th
5 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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5 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 M athematics (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.
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Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. S.
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Use appropriate tools strategically. Attend to precision. Look for and make use of structure. 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. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Measurement and Data – 5.MD www.youtube.com C. Geometric measurement: understand concepts of volume and relate volume to Math Fact Fluency Practice multiplication and to addition. www.mathwire.com 3. Recognize volume as an attribute of solid www.oswego.org/ocsd-­‐
figures and understand concepts of volume web/games/ measurement. http://mathfactspro.com/mathflu
encygame.html#/math-­‐facts-­‐
a. A cube with side length 1 unit, called a 5.MD.3a.1 Understand that unit cubes are used addition-­‐games to measure volume of solid figures. "unit cube," is said to have "one cubic http://jerome.northbranfordscho
unit" of volume, and can be used to ols.org/Content/Math_Fact_Flue
measure volume. ncy_Practice_Sheets.asp 5.MD.3b.1 Understand that unit cubes cannot http://www.mathfactcafe.com/ b. A solid figure which can be packed have gaps or overlap. www.factmonster.com without gaps or overlaps using n unit cubes is said to have a volume Lessons/Activities/Games of n cubic units. 5.MD.4.1 Use a visual model to measure Students decompose and https://www.illustrativemathema
volume by counting unit cubes. 4. Measure volumes by counting unit cubes, recompose geometric figures to tics.org/5 5.MD.4.2 Measure volume by counting unit using cubic cm, cubic in, cubic ft, and make sense of the spatial structure cubes. improvised units. https://learnzillion.com/ of volume (MP.7). In particular, Common Core State Standards for students explain their thinking and Mathematical Practice www.AECSD5thGradeMathemati
analyze others' reasoning as they csdoc practice partitioning figures into 3. Construct viable arguments and critique the layers a
nd each layer into rows reasoning of others. http://maccss.ncdpi.wikispaces.n
and each row into c
ubes (MP.3).
7. Look for and make use of structure. et/Fifth+Grade www.dpi.state.nc.us http://harcourtschool.com/searc
h/search.html www.tucerton.k12.nj.us/tes_curri
culum/mathematics_2/curriculu
m-­‐math-­‐grade-­‐five http://www.onlinemathlearning.c
om/common-­‐core-­‐grade5.html http://www.mathgoodies.com/st
andards/alignments/grade5.html http://www.k-­‐
5mathteachingresources.com/5th
-­‐grade-­‐number-­‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.co
m/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/ind
ex.php/common-­‐core-­‐standards http://map.mathshell.org/materi
als/stds.php#standard1159 Vocabulary
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Measurement Unit Unit Cubes Volume Essential Questions
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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 •
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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 Learning Targets Notes/Comments Unit Materials and Resources 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. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Number and Operations in Base Ten - 5.NBT www.youtube.com 5.NBT.5.1 Fluently multiply multi-­‐digit whole B. Perform operations with multi-­‐digit whole In this unit 5.NBT.B.5 and numbers. numbers and with decimals to hundredths. Math Fact Fluency Practice 5.NBT.B.6 will focus on operations 5. Fluently multiply multi-­‐digit whole numbers www.mathwire.com with whole numbers only. using the standard algorithm. www.oswego.org/ocsd-­‐
Operations with decimals will be web/games/ introduced in unit 10. These http://mathfactspro.com/mathflu
6. Find whole-­‐number quotients of whole standards will be finalized in unit 5.NBT.6.1 Determine the quotient of whole encygame.html#/math-­‐facts-­‐
numbers with up to four-­‐digit dividends 15, but should be practiced numbers with up to 3 digit dividends addition-­‐games and 1 digit divisors, which are throughout the year to provide and two-­‐digit divisors, using strategies http://jerome.northbranfordscho
5.NBT.6.2 multiples of ten. opportunities for students to based on place value, the properties of ols.org/Content/Math_Fact_Flue
Determine the quotient of whole develop proficiency with these operations, and/or the relationship ncy_Practice_Sheets.asp numbers with up to 4 digit dividends operations. between multiplication and division. http://www.mathfactcafe.com/ and 1 digit divisors, which are Illustrate and explain the calculation by www.factmonster.com multiples of ten. using equations, rectangular 5.NBT.6.3 Determine the quotient of whole arrays, and/or area models. Lessons/Activities/Games numbers with up to 4 digit dividends https://www.illustrativemathema
and 2 digit divisors. tics.org/5 5.NBT.6.4 Illustrate and explain division using equations, rectangular arrays, and/or https://learnzillion.com area models. Students look for regularity in their Common Core State Standards for www.AECSD5thGradeMathemati
work with multiplication and Mathematical Practice division use their understanding of csdoc http://maccss.ncdpi.wikispaces.n
1. Make sense of problem and persevere in the structure (MP.8) to make et/Fifth+Grade solving them. sense of their solutions and 8. Look for and express regularity in repeated understand the approaches of www.dpi.state.nc.us reasoning. other students (MP.1). Vocabulary
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Algorithims Dividend Equal groups Factor Multiples Multiplication Partial products Product Multiplication symbols Essential Questions
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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 •
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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 Learning Targets Notes/Comments Unit Materials and Resources 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. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Number and Operations-­‐Fractions - 5.NF 5.NF.1.1 Rewrite fractions as equivalent fractions. www.youtube.com In this unit, 5.NF.A.1 involves A. Use equivalent fractions as a strategy to add 5.NF.1.2 Write my answer in simplest form. students using the same method and subtract fractions. 5.NF.1.3 Add or subtract two fractions with unlike Math Fact Fluency Practice from Grade 4 to generate 1. Add and subtract fractions with unlike denominators (2, 4, 5, or 10). www.mathwire.com equivalent fractions (4.NF.A.1). In denominators (including mixed numbers) by 5.NF.1.4 Add or subtract two mixed numbers with www.oswego.org/ocsd-­‐
unit 7 students will extend this replacing given fractions with equivalent unlike denominators (2, 4, 5, or 10). web/games/ understanding of equivalency to 5.NF.1.5 Add and subtract two fractions with unlike http://mathfactspro.com/mathflu
fractions in such a way as to produce an understand that multiplying by a denominators (any denominator). encygame.html#/math-­‐facts-­‐
equivalent sum or difference of fractions fraction equivalent to 1 (e.g. 4/4) 5.NF.1.6 Add and subtract two mixed numbers with addition-­‐games with like denominators. For example, 2/3 + will result in an equivalent fraction unlike denominators (any denominator). http://jerome.northbranfordscho
5/4 = 8/12 + 15/12 = 23/12. (In general, a/b 4
(5.NF.B.5b). 5.NF.1.7 Add and subtract any number of fractions ols.org/Content/Math_Fact_Flue
+ c/d = (ad + bc)/bd.) with u
nlike d
enominators (
any ncy_Practice_Sheets.asp denominator). http://www.mathfactcafe.com/ 5.NF.1.8 Add and subtract any number of mixed www.factmonster.com numbers with unlike denominators (any denominator). Lessons/Activities/Games 5.NF.1.9 Add and subtract fractions within the same https://www.illustrativemathema
expression tics.org/5 https://learnzillion.com www.AECSD5thGradeMathemati
csdoc http://maccss.ncdpi.wikispaces.n
et/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. Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively. 4. Model with mathematics. 5.NF.2.1 5.NF.2.2 5.NF.2.3 5.NF.2.4 5.NF.2.5 Use benchmark numbers (0, ¼, ½, ¾, 1) to estimate sums and differences of fractions. Relate estimation to my answers to see if they make sense. 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/searc
h/search.html www.tucerton.k12.nj.us/tes_curri
culum/mathematics_2/curriculu
m-­‐math-­‐grade-­‐five http://www.onlinemathlearning.c
om/common-­‐core-­‐grade5.html http://www.mathgoodies.com/st
andards/alignments/grade5.html http://www.k-­‐
5mathteachingresources.com/5th
-­‐grade-­‐number-­‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.co
m/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/ind
ex.php/common-­‐core-­‐standards Vocabulary
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Benchmark Fraction Common Denominator Denominator Difference Equivalent Fraction Fraction Bar Improper Fraction Like Denominator Mixed Number Number Lines Numerator Proper Fraction Sum Unlike Denominator Essential Questions
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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. Formative Assessment Strategies • 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 Learning Targets Notes/Comments Unit Materials and Resources 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. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Number and Operations in Base Ten – 5.NBT www.youtube.com 5.NBT.1.1 Represent place values of whole A. Understand the place value system. numbers through 100,000,000 and 1. Recognize that in a multi-­‐digit Math Fact Fluency Practice decimals to the thousandths with number, a digit in one place www.mathwire.com manipulatives or visual models. represents 10 times as much as it www.oswego.org/ocsd-­‐
5.NBT.1.2 Recognize that in a multi-­‐digit web/games/ represents in the place to its right number, the digit to the left is 10x http://mathfactspro.com/mathflu
and 1/10 of what it represents in larger and the right is 1/10 smaller. encygame.html#/math-­‐facts-­‐
the place to its left. addition-­‐games http://jerome.northbranfordscho
5.NBT.2.1 Show repeated multiplication of tens Powers of 10 is a fundamental 2. Explain patterns in the number of zeros of ols.org/Content/Math_Fact_Flue
