4 grade math unit guide core/pacing guides...4th grade math unit guide 2014-2015 . ... ......

4th Grade

Math Unit Guide

2014-2015

Jackson County School District Year At A Glance 4th Grade Math

Unit 1 Applying place value concepts in whole number addition and subtraction 10 Days Unit 2 Exploring multiples and factors 7 Days Unit 3 Using multiplication and division strategies with larger numbers 25 Days Unit 4 Decomposing and composing fractions for addition and subtraction 8 Days Unit 5 Understanding fraction equivalence and comparison 12 Days Unit 6 Introducing measurement conversions 10 Days Unit 7 Solving problems using multiplicative comparison 10 Days Unit 8 Solving measurement problems using the four operations 8 Days Unit 9 Solving addition and subtraction word problems involving fractions 10 Days and mixed numbers Unit 10 Angle measurement 12 Days Unit 11 Multiplying fractions by whole numbers 10 Days Unit 12 Comparing decimal fractions and understanding notation 10 Days Unit 13 Recognizing and analyzing attributes of 2-dimensional shapes 10 Days Unit 14 Problem solving with whole numbers 15 Days

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4th Grade Math

Sequenced Units for the Common Core State Standards in Mathematics Grade 4 In the years prior to Grade 4, students gained an understanding of multiplication and division of whole numbers, generalized strategies for addition and subtraction to multi-‐digit numbers, developed understanding of fractions as numbers, and reasoned with shapes and their attributes. They used arrays and the concept of area to develop computational strategies for multiplication and division.

Throughout Grade 4, students continue to develop their understanding of number. They generalize their understanding of place value to 1,000,000. Students extend their understanding of the four operations to include multiplicative compare problems, operations with multi-‐digit numbers, and multiplying fractions by whole numbers. Students further develop their understanding of fractions to include addition of fractions with like denominators and comparison and ordering of fractions with either like numerators or like denominators. The geometry focus in Grade 4 is on reasoning about angle measurement and lines.

This document reflects our current thinking related to the intent of the Common Core State Standards for Mathematics (CCSSM) and assumes 160 days for instruction, divided among 14 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 4, students should continue to develop proficiency with the Common Core's eight Standards for Mathematical Practice:

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

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

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

Unit 1: Applying place value concepts in whole number addition and subtraction Suggested number of days: 10

Learning Targets Notes/Comments Unit Materials and Resources

Unit Overview: The focus of this unit is to provide students time to develop and practice efficient addition and subtraction of multi-‐digit whole numbers while developing place value concepts. Common Core State Standards for Mathematical Content

Number and Operations in Base Ten2 - 4.NBT A. Generalize place value understanding for multi-‐digit whole numbers. (M) 1. Recognize that in a multi-‐digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

2. Read and write multi-‐digit whole numbers using base-‐ten numerals, number names, and expanded form. Compare two multi-‐digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

4.NBT.1.1 4.NBT.1.2 4.NBT.1.3

Recognize a digit in one place represents 10 times as much as it represents in the place to the right (3 digit numbers). Recognize a digit in one place represents 10 times as much as it represents in the place to the right (4 digit numbers). Recognize a digit in one place represents 10 times as much as it represents in the place to the right (multi-‐digit numbers).

4.NBT.2.1 4.NBT.2.2

Read, write, and compare multi-‐digit numbers in expanded form. (MS) Read, write, and compare multi-‐digit numbers using base ten numerals (standard form). (MS)

4.NBT.2.3 Read, write, and compare multi-‐digit numbers in word form. (MS)

4.NBT.2.4 Compare multi-‐digit numbers using <, >, =. (MS)

4.NBT.A.1 will be revisited in unit 6 connected to conversions within the metric system of measurement.

Videos www.khanacademy.org www.teachingchannel.org www.youtube.com Math Fact Fluency Practice www.mathwire.com www.oswego.org/ocsd-‐web/games/ http://mathfactspro.com/mathfluencygame.html#/math-‐facts-‐addition-‐games http://jerome.northbranfordschools.org/Content/Math_Fact_Fluency_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathematics.org/4 https://learnzillion.com/ www.AECSD4thGradeMathematicsdoc http://maccss.ncdpi.wikispaces.net/Fourth+Grade www.dpi.state.nc.us http://harcourtschool.com/search/search.html

3. Use place value understanding to round multi-‐digit whole numbers to any place.

B. Use place value understanding and properties of operations to perform multi-‐digit arithmetic. (M)

4. Fluently add and subtract multi-‐digit whole numbers using the standard algorithm. NOTE: 2Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

Common Core State Standards for Mathematical Practice 6. Attend to precision. 8. Look for and express regularity in repeated reasoning.

4.NBT.3.1 Round multi-‐digit numbers up to the millions place. (MS)

4.NBT.4.1 Fluently add multi-‐digit numbers up to millions place. (MS)

4.NBT.4.2 Fluently subtract multi-‐digit numbers up to millions place. (MS)

4.NBT.A.3 will be revisited in unit 7 with multiplication and division as a context.

4.NBT.B.4 will be revisited in unit 8 and finalized in unit 14 for fluency in addition and subtraction of multi-‐digit whole numbers.

Students use the structure of the base-‐ten system to generalize their strategies and to discuss reasonableness of their computations and work towards fluency (MP.6, MP.8).

www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐four http://www.onlinemathlearning.com/common-‐core-‐grade4.html http://www.mathgoodies.com/standards/alignments/grade4.html http://www.k-‐5mathteachingresources.com/4th-‐grade-‐number-‐activities.html http://illuminations.nctm.org

Vocabulary Essential Questions • Base ten numeral • Compare • Digit • Equal to • Estimate • Exact • Expanded form • Fluent • Greater than • Inverse operation • Less than • Millions • Multi digit • Number names • Place value • Round

• How would you explain the base-‐ten place value system to another student? • What are different ways to represent multi-‐digit whole numbers up to one

million? • How can you use place value to compare two multi-‐digit whole numbers and why? • How is rounding useful in our everyday lives?

Formative Assessment Strategies

• 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 2: Exploring multiples and factors Suggested number of days: 7


Unit Overview: In this unit students develop understanding of multiples and factors, applying their understanding of multiplication from the previous year. This understanding lays a strong foundation for generalizing strategies learned in previous grades to develop, discuss, and use efficient, accurate, and generalizable computational strategies involving multi-‐digit numbers. These concepts and the terms "prime" and "composite" are new to Grade 4, so they are introduced early in the year to give students ample time to develop and apply this understanding. Common Core State Standards for Mathematical Content 0perations and Algebraic Thinking - 4.0A B. Gain familiarity with factors and multiples. (S) 4. Find all factor pairs for a whole number in the range 1-‐100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-‐100 is a multiple of a given one-‐digit number. Determine whether a given whole number in the range 1-‐100 is prime or composite.

C. Generate and analyze patterns. (A) 5. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule "Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

4.OA.4.1 Determine if a whole number

(1-‐100) is a multiple of a given 1 digit number (ex. – Is 56 a multiple of 7? Is 45 a multiple of 2) (MS)

4.OA.4.2 Find all factor pairs for a whole number up to 100 (ex. 56 = __ x __) (MS)

4.OA.4.3 Determine if a whole number (1-‐100) is prime or composite.

4.OA.4.4 Recognize that a whole number (1-‐100) is a multiple of each of its factors. (MS)

4.OA.5.1 Use a rule to create a number or shape pattern. (MS)

4.OA.5.2 Determine if there are other

relationships within a pattern (ex.4, 8, 16, 32… -‐ all even. 5, 12, 19, 26… -‐ odd/even).

4.OA.5.3 Express a pattern using a formula.

While working on 4.0A.C.5, students use manipulatives to determine whether a number is prime or composite. Although there are shape patterns in arrays, the focus of this unit is number patterns. 4.0A.C.5 is repeated in unit 13, where the focus will be on identifying shape patterns.

Videos www.khanacademy.org www.teachingchannel.org www.youtube.com Math Fact Fluency Practice www.mathwire.com www.oswego.org/ocsd-‐web/games/ http://mathfactspro.com/mathfluencygame.html#/math-‐facts-‐addition-‐games http://jerome.northbranfordschools.org/Content/Math_Fact_Fluency_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathematics.org/4 https://learnzillion.com/ www.AECSD4thGradeMathematicsdoc http://maccss.ncdpi.wikispaces.net/Fourth+Grade www.dpi.state.nc.us http://harcourtschool.com/searc

Common Core State Standards for Mathematical Practice 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure.

The focus of this unit is not necessarily to become fluent in finding all factor pairs, but to use student's understanding of the concept and language to discuss the structure of multiples and factors (MP.3, MP.7).

h/search.html www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐four http://www.onlinemathlearning.com/common-‐core-‐grade4.html http://www.mathgoodies.com/standards/alignments/grade4.html http://www.k-‐5mathteachingresources.com/4th-‐grade-‐number-‐activities.html http://illuminations.nctm.org

Vocabulary Essential Questions • Addition • Factor • Formula • Multi-‐step • Multiple • Multiplication • Pattern • Subtraction • Symbol

• 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 place value patterns repeated in large numbers? • How can a number be broken down into its smallest factors? • How are the four basic operations related to one another? • How do number properties assist in computation? • Is estimation more appropriate than finding an exact answer? • How do we use ordinal numbers in everyday life? • Where do we see numerals in the real world?

• What do numerals represent?


• One Sentence Summary -‐ Students are asked to write a summary sentence that answers the “who, what where, when, why, how” questions about the topic. • Summary Frames -‐ Description: A ___________ is a kind of____________ that ... Compare/Contrast, Problem/Solution, Cause/Effect. • One Word Summary -‐ Select (or invent) one word which best summarizes a topic. • Think/Pair/Share and Turn to your partner -‐ Teacher gives direction to students. Students formulate individual response, and then turn to a partner to share their answers. Teacher

calls on several random pairs to share their answers with the class.

Unit 3: Using multiplication and division strategies with larger numbers Suggested number of days: 25


Unit Overview: In this unit students continue using computational and problem-‐solving strategies, with a focus on building conceptual understanding of multiplication of larger numbers and division with remainders. Area and perimeter of rectangles provide one context for developing such understanding. Common Core State Standards for Mathematical Content Operations and Algebraic Thinking - 4.OA A. Use the four operations with whole numbers to solve problems. (M) 3. Solve multistep word problems posed with whole numbers and having whole-‐number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

4.OA.3.1 Solve multi-‐step word problems using

the four operations with whole numbers (3 or 4 digits by a 1 digit number or two two-‐digit numbers)

4.0A.3.2 Interpret remainders in various situations.

