common core math grade 4 fraction addition subtraction

4

Common Core2014

Mathematics Teacher Resource Book

Sa

mpler IncludesTeacher Resource Book• Table of Contents• Pacing Guides• Correlation Charts• Sample Lessons

For a complete Teacher Resource Book

call 800-225-0248

Table of Contents

CCSS Emphasis

Ready® Common Core Program Overview A6

Supporting the Implementation of the Common Core A7

Answering the Demands of the Common Core with Ready A8

The Standards for Mathematical Practice A9

Depth of Knowledge Level 3 Items in Ready Common Core A10

Cognitive Rigor Matrix A11

Using Ready Common Core A12

Teaching with Ready Common Core Instruction A14

Content Emphasis in the Common Core Standards A16

Connecting with the Ready® Teacher Toolbox A18

Using i-Ready® Diagnostic with Ready Common Core A20

Features of Ready Common Core Instruction A22

Supporting Research A38

Correlation ChartsCommon Core State Standards Coverage by Ready Instruction A42

Interim Assessment Correlations A46

Lesson Plans (with Answers)

Unit 1: Number and Operations in Base Ten, Part 1 1

Lesson 1 Understand Place Value 3 M CCSS Focus - 4.NBT.A.1, 2 Embedded SMPs - 2, 4, 6, 7

Lesson 2 Compare Whole Numbers 11 M CCSS Focus - 4.NBT.A.2 Embedded SMPs - 2, 4, 6–8

Lesson 3 Add and Subtract Whole Numbers 19 M CCSS Focus - 4.NBT.B.4 Embedded SMPs - 2, 5, 7, 8

Lesson 4 Round Whole Numbers 29 M CCSS Focus - 4.NBT.A.3 Embedded SMPs - 1, 2, 4, 6

Unit 1 Interim Assessment 37

Unit 2: Operations and Algebraic Thinking 40

Lesson 5 Understand Multiplication 42 M CCSS Focus - 4.OA.A.1 Embedded SMPs - 2–4

Lesson 6 Multiplication and Division in Word Problems 50 M CCSS Focus - 4.OA.A.2 Embedded SMPs - 2–5, 7

M = Lessons that have a major emphasis in the Common Core StandardsS/A = Lessons that have supporting/additional emphasis in the Common Core Standards

CCSS EmphasisUnit 2: Operations and Algebraic Thinking (continued)

Lesson 7 Multiples and Factors 60 S/A CCSS Focus - 4.OA.B.4 Embedded SMPs - 2, 5, 7

Lesson 8 Number and Shape Patterns 72 S/A CCSS Focus - 4.OA.C.5 Embedded SMPs - 2–5, 7

Lesson 9 Model Multi-Step Problems 82 M CCSS Focus - 4.OA.A.3 Embedded SMPs - 1, 2, 4–7

Lesson 10 Solve Multi-Step Problems 90 M CCSS Focus - 4.OA.A.3 Embedded SMPs - 1, 2, 4–7


Unit 3: Number and Operations in Base Ten, Part 2 102

Lesson 11 Multiply Whole Numbers 105 M CCSS Focus - 4.NBT.B.5 Embedded SMPs - 1–5, 7

Lesson 12 Divide Whole Numbers 115 M CCSS Focus - 4.NBT.B.6 Embedded SMPs - 2–5, 7


Unit 4: Number and Operations—Fractions 130

Lesson 13 Understand Equivalent Fractions 132 M CCSS Focus - 4.NF.A.1 Embedded SMPs - 2–4, 7, 8

Lesson 14 Compare Fractions 140 M CCSS Focus - 4.NF.A.2 Embedded SMPs - 1, 2, 4, 5, 7

Lesson 15 Understand Fraction Addition and Subtraction 150 M CCSS Focus - 4.NF.B.3a, 3b Embedded SMPs - 1–8

Lesson 16 Add and Subtract Fractions 158 M CCSS Focus - 4.NF.B.3a, 3d Embedded SMPs - 1, 2, 4–8

Lesson 17 Add and Subtract Mixed Numbers 168 M CCSS Focus - 4.NF.B.3b, 3c, 3d Embedded SMPs - 1–8

Lesson 18 Understand Fraction Multiplication 178 M CCSS Focus - 4.NF.B.4a, 4b Embedded SMPs - 1–8

Lesson 19 Multiply Fractions 186 M CCSS Focus - 4.NF.B.4c Embedded SMPs - 1, 2, 4–8

Lesson 20 Fractions as Tenths and Hundredths 194 M CCSS Focus - 4.NF.C.5 Embedded SMPs - 1, 2, 4, 5, 7


CCSS EmphasisUnit 4: Number and Operations—Fractions (continued)

Lesson 21 Relate Decimals and Fractions 202 M CCSS Focus - 4.NF.C.6 Embedded SMPs - 2, 4–7

Lesson 22 Compare Decimals 212 M CCSS Focus - 4.NF.C.7 Embedded SMPs - 2, 4, 5, 7, 8


Unit 5: Measurement and Data 226

Lesson 23 Convert Measurements 229 S/A CCSS Focus - 4.MD.A.1 Embedded SMPs - 2, 5, 6, 8

Lesson 24 Time and Money 239 S/A CCSS Focus - 4.MD.A.2 Embedded SMPs - 1, 2, 4–6

Lesson 25 Length, Liquid Volume, and Mass 249 S/A CCSS Focus - 4.MD.A.2 Embedded SMPs - 1, 2, 4–6

Lesson 26 Perimeter and Area 261 S/A CCSS Focus - 4.MD.A.3 Embedded SMPs - 1, 2, 4–7

Lesson 27 Line Plots 271 S/A CCSS Focus - 4.MD.B.4 Embedded SMPs - 2, 4–7

Lesson 28 Understand Angles 283 S/A CCSS Focus - 4.MD.C.5a, 5b Embedded SMPs - 6, 7

Lesson 29 Measure and Draw Angles 291 S/A CCSS Focus - 4.MD.C.6 Embedded SMPs - 2, 3, 5, 6

Lesson 30 Add and Subtract With Angles 301 S/A CCSS Focus - 4.MD.C.7 Embedded SMPs - 1–6


Unit 6: Geometry 314

Lesson 31 Points, Lines, Rays, and Angles 316 S/A CCSS Focus - 4.G.A.1 Embedded SMPs - 1, 3, 4–6

Lesson 32 Classify Two-Dimensional Figures 328 S/A CCSS Focus - 4.G.A.2 Embedded SMPs - 3, 5, 8

Lesson 33 Symmetry 340 S/A CCSS Focus - 4.G.A.3 Embedded SMPs - 1, 4–7



©Curriculum Associates, LLC Copying is not permitted.A8

Answering the Demands of the Common Core with Ready®

THE DEMANDS OF THE COMMON CORE HOW READY® DELIVERS

Focus: The Common Core Standards for Mathematics focus on fewer topics each year, allowing more time to truly learn a topic. Lessons need to go into more depth to help students to build better foundations and understanding.

Ready lessons reflect the same focus as the Common Core standards. In fact, the majority of the lessons in each grade directly address the major focus of the year. Furthermore, each lesson was newly-written specifically to address the Common Core Standards. There is at least one lesson for each standard and only lessons that address the Common Core Standards are included.

Coherent Connections (Building on Prior Knowledge): Instruction needs to provide logical ways for students to make connections between topics within a grade as well as across multiple grades. Instruction must build on prior knowledge and be organized to take advantage of the natural connections among standards within each cluster as well as connections across clusters or domains. This coherence is required for students to make sense of mathematics.

Ready units are organized by domains following the cluster headings of the Common Core. Each lesson starts by referencing prior knowledge and making connections to what students already know, particularly reinforcing algebraic thinking and problem-solving. These connections are highlighted for teachers in the Learning Progressions of the Teachers Resource Book so teachers can see at a glance how the lesson connects to previous and future learning.

Rigor and Higher-Order Thinking: To meet the Standards, equal attention must be given to conceptual understanding, procedural skill and fluency, and applications in each grade. Students need to use strategic thinking in order to answer questions of varying difficulty requiring different cognitive strategies and higher-order thinking skills.

Ready lessons balance conceptual understanding, skill and procedural fluency, and applications. Students are asked higher-order thinking questions throughout the lessons. They are asked to understand, interpret, or explain concepts, applications, skills and strategies. Practice questions match the diversity and rigor of the Common Core standards.

Conceptual Understanding: In the past, a major emphasis in mathematics was on procedural knowledge with less attention paid to understanding math concepts. The Common Core explicitly identifies standards that focus on conceptual understanding. Conceptual understanding allows students to see math as more than just a set of rules and isolated procedures and develop a deeper knowledge of mathematics.

Ready includes conceptual understanding in every lesson through questions that ask students to explain models, strategies, and their mathematical thinking. In addition, a “Focus on Math Concepts” lesson is included for every Common Core standard that focuses on conceptual development—those standards that begin with the word “understand.”

Mathematical Practices: The Standards for Mathematical Practice (SMP) must support content standards and be integrated into instruction. The content standards must be taught through intentional, appropriate use of the practice standards.

The Standards for Mathematical Practice are fully integrated in an age-appropriate way throughout each lesson. The Teachers Resource Book includes SMP Tips that provide more in-depth information for select practice standards addressed in the lesson. See pages A9 and A26 for more details.

Mathematical Reasoning: Mathematical reasoning must play a major role in student learning. Students must be able to analyze problems, determine effective strategies to use to solve them, and evaluate the reasonableness of their solutions. They must be able to explain their thinking, critique the reasoning of others, and generalize their results.

Ready lessons build on problem-solving as a main component of instruction. Students work through a problem, discuss it, draw conclusions, make generalizations, and determine the reasonableness of their solutions. Guided Practice problems ask students to critique arguments presented by fictional characters and justify their own solutions.


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

Try more than one approach, think strategically, and succeed in solving problems that seem very difficult.

Each Ready lesson leads students through new problems by using what they already know, demonstrates multiple approaches and access points, and gives encouraging tips and opportunities for cooperative dialogue.

2. Reason abstractly and quantitatively:

Represent a word problem with an equation, or other symbols, solve the math, and then interpret the solution to answer the question posed.

Ready lessons lead students to see mathematical relationships connecting equations, visual representations, and problem situations. Each lesson challenges students to analyze the connection between an abstract representation and pictorial or real-world situations.

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

Discuss, communicate reasoning, create explanations, and critique the reasoning of others.

In Ready, the teacher-led Mathematical Discourse feature guides students through collaborative reasoning and the exchange of ideas and mathematical arguments. Ready lessons also provide error-analysis exercises that ask students to examine a fictional student’s wrong answer, as well as multiple opportunities to explain and communicate reasoning.

