused question with a specific goal that can, in fact, be answered within a minute or two. • Analogy Prompt -‐ Present students with an analogy prompt: (A designated concept, principle, or process) is like ___________ because___________. • Misconception Check -‐ Present students with common or predictable misconceptions about a designated concept, principle, or process. Ask them whether they agree or disagree

and explain why. The misconception check can also be presented in the form of a multiple-‐choice or true-‐false quiz.

Unit 10: Performing operations with decimals Suggested number of days: 12


Unit Overview:

Measurement is used in this unit as a context for operations with decimals. Students' previous experiences with decimal fractions and fraction computations are applied here to provide multiple ways of thinking about operations with decimals. Students can use their understanding of decimal-‐fraction equivalencies, concrete or visual models, and place value to reason about decimal quantities and operations. Common Core State Standards for Mathematical Content

Number and Operations in Base Ten - 5.NBT B. Perform operations with multi-‐digit whole numbers and with decimals to hundredths. 7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

5.NBT.7.1 Add or subtract two decimal numbers

to the hundredths without regrouping.

5.NBT.7.2 Add and subtract two decimal numbers to hundredths with regrouping.

5.NBT.7.3 Multiply tenths by tenths. 5.NBT.7.4 Multiply tenths by hundredths.

5.NBT.7.5 Divide decimals with tenths.

5.NBT.7.6 Divide decimals with tenths and/or hundredths.

5.NBT.7.7 Relate the strategy used to a written method and explain the reasoning used.

5.NBT.7.8 Demonstrate computations by using models and drawings.


Measurement and Data -‐ 5.MD A. Convert like measurement units within a given measurement system. 1. Convert among different-‐sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-‐step, real world problems.


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

5.MD.1.1 Recall customary units of measurements.

5.MD.1.2 Recall metric units of measurements. 5.MD.1.3 Use manipulatives or visual models to

convert different measurement units (customary & metric) within a given measurement system to solve single step problems.

5.MD.1.4 Convert different measurement units (customary & metric) within a given measurement system to solve real world single step problems.

5.MD.1.5 Convert different measurement units (customary & metric) within a given measurement system to solve and create real world multi-‐step problems.

5.MD.1.6 Choose the appropriate measurement unit based on the given context.

5.MD.A.1 provides measurement conversion as a context for not only working with decimals but a deeper understanding for place value and the connection to the metric system.13 Instead of just computing answers, students reason about both the relationship between fraction and decimal operations and the relationship between whole number computation and fractional/decimal computation (MP.2, MP.3).



• Addition • Customary • Decimals • Division • Measurement • Metric • Multiplication • Subtraction


• How can I use decimals in real life? • How can decimals be rounded to the nearest whole number? • How can models help us understand the addition and subtraction of decimals? • How can model help us understand the multiplication and division of decimals? • How do I know if a decimal is repeating? • How are the units of measure within a standard system related? • How do you decide which unit of measurement to use? • How can I measure length, mass and capacity by using non-‐standard units? • How do I measure accurately* to the nearest inch? Nearest centimeter? • How do I choose the appropriate tool and unit when measuring? • How do I estimate and measure?


• Tic-‐Tac-‐Toe/Think-‐Tac-‐Toe -‐ A collection of activities from which students can choose to do to demonstrate their understanding. It is presented in the form of a nine square grid similar to a tic-‐tac-‐toe board and students may be expected to complete from one to “three in a row”. The activities vary in content, process, and product and can be tailored to address DOK levels.

• Four Corners -‐ Students choose a corner based on their level of expertise of a given subject. Based on your knowledge of _________________, which corner would you choose? Corner 1: The Dirt Road – (There’s so much dust, I can’t see where I’m going! Help!!), Corner 2: The Paved Road (It’s fairly smooth, but there are many potholes along the way.), Corner 3: The Highway (I feel fairly confident but have an occasional need to slowdown.) Corner 4: The Interstate (I ’m traveling along and could easily give directions to someone else.) Once students are in their chosen corners, allow students to discuss their progress with others. Questions may be prompted by teacher. Corner One will pair with Corner Three; Corner Two will pair with Corner four for peer tutoring.

Unit 11: Classifying two-‐dimensional geometric figures Suggested number of days: 10


Unit Overview:

In this unit the emphasis is on the hierarchical relationship among 2-‐dimensional geometric figures. Students have had previous experience classifying shapes using defining attributes, and this unit extends this concept to set a foundation for understanding the propagation of properties. Common Core State Standards for Mathematical Content

Geometry -‐ 5.G B. Classify two-‐dimensional figures into categories based on their properties. 3. Understand that attributes belonging to a category of two-‐ dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

4. Classify two-‐dimensional figures in a hierarchy based on properties.


3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure.

5.G.3.1 Understand that shared attributes

categorize two-‐dimensional figures.

5.G.4.1 Classify two-‐dimensional figures based on properties.

5.G.4.2 Classify two-‐dimensional figures in a hierarchy based on properties.

5.G.4.3 Demonstrate that attributes belonging to a category of two-‐dimensional also belong to all subcategories of that category.

5.G.4.4 Use appropriate tools to determine similarities and differences between categories and subcategories.

Students make use of structure to build a logical progression of statements and explore hierarchical relationships among 2-‐dimensional shapes (MP.3, MP.7).



• Angle • Attributes • Rectangle • Shapes • Square • Two Dimensional figures

• Where in the real world can I find shapes? • Where would you find symmetry? • How can objects be represented and compared using geometric attributes? • Is geometry more like map-‐making and using a map, or inventing and playing

games like chess? • How can I identify and describe solid figures by describing the faces, edges, and

sides? • In what ways can I match solid geometric figures to real-‐life objects? • How can I put shapes together and take them apart to form other shapes?


• Observation – Walking around classroom and observe for understanding. Anecdotal records, conferences, checklists. • 3-‐2-‐1 – 3 things you found out, 2 interesting things and 1 question you still have. • Exit Cards -‐ Exit cards are written student responses to questions posed at the end of a class or learning activity or at the end of a day. • Student Data Notebooks -‐ A tool for students to track their learning: Where am I going? Where am I now? How will I get there?

Unit 12: Solving problems with fractional quantities Suggested number of days: 12


Unit Overview:

In this unit students use data and other contexts to solve real world problems involving fractional computations. All of the different problem types in Tables 1 and 2 in the Common Core State Standards for Mathematics should be addressed in this unit. Common Core State Standards for Mathematical Content

Number and Operations-‐Fractions - 5.NF B. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1 c. Solve real world problems involving

division of unit fractions by non-‐zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-‐cup servings are in 2 cups of raisins?

NOTE: 1 Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.

5.NF.7c.1 Solve real-‐world problems involving

division of unit fractions and whole numbers.


Measurement and Data -‐ 5.MD B. Represent and interpret data. 2. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.


2. Reason abstractly and quantitatively. 5. Use appropriate tools strategically.

5.MD.2.1 Create a line plot to display a data set of measurements in fraction form with denominators of 2 and 4.

5.MD.2.2 Create a line plot to display a data set of measurements in fraction form with denominators of 2, 4 and 8.

5.MD.B.2 is included here so measurement line plots can be used as a context for students to apply fraction computation strategies.

Students use line plots and other tools/technology to reason about problem situations (MP.5). Students attend to the underlying meaning of the quantities and operations when solving problems rather than just how to compute answers (MP.2).



