Teacher ediTion and assessmenT Guide sampler On … · Teacher ediTion and assessmenT Guide sampler Teacher Edition and Assessment Guide Sampler includes: - On Core Program Overview

On Core MathematicsGrade 6

Teacher ediTion and

assessmenT Guide sampler

Teacher Edition and Assessment Guide Sampler includes:

- On Core Program Overview

- Table of Contents for Grade 6

- Teaching Support and Student Lessons

- Assessments

Bridge the gap between your program and the Common Core State Standards. A complete program of activities, practice, and assessment for each Common Core State Mathematics Standard.

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Ratios and Proportional RelationshipsUnderstand ratio concepts and use ratio reasoning to solve problems.

Lesson 1 6.RP.1 Investigate • Model Ratios . . . . . . . . . . . . . . . . 1Lesson 2 6.RP.1 Ratios and Rates . . . . . . . . . . . . . . . . . . . . . . 3Lesson 3 6.RP.2 Find Unit Rates . . . . . . . . . . . . . . . . . . . . . . . 5Lesson 4 6.RP.3a Equivalent Ratios and Multiplication Tables . . . . . . . . . 7Lesson 5 6.RP.3a Problem Solving • Use Tables to Compare Ratios . . . . . 9Lesson 6 6.RP.3a Algebra • Use Equivalent Ratios . . . . . . . . . . . . . 11Lesson 7 6.RP.3a Algebra • Equivalent Ratios and Graphs . . . . . . . . . 13Lesson 8 6.RP.3b Algebra • Use Unit Rates . . . . . . . . . . . . . . . . 15Lesson 9 6.RP.3c Investigate • Model Percents . . . . . . . . . . . . . . 17Lesson 10 6.RP.3c Write Percents as Fractions and Decimals . . . . . . . . 19Lesson 11 6.RP.3c Write Fractions and Decimals as Percents . . . . . . . . 21Lesson 12 6.RP.3c Percent of a Quantity. . . . . . . . . . . . . . . . . . . 23Lesson 13 6.RP.3c Problem Solving • Percents . . . . . . . . . . . . . . . 25Lesson 14 6.RP.3c Find the Whole from a Percent . . . . . . . . . . . . . . 27Lesson 15 6.RP.3d Convert Units of Length . . . . . . . . . . . . . . . . . 29Lesson 16 6.RP.3d Convert Units of Capacity . . . . . . . . . . . . . . . . 31Lesson 17 6.RP.3d Convert Units of Weight and Mass . . . . . . . . . . . 33Lesson 18 6.RP.3d Transform Units . . . . . . . . . . . . . . . . . . . . . 35Lesson 19 6.RP.3d Problem Solving • Distance, Rate, and Time

Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 37

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On Core Mathematics is a comprehensive, ready-made resource providing instruction, practice and assessment for each Common Core State Mathematics Standard at your grade level. Designed to be used hand-in-hand with your current elementary math series, On Core offers you a flexible way to fill in any gaps between your series and the new standards. Whether you use just the lessons you need, or decide use the entire student workbook for comprehensive Common Core coverage, On Core provides a complete Common Core solution in just four components:

student edition: provides a searchable database of additional worksheets, projects, and hands-on activities correlated to the Common Core State Standards. Helps teachers focus on the mathematical practices.

Teacher edition: Instructional support for each Common Core Standards lesson. The three part, research-based lesson plan (Introduce, Teach, and Practice), that uses manipulatives and powerful visual models, provides everything needed to use the content.

assessment Guide: One page of assessment for each standard in multiple-choice, free-response and constructed response formats.

Exam View® online assessment: Administer premade print or online assessments or create your own with this powerful online tool aligned to the Common Core Standards.

What isOn Core Mathematics?

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nyRatios and Proportional RelationshipsUnderstand ratio concepts and use ratio reasoning to solve problems.


Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 37

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Ratios and Proportional RelationshipsUnderstand ratio concepts and use ratio reasoning to solve problems.


Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 37

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Grade 6 Table of Contents

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The Number SystemApply and extend previous understandings of multiplication and division to divide fractions by fractions.

