2016/2017
2nd Quarter Elementary Mathematics
Weather Revision Curriculum Guide
Grade 4 Page 1 of 60 4th Grade Year at a Glance Math Florida Standards Modules Duration Quarter 1 8/15/16 ‐ 9/15/16 23 Days Module 1: Place Value, Rounding, and Algorithms for Addition and Subtraction Operations and Algebraic Thinking 4.OA.1.3 4.OA.1.1 Number and Operations in Base Ten 4.NBT.1.1 4.NBT.1.2 4.NBT.1.3 4.NBT.2.4 9/16/16 ‐ 9/23/16 6 Days Module 2: Unit Conversions and Problem Solving Measurement and Data 4.MD.1.1 4.MD.1.2 Quarter 2 9/26/16 ‐ 10/14/16 15 Days Quarter 3
Quarter 4
10/17/16 ‐
11/28/16 22 Days Module 3:
Multi‐Digit Multiplication and Division 11/29/16 ‐
1/5/17 ‐
12/22/16 2/6/17 22 Days 18 Days Module 5:
Fraction Equivalence, Ordering, and Operations 2/7/17 ‐
3/3/17 18 Days Module 6: Decimal Fractions 3/6/17 ‐ 3/27/17‐
3/16/17
4/4/17 10 Days 6 Days Module 4:
Angle Measure and Plane Figures Operations and Algebraic Thinking 4.OA.1.1 4.OA.1.2 4.OA.1.3 4.OA.2.4 4.OA.3.5 4.OA.1.a* 4.OA.1.b* Number and Operations in Base Ten 4.NBT.2.5 4.NBT.2.6 4.NBT.1.1 Measurement and Data 4.MD.1.3 Number and Operations ‐
Fractions 4.NF.1.1 4.NF.1.2 4.NF.2.3 4.NF.2.4 Measurement and Data 4.MD.2.4 4.MD.1.2 Number and Operations in Base Ten 4.NBT.1.1 Number and Operations ‐ Fractions 4.NF.3.5 4.NF.3.6 4.NF.3.7 4.NF.1.1 4.NF.2.3 Measurement and Data 4.MD.1.1 4.MD.1.2 Measurement and Data 4.MD.3.5 4.MD.3.6 4.MD.3.7 Geometry 4.G.1.1 4.G.1.2 4.G.1.3 4/5/17 –
5/1/17 17 Days Module 7:
Exploring Measurement with Multiplication Operations and Algebraic Thinking 4.OA.1.1 4.OA.1.2 4.OA.1.3 Number and Operations in Base Ten 4.NBT.2.5 4.NBT.2.6 Measurement and Data 4.MD.1.1 4.MD.1.2 5/2/17 – 6/2/17 23 Days Deepening Understanding: Reteach/ Enrich 4th Grade Major Work Operations and Algebraic Thinking 4.OA.1.3 4.OA.1.a* 4.OA.1.b* 4.OA.3.5 4.OA.1.2 4.OA.2.4 Number and Operations ‐ Fractions 4.NF.1.1 4.NF.2.3 4.NF.2.4 Measurement and Data 4.MD.2.4 4.MD.1.3 Page 2 of 60 Elementary Mathematics 4th Grade Testing Calendar Required (The following assessments have been scheduled for you in your grade level curriculum guides. Please note: these assessments are required to be administered within the testing window provided. Please refer to the Optional Testing calendar for additional assessments that have been developed for you to administer should you choose to. Required assessments are subject to change. Please refer to the district testing calendar for testing window and details.) Quarter Assessment Approximate Testing Dates Duration 1 i‐Ready Fall 8/17 – 8/31 2 days 2 Mid‐Year Scrimmage 12/1 – 12/18 2 days i‐Ready Winter 12/5 – 12/16 2 days 3 FSA 4/10 – 5/5 2 days 4 i‐Ready Spring 5/1 – 5/12 2 days Page 3 of 60 Elementary Mathematics 4th Grade Testing Calendar Optional (The following assessments have been developed and scheduled in your grade level Curriculum Guides. Please note: these assessments are optional, should you choose to administer them. Please refer to the required assessment calendar for required testing for your grade level.) Quarter Assessment 1 Assess/Performance Task 4.NBT.1.3 Mid‐Module 1 2 3 Approximate Testing Dates 8/29 Duration 9/1 1 day Assess/Performance Task 4.NBT.2.4 End of Module 1 9/9 9/14 1 day End of Module 2 9/23 1 day Assess/Performance Task 4.NBT.2.5 Mid‐Module 3 10/7 10/12 1 day 10/31 11/10 11/22 1 day Assess/Performance Task 4.NF.1.1 Mid‐Module 5 12/15 1/10 1 day Performance Task 4.NF.2.3 End of Module 5 1/26 2/3 1 day Mid‐Module 6 2/16 1 day Performance Task 4.NF.3.7 End of Module 6 2/21 3/2 1 day Assess/Performance Task 4.NBT.2.6 Assess/Performance Task 4.NBT.2.5 End of Module 3 Page 4 of 60 Performance Task 4.G.1.1 Mid‐Module 4 4 3/7 3/15 1 day Performance Task 4.MD.3.7 Performance Task 4.G.1.2 End of Module 4 3/28 3/29 4/3 1 day Performance Task 4.MD.1.1 Performance Task 4.MD.1.2 End of Module 7 4/6 4/24 5/1 1 day 5/4 5/12 Performance Task: 4.OA.3.5 Performance Task 4.OA.1.3 Page 5 of 60 November
Notes
Mon
Tues
14
Day 59
Module 3
Lesson 34
15
Day 60
Module 3
Lesson 35
21
Day 64
Teacher
Toolbox
Lesson 8b
28
Day 66
Review and
remediate
previously
taught
standards
22
Day 65
End-of-Module
3 Assessment
29
Day 67
Mod 5
Lessons 1&2
4th Grade Wed
Thu
Fri
Notes
18
Day 63
Teacher
Toolbox
Lesson 8a
23
24
25
Schools Closed Schools Closed Schools Closed
16
ERD
Day 61
Module 3
Lesson 36
17
Day 62
Module 3
Lesson 37
30
ERD
Day 68
Mod 5
Lesson 3
*Please Note: This calendar was developed with recommendations from ‘the CG. Teachers may not be on the exact day, based on class
needs.
Page 6 of 60 December
Notes
Mon
Tues
4th Grade Wed
Thu
Fri
1
Day 69
Mid-Year
Scrimmage
2
Day 70
Mid-Year
Scrimmage
5
Day 71
Mod 5
Lesson 5
6
Day 72
i-Ready
Diagnostic
7
Day 73
i-Ready
Diagnostic
8
Day 74
Mod 5
Lesson 6
9
Day 75
Mod 5
Lesson 7
12
Day 76
Mod 5
Lesson 8
13
Day 77
Mod 5
Lesson 9
14
ERD
Day 78
Mod 5
Lesson 10
15
Day 79
Mod 5
Lesson 11
16
Day 80
Mod 5
Lesson 12
19
Day 81
Mod 5
Lesson 13
20
Day 82
Mod 5
Lesson 14
21
Day 83
Mod 5
Lesson 15
22
Day 84
Mod 5
Lesson 16
23
Schools Closed
26
Schools Closed
27
Schools Closed
28
Schools Closed
29
Schools Closed
30
Schools Closed
i-Ready testing
window begins on this
day. There are two
days allotted as “flex”
for accommodating
testing days.
Notes
*Please Note: This calendar was developed with recommendations from the CG. Teachers may not be on the exact day, based on class
needs.
Page 7 of 60 Quarter 2 Module 3 (Part 2) DUVAL COUNTY PUBLIC SCHOOLS
Math Curriculum Lesson Guide
Course: Unit # Dates/Pacing: Grade 4 Math Module 3: Multi‐Digit Multiplication and Division 10/17/16 ‐‐ 11/28/16, 22 days Operations and Algebraic Thinking 4.OA.1.1: 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. Cognitive Complexity: Level 1 Recall 4.OA.1.2: 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. Cognitive Complexity: Level 2 Basic Application of Skills and Concepts 4.OA.1.3: 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. Cognitive Complexity: Level 2 Basic Application of Skills and Concepts 4.OA.2.4: Investigate factors and multiples. a. Find all factor pairs for a whole number in the range 1–100. b. 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 MAFS multiple of a given one‐digit number. c. Determine whether a given whole number in the range 1–100 is prime or composite. Cognitive Complexity: Level 2 Basic Application of Skills and Concepts 4.OA.3.5: 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. Cognitive Complexity: Level 2 Basic Application of Skills and Concepts 4.OA.1.a: Determine whether an equation is true or false by using comparative relational thinking. For example, without adding 60 and 24, determine whether the equation 60 + 24 = 57 + 27 is true or false. Cognitive Complexity: Level 3: Strategic Thinking & Complex Reasoning 4.OA.1.b: Determine the unknown whole number in an equation relating four whole numbers using comparative relational thinking. For example, solve 76 + 9 = n + 5 for n arguing that nine is four more than five, so the unknown number must be four greater than 76. Cognitive Complexity: Level 3: Strategic Thinking & Complex Reasoning Page 8 of 60 Number and Operations in Base Ten 4.NBT.2.5: 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. Cognitive Complexity: Level 2 Basic Application of Skills and Concepts 4.NBT.2.6: 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. Cognitive Complexity: Level 2 Basic Application of Skills and Concepts 4.NBT.1.1: 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. Cognitive Complexity: Level 1 Recall Measurement and Data 4.MD.1.3: 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. Cognitive Complexity: Level 2 Basic Application of Skills and Concepts Page 9 of 60 Standards for Mathematical Practices
Mathematically proficient students in grade 4 know that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Fourth graders may 1. Make sense of problems and use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking persevere in solving them. themselves, “Does this make sense?” They listen to the strategies of others and will try different approaches. They often will use another method to check their answers. Mathematically proficient fourth grade students should recognize that a number represents a specific quantity. They connect the quantity to written symbols and create a logical representation of the problem at hand, considering both the appropriate 2. Reason abstractly and units involved and the meaning of quantities. They extend this understanding from whole numbers to their work with quantitatively. fractions and decimals. Students write simple expressions, record calculations with numbers, and represent or round numbers using place value concepts. In fourth grade mathematically proficient students may construct arguments using concrete referents, such as objects, 3. Construct viable arguments pictures, and drawings. They explain their thinking and make connections between models and equations. They refine their and critique the reasoning of mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get others. that?” and “Why is that true?” They explain their thinking to others and respond to others’ thinking. Mathematically proficient fourth grade students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, making a chart, list, or graph, creating equations, 4. Model with mathematics. etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed. Fourth graders should evaluate their results in the context of the situation and reflect on whether the results make sense. Mathematically proficient fourth grader students consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they may use graph paper or a number 5. Use appropriate tools line to represent and compare decimals and protractors to measure angles. They use other measurement tools to strategically. understand the relative size of units within a system and express measurements given in larger units in terms of smaller units. As fourth grader students develop their mathematical communication skills, they try to use clear and precise language in 6. Attend to precision. their discussions with others and in their own reasoning. They are careful about specifying units of measure and state the meaning of the symbols they choose. For instance, they use appropriate labels when creating a line plot. In fourth grade mathematically proficient students look closely to discover a pattern or structure. For instance, students use 7. Look for and make use of properties of operations to explain calculations (partial products model). They relate representations of counting problems structure. such as tree diagrams and arrays to the multiplication principal of counting. They generate number or shape patterns that follow a given rule. Students in fourth grade should notice repetitive actions in computation to make generalizations. Students use models to 8. Look for and express explain calculations and understand how algorithms work. They also use models to examine patterns and generate their own regularity in repeated reasoning. algorithms. For example, students use visual fraction models to write equivalent fractions. Page 10 of 60 Test Item Specifications Content Standard MAFS.4.OA.1.1 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.