as an exponent. aspect of the base-­‐ten system, the product when multiplying a number by ncy_Practice_Sheets.asp thus 5.NBT.A.2 can help 5.NBT.2.2 Use manipulatives to explain patterns powers of 10, and explain patterns in the http://www.mathfactcafe.com/ students extend their in the number of zeros of the product placement of the decimal point when a www.factmonster.com when multiplying a number by powers understanding of place decimal is multiplied or divided by a power of 10. value to incorporate decimals to of 10. Use whole-­‐number exponents to Lessons/Activities/Games 6
5.NBT.2.3 Explain patterns in the number of hundredths. denote powers of 10. https://www.illustrativemathema
zeros of the product when multiplying tics.org/5 a number by powers of 10. https://learnzillion.com 5.NBT.2.4 Use manipulatives to explain patterns in the placement of the decimal point www.AECSD5thGradeMathemati
when a decimal is multiplied or csdoc divided by a power of 10. http://maccss.ncdpi.wikispaces.n
5.NBT.2.5 Explain patterns in the placement of et/Fifth+Grade the decimal point when a decimal is multiplied or divided by a power of 10. 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). Common Core State Standards for Mathematical Practice 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_curri
culum/mathematics_2/curriculu
m-­‐math-­‐grade-­‐five http://www.onlinemathlearning.c
om/common-­‐core-­‐grade5.html http://www.mathgoodies.com/st
andards/alignments/grade5.html http://www.k-­‐
5mathteachingresources.com/5th
-­‐grade-­‐number-­‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.co
m/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/ind
ex.php/common-­‐core-­‐standards http://map.mathshell.org/materi
als/stds.php#standard1159 Vocabulary
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Algorithim Equal Group Factor Multiples Multiplication Partial Products Product Symbols Essential Questions
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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? •
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Formative Assessment Strategies •
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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 Learning Targets Notes/Comments Unit Materials and Resources 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. 5.NBT.3b.1 5.NBT.3b.2 5.NBT.3b.3 5.NBT.3b.1 4. Use place value understanding to round decimals to any place. 5.NBT.4.1 Common Core State Standards for Mathematical Practice 6. Attend to precision. 7. Look for and make use of structure. Compare decimals to the tenths place using inequality symbols (<,>,=). Compare decimals to the hundredths place using inequality symbols (<,>,=). Compare decimals to any place using inequality symbols (<,>,=). Compare decimals to the tenths place using inequality symbols (<,>,=). 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). 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/mathflu
encygame.html#/math-­‐facts-­‐
addition-­‐games http://jerome.northbranfordscho
ols.org/Content/Math_Fact_Flue
ncy_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathema
tics.org/5 https://learnzillion.com www.AECSD5thGradeMathemati
csdoc http://maccss.ncdpi.wikispaces.n
et/Fifth+Grade www.dpi.state.nc.us Vocabulary
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Essential Questions
Decimal Division Hundredths Inequality Place Value •
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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? Formative Assessment Strategies •
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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 Learning Targets Notes/Comments Unit Materials and Resources 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. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Number and Operations-­‐Fractions – 5.NF www.youtube.com B. Apply and extend previous understandings of multiplication Math Fact Fluency Practice and division to multiply and divide www.mathwire.com fractions. www.oswego.org/ocsd-­‐
web/games/ 3. Interpret a fraction as division of the 5.NF.3.1 Interpret a fraction as a division http://mathfactspro.com/mathflu
numerator by the denominator (a/b = a ÷ problem. Ex. ¼ = 1 ÷ 4 encygame.html#/math-­‐facts-­‐
b). Solve word problems involving division 5.NF.3.2 Interpret a division problem as a addition-­‐games of whole numbers leading to answers in fraction. Ex. 1 ÷ 4 = ¼ http://jerome.northbranfordscho
the form of fractions or mixed numbers, 5.NF.3.3 Solve division word problems and ols.org/Content/Math_Fact_Flue
e.g., by using visual fraction models or express the quotient as a fraction or ncy_Practice_Sheets.asp equations to represent the problem. For mixed number by using visual fraction http://www.mathfactcafe.com/ example, interpret 3/4 as the result of 5.NF.3.4 models. www.factmonster.com dividing 3 by 4, noting that 3/4 multiplied Solve division word problems and by 4 equals 3, and that when 3 wholes are express the quotient as a fraction or shared equally among 4 people each mixed number by using equations. Lessons/Activities/Games person has a share of size 3/4. If 9 people https://www.illustrativemathema
want to share a 50-­‐pound sack of rice tics.org/5 equally by weight, how many pounds of rice should each person get? Between https://learnzillion.com what two whole numbers does your www.AECSD5thGradeMathemati
answer lie? csdoc http://maccss.ncdpi.wikispaces.n
et/Fifth+Grade www.dpi.state.nc.us 5.NF.4a.1 4. Apply and extend previous u nderstandings 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.) 5.NF.4a.2 5.NF.4a.3 5.NF.4a.4 5.NF.4a.5 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. 5.NF.4b.1 5.NF.4b.2 5.NF.4b.3 5.NF.4b.4 Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 4. Model with mathematics. 5. Use appropriate tools strategically. Represent a whole number as a fraction. Multiply a fraction by a fraction. Multiply a fraction by a whole number. Use a visual fraction model to represent multiplication of fractions. Create a context for a problem involving multiplication of fractions. 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. 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/searc
h/search.html www.tucerton.k12.nj.us/tes_curri
culum/mathematics_2/curriculu
m-­‐math-­‐grade-­‐five http://www.onlinemathlearning.c
om/common-­‐core-­‐grade5.html http://www.mathgoodies.com/st
andards/alignments/grade5.html http://www.k-­‐
5mathteachingresources.com/5th
-­‐grade-­‐number-­‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.co
m/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/ind
ex.php/common-­‐core-­‐standards Vocabulary
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Area Division Fraction Mixed Numbers Side length Tile Unit Square Visual Model Whole Number Essential Questions
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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? Formative Assessment Strategies •
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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 Learning Targets Notes/Comments Unit Materials and Resources 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. and In this unit, 5.NF.B.5a 5.NF.B.5b involve only multiplication by fractions. Division by unit fractions will be introduced in unit 8. 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/mathflu
encygame.html#/math-­‐facts-­‐
addition-­‐games http://jerome.northbranfordscho
ols.org/Content/Math_Fact_Flue
ncy_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathema
tics.org/5 https://learnzillion.com www.AECSD5thGradeMathemati
csdoc http://maccss.ncdpi.wikispaces.n
et/Fifth+Grade 5.NF.5b.1 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. 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. 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. Solve real-­‐world problems involving multiplication of fractions and mixed numbers. 5.NF.5b.2 5.NF.5b.3 5.NF.5b.4 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. 5.NF.6.1 Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively. 4. Model with mathematics. 6. Attend to precision. In 5.NF.B.6 students should have opportunities to work with all problem types. abstractly and Students reason practice communicating their situations thinking in real world (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/searc
h/search.html www.tucerton.k12.nj.us/tes_curri
culum/mathematics_2/curriculu
m-­‐math-­‐grade-­‐five http://www.onlinemathlearning.c
om/common-­‐core-­‐grade5.html http://www.mathgoodies.com/st
andards/alignments/grade5.html http://www.k-­‐
5mathteachingresources.com/5th
-­‐grade-­‐number-­‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.co
m/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/ind
ex.php/common-­‐core-­‐standards Vocabulary
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Area Division Fraction Mixed Numbers Side length Tile Unit Square Visual Model Whole Number Essential Questions
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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? Formative Assessment Strategies •
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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 Learning Targets Notes/Comments Unit Materials and Resources 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). Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Number and Operations-­‐Fractions - 5.NF www.youtube.com B. Apply and extend previous understandings of multiplication Math Fact Fluency Practice and division to multiply and divide www.mathwire.com fractions. www.oswego.org/ocsd-­‐
web/games/ 7. Apply and extend previous 5.NF.7 Understand the relationship between understandings of division to divide unit http://mathfactspro.com/mathflu
multiplication and division. fractions by whole numbers and whole encygame.html#/math-­‐facts-­‐
numbers by unit fractions. addition-­‐games http://jerome.northbranfordscho
a. Interpret division of a unit fraction by ols.org/Content/Math_Fact_Flue
5.NF.7a.1 Use a visual fraction model to divide a a non-­‐zero whole number, and ncy_Practice_Sheets.asp unit fraction by a whole number. compute such quotients. For http://www.mathfactcafe.com/ example, create a story context for 5.NF.7a.2 Create a context for a problem www.factmonster.com (1/3) ÷ 4, and use a visual fraction involving division of a unit fraction by a model to show the quotient. Use the whole number. Lessons/Activities/Games relationship between multiplication https://www.illustrativemathema