4. Model with mathematics:

Use math to solve actual problems.

Students create a mathematical model using pictures, diagrams, tables, or equations to solve problems in each Ready lesson. In the Teacher Resource Book, the Real-World Connection feature adds another dimension to understanding application of a skill.

5. Use appropriate tools strategically:

Make choices about which tools, if any, to use to solve a problem.

Ready lessons model the use of a variety of tools, including diagrams, tables, or number lines; Guided Practice problems may be solved with a variety of strategies.

6. Attend to precision:

Explain and argue, draw, label, and compute carefully and accurately.

Ready lessons guide students to focus on precision in both procedures and communication, including special error-analysis tasks and group discussion questions that motivate students to employ precise, convincing arguments.

7. Look for and make use of structure:

Build mathematical understanding by recognizing structures such as place value, decomposition of numbers, and the structure of fractions.

Each Ready Focus on Math Concepts lesson builds understanding of new concepts by explicitly reviewing prior knowledge of mathematical structure.

8. Look for and express regularity in repeated reasoning:

Recognize regularity in repeated reasoning and make generalizations or conjectures about other situations.

Each Ready lesson leads students to focus attention on patterns that reflect regularity. Where appropriate, students draw a conclusion or make a generalization and explain their reasoning by referencing the observed pattern.

The Standards for Mathematical PracticeMastery of the Standards for Mathematical Practice (SMP) is vital for educating students who can recognize and be proficient in the mathematics they will encounter in college and careers. As the chart below shows, the SMPs are built into the foundation of Ready® Instruction.


Depth of Knowledge Level 3 Items in Ready® Common CoreThe following table shows the Ready® lessons and sections with higher-complexity items, as measured by Webb’s Depth of Knowledge index.

Depth of Knowledge Level 3 Items in Ready Common Core

Lesson Section Item Lesson Section Item

1 Guided Practice 14 18 Guided Practice 13


1 Performance Task 17 18 Performance Task 16




4 Common Core Practice 6 21 Common Core Practice 5

Unit 1 Interim Assessment 7 22 Guided Practice 17

Unit 1 Interim Assessment PT Unit 4 Interim Assessment PT



5 Guided Practice 13 24 Common Core Practice 5




8 Common Core Practice 2 28 Guided Practice 15



Unit 2 Interim Assessment PT 28 Performance Task 18



Unit 3 Interim Assessment 5 30 Guided Practice 17

Unit 3 Interim Assessment PT Unit 5 Interim Assessment PT



13 Performance Task 16 32 Guided Practice 21




17 Guided Practice 18 Unit 6 Interim Assessment PT


Cognitive Rigor MatrixThe following table combines the hierarchies of learning from both Webb and Bloom. For each level of hierarchy, descriptions of student behaviors that would fulfill expectations at each of the four DOK levels are given. For example, when students compare solution methods, there isn’t a lower-rigor (DOK 1 or 2) way of truly assessing this skill.

Depth of Thinking (Webb) 1

Type of Thinking (Revised Bloom)

DOK Level 1 Recall &

Reproduction

DOK Level 2 Basic Skills & Concepts

DOK Level 3 Strategic Thinking

& Reasoning

DOK Level 4 Extended Thinking

Remember• Recallconversations,

terms, facts

Understand

• Evaluateanexpression• Locatepointsonagrid

or number on number line

• Solveaone-stepproblem• Representmath

relationships in words, pictures, or symbols

• Specify,explainrelationships

• Makebasicinferencesorlogical predictions from data/observations

• Usemodels/diagramstoexplain concepts

• Makeandexplainestimates

• Useconceptstosolvenon-routine problems

• Usesupportingevidenceto justify conjectures, generalize, or connect ideas

• Explainreasoningwhenmore than one response is possible

• Explainphenomenainterms of concepts

• Relatemathematicalconcepts to other content areas, other domains

• Developgeneralizationsof the results obtained and the strategies used and apply them to new problem situations

Apply

• Followsimpleprocedures

• Calculate,measure,apply a rule (e.g.,rounding)

• Applyalgorithmorformula

• Solvelinearequations• Makeconversions

• Selectaprocedureandperform it

• Solveroutineproblemapplying multiple concepts or decision points

• Retrieveinformationtosolve a problem

• Translatebetweenrepresentations

• Designinvestigationfora specific purpose or researchquestion

• Usereasoning,planning,and supporting evidence

• Translatebetweenproblem and symbolic notation when not a direct translation

• Initiate,design,andconduct a project that specifies a problem, identifies solution paths, solves the problem, and reports results

Analyze

• Retrieveinformationfrom a table or graph to answeraquestion

• Identifyapattern/trend

• Categorizedata,figures• Organize,orderdata• Selectappropriategraph

and organize and display data

• Interpretdatafromasimple graph

• Extendapattern

• Compareinformationwithin or across data sets or texts

• Analyzeanddrawconclusions from data, citing evidence

• Generalizeapattern• Interpretdatafrom

complex graph

• Analyzemultiplesourcesof evidence or data sets

Evaluate

• Citeevidenceanddevelop a logical argument

• Compare/contrastsolution methods

• Verifyreasonableness

• Applyunderstandingina novel way, provide argument or justification for the new application

Create

• Brainstormideas,concepts, problems, or perspectives related to a topic or concept

• Generateconjecturesorhypotheses based on observations or prior knowledge and experience

• Developanalternativesolution

• Synthesizeinformationwithin one data set

• Synthesizeinformationacross multiple sources or data sets

• Designamodeltoinformand solve a practical or abstract situation

SBAC,2012;adaptedfromHessetal.,2009


Using Ready® Common Core

InstructTeach one Ready® Common Core Instruction lesson per week, using the Pacing Guides on pages A14 and A15 for planning.

Use the web-based, electronic resources found in the Teacher Toolbox to review prerequisite skills and access on-level lessons as well as lessons from previous grades. See pages A18 and A19 for more information.

Differentiate InstructionIdentify struggling students and differentiate instruction using the Assessment and Remediation pages at the end of each lesson in the Teacher Resource Book. See page A23 for a sample.

Access activities and prerequisite lessons (including lessons from other grades) in the Teacher Toolbox to reteach and support students who are still struggling. See pages A18 and A19 for more details.

Assess and Monitor Progress Assess student understanding using the Common Core Practice and Interim Assessments in Ready Common Core Instruction. See pages A29 and A46 for more information.

Monitor progress using the benchmark tests in Ready® Practice to assess cumulative understanding, identify student weaknesses for reteaching, and prepare for Common Core assessments.

Use Ready® with the i-Ready®Diagnostic

You can add the i-Ready Diagnostic as part of your Ready solution.

• Administer the i-Ready Diagnostic as a cross-grade-level assessment to pinpoint what students know and what they need to learn.

• Use the detailed individual and classroom diagnostic reports to address individual and classroom instructional needs using the lessons in Ready Common Core Instruction and the Teacher Toolbox.

See pages A20 and A21 for more information.

Use Ready® as Your Primary Instructional Program

Because every Common Core Standard is addressed with clear, thoughtful instruction and practice, you can use Ready® Common Core as your primary instructional program for a year-long mathematics course. The lesson sequence is based on the learning progressions of the Common Core to help students build upon earlier learning, develop conceptual understanding, use mathematical practices, and make connections among concepts.


Using Ready® to Supplement Your Current Math Program

If your instructional program was not written specifically to address the Common Core Standards, then your textbook likely does not include the concepts, skills, and strategies your students need to be successful. By supplementing with Ready® Common Core Instruction, you’ll be able to address these concerns:

•Fillinggapsinmathematicscontentthathasshiftedfromanothergrade

•IncorporatingCommonCoremodelsandstrategiesintoinstruction

•IntegratingthehabitsofmindthatareintheStandardsforMathematicalPractice

•Askingquestionsrequiringstudentstoengageinhigher-levelthinking,suchasquestionsthatask studentstoexplaineffectivestrategiesusedtosolveproblems,critiquethereasoningofothers,and generalize their results

•Includinglessonsandquestionsthatdevelopconceptualunderstanding

•ProvidingrigorousquestionsmodeledonthelatestCommonCoreassessmentframeworks

STEP 1

IDENTIFY CONTENT

NEEDS

How do I know what to teach?

•IdentifytheReady lessons you need to include in your instructional plan.

−FirstidentifytheReady lessons that address standards that are a major emphasis in the CommonCore.SeepageA16ortheTableofContentstoeasilyidentifytheseReady lessons.

−Next,identifytheCommonCorestandardsinthetableonpageA17thatarenotaddressedin your current math program.

•IdentifytheplaceinyourscopeandsequencetoinserttheReadylessons.“FocusonMathConcepts” lessons should come before the lesson in your current book.

STEP 2

INTEGRATE READY

How do I make time to teach the Ready lessons?

•RemovelessonsorunitsfromyourcurrentinstructionalplanthatarenolongercoveredintheCommonCorestandardsatthatgradelevel.

•Replacelessonsorunitsthatdonotteachtopicsusingthemodels,strategies,andrigoroftheCommon Core with the appropriate Ready lessons.

STEP 3

MEASURE STUDENT PROGRESS

How can I address gaps in student knowledge?

•UsetheInterimAssessmentsinReady to make sure your students are successfully able to meet the rigorous demands of the Common Core.

•UsethebenchmarktestsinReady® Practice to identify student weaknesses and gaps in students’ knowledge.

•UsetheReady® Teacher Toolboxtoaccessactivities,on-levellessons,andlessonsfromothergradestoaddressgapsinstudents’backgroundandlearning.SeepagesA18andA19formoreon the Teacher Toolbox.