Items may not require students to solve for unknown factors that exceed 10 x 10 multiplication facts. Assessment Limits Item must include a verbal description of an equation or a multiplication equation. Multiplication situations must be a comparison (e.g., times as many). Equation Editor Multiple Choice Item Types GRID Multiselect Matching Item Open Response Allowable
Context Level 1: Recall Complexity Level Achievement Level Descriptions Level 3 Level 4 creates a context for a multiplicative recognizes that any two factors and comparison problem given an their product can be read as a equation comparison; represents those comparisons as equations Level 2 [intentionally left blank] Level 5 [intentionally left blank] Page 11 of 60 Content Standard MAFS.4.OA.1.2 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. Multiplication situation must be a comparison (e.g., times as many). Assessment Limits Limit multiplication and division to 2‐digit by 1‐digit or a multiple of 10 by a 1‐digit. Equation Editor Multiple Choice Item Types GRID Multiselect Required Context Level 2: Basic Application of Skills & Concepts Complexity Level Achievement Level Descriptions Level 2 Level 3 multiplies or divides to solve word problems involving multiplicative comparison (where the unknown is the product or quotient) multiplies or divides to solve word problems involving multiplicative comparison (where the unknown is in a variety of positions) Level 4 Level 5 creates and solves a multiplication [intentionally left blank] equation with a symbol for the unknown number to represent a word problem involving multiplicative comparison Page 12 of 60 Content Standard MAFS.4.OA.1.3 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. Items requiring precise or exact solutions are limited to: Assessment Limits  addition and subtraction within 1,000  multiplication of 2‐digit by 1‐digit or a multiple of 10 by a 1‐digit  division of 2‐digit by 1‐digit Items may contain a maximum of 3 steps. Items involving remainders must require the students to interpret and/or use the remainder with respect to the context. Variables must be represented by a letter, and variables must be defined or described in the context. Equation Editor Multiselect Item Types Multiple Choice Open Response Required Context Level 2: Basic Application of Skills & Concepts Complexity Level Achievement Level Descriptions Level 2 solves one‐step word problems (which do not include remainders) using the four operations with simple context and scaffolding where the sum, difference, product, or quotient is always the unknown Level 3 solves two‐step word problems (including interpreting remainders) using the four operations, where the unknown is in a variety of positions, and can be represented by a symbol/letter Level 4 solves three‐step word problems using the four operations; recognizes the reasonableness of answers using mental computation and estimation strategies Level 5 solves multistep word problems with multiple possible solutions and determines which would be the most reasonable based upon given criteria Page 13 of 60 Content Standard MAFS.4.OA.1a (assessed with MAFS.OA.1b) Determine whether an equation is true or false using comparative relational thinking. For example, without adding 60 and 24, determine whether the equation 60 + 24 = 57 + 27 is true or false. Whole number equations are limited to: Assessment Limits  addition and subtraction within 1,000  multiplication of 2‐digit by 1‐digit or a multiple of 10 by a 1‐digit  division of 2‐digit by 1‐digit Variables represented by a letter are allowable. Editing Task Choice Multiple Choice Item Types Equation Editor Multiselect GRID Open Response Hot Text Allowable Context Level 3: Strategic Thinking & Complex Reasoning Complexity Level Achievement Level Descriptions Level 2 Level 3
determines whether an equation is true or false; identifies true and false equations that use comparative relational thinking determines whether an equation is true or false, where addition or subtraction is used on both sides of the equal sign, and justifies by using comparative relational thinking Level 4
Level 5
determines whether an equation is true or false, where multiplication or division is used on both sides of the equal sign, and justifies by using comparative relational thinking determines whether an equation is true or false, where different operations are used on either side of the equal sign, and justifies by using comparative relational thinking Page 14 of 60 Content Standard MAFS.4.OA.1b (Also assesses MAFS.OA.1a) Determine the unknown whole number in an equation relating four whole numbers using comparative relational thinking. For example, solve 76 + 9 = n + 5 for n arguing that nine is four more than five, so the unknown number must be four greater than 76. Whole number equations are limited to: Assessment Limits  addition and subtraction within 1,000  multiplication of 2‐digit by 1‐digit or a multiple of 10 by a 1‐digit  division of 2‐digit by 1‐digit Variables represented by a letter are allowable. Editing Task Choice Multiple Choice Item Types Equation Editor Multiselect GRID Open Response Hot Text Allowable Context Level 3: Strategic Thinking & Complex Reasoning
Complexity Level Achievement Level Descriptions Level 2 Level 3
[intentionally left blank] determines the unknown number in an equation relating four whole numbers, where addition or subtraction is used on both sides of the equal sign, and justifies using comparative relational thinking Level 4
determines the unknown number in an equation relating four whole numbers, where multiplication or division is used on both sides of the equal sign, and justifies using comparative relational thinking Level 5
determines the unknown number in an equation relating four whole numbers, where different operations are used on either side of the equal sign, and justifies using comparative relational thinking Page 15 of 60 Content Standard MAFS.4.OA.2.4 Investigate factors and multiples. a. Find all factor pairs for a whole number in the range 1–100. b. 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. c. Determine whether a given whole number in the range 1–100 is prime or composite. Items may only contain whole numbers between 1‐100. Assessment Limits Vocabulary may include prime, composite, factor, or multiple. Equation Editor Multiple Choice Item Types GRID Multiselect Matching Item Table Item Allowable Context Level 2: Basic Application of Skills & Concepts Complexity Level Achievement Level Descriptions Level 2 Level 3 Level 4 Level 5 finds factor pairs for numbers in the range of 1 to 100, and determines whether a whole number in the range of 1 to 100 is prime or composite, given visual representations finds all factor pairs for whole numbers in the range of 1 to 100; recognizes that a whole number is a multiple of each of its factors; determines whether a whole number in the range of 1 to 100 is prime or composite determines common factors and multiples of numbers in the range of 1 to 100 applies the concepts of both factors, multiples, and prime and composite numbers in problem‐solving contexts Page 16 of 60 Content Standard MAFS.4.OA.3.5 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 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.
Items may only contain whole numbers from 1 to 1,000. Assessment Limits Operations in rules are limited to addition, subtraction, multiplication, and division. Items may not contain rules that exceed two procedural operations. Division rules may not require fractional responses. Rules may not be provided algebraically (e.g. x + 5). Items must provide the rule. Editing Task Choice Multiple Choice Item Types Equation Editor Multiselect GRID Open Response Hot Text Table Item Matching Item Allowable Context Level 2: Basic Application of Skills & Concepts Achievement Level Descriptions Complexity Level Level 2 extends a number or shape pattern that follows a given one‐step rule Level 3 generates a number or shape pattern that follows a given one‐step rule Level 4 generates a number or shape pattern that follows a given two‐step rule Level 5 identifies and/or explains apparent features that are not explicit in the rule from an observed pattern Page 17 of 60 Content Standard MAFS.4.NBT.2.5 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. Items may require multiplying: four digit by one digit, three digits by one‐digit, two digits by one digit, or two digits by Assessment Limits two digits. Equation Editor Multiselect Item Types GRID Open Response Multiple Choice No context Context Level 2: Basic Application of Skills & Concepts Complexity Level Achievement Level Descriptions Level 2 multiplies a whole number (of up to three digits) by a single‐digit whole number, including the use of strategies based on place value and visual models Level 3 multiplies a whole number up to four digits by a single‐digit whole number and two two‐digit whole numbers, using strategies based on place value; illustrates and explains calculations by using equations, rectangular arrays, and/or area models Level 4 determines the equation that represents a base‐ten model; makes connections between different multiplication strategies Level 5 analyzes and describes an error in a strategy and shows the correct solution Page 18 of 60 Content Standard MAFS.4.NBT.2.6 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. Items may not require finding a quotient within the factor pairs of 10 x 10. Assessment Limits Equation Editor Multiple Choice GRID Multiselect No context Item Types Context Complexity Level Level 2: Basic Application of Skills & Concepts Achievement Level Descriptions Level 2 divides a whole number (of up to three digits) by a single‐digit whole number, using strategies based on place value Level 3 divides a whole number up to four digits by a single‐digit whole number (including remainders), using strategies based on place value, properties of operations, and/or the relationship between multiplication and division; illustrates and explains calculations by using equations, rectangular arrays, and/or area models Level 4 determines the equation that represents a base‐ten model; makes connections between different division strategies Level 5 analyzes and describes an error in a strategy and shows the correct solution Page 19 of 60 Content Standard MAFS.4.NBT.1.1 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.
Items may contain whole numbers within 1,000,000. Assessment Limits Items may not compare digits across more than 1 place value. Editing Task Choice Multiple Choice Item Types Equation Editor Multiselect Hot Text Open Response No context Context Complexity Level Level 1: Recall Achievement Level Descriptions Level 2 Level 3 Level 4 Level 5 [intentionally left blank] recognizes that a digit in one place recognizes that a digit in one place recognizes that a digit in one place represents 10 times as much as it represents 10 times as much as it represents 10 times as much as it represents in the place to its right (for represents in the place to its right (for represents in the place to its right (for numbers up to and including 10,000), numbers up to and including 100,000) numbers up to and including 1,000,000) with visual representations Page 20 of 60 Content Standard MAFS.4.MD.1.3 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.