and division to explain that (1/3) ÷ 4 = tics.org/5 1/12 because (1/12) x 4 = 1/3. https://learnzillion.com www.AECSD5thGradeMathemati
csdoc http://maccss.ncdpi.wikispaces.n
et/Fifth+Grade www.dpi.state.nc.us http://harcourtschool.com/searc
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. Common Core State Standards for Mathematical Practice 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). h/search.html www.tucerton.k12.nj.us/tes_curri
culum/mathematics_2/curriculu
m-­‐math-­‐grade-­‐five http://www.onlinemathlearning.c
om/common-­‐core-­‐grade5.html http://www.mathgoodies.com/st
andards/alignments/grade5.html http://www.k-­‐
5mathteachingresources.com/5th
-­‐grade-­‐number-­‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.co
m/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/ind
ex.php/common-­‐core-­‐standards http://map.mathshell.org/materi
als/stds.php#standard1159 Vocabulary
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Division Fraction Mixed Numbers Multiplication Unit Fraction Visual Model Whole Number Essential Questions
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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? Formative Assessment Strategies •
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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 Learning Targets Notes/Comments Unit Materials and Resources 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. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Measurement and Data -­‐ 5.MD www.youtube.com 5.MD.5a.1 Find the volume of a right rectangular C. Geometric measurement: understand prism with whole-­‐number side concepts of volume and relate volume to Math Fact Fluency Practice lengths by packing it with unit cubes. multiplication and to addition. www.mathwire.com 5. Relate volume to the operations of www.oswego.org/ocsd-­‐
5.MD.5a.2 Relate volume to the operation of multiplication and addition and solve real web/games/ multiplication and addition. world and mathematical problems involving http://mathfactspro.com/mathflu
5.MD.5a.3 Measure volume with unit cubes and volume. encygame.html#/math-­‐facts-­‐
show that it is the same as: addition-­‐games a. Find the volume of a right rectangular multiplying the side lengths, http://jerome.northbranfordscho
prism with whole-­‐number side lengths multiplying the edge lengths, and ols.org/Content/Math_Fact_Flue
by packing it with unit cubes, and show multiplying the height by the area of ncy_Practice_Sheets.asp that the volume is the same as would the base. http://www.mathfactcafe.com/ be found by multiplying the edge www.factmonster.com lengths, equivalently by multiplying the height by the area of the base. Lessons/Activities/Games Represent threefold whole-­‐number https://www.illustrativemathema
products as volumes, e.g., to represent tics.org/5 the associative property of multiplication. https://learnzillion.com www.AECSD5thGradeMathemati
csdoc http://maccss.ncdpi.wikispaces.n
et/Fifth+Grade www.dpi.state.nc.us http://harcourtschool.com/searc
h/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. 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. Recognize volume is additive by finding the volume of solid figures of two non-­‐overlapping parts. Recognize volume is additive by finding the volume of solid figures of two or more non-­‐overlapping parts. 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. Common Core State Standards for Mathematical Practice 5. Use appropriate tools strategically. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. 5.MD.5c.1 5.MD.5c.2 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_curri
culum/mathematics_2/curriculu
m-­‐math-­‐grade-­‐five http://www.onlinemathlearning.c
om/common-­‐core-­‐grade5.html http://www.mathgoodies.com/st
andards/alignments/grade5.html http://www.k-­‐
5mathteachingresources.com/5th
-­‐grade-­‐number-­‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.co
m/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/ind
ex.php/common-­‐core-­‐standards http://map.mathshell.org/materi
als/stds.php#standard1159 Vocabulary
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Formula Rectangular Prism Solid Figures Unit Cube Volume Essential Questions
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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? Formative Assessment Strategies •
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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 Learning Targets Notes/Comments Unit Materials and Resources 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. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Number and Operations in Base Ten - 5.NBT www.youtube.com B. Perform operations with multi-­‐digit whole 5.NBT.7.1 Add or subtract two decimal numbers numbers and with decimals to hundredths. Math Fact Fluency Practice to the hundredths without regrouping. 7. Add, subtract, multiply, and divide www.mathwire.com 5.NBT.7.2 Add and subtract two decimal decimals to hundredths, using concrete www.oswego.org/ocsd-­‐
numbers to hundredths with models or drawings and strategies based web/games/ regrouping. http://mathfactspro.com/mathflu
on place value, properties of operations, 5.NBT.7.3 Multiply tenths by tenths. encygame.html#/math-­‐facts-­‐
and/or the relationship between 5.NBT.7.4 Multiply tenths by hundredths. addition-­‐games addition and subtraction; relate the http://jerome.northbranfordscho
strategy to a written method and explain 5.NBT.7.5 Divide decimals with tenths. ols.org/Content/Math_Fact_Flue
the reasoning used. ncy_Practice_Sheets.asp 5.NBT.7.6 Divide decimals with tenths and/or http://www.mathfactcafe.com/ hundredths. www.factmonster.com 5.NBT.7.7 Relate the strategy used to a written Lessons/Activities/Games method and explain the reasoning https://www.illustrativemathema
used. tics.org/5 5.NBT.7.8 Demonstrate computations by using models and drawings. https://learnzillion.com www.AECSD5thGradeMathemati
csdoc http://maccss.ncdpi.wikispaces.n
et/Fifth+Grade www.dpi.state.nc.us http://harcourtschool.com/searc
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. 5.MD.1.1 Recall customary units of measurements. 5.MD.1.2 5.MD.1.3 Recall metric units of measurements. Use manipulatives or visual models to convert different measurement units (customary & metric) within a given measurement system to solve single step problems. Convert different measurement units (customary & metric) within a given measurement system to solve real world single step problems. Convert different measurement units (customary & metric) within a given measurement system to solve and create real world multi-­‐step problems. Choose the appropriate measurement unit based on the given context. 5.MD.1.4 5.MD.1.5 5.MD.1.6 Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 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.
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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). h/search.html www.tucerton.k12.nj.us/tes_curri
culum/mathematics_2/curriculu
m-­‐math-­‐grade-­‐five http://www.onlinemathlearning.c
om/common-­‐core-­‐grade5.html http://www.mathgoodies.com/st
andards/alignments/grade5.html http://www.k-­‐
5mathteachingresources.com/5th
-­‐grade-­‐number-­‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.co
m/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/ind
ex.php/common-­‐core-­‐standards http://map.mathshell.org/materi
als/stds.php#standard1159 Vocabulary
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What are the mathematical properties that govern addition and multiplication? How would you use them? 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? Formative Assessment Strategies •
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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 Learning Targets Notes/Comments Unit Materials and Resources 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. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Geometry -­‐ 5.G www.youtube.com 5.G.3.1 Understand that shared attributes B. Classify two-­‐dimensional figures into categorize two-­‐dimensional figures. categories based on their properties. Math Fact Fluency Practice 3. Understand that attributes belonging www.mathwire.com to a category of two-­‐ dimensional www.oswego.org/ocsd-­‐
figures also belong to all subcategories web/games/ of that category. For example, all http://mathfactspro.com/mathflu
encygame.html#/math-­‐facts-­‐
rectangles have four right angles and addition-­‐games squares are rectangles, so all squares http://jerome.northbranfordscho
have four right angles. ols.org/Content/Math_Fact_Flue
ncy_Practice_Sheets.asp http://www.mathfactcafe.com/ 4. Classify two-­‐dimensional figures in a 5.G.4.1 Classify two-­‐dimensional figures based www.factmonster.com hierarchy based on properties. on properties. Lessons/Activities/Games 5.G.4.2 Classify two-­‐dimensional figures in a https://www.illustrativemathema
hierarchy based on properties. tics.org/5 5.G.4.3 Demonstrate t
hat a
ttributes b
elonging https://learnzillion.com to a
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ubcategories o
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Students make use of structure to category. Common Core State Standards for csdoc 5.G.4.4 Use appropriate tools to determine build a logical progression of Mathematical Practice similarities and differences between statements and explore http://maccss.ncdpi.wikispaces.n
categories and subcategories. hierarchical relationships among 3. Construct viable arguments and critique the et/Fifth+Grade reasoning of others. 2-­‐dimensional shapes (MP.3, 7. Look for and make use of structure. MP.7). Vocabulary
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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? •
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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 Learning Targets Notes/Comments Unit Materials and Resources 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. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Number and Operations-­‐Fractions - 5.NF www.youtube.com 5.NF.7c.1 Solve real-­‐world problems involving B. Apply and extend previous division of unit fractions and whole understandings of multiplication Math Fact Fluency Practice numbers. and division to multiply and divide www.mathwire.com fractions. www.oswego.org/ocsd-­‐