Step-by-Step Implementation Plan


Ready Instruction Year-Long Pacing Guide Week Ready® Common Core Instruction Lesson Days Minutes/day

1 Practice Test 1 or i-Ready Baseline Diagnostic 3 60 2 L1: Understand Place Value 5 30–45 3 L2: Compare Whole Numbers 5 30–45 4 L3: Add and Subtract Whole Numbers 5 30–45 5 L4: Round Whole Numbers 5 30–45 Unit 1 Interim Assessment 1 30–45 6 L5: Understand Multiplication 5 30–45 7 L6: Multiplication and Division in Word Problems 5 30–45 8 L7: Multiples and Factors 5 30–45 9 L8: Number and Shape Patterns 5 30–45 10 L9: Model Multi-Step Problems 5 30–45 11 L10: Solve Multi-Step Problems 5 30–45 Unit 2 Interim Assessment 1 30–45 12 L11: Multiply Whole Numbers 5 30–45 13 L12: Divide Whole Numbers 5 30–45 Unit 3 Interim Assessment 1 30–45 14 L13: Understand Equivalent Fractions 5 30–45 15 L14: Compare Fractions 5 30–45 16 L15: Understand Fraction Addition and Subtraction 5 30–45 17 L16: Add and Subtract Fractions 5 30–45 18 L17: Add and Subtract Mixed Numbers 5 30–45 19 L18: Understand Fraction Multiplication 5 30–45 20 L19: Multiply Fractions 5 30–45 21 L20: Fractions as Tenths and Hundredths 5 30–45 22 L21: Relate Decimals and Fractions 5 30–45 23 L22: Compare Decimals 5 30–45 Unit 4 Interim Assessment 1 30–45 24 Practice Test 2 or i-Ready Interim Diagnostic 3 60 25 L23: Convert Measurements 5 30–45 26 L24: Time and Money 5 30–45 27 L25: Length, Liquid Volume, and Mass 5 30–45 28 L26: Perimeter and Area 5 30–45 29 L27: Line Plots 5 30–45 30 L28: Understand Angles 5 30–45 31 L29: Measure and Draw Angles 5 30–45 32 L30: Add and Subtract With Angles 5 30–45 Unit 5 Interim Assessment 1 30–45 33 L31: Points, Lines, Rays, and Angles 5 30–45 34 L32: Classify Two-Dimensional Figures 5 30–45 35 L33: Symmetry 5 30–45 Unit 6 Interim Assessment 1 30–45

36 Practice Test 3 or i-Ready Year-End Diagnostic 3 60

Teaching with Ready® Common Core Instruction


Day 1 Day 2 Day 3 Day 4 Day 5

Introduction Modeled/Guided Instruction

Modeled/Guided Instruction Guided Practice Common Core

Practice

Whole Class

Introduction, including Vocabulary(30 minutes)

Mathematical Discourse (10 min)

Discuss graphic and verbal representations of a problem.

Visual Support (15 minutes)

Discuss graphic and verbal representations of a problem.

Concept Extension(15 minutes)

Discuss a sample problem.(10 minutes)

Small Group/

Independent

Hands-On Activity (where applicable)

Work the math with a symbolic representation and practice with Try It problems. (20 minutes)

Work the math with a symbolic representation and practice with Try It problems. (20 minutes)

Work three problems independently, then Pair/Share.(20 minutes)

Solve problems in test format or complete a Performance Task.(30 minutes)

Assessment

Discuss answer to the Reflect question.(5 minutes)

Discuss solutions to the Try It problems.(10 minutes)

Discuss solutions to the Try It problems.(10 minutes)

Check solutions and facilitate Pair/Share.(15 minutes)

Review solutions and explanations.(15 minutes)

Assessment and Remediation (time will vary)

Ready® Instruction Weekly Pacing (One Lesson a Week)Use Ready Common Core Instruction as the foundation of a year-long mathematics program. The Year-Long Sample Week (below) shows a recommended schedule for teaching one lesson per week. Each day is divided into periods of direct instruction, independent work, and assessment. Use the Year-Long Pacing Guide on page A14 for a specific week-to-week schedule.

Ready Instruction Weekly Pacing (Two Lessons a Week)Target Ready Common Core Instruction lessons based on Ready Common Core Practice results to focus learning in a compressed time period. The chart below models teaching two lessons per week. The two lessons are identified as Lesson A and Lesson B in the chart below.

Day 1 Day 2 Day 3 Day 4 Day 5

In Class

Lesson A

Introduction(15 minutes)

Modeled Instruction(30 minutes)

Lesson A

Guided Instruction(15 minutes)

Guided Practice(30 minutes)

Lesson B

Introduction(15 minutes)

Modeled Instruction(30 minutes)

Lesson B

Guided Instruction(15 minutes)

Guided Practice(30 minutes)

Lesson AReview concepts and skills (20 minutes)

Lesson BReview concepts and skills (20 minutes)

Homework (optional)

Lesson A

Common Core Practice

Lesson B

Common Core Practice


Correlation Charts

Common Core State Standards © 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

The Standards for Mathematical Practice are integrated throughout the instructional lessons.

Common Core State Standards Coverage by Ready® InstructionThe table below correlates each Common Core State Standard to the Ready® Common Core Instruction lesson(s) that offer(s) comprehensive instruction on that standard. Use this table to determine which lessons your students should complete based on their mastery of each standard.

Common Core State Standards for Grade 4 — Mathematics Standards

Content Emphasis

Ready® Common Core

Instruction Lesson(s)

Operations and Algebraic Thinking

Use the four operations with whole numbers to solve problems.

4.OA.A.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 3 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.

Major 5

4.OA.A.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.

Major 6

4.OA.A.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.

Major 9, 10

Gain familiarity with factors and multiples.

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

Supporting/Additional

7

Generate and analyze patterns.

4.OA.C.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.


8

Number and Operations in Base Ten

Generalize place value understanding for multi-digit whole numbers.

4.NBT.A.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 4 70 5 10 by applying concepts of place value and division.

Major 1

4.NBT.A.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using ., 5, and , symbols to record the results of comparisons.

Major 1, 2

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

Use place value understanding and properties of operations to perform multi-digit arithmetic.

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm. Major 3

4.NBT.B.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.

Major 11



Content Emphasis

Ready® Common Core


Number and Operations in Base Ten (continued)

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

4.NBT.B.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.

Major 12

Number and Operations—Fractions

Extend understanding of fraction equivalence and ordering.

4.NF.A.1 Explain why a fraction a ·

b is equivalent to a fraction (n 3 a)

······

(n 3 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.

Major 13

4.NF.A.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 ., 5, or ,, and justify the conclusions, e.g., by using a visual fraction model.

Major 14

Build fractions from unit fractions.

4.NF.B.3 Understand a fraction a ·

b with a . 1 as a sum of fractions 1

··

b . Major 15, 16, 17

4.NF.B.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Major 15, 16

4.NF.B.3b 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 5 1

··

8 1 1

··

8 1 1

··

8 ; 3

··

8 5 1

··

8 1 2

··

8 ; 2 1

··

8 5 1 1 1 1 1

··

8 5 8

··

8 1 8

··

8 1 1

··

8 .

Major 15, 17

4.NF.B.3c 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.

Major 17

4.NF.B.3d 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.

Major 16, 17

4.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. Major 18, 19

4.NF.B.4a 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 3 1 1

··

4 2 , recording the conclusion

by the equation 5 ··

4 5 5 3 1 1

··

4 2 .

Major 18

4.NF.B.4b 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 3 1 2

··

5 2 as 6 3 1 1

··

5 2 , recognizing this product as 6

··

5 .

(In general, n 3 1 a ·

b 2 5 (n 3 a)

······

b .)

Major 18

4.NF.B.4c 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?

Major 19



Content Emphasis

Ready® Common Core


Number and Operations—Fractions (continued)

Understand decimal notation for fractions, and compare decimal fractions.

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

··

10 as 30

···

100 , and add 3

··

10 1 4

···

100 5 34

···

100 .

Major 20

4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62

···

100 ; describe a length as 0.62 meters; locate 0.62 on a number line diagram. Major 21

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols ., 5, or ,, and justify the conclusions, e.g., by using a visual model.

Major 22

Measurement and Data

Solve problems involving measurement and conversion of measurements.

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


23

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


24, 25

4.MD.A.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.


26

Represent and interpret data.

4.MD.B.4 Make a line plot to display a data set of measurements in fractions of a unit 1 1 ··

2 , 1

··

4 , 1

··

8 2 .

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.


27

Geometric measurement: understand concepts of angle and measure angles.

4.MD.C.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:


28

4.MD.C.5a An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1

···

360 of a circle is called a “one-degree angle,” and can

be used to measure angles.


28



Content Emphasis

Ready® Common Core


Measurement and Data (continued)

Geometric measurement: understand concepts of angle and measure angles. (continued)

4.MD.C.5b An angle that turns through n one-degree angles is said to have an angle measure of n degrees.


28

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


29

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


30

Geometry

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

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


31

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


32

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


33


Ready® Common Core Interim Assessment Correlations

Unit 1: Number and Operations in Base Ten, Part 1

Question DOK1 Standard(s)Ready® Common Core

Student Lesson(s)

1 1 4.NBT.A.3 4

2 1 4.NBT.A.2 1

3 1 4.NBT.B.4 3

4 1 4.NBT.A.2 1

5 1 4.NBT.A.2, 4.NBT.B.4 2, 3

6 2 4.NBT.A.1, 4.NBT.A.2 1

7 3 4.NBT.A.2, 4.NBT.A.3 2, 4

PT 3 4.NBT.A.1, 4.NBT.A.2, 4.NBT.A.3, 4.NBT.B.4 1, 2, 3, 4

Unit 2: Operations and Algebraic Thinking

Question DOK Standard(s)Ready® Common Core

Student Lesson(s)

1 1 4.OA.B.4 7

2 2 4.OA.A.3 9

3 2 4.OA.A.3 9, 10

4 1 4.OA.C.5 8

5 2 4.OA.A.3 9, 10

6 2 4.OA.B.4 7

PT 3 4.OA.A.3, 4.OA.B.4, 4.NBT.B.4, 4.NBT.B.5 3, 7, 9, 10, 11

Unit 3: Number and Operations in Base Ten, Part 2


Student Lesson(s)

1 1 4.NBT.B.5 11

2 2 4.NBT.B.6 12

3 2 4.NBT.B.5 11

4 2 4.NBT.B.5 11

5 3 4.NBT.B.5 11

6 1 4.NBT.B.6 12

PT 3 4.NBT.B.4, 4.NBT.B.5, 4.NBT.B.6, 4.OA.A.3 3, 9, 10, 11, 12

Interim Assessment CorrelationsThe tables below show the depth-of-knowledge (DOK) level for the items in the Interim Assessments, as well as the standard(s) addressed, and the corresponding Ready® Instruction lesson(s) being assessed by each item. Use this information to adjust lesson plans and focus remediation.

1Depth of Knowledge levels:1. The item requires superficial knowledge of the standard.2. The item requires processing beyond recall and observation.3. The item requires explanation, generalization, and connection to other ideas.