Figures are limited to rectangles or composite figures composed of rectangles. Assessment Limits Fractions are limited to like denominators. Limit multiplication and division to 2‐digit by 1‐digit or a multiple of 10 by 1‐digit. Quotients may only be whole numbers. Limit addition and subtraction to solutions within 1,000. When constructing rectangles, one grid must be labeled with the appropriate dimension. That dimension must be “1 ____,” as items at this standard may not assess scale. Equation Editor Multiple Choice Item Types GRID Multiselect Allowable Context Level 2: Basic Application of Skills & Concepts Complexity Level Achievement Level Descriptions Level 2 Level 3 uses place value understanding to round multi‐digit whole numbers to any place within 1,000 uses place value understanding to round multi‐digit whole numbers to any place within 1,000,000 Level 4 uses place value understanding to round whole numbers up to any place where the digit to the left is also affected (e.g., round 199 to the nearest ten) Level 5 determines a number that falls between two numbers of different place values Page 21 of 60 Teacher Academic Language New or Recently Introduced Terms  Associative property [96 = 3 × (4 × 8) = (3 × 4) × 8]  Composite number (positive integer having three or more whole number factors)  Distributive Property [64 × 27 = (60 × 20) + (60 × 7) + (4 × 20) + (4 × 7)]  Divisor (the number by which another number is divided)  Partial product (e.g. 24 × 6 = (20 × 6) + (4 × 6) = 120 + 24)  Prime number (positive integer only having whole number factors of one and itself)  Remainder (the number left over when one integer is divided by another) Familiar Terms and Symbols  Algorithm (steps for base ten computations with the four operations)  Area (the amount of two‐dimensional space in a bounded region)  Area model (a model for multiplication problems, in which the length and width of a rectangle represent the factors)  Bundling, grouping, renaming, changing  Compare (to find the similarity or dissimilarity between)  Distribute (decompose an unknown product in terms of two known products to solve)  Divide/Division (e.g., 15 ÷ 5 = 3)  Equation (a statement that the values of two mathematical expressions are equal using the = sign)  Factors (numbers that can be multiplied together to get other numbers)  Mixed units (e.g., 1 ft 3 in, 4 lb 13 oz)  Multiple (product of a given number and any other whole number)  Multiply/Multiplication (e.g., 5 × 3 =15)  Perimeter (length of a continuous line forming the boundary of a closed geometric figure)  Place value (the numerical value that a digit has by virtue of its position in a number)  Product (the result of multiplication)  Quotient (the result of division)  Rectangular array (an arrangement of a set of objects into rows and columns)  ___ times as many ___ as ___ (sentence frame) Page 22 of 60 Multi‐Digit Multiplication and Division
OVERVIEW In this 41‐day module, students use place value understanding and visual representations to solve multiplication and division problems with multi‐digit numbers. As a key area of focus for Grade 4, this module moves slowly but comprehensively to develop students’ ability to reason about the methods and models chosen to solve problems with multi‐digit factors and dividends. Students begin in Topic A by investigating the formulas for area and perimeter. They then solve multiplicative comparison problems including the language of times as much as with a focus on problems using area and perimeter as a context (e.g., “A field is 9 feet wide. It is 4 times as long as it is wide. What is the perimeter of the field?”). Students create diagrams to represent these problems as well as write equations with symbols for the unknown quantities (4.OA.1.1). This is foundational for understanding multiplication as scaling in Grade 5 and sets the stage for proportional reasoning in Grade 6. This Grade 4 module, beginning with area and perimeter, allows for new and interesting word problems as students learn to calculate with larger numbers and interpret more complex problems (4.OA.1.2, 4.OA.1.3, 4.MD.1.3). In Topic B, students use place value disks to multiply single‐digit numbers by multiples of 10, 100, and 1,000 and two‐digit multiples of 10 by two‐digit multiples of 10 (4.NBT.2.5). Reasoning between arrays and written numerical work allows students to see the role of place value units in multiplication (as pictured below). Students also practice the language of units to prepare them for multiplication of a single‐digit factor by a factor with up to four digits and multiplication of two two‐digit factors. In preparation for two‐digit by two‐digit multiplication, students practice the new complexity of multiplying two two‐digit multiples of 10. For example, students have multiplied 20 by 10 on the place value chart and know that it shifts the value one place to the left, 10 × 20 = 200. To multiply 20 by 30, the associative property allows for simply tripling the product, 3 × (10 × 20), or multiplying the units, 3 tens × 2 tens = 6 hundreds (alternatively, (3 × 10) × (2 × 10) = (3 × 2) × (10 × 10)). Introducing this early in the module allows students to practice during fluency so that, by the time it is embedded within the two‐digit by two‐digit multiplication in Topic H, understanding and skill are in place. Page 23 of 60 Building on their work in Topic B, students begin in Topic C decomposing numbers into base ten units in order to find products of single‐digit by multi‐digit numbers. Students use the distributive property and multiply using place value disks to model. Practice with place value disks is used for two‐, three‐, and four‐digit by one‐digit multiplication problems with recordings as partial products. Students bridge partial products to the recording of multiplication via the standard algorithm.1 Finally, the partial products method, the standard algorithm, and the area model are compared and connected by the distributive property (4.NBT.2.5). 1,423 x 3 Topic D gives students the opportunity to apply their new multiplication skills to solve multi‐step word problems (4.OA.1.3, 4.NBT.2.5) and multiplicative comparison problems (4.OA.1.2). Students write equations from statements within the problems (4.OA.1.1) and use a combination of addition, subtraction, and multiplication to solve. In Topic E, students synthesize their Grade 3 knowledge of division types (group size unknown and number of groups unknown) with their new, deeper understanding of place value. 1 Students become fluent with the standard algorithm for multiplication in Grade 5 (5.NBT.5). Grade 4 students are introduced to the standard algorithm in preparation for fluency and as a general method for solving multiplication problems based on place value strategies, alongside place value disks, partial products, and the area model. Students are not assessed on the standard algorithm in Grade 4. Page 24 of 60 Students focus on interpreting the remainder within division problems, both in word problems and long division (4.OA.1.3). A remainder of 1, as exemplified below, represents a leftover flower in the first situation and a remainder of 1 ten in the second situation.2 While we have no reason to subdivide a remaining flower, there are good reasons to subdivide a remaining ten. Students apply this simple idea to divide two‐
digit numbers unit by unit: dividing the tens units first, finding the remainder (the number of tens unable to be divided), and decomposing remaining tens into ones to then be divided. Students represent division with single‐digit divisors using arrays and the area model before practicing with place value disks. The standard division algorithm3 is practiced using place value knowledge, decomposing unit by unit. Finally, students use the area model to solve division problems, first with and then without remainders (4.NBT.2.6).
In Topic F, armed with an understanding of remainders, students explore factors, multiples, and prime and composite numbers within 100 (4.OA.2.4), gaining valuable insights into patterns of divisibility as they test for primes and find factors and multiples. This prepares them for Topic G’s work with multi‐digit dividends. Topic G extends the practice of division with three‐ and four‐digit dividends using place value understanding. A connection to Topic B is made initially with dividing multiples of 10, 100, and 1,000 by single‐digit numbers. Place value disks support students visually as they decompose each unit before dividing. Students then practice using the standard algorithm to record long division. They solve word problems and make connections to the area model as was done with two‐digit dividends (4.NBT.2.6, 4.OA.1.3). The module closes as students multiply two‐digit by two‐digit numbers. Students use their place value understanding and understanding of the area model to empower them to multiply by larger numbers (as pictured to the right). Topic H culminates at the most abstract level by explicitly connecting the partial products 2 Note that care must be taken in the interpretation of remainders. Consider
the fact that 7 ÷ 3 is not equal to 5 ÷ 2 because the remainder of 1 is in reference to a different whole amount (2 is not
equal to 2 ). 3 Students become fluent with the standard division algorithm in Grade 6 (6.NS.2). For adequate practice in reaching fluency, students are introduced to, but not assessed on, the division algorithm in Grade 4 as a general method for solving division problems. Page 25 of 60 appearing in the area model to the distributive property and recording the calculation vertically (4.NBT.2.5). Students see that partial products written vertically are the same as those obtained via the distributive property: 4 twenty‐sixes + 30 twenty‐sixes = 104 + 780 = 884. As students progress through this module, they are able to apply the multiplication and division algorithms because of their in‐depth experience with the place value system and multiple conceptual models. This helps to prepare them for fluency with the multiplication algorithm in Grade 5 and the division algorithm in Grade 6. Students are encouraged in Grade 4 to continue using models to solve when appropriate. Suggested Methods of Instructional Delivery Directions for Administration of Sprints Sprints are designed to develop fluency. They should be fun, adrenaline‐rich activities that intentionally build energy and excitement. A fast pace is essential. During Sprint administration, teachers assume the role of athletic coaches. A rousing routine fuels students’ motivation to do their personal best. Student recognition of increasing success is critical, and so every improvement is celebrated. One Sprint has two parts with closely related problems on each. Students complete the two parts of the Sprint in quick succession with the goal of improving on the second part, even if only by one more. With practice, the following routine takes about 9 minutes. Sprint A Pass Sprint A out quickly, face down on student desks with instructions to not look at the problems until the signal is given. (Some Sprints include words. If necessary, prior to starting the Sprint, quickly review the words so that reading difficulty does not slow students down.) T: You will have 60 seconds to do as many problems as you can. I do not expect you to finish all of them. Just do as many as you can, your personal best. (If some students are likely to finish before time is up, assign a number to count by on the back.) T: Take your mark! Get set! THINK! Students immediately turn papers over and work furiously to finish as many problems as they can in 60 seconds. Time precisely. T: T: S: T: S: Stop! Circle the last problem you did. I will read just the answers. If you got it right, call out “Yes!” If you made a mistake, circle it. Ready? (Energetically, rapid‐fire call the first answer.) Yes! (Energetically, rapid‐fire call the second answer.) Yes! Repeat to the end of Sprint A or until no student has a correct answer. If needed, read the count‐by answers in the same way you read Sprint answers. Each Page 26 of 60 number counted‐by on the back is considered a correct answer.
T: Fantastic! Now, write the number you got correct at the top of your page. This is your personal goal for Sprint B. T: How many of you got one right? (All hands should go up.) T: Keep your hand up until I say the number that is one more than the number you got correct. So, if you got 14 correct, when I say 15, your hand goes down. Ready? T: (Continue quickly.) How many got two correct? Three? Four? Five? (Continue until all hands are down.) If the class needs more practice with Sprint A, continue with the optional routine presented below. T: I’ll give you one minute to do more problems on this half of the Sprint. If you finish, stand behind your chair. As students work, the student who scored highest on Sprint A might pass out Sprint B. T: Stop! I will read just the answers. If you got it right, call out “Yes!” If you made a mistake, circle it. Ready? (Read the answers to the first half again as students stand.) Movement To keep the energy and fun going, always do a stretch or a movement game in between Sprints A and B. For example, the class might do jumping jacks while skip‐counting by 5 for about 1 minute. Feeling invigorated, students take their seats for Sprint B, ready to make every effort to complete more problems this time. Sprint B Pass Sprint B out quickly, face down on student desks with instructions to not look at the problems until the signal is given. (Repeat the procedure for Sprint A up through the show of hands for how many right.) T: Stand up if you got more correct on the second Sprint than on the first. S: (Stand.) T: Keep standing until I say the number that tells how many more you got right on Sprint B. If you got three more right on Sprint B than you did on Sprint A, when I say three, you sit down. Ready? (Call out numbers starting with one. Students sit as the number by which they improved is called. Celebrate the students who improved most with a cheer.) T: Well done! Now, take a moment to go back and correct your mistakes. Think about what patterns you noticed in today’s Sprint. T: How did the patterns help you get better at solving the problems? T: Rally Robin your thinking with your partner for 1 minute. Go! Rally Robin is a style of sharing in which partners trade information back and forth, one statement at a time per person, for about 1 minute. This is an especially valuable part of the routine for students who benefit from their friends’ support to identify patterns and try new strategies. Students may take Sprints home. Page 27 of 60 RDW or Read, Draw, Write (an Equation and a Statement) Mathematicians and teachers suggest a simple process applicable to all grades: 1) Read. 2) Draw and Label. 3) Write an equation. 4) Write a word sentence (statement). The more students participate in reasoning through problems with a systematic approach, the more they internalize those behaviors and thought processes. 