web/games/ 7. Apply and extend previous understandings of division to divide unit http://mathfactspro.com/mathflu
fractions by whole numbers and whole encygame.html#/math-­‐facts-­‐
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numbers by unit fractions. addition-­‐games c. Solve real world problems involving http://jerome.northbranfordscho
division of unit fractions by non-­‐zero ols.org/Content/Math_Fact_Flue
whole numbers and division of whole ncy_Practice_Sheets.asp http://www.mathfactcafe.com/ numbers by unit fractions, e.g., by using www.factmonster.com visual fraction models and equations to represent the problem. For example, Lessons/Activities/Games how much chocolate will each person get https://www.illustrativemathema
if 3 people share 1/2 lb of chocolate tics.org/5 equally? How many 1/3-­‐cup servings are in 2 cups of raisins? 1 https://learnzillion.com NOTE: Students able to multiply fractions in general can develop strategies to divide www.AECSD5thGradeMathemati
fractions in general, by reasoning about the csdoc relationship between multiplication and division. But division of a fraction by a http://maccss.ncdpi.wikispaces.n
fraction is not a requirement at this grade. et/Fifth+Grade www.dpi.state.nc.us http://harcourtschool.com/searc
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. Common Core State Standards for Mathematical Practice 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). h/search.html www.tucerton.k12.nj.us/tes_curri
culum/mathematics_2/curriculu
m-­‐math-­‐grade-­‐five http://www.onlinemathlearning.c
om/common-­‐core-­‐grade5.html http://www.mathgoodies.com/st
andards/alignments/grade5.html http://www.k-­‐
5mathteachingresources.com/5th
-­‐grade-­‐number-­‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.co
m/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/ind
ex.php/common-­‐core-­‐standards http://map.mathshell.org/materi
als/stds.php#standard1159 Vocabulary
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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? Formative Assessment Strategies •
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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 Learning Targets Notes/Comments Unit Materials and Resources 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. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org The expressions described in 5.OA.1.1 Use the order of operations to Operations and Algebraic Thinking - 5.OA www.youtube.com 5.OA.A.1 include the use of evaluate numerical expressions. A. Write and interpret numerical expressions. parentheses but should not 5.OA.1.2 Explain the order of operations. 1. Use parentheses, brackets, or braces in Math Fact Fluency Practice contain nested grouping numerical expressions, and evaluate www.mathwire.com symbols. expressions with these symbols. www.oswego.org/ocsd-­‐
5.OA.1.3 Apply the order of operations to web/games/ evaluate expressions. http://mathfactspro.com/mathflu
encygame.html#/math-­‐facts-­‐
addition-­‐games 2. Write simple expressions that record The expressions described in http://jerome.northbranfordscho
5.OA.2.1 Interpret a numerical expression into calculations with numbers, and interpret 5.OA.A.2 should be no more ols.org/Content/Math_Fact_Flue
words (without evaluating the numerical expressions without evaluating complex than the expressions ncy_Practice_Sheets.asp expression). them. For example, express the calculation http://www.mathfactcafe.com/ one f
inds in an application of the 5.OA.2.2 Write simple expressions that record "add 8 and 7, then multiply by 2” as 2 x www.factmonster.com associative or d
istributive calculations with numbers {8 + 7}. Recognize that 3 x {18932 + property. 921} is three times as large as 18932 + Lessons/Activities/Games 921, without having to calculate the https://www.illustrativemathema
tics.org/5 indicated sum or product. Students discuss the meaning of https://learnzillion.com symbols and interpret numerical Common Core State Standards for expressions precisely (MP.6). Mathematical Practice www.AECSD5thGradeMathemati
csdoc 6. Attend to precision. http://maccss.ncdpi.wikispaces.n
et/Fifth+Grade www.dpi.state.nc.us Vocabulary
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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? •
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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 Learning Targets Notes/Comments Unit Materials and Resources 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. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Operations and Algebraic Thinking - 5.OA www.youtube.com 5.OA.3.1 Create two numerical patterns from B. Analyze patterns and relationships. two given rules. 3. Generate two numerical patterns using Math Fact Fluency Practice 5.OA.3.2 Identify relationships between two given rules. Identify apparent www.mathwire.com corresponding terms in a pattern. relationships between corresponding www.oswego.org/ocsd-­‐
5.OA.3.3 Extend a numerical pattern from a web/games/ terms. Form ordered pairs consisting of given rule. http://mathfactspro.com/mathflu
corresponding terms from the two 5.OA.3.4 Determine a rule from a given encygame.html#/math-­‐facts-­‐
patterns, and graph the ordered pairs on a numerical pattern. addition-­‐games coordinate plane. For example, given the 5.OA.3.5 Create ordered pairs of the http://jerome.northbranfordscho
rule "Add 3” and the starting number 0, corresponding terms from two ols.org/Content/Math_Fact_Flue
and given the rule "Add 6” and the starting patterns. ncy_Practice_Sheets.asp number 0, generate terms in the resulting 5.OA.3.6 Plot an ordered pair on a coordinate http://www.mathfactcafe.com/ sequences, and observe that the terms in plane from 2 patterns and justify. www.factmonster.com one sequence are twice the corresponding terms in the other sequence. Explain Lessons/Activities/Games informally why this is so. https://www.illustrativemathema
tics.org/5 https://learnzillion.com www.AECSD5thGradeMathemati
csdoc http://maccss.ncdpi.wikispaces.n
et/Fifth+Grade www.dpi.state.nc.us 5.G.1.1 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). 5.G.1.2 5.G.1.3 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. 5.G.2.1 5.G.2.2 5.G.2.3 Common Core State Standards for Mathematical Practice 4. Model with mathematics. 6. Attend to precision. 5.G.2.4 Label the axes, quadrants and origin on the coordinate plane. Identify ordered pairs. Plot points on the coordinate plane. Represent real-­‐world and mathematical problems by locating or graphing points in the first quadrant of a coordinate plane. Represent real-­‐world and mathematical problems by locating and graphing points in the first quadrant of a coordinate plane. Interpret coordinate values in the context of the situation. 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/searc
h/search.html www.tucerton.k12.nj.us/tes_curri
culum/mathematics_2/curriculu
m-­‐math-­‐grade-­‐five http://www.onlinemathlearning.c
om/common-­‐core-­‐grade5.html http://www.mathgoodies.com/st
andards/alignments/grade5.html http://www.k-­‐
5mathteachingresources.com/5th
-­‐grade-­‐number-­‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.co
m/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/ind
ex.php/common-­‐core-­‐standards http://map.mathshell.org/materi
als/stds.php#standard1159 Vocabulary
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Axes Coordinate Plane Graphing Origin Patterns Rules Quadrant Essential Questions
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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? •
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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 Learning Targets Notes/Comments Unit Materials and Resources 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. Videos Common Core State Standards for www.khanacademy.org Mathematical Content www.teachingchannel.org Number and Operations in Base Ten - 5.NBT www.youtube.com B. Perform operations with multi-­‐digit whole numbers and with decimals to hundredths. Math Fact Fluency Practice 5.NBT.5.1 Fluently multiply multi-­‐digit whole www.mathwire.com 5. Fluently multiply multi-­‐digit whole numbers numbers. www.oswego.org/ocsd-­‐
using the standard algorithm. web/games/ http://mathfactspro.com/mathflu
encygame.html#/math-­‐facts-­‐
addition-­‐games 6. Find whole-­‐number quotients of whole 5.NBT.6.1 Determine the quotient of whole 5.NBT.B.6 is a milestone along the http://jerome.northbranfordscho
way to reaching fluency with the numbers with up to 3 digit dividends numbers with up to four-­‐digit dividends ols.org/Content/Math_Fact_Flue
standard algorithm in Grade 6 and 1 digit divisors, which are and two-­‐digit divisors, using strategies ncy_Practice_Sheets.asp (6.NS.B.2). 5.NBT.6.2 multiples of ten. based on place value, the properties of http://www.mathfactcafe.com/ Determine the quotient of whole operations, and/or the relationship www.factmonster.com numbers with up to 4 digit dividends between multiplication and division. and 1
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hich a
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using equations, rectangular arrays, and/or 5.NBT.6.3 Determine t
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hole tics.org/5 area models. numbers with up to 4 digit dividends and 2 digit divisors. https://learnzillion.com Common Core State Standards for 5.NBT.6.4 Illustrate and explain division using Mathematical Practice equations, rectangular arrays, and/or www.AECSD5thGradeMathemati
Students use efficient strategies area models. csdoc and look for shortcuts to multiply 1. Make sense of problems and persevere in and divide whole numbers with solving them. http://maccss.ncdpi.wikispaces.n
accuracy (MP.1, MP.8). 8. Look for and express regularity in repeated et/Fifth+Grade reasoning. Vocabulary
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What are the mathematical properties that govern addition and multiplication? How would you use them? 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? Formative Assessment Strategies •
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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. 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.
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.
3.
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.
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
a.
Key:
Major Clusters;
Supporting Clusters;
Additional Clusters