4.OA.3.3 Find whole number quotients without remainders (3 digit dividends and one digit divisor). (MS)

4.OA.3.4 Find whole number quotients with remainders (3 digit dividends and one digit divisor). (MS)



4.OA.3.7 Justify an answer based upon the interpretation of remainders.

4.OA.3.8

Justify an answer using mental math and estimation.

4.OA.A.3 is the first time students are expected to interpret remainders based upon the context. All four operations will be addressed in unit 8, and the standard will be finalized in unit 14.


Number and Operations in Base Ten2 - 4.NBT B. Use place value understanding and properties of operations to perform multi-‐digit arithmetic. (M) 5. Multiply a whole number of up to four digits by a one-‐digit whole number, and multiply two two-‐digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

6. Find whole-‐number quotients and remainders with up to four-‐digit dividends and one-‐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. NOTE: 2Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

4.NBT.5.1 Multiply a 4 digit number by a 1 digit number.

4.NBT.5.2 Illustrate and explain multiplication using rectangular arrays.

4.NBT.5.3 Illustrate and explain multiplication using area models.

4.NBT.5.4 Apply the properties of operations to multiply numbers. (MS)

4.NBT.5.5 Multiply 2, two digit numbers (ex. 23 x 45). (MS)

4.NBT.5.6 Multiply numbers using written equations. (MS)

4.NBT.6.1 Divide up to 4 digit number by a 1 digit

divisor. (MS)

4.NBT.6.2 Apply the properties of operations to divide 4 digit numbers. (MS)

4.NBT.6.3 Apply strategies based on place value to divide up to 4 digit number by a 1 digit divisor. (MS)

4.NBT.6.4 Explore different strategies for the division of 4 digit dividends and 1 digit divisors. (MS)

4.NBT.6.5 Illustrate and explain division with a rectangular array.

4.NBT.6.6 Illustrate and explain division with an area model.

4.NBT.6.7 Illustrate and explain division with an equation.

4.NBT.6.8 Explore the relationship between multiplication and division. (MS)

www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐four http://www.onlinemathlearning.com/common-‐core-‐grade4.html http://www.mathgoodies.com/standards/alignments/grade4.html http://www.k-‐5mathteachingresources.com/4th-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/ http://www.mathplayground.com/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/index.php/common-‐core-‐standards http://map.mathshell.org/materials/stds.php#standard1159

Measurement and Data - 4.MD A. Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. (S) 3. Apply the area and perimeter formulas for

rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

Common Core State Standards for Mathematical Practice

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

2. Reason abstractly and quantitatively. 8. Look for and express regularity in repeated reasoning.

4.MD.3.1 4.MD.3.2

Calculate the area of a rectangle using the formula A=L x W or A= B x H when side lengths are given. Solve for the missing side length of a rectangle using the formula A=L x W or A=B x H when the area is given along with one other dimension.

4.MD.3.3 4.MD.3.4

Calculate the perimeter of a rectangle using the formula P=S+S+S+S or P=2L + 2W when side lengths are given. Solve for the missing side length of a rectangle using the formula P=S+S+S+S or P=2L + 2W when the perimeter is given along with one other dimension.

4.MD.3.5 Apply the area and perimeter formula to solve real-‐world problems.

4.MD.A.3 provides the context of area and perimeter of rectangles to use for problem solving. Students are first introduced to formulas in this unit and make sense of the formulas using their prior work with area and perimeter. Students make sense of multi-‐step problems (MP.1) and reason about how the formulas connect to the context (MP.2). The use of generalized strategies and formulas provides an opportunity to investigate and use regularity in repeated reasoning (MP.8).

Vocabulary Essential Questions • Area Model • Array • Difference • Dimension • Division • Divisor • Equation • Estimation • Formula • Mental math • Multi step word problem • Perimeter • Quotient • Rectangular array • Remainder • Sum • Total • Unit

• How are perimeter and area similar? • How would you determine if an answer is reasonable? • How is your strategy for solving perimeter connected to a formula? • How do you use weight and measurement in your life? • 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?


• 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 4: Decomposing and composing fractions for addition and subtraction Suggested number of days: 8


Unit Overview: In this unit students extend their prior knowledge of unit fractions with denominators of 2, 3, 4, 6, and 8 from Grade 3 to include denominators of 5, 10, 12, and 100. In Grade 4, they use their understanding of partitioning to find unit fractions to compose and decompose fractions in order to add fractions with like denominators. This is foundational for further work with fractions later in the year, such as comparing fractions and multiplying fractions by a whole number. Common Core State Standards for Mathematical Content

Number and Operations-‐Fractions3 - 4.NF B. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. (M) 3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

b. Decompose a fraction into a sum of

fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

NOTE: 3Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

4.NF.3a.1 Add or subtract fractions with like

denominators using manipulatives or visual models. (MS)

4.NF.3a.2 Add and subtract improper fractions with like denominators using manipulatives or visual models.

4.NF.3b.1 Decompose a fraction into a sum of fractions with the same denominator in more than one way.

4.NF.3b.2 Create a visual model to justify decompositions.

Videos www.khanacademy.org www.teachingchannel.org www.youtube.com Math Fact Fluency Practice www.mathwire.com www.oswego.org/ocsd-‐web/games/ http://mathfactspro.com/mathfluencygame.html#/math-‐facts-‐addition-‐games http://jerome.northbranfordschools.org/Content/Math_Fact_Fluency_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathematics.org/4 https://learnzillion.com/ www.AECSD4thGradeMathematicsdoc http://maccss.ncdpi.wikispaces.net/Fourth+Grade

Common Core State Standards for Mathematical Practice 4. Model with mathematics.

Students use visual and concrete models to represent a fractional situation in order to add and subtract fractions (MP.4).

Vocabulary Essential Questions • Decompose • Decomposition • Denominator • Fractional form • Improper fraction • Mixed numbers • Numerator • Sum • Unit fraction

• Why is it important that fractions refer to the same whole when solving problems involving addition and subtraction?

• What are multiple ways to separate (decompose) a fraction? • How does replacing a mixed number with an equivalent fraction help to add and

subtract fractions? • How is adding fractions similar to adding whole numbers? How are they different? • How is subtracting fractions similar to subtracting whole numbers? How are they

different?


• 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? • 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 5: Understanding fraction equivalence and comparison Suggested number of days: 12


Unit Overview:

In this unit students develop an understanding of fraction equivalence and various methods for comparing fractions. Students should understand that when comparing fractions, it is not always necessary to generate equivalent fractions. Other methods, such as comparing fractions to a benchmark, can be used to discuss relative sizes. The justification of comparing or generating equivalent fractions using visual models is an emphasis of this unit.1 Common Core State Standards for Mathematical Content

Number and Operations-‐Fractions3 - 4.NF A. Extend understanding of fraction equivalence and ordering. (M) 1. Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. NOTE: 3 Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

4.NF.1.1 Identify equivalent fractions. 4.NF.1.2 Create visual fraction models to

show why fractions are equal. 4.NF.1.3 Explain why two fractions are

equivalent, but have different denominators.

4.NF.1.4 Create equivalent fractions in number form (ie. ½ = 6/12) by multiplying or dividing the numerator and denominator by the same number.

4.NF.2.1 Compare a fraction to a benchmark fraction such as 1/2, using a visual model.

4.NF.2.2 Compare fractions to a benchmark fraction such as 1/2, using numerical comparison. (ie. 3/6 ____ 7/12)

4.NF.2.3 Use multiples to find a LCD. 4.NF.2.4 Compare fractions using symbols (<,

>, =). 4.NF.2.5 Compare two fractions with different

numerators (like denominators). 4.NF.2.6 Compare two unlike fractions by

creating like denominators.


Common Core State Standards for Mathematical Practice 3. Construct viable arguments and critique the reasoning of others. 5. Use appropriate tools strategically.

4.NF.2.7 Explain that the size of the whole matters when comparing fractions (ie. ½ of a medium pizza is not equal to ½ of a large pizza).

4.NF.2.8 Justify comparisons by using a visual fraction model.

4.NF.2.9 Create a visual model to explain the comparison of fractions.

4.NF.2.10 Compare two unlike fractions using a variety of methods.

Students justify their methods for generating equivalent fractions and comparing fractions by using their conceptual understanding and models (MP.3, MP.5).

h/search.html www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐four http://www.onlinemathlearning.com/common-‐core-‐grade4.html http://www.mathgoodies.com/standards/alignments/grade4.html http://www.k-‐5mathteachingresources.com/4th-‐grade-‐number-‐activities.html http://illuminations.nctm.org/ http://www.coolmath.com/

Vocabulary Essential Questions • Benchmark fractions • Circle • Compare • Denominator • Equivalent • Equivalent fractions • Fourths • Halves • Identity property of multiplication • Number • Numerator • Part • Square • Visual models • Whole

• How can benchmark numbers be used to compare fractions? • What is your strategy for comparing these two fractions? What other strategy can

you use to check your reasoning? • How are two equivalent fractions the same? How do they differ? • Given two equivalent fractions, what is the relationship between the number and

size of the pieces? • When comparing fractions, why do both fractions need to refer to the same

whole? • Why does (doubling, tripling) both the numerator and the denominator of a

fraction produce an equivalent fraction?


• 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.

Unit 6: Introducing measurement conversions Suggested number of days: 10


Unit Overview:

In this unit students build a conceptual understanding of the relative sizes of units of measure within a single system of measurement. Measurement conversions are used to introduce multiplication as a comparison. The concepts in this unit are foundational for the concepts in unit 7 and unit 8. Common Core State Standards for Mathematical Content Operations and Algebraic Thinking - 4.OA A. Use the four operations with whole numbers to solve problems. (M) 1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

Number and Operations in Base Ten2 - 4.NBT A. Generalize place value understanding for multi-‐digit whole numbers. (M) 1. Recognize that in a multi-‐digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division. NOTE: 2Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

4.OA.1.1 Multiply two given numbers (one and

two digits).

4.OA.1.2 Write/compose a multiplication equation.