• Division • Fractions • Unit Fractions

• How can I use fractions in real life? • How do I explain the meaning of a fraction and its numerator and denominator,

and use my understanding to represent and compare fractions? • How do I explain how changing the size of the whole affects the size or amount of

a fraction? • How do you know if a number is divisible by 2, 3, 5, and 10? • What strategies aid in mastering division facts? • How can I use what I know about repeated subtraction, equal sharing, and

forming equal groups to solve division problems? • How does my knowledge about division facts help me to solve problems?


• Take and Pass -‐ Cooperative group activity used to share or collect information from each member of the group; students write a response, then pass to the right, add their response to next paper, continue until they get their paper back, then group debriefs.

• Slap It -‐ Students are divided into two teams to identify correct answers to questions given by the teacher. Students use a fly swatter to slap the correct response posted on the wall.

• Numbered Heads Together -‐ Students sit in groups and each group member is given a number. The teacher poses a problem and all four students discuss. The teacher calls a number and that student is responsible for sharing for the group.

• Circle, Triangle, Square -‐ Something that is still going around in your head (Triangle) Something pointed that stood out in your mind (Square) Something that “Squared” or agreed with your thinking

Unit 13: Representing algebraic thinking Suggested number of days: 10


Unit Overview: In this unit students explore algebraic expressions more formally to represent and interpret calculations involving whole numbers, fractions, and decimals. They apply their understanding of the different algebraic properties of operations and explain the relationships between the quantities with the written expressions. This unit includes opportunities to both evaluate expressions and reason about expressions without calculating a solution. This is foundational for further work with number in later grades. Common Core State Standards for Mathematical Content

Operations and Algebraic Thinking - 5.OA A. Write and interpret numerical expressions. 1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2” as 2 x {8 + 7}. Recognize that 3 x {18932 + 921} is three times as large as 18932 + 921, without having to calculate the indicated sum or product.


6. Attend to precision.

5.OA.1.1 Use the order of operations to

evaluate numerical expressions.

5.OA.1.2 Explain the order of operations.

5.OA.1.3 Apply the order of operations to evaluate expressions.

5.OA.2.1 Interpret a numerical expression into words (without evaluating the expression).

5.OA.2.2 Write simple expressions that record calculations with numbers

The expressions described in 5.OA.A.1 include the use of parentheses but should not contain nested grouping symbols.

The expressions described in 5.OA.A.2 should be no more complex than the expressions one finds in an application of the associative or distributive property.

Students discuss the meaning of symbols and interpret numerical expressions precisely (MP.6).



• Addition • Division • Multiplication • Properties • Order of Operations • Subtraction

• When is the “correct” answer not the best solution? • What information and strategies would you use to solve a multi-‐step word

problem? • When should you use mental computation? • When should you use pencil computation? • When should you use a calculator? • What number or symbol is needed to make number sentences true? • How are the four basic operations related to one another? • How do number properties assist in computation? • Why is the order of operations so important?


• Flag It – Students use “flags” (sticky notes) to flag important information presented in class or while working problems. • Triangular Prism (Red, Yellow, Green) -‐ Students give feedback to teacher by displaying the color that corresponds to their level of understanding. • Word Sort -‐ Given a set of vocabulary terms, students sort in to given categories or create their own categories for sorting. • Cubing -‐ Display 6 questions from the lesson Have students in groups of 4. Each group has 1 die. Each student rolls the die and answers the question with the corresponding

number. If a number is rolled more than once the student may elaborate on the previous response or roll again. • Tic-‐Tac-‐Toe/Think-‐Tac-‐Toe -‐ A collection of activities from which students can choose to do to demonstrate their understanding. It is presented in the form of a nine square grid

similar to a tic-‐tac-‐toe board and students may be expected to complete from one to “three in a row”. The activities vary in content, process, and product and can be tailored to address DOK levels.

• Four Corners -‐ Students choose a corner based on their level of expertise of a given subject. Based on your knowledge of _________________, which corner would you choose? Corner 1: The Dirt Road – (There’s so much dust, I can’t see where I’m going! Help!!), Corner 2: The Paved Road (It’s fairly smooth, but there are many potholes along the way.), Corner 3: The Highway (I feel fairly confident but have an occasional need to slowdown.) Corner 4: The Interstate (I ’m traveling along and could easily give directions to someone else.) Once students are in their chosen corners, allow students to discuss their progress with others. Questions may be prompted by teacher. Corner One will pair with Corner Three; Corner Two will pair with Corner four for peer tutoring.

Unit 14: Exploring the coordinate plane Suggested number of days: 10


Unit Overview:

In this unit students are introduced to the coordinate plane, applying their knowledge of the number line to understand the relationship of the two dimensions of a point in the coordinate plane. Students connect their work with numerical patterns to form ordered pairs and graph these ordered pairs in the first quadrant of a coordinate plane. Students use this model to make sense of and explain the relationships within the numerical patterns they generate. This prepares students for future work with functions and proportional relationships in the middle grades. Common Core State Standards for Mathematical Content

Operations and Algebraic Thinking - 5.OA B. Analyze patterns and relationships. 3. Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3” and the starting number 0, and given the rule "Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

5.OA.3.1 Create two numerical patterns from

two given rules. 5.OA.3.2 Identify relationships between

corresponding terms in a pattern. 5.OA.3.3 Extend a numerical pattern from a

given rule. 5.OA.3.4 Determine a rule from a given

numerical pattern.

5.OA.3.5 Create ordered pairs of the corresponding terms from two patterns.

5.OA.3.6 Plot an ordered pair on a coordinate plane from 2 patterns and justify.


Geometry -‐ 5.G A. Graph points on the coordinate plane to solve real-‐world and mathematical problems. 1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-‐axis and x-‐coordinate, y-‐axis and y-‐ coordinate).

2. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.


4. Model with mathematics. 6. Attend to precision.

5.G.1.1 Label the axes, quadrants and origin on the coordinate plane.

5.G.1.2 Identify ordered pairs.

5.G.1.3 Plot points on the coordinate plane.

5.G.2.1 Represent real-‐world and mathematical problems by locating or graphing points in the first quadrant of a coordinate plane.

5.G.2.2 Represent real-‐world and mathematical problems by locating and graphing points in the first quadrant of a coordinate plane.

5.G.2.3 Interpret coordinate values in the context of the situation.

5.G.2.4 Create real-‐world and mathematical problems that require locating and graphing points in the first quadrant of the coordinate plane.

Students precisely describe the coordinates of points and the relationship of the coordinate plane to the number line (MP.6). Students both generate and identify relationships in numerical patterns, using the coordinate plane as a way of representing these relationships and patterns (MP.4).

http://harcourtschool.com/search/search.html www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐five http://www.onlinemathlearning.com/common-‐core-‐grade5.html http://www.mathgoodies.com/standards/alignments/grade5.html http://www.k-‐5mathteachingresources.com/5th-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.com/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/index.php/common-‐core-‐standards http://map.mathshell.org/materials/stds.php#standard1159


• Axes • Coordinate Plane • Graphing • Origin • Patterns • Rules • Quadrant

• Are patterns important in the world today? • When are algebraic and numeric expressions used? • What patterns or relationships do we see in each type of mathematics? • What are the different ways to represent the patterns or relationships? • What different interpretations can be obtained from a particular pattern or

relationship? • What predictions can the patterns or relationships support? • Where in the real world would I find patterns? • What strategies can be used to solve for unknowns in algebraic When solving

multi-‐step word problems using charts, tables, and graphs, how can you tell if the information is sufficient?