Lesson 20 6.NS.1 Investigate • Model Fraction Division . . . . . . . . . . 39Lesson 21 6.NS.1 Estimate Quotients . . . . . . . . . . . . . . . . . . . . 41Lesson 22 6.NS.1 Divide Fractions . . . . . . . . . . . . . . . . . . . . . 43Lesson 23 6.NS.1 Investigate • Model Mixed Number Division . . . . . . 45Lesson 24 6.NS.1 Divide Mixed Numbers . . . . . . . . . . . . . . . . . . 47Lesson 25 6.NS.1 Problem Solving • Fraction Operations . . . . . . . . . 49

Compute fl uently with multi-digit numbers and fi nd common factorsand multiples.

Lesson 26 6.NS.2 Divide Multi-Digit Numbers. . . . . . . . . . . . . . . . 51Lesson 27 6.NS.3 Add and Subtract Decimals. . . . . . . . . . . . . . . . 53Lesson 28 6.NS.3 Multiply Decimals . . . . . . . . . . . . . . . . . . . . 55Lesson 29 6.NS.3 Divide Decimals by Whole Numbers . . . . . . . . . . . 57Lesson 30 6.NS.3 Divide with Decimals . . . . . . . . . . . . . . . . . . . 59Lesson 31 6.NS.4 Prime Factorization . . . . . . . . . . . . . . . . . . . . 61Lesson 32 6.NS.4 Least Common Multiple . . . . . . . . . . . . . . . . . 63Lesson 33 6.NS.4 Greatest Common Factor. . . . . . . . . . . . . . . . . 65Lesson 34 6.NS.4 Problem Solving • Apply the Greatest

Common Factor . . . . . . . . . . . . . . . . . . . . . 67Lesson 35 6.NS.4 Multiply Fractions . . . . . . . . . . . . . . . . . . . . 69Lesson 36 6.NS.4 Simplify Factors. . . . . . . . . . . . . . . . . . . . . . 71

Apply and extend previous understandings of numbers to the systemof rational numbers.

Lesson 37 6.NS.5 Understand Positive and Negative Numbers . . . . . . . 73Lesson 38 6.NS.6a Rational Numbers and the Number Line . . . . . . . . . 75Lesson 39 6.NS.6b Ordered Pair Relationships . . . . . . . . . . . . . . . . 77Lesson 40 6.NS.6c Fractions and Decimals . . . . . . . . . . . . . . . . . . 79Lesson 41 6.NS.6c Compare and Order Fractions and Decimals . . . . . . . 81Lesson 42 6.NS.6c Rational Numbers and the Coordinate Plane . . . . . . . 83Lesson 43 6.NS.7a Compare and Order Integers . . . . . . . . . . . . . . . 85Lesson 44 6.NS.7a Compare and Order Rational Numbers . . . . . . . . . . 87 6.NS.7bLesson 45 6.NS.7c Absolute Value . . . . . . . . . . . . . . . . . . . . . . 89Lesson 46 6.NS.7d Compare Absolute Values . . . . . . . . . . . . . . . . 91Lesson 47 6.NS.8 Distance on the Coordinate Plane . . . . . . . . . . . . 93Lesson 48 6.NS.8 Problem Solving • The Coordinate Plane . . . . . . . . 95

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Expressions and EquationsApply and extend previous understandings of arithmetic to algebraicexpressions.

Lesson 49 6.EE.1 Exponents . . . . . . . . . . . . . . . . . . . . . . . . 97Lesson 50 6.EE.1 Evaluate Expressions Involving Exponents. . . . . . . . . 99Lesson 51 6.EE.2a Write Algebraic Expressions . . . . . . . . . . . . . . .101Lesson 52 6.EE.2b Identify Parts of Expressions . . . . . . . . . . . . . . .103Lesson 53 6.EE.2c Evaluate Algebraic Expressions and Formulas . . . . . . .105Lesson 54 6.EE.3 Problem Solving • Combine Like Terms . . . . . . . .107Lesson 55 6.EE.3 Generate Equivalent Expressions . . . . . . . . . . . . .109Lesson 56 6.EE.4 Identify Equivalent Expressions . . . . . . . . . . . . . .111

Reason about and solve one-variable equations and inequalities.