4.OA.1.3 Interpret a verbal comparison into an equation.

4.OA.1.4 Compare amounts using multiplication. (for example – 5 times as many…)

4.OA.1.5 Change a number sentence into a word sentence.

4.OA.1.6 Translate (give an example of) verbal statements as multiplication statement.

4.NBT.1.1 Identify place values up to 1,000,000. 4.NBT.1.2 Use multiplication by a power of 10 to

determine the value of a digit in a multi-‐digit whole number.

4.OA.A.1 is repeated in unit 11, in which the focus is on multiplication of fractions.2

4.NBT.A.1 was addressed in unit 4, in which the focus was on addition and subtraction. In this unit, metric measurement provides an opportunity to deepen the students' understanding of place value in relation to multiples of 10.


Measurement and Data - 4.MD A. Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. (S)

1. Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-‐ column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...

Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively.

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

4.MD.1.1 Identify and associate units of measurements used to measure length.

4.MD.1.2 Identify and associate units of measurements used to measure capacity.

4.MD.1.3 Identify and associate units of measurements used to measure weight.

4.MD.1.4 Identify and associate units of measurements used to measure time.

4.MD.1.5 Compare units of measurement within a given system (ie. 1 inch < 1 foot).

4.MD.1.6 Convert (change) from a larger unit to a smaller unit.

4.MD.1.7 Create a table to record equivalent measures listing number pairs.

4.MD.A.1 introduces units of measure new to Grade 4.

In this unit students look for patterns in different measurement systems (MP.2, MP.7) and discuss precisely how many times larger one unit is than another (MP.6)

www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐four http://www.onlinemathlearning.com/common-‐core-‐grade4.html http://www.mathgoodies.com/standards/alignments/grade4.html http://www.k-‐5mathteachingresources.com/4th-‐grade-‐number-‐activities.html http://illuminations.nctm.org http://www.coolmath.com/

Vocabulary Essential Questions • Centimeter • Composite • Factors • Foot • Inch • Measurement • Meters • Number line diagram • Prime • Yards

• What is the relationship between factors and their multiples? • How would you identify the multiples of any one-‐digit number? • How do you classify numbers as prime or composite? • How is knowing multiples and factors of a number related to conversion of

measurement? • What is the difference between two times a number and two more than a

number? • What is your strategy for expressing meters to centimeters (e.g., yards to inches,

hours to seconds, etc.)? • How would you represent measurement quantities using a scaled number-‐line

diagram?


• 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.

• 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 7: Solving problems using multiplicative comparison Suggested number of days: 10


Unit Overview:

In this unit students are introduced to multiplicative compare problems, extending their conceptual work with multiplicative comparison from unit 6. For students to develop this concept, they must be provided rich problem situations that encourage them to make sense of the relationships among the quantities involved, model the situation, and check their solution using a different method. CCSSM Table 2 is an important resource for understanding multiplicative comparison problems, which are new to Grade 4 students. Common Core State Standards for Mathematical Content Operations and Algebraic Thinking - 4.OA A. Use the four operations with whole numbers to solve problems. (M) 2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

4.OA.2.1 Identify variables, symbols, &

unknown numbers.

4.OA.2.2 Identify key words and relate words to operations.

4.OA.2.3 Represent word problems and/or equations with pictures and symbols.

4.OA.2.4 Compare amounts with multiplication. (for example – 7 times as many as…) *this is not repeated addition.

4.OA.2.5 Compare amounts with addition (7 more than).

4.OA.2.6 Solve word problems using multiplication (4 digits by 1 digit or 2 digits by 2 digits).

4.OA.2.7 Solve word problems using division (4 digit dividends by 1 digit divisor).

4.OA.2.8 Identify multiplicative comparisons from additive comparisons.

4.OA.A.2 is also addressed in unit 14 because of the time it takes to master the concepts and its importance to future mathematics.4


Number and Operations in Base Ten2 - 4.NBT A. Generalize place value understanding for multi-‐digit whole numbers. (M) 3. Use place value understanding to round multi-‐digit whole numbers to any place. NOTE: 2Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000

Measurement and Data - 4.MD A. Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. (S) 2. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them.

4.NBT.3.1 Round numbers up to the millions place.

4.NBT.3.2 Explain why a number is rounded to a given place.

4.NBT.3.3 Demonstrate understanding of place value using a drawing, chart, table, diagram, etc…

4.MD.2.1 Represent measurement quantities

using diagrams with a measurement scale.

4.MD.2.2 Apply the four operations to solve word problems involving distance.

4.MD.2.3 Apply the four operations to solve word problems involving elapsed time.

4.MD.2.4 Apply the four operations to solve word problems involving liquid volume.

4.MD.2.5 Apply the four operations to solve word problems involving mass.

4.MD.2.6 Apply the four operations to solve word problems involving money.

*Note:

− These problems are limited to converting larger to smaller units.

− These problems include whole numbers, fractions, and decimals.

4.NBT.A.3 was addressed in unit 4 with a focus on addition and subtraction. In this unit, the focus is on multiplication and division.

4.MD.A.2 is used as a context for multiplicative compare problems with whole numbers only. This standard is revisited in unit 8 to include the four operations, and addressed in unit 12 with decimal fractions. Students use charts and diagrams to explain their own methods as well make sense of approaches taken by others (MP.1).

h/search.html www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐four http://www.onlinemathlearning.com/common-‐core-‐grade4.html http://www.mathgoodies.com/standards/alignments/grade4.html http://www.k-‐5mathteachingresources.com/4th-‐grade-‐number-‐activities.html http://illuminations.nctm.org http://www.coolmath.com/

Vocabulary Essential Questions • Comparison • Decimal • Diagram • Equation • Fraction • Place Value • Quantity • Rounding • Symbols • Variable

• Why do I need to add? • Why do I need to subtract? • How can knowing the addition and subtraction facts help me? • How can I use what I know about tens and ones to add and subtract two-‐digit

numbers? • How can I measure length, mass and capacity by using non-‐standard units? • What is perimeter and how is it measured? • 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?

• What benchmarks do I use to estimate the weight of common objects?


• 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 8: Solving measurement problems using the four operations Suggested number of days: 8


Unit Overview:

In this unit students combine competencies from different domains to solve measurement problems using the four operations.S Measurement is included in this unit to provide a context for problem solving. All of the problem types in Table 1 and Table 2 on pages 88 and 89 of the Common Core State Standards for Mathematics should be addressed in this unit. Common Core State Standards for Mathematical Content Operations and Algebraic Thinking - 4.OA A. Use the four operations with whole numbers to solve problems. (M) 3. Solve multistep word problems posed with whole numbers and having whole-‐number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Number and Operations in Base Ten2 - 4.NBT B. Use place value understanding and properties

4.OA.3.1 Add, subtract, multiply and divide with

or without remainders.

4.OA.3.2 Identify key words to decide which operation(s) to use to solve a word problem.

4.OA.3.3 Explain what a remainder is and how it will affect a given problem.

4.OA.3.4 Determine if an answer makes sense, based on the problem.

4.OA.3.5 Justify my answers using mental math and estimation.

4.OA.3.6 Write an equation to solve the word problem using a letter to represent the missing number

4.OA.3.7 Solve multistep word problems with whole numbers.

4.OA.3.8 Calculate long division with remainders.

4.NBT.4.1 Add numbers up to millions place value.

4.NBT.4.2 Subtract numbers up to millions place

4.OA.A.3 and 4.NBT.B.4 are repeated here to include all four operations and will be finalized in unit 14. Repeating these standards throughout the year provides students multiple opportunities to develop these skills-‐which are major areas of focus for this grade level.


of operations to perform multi-‐digit arithmetic. (M) 4. Fluently add and subtract multi-‐digit whole numbers using the standard algorithm.

Measurement and Data - 4.MD A. Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. (S) 2. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively. 6. Attend to precision.

value. 4.NBT.4.3 Justify an answer by using the

relationship between addition and subtraction (inverse operations).

4.MD.2.1 Represent measurement quantities using diagrams with a measurement scale.






*Note:



4.MD.A.2 is repeated from the previous unit, but in this unit the emphasis is on using the four operations and all problem types. This standard will be finalized in unit 12 to include decimal fractions. Students use various diagrams and precise language to solve measurement problems and explain their strategies (MP.1, MP.6). They make connections between abstract representations and the problem situations (MP.2).


Vocabulary Essential Questions • Decimals • Diagram • Distance • Estimation • Equations • Intervals • Liquid Volume • Mass • Measurement • Mental Computation • Money • Multistep Problems • Number Line • Quantity • Remainder • Scale • Unit

• How would you explain the base-‐ten place value system to another student? • What are different ways to represent multi-‐digit whole numbers up to one

million? • How can you use place value to compare two multi-‐digit whole numbers and why? • How is rounding useful in our everyday lives? • What is the relationship between factors and their multiples? • How would you identify the multiples of any one-‐digit number? • How do you classify numbers as prime or composite? • How is knowing multiples and factors of a number related to conversion of

measurement? • What is the difference between two times a number and two more than a

number? • What is your strategy for expressing meters to centimeters (e.g., yards to inches,

hours to seconds, etc.)? • How would you represent measurement quantities using a scaled number-‐line

diagram?


• 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 addition and subtraction word problems involving fractions and mixed numbers Suggested number of days: 10


Unit Overview:

In this unit students will use their understanding of adding and subtracting fractions and generating equivalent fractions to solve problems involving fractions and mixed numbers. Students rely on their previous work with whole numbers as fractions to compose and decompose whole numbers into fractional quantities.6 Data is used in this unit to support

students' understanding of fractional quantities both smaller and larger than 1.7 Common Core State Standards for Mathematical Content

Number and Operations-‐Fractions - 4.NF B. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. (M)

3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

NOTE: Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

4.NF.3c.1 Justify that a mixed number is a whole

number and a fraction by using a visual model.

4.NF.3c.2 Change a mixed number into an improper fraction.

4.NF.3c.3 Change an improper fraction into a mixed number.

4.NF.3d.1 Solve word problems using addition and subtraction of fractions with like denominators using visual models and equations.