FIFTH GRADE CRITICAL AREAS OF FOCUS
CRITICAL AREA OF F OCUS #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.
Key:
Major Clusters;
Supporting Clusters;
Additional Clusters
FIFTH GRADE CRITICAL AREAS OF FOCUS
CRITICAL AREA OF F OCUS #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.
Key:
Major Clusters;
Supporting Clusters;
Additional Clusters
FIFTH GRADE CRITICAL AREAS OF FOCUS
CRITICAL AREA OF F OCUS #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.
Key:
Major Clusters;
Supporting Clusters;
Additional Clusters
FIFTH GRADE CRITICAL AREAS OF FOCUS
CRITICAL AREA OF F OCUS #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 nonoverlapping 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.)
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.
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.
Key:
Major Clusters;
Supporting Clusters;
Additional Clusters
FIFTH GRADE CRITICAL AREAS OF FOCUS
CRITICAL AREA OF F OCUS #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.
Key:
Major Clusters;
Supporting Clusters;
Additional Clusters
FIFTH GRADE CRITICAL AREAS OF FOCUS
S TANDARDS AND C LUSTERS B EYOND THE C RITICAL AREAS OF F OCUS
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, yaxis and y-coordinate).
2.
Key:
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.
Major Clusters;
Supporting Clusters;
Additional Clusters
Performance Level Descriptors – Grade 5 Mathematics
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
Level 3: Moderate
Level 2: Partial
Level 4: Strong Command
Command
Command
Command
Addition and
Subtraction
Operations with
Decimals
5.NBT.7-1
5.NBT.7-2
Adding and
Subtracting in
Context with
Fractions
5.NF.2-1
5.NF.2-2
July 2013
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 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.
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.
Relates the strategy to a
written method and
explain the reasoning used.
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 and justifies
reasonableness using
benchmark fractions and
number sense of fractions.
Assesses reasonableness
using benchmark fractions
and number sense of
fractions.
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.
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.
Page 1 of 20
Performance Level Descriptors – Grade 5 Mathematics
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
Level 3: Moderate
Level 2: Partial
Level 4: Strong Command
Command
Command
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
July 2013
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.
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.
Page 2 of 20
Performance Level Descriptors – Grade 5 Mathematics
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
Level 3: Moderate
Level 2: Partial
Level 4: Strong Command
Command
Command
Command
appropriate.
Relates the strategy to a
written method.
applying place value
strategies when
appropriate.
Relates the strategy to a
Applies this concept in the
written method.
context of metric
measurement (e.g., find the
area of a rectangle with
length=0.7cm and
width=0.4cm.)
Multiply with
Whole Numbers
5.NBT.5-1
5.Int.1
5.Int.2
5.NBT.Int.1
July 2013
Solves multi-step
unscaffolded word
problems involving
multiplication and
multiplies three-digit by
two-digit whole numbers
using the standard
algorithm.
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.
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 threedigit by a one-digit whole
number.
Solves one-step word
problems involving
multiplication.
Page 3 of 20
Performance Level Descriptors – Grade 5 Mathematics
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
Level 3: Moderate
Level 2: Partial
Level 4: Strong Command
Command
Command
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.
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.
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.
July 2013
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.
Illustrates and explains the
calculations by using
equations, rectangular
arrays, and area models.
Checks reasonableness of
answers by using
multiplication or
estimation.
Page 4 of 20
Performance Level Descriptors – Grade 5 Mathematics
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
Level 3: Moderate
Level 2: Partial
Level 4: Strong Command
Command
Command
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
Interpreting
Fractions
5.NF.3-1
5.NF.3-2
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.
Solves word problems
involving division of whole
numbers leading to answers
in the form of fractions or
mixed numbers.
Solves word problems
involving division of whole
numbers leading to answers
in the form of fractions or
mixed numbers.
Solves word problems
involving division of whole
numbers leading to answers
in the form of fractions or
mixed numbers.
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.
Interprets the fraction as
Interprets the fraction as
division of the numerator by division of the numerator by
the denominator.
the denominator.
Creates a model
representing the situation.
July 2013
Interprets the fraction as
division of the numerator
by the denominator.
Identifies a simple model
representing the situation.
Page 5 of 20
Performance Level Descriptors – Grade 5 Mathematics
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
Level 3: Moderate
Level 2: Partial
Level 4: Strong Command
Command
Command
Command
Recognizing
Volume
5.MD.3
5.MD.4
Finding Volume
5.MD.5b
5.MD.5c
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
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.
Represents the volume of a
solid figure as “n” cubic
units.
Creates an equation that
illustrates the unit cube
pattern.
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 nonoverlapping parts.
Creates real-world and
mathematical problems that
July 2013
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.
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.
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).
Page 6 of 20
Performance Level Descriptors – Grade 5 Mathematics
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
Level 3: Moderate
Level 2: Partial
Level 4: Strong Command
Command
Command
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
Place Value
5.NBT.1
5.NBT.2-2
5.NBT.A.Int.1
July 2013
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.
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.
Performs exact and
approximate
multiplications and
divisions by mentally
applying place value
strategies when
appropriate.
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.
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.
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.
Page 7 of 20
Performance Level Descriptors – Grade 5 Mathematics
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
Level 3: Moderate
Level 2: Partial
Level 4: Strong Command
Command
Command
Command
uses symbols to compare
two powers of 10 expressed
exponentially (compare 102
to 105).
Multiplication
Scaling
5.NF.5a
Write and
Interpret
Numerical
Expressions
5.OA.1
5.OA.2-1
5.OA.2-2
July 2013
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 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.
Uses parentheses, brackets,
or braces with no greater
depth than two, to write,
evaluate and create
numerical expressions.
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.
Uses parentheses, brackets,
or braces with no greater
depth than two, to write
and evaluate numerical
expressions.
Uses parentheses, brackets,
or braces to write
numerical expressions.
Uses parentheses, brackets,
or braces to write simple
numerical expressions.
Interprets numerical
expressions without
evaluating them.
Interprets numerical
expressions without
evaluating them.
Interprets simple numerical
expressions without
evaluating them.
Page 8 of 20
Performance Level Descriptors – Grade 5 Mathematics
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.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Graphing on the
Coordinate Plane
5.G.1
5.G.2
5.OA.3
Two-Dimensional
Figures
5.G.3
5.G.4
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.
Classifies two-dimensional
figures in a hierarchy based
on properties.
Classifies two-dimensional
figures in a hierarchy based
on properties.
Classifies two-dimensional
figures in a hierarchy based
on properties.
Classifies two-dimensional
figures based on properties.
Demonstrates that
attributes belonging to a
category of twodimensional figures also
belong to all subcategories
of that category.
Understands that attributes
belonging to a category of
two-dimensional figures
also belong to all
subcategories of that
category.
Understands that shared
attributes categorize twodimensional figures.
Uses appropriate tools to
determine similarities and
differences between
categories and
subcategories.
July 2013
Page 9 of 20
Performance Level Descriptors – Grade 5 Mathematics
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.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Conversions
5.MD.1-1
5.MD.1-2
Converts among differentsized standard
measurement units within a
given measurement system
and uses these conversions
to create real-world, multistep problems.
Converts among differentsized standard
measurement units within a
given measurement system
and uses these conversions
to solve real-world, multistep problems.
Converts among differentsized standard
measurement units within a
given measurement system
and uses these conversions
to solve real-world, singlestep problems.
Converts among differentsized standard
measurement units within a
given measurement system
and solves single-step
problems by using
manipulatives or visual
models.
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.
Chooses the appropriate
measurement unit based on
the given context.
Data Displays
5.MD.2-1
5.MD.2-2
July 2013
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.
Page 10 of 20
Performance Level Descriptors – Grade 5 Mathematics
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.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
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 wellorganized and complete
written response based on
explanations/reasoning
using the:
Constructs and
communicates a wellorganized and complete
written response based on
explanations/reasoning
using the:
Constructs and
communicates a complete
written response based on
explanations/reasoning
using the:
Constructs and
communicates an
incomplete written
response based on
explanations/reasoning
using the:










properties of operations
relationship between
addition and
subtraction
relationship between
multiplication and
division

properties of operations
relationship between
addition and
subtraction
relationship between
multiplication and
division

properties of operations
relationship between
addition and
subtraction
relationship between
multiplication and
division
Response may include:
Response may include:
Response may include:
Response may include:





July 2013
properties of operations
relationship between
addition and
subtraction
relationship between
multiplication and
division
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


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



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



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-
Page 11 of 20
Performance Level Descriptors – Grade 5 Mathematics
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.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command





July 2013
justification
precision of calculation
correct use of gradelevel 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.




correct use of gradelevel 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).


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
partial justification of a
conclusion based on
own calculations
Page 12 of 20
Performance Level Descriptors – Grade 5 Mathematics
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.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Place Value
5.C.3
Clearly constructs and
communicates a wellorganized and complete
response based on place
value system including:
Clearly constructs and
communicates a wellorganized and complete
response based on place
value system including:
Constructs and
communicates a complete
response based on place
value system including:
Constructs and
communicates an
incomplete response based
on place value system which
may include:









July 2013
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
precision of calculation
correct use of gradelevel vocabulary,
symbols and labels
justification of a
conclusion
evaluation of whether
an argument or
conclusion is
generalizable






a logical approach
based on a conjecture
and/or stated
assumptions, utilizing
mathematical
connections (when
appropriate)
a logical progression of
steps
precision of calculation
correct use of gradelevel vocabulary,
symbols and labels
justification of a
conclusion
evaluation of whether
an argument or
conclusion is
generalizable
evaluating, interpreting
and critiquing 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
and labels
partial justification of a
conclusion based on
own calculations
evaluating the validity
of other’s responses,
approaches and
conclusions.