• How do you collect data? • Why are graphs helpful? • What kinds of questions can be answered using different data displays? • What data display is appropriate for a given set of data?


• Think-‐Write-‐Pair-‐Share -‐ Students think individually, write their thinking, pair and discuss with partner, then share with the class. • Choral Response -‐ In response t o a cue, all students respond verbally at the same time. The response can be either to answer a question or to repeat something the teacher has

said. • Self Assessment -‐ process in which students collect information about their own learning, analyze what it reveals about their progress toward the intended learning goals and plan

the next steps in their learning. • Web or Concept Map -‐ Any of several forms of graphical organizers which allow learners to perceive relationships between concepts through diagramming key words representing

those concepts. http://www.graphic.org/concept.html

Unit 15: Finalizing multiplication and division with whole numbers Suggested number of days: 10


Unit Overview:

These standards were introduced in Unit 2 to provide opportunities throughout the year for students to work towards fluency. In this unit students demonstrate fluency in multiplication with whole numbers and continue to practice division with whole numbers using various strategies. Common Core State Standards for Mathematical Content

Number and Operations in Base Ten - 5.NBT B. Perform operations with multi-‐digit whole numbers and with decimals to hundredths.

5. Fluently multiply multi-‐digit whole numbers using the standard algorithm.

6. Find whole-‐number quotients of whole numbers with up to four-‐digit dividends and two-‐digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.


1. Make sense of problems and persevere in solving them. 8. Look for and express regularity in repeated reasoning.

5.NBT.5.1 Fluently multiply multi-‐digit whole

numbers.

5.NBT.6.1 5.NBT.6.2




5.NBT.B.6 is a milestone along the way to reaching fluency with the standard algorithm in Grade 6 (6.NS.B.2).

Students use efficient strategies and look for shortcuts to multiply and divide whole numbers with accuracy (MP.1, MP.8).



• Area Models • Array • Dividend • Divisor • Division • Equation • Multiplication • Rectangular Array


• How do you know if a number is divisible by 2, 3, 5, and 10? • How can multiples be used to solve problems? • What strategies aid in mastering multiplication and division facts? • How can numbers be broken down into its smallest factors? • How can multiples be used to solve problems? • How do you find the prime factors and multiples of a number? • How can multiples be used to solve problems? • How can I use the array model to explain multiplication? • How can I relate what I know about skip counting to help me learn the multiples of

2,5,10? • How are repeated addition and multiplication related? • How can I use what I know about repeated subtraction, equal sharing, and

forming equal groups to solve division problems? • How does my knowledge about multiplication facts help me to solve problems?


• Index Card Summaries/Questions -‐ Periodically, distribute index cards and ask students to write on both sides, with these instructions: (Side 1) Based on our study of (unit topic), list a big idea that you understand and word it as a summary statement. (Side 2) Identify something about (unit topic) that you do not yet fully understand and word it as a statement or question.

• Hand Signals -‐ Ask students to display a designated hand signal to indicate their understanding of a specific concept, principal, or process: -‐ I understand____________ and can explain it (e.g., thumbs up). -‐ I do not yet understand ____________ (e.g., thumbs down). -‐ I’m not completely sure about ____________ (e.g., wave hand).

• One Minute Essay -‐ A one-‐minute essay question (or one-‐minute question) is a focused question with a specific goal that can, in fact, be answered within a minute or two. • Analogy Prompt -‐ Present students with an analogy prompt: (A designated concept, principle, or process) is like ___________ because___________. • Misconception Check -‐ Present students with common or predictable misconceptions about a designated concept, principle, or process. Ask them whether they agree or disagree

and explain why. The misconception check can also be presented in the form of a multiple-‐choice or true-‐false quiz.

Key: Major Clusters; Supporting Clusters; Additional Clusters

FIFTH GRADE CRITICAL AREAS OF FOCUS CRITICAL AREA OF FOCUS #1 Developing fluency with addition and subtraction of fractions and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions) Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them. Students also use the meaning of fractions, of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense. (Note: this is limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.)

Number and Operations - Fractions 5.NF Use equivalent fractions as a strategy to add and subtract fractions.

1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result

2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the

result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)


5. Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other

factor, without performing the indicated multiplication. b. Explaining why multiplying a given number by a fraction greater than 1 results in a product

greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1


FIFTH GRADE CRITICAL AREAS OF FOCUS CRITICAL AREA OF FOCUS #1, CONTINUED

results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Measurement and Data 5.MD Represent and interpret data.

2. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.


FIFTH GRADE CRITICAL AREAS OF FOCUS CRITICAL AREA OF FOCUS #2 Extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.

Operations and Algebraic Thinking 5.O Write and interpret numerical expressions.

1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Number and Operations in Base Ten 5.NBT Understand the place v alue system.

1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

3. Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base-ten numerals, number names, and

expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).

b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

4. Use place value understanding to round decimals to any place. Perform operations with multi-digit whole numbers and with decimals to hundredths.

5. Fluently multiply multi-digit whole numbers using the standard algorithm. 6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit

divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.


FIFTH GRADE CRITICAL AREAS OF FOCUS

CRITICAL AREA OF FOCUS #2, CONTINUED

Measurement and Data 5.MD Convert like measurement units within a given measurement system.

1. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.



CRITICAL AREA OF FOCUS #3 Developing understanding of volume Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the total number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in order to determine volumes to solve real world and mathematical problems.

Measurement and Data 5.MD Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume,

and can be used to measure volume. b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to

have a volume of n cubic units. 4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 5. Relate volume to the operations of multiplication and addition and solve real world and

mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole-number side lengths by packing it

with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

c. Recognize volume as additive. Find volumes of solid figures composed of two non- overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

Number and Operations—Fractions 5.NF Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as

the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)


5. Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other

factor, without performing the indicated multiplication.



CRITICAL AREA OF FOCUS #3, CONTINUED

Geometry 5.G Classify two-dimensional figures into categories based on their properties.

3. Understand that attributes belonging to a category of two dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

4. Classify two-dimensional figures in a hierarchy based on properties.


FIFTH GRADE CRITICAL AREAS OF FOCUS STANDARDS AND CLUSTERS BEYOND THE CRITICAL AREAS OF FOCUS Modeling numerical relationships with the coordinate plane Based on previous work with measurement and number lines, students develop understanding of the coordinate plane as a tool to model numerical relationships. These initial understandings provide the foundation for work with negative numbers, and ratios and proportional relationships in Grade Six and functional relationships in further grades.

Operations and Algebraic Thinking 5.OA Analyze patterns and relationships.

3. Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

Geometry 5.G Graph points on the coordinate plane to solv e real-world and mathematical problems.

1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y- axis and y-coordinate).

2. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Performance Level Descriptors – Grade 5 Mathematics

July 2013 Page 1 of 20

Grade 5 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with connections to the Standards

for Mathematical Practice.

Level 5: Distinguished Command

Level 4: Strong Command Level 3: Moderate

Command Level 2: Partial

Command

Addition and Subtraction Operations with Decimals 5.NBT.7-1 5.NBT.7-2

Adds or subtracts two decimals to hundredths using concrete models, drawings or strategies based on place value, properties of operations and/or the relationship between addition and subtraction. Applies this concept to a real-world context, relates the strategy to a written method and explains the reasoning used.

Adds or subtracts two decimals to hundredths using concrete models, drawings or strategies based on place value, properties of operations and/or the relationship between addition and subtraction. Relates the strategy to a written method and explain the reasoning used.