Videos www.khanacademy.org www.teachingchannel.org www.youtube.com Math Fact Fluency Practice www.mathwire.com www.oswego.org/ocsd-‐web/games/ http://mathfactspro.com/mathfluencygame.html#/math-‐facts-‐addition-‐games http://jerome.northbranfordschools.org/Content/Math_Fact_Fluency_Practice_Sheets.asp http://www.mathfactcafe.com/ www.factmonster.com Lessons/Activities/Games https://www.illustrativemathematics.org/4 https://learnzillion.com/ www.AECSD4thGradeMathematicsdoc http://maccss.ncdpi.wikispaces.net/Fourth+Grade

Measurement and Data - 4.MD B. Represent and interpret data. (S) 4. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.

Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively. 4. Model with mathematics.

4.MD.4.1 Answer questions about data measured on a line plot.

4.MD.4.2 Create a line plot to display (show) a data set that includes fractions or mixed numbers.

4.MD.4.3 Add and subtract fractions using information from a line plot.

4.MD.B.4 extends students' work from Grade 3 with simple fractions on a line plot (3.MD.B.4) to include eighths and to now solve addition and subtraction problems using the data. Students reason about fractions by using abstract models to represent both the data and the fractional quantities (MP.2, MP.4).

www.dpi.state.nc.us http://harcourtschool.com/search/search.html www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐four http://www.onlinemathlearning.com/common-‐core-‐grade4.html http://www.mathgoodies.com/standards/alignments/grade4.html http://www.k-‐5mathteachingresources.com/4th-‐grade-‐number-‐activities.html http://illuminations.nctm.org

Vocabulary Essential Questions • Data • Decompose • Fractional form • Improper fraction • Length • Line Plot • Measurement • Mixed number • Sum • Unit Fraction

• Why is it important that fractions refer to the same whole when solving problems involving addition and subtraction?

• What are multiple ways to separate (decompose) a fraction? • How does replacing a mixed number with an equivalent fraction help to add and

subtract fractions? • How is adding fractions similar to adding whole numbers? How are they different? • How is subtracting fractions similar to subtracting whole numbers? How are they

different? • What is the purpose of using line plots?


• One Sentence Summary -‐ Students are asked to write a summary sentence that answers the “who, what where, when, why, how” questions about the topic. • Summary Frames -‐ Description: A ___________ is a kind of____________ that ... Compare/Contrast, Problem/Solution, Cause/Effect. • One Word Summary -‐ Select (or invent) one word which best summarizes a topic. • Think/Pair/Share and Turn to your partner -‐ Teacher gives direction to students. Students formulate individual response, and then turn to a partner to share their answers. Teacher

calls on several random pairs to share their answers with the class.

Unit 10: Angle measurement Suggested number of days: 12


Unit Overview:

This unit is an introduction to angles and angle measurement. Students start this unit drawing points, lines, segments, rays and angles since it is foundational to the other standards in this unit. Students use their understanding of equal partitioning and unit measurement to understand angle and turn measure. Common Core State Standards for Mathematical Content Measurement and Data - 4.MD C. Geometric measurement: understand concepts of angle and measure angles. (A) 5. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:

a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a "one-‐degree angle," and can be used to measure angles.

b. An angle that turns through n one-‐degree angles is said to have an angle measure of n degrees.

6. Measure angles in whole-‐number degrees using a protractor. Sketch angles of specified measure.

4.MD.5.1 Identify an angle. 4.MD.5a.1 Recognize that a circle has 360

degrees. 4.MD.5a.2 Explain that an angle measurement is

a fraction of a circle.

4.MD.5b.1 Recognize that angles are measured in degrees within a circle.

4.MD.6.1 Identify benchmark angles (90º, 180º, 270º, 360º).

4.MD.6.2 Measure angles using a protractor. 4.MD.6.3 Sketch angles of a given

measurement (degree) using a protractor.


Geometry - 4.G A. Draw and identify lines and angles, and classify shapes by properties of their lines and angles. (A) 1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-‐dimensional figures.

Common Core State Standards for Mathematical Practice 4. Model with mathematics. 5. Use appropriate tools strategically.

4.G.1.1 Identify points, lines, line segments, rays, angles, perpendicular, and parallel lines in two dimensional figures.

4.G.1.2 Draw points, lines, line segments, rays, angles, perpendicular, and parallel lines in two dimensional figures

4.G.1.3 Identify types of angles (right, acute, obtuse) in two-‐dimensional figures.

In this unit, 4.G.A.1 focuses on drawing points, lines, line segments, rays, and different types of angles. The standard will be addressed in its entirety in unit 13. Students select and use a protractor to measure angles and represent the angles with drawings (MP.4, MP.S).


Vocabulary Essential Questions • Acute • Angle • Lines • Line Segments • Obtuse • Parallel • Perpendicular • Points • Rays • Right Angle • Segments • Two dimensional figures

• What is a distinguishing feature of this pattern? • What would a pattern look like using this rule? • How can you classify a 2-‐D figure based on its lines and angle size? • How are angles classified? • How are angles measured? • How are points, lines, line segments, rays, and angles related?


• 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 11: Multiplying fractions by whole numbers Suggested number of days: 10


Unit Overview:

In this unit students apply their understanding of composing and decomposing fractions to develop a conceptual understanding of multiplication of a fraction by a whole number. Students also use and extend their previous understandings of operations with whole numbers and relate that understanding to fractions. Common Core State Standards for Mathematical Content Operations and Algebraic Thinking - 4.OA A. Use the four operations with whole numbers to solve problems. (M) 1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

Number and Operations-‐Fractions - 4.NF B. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. (M) 4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4).

4.OA.1.1 Multiply two given numbers (one and

two digits).

4.OA.1.2 Write/compose a multiplication equation.

4.OA.1.3 Interpret a verbal comparison into an equation.

4.OA.1.4 Compare amounts using multiplication. (for example – 5 times as many…)

4.OA.1.5 Change a number sentence into a word sentence.

4.OA.1.6 Translate (give an example of) verbal statements as multiplication statement.

4.NF.4a.1 Multiply a whole number by a fraction by changing the whole number into a fraction (ie. 5 x ¾ = 5/1 x ¾).

4.OA.A.1 is readdressed in this unit to include multiplication of fractions and apply the understanding of "times as much" (i.e. multiplication as comparison) to multiplying a fraction by a whole number.


b. Understand a multiple of a/b as a multiple of 1lb, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 x (2/5) as 6 x (1/5), recognizing this product as 6/5. (In general, n x (a/b) = (n x a)/b.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

NOTE: Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 6. Attend to precision.

4.NF.4b.1 Create a fraction from a whole number by placing the whole number over 1 (ie. 5 = 5/1).

4.NF.4b.2 Use models to represent a fraction times a whole number.

4.NF.4c.1 Solve multiplication problems by multiplying the whole number by the numerator.

4.NF.4c.2 Solve multiplication word problems involving fractions and whole numbers using visual models.

4.NF.4c.3 Solve multiplication word problems involving fractions and whole numbers using equations.

Students use precise language to communicate their comprehension of the problem situations and defend their various solution methods (MP.1, MP.6)


Vocabulary Essential Questions • Benchmark fractions • Compare • Denominator • Equivalent • Fractions • Identity property of multiplication • Numerator • Part • Visual Models

• What is the relationship between factors and their multiples? • How would you identify the multiples of any one-‐digit number? • What is your strategy for comparing these two fractions? What other strategy can

you use to check your reasoning? • How are two equivalent fractions the same? How do they differ? • Given two equivalent fractions, what is the relationship between the number and

size of the pieces? • When comparing fractions, why do both fractions need to refer to the same

whole? • Why does (doubling, tripling) both the numerator and the denominator of a

fraction produce an equivalent fraction? Formative Assessment Strategies

• 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 12: Comparing decimal fractions and understanding notation Suggested number of days: 10


Unit Overview:

In this unit of study students use their previous work with fractions to represent special fractions in a new way. Students use their understanding of equivalent fractions to begin to use decimal notation-‐however, it is not the intent at this grade level to connect this notation to the base-‐ten system. The focus is on solving word problems involving simple fractions or decimals. Work with money can support this work with decimal fractions. Common Core State Standards for Mathematical Content

Number and Operations-‐Fractions - 4.NF C. Understand decimal notation for fractions, and compare decimal fractions. (M) 5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. NOTE: Students who can generate

equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.

6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

7. Compare two decimals to hundredths by reasoning about their size. Recognize that

4.NF.5.1 Use base ten models to represent

fractions. 4.NF.5.2 Convert unlike denominators to like

denominators (10,100) and add fractions.

4.NF.5.3 Change a fraction with a denominator of 10 into a fraction with a denominator of 100 by multiplying the numerator and denominator by 10.

4.NF.6.1 Write a fraction as a decimal to represent its place value.

4.NF.6.2 Convert between decimals and fractions (ie. 0.62 = 62/100).

4.NF.6.3 Locate fractions and decimals on a number line (tenths and hundredths).


comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. NOTE: Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Measurement and Data - 4.MD A. Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. (M) 2. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

Common Core State Standards for Mathematical Practice 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure.

4.NF.7.1 Compare two decimals to the hundredth place using a hundreds grid and using symbols (<,>,=).

4.NF.7.2 Recognize that in order to compare decimals, they must have the same place value.

4.NF.7.3 Create a model to justify an answer.

4.MD.2.1 Represent measurement quantities using diagrams with a measurement scale.






*Note:



4.MD.A.2 was addressed in unit 7. It is important to note that students are not expected to do computations with quantities in decimal notation. Students can use visual fraction models to solve problems involving simple fractions or decimals. Students compare decimals fractions and justify their comparisons using either a fraction model or their understanding of the notation (MP.3, MP.7).


Vocabulary Essential Questions • Denominator • Distance • Equivalent • Estimate • Fractions • Mass • Measure • Numerator • Standard • Units • Volume • Weight

• 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 can models help us understand the addition and subtraction of decimals? • How many ways can we use models to determine and compare equivalent

fractions? • How do you use weight and measurement in your life? • 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?


• 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 13: Recognizing and analyzing attributes of 2-‐dimensional shapes Suggested number of days: 10


Unit Overview:

In this unit students develop their spatial reasoning skills by using a wide variety of attributes to talk about 2-‐dimensional shapes. Students analyze geometric figures based on angle measurement, parallel and perpendicular lines, and symmetry. Common Core State Standards for Mathematical Content 0perations and Algebraic Thinking - 4.0A C. Generate and analyze patterns. (A) 5. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule "Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

Measurement and Data - 4.MD C. Geometric measurement: understand concepts of angle and measure angles. (A) 7. Recognize angle measure as additive. When an angle is decomposed into non-‐overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

Geometry - 4.G A. Draw and identify lines and angles, and classify shapes by properties of their lines and angles. (A)

4.OA.5.1 Use rules to explain a pattern. 4.OA.5.2 Create a number or shape pattern. 4.OA.5.3 Create and explain a number or shape

pattern.