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What do I see? Can I draw something? What conclusions can I make from my drawing? Page 28 of 60 Module Specific Notes: Two additional lessons were added to the end of this module to address 4.OA.1a, 4.OA.1b, and 4.OA.3.5. These are the only lessons that address these standards prior to FSA. If students struggle, seek out further resources from CPALMS to address these standards. Resources Module Materials Math Studio Talk: Common Core Instruction for 4.OA https://www.engageny.org/resource/math‐studio‐
talk‐common‐core‐instruction‐4oa Grade 4 Math: Represent and solve division problems with up to a three‐digit dividend 4.NBT.6 https://www.engageny.org/resource/grade‐4‐math‐
represent‐and‐solve‐division‐problems‐a‐three‐digit‐
dividend‐4nbt6 
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Place value disks: 18 ones, 18 tens, 18 hundreds, 18 thousands, 1 ten thousand (1 set per student/pair) Square inch tiles (about 30 per student/pair) Personal white boards (class set) Grid paper Crayons: red, green, blue, orange (1 per student/pair) Additional Aligned Activities Investigations in Numbers, Data, and Space Unit 1 Session 1.4 Factor Pairs Unit 1 Session 2.3 Multiple Turn Over Unit 1 Session 3.1 Finding Factor of 100 enVision Math Common Core Topic 7‐ 7.2** Clip and Cover Topic 8‐8.1** Teamwork Topic 8‐ 8.2* Toss and Talk Topic 8‐ 8.2** Toss and Talk CPALMS Searching for Primes http://www.cpalms.org/Public/PreviewResourc
eLesson/Preview/46825 Using Rectangles to Find Prime and Composite Numbers http://www.cpalms.org/Public/PreviewResourc
eLesson/Preview/73111 Page 29 of 60 Standards 4.NBT.2.6 4.OA.1.3 Topics and Objectives E Division of Tens and Ones with Successive Remainders Lesson 16: Understand and solve two‐digit dividend division problems with a remainder in the ones place by using place value disks. Lesson 17: Represent and solve division problems requiring decomposing a remainder in the tens. Lesson 18: Find whole number quotients and remainders. *Modified lesson and student problem set by Duval Math. Lesson 19: Explain remainders by using place value understanding and models. Omitted from Duval Math Curriculum Guide. This lesson was moved to Quarter 4. Lesson 20: Solve division problems without remainders using the area model. Lesson 21: Solve division problems with remainders using the area model. Omitted from Duval Math Curriculum Guide. Solving division problems with remainders using the area model is beyond the content limits of the 4th grade test item specifications. 4.OA.2.4 F Reasoning with Divisibility Lesson 22: Find factor pairs for numbers to 100, and use understanding of factors to define prime and composite. Lesson 23: Use division and the associative property to test for factors and observe patterns. Lesson 24: Determine if a whole number is a multiple of another number. Lesson 25: Explore properties of prime and composite numbers to 100 by using multiples. Page 30 of 60 4.OA.1.3 4.NBT.2.6 4.NBT.1.1 4.NBT.2.5 4.OA.1.3 4.MD.1.3 G H Division of Thousands, Hundreds, Tens, and Ones Lesson 26: Divide multiples of 10, 100, and 1,000 by single‐digit numbers. Lesson 27: Represent and solve division problems with up to a three‐digit dividend numerically and with place value disks requiring decomposing a remainder in the hundreds place. Lesson 28: Represent and solve three‐digit dividend division with divisors of 2, 3, 4, and 5 numerically. Lesson 29: Represent numerically four‐digit dividend division with divisors of 2, 3, 4, and 5, decomposing a remainder up to three times. Lesson 30: Solve division problems with a zero in the dividend or with a zero in the quotient. Lesson 31: Interpret division word problems as either number of groups unknown or group size unknown. Omitted from Duval Math Curriculum Guide. Embed analysis of division situations throughout later lessons. Lesson 32: Interpret and find whole number quotients and remainders to solve one‐step division word problems with larger divisors of 6, 7, 8, and 9. Lesson 33: Explain the connection of the area model of division to the long division algorithm for three‐ and four‐digit dividends. Omitted from Duval Math Curriculum Guide. Embed this discussion into Lesson 30 rather. Multiplication of Two‐Digit by Two‐Digit Numbers Lesson 34: Multiply two‐digit multiples of 10 by two‐digit numbers using a place value chart. Lesson 35: Multiply two‐digit multiples of 10 by two‐digit numbers using the area model. Lesson 36: Multiply two‐digit by two‐digit numbers using four partial products. Lessons 37–38: Transition from four partial products to the standard algorithm for two‐digit by two‐digit multiplication. CPALMS Lesson: Rules, Numbers Patterns, and Tables *Additional lesson added to address 4.OA.3.5 End‐of‐Module Assessment: Topics A–H Page 31 of 60 Topic H: Multiplication of Two‐Digit by Two‐Digit Numbers Module 3 closes with Topic H as students multiply two‐digit by two‐digit numbers. Lesson 34 begins this topic by having students use the area model to represent and solve the multiplication of two‐digit multiples of 10 by two‐digit numbers using a place value chart. Practice with this model helps to prepare students for two‐digit by two‐digit multiplication and builds the understanding of multiplying units of 10. In Lesson 35, students extend their learning to represent and solve the same type of problems using area models and partial products. In Lesson 36, students make connections to the distributive property and use both the area model and four partial products to solve problems. Lesson 37 deepens students’ understanding of multi‐digit multiplication by transitioning from four partial products with representation of the area model to two partial products with representation of the area model and finally to two partial products without representation of the area model. Topic H culminates at the most abstract level with Lesson 38 as students are introduced to the multiplication algorithm for two‐digit by two‐digit numbers. Knowledge from Lessons 34–37 provides a firm foundation for understanding the process of the algorithm as students make connections from the area model to partial products to the standard algorithm (4.NBT.2.5). Students see that partial products written vertically are the same as those obtained via the distributive property: 4 twenty‐sixes + 30 twenty‐
sixes = 104 + 780 = 884. Date/ Day of the school year Approx. Date: 11/14/2016 Day: 59 Objective/ Essential Question OBJ: Students will multiply two‐digit multiples of 10 by two‐digit numbers using a place value chart. EQ: How can I multiply two‐
digit multiples of 10 by two digit numbers using a place value chart? Math Florida Standards/ Assessment / Academic Language Lesson Mathematical Performance Tasks Practice Target
area models Module 3
4.NBT.2.5 expression Topic H associative property Lesson 34: Multiply two‐digit multiples of 10 by two‐digit Embedded numbers using a place value chart. 4.NBT.1.1 4.MD.1.3 Lesson Materials: (T) Thousands place value chart for dividing (Lesson 26 Template) (S) Personal white board, thousands place value chart for Fluency dividing (Lesson 26 Template) 4.NBT.2.6 3.G.1.2 Mathematical Practices Notes: Topic G, Lesson 33: Explain the connection of the area model of division to the long division algorithm written method for three‐ and four‐digit dividends was omitted from the curriculum. This content should have been embedded in Lesson 30. Page 32 of 60 Approx. Date: 11/15/2016 Day: 60 OBJ: Students will multiply two‐digit multiples of 10 by two‐digit numbers using the area model. EQ: How can I multiply two‐
digit multiples of 10 by two‐digit numbers using the area model? Target:
4.NBT.2.5 Embedded 4.NBT.1.1 Fluency 4.NBT.2.6 3.G.1.2 Mathematical Practices MP8 Notes: arrays equal groups partial products regular products vertically horizontally Module 3
Topic H Lesson 35: Multiply two‐digit multiples of 10 by two‐digit numbers using the area model. Lesson Materials: (S) Personal white board, thousands place value chart (Lesson 4 Template) Approx. Date: 11/16/2016 Day: 61 OBJ: Students will multiply two‐digit by two‐digit numbers using four partial products. EQ: How can I multiply two‐
digit numbers using four partial products? Target
4.NBT.2.5 Embedded 4.NBT.2.4 4.MD.1.3 Fluency 4.NBT.2.6 3.G.1.2 Mathematical Practices MP4 Notes: distributive property area model visualize rectangle Module 3
Topic H Lesson 36: Multiply two‐digit by two‐digit numbers using four partial products. Lesson Materials: (S) Personal white board Approx. Date: 11/17/2016 Day: 62 OBJ: Students will transition from four partial products to the standard algorithm for two‐digit by two‐digit multiplication. EQ: How can I move from four partial products to Target
4.NBT.2.5 Embedded Fluency 4.MD.3.7 Mathematical Practices MP4 partial products mental math area models visualize Module 3
4.NBT.2.5 Topic H Performance Task The Lesson 37: Transition from four partial products to the Produce Shop standard algorithm for two‐digit by two‐digit multiplication. Lesson Materials: (S) Personal white board, paper and pencil Page 33 of 60 the standard algorithm for two‐digit multiplication? Approx. Date: 11/18/2016 Day: 63 Approx. Date: 11/21/2016 Day: 64 Notes:  Lesson 38: Transition from four partial products to the standard algorithm for two‐digit by two‐digit multiplication was omitted from the Curriculum Guide because it exceeds the content limits of the 4th grade item specification.  If more examples are needed, use Lesson 38 Concept Development and/or Problem Set.  Solving multiplication problems using the standard algorithm exceeds the content limits for 4th grade. Target
OBJ: equation
iReady Teacher Toolbox (www.teacher‐toolbox.com) Students will use 4th Grade 4.OA.1a distributive comparative thinking Lesson 8a Working with Equations 4.OA.1b property and numerical factor relationships to Lesson Materials: Embedded unknown determine if an equation 4.NBT.2.4 (S) Ready Instruction Book Student is true or false, or to find 4.OA.1.1 (T) Ready Instruction Book Teacher the value of the unknown. Fluency EQ: Mathematical How can I use Practices comparative thinking and numerical relationships to Notes: determine if an equation This lesson is the only lesson that addresses 4.OA.1a and 4.OA.1b prior to FSA. These standards will be addressed again in Quarter is true or false, or to find 4. If students struggle, seek out further resources from CPALMS to address these standards. the value of the unknown? OBJ: Target
pattern
iReady Teacher Toolbox (www.teacher‐toolbox.com) Students will follow a 4.OA.3.5 rule 4th Grade rule to generate a table Lesson 8b Number and Shape Patterns number pattern or Embedded shape pattern and then 4.OA.2.4 Lesson Materials: identify apparent (S) Ready Instruction Book Student features of the pattern. (T) Ready Instruction Book Teacher Fluency EQ: How can I follow a rule Mathematical to generate a number or Practices shape pattern and Notes: identify features of the This lesson is the only lesson that addresses 4.OA.3.5 prior to FSA. This standard will be addressed again in Quarter 4. If students pattern? struggle, seek out further resources from CPALMS to address this standard. Page 34 of 60 Approx. Date: 11/22/2016 Day: 65 OBJ: Students will show mastery of standards introduced in Module 3 Topics A‐H. EQ: How can I use strategies I’ve learned to help me as a mathematician? Approx. Date: 11/28/2016 Day: 66 OBJ: Students will show mastery of standards introduced in Module 3 Topics A‐H. EQ: How can I use strategies I’ve learned to help me as a mathematician? Target
4.OA.1.1 4.OA.1.2 4.OA.1.3 4.OA.1a 4.OA.1b 4.OA.2.4 4.OA.3.5 4.NBT.2.5 4.NBT.2.6 4.MD.1.3 Embedded 4.NBT.1.1 Notes: Target
4.OA.1.1 4.OA.1.2 4.OA.1.3 4.OA.1a 4.OA.1b 4.OA.2.4 4.OA.3.5 4.NBT.2.5 4.NBT.2.6 4.MD.1.3 Embedded 4.NBT.1.1 Notes:
Module 3
End‐of‐Module‐Assessment Topics A‐H Module 3
FLEX Day Use this time to review and remediate previously taught standards based on data. The recommendation is to use district supported materials. Module 3
End‐of‐Module‐
Assessment Topics A‐
H Review End of Module Assessment: Topics A‐
H Page 35 of 60 Quarter 2 Module 5 DUVAL COUNTY PUBLIC SCHOOLS
Math Curriculum Lesson Guide