Adds or subtracts two decimals to hundredths using concrete models, drawings or strategies based on place value, properties of operations and/or the relationship between addition and subtraction.

Adds or subtracts (without regrouping) two decimals to hundredths using concrete models, drawings or strategies based on place value and/or the relationship between addition and subtraction.

Adding and Subtracting in Context with Fractions 5.NF.2-1 5.NF.2-2

Creates word problems involving addition and subtraction of fractions, referring to the same whole in cases of unlike denominators by using visual fraction models and equations. Assesses and justifies reasonableness using benchmark fractions and number sense of fractions.

Solves word problems involving addition and subtraction of fractions, referring to the same whole in cases of unlike denominators by using visual fraction models or equations. Assesses reasonableness using benchmark fractions and number sense of fractions.

Solves word problems involving addition and subtraction of fractions, referring to the same whole in cases of unlike denominators by using visual fraction models or equations.

Solves word problems involving addition and subtraction of fractions using benchmark fractions with unlike denominators, referring to the same whole by using visual fraction models or equations.








Command

Fractions with Unlike Denominators 5.NF.1-1 5.NF.1-2 5.NF.1-3 5.NF.1-4 5.NF.1-5

Adds and subtracts more than three fractions and mixed numbers with unlike denominators in such a way as to produce an equivalent sum or difference with like denominators.

Adds and subtracts up to three fractions and adds and subtracts two mixed numbers with unlike denominators in such a way as to produce an equivalent sum or difference with like denominators.

Adds and subtracts two fractions or mixed numbers with unlike denominators in such a way as to produce an equivalent sum or difference with like denominators.

Adds or subtracts two fractions or mixed numbers with unlike denominators using only fractions with denominators of 2,4, 5 or 10 in such a way as to produce an equivalent sum or difference with like denominators.* *below grade level.

Multiplication and Division Operations with Decimals 5.NBT.7-3 5.NBT.7-4 5.NBT.Int.1

Multiplies tenths by tenths or tenths by hundredths and divides in problems involving tenths and/or hundredths using strategies based on place value, properties of operations and/or the relationship between addition and subtraction. Performs exact and approximate multiplications and divisions by mentally applying place value strategies when

Multiplies tenths by tenths or tenths by hundredths and divides in problems involving tenths and/or hundredths using concrete models or drawings and strategies based on place value, properties of operations and/or the relationship between addition and subtraction. Performs exact and approximate multiplications and divisions by mentally

Multiplies tenths by tenths and divides in problems involving tenths using concrete models or drawings and strategies based on place value, properties of operations and/or the relationship between addition and subtraction. Relates the strategy to a written method.

Multiplies tenths by tenths and divides in problems involving tenths using concrete models or drawings and strategies based on place value, properties of operations and/or the relationship between addition and subtraction.








Command appropriate. Relates the strategy to a written method. Applies this concept in the context of metric measurement (e.g., find the area of a rectangle with length=0.7cm and width=0.4cm.)

applying place value strategies when appropriate. Relates the strategy to a written method.

Multiply with Whole Numbers 5.NBT.5-1 5.Int.1 5.Int.2 5.NBT.Int.1

Solves multi-step unscaffolded word problems involving multiplication and multiplies three-digit by two-digit whole numbers using the standard algorithm. Performs exact and approximate multiplications and divisions by mentally applying place value strategies when appropriate.

Solves two-step unscaffolded word problems involving multiplication and multiplies three-digit by two-digit whole numbers using the standard algorithm. Performs exact and approximate multiplications and divisions by mentally applying place value strategies when appropriate.

Solves two-step scaffolded word problems involving multiplication of a three-digit by a one-digit whole number.

Solves one-step word problems involving multiplication.








Command

Quotients and Dividends 5.NBT.6 5.NBT.Int.1

Divides whole numbers up to four-digit dividends and two-digit divisors using strategies based on place value, the properties of operations and/or the relationship between multiplication and division. Performs exact and approximate multiplications and divisions by mentally applying place value strategies when appropriate. Illustrates and explains the calculations by using equations, rectangular arrays, and area models. Identifies correspondences between different approaches. Checks reasonableness of answers by using multiplication or estimation.

Divides whole numbers up to four-digit dividends and two-digit divisors using strategies based on place value, the properties of operations and/or the relationship between multiplication and division. Performs exact and approximate multiplications and divisions by mentally applying place value strategies when appropriate. Illustrates and explains the calculations by using equations, rectangular arrays, and area models. Checks reasonableness of answers by using multiplication or estimation.

Divides whole numbers up to four-digit dividends and one-digit divisors which are multiples of ten using strategies based on place value, the properties of operations and/or the relationship between multiplication and division.

Divides whole numbers up to three-digit dividends and one-digit divisors which are multiples of ten using strategies based on place value, the properties of operations and/or the relationship between multiplication and division.








Command

Multiplying and Dividing with Fractions 5.NF.4a-1 5.NF.4a-2 5.NF.4b-1 5.NF.6-1 5.NF.6-2 5.NF.7a 5.NF.7b 5.NF.7c

Creates real-world problems, by multiplying a mixed number by a fraction, a fraction by a fraction, and a whole number by a fraction; dividing a fraction by a whole number and a whole number by a fraction and creating context for the mathematics and equations.

Solves real-world problems, by multiplying a mixed number by a fraction, a fraction by a fraction and a whole number by a fraction; dividing a fraction by a whole number and a whole number by a fraction using visual fraction models and creating context for the mathematics, including rectangular areas; and interpreting the product and/or quotient.

Multiplies a fraction or a whole number by a fraction and divides a fraction by a whole number – or whole number by a fraction – using visual fraction models and creating context for the mathematics, including rectangular areas.

Multiplies a fraction or a whole number by a fraction and divide a fraction by a whole number or whole number by a fraction using visual fraction models.

Interpreting Fractions 5.NF.3-1 5.NF.3-2

Solves word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers. Interprets the fraction as division of the numerator by the denominator. Creates a model representing the situation.

Solves word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers. Interprets the fraction as division of the numerator by the denominator. Identifies a simple model representing the situation.

Solves word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers. Interprets the fraction as division of the numerator by the denominator.

Solves word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers by using manipulatives or visual models to identify between which two whole numbers the answer lies.








Command

Recognizing Volume 5.MD.3 5.MD.4

Recognizes volume as an attribute of solid figures and understands volume is measured using cubic units and can be found by packing a solid figure with unit cubes and counting them. Represents the volume of a solid figure as “n” cubic units. Creates an equation that illustrates the unit cube pattern.

Recognizes volume as an attribute of solid figures and understands volume is measured using cubic units and can be found by packing a solid figure with unit cubes and counting them. Represents the volume of a solid figure as “n” cubic units.

Recognizes volume as an attribute of solid figures and understands volume is measured using cubic units and can be found by packing a solid figure with unit cubes and counting them.

Recognizes volume as an attribute of solid figures and with a visual model understands that volume is measured using cubic units and can be found by packing a solid figure with unit cubes and counting them.

Finding Volume 5.MD.5b 5.MD.5c

Applies the formulas for volume, relates volume to the operations of multiplication and addition, and recognizes volume is additive by finding the volume of solid figures of two or more non-overlapping parts. Creates real-world and mathematical problems that

Solves real-world and mathematical problems by applying the formulas for volume, relating volume to the operations of multiplication and addition, and recognizing volume is additive by finding the volume of solid figures of two non-overlapping parts.