4.OA.5.4 Identify a pattern within a pattern. (ex. 1,4,7,10)

4.MD.7.1 Decompose (separate) angles into smaller angles.

4.MD.7.2 Add angle measures to make a larger angle.

4.MD.7.3 Use addition and subtraction to find unknown angles in real-‐world and mathematical problems.

4.MD.7.4 Use an equation with a symbol for the unknown angle measure.

4.G.1.1 Identify points, lines, line segments, rays, angles, perpendicular, and parallel lines in two dimensional figures.

In this unit, 4.0A.C.5 includes repeated and growing shape patterns.10

4.G.A.1 was first addressed in unit 10, and is addressed in its entirety


1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-‐dimensional figures.

2. Classify two-‐dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

3. Recognize a line of symmetry for a two-‐dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-‐symmetric figures and draw lines of symmetry.

Common Core State Standards for Mathematical Practice 3. Construct viable arguments and critique the reasoning of others.

5. Use appropriate tools strategically. 7. Look for and make use of structure.

4.G.1.2 Draw points, lines, line segments, rays, angles, perpendicular, and parallel lines in two dimensional figures

4.G.1.3 Identify types of angles (right, acute, obtuse) in two-‐dimensional figures.

4.G.2.1 Identify two dimensional shapes that have parallel or perpendicular lines.

4.G.2.2 Identify two dimensional shapes that have angles of a specific size.

4.G.2.3 Recognize and label a right triangle. 4.G.2.4 Classify two dimensional shapes on the

presence or absence of designated lines and angles.

4.G.3.1 Recognize a line of symmetry. 4.G.3.2 Draw a line of symmetry. 4.G.3.3 Recognize figures that have lines of

symmetry.

4.G.3.4 Draw figures that have lines of symmetry.

in this unit to include perpendicular and parallel lines. The concepts in this unit lend themselves to using technology applications (MP.S). Students understand that geometric figures can be classified by analyzing various properties (MP.7) and justify their conclusions by using viable arguments (MP.3).


Vocabulary Essential Questions • Angle • Lines • Parallel • Pattern

• Are patterns important in the world today? • Where in the real world will I find patterns? • How are angles measured? • How are angles classified?

• Perpendicular • Points • Segments • Shapes • Symmetry • Two dimensional shapes

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

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

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


• 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.

Unit 14: Problem solving with whole numbers Suggested number of days: 15


Unit Overview:

This is a culminating unit in which students focus on problem solving in order to demonstrate fluency with the standard algorithms in addition and subtraction. They demonstrate computational fluency with all problem types. All standards in this unit have been addressed in prior units. These concepts require greater emphasis due to the depth of the ideas, the time they take to master, and/or their importance to future mathematics. Common Core State Standards for Mathematical Content Operations and Algebraic Thinking - 4.OA A. Use the four operations with whole numbers to solve problems. (M) 2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. 1 NOTE: 1See Glossary, Table 2.

3. Solve multistep word problems posed with whole numbers and having whole-‐number

4.OA.2.1 Identify variables, symbols, & unknown

numbers.

4.OA.2.2 Identify key words and relate words to operations.

4.OA.2.3 Represent word problems and/or equations with pictures and symbols.

4.OA.2.4 Compare amounts with multiplication. (for example – 7 times as many as…) *this is not repeated addition.

4.OA.2.5 Compare amounts with addition (7 more than).

4.OA.2.6 Solve word problems using multiplication (4 digits by 1 digit or 2 digits by 2 digits).

4.OA.2.7 Solve word problems using division (4 digit dividends by 1 digit divisor).

4.OA.2.8 Identify multiplicative comparisons from additive comparisons.

4.OA.3.1 Add, subtract, multiply and divide with or without remainders.


answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Number and Operations in Base Ten - 4.NBT B. Use place value understanding and properties of operations to perform multi-‐digit arithmetic. (M) 4. Fluently add and subtract multi-‐digit whole numbers using the standard algorithm.

5. Multiply a whole number of up to four digits by a one-‐digit whole number, and multiply two two-‐digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the

4.OA.3.2 Identify key words to decide which operation(s) to use to solve a word problem.

4.OA.3.3 Explain what a remainder is and how it will affect a given problem.

4.OA.3.4 Determine if an answer makes sense, based on the problem.

4.OA.3.5 Justify my answers using mental math and estimation.

4.OA.3.6 Write an equation to solve the word problem using a letter to represent the missing number

4.OA.3.7 Solve multistep word problems with whole numbers.

4.OA.3.8 Calculate long division with remainders.

4.NBT.4.1 Add numbers up to millions place value.

4.NBT.4.2 Subtract numbers up to millions place value.

4.NBT.4.3 Justify an answer by using the relationship between addition and subtraction (inverse operations).

4.NBT.5.1 Multiply a 4 digit number by a 1 digit number.

4.NBT.5.2 Illustrate and explain multiplication using rectangular arrays.

www.tucerton.k12.nj.us/tes_curriculum/mathematics_2/curriculum-‐math-‐grade-‐four http://www.onlinemathlearning.com/common-‐core-‐grade4.html http://www.mathgoodies.com/standards/alignments/grade4.html http://www.k-‐5mathteachingresources.com/4th-‐grade-‐number-‐activities.html http://illuminations.nctm.org http://www.coolmath.com/ http://www.mathplayground.com/ http://www.funbrain.com/ http://www.aaamath.com/ http://insidemathematics.org/index.php/common-‐core-‐standards http://map.mathshell.org/materials/stds.php#standard1159

calculation by using equations, rectangular arrays, and/or area models.

6. Find whole-‐number quotients and remainders with up to four-‐digit dividends and one-‐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.

NOTE: Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

Common Core State Standards for Mathematical Practice 2. Reason abstractly and quantitatively. 8. Look for and express regularity in repeated reasoning.


4.NBT.5.4 Apply the properties of operations to multiply numbers.

4.NBT.5.5 Multiply 2, two digit numbers (ex. 23 x 45).

4.NBT.5.6 Multiply numbers using written equations.

4.NBT.6.1 Divide up to 4 digit number by a 1 digit divisor.

4.NBT.6.2 Apply the properties of operations to divide 4 digit numbers.

4.NBT.6.3 Apply strategies based on place value to divide up to 4 digit number by a 1 digit divisor.

4.NBT.6.4 Explore different strategies for the division of 4 digit dividends and 1 digit divisors.




4.NBT.6.8 Explore the relationship between multiplication and division.

In demonstrating fluency, students explain and flexibly use properties of operations and place value to solve problems, looking for shortcuts and applying generalized strategies (MP.2, MP.8).

Vocabulary Essential Questions • Array • Dividend • Division • Equation • Equivalent • Multiply • Multiplication • Place Value • Properties • Rectangular Array

• 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?


• 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.

Key: Major Clusters; Supporting Clusters; Additional Clusters

FOURTH GRADE CRITICAL AREAS OF FOCUS CRITICAL AREA OF FOCUS #1 Developing an understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends Students generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in each place. They apply their understanding of models for multiplication (equal-sized groups, arrays, area models), place value, and properties of operations, in particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalizable methods to compute products of multi-digit whole numbers. Depending on the numbers and the context, they select and accurately apply appropriate methods to estimate or mentally calculate products. They develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to solve problems. Students apply their understanding of models for division, place value, properties of operations, and the relationship of division to multiplication as they develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multi-digit dividends. They select and accurately apply appropriate methods to estimate and mentally calculate quotients, and interpret remainders based upon the context.

Operations and Algebraic Thinking 4.OA Use the four operations with whole numbers to solve problems.

1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Gain familiarity with factors and multiples. 4. Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a

multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1– 100 is prime or composite.

Number and Operations in Base Ten 4.NBT Generalize place value understanding for multi-digit whole numbers.

1. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

2. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

3. Use place value understanding to round multi-digit whole numbers to any place. Use place value understanding and properties of operations to perform multi-digit arithmetic.

5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two- digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.


FOURTH GRADE CRITICAL AREAS OF FOCUS

CRITICAL AREA OF FOCUS #1, CONTINUED

6. Find whole-number quotients and remainders with up to four-digit dividends and one-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.

Measurement and Data 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

2. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

3. Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.


FOURTH GRADE CRITICAL AREAS OF FOCUS CRITICAL AREA OF FOCUS #2 Developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions. Students extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number.

Number and Operations—Fractions 4.NF Extend understanding of fraction equivalence and ordering.

1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b. a. Understand addition and subtraction of fractions as joining and separating parts referring to

the same whole. b. Decompose a fraction into a sum of fractions with the same denominator in more than one

way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 1. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to

represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

2. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

3. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?


FOURTH GRADE CRITICAL AREAS OF FOCUS CRITICAL AREA OF FOCUS #2, CONTINUED Understand decimal notation for fractions, and compare decimal fractions.

5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.

6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

Measurement and Data 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

1. Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), …

2. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

Represent and interpret data. 4. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve

problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.


FOURTH GRADE CRITICAL AREAS OF FOCUS CRITICAL AREA OF FOCUS #3 Understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry. Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing two-dimensional shapes, students deepen their understanding of properties of two- dimensional objects and the use of them to solve problems involving symmetry.

Measurement and Data 4.MD Geometric measurement: understand concepts of angle and measure angles.

5. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: a. An angle is measured with reference to a circle with its center at the common endpoint of the

rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.

b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

6. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

7. Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

Geometry 4.G Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

2. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

3. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.


FOURTH GRADE CRITICAL AREAS OF FOCUS STANDARDS AND CLUSTERS BEYOND THE CRITICAL AREAS OF FOCUS Analyzing patterns Students create patterns with numbers (or shapes) that satisfy a given rule. They analyze these patterns for their characteristics.

Operations and Algebraic Thinking 4.OA Generate and analyze patterns.

5. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

Adding and subtracting multi-digit whole numbers Students efficiently and effectively add and subtract multi-digit whole numbers.

Number and Operations in Base Ten 4.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic.

4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.

Performance Level Descriptors – Grade 4 Mathematics

July 2013 Page 1 of 18

Grade 4 Math : Sub-Claim A The student solves problems involving the Major Content for the grade/course with connections to the

Standards for Mathematical Practice.

Level 5: Distinguished

Command Level 4: Strong Command

Level 3: Moderate Command

Level 2: Partial Command

Fractions and Decimals 4.NF.1-2 4.NF.2-1 4.NF.A.Int.1 4.NF.5 4.NF.6 4.NF.7

Compares decimals to hundredths; uses decimal notations for fractions (tenths and hundredths); compares fractions, with like or unlike numerators and denominators, by creating equivalent fractions with common denominators, comparing to a benchmark fraction and generating equivalent fractions. Recognizes that decimals and fractions must refer to the same whole in order to compare. Shows results using symbols. Demonstrates the use of conceptual understanding of fractional equivalence and ordering when solving simple word problems requiring fraction

Compares decimals to hundredths; uses decimal notations for fractions (tenths and hundredths); compares fractions, with like or unlike numerators and denominators, by creating equivalent fractions with common denominators, comparing to a benchmark fraction and generating equivalent fractions. Recognizes that decimals and fractions must refer to the same whole in order to compare. Shows results using symbols. Demonstrates the use of conceptual understanding of fractional equivalence and ordering when solving simple word problems requiring fraction

Given a visual model and/or manipulatives, compares decimals to hundredths; uses decimal notations for fractions (tenths and hundredths); compares fractions, with like or unlike numerators and denominators, by creating equivalent fractions with common denominators and comparing to a benchmark fraction. Recognizes that decimals and fractions must refer to the same whole in order to compare. Shows results using symbols. Solves simple word problems requiring fraction comparison.

Given a visual model and/or manipulatives, compares decimals to hundredths; uses decimal notations for fractions (tenths and hundredths); compares fractions, with like or unlike numerators and denominators by comparing to a benchmark fraction. Recognizes that decimals and fractions must refer to the same whole in order to compare. Shows results using symbols. Solves simple word problems requiring fraction comparison with scaffolding.









comparison. Converts a simple fraction to a denominator of 10 or 100 and rewrites as a decimal (e.g.,1/2, ¼,1/20). Adds fractions with denominators of 10 and 100.

comparison. Adds fractions with denominators of 10 and 100.

Building Fractions 4.NF.3a 4.NF.3b-1 4.NF.3c 4.NF.3d

Creates and solves mathematical and real-world problems involving the addition and subtraction of fractions and mixed numbers with like denominators by joining and separating parts referring to the same whole. Decomposes a fraction into a sum of fractions with the same denominator in more than one way and records the decomposition using an equation.

Understands and solves mathematical and real-world problems involving the addition and subtraction of fractions and mixed numbers with like denominators by joining and separating parts referring to the same whole. Decomposes a fraction into a sum of fractions with the same denominator in more than one way and records the decomposition using an equation.

Using visual models and/or manipulatives, solves mathematical and word problems involving the addition and subtraction of fractions and mixed numbers with like denominators by joining and separating parts referring to the same whole. Decomposes a fraction into a sum of fractions with the same denominator in more than one way and records the decomposition using an equation.

Using visual models and/or manipulatives, solves mathematical problems involving the addition and subtraction of fractions with like denominators by joining and separating parts referring to the same whole. Decomposes a fraction into a sum of fractions with the same denominator in more than one way and records the decomposition using an equation.









Multiplying Fractions 4.NF.4a 4.NF.4 b-1 4.NF.4 b-2 4.NF.4 c

Creates a visual fraction model and solves mathematical and real-world problems by recognizing that fraction a/b is a multiple of 1/b and uses that construct to multiply a fraction by a whole number.

Understands and solves mathematical and real-world problems by recognizing that fraction a/b is a multiple of 1/b and uses that construct to multiply a fraction by a whole number.

Using visual models and/or manipulatives, solves mathematical and real-world problems by recognizing that fraction a/b is a multiple of 1/b and uses that construct to multiply a fraction by a whole number.

Using visual models and/or manipulatives, solves mathematical problems by recognizing that fraction a/b is a multiple of 1/b and uses that construct to multiply a fraction by a whole number.

Solving with Multiplication 4.OA.1-1, 4.OA. 1-2 4.OA.2

Interprets multiplication equations as comparisons and represents statements of multiplicative comparisons as multiplicative equations. Distinguishes multiplicative comparisons. Uses multiplication or division to solve word problems involving multiplicative comparisons. Uses a symbol for the unknown number. Creates real-world

Interprets multiplication equations as comparisons and represents statements of multiplicative comparisons as multiplicative equations. Distinguishes multiplicative comparisons. Uses multiplication or division to solve word problems involving multiplicative comparisons. Uses a symbol for the unknown number.

Interprets multiplication equations as comparisons or represents statements of multiplicative comparisons as multiplicative equations. Uses multiplication or division to solve word problems involving multiplicative comparisons.

Interprets multiplication equations as comparisons or represents statements of multiplicative comparisons as multiplicative equations. Uses multiplication or division to solve scaffolded problems involving multiplicative comparisons.









problems that would be solved using multiplicative comparison.

Multi-step Problems 4.OA.3-1 4.OA. 3-2 4.NBT.5-1 4.NBT. 5-2 4.NBT.6-1 4.NBT.6-2

Solves multiple-step word and other problems using the four operations with whole numbers: in multiplying a three- or four-digit by a one-digit number or two two-digit numbers. Finds whole number quotients and remainders with up to four-digit dividends and one-digit divisors and interprets remainders as appropriate. Chooses from a variety of strategies to solve these problems and selects an appropriate context for the task.

Solves multiple-step word and other problems using the four operations with whole numbers: in multiplying a three- or four-digit by a one-digit number or two two-digit numbers. Finds whole number quotients and remainders with up to four-digit dividends and one-digit divisors and interprets remainders as appropriate. Chooses from a variety of strategies to solve these problems.

Solves two-step word and other problems using the four operations with whole numbers: in multiplying a three-digit by a one-digit number or two two-digit numbers. Finds whole number quotients and remainders with up to three-digit dividends and one-digit divisors and interprets remainders as appropriate. Chooses from a variety of strategies to solve these problems.

Solves one-step word and other problems using the four operations with whole numbers: in multiplying a three-digit by a one-digit number or two two-digit numbers. Finds whole number quotients and remainders with up to three-digit dividends and one-digit divisors. Chooses from a variety of strategies to solve these problems. Can only solve two-step problems when scaffolding is provided for each step.









Place Value 4.NBT.1 4.NBT.2 4.NBT.3 4.NBT.Int.1

In any multi-digit whole number, recognizes a digit in one place represents 10 times as much as it represents in the place to its right. Reads, writes and compares multi-digit whole numbers using base-10 numerals, number names in expanded form and inequality symbols (>, <, =), and rounds to any place and chooses appropriate context given a rounded number. Performs computations by applying conceptual understanding of place value, rather than by applying multi-digit algorithms.

In any multi-digit whole number, recognizes a digit in one place represents 10 times as much as it represents in the place to its right. Reads, writes and compares multi-digit whole numbers using base-10 numerals, number names in expanded form and inequality symbols (>, <, =), and rounds to any place. Performs computations by applying conceptual understanding of place value, rather than by applying multi-digit algorithms.

In any four-digit whole number, recognizes a digit in one place represents 10 times as much as it represents in the place to its right. Reads, writes and compares four-digit whole numbers using base-10 numerals, number names in expanded form and inequality symbols (>, <, =), and rounds to any place

In any three-digit whole number, recognizes a digit in one place represents 10 times as much as it represents in the place to its right. Reads, writes and compares three-digit whole numbers using base-10 numerals, number names in expanded form and inequality symbols (>, <, =), and rounds to any place with scaffolding.



Grade 4 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 Command

Level 4: Strong Command Level 3: Moderate

Command Level 2: Partial Command

Operations and Factors 4.OA.4-1 4.OA.4-2 4.OA.4-3 4.OA.4-4

Recognizes that a whole number is a multiple of each of its factors, and within the range of 1-100 finds all factor pairs and determines multiples of whole numbers. Determines whether a whole number in the range 1-100 is prime or composite.

Recognizes that a whole number is a multiple of each of its factors, and within the range of 1-100 finds factor pairs and determines multiples of whole numbers. Determines whether a whole number in the range 1-100 is prime or composite.

Recognizes that a whole number is a multiple of each of its factors, and within the range of 1-100 finds factor pairs or determines multiples of whole numbers. Determines whether a whole number in the range 1-100 is prime or composite.

Recognizes that a whole number is a multiple of each of its factors, and within the range of 1-100 finds factor pairs or determines multiples of whole numbers using a hundreds chart. Determines whether a whole number in the range 1-100 is prime or composite.

Measurement and Conversion 4.MD.1 4.MD.2-1 4.MD.2-2 4.MD.3

Solves problems which include calculating area and perimeter – including those in which side lengths are missing – and simple fractions and decimals using all four operations. Records measurement equivalents in a two-column table. Uses knowledge of measurement units within one system to solve word

Solves problems which include calculating area and perimeter – including those in which side lengths are missing – and simple fractions and decimals using all four operations. Records measurement equivalents in a two-column table. Uses knowledge of measurement units within one system to solve word

Solves problems which include calculating area and perimeter – when information about side lengths is provided – and simple fractions and decimals using all four operations. Records measurement equivalents in a two-column table. Uses knowledge of measurement units within one system to solve word

Solves mathematical problems which include use of conversions of simple fractions or decimals using all four operations. Records measurement equivalents in a two-column table. Uses knowledge of measurement units within one system to convert from








problems, real-world problems, and mathematical problems involving converting from larger units to smaller units, Represents measurement quantities using diagrams such as number line diagrams that require students to provide the appropriate measurement scale given the context.

problems, real-world problems, and mathematical problems involving converting from larger units to smaller units. Represents measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

problems, real-world problems and mathematical problems involving converting from larger units to smaller units. Represents measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

larger units to smaller units.

Represent and Interpret Data 4.MD.4-1 4.MD.4-2

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

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

Makes a line plot to display a data set of measurements in fractions of a unit with like denominators of 2 or 4.