Lesson 57 6.EE.5 Solutions of Equations . . . . . . . . . . . . . . . . . .113Lesson 58 6.EE.5 Solutions of Inequalities . . . . . . . . . . . . . . . . .115Lesson 59 6.EE.6 Use Algebraic Expressions . . . . . . . . . . . . . . . .117Lesson 60 6.EE.7 Write Equations . . . . . . . . . . . . . . . . . . . . .119Lesson 61 6.EE.7 Investigate • Model and Solve Addition Equations . . .121Lesson 62 6.EE.7 Solve Addition and Subtraction Equations . . . . . . . .123Lesson 63 6.EE.7 Investigate • Model and Solve Multiplication

Equations . . . . . . . . . . . . . . . . . . . . . . . .125Lesson 64 6.EE.7 Solve Multiplication and Division Equations. . . . . . . .127Lesson 65 6.EE.7 Problem Solving • Equations with Fractions. . . . . . .129Lesson 66 6.EE.8 Write Inequalities. . . . . . . . . . . . . . . . . . . . .131Lesson 67 6.EE.8 Graph Inequalities . . . . . . . . . . . . . . . . . . . .133

Represent and analyze quantitative relationships between dependent and independent variables.

Lesson 68 6.EE.9 Independent and Dependent Variables . . . . . . . . . .135Lesson 69 6.EE.9 Equations and Tables . . . . . . . . . . . . . . . . . .137Lesson 70 6.EE.9 Problem Solving • Analyze Relationships . . . . . . . .139Lesson 71 6.EE.9 Graph Relationships . . . . . . . . . . . . . . . . . . .141Lesson 72 6.EE.9 Equations and Graphs . . . . . . . . . . . . . . . . . .143

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nyExpressions and EquationsApply and extend previous understandings of arithmetic to algebraicexpressions.

Lesson 49 6.EE.1 Exponents . . . . . . . . . . . . . . . . . . . . . . . . 97Lesson 50 6.EE.1 Evaluate Expressions Involving Exponents. . . . . . . . . 99Lesson 51 6.EE.2a Write Algebraic Expressions . . . . . . . . . . . . . . .101Lesson 52 6.EE.2b Identify Parts of Expressions . . . . . . . . . . . . . . .103Lesson 53 6.EE.2c Evaluate Algebraic Expressions and Formulas . . . . . . .105Lesson 54 6.EE.3 Problem Solving • Combine Like Terms . . . . . . . .107Lesson 55 6.EE.3 Generate Equivalent Expressions . . . . . . . . . . . . .109Lesson 56 6.EE.4 Identify Equivalent Expressions . . . . . . . . . . . . . .111

Reason about and solve one-variable equations and inequalities.

Lesson 57 6.EE.5 Solutions of Equations . . . . . . . . . . . . . . . . . .113Lesson 58 6.EE.5 Solutions of Inequalities . . . . . . . . . . . . . . . . .115Lesson 59 6.EE.6 Use Algebraic Expressions . . . . . . . . . . . . . . . .117Lesson 60 6.EE.7 Write Equations . . . . . . . . . . . . . . . . . . . . .119Lesson 61 6.EE.7 Investigate • Model and Solve Addition Equations . . .121Lesson 62 6.EE.7 Solve Addition and Subtraction Equations . . . . . . . .123Lesson 63 6.EE.7 Investigate • Model and Solve Multiplication

Equations . . . . . . . . . . . . . . . . . . . . . . . .125Lesson 64 6.EE.7 Solve Multiplication and Division Equations. . . . . . . .127Lesson 65 6.EE.7 Problem Solving • Equations with Fractions. . . . . . .129Lesson 66 6.EE.8 Write Inequalities. . . . . . . . . . . . . . . . . . . . .131Lesson 67 6.EE.8 Graph Inequalities . . . . . . . . . . . . . . . . . . . .133

Represent and analyze quantitative relationships between dependent and independent variables.

Lesson 68 6.EE.9 Independent and Dependent Variables . . . . . . . . . .135Lesson 69 6.EE.9 Equations and Tables . . . . . . . . . . . . . . . . . .137Lesson 70 6.EE.9 Problem Solving • Analyze Relationships . . . . . . . .139Lesson 71 6.EE.9 Graph Relationships . . . . . . . . . . . . . . . . . . .141Lesson 72 6.EE.9 Equations and Graphs . . . . . . . . . . . . . . . . . .143

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GeometrySolve real-world and mathematical problems involving area, surface area, and volume.