Ready® Common Core Interim Assessment Correlations (continued)

Unit 4: Number and Operations—Fractions


Student Lesson(s)

1 1 4.NF.B.4b 18

2 1 4.NF.C.5 20

3 2 4.NF.B.3 15, 16

4 2 4.NF.A.2 14

5 2 4.NF.A.1 13

PT 3 4.NF.B.3, 4.NF.B.4, 4.NBT.B.5, 4.NBT.B.6 11, 12, 15, 16, 18, 19

Unit 5: Measurement and Data


Student Lesson(s)

1 1 4.MD.C.7 30

2 2 4.MD.A.1 23

3 1 4.MD.A.3 26

4 2 4.MD.A.1 23

5 1 4.MD.C.5a 28

PT 3 4.MD.A.1, 4.MD.A.2, 4.OA.A.2, 4.NF.B.3a 6, 15, 16, 23, 24, 25

Unit 6: Geometry


Student Lesson(s)

1 1 4.G.A.3 33

2 1 4.G.A.2 32

3 2 4.G.A.2, 4.G.A.3 32, 33

4 2 4.G.A.3 33

5 1 4.G.A.2 32

6 2 4.G.A.3 33

PT 3 4.G.A.1, 4.G.A.2, 4.G.A.3 31, 32, 33

Understand Fraction Addition and SubtractionLesson 15

Focus on Math Concepts

L15: Understand Fraction Addition and Subtraction150©Curriculum Associates, LLC Copying is not permitted.

(Student Book pages 136–141)

LESSON OBJECTIVES

•Understandadditionasjoiningparts.

•Understandsubtractionasseparatingparts.

•Extendtheirunderstandingofadditionandsubtractionofwholenumberstoadditionandsubtractionoffractions.

•Usefractionmodelstoaddandsubtractfractionswithlikedenominators.

PREREQUISITE SKILLS

Inordertobeproficientwiththeconceptsinthislesson,studentsshould:

•Knowadditionandsubtractionbasicfacts.

•Understandthemeaningoffractions.

•Identifynumeratorsanddenominators.

•Writewholenumbersasfractions.

VOCABULARY

Thereisnonewvocabulary.Reviewthefollowingkeyterms.

numerator:thetopnumberinafraction;ittellsthenumberofequalpartsthatarebeingdescribed

denominator:thebottomnumberinafraction;ittellsthetotalnumberofequalpartsinthewhole

THE LEARNING PROGRESSION

OnegoaloftheCommonCoreistodevelopadeeperunderstandingoffractionsbyusingaprogressionofconceptsfromsimpletocomplex.Thislessonpreparesstudentsfortheconceptualshiftinvolvedinprogressingfromaddingandsubtractingwholenumberstoaddingandsubtractingfractions.Studentsareguidedtothinkofoperationswithfractionsasverymuchlikeoperationswithwholenumbers.

Studentsseethatyoucancountwithunitfractions

justasyoucountwithwholenumbers.Andbecause

youcancountwithunitfractions,youcanalso

doarithmeticwiththem.Ifyouwalked2··

5ofa

mile(2 fifths)yesterdayand4··

5ofamile(4fifths)today,

altogetheryouwalked6··

5ofamile(6fifths;because

2 thingsplus4moreofthosethingsis6ofthose

things).

Studentsusethemeaningoffractionsandthemeaningsofadditionandsubtractionthatwerebuiltinearliergradestounderstandwhytheproceduresforaddingandsubtractingfractionsmakesense.

Teacher Toolbox Teacher-Toolbox.com

✓ ✓ ✓

✓

Prerequisite Skills

4.NF.B.3a 4.NF.B.3b

Ready Lessons

Tools for Instruction

Interactive Tutorials ✓

CCSS Focus

4.NF.B.3 Understandafractiona ·

b witha.1asasumoffractions1

··

b .

a. Understandadditionandsubtractionoffractionsasjoiningandseparatingpartsreferringtothesamewhole.

b. Decomposeafractionintoasumoffractionswiththesamedenominatorinmorethanoneway,recordingeachdecompositionbyanequation.Justifydecompositions,e.g.,byusingavisualfractionmodel.

STANDARDS FOR MATHEMATICAL PRACTICE:SMP 1–8(See page A9 for full text.)

L15: Understand Fraction Addition and Subtraction 151©Curriculum Associates, LLC Copying is not permitted.

Part 1: Introduction Lesson 15

AT A GLANCE

Students explore the idea that adding fractions is not

essentially different from adding whole numbers.

A number line diagram gives meaning to the

expression 2 ··

4 1 3

··

4 .

STEP BY STEP

•IntroducetheQuestionatthetopofthepage.

•Helpstudentsrelatethenumberlinediagramtothesum 2 1 3.

•ReadThinkwithstudents.Reinforcetheideathatfractions are numbers.

•Guidestudentstorecognizethatjustasthe

number 5 is made up of 5 ones, the number 5 ··

4 is

made up of 5 one-fourths.

•Ifstudentsneedadditionalsupportwithlocating

fractions on a number line, have them build a

number line by putting 1 ··

4 fraction strips end-to-end,

creating a concrete model to show 2 ··

4 1 3

··

4 .

Toextendstudents’understandingofdecomposingfractions, follow these steps:

•Drawandlabelanumberlineontheboardfrom0 to2liketheoneonthepageshowingfourths.

•Askstudentstothinkoftwodifferentfractions

that you could put together that would give you

the same sum as adding 2 ··

4 and 3

··

4 .

•Haveavolunteergototheboardtoshowthe

two fractions on the number line. 3 1 ··

4 and 4

··

4 in

either order 4

Concept Extension•How would you explain adding in your own words?

Responsesshouldincludephrasessuchas“join”or“puttogether.”

•How is adding fractions like adding whole numbers?

Students may mention that, in both cases, you are putting things together.

•Can you think of another way to explain adding fractions?

Students may suggest that you can count on withfractionsjustlikeyoucountonwithwhole numbers.

Mathematical Discourse

©Curriculum Associates, LLC Copying is not permitted.L15: Understand Fraction Addition and Subtraction136

Understand Fraction Addition and SubtractionLesson 15 Part 1: Introduction

Focus on Math Concepts

What’s really going on when we add numbers?

Adding means joining or putting things together.

Think about how you could explain adding 2 1 3 to a first grader. You could start at 2, count on 3 more, and see where you end up: 2 . . . 3 . . . 4 . . . 5.

Or, you could put a segment with a length of 2 and a segment with a length of 3 next to each other on a number line to show 2 1 3.

0 1 2 3 4 5 6 7 8 9 10

1 1 1 1 1

When you add 2 1 3, you are putting ones together.

Think

Adding fractions means joining or putting together parts of the same whole.

You can put a segment with a length of 2 ·· 4 and a segment

with a length of 3 ·· 4 next to each other to show 2 ·· 4 1 3 ·· 4 .

0 1 2

14

04

44

84

24

34

54

64

74

14

14

14

14

14

When you add 2 ·· 4 1 3 ·· 4 , you are putting one-fourths together.

Underline the sentence that explains what adding fractions means.

CCSS4.NF.B.3a

4.NF.B.3b

L15: Understand Fraction Addition and Subtraction152©Curriculum Associates, LLC Copying is not permitted.


AT A GLANCE

Students explore the idea that subtracting fractions is

not essentially different from subtracting whole

numbers. A number line diagram gives meaning to the

expression 5 ··

4 2 2

··

4 .

STEP BY STEP

•ReadThinkwithstudents.

•Discusshowthenumberlinerepresentstheproblem5 2 2. Show how to subtract on the number line. (startat5andcountback2)

•Askavolunteertoexplainhowtousethenumber

line to find 5 ··

4 2 2

··

4 . Provide 1

··

4 fraction strips for

students who need more support.

•HavestudentsreadandreplytotheReflectdirective.

•Tellstudentsthatyouwilluseanumberlinetoshow 5

··

8 2 3

··

8 .

•Drawanumberlinefrom0to1ontheboard.

•Askstudentsforideasonhowtodividethelinesothat you can use it to help you solve the problem.

•Havestudentsexplainwhydividingthelineintoeighthsmakessense.

•Label0and1onthelineandhavestudentsprovidelabelsfortheothermarksasyoumoveyour finger along the line.

•Askavolunteertoshowhowtofindtheanswertothe problem using the number line.

Visual Model

SMP Tip: In the Visual Model activity, students areaskedtoreasonquantitativelyandexplainwhydividingthelineintoeighthsmakessense.(SMP 2)

•How would you explain subtracting in your own words?

Listenforphrasessuchas“takeapart”or “takeaway.”

•How is subtracting fractions like subtracting whole numbers?

Students may note that subtracting means takingaway.Itdoesn’tmatterwhatkindsofnumbersyou’resubtracting.

•Do you see a connection between the whole numbers and the numerators of the fractions on this page?

Students may mention that the whole numbers and the numerators of the fractions are the same numbers, and to answer both problems you subtract 2 from 5.


©CurriculumAssociates,LLC Copyingisnotpermitted.137L15: Understand Fraction Addition and Subtraction

Lesson 15Part 1: Introduction

Think

Subtracting means separating or taking away.

On a number line, you can start with a segment of length 5 and take away a segment of length 2 to show 5 2 2.

0 1 2 3 4 5 6 7 8 9 10

1 1 1 1 1

1 1

When you subtract 5 2 2, you are taking away ones.

You can show subtracting fractions on a number line. Start with a segment of

length 5 ·· 4 and take away a segment of length 2 ·· 4 to show 5 ·· 4 2 2 ·· 4 .

14

24

34

54

64

74

84

44

04

0 1 2

14

14

14

14

14

14

14

When you subtract 5 ·· 4 2 2 ·· 4 , you are taking away one-fourths.

Now you’ll have a chance to think more about how adding or subtracting fractions is like adding or subtracting whole numbers. You may find that using number lines or area models can help you explain your thinking.

Reflect

1 Use your own words to describe what you just learned about adding and subtracting fractions.

Look at the whole numbers. Now look at the numerators of the fractions. I think I see a connection.

Possible answer: I learned that adding and subtracting

fractions is just like adding and subtracting whole numbers. When the

denominators are the same, you can just add or subtract the numerators.


Part 2: Guided Instruction Lesson 15

AT A GLANCE

Students use number lines to answer questions, reinforcing the understanding that fractions are numbers.

STEP BY STEP

•TellstudentsthattheywillhavetimetoworkindividuallyontheExploreItproblemsonthispageandthensharetheirresponsesingroups.Youmaychoosetoworkthroughthefirstproblemtogetherasa class.

•Asstudentsworkindividually,circulateamongthem.Thisisanopportunitytoassessstudentunderstandingandaddressstudentmisconceptions.Use the Mathematical Discourse questions to engage studentthinking.

•Ifstudentsneedmoresupport,suggestthattheycountoutloudtohelpthemfillinthemissingnumbersinproblems2and3.