Course: Unit # Dates/Pacing: Grade 4 Math Module 5: Fraction Equivalence, Ordering, and Operations 11/29/16 – 12/22/16, 18 days Number and Operations ‐ Fractions 4.NF.1.1: 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. Cognitive Complexity: Level 3: Strategic Thinking and Complex Reasoning 4.NF.1.2: 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. Cognitive Complexity: Level 2 Basic Application of Skills and Concepts 4.NF.2.3: 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. MAFS 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 fraction models and equations to represent the problem. Cognitive Complexity: Level 2 Basic Application of Skills and Concepts 4.NF.2.4: 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? Cognitive Complexity: Level 2 Basic Application of Skills and Concepts Page 36 of 60 Measurement and Data 4.MD.1.2: Use the four operations to solve word problems1 involving distances, intervals of time, and money, including problems involving simple fractions or decimals2. Represent fractional quantities of distance and intervals of time using linear models. (1See glossary Table 1 and Table 2) (Computational fluency with fractions and decimals is not the goal for students at this grade level.) Cognitive Complexity: Level 2 Basic Application of Skills and Concepts 4.MD.2.4: 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. Cognitive Complexity: Level 2 Basic Application of Skills and Concepts Page 37 of 60 Test Item Specifications Content Standard MAFS.4.NF.1.1 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. Denominators of given fractions are limited to: 2, 3, 4, 5, 6, 8, 10, 12, 100. Assessment Limits For denominators of 10 and 100, focus may not be on equivalence between these 2 denominators, since this is addressed specifically in standards MAFS.4.NF.5 – 7, but should be more on equivalence between fractions with denominators of 2, 4, and 5, and fractions with denominators of 10 and 100, e.g. = , = Fractions must refer to the same whole, including in models. Fraction models are limited to number lines, rectangles, squares, and circles. Fractions can be fractions greater than 1 and students may not be guided to put fractions in lowest terms or to simplify. . Equivalent fractions also include fractions . Editing Task Choice Matching Item Equation Editor Multiple Choice GRID Multiselect Hot Text Open Response Allowable
Item Types Context Complexity Level Level 3: Strategic Thinking and Complex Reasoning Achievement Level Descriptions Level 2 uses visual fraction models to recognize equivalent fractions by partitioning unit fraction pieces into smaller equal pieces Level 3 uses visual fraction models to
generate and explain equivalent fractions by partitioning unit fraction pieces into smaller pieces (and understands that this is the same); generates and explains why fraction a/b is equivalent to a fraction (n x a)/(n x b), and multiplies by 1 represented as a fraction Level 4 uses a variety of strategies to
generate and justify why fraction a/b is equivalent to a fraction (n x a)/(n x b) Level 5 [intentionally left blank] Page 38 of 60 Content Standard MAFS.4.NF.1.2 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. Denominators of given fractions are limited to: 2, 3, 4, 5, 6, 8, 10, 12, 100. Assessment Limits Fractions may be fractions greater than 1 and students may not be guided to put fractions in lowest terms or to simplify.
Two fractions being compared must have both different numerator and different denominator. Editing Task Choice Matching Item Item Types Equation Editor Multiple Choice GRID Multiselect Hot Text Open Response Allowable Context Complexity Level Level 2: Basic Application of Skills & Concepts Achievement Level Descriptions Level 2 Level 3 uses visual fraction model to compare two fractions with different numerators and different denominators (2, 3, 4, 6, and 8), using <, >, and =, with the understanding that the fractions must refer to the same whole compares two fractions with different numerators and different denominators, using visual fraction models and <, >, and = Level 4 compares two fractions with different numerators and different denominators, using <, >, and =; justifies answers Level 5 [intentionally left blank] Page 39 of 60 Content Standard MAFS.4.NF.2.3 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. Denominators of given fractions are limited to: 2, 3, 4, 5, 6, 8, 10, 12, 100. Assessment Limits Mixed numbers and fractions must contain like denominators. Items must reference the same whole. Visual fraction models are limited to circular models, rectangular models, and number line models. Equation Editor Multiple Choice Item Types GRID Multiselect Matching Item Open Response Allowable. Required for MAFS.4.NF.2.3d Context Complexity Level Level 2: Basic Application of Skills & Concepts Achievement Level Descriptions Level 2 adds and subtracts 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 and represents the decomposition using an equation Level 3 adds and subtracts fractions
and/or mixed numbers with like denominators, in mathematical and real‐world context, by replacing each mixed number with an equivalent fraction, without regrouping, and by using the properties of operations and the relationship between addition and subtraction; decomposes a mixed number into a sum of fractions with the same denominator in more than one way and records and justifies the decomposition Level 4 adds and subtracts mixed
numbers with like denominators, in mathematical and real‐world context, by replacing each mixed number with an equivalent fraction, with regrouping, and by using the properties of operations and the relationship between addition and subtraction Level 5 solves multistep word problems
involving addition and subtraction of fractions and/or mixed numbers Page 40 of 60 Content Standard MAFS.4.NF.2.4 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? Fractions may only be multiplied by a whole number. Assessment Limits Denominators of given fractions are limited to: 2, 3, 4, 5, 6, 8, 10, 12, 100. Equation Editor Multiple Choice Item Types GRID Multiselect Allowable Context Level 2: Basic Application of Skills & Concepts
Complexity Level Achievement Level Descriptions Level 2 understands a fraction a/b as a multiple of 1/b including the use of visual fraction models or repeated addition Level 3 understands and solves one‐
step mathematical and real‐ world problems involving a fraction a/b as a multiple of 1/b, and uses this understanding to multiply a fraction by a whole number, using visual fraction model Level 4 Level 5 understands and solves word
solves multistep word problems 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 (in general, n x a/b is (n x a)/b ) Page 41 of 60 Content Standard MAFS.4.MD.2.4 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. Measurement units are limited to halves, quarters, and eighths. Assessment Limits Addition and subtraction of fractions is limited to fractions with like denominators. Limit addition and subtraction to solutions within 1,000. Equation Editor Multiple Choice Item Types GRID Multiselect Allowable Context Level 2: Basic Application of Skills & Concepts
Complexity Level Achievement Level Descriptions Level 2 Level 3 makes a line plot to display a data set uses addition and subtraction of of measurements in fractions of a unit fractions to solve problems by using (1/8, 1/4, 1/2) information from a line plot Level 4 uses addition and subtraction of fractions to solve two‐step problems by using information from a line plot Level 5 uses addition and subtraction of fractions to solve multistep problems by using information from a line plot; draws conclusions Page 42 of 60 Content Standard MAFS.4.MD.1.2 Use the four operations to solve word problems1 involving distances, intervals of time, and money, including problems involving simple fractions or decimals2. Represent fractional quantities of distance and intervals of time using linear models. (1See glossary Table 1 and Table 2) (Computational fluency with fractions and decimals is not the goal for students at this grade level.) Measurement conversions are from larger units to smaller units. Assessment Limits Calculations are limited to simple fractions or decimals. Operations may include addition, subtraction, multiplication, and division. Item contexts are not limited to distances, intervals of time, and money. Equation Editor Multiple Choice Item Types GRID Multiselect Required Context Level 2: Basic Application of Skills & Concepts Complexity Level Achievement Level Descriptions Level 2 uses the four operations to solve word problems (involving distance, intervals of time, and money) with context, including problems involving whole numbers Level 3 uses the four operations to
solve word problems (involving distance, intervals of time, and money) including problems involving simple fractions or decimals; represents measurement quantities using linear models Level 4 Level 5 uses the four operations to
solve word problems including problems involving simple fractions or decimals and problems that require expressing measurements given in a larger unit in terms of a smaller unit uses the four operations to
solve multistep word problems, including problems involving fractions or decimals and problems that require expressing measurements given in a larger unit in terms of a smaller unit Page 43 of 60 Teacher Academic Language New or Recently Introduced Terms 
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Benchmark (standard or reference point by which something is measured) Common denominator (when two or more fractions have the same denominator) Denominator (e.g., the 5 in names the fractional unit as fifths) Fraction greater than 1 (a fraction with a numerator that is greater than the denominator) Line plot (display of data on a number line, using an x or another mark to show frequency) Mixed number (number made up of a whole number and a fraction) Numerator (e.g., the 3 in indicates 3 fractional units are selected) Familiar Terms and Symbols4 
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=, <, > (equal to, less than, greater than) Compose (change a smaller unit for an equivalent of a larger unit, e.g., 2 fourths = 1 half, 10 ones = 1 ten; combining 2 or more numbers, e.g., 1 fourth + 1 fourth = 2 fourths, 2 + 2 + 1 = 5) Decompose (change a larger unit for an equivalent of a smaller unit, e.g., 1 half = 2 fourths, 1 ten = 10 ones; partition a number into 2 or more parts, e.g., 2 fourths = 1 fourth + 1 fourth, 5 = 2 + 2 + 1) Equivalent fractions (fractions that name the same size or amount) Fraction (e.g., , , , ) Fractional unit (e.g., half, third, fourth) Multiple (product of a given number and any other whole number) Non‐unit fraction (fractions with numerators other than 1) Unit fraction (fractions with numerator 1) Unit interval (e.g., the interval from 0 to 1, measured by length) Whole (e.g., 2 halves, 3 thirds, 4 fourths) 4
These are terms and symbols students have seen previously. Page 44 of 60 Fraction Equivalence, Ordering, and Operations
OVERVIEW In this 38‐day module, students build on their Grade 3 work with unit fractions as they explore fraction equivalence and extend this understanding to mixed numbers. This leads to the comparison of fractions and mixed numbers and the representation of both in a variety of models. Benchmark fractions play an important part in students’ ability to generalize and reason about relative fraction and mixed number sizes. Students then have the opportunity to apply what they know to be true for whole number operations to the new concepts of fraction and mixed number operations. Students begin Topic A by decomposing fractions and creating tape diagrams to represent them as sums of fractions with the same denominator in different 3
1
1
1
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2
ways (e.g., = + + = + ) (4.NF.2.3b). They proceed to see that representing a fraction as the repeated addition of a unit fraction is the same as 5
5
5
5
5
5
multiplying that unit fraction by a whole number. This is already a familiar fact in other contexts. 1
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For example, just as 3 twos = 2 + 2 + 2 = 3 × 2, so does 3 fourths= + + =3 × . 4