Geometric Measurement 4.MD.5

Recognizes how angles are formed and that angle measures are additive. Understands and applies

Recognizes how angles are formed and that angle measures are additive. Understands and applies

Understands and applies concepts of angle measurement.

Understands and applies concepts of angle measurement.








4.MD.6 4.MD.7

concepts of angle measurement recognizing that angles are measured in reference to a circle. Uses a protractor to measure and sketch angles. Solves mathematical and real world problems by composing and decomposing angles with equations. Creates or solves real-world problems using understanding of angles.

concepts of angle measurement recognizing that angles are measured in reference to a circle. Uses a protractor to measure and sketch angles. Solves mathematical and real world problems by composing and decomposing angles with equations.

Uses a protractor to measure and sketch angles. Solves mathematical and real world problems by composing and decomposing angles.

Uses a protractor to measure angles.

Lines, Angles and Shapes 4.G 1 4.G.2 4.G.3

Draws and identifies points, lines, line segments, rays, angles (right, obtuse and acute), perpendicular lines, parallel lines, lines of symmetry and right triangles, and use any of these to classify two-dimensional figures. Creates two-dimensional figures based on given properties.

Draws and identifies points, lines, line segments, rays, angles (right, obtuse and acute), perpendicular lines, parallel lines, lines of symmetry and right triangles, and use any of these to classify two-dimensional figures.

Draws and identifies points, lines, line segments, rays, angles (right, obtuse and acute), perpendicular lines, parallel lines, lines of symmetry and right triangles, and use some of these to classify two-dimensional figures.

Identifies points, lines, line segments, rays, angles (right, obtuse and acute), perpendicular lines, parallel lines, lines of symmetry and right triangles, and use some of these to classify quadrilaterals and triangles.








Generate and Analyze Patterns 4.OA.5

Generates a number or shape pattern that follows a given rule, identifies apparent features of the pattern that were not explicit in the rule itself and expresses the pattern using a formula.

Generates a number or shape pattern that follows a given rule and identifies apparent features of the pattern that were not explicit in the rule itself.

Generates a number or shape pattern that follows a given rule and identifies explicit features of the pattern.

Generates a number or shape pattern that follows a given rule.



Grade 4 Math: Sub-Claim C The student expresses grade/course-level appropriate mathematical reasoning by constructing viable

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




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

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

properties of operations

relationship between addition and subtraction

relationship between multiplication and division

identification of arithmetic patterns

Response may include:

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

an efficient and logical progression of steps

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






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

a logical progression of steps

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






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

a logical, but incomplete, progression of steps

minor calculation errors

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






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

an incomplete or illogical progression of steps








with appropriate justification

precision of calculation

correct use of grade-level vocabulary, symbols and labels

justification of a conclusion

evaluation of whether an argument or conclusion is generalizable

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





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

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 intrusive calculation error

limited use of grade-level vocabulary, symbols and labels


Concrete Referents and Diagrams

Clearly constructs and communicates a well-organized and complete response based on

Clearly constructs and communicates a well-organized and complete response based on

Constructs and communicates a complete response based on operations using concrete

Constructs and communicates an incomplete response based on operations using








4.C.4-1 4.C.4-2 4.C.4-3 4.C.4-4 4.C.4-5 4.C.7-1 4.C.7-2 4.C.7-3 4.C.7-4

operations using concrete referents such as diagrams – including number lines (whether provided in the prompt or constructed by the student) and connecting the diagrams to a written (symbolic) method, which may include:

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

an efficient and logical progression of steps with appropriate justification




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:






evaluation of whether an argument or

referents such as diagrams – including number lines (provided in the prompt) – connecting the diagrams to a written (symbolic) method, which may include:





partial justification of a conclusion based on own calculations.

concrete referents such as diagrams – including number lines (provided in the prompt) – connecting the diagrams to a written (symbolic) method, which may include:

a conjecture and/or stated or faulty assumptions













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

conclusion is generalizable

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

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

accepting the validity of other’s responses.

Distinguish Correct Explanation/ Reasoning from that which is Flawed 4.C.5-1 4.C.5-2 4.C.5-3 4.C.5-4 4.C.5-5 4.C.6-1 4.C.6-2 4.C.6-3 4.C.8

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

presenting 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

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

presenting 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

Constructs and communicates a complete response by:

presenting 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

Constructs and communicates an incomplete response by:

presenting 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








argument

presenting and defending corrected reasoning



an efficient and logical progression of steps with appropriate justification




evaluation of whether an

flawed

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

presenting corrected reasoning







evaluation of whether an argument or conclusion

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

presenting corrected reasoning







evaluating the validity of other’s responses,

from that which is flawed

identifying an error in reasoning


a conjecture based on faulty assumptions





accepting the validity of other’s responses.








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.

is generalizable

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

approaches and conclusions.



Grade 4 Math: Sub-Claim D The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the current grade/course (or for more complex problems,

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

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





Modeling 4.D.1 4.D.2

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

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

analyzing and/or creating constraints, relationships and goals

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

analyzing relationships mathematically between important quantities to draw conclusions


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

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


interpreting mathematical results in the context of the


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

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


interpreting mathematical results in a simplified context

reflecting on whether


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

identifying important quantities

using provided tools to create models

analyzing relationships mathematically to draw conclusions

writing an arithmetic expression or equation to describe a situation



Grade 4 Math: Sub-Claim D The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the current grade/course (or for more complex problems,

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

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





justifying and defending models which lead to a conclusion

interpreting mathematical results in the context of the situation

reflecting on whether the results make sense

improving the model if it has not served its purpose

writing a concise arithmetic expression or equation to describe a situation

situation

reflecting on whether the results make sense

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


the results make sense

modifying the model if it has not served its purpose




Grade 4 Math: Sub-Claim E The student demonstrates fluency in areas set forth in the Standards for Content in grades 3-6.





Fluency 4.NBT.4-1 4.NBT.4-2

Accurately and quickly adds and subtracts multi-digit whole numbers using the standard algorithm. Knows from memory 100 percent of the sums and differences on items in less than the allotted time on items which are timed.

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

Accurately adds and subtracts multi-digit whole numbers using the standard algorithm. Knows from memory more than 80 percent and less than 100 percent of the sums and differences on items which are timed.

Adds and subtracts multi-digit whole numbers using the standard algorithm with some level of accuracy. Knows from memory greater than or equal to 70 percent and less than or equal to 80 percent of the sums and differences on items which are timed.

Latest Revision 6/17/2013 1

Bailey●Kirkland Education Group, LLC Common Core State Standard I Can Statements

4th

Grade Mathematics 6/18/2013 CCSS Key: PLD Key: Operations and Algebraic Thinking (OA) Partial Command Number and Operations in Base Ten (NBT) Moderate Command Numbers and Operations–Fractions (NF) Distinguished Command Measurement and Data (MD) Geometry (G)

Common Core State Standards for Mathematics (Outcome Based)

I Can Statements

Operations and Algebraic Thinking (OA) 4.OA.1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

I Can: 4.OA.1.1 Interpret a verbal comparison as a

multiplication equation (35 is 5 times as many as 7 and 7 times as many as 5 → 35=7x5 or Eric is nine years old).

4.OA.1.2 Interpret a multiplication equation as a verbal comparison (35=7x5 → 35 is 5 times as many as 7 and 7 times as many as 5).

4.OA.1.3 Identify which factor is being multiplied and which number tells how many times in a multiplication equation.

4.OA.1.4 4.OA.1.5

Verbalize which factor is being multiplied and which number tells how many times in a multiplication equation. Write a mulplication equation from a verbal statement that compares with multiplication.

4.OA.2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

I Can: 4.OA.2.1 Represent word problems and/or

equations with pictures and symbols to represent the unknown number.

4.OA.2.2 Solve one-step word problem using multiplication (3 digits by a 1 digit number or two two-digit numbers).

4.OA.2.3 Solve one-step word problem using division (3 digits dividends and 1 digit divisors).

4.OA.2.4 Solve two-step word problems using multiplication or division (3 digits by a 1 digit number or two two-digit numbers).

4.OA.2.5 Distinguish multiplication problems from addition problems.

4.OA.2.6 Create real-world problems that will be solved using multiplicative comparison.



I Can Statements

4.OA.3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

I Can: 4.OA.3.1 Solve multi-step word problems using

the four operations with whole numbers (3 or 4 digits by a 1 digit number or two two-digit numbers)

4.0A.3.2 Interpret remainders in various situations.





4.OA.3.7 Justify an answer based upon the interpretation of remainders.

4.OA.3.8

Justify an answer using mental math and estimation.

4.OA.4. Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.

I Can: 4.OA.4.1 Determine if a whole number (1-100) is

a multiple of a given 1 digit number (ex. – Is 56 a multiple of 7? Is 45 a multiple of 2) (MS)

4.OA.4.2 Find all factor pairs for a whole number up to 100 (ex. 56 = __ x __) (MS)

4.OA.4.3 Determine if a whole number (1-100) is prime or composite.

4.OA.4.4 Recognize that a whole number (1-100) is a multiple of each of its factors. (MS)

4.OA.5. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

I Can: 4.OA.5.1 Use a rule to create a number or shape

pattern. (MS)

4.OA.5.2 Determine if there are other relationships within a pattern (ex.4, 8, 16, 32… - all even. 5, 12, 19, 26… - odd/even).

4.OA.5.3 Express a pattern using a formula.



I Can Statements

Numbers and Operations–Fractions (NF) 4.NF.1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

I Can: 4.NF.1.1 Recognize equivalent fractions. (MS)

4.NF.1.2 Create visual fraction models to explain why fractions are equal. (MS)

4.NF.1.3 Use a visual model to explain that two fractions are equivalent even when the number and size of the parts are different. (MS)

4.NF.1.4

Create equivalent fractions in number form (ie. ½ = 6/12) by multiplying or dividing the numerator and denominator by the same number. (MS)

4.NF.2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

I Can: 4.NF.2.1 Compare a fraction to a benchmark fraction

such as 1/2, using a visual model. (MS)

4.NF.2.2 Compare fractions to a benchmark fraction such as 1/2, using numerical comparison. (ie. 3/6 ____ 7/12) (MS)

4.NF.2.3 Use multiples to find a LCD.