Lesson 73 6.G.1 Algebra • Area of Parallelograms . . . . . . . . . . . .145Lesson 74 6.G.1 Investigate • Explore Area of Triangles . . . . . . . . .147Lesson 75 6.G.1 Algebra • Area of Triangles . . . . . . . . . . . . . . .149Lesson 76 6.G.1 Investigate • Explore Area of Trapezoids . . . . . . . .151Lesson 77 6.G.1 Algebra • Area of Trapezoids . . . . . . . . . . . . . .153Lesson 78 6.G.1 Area of Regular Polygons. . . . . . . . . . . . . . . . .155Lesson 79 6.G.1 Composite Figures . . . . . . . . . . . . . . . . . . . .157Lesson 80 6.G.1 Problem Solving • Changing Dimensions . . . . . . . .159Lesson 81 6.G.2 Investigate • Fractions and Volume . . . . . . . . . . .161Lesson 82 6.G.2 Algebra • Volume of Rectangular Prisms. . . . . . . . .163Lesson 83 6.G.3 Figures on the Coordinate Plane . . . . . . . . . . . . .165Lesson 84 6.G.4 Three-Dimensional Figures and Nets . . . . . . . . . . .167Lesson 85 6.G.4 Investigate • Explore Surface Area Using Nets. . . . . .169Lesson 86 6.G.4 Algebra • Surface Area of Prisms . . . . . . . . . . . .171Lesson 87 6.G.4 Algebra • Surface Area of Pyramids . . . . . . . . . . .173Lesson 88 6.G.4 Problem Solving • Geometric Measurements . . . . . .175

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Statistics and ProbabilityDevelop understanding of statistical variability.

Lesson 89 6.SP.1 Recognize Statistical Questions . . . . . . . . . . . . . .177Lesson 90 6.SP.2 Describe Distributions . . . . . . . . . . . . . . . . . .179Lesson 91 6.SP.2 Problem Solving • Misleading Statistics . . . . . . . . .181Lesson 92 6.SP.3 Apply Measures of Center and Variability. . . . . . . . .183

Summarize and describe distributions.

Lesson 93 6.SP.4 Dot Plots and Frequency Tables . . . . . . . . . . . . . .185Lesson 94 6.SP.4 Histograms . . . . . . . . . . . . . . . . . . . . . . . .187Lesson 95 6.SP.4 Problem Solving • Data Displays . . . . . . . . . . . .189Lesson 96 6.SP.4 Box Plots . . . . . . . . . . . . . . . . . . . . . . . . .191Lesson 97 6.SP.5a Describe Data Collection . . . . . . . . . . . . . . . . .193

6.SP.5bLesson 98 6.SP.5c Investigate • Mean as Fair Share and Balance Point . . .195Lesson 99 6.SP.5c Measures of Center . . . . . . . . . . . . . . . . . . .197Lesson 100 6.SP.5c Patterns in Data . . . . . . . . . . . . . . . . . . . . .199Lesson 101 6.SP.5c Investigate • Mean Absolute Deviation . . . . . . . . .201Lesson 102 6.SP.5c Measures of Variability . . . . . . . . . . . . . . . . . .203Lesson 103 6.SP.5d Effects of Outliers . . . . . . . . . . . . . . . . . . . .205Lesson 104 6.SP.5d Choose Appropriate Measures of Center and

Variability. . . . . . . . . . . . . . . . . . . . . . . . .207

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nyStatistics and ProbabilityDevelop understanding of statistical variability.

Lesson 89 6.SP.1 Recognize Statistical Questions . . . . . . . . . . . . . .177Lesson 90 6.SP.2 Describe Distributions . . . . . . . . . . . . . . . . . .179Lesson 91 6.SP.2 Problem Solving • Misleading Statistics . . . . . . . . .181Lesson 92 6.SP.3 Apply Measures of Center and Variability. . . . . . . . .183

Summarize and describe distributions.