•Tohelpstudentsanswerproblem4,havethemput

theirfingeron6 ··

4 on the number line, then count on

by1 ··

4 .Similarly,toanswerproblem5,havethemput

theirfingeron3 ··

5 on the number line and count on

by1 ··

5 .

•Takenoteofstudentswhoarestillhavingdifficultyandwaittoseeiftheirunderstandingprogressesastheyworkintheirgroupsduringthenextpartof the lesson.

STUDENT MISCONCEPTION ALERT: Some studentsmaythinkthatafractionisalwayslessthan1.Ifthismisconceptionpersists,usefractionstripstodemonstratefractionslessthan,equalto,andgreaterthan1.Then,encouragestudentstousethefractionstripstoshowandnameotherfractionsgreater than 1.

•In which direction on the number line do you move when adding? Explain.

Responsesmightincludethefactthataddingmeansjoiningsoyouwillbegettingsegmentsthat are longer or an answer farther to the right thanthenumberyoustartedwith.

•For problem 5, will the answer change if you find

3 ··

5 more than 1

··

5 ? Explain.

Listenforresponsesthatdemonstrateanunderstandingthatyoucanaddtwonumbersinanyorderandgetthesamesum.



©CurriculumAssociates,LLC Copyingisnotpermitted.

L15: Understand Fraction Addition and Subtraction138

Explore It

Counting on and using a number line are two ways to think about adding fractions.

2 Count by fourths to fi ll in the blanks: 1 ·· 4 , 2 ·· 4 , , 4 ·· 4 , 5 ·· 4 , , , ,

Now label the number line.

0 1

424

44

54

10 4

3 Count by fi fths to fi ll in the blanks: 1 ·· 5 , 2 ·· 5 , , ,

Now label the number line.

0 1

525

65

Use the number lines above to answer numbers 4 and 5.

4 What is 1 ·· 4 more than 6 ·· 4 ?

5 What is 1 ·· 5 more than 3 ·· 5 ?

Now try these two problems.

6 Label the number line below and use it to show 2 ·· 4 1 1 ·· 4 .


3 ·· 4

3 ·· 4

6 ·· 4

6 ·· 4

3 ·· 5

3 ·· 5

4 ·· 5

4 ·· 5

5 ·· 5

5 ·· 5

4 ·· 5

7 ·· 4

7 ·· 4

7 ·· 4

8 ·· 4

8 ·· 4

9 ·· 4

9 ·· 4

14

04

44

84

24

34

54

64

74

14

04

44

84

24

34

54

64

74

154 L15: Understand Fraction Addition and Subtraction

©Curriculum Associates, LLC Copying is not permitted.


AT A GLANCE

Students use number lines to show subtracting fractions. Then they use models to show adding and subtracting fractions.

STEP BY STEP

•Organizestudentsinpairsorgroups.Youmaychoose to work through the first Talk About It problemtogetherasaclass.

•Walkaroundtoeachgroup,listento,andjoinin ondiscussionsatdifferentpoints.UsetheMathematicalDiscoursequestionstohelpsupportorextend students’ thinking.

•Whensharingideasaboutproblems9and10,besuretoemphasizethatwhenlabelingthenumberline,numeratorscountonbyones,butthedenominatorremains the same.

•Directthegroup’sattentiontoTryItAnotherWay.Haveavolunteerfromeachgroupcometotheboardtodrawthegroup’ssolutionstoproblems11and12.

SMP Tip:Duringthistime,youmaychoosetoaskaparticulargrouptopreparetosharetheirthinkingor solution. Encourage students to critique the group’sreasoning,especiallyifitshowsadifferentwaytoshoworthinkaboutoneoftheproblemsorshowsamisconceptionthatsurfacedduringthegroupworktime.(SMP 3)

Use fraction strips to subtract fractions.

Materials:stripsofpaper,markers,scissors

•Modelhowtofoldthestripofpaperinhalf,in halfagain,andinhalfathirdtime.Tellstudentstounfoldthestripsanduseamarkertoshowthe8 equal sections.

•Directstudentstocutouteachsection.Askthem

tonamethefractionthatrepresentseachsection.

3 1 ··

8 4 Have them label each section.

•Write7 ··

8 2 5

··

8 on the board. Have students use their

stripstoshowthatthedifferenceis2 ··

8 .

Hands-On Activity•What is another name for 8

··

8 ? 12

··

6 ? Explain your

thinking.

Studentsshouldrecognizethateight1 ··

8 pieces

makeup1wholeandthattwelve1 ··

6 piecesmake

up2wholes.

•Can you think of another way to show finding a difference on a number line?

Studentsmaymentionaddinguptosubtract.

Forexample,tofind7 ··

8 2 2

··

8 ,youmightstart

at 2 ··

8 andthink,“WhatdoIneedtoaddto

get to 7 ··

8 ?”




Talk About It

Solve the problems below as a group.

8 Look at your answers to problems 2 and 3. How is counting by fractions the same as counting with whole numbers?

How is it diff erent?



Try It Another Way

Work with your group to use the area models to show adding or subtracting fractions.

11 Show 1 ·· 8 1 2 ·· 8 .

12 Show 6 ·· 10 2 2 ·· 10 .

Possible answer: When you count with

whole numbers, you count by ones. When you count with fractions, the

numerator counts by ones as long as the denominators are the same.

Possible answer: When you count by fractions, you are counting by parts.

18

0 48

88

98

108

118

128

28

38

58

68

78

16

0 46

86

96

106

116

126

26

36

56

66

76


Part 3: Guided Practice Lesson 15

AT A GLANCE

Students demonstrate their understanding of adding and subtracting fractions as they talk through three problems.

STEP BY STEP

•DiscusseachConnectItproblemasaclassusingthediscussion points outlined below.

Compare:

•Youmaychoosetohavestudentsworkinpairstoencourage sharing ideas. Each partner draws a different model.

•Foraquickandeasyassessment,havestudentsdrawtheir models on small whiteboards or paper and hold themup.Chooseseveralpairstoexplaintheirmodels to the class.

•Usethefollowingtoleadtheclassdiscussion:

Explain how you knew the number of parts to draw in the whole.

How did you show subtraction in your model?

How are the models the same? How are they different?

Explain:

•Thesecondproblemfocusesontheimportanceofthe whole and the fact that you cannot add or subtract fractions unless they refer to the same whole.

•Readtheproblemtogetherasaclass.Askstudentstocontinue to work in pairs to discuss and write their responsesaboutwhatRobdidwrong.

•Beginthediscussionbyaskingquestions,suchas:

What fraction describes a slice of the larger pizza?

3 1 ··

4 4 What fraction describes a slice of the

smaller pizza? 3 1 ··

4 4

Are both 1 ··

4 s the same size? [no] Why not? [the whole

pizzas are not the same size]

Why doesn’t it make sense to add these two fractions? [the wholes are not the same]

Demonstrate:

•Thisdiscussiongivesstudentsanopportunitytothinkaboutproblemsthatinvolveaddingthreefractions.

•Discusshowyoucanaddthree(ormore)fractionsinthe same way as adding whole numbers as long as you are talking about the same type of fractions. Havestudentsexplainhowtheyusedthemodelstoshow the sum.

•Remindstudentstostartat0whenlabelingthenumber line.

SMP Tip:Askstudentstoshowhowtouseanumber line as a tool to model the sum of three whole numbers. (SMP 5)



L15: Understand Fraction Addition and Subtraction140

Connect It

Talk through these problems as a class, then write your answers below.

13 Compare: Draw two diff erent models to show 2 ·· 3 2 1 ·· 3 .

14 Explain: Rob had a large pizza and

a small pizza. He cut each pizza into

fourths. He took one fourth from each

pizza and used the following problem

to show their sum: 1 ·· 4 1 1 ·· 4 5 2 ·· 4 .

What did Rob do wrong?

15 Demonstrate: Think about how you would add three whole numbers. You add two of the numbers fi rst, and then add the third to that sum. You add three fractions the same way.

Use the number line and area model below to show 1 ·· 10 1 3 ·· 10 1 4 ·· 10 .

Possible answers:

0 110

210

310

410

510

610

710

810

910

1010

Possible answer: Rob’s addition is correct, but he cannot add one fourth of

the large pizza and one fourth of the small pizza in this way because the

wholes are not the same.

0 13

23

33

156 L15: Understand Fraction Addition and Subtraction


Part 4: Common Core Performance Task Lesson 15

AT A GLANCE

Students write two questions that can be answered using some or all of the given information about the problem situation. Then they answer one of the questions.

STEP BY STEP

•DirectstudentstocompletethePutItTogethertaskon their own.

•Explaintostudentsthatthequestionstheywritedonot have to use all of the given information.

•Asstudentsworkontheirown,walkaroundtoassesstheirprogressandunderstanding,toanswertheirquestions,andtogiveadditionalsupport,ifneeded.

•Iftimepermits,havestudentsshareoneoftheirquestions with a partner and show how to find the answer to their partner’s question using a visual model.

SCORING RUBRICS

See student facsimile page for possible student answers.

A Points Expectations

2 The response demonstrates the student’s mathematical understanding of adding and subtracting fractions. Both questions can be answered using the information given in the problem.

1 An effort was made to accomplish the task. The response demonstrates some evidence of verbal and mathematical reasoning, but the student’s questions may contain some misunderstandings.

0 There is no response or the response shows little or no understanding of the task.

B Points Expectations

2 Both a number line and an area model are correctly drawn and labeled to show the solution to the problem.

1 Only one model is correctly drawn and labeled or the models drawn may contain minor errors. Evidence in the response demonstrates that with feedback, the student can revise the work to accomplish the task.

0 There are no models drawn or the models show no evidence of providing visual support for solving the problem.

Part 4: Common Core Performance Task Lesson 15


Put It Together

16 Use what you have learned to complete this task.

Jen has 4 ··

10

ofakilogramofdogfood.Luishas3 ··

10

ofakilogramofdogfood.

Alargedogeats2 ··

10

ofakilograminonemeal.

A Write two diff erent questions about this problem that involve adding or subtracting fractions.

i

ii

B Choose one of your questions to answer. Circle the question you chose. Show how to fi nd the answer using a number line and an area model.

Possible answers:

0 110

210

310

410

510

610

710

810

910

1010

4 ··· 10 1 3 ··· 10

Possible answer: How much dog food do Jen and Luis have altogether?

Possible answer: How much more dog food does Jen have than Luis?

Differentiated Instruction Lesson 15

Challenge Activity

Intervention Activity On-Level Activity


Decompose fractions in more than one way.

In this activity, students think of multiple ways to decompose fractions.