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The introduction of multiplication as a record of the decomposition of a fraction (4.NF.2.4a) early in the module allows students to become familiar with the notation before they work with more complex problems. As students continue working with decomposition, they represent familiar unit fractions as the sum of smaller unit fractions. A folded paper activity allows them to see that, when the number of fractional parts in a whole increases, the size of the parts decreases. They proceed to investigate this concept with the use of tape diagrams and area models. Reasoning enables them to explain why two different fractions can represent the same portion of a whole (4.NF.1.1). In Topic B, students use tape diagrams and area models to analyze their work from earlier in the module and begin using multiplication to create an equivalent fraction that comprises smaller units, e.g., "2" /"3" " = " "2 × 4" /"3 × 4" " = " "8" /"12" (4.NF.1). Based on the use of multiplication, they reason that division can be used to create a fraction that comprises larger units (or a single unit) equivalent to a given fraction (e.g., "8" /"12" " = " "8 ÷ 4" /"12 ÷ 4" " = " "2" /"3"). Their work is justified using area models and tape diagrams and, conversely, multiplication is used to test for and/or verify equivalence. Students use the tape diagram to transition to modeling equivalence on the number line. They see that, by multiplying, any unit fraction length can be partitioned into n equal lengths and that doing so multiplies both the total number of fractional units (the denominator) and number of selected units (the numerator) by n. They also see that there are times when fractional units can be grouped together, or divided, into larger fractional units. When that occurs, both the total number of fractional units and number of selected units are divided by the same number. Page 45 of 60 In Grade 3, students compared fractions using fraction strips and number lines with the same denominators. In Topic C, they expand on comparing fractions by reasoning about fractions with unlike denominators. Students use the relationship between the numerator and denominator of a fraction to compare to a 1
known benchmark (e.g., 0, , or 1) on the number line. Alternatively, students compare using the same numerators. They find that the fraction with the 2
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1
5
10
greater denominator is the lesser fraction since the size of the fractional unit is smaller as the whole is decomposed into more equal parts (e.g., > , therefore 3
5
3
> ). Throughout the process, their reasoning is supported using tape diagrams and number lines in cases where one numerator or denominator is a factor 10
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of the other, such as and or and . When the units are unrelated, students use area models and multiplication, the general method pictured below to the 5
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left, whereby two fractions are expressed in terms of the same denominators. Students also reason that comparing fractions can only be done when referring to the same whole, and they record their comparisons using the comparison symbols <, >, and = (4.NF.1.2). Comparison Using Like Comparison Using Like Page 46 of 60 In Topic D, students apply their understanding of whole number addition (the combining of like units) and subtraction (finding an unknown part) to work with fractions (4.NF.2.3a). They see through visual models that, if the units are the same, computation can be performed immediately, e.g., 2 bananas + 3 bananas = 5 bananas and 2 eighths + 3 eighths = 5 eighths. They see that, when subtracting fractions from one whole, the whole is decomposed into the same units as 3
5
3
2
the part being subtracted, e.g., 1 – = – = . Students practice adding more than two fractions and model fractions in word problems using tape diagrams 5
5
5
5
(4.NF.2.3d). As an extension of the Grade 4 standards, students apply their knowledge of decomposition from earlier topics to add fractions with related units 1
1
using tape diagrams and area models to support their numerical work. To find the sum of and , for example, one simply decomposes 1 half into 2 smaller 2
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equal units, fourths, just as in Topics A and B. Now, the addition can be completed: + = . Though not assessed, this work is warranted because, in 4
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Module 6, students are asked to add tenths and hundredths when working with decimal fractions and decimal notation. At the beginning of Topic E, students use decomposition and visual models to add and subtract fractions less than 1 to or from whole numbers, e.g., 4+ 3
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3
3
3
4
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=4 and 4 – = 3 + 1 – . They use addition and multiplication to build fractions greater than 1 and represent them on the number line. Students then use these visual models and decompositions to reason about the various forms in which a fraction greater than or equal to 1 may be presented, both as fractions and mixed numbers. They practice converting between these forms and begin understanding the usefulness of each form in different situations. Through this understanding, the common misconception that every improper fraction must be converted to a mixed number is avoided. Next, students compare fractions greater than 1, building on their rounding skills and using understanding of benchmarks to reason about which of two fractions is greater (4.NF.1.2). This activity continues to build understanding of the relationship between the numerator and denominator of a fraction. Students progress to finding and using like denominators or numerators to compare and order mixed numbers. They apply their skills of comparing numbers greater than 1 by solving word problems (4.NF.2.3d) requiring the interpretation of data presented in line plots (4.MD.2.4). Students use addition and subtraction strategies to solve the problems, as well as decomposition and modeling to compare numbers in the data sets Page 47 of 60 In Topic F, students estimate sums and differences of mixed numbers, rounding before performing the actual operation to determine what a reasonable outcome will be. They proceed to use decomposition to add and subtract mixed numbers (4.NF.2.3c). This work builds on their understanding of a mixed number being the sum of a whole number and fraction. Using unit form, students add and subtract like units first (e.g., ones and ones, fourths and fourths). Students use decomposition, shown with number bonds, in mixed number addition to make one from fractional units before finding the sum. When subtracting, students learn to decompose the minuend or subtrahend when there are not enough fractional units from which to subtract. Alternatively, students can rename the subtrahend, giving more units to the fractional units, which connects to whole number subtraction when renaming 9 tens 2 ones as 8 tens 12 ones. In Topic G, students build on the concept of representing repeated addition as multiplication, applying this familiar concept to work with fractions (4.NF.2.4a, 4.NF.2.4b). They use the associative property and their understanding of decomposition. Just as with whole numbers, the unit remains unchanged. 3
1
1 4 × 3 12
4 × = 4 × 3 × = 4 × 3 × = = 5
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5
This understanding connects to students’ work with place value and whole numbers. Students proceed to explore the use of the distributive property to multiply a whole number by a mixed number. They recognize that they are multiplying each part of a mixed number by the whole number and use efficient strategies to do so. The topic closes with solving multiplicative comparison word problems involving fractions (4.NF.2.4c) as well as problems involving the interpretation of data presented on a line plot. Topic H comprises an exploration lesson where students find the sum of all 0
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like denominators from to . Students first work in teams with fourths, sixths, eighths, and tenths. For example, they might find the sum of all sixths from 6
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6
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to . Students discover that they can make pairs with a sum of 1 to add more efficiently, e.g., + , + , + , and there will be one fraction, , without a 6
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6 6
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pair. They then extend this to similarly find sums of thirds, fifths, sevenths, and ninths, observing patterns when finding the sum of odd and even denominators (4.OA.3.5). The Mid‐Module Assessment follows Topic D, and the End‐of‐Module Assessment follows Topic H. Page 48 of 60 Module Specific Notes: 
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During this module, many of the problems with the fraction unit use denominators that are beyond the content limits of the 4th grade Test Item Specifications. Please adjust the denominators in the problem sets and homework to meet the content limits. Module 5 will continue in Quarter 3.
Module 5 and 6 were moved ahead of Module 4 in order to teach the major work of the grade level earlier in the school year, as suggested by the Eureka pacing guide. Module 4 will follow Module 6, but will be completed before FSA. Resources Math Studio talk: Common Core Instruction for 4.NF Module Materials 
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Fraction strips (made from paper, folded, and used to model equivalent fractions)
Number line
Rulers
Colored pencils or markers (1 set per student/pair)
Index cards Scissors (one per student/pair)
Additional Aligned Activities Investigations in Numbers, Data and Space Unit 6: Fraction Cards and Decimal Squares Session 1.1 Fractions of an Area: Halves, Fourths, and Eighths (4.NF.1.1) Session 1.2 Fractions of an Area: Thirds and Sixths (4.NF.1.1) Session 2.1/2.2 Fraction Cards (4.NF.2.3) Session 2.3 Capture Fractions (4.NF.1.2) Session 2.4 Comparing Fractions to Landmark (4.NF.1.2, 4.NF.2.3) Session 2.5 Fractions on a Number Line (4.NF.1.2) enVision Math Common Core 11‐5 Teamwork (4.NF.1.1) 11‐6 Tic‐tac‐toe (4.NF.1.2) 11‐7 Clip and Cover (4.NF.1.2) 12‐1 Clip and Cover (4.NF.2.3a) 12‐2 Toss and Talk (4.NF.2.3a) 12‐3 Clip and Cover (4.NF.2.3a) 12‐4 Toss and Talk (4.NF.2.3a) 12‐6 Tic‐Tac‐Toe (4.NF.2.3a) 12‐7 Teamwork (4.NF.2.3c) 12‐8 Display the Digits (4.NF.2.3c) 12‐9 Teamwork (4.NF.2.3c) 12‐10 Clip and Cover (4.NF.2.3b) CPALMs Page 49 of 60 Lesson plans, virtual manipulatives and formative assessment tasks for 4.NF.1.1 Lesson plans, virtual manipulatives and formative assessment tasks for 4.NF.1.2 Lesson plans, virtual manipulatives, and formative assessment tasks for 4.NF.2.3 Lesson plans, virtual manipulatives, and formative assessment tasks for 4.NF.2.4 Page 50 of 60 Standards 4.NF.2.3b 4.NF.2.4a 4.NF.2.3a 4.NF.1.1 4.NF.2.3b Topics and Objectives A B Decomposition and Fraction Equivalence Lessons 1–2: Decompose fractions as a sum of unit fractions using tape diagrams. These lessons have been consolidated and will be taught in one day, in an effort to introduce all major concepts before FSA. Lesson 3: Decompose non‐unit fractions and represent them as a whole number times a unit fraction using tape diagrams. Lesson 4: Decompose fractions into sums of smaller unit fractions using tape diagrams. This lesson was omitted in an effort to introduce all major concepts before the FSA. Embed the contrast of the decomposition of a fraction using the tape vs. the area model in the upcoming Lesson 5. Lesson 5: Decompose unit fractions using area models to show equivalence. Lesson 6: Decompose fractions using area models to show equivalence. Fraction Equivalence Using Multiplication and Division Lessons 7–8: Use the area model and multiplication to show the equivalence of two fractions. Lessons 9‐10: Use the area model and division to show the equivalence of two fractions. . Lesson 11: Explain fraction equivalence using a tape diagram and the number line, and relate that to the use of multiplication and division.