Given a visual model, solves real-world and mathematical problems by applying the formulas for volume, relating volume to the operations of multiplication and addition, and recognizing volume is additive by finding the volume of solid figures of two non-overlapping parts.

Given a visual model and the formulas for finding volume, solves real-world and mathematical problems by applying the formulas for volume (V = l x w x h and V = B x h).








Command would be solved by finding volume.

Read, Write and Compare Decimals 5.NBT.3a 5.NBT.3b 5.NBT.4 5.NBT.Int.1

Reads, writes and compares decimals to any place using numerals and symbols and rounds to any place and chooses appropriate context given a rounded number. Performs exact and approximate multiplications and divisions by mentally applying place value strategies when appropriate.

Reads, writes and compares decimals to the thousandths using numerals, number names, expanded form and symbols (>, <, =) and rounds to any place. Performs exact and approximate multiplications and divisions by mentally applying place value strategies when appropriate.

Reads, writes and compares decimals to the hundredths using numerals, number names, expanded form and symbols (>, <, =), and rounds to any place

Reads, writes and compares decimals to the tenths using numerals, number names, expanded form and symbols (>, <, =), and rounds to any place with scaffolding.

Place Value 5.NBT.1 5.NBT.2-2 5.NBT.A.Int.1

In any multi-digit number, recognizes a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left, uses whole number exponents to denote powers of 10 and

In any multi-digit number, recognizes a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left and uses whole number exponents to denote powers of 10.

In any multi-digit number, recognizes a digit in one place represents 10 times as much as it represents in the place to its right or 1/10 of what it represents in the place to its left and uses whole number exponents to denote powers of 10.

In any multi-digit number, recognizes a digit in one place represents 10 times as much as it represents in the place to its right or 1/10 of what it represents in the place to its left by using manipulatives or visual models.








Command uses symbols to compare two powers of 10 expressed exponentially (compare 102 to 105).

Multiplication Scaling 5.NF.5a

Interprets multiplication scaling by comparing the size of the product to the size of one factor on the basis of the size of the second factor without performing the indicated multiplication with two fractions.

Interprets multiplication scaling by comparing the size of the product to the size of one factor on the basis of the size of the second factor without performing the indicated multiplication, focusing on one factor being a fraction greater than or less than one.

Interprets multiplication scaling by comparing the size of a product to the size of one factor on the basis of the size of the second factor by performing the indicated multiplication where one factor is a fraction less than one.

Interprets multiplication scaling by comparing the size of a product to the size of one factor on the basis of the size of the second factor by performing the indicated multiplication where one factor is a fraction less than one using manipulatives or visual models.

Write and Interpret Numerical Expressions 5.OA.1 5.OA.2-1 5.OA.2-2

Uses parentheses, brackets, or braces with no greater depth than two, to write, evaluate and create numerical expressions. Interprets numerical expressions without evaluating them.

Uses parentheses, brackets, or braces with no greater depth than two, to write and evaluate numerical expressions. Interprets numerical expressions without evaluating them.

Uses parentheses, brackets, or braces to write numerical expressions. Interprets simple numerical expressions without evaluating them.

Uses parentheses, brackets, or braces to write simple numerical expressions.



Grade 5 Math: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with

connections to the Standards for Mathematical Practice.



Command Level 2: Partial Command

Graphing on the Coordinate Plane 5.G.1 5.G.2 5.OA.3

Creates real-world and mathematical problems which require locating and graphing points in the first quadrant of a coordinate plane and interprets coordinate values of points in the context of the situation.

Represents real-world and mathematical problems by locating and graphing points in the first quadrant of a coordinate plane and interprets coordinate values of points in the context of the situation.

Represents real-world and mathematical problems by locating and graphing points in the first quadrant of a coordinate plane.

Represents real-world and mathematical problems by locating or graphing points in the first quadrant of a coordinate plane.

Two-Dimensional Figures 5.G.3 5.G.4

Classifies two-dimensional figures in a hierarchy based on properties. Demonstrates that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. Uses appropriate tools to determine similarities and differences between categories and subcategories.

Classifies two-dimensional figures in a hierarchy based on properties. Understands that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.

Classifies two-dimensional figures in a hierarchy based on properties. Understands that shared attributes categorize two-dimensional figures.

Classifies two-dimensional figures based on properties.



Grade 5 Math: Sub-Claim B The student solves problems involving the Additional and Supporting Content for the grade/course with

connections to the Standards for Mathematical Practice.




Conversions 5.MD.1-1 5.MD.1-2

Converts among different-sized standard measurement units within a given measurement system and uses these conversions to create real-world, multi-step problems. Chooses the appropriate measurement unit based on the given context.

Converts among different-sized standard measurement units within a given measurement system and uses these conversions to solve real-world, multi-step problems.

Converts among different-sized standard measurement units within a given measurement system and uses these conversions to solve real-world, single-step problems.

Converts among different-sized standard measurement units within a given measurement system and solves single-step problems by using manipulatives or visual models.

Data Displays 5.MD.2-1 5.MD.2-2

Makes a line plot to display a data set of measurements in fractions of a unit with denominators limited to 2, 4 and 8, uses operations on fractions to solve problems involving information in line plots and interprets the solution in relation to the data.

Makes a line plot to display a data set of measurements in fractions of a unit with denominators limited to 2, 4 and 8, and uses operations on fractions to solve problems involving information in line plots.

Makes a line plot to display a data set of measurements in fractions of a unit with denominators limited to 2 and 4, and uses operations on fractions with denominators of 2 and 4 to solve problems involving information in line plots.

Makes a line plot to display a data set of measurements in fractions of a unit with like denominators of 2 or 4, and uses operations on fractions with like denominators of 2 or 4 to solve problems involving information in line plots.



Grade 5 Math: Sub-Claim C The student expresses grade/course-level appropriate mathematical reasoning by constructing viable

arguments, critiquing the reasoning of others and/or attending to precision when making mathematical statements.




Properties of Operations 5.C.1-1 5.C.1-2 5.C.1-3 5.C.2-1 5.C.2-2 5.C.2-3 5.C.2-4

Constructs and communicates a well-organized and complete written response based on explanations/reasoning using the:

properties of operations

relationship between addition and subtraction

relationship between multiplication and division

Response may include:

a logical/defensible approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)

an efficient and logical progression of steps with appropriate

Constructs and communicates a well-organized and complete written response based on explanations/reasoning using the:





a logical/defensible approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)

a logical progression of steps

precision of calculation

Constructs and communicates a complete written response based on explanations/reasoning using the:





a logical approach based on a conjecture and/or stated assumptions

a logical, but incomplete, progression of steps

minor calculation errors

some use of grade-level vocabulary, symbols

Constructs and communicates an incomplete written response based on explanations/reasoning using the:





an approach based on a conjecture and/or stated or faulty assumptions

an incomplete or illogical progression of steps

an intrusive calculation error

limited use of grade-








justification


correct use of grade-level vocabulary, symbols and labels

justification of a conclusion

evaluation of whether an argument or conclusion is generalizable

evaluating, interpreting and critiquing the validity of other’s responses, reasonings, and approaches, utilizing mathematical connections (when appropriate). Provides a counter-example where applicable.




evaluating, interpreting and critiquing the validity of other’s responses, reasonings, and approaches, utilizing mathematical connections (when appropriate).

and labels

partial justification of a conclusion based on own calculations

evaluating the validity of other’s responses, approaches and conclusions.

level vocabulary, symbols and labels









Place Value 5.C.3

Clearly constructs and communicates a well-organized and complete response based on place value system including:

a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)

an efficient and logical progression of steps with appropriate justification





Clearly constructs and communicates a well-organized and complete response based on place value system including:

a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)






evaluating, interpreting and critiquing the

Constructs and communicates a complete response based on place value system including:




some use of grade-level vocabulary, symbols and labels



Constructs and communicates an incomplete response based on place value system which may include:

an approach based on a conjecture and/or stated or faulty assumptions



limited use of grade-level vocabulary, symbols and labels









evaluating, interpreting and critiquing the validity of other’s responses, approaches and reasoning, and providing a counter-example where applicable.

validity of other’s responses, approaches and reasoning.