Write the fraction 6 ··

5 on the board. Ask students to

think about 6 ··

5 as a sum. Ask them to find at least four

ways they can put fifths together to make 6 ··

5 . Provide

number lines, area models, or fraction strips for

students to use.

Provide at least one example: 6 ··

5 5 1

··

5 1 1

··

5 1 2

··

5 1 2

··

5

Note the methods students use. Are they systematic or do they just guess and check their answers? Do they find more than four ways?

If time permits, give students (or pairs or groups) practice decomposing other fractions, such as 5

··

8 or

5 ··

10

. This activity also gives students practice writing number sentences.

Use fraction strips to model adding and subtracting fractions.

Materials: fraction strips

Write an addition expression on the board, such as

2 ··

8 1 3

··

8 . Have students lay 1

··

8 fraction strips end-to-end

to show the sum. Ask them to tell you how many

1 ··

8 s there are in all. Continue with similar problems.

Include expressions whose sums are greater than

one, such as 3 ··

4 1 2

··

4 .

Write a subtraction expression on the board, such

as 5 ··

6 2 2

··

6 . Have students lay 1

··

6 fraction strips end-to-

end to show 5 ··

6 . Then have them “take away” 2

··

6 . Ask

them to tell you how many 1 ··

6 s are left. Continue with

similar problems. Be sure to provide expressions that

include fractions greater than one, such as 6 ··

5 2 3

··

5 .

Write a question for the answer given.

Write the following problem on the board: The answer is 7 ··

8 . What could the question be?

Encourage students to think about both addition and subtraction. Provide number lines, area models, or fraction strips for support as necessary.

Note the methods students use. Do they just guess, work out their problem, check to see if it’s correct, and then adjust their responses if necessary? Do they use a visual model or do they work symbolically?

If time permits, give students (or pairs or groups) practice with similar problems. You might ask them to write two questions for each answer you supply, one using addition and one using subtraction.

Add and Subtract FractionsLesson 16

L16: Add and Subtract Fractions158©Curriculum Associates, LLC Copying is not permitted.

Develop Skills and Strategies

LESSON OBJECTIVES

•Addfractionswithlikedenominators.

•Subtractfractionswithlikedenominators.

•Usefractionmodels,numberlines,andequationstorepresentwordproblems.

PREREQUISITE SKILLS

Inordertobeproficientwiththeconcepts/skillsinthislesson,studentsshould:

•Understandadditionasjoiningparts.

•Understandsubtractionasseparatingparts.

•Knowadditionandsubtractionbasicfacts.

•Understandthemeaningoffractions.

•Identifynumeratorsanddenominators.

•Writewholenumbersasfractions.

•Composeanddecomposefractions.

VOCABULARY

Thereisnonewvocabulary.Reviewthefollowingkeyterms.

numerator:thetopnumberinafraction;ittellsthenumberofequalpartsthatarebeingdescribed

denominator:thebottomnumberinafraction;ittellsthetotalnumberofequalpartsinthewhole

THE LEARNING PROGRESSION

InkeepingwiththeCommonCoregoalofdevelopingadeeperunderstandingoffractions,thislessonextendsstudents’understandingoffractionadditionandsubtraction.Studentsusevisualmodelsandequationstorepresentandsolvewordproblemsinvolvingtheadditionandsubtractionoffractionsreferringtothesamewholeandhavinglikedenominators.

(Student Book pages 142–151)

Teacher Toolbox Teacher-Toolbox.com

✓ ✓

✓

✓

✓

Prerequisite Skills

4.NF.B.3a 4.NF.B.3d

Ready Lessons

Tools for Instruction

Interactive Tutorials ✓

CCSS Focus

4.NF.B.3 Understandafractiona ·

b witha.1asasumoffractions1

··

b .

a. Understandadditionandsubtractionoffractionsasjoiningandseparatingpartsreferringtothesamewhole.

d. Solvewordproblemsinvolvingadditionandsubtractionoffractionsreferringtothesamewholeandhavinglikedenominators,e.g.,byusingvisualfractionmodelsandequationstorepresenttheproblem.

STANDARDS FOR MATHEMATICAL PRACTICE:SMP 1, 2, 4, 5, 6, 7, 8(See page A9 for full text.)

L16: Add and Subtract Fractions 159©Curriculum Associates, LLC Copying is not permitted.


AT A GLANCE

Students read a word problem and answer a series of questions designed to explore the connection between adding and subtracting fractions and adding and subtracting whole numbers.

STEP BY STEP

•Tellstudentsthatthispagemodelsbuildingthesolution to a problem one step at a time and writing to explain the solution.

•Havestudentsreadtheproblematthetopofthepage.

•WorkthroughExploreItasaclass.

•AskstudentstoexplainhowtheyfiguredouttheanswersforhowmanycardsLynnandPacoreceivedaltogether,andforhowmanycardsToddreceived.

•Guidestudentstounderstandthattheyneededto“join”and“takeaway”thenumbersofcardstoanswer the questions.

•Besuretopointoutthat41 3 1 5 equals the total number of cards, 12. Remind students that the whole isrepresentedbytheset,orpack,ofcards.

•Askstudentpairsorgroupstoexplaintheiranswersfor the remaining questions.

•Encouragestudentstoexplaintheconnectionbetween adding and subtracting fractions and whole numbers.[Whenaddingorsubtractingwholenumbers,youjoinorseparatewholenumbers.And,whenaddingorsubtractingfractions,youjoinorseparate parts of a set or whole.]

•What does the denominator of a fraction tell you?

Listenforresponsesthatincludethephrase“equal parts of a whole” or “equal parts of a set.”

•What does the numerator of a fraction tell you?

Students’ responses should indicate an understandingthatthenumeratortellsyouthenumberofequalpartsyouaretalkingabout.


©CurriculumAssociates,LLC Copyingisnotpermitted.L16: Add and Subtract Fractions142

Add and Subtract FractionsLesson 16 Part 1: Introduction

Develop Skills and Strategies

In Lesson 15, you learned that adding fractions is a lot like adding whole numbers. Take a look at this problem.

Lynn,Paco,andToddsplitapackof12baseballcards.Lynngot4 cards,Paco

got3 cards,andToddgottherestofthecards.Whatfractionofthepackdid

Toddget?

Explore It

Use the math you already know to solve the problem.

How many cards did Lynn and Paco get altogether?

How many cards did Todd get?

There are 12 cards in the pack. What fraction represents the whole pack of cards?

If Lynn got 4 cards out of 12, that means she got 4 ·· 12 of the pack. If Paco got 3 cards

out of 12, what fraction of the pack did he get?

What fraction of the pack did Lynn and Paco get altogether?

Explain how you could find the fraction of the pack that Todd got.

CCSS4.NF.B.3a

4.NF.B.3d

Possible answer: Todd got 5 cards. There are 12 cards in the pack. If the

numerator tells the number of cards Todd got, and the denominator tells

the number of cards in the pack, then Todd got 5 ··· 12 of the pack.

7 ··· 12

7

5

12 ··· 12

3 ··· 12

L16: Add and Subtract Fractions160©Curriculum Associates, LLC Copying is not permitted.


AT A GLANCE

Students use fraction models to review adding and subtracting fractions.

STEP BY STEP

•ReadFindOutMoreasaclass.

•Pointoutthatwhenyouhaveasetofobjects,thedenominatorrepresentsthetotalnumberofobjectsintheset.Sincethereare12baseballcardsinthepack,thatmeansthereare12partsintheset.Thenumberofcardsthateachpersonhasrepresentsthenumeratorofthefraction.

•Remindstudentsthatwhenyouhaveawholeobjectthatisdividedintoequalparts,thedenominatorshowsthetotalnumberofparts.

•Notethatthewholepizzawasdividedinto8equalslices,sothedenominatoris8.Ifthereare7slicesremaining,thenthenumeratorofthefractionis7.If2moreslicesaretakenaway,thenthereare5slicesleft,andthenumeratorofthefractionis5.

•HavestudentsreadandreplytotheReflectdirective.

Use models to add fractions.

Materials: Drawing paper and notebook paper

•Distributedrawingpaperandapieceofnotebookpapertoeachstudent.Tellstudentstousescissorstocutout12equal-sizedcards.Explaintostudentsthatthe12cardsrepresentonepackofcards,oronewholeset,andthatthereare12 partsintheset.

•Tellstudentstoholdup2cards.Havestudentswritethenameofthefractionrepresentedbythe 2cardsontheirpaper.Reviewthemeaningof thefraction.[2cardsoutof12]Then,repeatwith7cards.

•Tellstudentstoadd(join)thefractionsandwritethesumontheirpaper.Haveavolunteerexplainhowtheydeterminedtheiranswer.

•Iftimepermits,repeatforadditionalfractionpairs.

Hands-On ActivityEncouragestudentstothinkabouteverydayplacesorsituationswherepeoplemightneedtoaddorsubtractlikefractions.Havevolunteerssharetheirideas.

Examples:cooking,constructionsite,distanceson a map

Real-World Connection

©CurriculumAssociates,LLC Copyingisnotpermitted.143L16: Add and Subtract Fractions

Lesson 16Part 1: Introduction

Find Out More

We often use fractions in real life. Sometimes they refer to parts of a set of objects, like the baseball card problem. In that problem, the “whole” is the pack, and 12 cards means there are 12 parts of the whole.

Each person got baseball cards from the same pack, so each fraction refers to the same whole. When you add or subtract baseball cards, the whole will stay the same because the cards are all from the same pack of 12.

4

123

125

12

Fractions in real life can also refer to equal parts of a whole object. Lynn, Paco, and Todd might share a pizza cut into 8 slices. The “whole” is the pizza, and 8 slices means there are 8 equal parts of the whole. Even if a person takes away 1 slice or 3 slices from the pizza, the whole will stay the same.

Reflect

1 Describe another example of a set of objects or a whole object divided into fractions.

Possible answer: You can think of a full egg carton as a set of objects.

Each egg is 1 ··· 12 of the set.


Lesson 16Part 2: Modeled Instruction

AT A GLANCE

Students use models and number lines to review adding fractions.

STEP BY STEP

•Readtheproblematthetopofthepageasaclass.

•ReadPictureIt.Haveavolunteernamethe

denominatorofthefractionintheproblem.[10]

Pointoutthateachpotis1 ··

10

ofthetotalnumber

ofpots.

•GuidestudentstorecognizethatsinceJosie

painted3 ··

10

ofthepotsandMargopainted4 ··

10

,the

pictureisshadedtorepresentthenumberofpots

eachgirlpainted,3forJosieand4forMargo.Have

studentscountaloudtofindthesum.