4.NF.1.2 C Fraction Comparison Lessons 12–13: Reason using benchmarks to compare two fractions on the number line. Lessons 14–15: Find common units or number of units to compare two fractions.
4.NF.2.3ad 4.NF.1.1 4.MD.1.2 D Fraction Addition and Subtraction Lesson 16: Use visual models to add and subtract two fractions with the same units. Lesson 17: Use visual models to add and subtract two fractions with the same units, including subtracting from one whole. Lesson 18: Add and subtract more than two fractions. Lesson 19: Solve word problems involving addition and subtraction of fractions. Lessons 20–21: Use visual models to add two fractions with related units using the denominators 2, 3, 4, 5, 6, 8, 10, and 12. Omitted from Duval Math Curriculum Guide. The lessons exceed the content limits for 4th grade and has been moved to Quarter 4.
Mid‐Module Assessment: Topics A–D *End of Quarter 2. Module 5 is continued in Quarter 3. Page 51 of 60 Topic A: Decomposition and Fraction Equivalence Topic A builds on Grade 3 work with unit fractions. Students explore fraction equivalence through the decomposition of non‐unit fractions into unit fractions, as well as the decomposition of unit fractions into smaller unit fractions. They represent these decompositions, and prove equivalence, using visual models. In Lesson 1, students use paper strips to represent the decomposition of a whole into parts. In Lessons 1 and 2, students decompose fractions as unit fractions, drawing tape diagrams to represent them as sums of fractions with the same denominator in different ways, e.g., = + + = + . In Lesson 3, students see that representing a fraction as the repeated addition of a unit fraction is the same as multiplying that unit fraction by a whole number. This is already a familiar fact in other contexts. An example is as follows: 3 bananas = 1 banana + 1 banana + 1 banana = 3 × 1 banana 3 twos = 2 + 2 + 2 = 3 × 2 3 fourths = 1 fourth + 1 fourth + 1 fourth = 3 × 1 fourth = + + = 3 × By introducing multiplication as a record of the decomposition of a fraction early in the module, students are accustomed to the notation by the time they work with more complex problems in Topic G. Students continue with decomposition in Lesson 4, where they use tape diagrams to represent fractions, e.g. , , and , as the sum of smaller unit fractions. Students record the results as a number sentence, e.g., = + = ( + ) + ( + ) . In Lesson 5, this idea is further investigated as students represent the decomposition of unit fractions in area models. In Lesson 6, students use the area model for a second day, this time to represent fractions with different numerators. They explain why two different fractions represent the same portion of a whole. Date/ Day of the school year Approx. Date: 11/29/2016 Day: 67 Objective/ Essential Question OBJ: Students will decompose fractions as a sum of unit fractions using tape diagrams. EQ: How can I decompose fractions as a sum of unit fractions using tape diagrams? Math Florida Standards/ Mathematical Practice Target 4.NF.2.3b Embedded 4.NF.2.3a Fluency 3.OA.1.3 3.NF.1.1 3.NF.1.3 Mathematical Practices MP3 Academic Language thirds
sixths decompose decomposition number bond repeated addition sentence whole half diagonal equivalent unit fraction tape diagram fifths fourths brackets addends mixed number shade numerator denominator partitioned equal parts eighths Lesson Assessment / Performance Tasks Module 5
Topic A Lesson 1 and Lesson 2: Decompose fractions as a sum of unit fractions using tape diagrams. Lesson Materials: (S) 1 index card with diagonals drawn, pair of scissors (1 per pair of students) (T) 5 strips of paper and markers (S) 5 strips of paper, colored markers or colored pencils, personal white board Notes:  Ensure students are presented with fractions using both shaded and unshaded visual models. Page 52 of 60 
Approx. Date: 11/30/2016 Day: 68 Approx. Date: 12/01/2016 Day: 69 Approx. Date: 12/02/2016 Day: 70 Approx. Date: 12/05/2016 Day: 71 The language numerator and denominator are not mentioned within the script of lessons 1‐6, but it is important that this terminology be reviewed and used throughout these lessons. Target OBJ: tenths
Module 5
Students will 4.NF.2.4a shaded Topic A decompose non‐unit unshaded Lesson 3: Decompose non‐unit fractions and represent fractions and Embedded multiplication them as a whole number times a unit fraction using tape represent them as a 4.NF.2.3b number sentence diagrams. whole number times a unit fraction using Fluency Lesson Materials: tape diagrams. 4.OA.2.4 (S) Personal white board EQ: Mathematical How can I decompose Practices non‐unit fractions and Notes: represent them as a  This lesson goes beyond the content limits of the 4th grade test item specifications – some denominators are not whole number times a represented. Please look at the denominators carefully (2, 3, 4, 5, 6, 8, 10, 12,100). unit fraction using a  Omit problem #3c on the problem set has 9 as a denominator – ninths are not assessed. tape diagram? OBJ: Flex Day: Mid‐Year Scrimmage Testing. This Day is added Students will as a flex day to accommodate for Mid‐ Year FSA testing. complete the Mid‐
Please move this date to accommodate for testing window Year Scrimmage. and continue to move forward with the pacing of the curriculum guide as necessary. OBJ: Flex Day: Mid‐Year Scrimmage Testing. This Day is added Students will as a flex day to accommodate for Mid‐ Year FSA testing. complete the Mid‐
Please move this date to accommodate for testing window Year Scrimmage. and continue to move forward with the pacing of the curriculum guide as necessary. Target OBJ: area model
Module 5
4.NF.2.3b Students will region Topic A 4.NF.1.1 decompose unit twelfth Lesson 5: Decompose unit fractions using area models to fractions using area show equivalence. Embedded models to show 4.NF.2.4a equivalence. Lesson Materials: 4.NF.2.4b (S) personal white boards EQ: How can I decompose Fluency unit fractions using 3.NF.1.3 area models to show equivalence? Mathematical Practices MP2 Page 53 of 60 Date 12/06/2016 Day: 72 OBJ: Students will complete i‐Ready Diagnostic. Date 12/07/2016 Day: 73 OBJ: Students will complete i‐Ready Diagnostic. Approx. Date: 12/08/2016 Day: 74 OBJ: Students will decompose unit fractions using area models to show equivalence. EQ: How can I decompose unit fractions use area models to show equivalence? Notes: Topic A, Lesson 4: Decompose fractions into sums of smaller unit fractions using tape diagrams was omitted from the curriculum guide. Please embed the contrast of the decomposition of a fraction using the tape vs. the area model in the coming Lesson 5. Flex Day: i‐Ready Diagnostic Testing. This Day is added as i‐Ready Diagnostic a flex day to accommodate for I‐ready testing. Please move this date to accommodate for testing window and continue to move forward with the pacing of the curriculum guide as necessary. Recommendation: Curriculum associates recommends 4th grade administer in 40 minute sessions for completion, not a continuous 80‐90 minutes. Flex Day: i‐Ready Diagnostic Testing. This Day is added as i‐Ready Diagnostic a flex day to accommodate for I‐ready testing. Please move this date to accommodate for testing window and continue to move forward with the pacing of the curriculum guide as necessary. Recommendation: Curriculum associates recommends 4th grade administer in 40 minute sessions for completion, not a continuous 80‐90 minutes. Target equivalence
Module 5
4.NF.1.1 vertical line Topic A horizontal line Lesson 6: Decompose unit fractions using area models to Embedded equivalent fraction show equivalence. 4.NF.2.3 portion 4.NF.2.4 fractional unit Lesson Materials: S) Multiply whole numbers times fractions sprint; personal Fluency white boards Mathematical Practices Notes: Problems #1c of page 1 and page 3 of the homework include problems that are beyond the content limits of the 4th grade test item specifications. Page 54 of 60 Topic B: Fraction Equivalence Using Multiplication and Division In Topic B, students begin generalizing their work with fraction equivalence. In Lessons 7 and 8, students analyze their earlier work with tape diagrams and the area model in Lessons 3 through 5 to begin using multiplication to create an equivalent fraction that comprises smaller units, e.g., = = . Conversely, students reason, in Lessons 9 and 10, that division can be used to create a fraction that comprises larger units (or a single unit) equivalent to a given fraction, e.g., = 8 ÷ 4 = . The numerical work of Lessons 7 through 10 is introduced and supported using area models and tape diagrams. In Lesson 11, students use tape diagrams to transition their knowledge of fraction equivalence to the number line. They see that any unit fraction length can be partitioned into n equal lengths. For example, each third in the interval from 0 to 1 may be partitioned into 4 equal parts. Doing so multiplies both the total number of fractional units (the denominator) and the number of selected units (the numerator) by 4. Conversely, students see that, in some cases, fractional units may be grouped together to form some number of larger fractional units. For example, when the interval from 0 to 1 is partitioned into twelfths, one may group 4 twelfths at a time to make thirds. By doing so, both the total number of fractional units and number of selected units are divided by 4. Date/ Day of the school year Approx. Date: 12/09/2016 Day: 75 Approx. Date: 12/12/2016 Day: 76 Math Florida Standards/ Assessment / Academic Language Lesson Mathematical Performance Tasks Practice Target OBJ: doubling Module 5
Students will use the 4.NF.1.1 quadrupling Topic B area model and triple Lesson 7: Use the area model and multiplication to show multiplication to show Embedded relationship the equivalence of two fractions. the equivalence of 4.NF.2.3 pattern two fractions. Lesson Materials: Fluency (S) personal white boards EQ: 3.NF.1.3 How can I use the area model and Mathematical multiplication to show Practices the equivalence of MP7 two fractions? Notes: On the problem set, question 3b uses a denominator of ninths. Ninths is beyond content limits of the 4th grade test item specifications. The homework includes denominator of twelfths which is beyond 4th grade content limits. Please omit or change the denominator for these problems. OBJ: Target equivalent
Module 5
Students will use the 4.NF.1.1 numerator Topic B area model and denominator Lesson 8: Use the area model and multiplication to show multiplication to show Embedded multiply the equivalence of two fractions. the equivalence of 4.OA.2.4 divide Lesson Materials: two fractions. 3.NF.1.3 unit fraction (S) Personal white board area model Fluency double triple Objective/ Essential Question Page 55 of 60 Approx. Date: 12/13/2016 Day: 77 EQ: How can I use the area model and multiplication to show the equivalence of two fractions? OBJ: Students will use the area model and division to show the equivalence of two fractions. EQ: How can I use the area model and division to show the equivalence of two fractions? Mathematical Practices MP7 quadruple
Notes: Target 4.NF.1.1 Embedded Fluency 4.NBT.2.4 Mathematical Practices MP2, MP7 simplify
equivalent compose decompose numerator denominator area model multiply divide factors multiples units Module 5