Concrete Referents and Diagrams 5.C.4-1 5.C.4-2 5.C.4-3 5.C.4-4 5.C.5-1 5.C.5-2 5.C.5-3 5.C.6

Clearly constructs and communicates a well-organized and complete response based on operations using concrete referents such as diagrams – including number lines (whether provided in the prompt or constructed by the student) and connecting the diagrams to a written (symbolic) method, which may include:

a logical approach based on a conjecture and/or stated assumptions, utilizing

Clearly constructs and communicates a well-organized and complete response based on operations using concrete referents such as diagrams – including number lines (whether provided in the prompt or constructed by the student) and connecting the diagrams to a written (symbolic) method, which may include:

a logical approach based on a conjecture and/or stated assumptions, utilizing

Constructs and communicates a complete response based on operations using concrete referents such as diagrams – including number lines (provided in the prompt) – connecting the diagrams to a written (symbolic) method, which may include:


Constructs and communicates an incomplete response based on operations using concrete referents such as diagrams – including number lines (provided in the prompt) – connecting the diagrams to a written (symbolic) method, which may include:

a conjecture and/or stated or faulty assumptions








mathematical connections (when appropriate)






evaluating, interpreting, and critiquing the validity of other’s responses, approaches, and reasoning, and providing a counter-example where applicable.

mathematical connections (when appropriate)






evaluating, interpreting, and critiquing the validity of other’s responses, approaches, and reasoning.




partial justification of a conclusion based on own calculations.

evaluating the validity of other’s responses, approaches and conclusions





accepting the validity of other’s responses








Distinguish Correct Explanation/ Reasoning from that which is Flawed 5.C.7-1 5.C.7-2 5.C.7-3 5.C.8-1 5.C.8-2 5.C.9

Clearly constructs and communicates a well-organized and complete response by:

analyzing and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately

evaluating explanation/reasoning if there is a flaw in the argument

presenting and defending corrected reasoning


a logical approach based on a conjecture

Clearly constructs and communicates a well-organized and complete response by:

analyzing and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately

distinguishing correct explanation/reasoning from that which is flawed

identifying and describing the flaw in reasoning or describing errors in solutions to multi-step problems

presenting corrected reasoning


a logical approach

Constructs and communicates a complete response by:

analyzing solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately


identifying and describing the flaw in reasoning or describing errors in solutions to multi-step problems

presenting corrected reasoning


a logical approach

Constructs and communicates an incomplete response by:

analyzing solutions to scaffolded two-step problems in the form of valid chains of reasoning, sometimes using symbols such as equal signs appropriately


identifying an error in reasoning


a conjecture based on








and/or stated assumptions, utilizing mathematical connections (when appropriate)






evaluating, interpreting and critiquing the validity of other’s responses, approaches and reasoning, and providing a counter-example where applicable

based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)






evaluating, interpreting and critiquing the validity of other’s responses, approaches and reasoning

based on a conjecture and/or stated assumptions






faulty assumptions





accepting the validity of other’s responses



Grade 5 Math: Sub-Claim D The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the current grade/course (or for more complex problems,

knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure

and/or looking for and expressing regularity in repeated reasoning.




Modeling 5.D.1 5.D.2

Devises a plan and applies mathematics to solve multi-step, real-world contextual word problems by:

using stated assumptions or making assumptions and using approximations to simplify a real-world situation

analyzing and/or creating constraints, relationships and goals

mapping relationships between important quantities by selecting appropriate tools to create models

analyzing relationships mathematically between important quantities to draw conclusions


using stated assumptions or making assumptions and using approximations to simplify a real-world situation

mapping relationships between important quantities by selecting appropriate tools to create models


interpreting mathematical results in the context of the


using stated assumptions and approximations to simplify a real-world situation

illustrating relationships between important quantities by using provided tools to create models


interpreting mathematical results in a simplified context


using stated assumptions and approximations to simplify a real-world situation

identifying important quantities

using provided tools to create models

analyzing relationships mathematically to draw conclusions

writing an arithmetic expression or equation to describe a situation



Grade 5 Math: Sub-Claim D The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the current grade/course (or for more complex problems,

knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure

and/or looking for and expressing regularity in repeated reasoning.




justifying and defending models which lead to a conclusion

interpreting mathematical results in the context of the situation

reflecting on whether the results make sense

improving the model if it has not served its purpose

writing a concise arithmetic expression or equation to describe a situation

situation


modifying and/or improving the model if it has not served its purpose



modifying the model if it has not served its purpose




Grade 5 Math: Sub-Claim E The student demonstrates fluency in areas set forth in the Standards for Content in grades 3-6.




Fluency 5.NBT.5

Accurately and quickly multiplies whole numbers and decimals to hundredths using the standard algorithm and assesses reasonableness of the product. Knows from memory 100 percent of the products on items in less than the allotted time on items which are timed.

Accurately and in a timely manner multiplies multi-digit whole numbers using the standard algorithm. Knows from memory 100 percent of the products on items in the allotted time on items which are timed

Accurately multiplies multi-digit whole numbers using the standard algorithm. Knows from memory more than 80 percent and less than 100 percent of the multiplication and division facts within 100 on items which are timed.

Multiplies multi-digit whole numbers using the standard algorithm with some level of accuracy. Knows from memory greater than or equal to 70 percent and less than or equal to 80 percent of the multiplication and division facts within 100 on items which are timed.

Latest Revision 6/18/2013 I Can Statements are in draft form due to the iterative nature of the item development process. 1

Bailey Education Group, LLC

Common Core State Standard I Can Statements 5th Grade Mathematics 6/18/2013

CCSS Key: PLD Key: Operations and Algebraic Thinking (OA) Partial Command Number and Operations in Base Ten (NBT) Moderate Command Numbers and Operations–Fractions (NF) Distinguished Command Measurement and Data (MD) Geometry (G) Common Core State Standards for

Mathematics (Outcome Based) “I Can” Statements

Operations and Algebraic Thinking (OA) 5.OA.1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

I Can: 5.OA.1.1 Use the order of operations to evaluate numerical

expressions. 5.OA.1.2 Explain the order of operations. 5.OA.1.3 Apply the order of operations to evaluate

expressions. *5.OA.1 is part of an Additional Cluster and will only be assessed on the EOY Assessment. This domain is not explicitly addressed in the Performance Level Descriptors.

5.OA.2. Write simple expressions that record calculations with numbers and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

I Can: 5.OA.2.1 Interpret a numerical expression into words (without

evaluating the expression). 5.OA.2.2 Write simple expressions that record calculations

with numbers *5.OA.2 is part of an Additional Cluster and will only be assessed on the EOY Assessment. This domain is not explicitly addressed in the Performance Level Descriptors.