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 gradelevel vocabulary,
symbols and labels
partial justification of a
conclusion based on
own calculations
Page 13 of 20
Performance Level Descriptors – Grade 5 Mathematics
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.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command

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
July 2013
validity of other’s
responses, approaches
and reasoning.
evaluating, interpreting
and critiquing the
validity of other’s
responses, approaches
and reasoning, and
providing a counterexample where
applicable.
Clearly constructs and
communicates a wellorganized 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:
Clearly constructs and
communicates a wellorganized 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:
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 logical approach
based on a conjecture
and/or stated
assumptions, utilizing
a logical approach
based on a conjecture
and/or stated
assumptions, utilizing
a logical approach
based on a conjecture
and/or stated
assumptions
a conjecture and/or
stated or faulty
assumptions
Page 14 of 20
Performance Level Descriptors – Grade 5 Mathematics
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.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command






July 2013
mathematical
connections (when
appropriate)
an efficient and logical
progression of steps
with appropriate
justification
precision of calculation
correct use of gradelevel 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, approaches,
and reasoning, and
providing a counterexample where
applicable.






mathematical
connections (when
appropriate)
a logical progression of
steps
precision of calculation
correct use of gradelevel 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, approaches,
and reasoning.





a logical, but
incomplete, progression
of steps
minor calculation errors
some use of grade-level
vocabulary, symbols
and labels
partial justification of a
conclusion based on
own calculations.
evaluating the validity
of other’s responses,
approaches and
conclusions





an incomplete or
illogical progression of
steps
an intrusive calculation
error
limited use of gradelevel vocabulary,
symbols and labels
partial justification of a
conclusion based on
own calculations
accepting the validity of
other’s responses
Page 15 of 20
Performance Level Descriptors – Grade 5 Mathematics
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.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
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 wellorganized and complete
response by:



Clearly constructs and
communicates a wellorganized 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

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
Constructs and
communicates a complete
response by:
Constructs and
communicates an
incomplete response by:





analyzing 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


analyzing solutions to
scaffolded two-step
problems in the form of
valid chains of
reasoning, sometimes
using symbols such as
equal signs
appropriately
distinguishing correct
explanation/reasoning
from that which is
flawed
identifying an error in
reasoning
Response may include:

July 2013
a logical approach
based on a conjecture
Response may include:
Response may include:
Response may include:



a logical approach
a logical approach
a conjecture based on
Page 16 of 20
Performance Level Descriptors – Grade 5 Mathematics
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.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command






July 2013
and/or stated
assumptions, utilizing
mathematical
connections (when
appropriate)
an efficient and logical
progression of steps
with appropriate
justification
precision of calculation
correct use of gradelevel 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,
approaches and
reasoning, and providing
a counter-example
where applicable






based on a conjecture
and/or stated
assumptions, utilizing
mathematical
connections (when
appropriate)
a logical progression of
steps
precision of calculation
correct use of gradelevel 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, approaches
and reasoning





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
and labels
partial justification of a
conclusion based on
own calculations
evaluating the validity
of other’s responses,
approaches and
conclusions.





faulty assumptions
an incomplete or
illogical progression of
steps
an intrusive calculation
error
limited use of gradelevel vocabulary,
symbols and labels
partial justification of a
conclusion based on
own calculations
accepting the validity of
other’s responses
Page 17 of 20
Performance Level Descriptors – Grade 5 Mathematics
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.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
Modeling
5.D.1
5.D.2
Devises a plan and applies
mathematics to solve multistep, real-world contextual
word problems by:
Devises a plan and applies
mathematics to solve multistep, real-world contextual
word problems by:
Devises a plan and applies
mathematics to solve multistep, real-world contextual
word problems by:
Devises a plan and applies
mathematics to solve multistep, real-world contextual
word problems by:







July 2013
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
analyzing relationships
mathematically
between important
quantities to draw
conclusions
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
analyzing relationships
mathematically
between important
quantities to draw
conclusions
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
Page 18 of 20
Performance Level Descriptors – Grade 5 Mathematics
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.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command





July 2013
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
reflecting on whether
the results make sense
modifying and/or
improving the model if
it has not served its
purpose
writing an arithmetic
expression or equation
to describe a situation