•DirectstudentstolookatthenumberlineinModelIt.Emphasizethatthenumberlineisdividedintotenthstorepresentthetotalnumberofpots.

•Youmaywishtodrawthenumberlineontheboard

andhaveavolunteerdemonstratethe4jumpstothe

righttoadd4tenthsto3 ··

10

.

SMP Tip:Helpstudentsmakesenseoftheproblemandgeneralizethatthesamepropertiesthatapplytowholenumbersapplytofractions.(SMP 1)

Illustrate the commutative property of addition.

•Ask,What if I drew the starting point at 4 ··

10

instead

of 3 ··

10

? Could I still solve the problem?

•Toemphasizethepoint,drawanumberlineon

theboardwithapointat4 ··

10

.Then,havestudents

explainhowtocountonfrom4 ··

10

tofindthe

answer.Encourageavolunteertocometothe

boardanddemonstratehowtofindthesum.

Concept Extension

•How could you use fractions to label 0 and 1 on the number line?

Studentsmaysuggestthatyoucanwriteboth

asanumberoutof10,so0 ··

10

and 10 ··

10

.

•What is another way you could solve the problem?

Responsesmaymentionusingfractionstrips.

Youcouldlineupthree1 ··

10

stripsandfour

1 ··

10

stripsinasinglerow.Then,youcould

counthowmanytenthsyouhavealtogether.


Lesson 16


L16: Add and Subtract Fractions144

Part 2: Modeled Instruction

Read the problem below. Then explore different ways to understand it.

JosieandMargomade10claypotsinartclass.Josiepainted3 ··

10

ofthepots.

Margopainted4 ··

10

ofthepots.Whatfractionoftheclaypotsdidtheypaint?

Picture It

You can use models to help understand the problem.

The following model shows the pots. Each pot is 1 ·· 10 of the total number of pots.

Josie painted 3 pots, and Margo painted 4 pots. They painted a total of 7 pots.

1

3 tenths 4 tenths 7 tenths

5J J J J J JJ J

M M M M

J M M M M

Model It

You can also use a number line to help understand the problem.

The following number line is divided into tenths, with a point at 3 ·· 10 .

0 11

10210

310

410

510

610

710

810

910

Start at 3 ·· 10 and count 4 tenths to the right to add 4 ·· 10 .

0 11

102

103

104

105

106

107

108

109

10

162 L16: Add and Subtract Fractions


Lesson 16Part 2: Guided Instruction

AT A GLANCE

Students revisit the problem on page 144 to learn how to add fractions using equations. Then, students solve addition word problems.

STEP BY STEP

•ReadConnectItasaclass.Besuretopointoutthatthe questions refer to the problem on page 144.

•Reviewthemeaningsofnumerator(thenumberofequalpartsofasetyouhave)anddenominator(thetotal number of equal parts the set is divided into).

•Ask,If Josie and Margo only made 8 pots, what fraction

would represent 1 of the pots? 3 1 ··

8 4

•Emphasizethataddingfractionsislikeaddingwholenumbers. Say, When finding the number of pots Josie and Margo painted altogether, you add the numerators of the fractions and write that sum over the denominator.

TRY IT SOLUTIONS

7 Solution: 2 ··

3 ; Students may show 1

··

3 on a number line

dividedintothirdsandcount1marktotheright.

They also may write the equation 1 ··

3 1 1

··

3 5 2

··

3 .

8 Solution: 4 ··

5 of a meter; Students may show 1

··

5 on a

numberlinedividedintofifthsandcount3marks

to the right. They also may write the equation

1 ··

5 1 3

··

5 5 4

··

5 .

•Writethewordtenthsontheboard.Circletheletters that spell ten in the word and write the number 10 below it.

•Repeatusingthewordeighths.

•Havestudentswritetenths and eighths on a piece of paper. Next to the words, have them write fractions associated with the words.

•Iftimeallows,repeatwithotherfractionwords.

ELL Support

ERROR ALERT: Students who wrote 4 ··

10

( or 2 ··

5 ) added

both the numerators and the denominators.

Lesson 16


Part 2: Guided Instruction

Connect It

Now you will solve the problem from the previous page using equations.

2 How do you know that each pot is 1 ·· 10 of the total number of pots?

3 What do the numerators, 3 and 4, tell you?

4 How many clay pots did Josie and Margo paint altogether?

5 Write equations to show what fraction of the clay pots Josie and Margo painted altogether.

Use words: 3 tenths 1 4 tenths 5 tenths

Use fractions: 3 ·· 10 1 4 ·· 10 5 ···· 10

6 Explain how you add fractions with the same denominator.

Try It

Use what you just learned to solve these problems. Show your work on a separate sheet of paper.

7 Lita and Otis are helping their mom clean the house. Lita cleaned 1 ·· 3 of the rooms.

Otis cleaned 1 ·· 3 of the rooms. What fraction of the rooms did Lita and Otis clean

altogether?

8 Mark’s string is 1 ·· 5 of a meter long. Bob’s string is 3 ·· 5 of a meter long. How long are

the two strings combined? of a meter

Possible answer: The denominator tells the total number of pots.

The numerator tells the number of pots that you are talking about.

Possible answer: Add the numerators and leave the denominator as is.

Possible answer: 3 tells the number

of pots that Josie painted. 4 tells the number of pots that Margo painted.

7

2 ·· 3

4 ·· 5

7

7


Lesson 16Part 3: Modeled Instruction

AT A GLANCE

Students use models and number lines to review subtracting fractions.

STEP BY STEP

•Readtheproblematthetopofthepageasaclass.

•ReadPictureIt.GuidestudentstorecognizethatAlberto’swaterbottleisdividedinto6equalparts.Ask, What do the 6 equal parts represent?(thedenominator) What do the 5 shaded parts represent? (thenumerator,orhowmuchwaterisinthebottle)

•Pointoutthat4sixthsarebeingtakenawaysinceAlbertodrank4partsofthewaterbottle.Ask, What is 5 2 4?[1]Say,So, 1 sixth of Alberto’s water bottle still has water in it.

•TellstudentstolookatthenumberlineinModelIt.Pointoutthatthenumberlineisdividedintosixthstorepresentthe6equalpartsofAlberto’swaterbottle.

•Haveavolunteercount4jumpstotheleftfrom5 ··

6 to

subtract4sixths.Ask,What number did [volunteer’s

name] land on? 3 1 ··

6 4Say,So, both the model and number

line show that 1 sixth of Alberto’s water bottle still has

water in it.

Help students see the relationship between the picture and the number line.

•Drawthenumberlineontheboard.Then,drawthe5

··

6 -fullwaterbottleturnedonitssideabovethe

numberline,makingsureeachpartofthewaterbottleislinedupwithitstickmarkonthe number line.

•Pointoutthat5 ··

6 onthenumberlinelinesupwith

thetopofthewaterbottle.

•Then,crossout(orerase)onepartofthewaterbottleatatime,movingfromrighttoleftalongthenumberline.After4partsarecrossedout(orerased)toshowthewaterAlbertodrank,pointouttostudentsthattheremainingwaterislinedupwiththe1

··

6 -markonthenumberline.

Concept Extension•What is the difference between adding fractions and

subtracting fractions on a number line?

Responsesmayindicatedirection,movingtotherighttoaddandmovingtothelefttosubtract.

•What is another way to solve this problem?

Studentsmaymentionusingfractionstripsorwriting an equation.


Lesson 16



Read the problem below. Then explore different ways to understand it.

Alberto’swaterbottlehad5 ··

6 ofaliterinit.Hedrank4

··

6 ofaliter.Whatfraction

ofthebottlestillhaswaterinit?

Picture It

You can use models to help understand the problem.

The following model shows the water bottle divided into 6 equal parts. Each part

is 1 ·· 6 of a liter. Five shaded parts show how much water is in the bottle.

Alberto drank 4 parts of the water in the bottle, so take away 4 shaded parts of the bottle. There is 1 part of the bottle left with water in it.

2 5

5 sixths 4 sixths 1 sixth

Model It

You can use a number line to help understand the problem.

The following number line is divided into sixths, with a point at 5 ·· 6 .

0 11

626

36

46

56

Start at 5 ·· 6 and count 4 sixths to the left to subtract 4 ·· 6 .

0 11

626

36

46

56

Part 3: Modeled Instruction



Lesson 16Part 3: Guided Instruction

AT A GLANCE

Students revisit the problem on page 146 to learn how to subtract fractions using equations. Then, students solve subtraction word problems.

STEP BY STEP

•Readpage147asaclass.BesuretopointoutthatConnect It refers to the problem on page 146.

•Remindstudentsthatsubtractingfractionsislikesubtracting whole numbers. Say, When finding the number of parts of the water bottle that still have water, you subtract the numerators of the fractions and write the difference over the denominator.

TRY IT SOLUTIONS

14 Solution: 1 ··

4 ; Students may show 3

··

4 on a number line

dividedintofourthsandcount2 markstotheleft.

They also may write the equation 3 ··

4 2 2

··

4 5 1

··

4 .

15 Solution: 3 ··

10

; Students may show 8 ··

10

on a number

linedividedintotenthsandcount5markstothe

left. They also may write the equation 8 ··

10

2 5 ··

10

5 3 ··

10

.

SMP Tip: Discuss with students how important it is to communicate clearly and precisely by reviewing the meanings of numerator (the number of equal partsyou’retalkingabout)anddenominator(thetotalnumberofequalpartsinthewhole).Ask,If Alberto’s water bottle was divided into 3 equal parts, what fraction would represent 1 of those parts? 3 1

··

3 4 (SMP 6)

Use paper plates to subtract fractions.

Materials:paperplates,markers,rulers,scissors

•Distributepaperplates,markers,andscissorstoeach student. Model how to use the ruler to divide the plate into 8 equal sections. Students should draw 4 lines.

•Directstudentstocolor5 ··

8 of the plate and then

cutoutthatfractionoftheplate.Askstudentsto

name the fraction of the plate they have. 3 5 ··

8 4

•Tellstudentstosubtract2moreeighths.Guidestudents to cut 2 more sections from the color portion of the plate they are holding.

•Askstudentstonamethefractionoftheplate

they are left with. 3 3 ··

8 4

•Write5 ··

8 2 2

··

8 5 3

··

8 on the board.

•Iftimeallows,repeatforothersubtractionproblems.

Hands-On Activity

ERROR ALERT: Students who wrote 2 ··

4 ( or 1

··

2 )

subtracted from a full carton of eggs ( 4 ··

4 ) rather than

the 3 ··

4 ofacartonthatMrs.Kirkhad.

Lesson 16


Connect It

Now you will solve the problem from the previous page using equations.