4.NF.2.4 Show results of compared fractions using symbols (<, >, =). (MS)

4.NF.2.5 Compare two fractions with different numerators (like denominators). (MS)

4.NF.2.6 Compare two unlike fractions by creating like denominators. (MS)

4.NF.2.7 Explain that the size of the whole matters when comparing fractions (ie. ½ of a medium pizza is not equal to ½ of a large pizza).

4.NF.2.8 Justify comparisons by using a visual fraction model.

4.NF.2.9 Create a visual model to explain the comparison of fractions.



I Can Statements

4.NF.3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

I Can: 4.NF.3a.1 Add or subtract fractions with like

denominators using manipulatives or visual models. (MS)

4.NF.3a.2 Add and subtract improper fractions with like denominators using manipulatives or visual models.

4.NF.3b.1 Decompose a fraction into a sum of fractions with the same denominator in more than one way.

4.NF.3b.2 Create a visual model to justify decompositions.

4.NF.3c.1 Add and subtract mixed numbers with like denominators by using a visual model.

4.NF.3c.2 Convert a mixed number into an improper fraction.

4.NF.3c.3 Convert an improper fraction into a mixed number.

4.NF.3d.1 4.NF.3d.2

Solve word problems using addition and subtraction of fractions with like denominators using visual models and equations. (MS) Solve word problems using addition and subtraction of fractions with like denominators. (MS)

4.NF.4

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

I Can: 4.NF.4a.1 Use a visual fraction model to represent

a/b as the product of a and 1/b.

4.NF.4b.1 4.NF.4b.2

Use a visual fraction model to represent a fraction times a whole number. Create a visual fraction model to represent a fraction times a whole number.

4.NF.4c.1 Solve multiplication word problems involving fractions and whole numbers using visual models.

4.NF.4c.2 Solve multiplication word problems involving fractions and whole numbers using equations.



I Can Statements

4.NF.5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.4 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.

I Can: 4.NF.5.1 Use base ten models to represent fractions.

4.NF.5.2 Convert unlike denominators to like denominators (10,100) and add fractions.

4.NF.6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

I Can: 4.NF.6.1 Convert between decimals and fractions

with the denominator of 10 or 100 (ie. 0.62 = 62/100). (MS)

4.NF.6.2 Locate fractions and decimals on a number line and meter stick (tenths and hundredths).

4.NF.7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

I Can: 4.NF.7.1 4.NF.7.2

Compare two decimals to the hundredth place using a hundreds grid and using symbols (<,>,=). Compare two decimals to the hundredth place using symbol (<,>,=)

4.NF.7.3 Recognize that comparisons are valid only when the two decimals refer to the same whole.

4.NF.7.4 Create a model to justify an answer.

Number and Operations in Base Ten (NBT)

4.NBT.1. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

I Can: 4.NBT.1.1 4.NBT.1.2 4.NBT.1.3

Recognize a digit in one place represents 10 times as much as it represents in the place to the right (3 digit numbers). Recognize a digit in one place represents 10 times as much as it represents in the place to the right (4 digit numbers). Recognize a digit in one place represents 10 times as much as it represents in the place to the right (multi-digit numbers).

4.NBT.2. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

I Can: 4.NBT.2.1 4.NBT.2.2

Read, write, and compare multi-digit numbers in expanded form. (MS) Read, write, and compare multi-digit numbers using base ten numerals (standard form). (MS)

4.NBT.2.3 Read, write, and compare multi-digit numbers in word form. (MS)

4.NBT.2.4 Compare multi-digit numbers using <, >, =. (MS)

4.NBT.3. Use place value understanding to round multi-digit whole numbers to any place.

I Can: 4.NBT.3.1 Round multi-digit numbers up to the

millions place. (MS)



I Can Statements

4.NBT.4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.

I Can: 4.NBT.4.1 Fluently add multi-digit numbers up to

millions place. (MS) 4.NBT.4.2 Fluently subtract multi-digit numbers up to

millions place. (MS)

4.NBT.5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

I Can: 4.NBT.5.1 Multiply a 4 digit number by a 1 digit

number. 4.NBT.5.2 Illustrate and explain multiplication using

rectangular arrays.


4.NBT.5.4 Apply the properties of operations to multiply numbers. (MS)

4.NBT.5.5 Multiply 2, two digit numbers (ex. 23 x 45). (MS)

4.NBT.5.6 Multiply numbers using written equations. (MS)

4.NBT.6. Find whole-number quotients and remainders with up to four-digit dividends and one-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: 4.NBT.6.1 Divide up to 4 digit number by a 1 digit

divisor. (MS)

4.NBT.6.2 Apply the properties of operations to divide 4 digit numbers. (MS)

4.NBT.6.3 Apply strategies based on place value to divide up to 4 digit number by a 1 digit divisor. (MS)

4.NBT.6.4 Explore different strategies for the division of 4 digit dividends and 1 digit divisors. (MS)




4.NBT.6.8 Explore the relationship between multiplication and division. (MS)



I Can Statements

Measurement and Data (MD)

4.MD.1. Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36)...

I Can: 4.MD.1.1 Identify and associate units of

measurements used to measure length. (MS)

4.MD.1.2 Identify and associate units of measurements used to measure capacity. (MS)

4.MD.1.3 Identify and associate units of measurements used to measure weight. (MS)

4.MD.1.4 Identify and associate units of measurements used to measure time. (MS)

4.MD.1.5 Compare units of measurement within a given system (ie. 1 inch < 1 foot). (MS)

4.MD.1.6 Convert (change) from a larger unit to a smaller unit. (MS)

4.MD.1.7 Create a table to record equivalent measures listing number pairs.

4.MD.2. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

I Can: 4.MD.2.1 Represent measurement quantities using

diagrams with a measurement scale.

4.MD.2.2 Apply the four operations to solve word problems involving distance. (MS)

4.MD.2.3 Apply the four operations to solve word problems involving elapsed time. (MS)

4.MD.2.4 Apply the four operations to solve word problems involving liquid volume. (MS)

4.MD.2.5 Apply the four operations to solve word problems involving mass. (MS)


*Note:

These problems are limited to converting larger to smaller units.

These problems include whole numbers, fractions, and decimals.

4.MD.3. Apply the area and perimeter formulas for rectangles in real-world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

I Can: 4.MD.3.1 4.MD.3.2

Calculate the area of a rectangle using the formula A=L x W or A= B x H when side lengths are given. Solve for the missing side length of a rectangle using the formula A=L x W or A=B x H when the area is given along with one other dimension.

4.MD.3.3 4.MD.3.4

Calculate the perimeter of a rectangle using the formula P=S+S+S+S or P=2L + 2W when side lengths are given. Solve for the missing side length of a rectangle using the formula P=S+S+S+S or P=2L + 2W when the perimeter is given along with one other dimension.

4.MD.3.5 Apply the area and perimeter formula to solve real-world problems.



I Can Statements

4.MD.4. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.

I Can: 4.MD.4.1 Answer questions about data displayed on

a line plot.

4.MD.4.2 Create a line plot to display a data set that includes fractions with denominators 2 or 4.

4.MD.4.3 Create a line plot to display a data set that includes fractions with denominators 2, 4, and 8.

4.MD.4.3 Add and subtract fractions using information from a line plot.

4.MD.4.4 Evaluate solutions in relation to data.

4.MD.5. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles. b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

I Can: 4.MD.5a.1 Recognize that a circle has 360 degrees.

4.MD.5a.2 Explain that an angle measurement is a fraction of a circle.

4.MD.5b.1 Recognize that angles are measured in degrees within a circle.

4.MD.6. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

I Can: 4.MD.6.1 Identify benchmark angles (90º, 180º, 270º,

360º).

4.MD.6.2 Measure angles using a protractor.

4.MD.6.3 Sketch angles of a given measurement (degree) using a protractor.

4.MD.7. Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

I Can: 4.MD.7.1 Decompose angles into smaller angles.

4.MD.7.2 Add angle measures to make a larger angle.

4.MD.7.3 Use addition and subtraction to find unknown angles in real-world and mathematical problems.

4.MD.7.4 Use an equation with a symbol for the unknown angle measure.



I Can Statements

Geometry (G)

4.G.1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

I Can: 4.G.1.1 Identify points, lines, line segments, rays,

angles, perpendicular, and parallel lines in two dimensional figures. (MS)

4.G.1.2 Draw points, lines, line segments, rays, angles, perpendicular, and parallel lines in two dimensional figures. (MS)

4.G.1.3 Identify types of angles (right, acute, obtuse) in two-dimensional figures. (MS)

4.G.2. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

I Can: 4.G.2.1 Classify two dimensional shapes based on

parallel or perpendicular lines. (MS)

4.G.2.2 Classify two dimensional shapes based on types of angles. (MS)

4.G.2.3 Classify quadrilaterals and triangles based on parallel or perpendicular lines. (MS)

4.G.2.4 Classify quadrilaterals and triangles based on types of angles. (MS)

4.G.2.5 Recognize and label a right triangle. (MS)

4.G.2.6 Create a two dimensional shapes when given the properties.

4.G.3. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

I Can: 4.G.3.1 Recognize a line of symmetry.

4.G.3.2 Draw a line of symmetry.

4.G.3.3 Identify lines of symmetry in two dimensional figures.

4.G.3.4 Draw figures that have lines of symmetry.

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

1

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

The Standards: 1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

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

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

1. Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

2. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

3. Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects,

Standards for Mathematical Practice

2

drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

4. Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

5. Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

6. Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

7. Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x 8 equals the well-remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

8. Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x +1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

CCSS Standards for Mathematical Practice

Questions for Teachers to Ask 1.Make sense of problems and persevere in

solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics

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

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

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

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

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

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

Teachers ask: • What does the number ____ represent in

the problem? • How can you represent the problem with

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

Teachers ask: • How is your answer different than

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

correct? • What examples could prove or disprove

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

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

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

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

Teachers ask: • Write a number sentence to describe this

situation. • How could we use symbols to represent

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

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

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

Teachers ask: • How could you use manipulatives or a

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

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

other one would you have chosen?

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

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

did you use them?

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

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

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

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

problem? How would your shortcut make the problem easier?

• How could this problem help you solve another problem?

*inductive reasoning (moving from specific to general)

4 grade math unit guide core/pacing guides...4th grade math unit guide 2014-2015 . ... ......

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