Lesson 93 6.SP.4 Dot Plots and Frequency Tables . . . . . . . . . . . . . .185Lesson 94 6.SP.4 Histograms . . . . . . . . . . . . . . . . . . . . . . . .187Lesson 95 6.SP.4 Problem Solving • Data Displays . . . . . . . . . . . .189Lesson 96 6.SP.4 Box Plots . . . . . . . . . . . . . . . . . . . . . . . . .191Lesson 97 6.SP.5a Describe Data Collection . . . . . . . . . . . . . . . . .193

6.SP.5bLesson 98 6.SP.5c Investigate • Mean as Fair Share and Balance Point . . .195Lesson 99 6.SP.5c Measures of Center . . . . . . . . . . . . . . . . . . .197Lesson 100 6.SP.5c Patterns in Data . . . . . . . . . . . . . . . . . . . . .199Lesson 101 6.SP.5c Investigate • Mean Absolute Deviation . . . . . . . . .201Lesson 102 6.SP.5c Measures of Variability . . . . . . . . . . . . . . . . . .203Lesson 103 6.SP.5d Effects of Outliers . . . . . . . . . . . . . . . . . . . .205Lesson 104 6.SP.5d Choose Appropriate Measures of Center and

Variability. . . . . . . . . . . . . . . . . . . . . . . . .207

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Ratios and Proportional Relationships 15

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CC.6.RP.3c

Lesson 14

Find the Whole from a Percent

Find the unknown value.

1. 9 is 15% of n

____

2. 54 is 75% of n

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3. 12 is 2% of n

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4. 18 is 50% of n

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5. 16 is 40% of n

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6. 56 is 28% of n

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7. 5 is 10% of n

____

8. 24 is 16% of n

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9. 15 is 25% of n

____

10. 11 is 44% of n

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11. 19 is 95% of n

____

12. 10 is 20% of n

____

13. Michaela is hiking on a weekend camping trip. She has walked 6 miles so far. This is 30% of the total distance. What is the total number of miles she will walk?

14. A customer placed an order with a bakery for cupcakes. The baker has completed 37.5% of the order after baking 81 cupcakes. How many cupcakes did the customer order?

60

15 ___ 100

5 9 __ n

15 4 5 ______ 100 4 5

5 3 3 3 _____ 20 3 3

5 9 __ 60

72 600

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20 miles 216 cupcakes

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COMMON CORE STANDARD CC.6.RP.3c

Lesson 14


You can use equivalent ratios to find the whole, given a part and the percent.

percent 5 part _____ whole

60% 5 54 __ j

60 4 20 ________ 100 4 20

5 54 __ j

3 _ 5 5 54 __

n

3 3 18 ______ 5 3 18

5 54 __ n

54 __ 90

5 54 __ n

54 is 60% of what number?

Step 1 Write the relationship among the percent, part, and whole. The percent is 60%. The part is 54. The whole is unknown.

Step 2 Write the percent as a ratio.

Step 3 Simplify the known ratio.

• Find the GCF of the numerator and denominator.

60 5 2 3 2 3 3 3 5

100 5 2 3 2 3 5 3 5

• Divide both the numerator and denominator by the GCF.

Step 4 Write an equivalent ratio.

• Look at the numerators. Think: 3 3 18 5 54

• Multiply the denominator by 18 to fi nd the whole.

So, 54 is 60% of 90.


60 ___ 100

5 54 __ j

Lesson Objective: Find the whole given a part and the percent.

GCF 5 2 3 2 3 5 5 20

1. 12 is 40% of n

2. 15 is 25% of n

3. 24 is 20% of n

4. 36 is 50% of n

5. 4 is 80% of n

6. 12 is 15% of n

7. 36 is 90% of n

8. 12 is 75% of n

9. 27 is 30% of n

30 60 120

72 5 80

40 16 90

About the MathThe relationship among the part, percent, and whole can be used to find the whole when the percent and part are given. Students will support their learning when they look for and express regularity in repeated reasoning.

The LessonIntroduce Remind students that they have used equivalent ratios to find the part, given the percent and the whole. In this lesson they will use equivalent ratios to find the whole, given the percent and the part.