9 How do you know that each part is 1 ·· 6 of a liter?

10 What do the numerators, 5 and 4, tell you?

11 How many parts of water are left in the bottle after Alberto drank 4 parts?

12 Write equations to show what fraction of the bottle has water left in it.

Use words: 5 sixths 2 4 sixths 5 sixth

Use fractions: 5 ·· 6 2 4 ·· 6 5 ···· 6

13 Explain how you subtract fractions with the same denominator.

Try It

Use what you just learned to solve these problems. Show your work on a separate sheet of paper.

14 Mrs. Kirk had 3 ·· 4 of a carton of eggs. She used 2 ·· 4 of the carton to make breakfast.

What fraction of the carton of eggs does Mrs. Kirk have left?

15 Carmen had 8 ·· 10 of the yard left to mow. She mowed 5 ·· 10 of the yard. What fraction

of the yard is left to mow?

Part 3: Guided Instruction

Possible answer: The denominator tells the number of equal parts the bottle is

divided into. The numerator tells the number of parts you are talking about.

Possible answer: 5 tells the

number of parts that have water. 4 tells the number of parts that

Alberto drank.

Possible answer: Subtract the numerators and leave the denominator as is.

1 ·· 4

3 ··· 10

1

1

1


Lesson 16Part 4: Guided Practice

AT A GLANCE

Students use models, number lines, or equations to solve word problems involving addition and subtraction of fractions.

STEP BY STEP

•Askstudentstosolvetheproblemsindividuallyandlabelfractionsintheirdrawings.

•Whenstudentshavecompletedeachproblem,havethemPair/Sharetodiscusstheirsolutionswithapartner or in a group.

SOLUTIONS

Ex Anumberlineisshownasonewaytosolvetheproblem.Studentscouldalsosolvetheproblembydrawingamodelthatisdividedintofifthsandshading4sections(2sectionsoutof5plus2 sectionsoutof5).

16 Solution: 2 ··

3 ofasmoothie;Studentscouldsolvethe

problembyusingtheequation3 ··

3 2 1

··

3 5 2

··

3 . (DOK 2)

17 Solution: 5 ··

10

;Studentscouldsolvetheproblemby

drawing a picture of 10 balloons and labeling 3 as

redand2asblue. (DOK 2)

18 Solution: C;Robaddedthenumeratorscorrectly,buthemistakenlyaddedthedenominatorstogether,too.

Explaintostudentswhytheothertwoanswerchoicesarenotcorrect:Aisnotcorrectbecauseyouarenotsubtracting 1 ··

6 from 2

··

6 ;thisisanadditionproblem.

B is not correct because 1 ··

3 is not equivalent to 3

··

6 .

(DOK 3)



I think that there are at least two different steps to solve this problem.

How is this problem different from the others you’ve seen in this lesson?

Pair/Share

To find the fraction of the bag Emily and Nick ate altogether, should you add or subtract?

Does Rob’s answer make sense?

Pair/Share

17 Mr. Chang has a bunch of balloons. 3 ·· 10 of the balloons are red.

2 ·· 10 of the balloons are blue. What fraction of the balloons are

neither red nor blue?

Show your work.

Solution:

18 Emily ate 1 ·· 6 of a bag of carrots. Nick ate 2 ·· 6 of the bag of carrots.

What fraction of the bag of carrots did Emily and Nick eat

altogether? Circle the letter of the correct answer.

A 1 ·· 6

B 1 ·· 3

C 3 ·· 6

D 3 ·· 12

Rob chose D as the correct answer. How did he get that answer?

Rob added both the numerators and the denominators.

5 ··· 10

Possible student work using a model:

red

3 ··· 10

blue

2 ··· 10

neither red nor blue

5 ··· 10

r r r b b

Student Model




How did you and your partner decide what fraction to start with?

Pair/Share

What fraction represents the whole fruit smoothie?

How else could you solve this problem?

Pair/Share

The student used labels and “jump” arrows to show each part of the hike on a number line. It is just like adding whole numbers!

Study the model below. Then solve problems 16–18.

Jessicahiked2 ··

5 mileonatrailbeforeshestoppedtogetadrink

ofwater.Afterherdrink,Jessicahikedanother2 ··

5 mile. How far

didJessicahikeinall?

Look at how you could show your work using a number line.

0 115

25

35

45

25

25

before drink after drink

Solution:

16 Ruth made a fruit smoothie. She drank 1 ·· 3 of it. What fraction of the

fruit smoothie is left?

Show your work.

Solution:

4 ·· 5 mile

2 ·· 3 of a smoothie

Possible student work using an equation:

3 ·· 3 2 1 ·· 3 5 2 ·· 3



Lesson 16Part 5: Common Core Practice

AT A GLANCE

Students add and subtract fractions to solve word problems that might appear on a mathematics test.

SOLUTIONS

1 Solution: C; Possible student work using

an equation: 5 ··

8 1 2

··

8 5 7

··

8 (DOK 1)

2 Solution: C; Possible student work using equations:

2 ··

12

1 3 ··

12

5 5 ··

12

; 12 ··

12

2 5 ··

12

5 7 ··

12

(DOK 1)

3 Solution: 1 ··

3 cup; Possible student work using

an equation: 2 ··

3 2 1

··

3 5 1

··

3 (DOK 1)

4 Solution: A; The model shows 2 ··

8 shaded in light gray

for Lucy’s sections and 4 ··

8 shaded in dark gray for

Margot’s sections. The total shaded sections

represent the total fraction of the lawn they mowed.

D; The number line starts at Margot’s fraction ( 4 ··

8 )

and adds 2 ··

8 for Lucy’s fraction, for a total of 6

··

8 .

(DOK 2)

5 Solution: 6 ··

10

; Possible student work using

an equation: 9 ··

10

2 3 ··

10

5 6 ··

10

(DOK 1)

6 Solution: Possible student work using equations:

0 ··

8 1 5

··

8 5 5

··

8 , 1

··

8 1 4

··

8 5 5

··

8 , 2

··

8 1 3

··

8 5 5

··

8 , 3

··

8 1 2

··

8 5 5

··

8 ,

4 ··

8 1 1

··

8 5 5

··

8 , 5

··

8 1 0

··

8 5 5

··

8 (DOK 2)

Part 5: Common Core Practice Lesson 16

©Curriculum Associates, LLC Copying is not permitted.151L16: Add and Subtract Fractions

4 Lucy and Margot are mowing the lawn. They divided the lawn into 8 equal sections. Lucy mowed 2 sections and Margot mowed 4 sections. Which model can be used to fi nd the total fraction of the lawn they mowed? Circle the letter of all that apply.

A

B

C 0 11

828

38

48

58

68

78

D 0 11

828

38

48

58

68

78

5 In all, Cole and Max picked 9 }} 10

of a bucket of blueberries. Cole picked 3 }} 10

of the

bucket of blueberries. What fraction of the bucket of blueberries did Max pick?

Show your work.

Answer Max picked of the bucket of blueberries.

6 A pizza is cut into 8 equal slices. Together, Regan and Juanita will eat 5 } 8 of the pizza.

What is one way the girls could eat the pizza?

Show your work.

Answer Regan could eat of the pizza, and

Juanita could eat of the pizza.

Go back and see what you can check off on the Self Check on page 119.Self Check

Possible student work using a model:

RRJ

JJ

2 ·· 8

3 ·· 8

Possible student work using a number line:

0 1110

210

310

410

510

610

710

810

910

6 ··· 10

Part 5: Common Core Practice Lesson 16



Solve the problems.

1 Liang bought some cloth. He used 5 } 8 of a yard for a school project. He has 2 }

8 of a yard

left. How much cloth did Liang buy?

A 3 } 8 of a yard

B 7 }} 16

of a yard

C 7 } 8 of a yard

D 8 } 8 of a yard

2 Carmela cut a cake into 12 equal-sized pieces. She ate 2 }} 12

of the cake, and her brother

ate 3 }} 12

of the cake. What fraction of the cake is left?

A 1 }} 12

of the cake

B 5 }} 12

of the cake

C 7 }} 12

of the cake

D 12 }} 12

of the cake

3 Lee’s muffi n mix calls for 2 } 3 cup of milk, 1 }

3 cup of oil, and 1 }

3 cup of sugar. How much

more milk than oil does she need for the muffi n mix?

1 ·· 3 cup

Differentiated Instruction Lesson 16

Assessment and Remediation

Hands-On Activity Challenge Activity


Write a problem for a given sum.

Tell students that the sum of two fractions is 2 ··

5 .

However, the original fractions did not have

denominators of 5. Challenge students to write a

fraction addition problem that has a sum of 2 ·· 5 .

3 Possible answer: 3 ··· 10 1 1 ··· 10 4

Use fraction strips to add fractions.

Materials: strips of paper, markers

Distribute paper and markers to each student. Direct

students to fold a strip of paper in half, and then in

half again. Tell them to unfold the strips and use the

marker to show the 4 equal sections. Tell students to

color 1 ··

4 of the strip. Then have them color another 1

··

4

of the strip. Write 1 ··

4 1 1

··

4 on the board. Challenge

them to use their fraction strips to show that the sum

is 2 ··

4 or 1

··

2 . If time allows, repeat for other

denominators by folding another strip of paper three

or four times.

•Askstudentstofind4 ··

10

1 2 ··

10

. 3 6 ··

10

or 3 ··

5 4

•Forstudentswhoarestillstruggling,usethechartbelowtoguideremediation.

•Afterprovidingremediation,checkstudents’understanding.Askstudentstoexplain theirthinkingwhilefinding2

··

5 1 3

··

5 . 3 5

··

5 or 1 4

•Ifastudentisstillhavingdifficulty,useReady Instruction, Level 4, Lesson 15.

If the error is . . . Students may . . . To remediate . . .

6 ·· 20 have added both the numerators and the denominators.

Remind students that the denominator tells the kind of parts you are adding. Explain that just as 4 apples 1 2 apples 5 6 apples, 4 tenths 1 2 tenths 5 6 tenths.

3 ·· 10 have added numerators, added denominators, and then simplified.

Remind students that the denominator tells the kind of parts you are adding. Explain that just as 4 apples + 2 apples = 6 apples, 4 tenths + 2 tenths = 6 tenths.

2 ·· 10 have subtracted the fractions.

Remind students to read the problem carefully to be sure they’re using the correct operation.

1 ·· 5 have subtracted the fractions and simplified.

Remind students to read the problem carefully to be sure they’re using the correct operation.

Built for the Common Core

True to the details and intent of the new standards, this rigorous instruction and practice program guarantees students and teachers will be Common Core-ready.

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