5.OA.3. Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

I Can: 5.OA.3.1 Create two numerical patterns from two given rules. 5.OA.3.2 Identify relationships between corresponding terms

in a pattern. 5.OA.3.3 Extend a numerical pattern from a given rule. 5.OA.3.4 Determine a rule from a given numerical pattern. 5.OA.3.5 Create ordered pairs of the corresponding terms

from two patterns. 5.OA.3.6 Plot an ordered pair on a coordinate plane from 2

patterns and justify. *5.OA.3 is part of an Additional Cluster and will only be assessed on the EOY Assessment. This domain is not explicitly addressed in the Performance Level Descriptors.


Common Core State Standards for Mathematics (Outcome Based) “I Can” Statements

Numbers and Operations–Fractions (NF)

5.NF.1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

I can: 5.NF.1.1 Rewrite fractions as equivalent fractions. 5.NF.1.2 Write my answer in simplest form. 5.NF.1.3 Add or subtract two fractions with unlike denominators (2, 4, 5, or 10). 5.NF.1.4 Add or subtract two mixed numbers with unlike denominators (2, 4, 5, or 10). 5.NF.1.5 Add and subtract two fractions with unlike denominators (any denominator). 5.NF.1.6 Add and subtract two mixed numbers with unlike denominators (any denominator). 5.NF.1.7 Add and subtract any number of fractions with unlike denominators (any denominator). 5.NF.1.8 Add and subtract any number of mixed numbers with unlike denominators (any denominator). 5.NF.1.9 Add and subtract fractions within the same expression.

5.NF.2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

I can: 5.NF.2.1 Use benchmark numbers (0, ¼, ½, ¾, 1) to estimate

sums and differences of fractions. 5.NF.2.2 Relate estimation to my answers to see if they make

sense. 5.NF.2.3 5.NF.2.4 5.NF.2.5

Create a visual fraction model to represent the fractions in a word problem. Create an equation to represent a word problem. Create a word problem involving addition and subtraction of fractions.

5.NF.3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie.

I can: 5.NF.3.1 Interpret a fraction as a division problem. Ex. ¼ = 1

÷ 4 5.NF.3.2 Interpret a division problem as a fraction. Ex. 1 ÷ 4 =

¼ 5.NF.3.3 5.NF.3.4

Solve division word problems and express the quotient as a fraction or mixed number by using visual fraction models. Solve division word problems and express the quotient as a fraction or mixed number by using equations.



5.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

I can: 5.NF.4a.1 Represent a whole number as a fraction. 5.NF.4a.2 5.NF.4a.3

Multiply a fraction by a fraction. Multiply a fraction by a whole number.

5.NF.4a.4 5.NF.4a.5

Use a visual fraction model to represent multiplication of fractions. Create a context for a problem involving multiplication of fractions.

5.NF.4b.1 5.NF.4b.2 5.NF.4b.3

Find the area of a rectangle with fractional side lengths by tiling it with unit squares. Relate different strategies for calculating the area of a rectangle. (tiling vs. formula) Multiply fractional side lengths to find areas of rectangles.

5.NF.4b.4 Apply my understanding of the area of rectangles to include fractional units.

5.NF.5. Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

I can: 5.NF.5a.1 5.NF.5a.2 5.NF.5a.3 5.NF.5a.4

Use visual models or manipulatives to interpret multiplication scaling and correctly perform the indicated multiplication. Interpret multiplication scaling by performing the indicated multiplication where one factor is a fraction. Interpret multiplication scaling without performing the indicated multiplication where one factor is a mixed number. Interpret multiplication scaling without performing the indicated multiplication where both factors are fractions.

5.NF.5b.1 Predict the size of the product based on the size of the factors. Ex: fraction x fraction = smaller fraction, fraction x whole number = a fraction of the whole number.

5.NF.5b.2 5.NF.5b.3 5.NF.5b.4

Use visual models or manipulatives to explain when multiplying by a fraction greater than one, the number increases and when multiplying by a number less than one, the number decreases. Explain when multiplying by a fraction greater than one, the number increases and when multiplying by a number less than one, the number decreases. Explain that when multiplying the numerator and denominator by the same number is the same as multiplying by one.



5.NF.6. Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

I Can: 5.NF.6.1 Solve real-world problems involving multiplication of

fractions and mixed numbers.

5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.1

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4. c. Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

I Can: 5.NF.7 Understand the relationship between multiplication

and division.

5.NF.7a.1 Use a visual fraction model to divide a unit fraction by a whole number.

5.NF.7a.2 Create a context for a problem involving division of a unit fraction by a whole number.

5.NF.7b.1 Use a visual fraction model to divide a whole number by a unit fraction.

5.NF.7b.2 Create a context for a problem involving division of a whole number by a unit fraction.

5.NF.7c.1 Solve real-world problems involving division of unit fractions and whole numbers.

Number and Operations in Base Ten (NBT) 5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

I Can: 5.NBT.1.1 Represent place values of whole numbers through

100,000,000 and decimals to the thousandths with manipulatives or visual models.

5.NBT.1.2 Recognize that in a multi-digit number, the digit to the left is 10x larger and the right is 1/10 smaller.



5.NBT.2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

I Can: 5.NBT.2.1 Show repeated multiplication of tens as an

exponent. 5.NBT.2.2 Use manipulatives to explain patterns in the

number of zeros of the product when multiplying a number by powers of 10.

5.NBT.2.3 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10.

5.NBT.2.4 Use manipulatives to explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.

5.NBT.2.5 Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.

5.NBT.3. Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000). b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

I Can: 5.NBT.3a.1 Read and write decimals to the tenths place using

numerals, number names, and expanded form. 5.NBT.3a.2 Read and write decimals to the hundredth place

using numerals, number names, and expanded form.

5.NBT.3a.3 Read and write decimals to any place using numerals, number names, and expanded form.

5.NBT.3b.1 Compare decimals to the tenths place using inequality symbols (<,>,=).

5.NBT.3b.2 Compare decimals to the hundredths place using inequality symbols (<,>,=).

5.NBT.3b.3 Compare decimals to any place using inequality symbols (<,>,=).

5.NBT.4. Use place value understanding to round decimals to any place.

I Can: 5.NBT.4.1 Use place value understanding to round decimals

to any place.

5.NBT.5. Fluently multiply multi-digit whole numbers using the standard algorithm.

I Can: 5.NBT.5.1 Fluently multiply multi-digit whole numbers.

5.NBT.6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

I Can: 5.NBT.6.1 5.NBT.6.2






5.NBT.7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

I Can: 5.NBT.7.1 Add or subtract two decimal numbers to the

hundredths without regrouping. 5.NBT.7.2 Add and subtract two decimal numbers to

hundredths with regrouping. 5.NBT.7.3 Multiply tenths by tenths. 5.NBT.7.4 Multiply tenths by hundredths. 5.NBT.7.5 Divide decimals with tenths. 5.NBT.7.6 Divide decimals with tenths and/or hundredths. 5.NBT.7.7 Relate the strategy used to a written method and

explain the reasoning used. 5.NBT.7.8 Demonstrate computations by using models and

drawings.

Measurement and Data (MD) 5.MD.1. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real-world problems.

I Can: 5.MD.1.1 Recall customary units of measurements. 5.MD.1.2 Recall metric units of measurements. 5.MD.1.3 Use manipulatives or visual models to convert

different measurement units (customary & metric) within a given measurement system to solve single step problems.