reflecting on whether
the results make sense
modifying the model if
it has not served its
purpose
writing an arithmetic
expression or equation
to describe a situation
Page 19 of 20
Performance Level Descriptors – Grade 5 Mathematics
Grade 5 Math: Sub-Claim E
The student demonstrates fluency in areas set forth in the Standards for Content in grades 3-6.
Level 5: Distinguished
Level 3: Moderate
Level 4: Strong Command
Level 2: Partial Command
Command
Command
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.
July 2013
Accurately and in a timely
manner multiplies multidigit whole numbers using
the standard algorithm.
Accurately multiplies multidigit whole numbers using
the standard algorithm.
Multiplies multi-digit whole
numbers using the standard
algorithm with some level
of accuracy.
Knows from memory 100
percent of the products on
items in the allotted time
on items which are timed
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.
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.
Page 20 of 20
Bailey Education Group, LLC
Common Core State Standard I Can Statements
5th Grade Mathematics 6/18/2013
CCSS Key:
Operations and Algebraic Thinking (OA)
Number and Operations in Base Ten (NBT)
Numbers and Operations–Fractions (NF)
Measurement and Data (MD)
Geometry (G)
PLD Key:
Partial Command
Moderate Command
Distinguished Command
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.
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.
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.1.1
5.OA.1.2
5.OA.1.3
Use the order of operations to evaluate numerical
expressions.
Explain the order of operations.
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.
I Can:
5.OA.2.1
5.OA.2.2
Interpret a numerical expression into words (without
evaluating the expression).
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.
I Can:
5.OA.3.1
5.OA.3.2
5.OA.3.3
5.OA.3.4
5.OA.3.5
5.OA.3.6
Create two numerical patterns from two given rules.
Identify relationships between corresponding terms
in a pattern.
Extend a numerical pattern from a given rule.
Determine a rule from a given numerical pattern.
Create ordered pairs of the corresponding terms
from two patterns.
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.
Latest Revision 6/18/2013
I Can Statements are in draft form due to the iterative nature of the item development process.
1
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.
I can:
5.NF.2.
5.NF.2.1
Solve word problems involving addition and
subtraction of fractions referring to the
same whole, including cases of unlike
5.NF.2.2
denominators, e.g., by using visual fraction
models or equations to represent the
5.NF.2.3
problem. Use benchmark fractions and
number sense of fractions to estimate
5.NF.2.4
mentally and assess the reasonableness of
5.NF.2.5
answers. For example, recognize an
incorrect result 2/5 + 1/2 = 3/7, by observing
that 3/7 < 1/2.
I can:
5.NF.3.
5.NF.3.1
Interpret a fraction as division of the
numerator by the denominator (a/b = a ÷ b).
Solve word problems involving division of
5.NF.3.2
whole numbers leading to answers in the
form of fractions or mixed numbers,
5.NF.3.3
e.g., by using visual fraction models or
equations to represent the problem. For
example, interpret 3/4 as the result of
5.NF.3.4
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.
Use benchmark numbers (0, ¼, ½, ¾, 1) to estimate
sums and differences of fractions.
Relate estimation to my answers to see if they make
sense.
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.
Interpret a fraction as a division problem. Ex. ¼ = 1
÷4
Interpret a division problem as a fraction. Ex. 1 ÷ 4 =
¼
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.
Latest Revision 6/18/2013
I Can Statements are in draft form due to the iterative nature of the item development process.
2
Common Core State Standards for
Mathematics (Outcome Based)
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.
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” Statements
I can:
5.NF.4a.1
5.NF.4a.2
5.NF.4a.3
5.NF.4a.4
Represent a whole number as a fraction.
Multiply a fraction by a fraction.
Multiply a fraction by a whole number.
Use a visual fraction model to represent
multiplication of fractions.
5.NF.4a.5 Create a context for a problem involving
multiplication of fractions.
5.NF.4b.1 Find the area of a rectangle with fractional side
lengths by tiling it with unit squares.
5.NF.4b.2 Relate different strategies for calculating the area of
a rectangle. (tiling vs. formula)
5.NF.4b.3 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.
I can:
5.NF.5a.1 Use visual models or manipulatives to interpret
multiplication scaling and correctly perform the
indicated multiplication.
5.NF.5a.2 Interpret multiplication scaling by performing the
indicated multiplication where one factor is a
fraction.
5.NF.5a.3 Interpret multiplication scaling without performing
the indicated multiplication where one factor is a
mixed number.
5.NF.5a.4 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 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.
5.NF.5b.3 Explain when multiplying by a fraction greater than
one, the number increases and when multiplying by
a number less than one, the number decreases.
5.NF.5b.4 Explain that when multiplying the numerator and
denominator by the same number is the same as
multiplying by one.
Latest Revision 6/18/2013
I Can Statements are in draft form due to the iterative nature of the item development process.
3
Common Core State Standards for
Mathematics (Outcome Based)
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.
5.NF.7.
Apply and extend previous understandings
of division to divide unit fractions by whole
numbers and whole numbers by unit
fractions.
“I Can” Statements
I Can:
5.NF.6.1
Solve real-world problems involving multiplication of
fractions and mixed numbers.
I Can:
5.NF.7
Understand the relationship between multiplication
and division.
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
5.NF.7a.1 Use a visual fraction model to divide 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 ×
(1/5) = 4.
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.
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?
5.NF.7c.1
5.NF.7a.2 Create a context for a problem involving division of
a unit fraction by a whole number.
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.
Latest Revision 6/18/2013
I Can Statements are in draft form due to the iterative nature of the item development process.
4
Common Core State Standards for
Mathematics (Outcome Based)
“I Can” Statements
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.3.
Read, write, and compare decimals to
thousandths.
I Can:
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.
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.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.
5.NBT.5.
Fluently multiply multi-digit whole numbers
using the standard algorithm.
I Can:
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.4.1 Use place value understanding to round decimals
to any place.
I Can:
5.NBT.5.1 Fluently multiply multi-digit whole numbers.
5.NBT.6.1 Determine the quotient of whole numbers with up
to 3 digit dividends and 1 digit divisors, which are
multiples of ten.
5.NBT.6.2 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.
Latest Revision 6/18/2013
I Can Statements are in draft form due to the iterative nature of the item development process.
5
Common Core State Standards for
Mathematics (Outcome Based)
“I Can” Statements
5.NBT.6.4 Illustrate and explain division using equations,
rectangular arrays, and/or area models.
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
5.MD.1.2
5.MD.1.3
5.MD.1.4
5.MD.1.5
5.MD.1.6
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.
Recall customary units of measurements.
Recall metric units of measurements.
Use manipulatives or visual models to convert
different measurement units (customary & metric)
within a given measurement system to solve single
step problems.
Convert different measurement units (customary &
metric) within a given measurement system to solve
real world single step problems.
Convert different measurement units (customary &
metric) within a given measurement system to solve
and create real world multi-step problems.
Choose the appropriate measurement unit based on
the given context.
I Can:
5.MD.2.1
5.MD.2.2
5.MD.2.3
5.MD.2.4
Create a line plot to display a data set of
measurements in fraction form with denominators of
2 and 4.
Create a line plot to display a data set of
measurements in fraction form with denominators of
2, 4 and 8.
Use operations on fractions to solve problems
involving information presented in line plots.
Interpret the solution in relation to the data.
Latest Revision 6/18/2013
I Can Statements are in draft form due to the iterative nature of the item development process.
6
Common Core State Standards for
Mathematics (Outcome Based)
5.MD.3.
Recognize volume as an attribute of solid
figures and understand concepts of volume
measurement.
“I Can” Statements
I Can:
5.MD.3a.1 Understand that unit cubes are used to measure
volume of solid figures.
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.
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.5.
Relate volume to the operations of
multiplication and addition and solve real
world and mathematical problems involving
volume.
I Can:
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.4.1
5.MD.4.2
5.MD.4.3
Use a visual model to measure volume by counting
unit cubes.
Measure volume by counting unit cubes.
Represent the volume of a solid figure as “n” cubic
units. (cm3, in3, ft3, units3).
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.
b. Apply the formulas V = l × w × h and V =
b × h for rectangular prisms to find volumes
of right rectangular prisms with wholenumber 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
5.MD.5c.1
Recognize volume is additive by finding the volume
of solid figures of two non-overlapping parts.
Latest Revision 6/18/2013
I Can Statements are in draft form due to the iterative nature of the item development process.
7
Common Core State Standards for
Mathematics (Outcome Based)
adding the volumes of the non-overlapping
parts, applying this technique to solve real
world problems.
“I Can” Statements
5.MD.5c.2
Recognize volume is additive by finding the volume
of solid figures of two or more non-overlapping
parts.
Geometry (G)
I Can:
5.G.1.
5.G.1.1
Use a pair of perpendicular number lines,
called axes, to define a coordinate system,
with the intersection of the lines (the origin)
5.G.1.2
arranged to coincide with the 0 on each line
5.G.1.3
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.2.
5.G.2.1
Represent real-world and mathematical
problems by graphing points in the first
quadrant of the coordinate plane, and
5.G.2.2
interpret coordinate values of points in the
context of the situation.
5.G.2.3
5.G.2.4
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.
5.G.4.
Classify two-dimensional figures in a
hierarchy based on properties.
Label the axes, quadrants and origin on the
coordinate plane.
Identify ordered pairs.
Plot points on the coordinate plane.
Represent real-world and mathematical problems by
locating or graphing points in the first quadrant of a
coordinate plane.
Represent real-world and mathematical problems by
locating and graphing points in the first quadrant of
a coordinate plane.
Interpret coordinate values in the context of the
situation.
Create real-world and mathematical problems that
require locating and graphing points in the first
quadrant of the coordinate plane.
I Can:
5.G.3.1
Understand that shared attributes categorize twodimensional figures.
I Can:
5.G.4.1
5.G.4.2
5.G.4.3
Classify two-dimensional figures based on
properties.
Classify two-dimensional figures in a hierarchy
based on properties.
Demonstrate that attributes belonging to a category
of two-dimensional also belong to all subcategories
of that category.
Latest Revision 6/18/2013
I Can Statements are in draft form due to the iterative nature of the item development process.
8
Common Core State Standards for
Mathematics (Outcome Based)
“I Can” Statements
5.G.4.4
Use appropriate tools to determine similarities and
differences between categories and subcategories.
Latest Revision 6/18/2013
I Can Statements are in draft form due to the iterative nature of the item development process.
9
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
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,
1
Standards for Mathematical Practice
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.
2
CCSS Standards for Mathematical Practice
Questions for Teachers to Ask
1.Make sense of problems and persevere in
solving them
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?
2. Reason abstractly and quantitatively
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.
3. Construct viable arguments and critique
the reasoning of others
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?
4. Model with mathematics
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
*It is important that the teacher poses tasks that
involve arguments or critiques
5. Use appropriate tools strategically
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?
6. Attend to precision
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?
7. Look for and make use of structure
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)
8. Look for and express regularity in
repeated reasoning
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)