Topic B Lesson 9: Use the area model and division to show the equivalence of two fractions. Lesson Materials: (S) Personal white board Notes: Fraction bars can be used in this lesson as manipulatives to assist students with seeing the equivalency of the fractions presented. Approx. Date: 12/14/2016 Day: 78 Approx. Date: 12/15/2016 Day: 79 OBJ: Students will use the area model and division to show the equivalence of two fractions. EQ: How can I use the area model and division to show the equivalence of two fractions? Target 4.NF.1.1 Embedded Fluency 3.NF.1.3 4.NF.1.1 Mathematical Practices Notes: Fraction bars can be used in this lesson as manipulatives to assist students with seeing the equivalency of the fractions presented. OBJ: Students will explain fraction equivalence using a tape diagram and the number line, and relate that to the Target 4.NF.1.1 Embedded 4.NBT.2.6 simplify
equivalent compose decompose numerator denominator area model multiply divide factors multiples units tape diagram
number line equivalent fractions compose decompose numerator Module 5
Topic B Lesson 10: Use area model and division to show the equivalence of two fractions. Lesson Materials: (S) Personal white board Module 5
Topic B Lesson 11: Explain fraction equivalence using a tape diagram and the number line, and relate that to the use of multiplication and division. 4.NF.1.1 Equivalent Fractions on a Number Line (CPALMS) Page 56 of 60 use of multiplication and division. EQ: How can I explain fraction equivalence using a tape diagram and the number line, and relate that to the use of multiplication and division? Fluency
denominator
Lesson Materials: multiply (S) Personal white board, ruler Mathematical divide Practices MP3 Notes:  Topic B, Lesson 10: Use the area model and division to show the equivalence of two fractions was omitted from the curriculum guide in an effort to introduce all concepts before FSA. Lesson 10 reviews and reinforces the concepts taught in this lesson. If time permits, consider teaching Lesson 10 based on the needs of your students.  The second part of the concept development can be done as a We Do in students’ Interactive Journals. Topic C: Fraction Comparison In Topic C, students use benchmarks and common units to compare fractions with different numerators and different denominators. The use of benchmarks is the focus of Lessons 12 and 13 and is modeled using a number line. Students use the relationship between the numerator and denominator of a fraction to compare to a known benchmark (e.g., 0, , or 1) and then use that information to compare the given fractions. For example, when comparing and , students reason that 4 sevenths is more than 1 half, while 2 fifths is less than 1 half. They then conclude that 4 sevenths is greater than 2 fifths. In Lesson 14, students reason that they can also use like numerators based on what they know about the size of the fractional units. They begin at a simple level by reasoning, for example, that 3 fifths is less than 3 fourths because fifths are smaller than fourths. They then see, too, that it is easy to make like numerators at times to compare, e.g., < , because = , and < because < . Using their experience with fractions in Grade 3, they know the larger the denominator of a unit fraction, the smaller the size of the fractional unit. Date/ Day of the school year Approx. Date: 12/16/2016 Day: 80 Math Florida Standards/ Assessment / Academic Language Lesson Mathematical Performance Tasks Practice Target OBJ: benchmark
Module 5
Students will reason 4.NF.1.2 plot Topic C using benchmarks to a little more than Lesson 12: Reason using benchmarks to compare two compare two fractions Embedded ____ fractions on the number line. on the number line. 4.NF.1.1 a little less than ____ Lesson Materials: EQ: Fluency compare (S) Personal white board, number line (Template) How can I reason 4.NBT.2.4 partition using benchmarks to 4.NF.2.3 compare two fractions on the number line? Mathematical Practices MP4 Notes:  This lesson is only addressing part of the standard 4.NF.1.2. Students will be comparing fractions by comparing to a benchmark fraction. The other comparison strategies included in the standard will be addressed in a later lesson. Objective/ Essential Question Page 57 of 60 
Approx. Date: 12/19/2016 Day: 81 Approx. Date: 12/20/2016 Day: 82 Emphasize that comparisons are valid only when the two fractions refer to the same whole. For example, ½ of a small pizza is not equivalent to ½ of a large pizza.  Always refer back to the same size whole. Target OBJ: number bond
Module 5
Students will reason 4.NF.1.2 discuss the Topic C using benchmarks to relationship Lesson 13: Reason using benchmarks to compare two compare two fractions Embedded “in relation to” fractions on the number line. on the number line. 4.NF.2.3 4.NF.1.1 Lesson Materials: EQ: (S) Personal white board, blank number lines with How can I reason Fluency midpoint (Template) using benchmarks to 4.NBT.2.6 compare two fractions 3.NF.2.3 on the number line? Mathematical Practices MP2 Notes:  This lesson is only addressing part of the standard 4.NF.1.2. Students will be comparing fractions by comparing to a benchmark fraction. The other comparison strategies included in the standard will be addressed in a later lesson.  Emphasize that comparisons are valid only when the two fractions refer to the same whole. For example, ½ of a small pizza is not equivalent to ½ of a large pizza. OBJ: Students will find common units or number of units to compare two fractions. EQ: How can I find common units to compare two fractions? Target common Module 5
4.NF.1.2 denominator Topic C common numerator Lesson 14: Find common units or number of units to Embedded compare two fractions. 4.NF.1.1 Lesson Materials: Fluency (S) Personal white board 4.NBT.2.4 4.NF.2.3 Mathematical Practices MP7 Notes:  This lesson is only addressing part of the standard 4.NF.1.2. Students will be comparing fractions by creating common numerators or denominators. The other comparison strategies included in the standard were addressed in previous lessons.  Emphasize that comparisons are valid only when the two fractions refer to the same whole. For example, ½ of a small pizza is not equivalent to ½ of a large pizza. Page 58 of 60 
Approx. Date: 12/21/2016 Day: 83 OBJ: Students will find common units or number of units to compare two fractions. EQ: How can I find common units to compare two fractions? Many of the problems with the fraction unit use denominators that are beyond the content limits of the 4th grade Test Item Specifications. Please adjust the denominators in the problem sets and homework to meet the content limits. Target like units Module 5
4.NF.1.2 vertical lines Topic C horizontal lines Lesson 15: Find common units or number of units to Embedded compare two fractions. 4.NF.1.1 Lesson Materials: Fluency (S) Personal white board Mathematical Practices MP7 Notes:  Within the Concept Development piece of the lesson, problems 1 and 3 exceed the item specification content limits. The problem involves creating equivalent fractions using the denominators 20 and 60. Students will not be assessed using a denominator of 20 or 60. Please adjust denominators to fit within the content limits.  Within the Problem Set, questions 1c, 1d, and 1e exceed the item specifications content limits. Please adjust denominators to fit within the content limits.  Within the homework problem set, questions 3a, 3b, and 3d exceed the item specifications content limits. These problems include denominators that will not be assessed. Please adjust denominators to fit within the content limits. Topic D: Fraction Addition and Subtraction Topic D bridges students’ understanding of whole number addition and subtraction to fractions. Everything that they know to be true of addition and subtraction with whole numbers now applies to fractions. Addition is finding a total by combining like units. Subtraction is finding an unknown part. Implicit in the equations 3 + 2 = 5 and 2 = 5 – 3 is the assumption that the numbers are referring to the same units. In Lessons 16 and 17, students generalize familiar facts about whole number addition and subtraction to work with fractions. Just as 3 apples – 2 apples = 1 apple, students note that 3 fourths – 2 fourths = 1 fourth. Just as 6 days + 3 days = 9 days = 1 week 2 days, students note that + = = + = 1 . In Lesson 17, students decompose a whole into a fraction having the same denominator as the subtrahend. For example, 1 – 4 fifths becomes 5 fifths – 4 fifths = 1 fifth, connecting with Topic B skills. They then see that, when solving 1 – they have a choice of subtracting from or from 1 (as pictured to the right). Students model with tape diagrams and number lines to understand and then verify their numerical work. In Lesson 18, students add more than two fractions and see sums of more than one whole, such as + + , = . As students move into problem solving in Lesson 19, they create tape diagrams or number lines to represent and solve fraction addition and subtraction word problems (see the example below). These problems bridge students into work with mixed numbers, which follows the Mid‐Module Assessment. Mary mixed cup of wheat flour, cup of rice flour, and cup of oat flour for her bread dough. How many cups of flour did she put in her bread in all? In Lessons 20 and 21, students add fractions with related units, where one denominator is a multiple (or factor) of the other. To add such fractions, a decomposition is necessary. Decomposing one unit into another is familiar territory: Students have had ample practice composing and decomposing in Topics A and B when working with place value units, converting units of measurement, and using the distributive property. For example, they have converted between equivalent measurement units (e.g., 100 cm = 1 m), and they have used such conversions to do arithmetic (e.g., 1 meter – 54 centimeters). With fractions, the concept is the same. To find the sum of and one simply renames (converts, decomposes) as and adds: + = . All numerical work is accompanied by visual models that allow students to use and apply their known skills and understandings. The addition of Page 59 of 60 fractions with related units is also foundational to decimal work when adding tenths and hundredths in Module 6. Please note that addition of fractions with related denominators is not assessed. Date/ Day of the school year Approx. Date: 12/22/2016 Day: 84 Objective/ Essential Question OBJ: Students will use visual models to add and subtract two fractions with the same units. EQ: How can I use visual models to add and subtract two fractions with the same units? Math Florida Standards/ Assessment / Academic Language Lesson Mathematical Performance Tasks Practice Target number sentence
Module 5
4.NF.2.3a units Topic D 4.NF.2.3d number line Lesson 16: Use visual models to add and subtract two endpoints fractions with the same units. Embedded partition add Lesson Materials: Fluency subtract (S) Personal white board, blank number lines (Template) 4.NF.1.1 mixed number 4.NF.1.2 sum difference Mathematical number bonds Practices MP2 Notes: On the Problem Set, problem 6c is beyond the content limits using ninths as the denominator. On the Exit Ticket, problem 1 is beyond the content limits using ninths as the denominator. On the Homework, problems 2b, 3e, 5a and 7a are beyond content limits using denominators sevenths, ninths and elevenths. Please adjust the denominators to meet the content limits. Page 60 of 60