5.MD.1.4 Convert different measurement units (customary & metric) within a given measurement system to solve real world single step problems.

5.MD.1.5 Convert different measurement units (customary & metric) within a given measurement system to solve and create real world multi-step problems.

5.MD.1.6 Choose the appropriate measurement unit based on the given context.

5.MD.2. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

I Can: 5.MD.2.1 Create a line plot to display a data set of

measurements in fraction form with denominators of 2 and 4.

5.MD.2.2 Create a line plot to display a data set of measurements in fraction form with denominators of 2, 4 and 8.

5.MD.2.3 Use operations on fractions to solve problems involving information presented in line plots.

5.MD.2.4 Interpret the solution in relation to the data.



5.MD.3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume. b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

I Can: 5.MD.3a.1 Understand that unit cubes are used to measure

volume of solid figures.

5.MD.3b.1 Understand that unit cubes cannot have gaps or overlap.

5.MD.4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

I Can: 5.MD.4.1 Use a visual model to measure volume by counting

unit cubes. 5.MD.4.2 Measure volume by counting unit cubes. 5.MD.4.3 Represent the volume of a solid figure as “n” cubic

units. (cm3, in3, ft3, units3).

5.MD.5. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole- number edge lengths in the context of solving real world and mathematical problems. c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by

I Can: 5.MD.5a.1 Find the volume of a right rectangular prism with

whole-number side lengths by packing it with unit cubes.

5.MD.5a.2 Relate volume to the operation of multiplication and addition.

5.MD.5a.3 Measure volume with unit cubes and show that it is the same as: multiplying the side lengths, multiplying the edge lengths, and multiplying the height by the area of the base.

5.MD.5b.1 Apply the formulas (V= l x w x h) & (V = B x h) to solve real world and mathematical problems.

5.MD.5b.2 Create real-world and mathematical problems that would be solved by finding volume.

5.MD.5c.1 Recognize volume is additive by finding the volume of solid figures of two non-overlapping parts.



adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

5.MD.5c.2 Recognize volume is additive by finding the volume of solid figures of two or more non-overlapping parts.

Geometry (G) 5.G.1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).

I Can: 5.G.1.1 Label the axes, quadrants and origin on the

coordinate plane. 5.G.1.2 Identify ordered pairs. 5.G.1.3 Plot points on the coordinate plane.

5.G.2. Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

I Can: 5.G.2.1 Represent real-world and mathematical problems by

locating or graphing points in the first quadrant of a coordinate plane.

5.G.2.2 Represent real-world and mathematical problems by locating and graphing points in the first quadrant of a coordinate plane.

5.G.2.3 Interpret coordinate values in the context of the situation.

5.G.2.4 Create real-world and mathematical problems that require locating and graphing points in the first quadrant of the coordinate plane.

5.G.3. Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

I Can: 5.G.3.1 Understand that shared attributes categorize two-

dimensional figures.

5.G.4. Classify two-dimensional figures in a hierarchy based on properties.

I Can: 5.G.4.1 Classify two-dimensional figures based on

properties. 5.G.4.2 Classify two-dimensional figures in a hierarchy

based on properties. 5.G.4.3 Demonstrate that attributes belonging to a category

of two-dimensional also belong to all subcategories of that category.



5.G.4.4 Use appropriate tools to determine similarities and differences between categories and subcategories.

Common Core “Shifts” in Mathematics There are six shifts in Mathematics that the Common Core requires of us if we are to be truly

aligned with it in terms of curricular materials and classroom instruction. Shift 1 - Focus Teachers use the power of the eraser and significantly narrow and deepen the scope of how time and energy is spent in the math classroom. They do so in order to focus deeply on only the concepts that are prioritized in the standards so that students reach strong foundational knowledge and deep conceptual understanding and are able to transfer mathematical skills and understanding across concepts and grades. Shift 2 - Coherence Principals and teachers carefully connect the learning within and across grades so that, for example, fractions or multiplication spiral across grade levels and students can build new understanding onto foundations built in previous years. Teachers can begin to count on deep conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning. Shift 3 - Fluency Students are expected to have speed and accuracy with simple calculations; teachers structure class time and/or homework time for students to memorize, through repetition, core functions (found in the attached list of fluencies) such as multiplication tables so that they are more able to understand and manipulate more complex concepts. Shift 4 - Deep Understanding Teachers teach more than “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of core math concepts by applying them to new situations, as well as writing and speaking about their understanding. Shift 5 – Application Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so. Teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations. Teachers in content areas outside of math, particularly science, ensure that students are using math – at all grade levels – to make meaning of and access content. Shift 6 - Dual Intensity Students are practicing and understanding. There is more than a balance between these two things in the classroom – both are occurring with intensity. Teachers create opportunities for students to participate in “drills” and make use of those skills through extended application of math concepts. The amount of time and energy spent practicing and understanding learning environments is driven by the specific mathematical concept and therefore, varies throughout the given school year.

Standards for Mathematical Practice

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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).

The Standards: 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. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

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,

Standards for Mathematical Practice

2

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 x 8 equals the well-remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 x 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.

CCSS Standards for Mathematical Practice

Questions for Teachers to Ask 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

Teachers ask: • What is this problem asking? • How would you describe the problem in

your own words? • Could you try this with simpler numbers?

Fewer numbers? • How could you start this problem? • Would it help to create a diagram? Make

a table? Draw a picture? • How is ___’s way of solving the problem

like/different from yours? • Does your plan make sense? Why or why

not? • What are you having trouble with? • How can you check this?

Teachers ask: • What does the number ____ represent in

the problem? • How can you represent the problem with

symbols and numbers? • Create a representation of the problem.

Teachers ask: • How is your answer different than

_____’s? • What do you think about what _____ said? • Do you agree? Why/why not? • How can you prove that your answer is

correct? • What examples could prove or disprove

your argument? • What do you think about _____’s

argument? • Can you explain what _____ is saying? • Can you explain why his/her strategy

works? • How is your strategy similar to _____? • What questions do you have for ____? • Can you convince the rest of us that your

answer makes sense? *It is important that the teacher poses tasks that involve arguments or critiques

Teachers ask: • Write a number sentence to describe this

situation. • How could we use symbols to represent

what is happening? • What connections do you see? • Why do the results make sense? • Is this working or do you need to change

your model? *It is important that the teacher poses tasks that involve real world situations

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

Teachers ask: • How could you use manipulatives or a

drawing to show your thinking? • How did that tool help you solve the

problem? • If we didn’t have access to that tool, what

other one would you have chosen?

Teachers ask: • What does the word ____ mean? • Explain what you did to solve the problem. • Can you tell me why that is true? • How did you reach your conclusion? • Compare your answer to _____’s answer • What labels could you use? • How do you know your answer is

accurate? • What new words did you use today? How

did you use them?

Teachers ask: • Why does this happen? • How is ____ related to ____? • Why is this important to the problem? • What do you know about ____ that you

can apply to this situation? • How can you use what you know to

explain why this works? • What patterns do you see? *deductive reasoning (moving from general to specific)

Teachers ask: • What generalizations can you make? • Can you find a shortcut to solve the

problem? How would your shortcut make the problem easier?

• How could this problem help you solve another problem?

*inductive reasoning (moving from specific to general)

5 grade math unit guide core/pacing guides... · jackson county school district year at a glance...

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