Teach Point out to students that both ratios in Step 2 represent part ____ whole . A percent

is part out of 100, so 60 is the part and 100 is the whole. In the second ratio, 54 is the part and the whole is unknown.

Explain to students that simplifying the known ratio makes it easier to find the equivalent ratio.

Extend the process of finding the whole given a part and the percent by having students estimate first. Then have them check to see if their answers are reasonable.

Practice Have students complete page 28. You may wish to review the process by discussing the first exercise.


COMMON CORE STANDARDCC.6.RP.3c

OBJECTIVEFind the whole given a part and the percent.

ESSENTIAL QUESTIONHow can you find the whole given a part and the percent?

VOCABULARY

MATERIALS

PREREQUISITESWrite equivalent fractions.

Write a percent as ratio out of 100.

LESSON 14Pages 27–28Page 14, Assessment Guide

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Lesson 14




60% 5 54 __ j

60 4 20 ________ 100 4 20

5 54 __ j

3 _ 5 5 54 __

n

3 3 18 ______ 5 3 18

5 54 __ n

54 __ 90

5 54 __ n






60 5 2 3 2 3 3 3 5

100 5 2 3 2 3 5 3 5





So, 54 is 60% of 90.


60 ___ 100

5 54 __ j


GCF 5 2 3 2 3 5 5 20

1. 12 is 40% of n

2. 15 is 25% of n

3. 24 is 20% of n

4. 36 is 50% of n

5. 4 is 80% of n

6. 12 is 15% of n

7. 36 is 90% of n

8. 12 is 75% of n

9. 27 is 30% of n

30 60 120

72 5 80

40 16 90

6_MNLAEWB575247_SE_L014R.indd 27 12/9/10 11:12:05 PM

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Lesson 14




60% 5 54 __ j

60 4 20 ________ 100 4 20

5 54 __ j

3 _ 5 5 54 __

n

3 3 18 ______ 5 3 18

5 54 __ n

54 __ 90

5 54 __ n






60 5 2 3 2 3 3 3 5

100 5 2 3 2 3 5 3 5





So, 54 is 60% of 90.


60 ___ 100

5 54 __ j


GCF 5 2 3 2 3 5 5 20

1. 12 is 40% of n

2. 15 is 25% of n

3. 24 is 20% of n

4. 36 is 50% of n

5. 4 is 80% of n

6. 12 is 15% of n

7. 36 is 90% of n

8. 12 is 75% of n

9. 27 is 30% of n

30 60 120

72 5 80

40 16 90


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CC.6.RP.3c

Lesson 14



1. 9 is 15% of n

____

2. 54 is 75% of n

____

3. 12 is 2% of n

____

4. 18 is 50% of n

____

5. 16 is 40% of n

____

6. 56 is 28% of n

____

7. 5 is 10% of n

____

8. 24 is 16% of n

____

9. 15 is 25% of n

____

10. 11 is 44% of n

____

11. 19 is 95% of n

____

12. 10 is 20% of n

____

13. Michaela is hiking on a weekend camping trip. She has walked 6 miles so far. This is 30% of the total distance. What is the total number of miles she will walk?

14. A customer placed an order with a bakery for cupcakes. The baker has completed 37.5% of the order after baking 81 cupcakes. How many cupcakes did the customer order?

60

15 ___ 100

5 9 __ n

15 4 5 ______ 100 4 5

5 3 3 3 _____ 20 3 3

5 9 __ 60

72 600

36

50

25

20 miles 216 cupcakes

150

20

40 200

60

50

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10 52 Expressions and Equations

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Expressions and Equations 97

Lesson Objective: Write and evaluate expressions involving exponents.

Lesson 49COMMON CORE STANDARD CC.6.EE.1

Exponents

An exponent tells how many times a number is used as a factor.

The base is the number being multiplied repeatedly.

For example, in 2 5 , 5 is the exponent and 2 is the base. 2 5 5 2 3 2 3 2 3 2 3 2 5 32

Write the expression 4 5 using equal factors. Then fi nd the value.

Step 1 Identify the base. The base is 4.

Step 2 Identify the exponent. The exponent is 5.

Step 3 Write the base as many times as the 4 3 4 3 4 3 4 3 4 exponent tells you. Place a multiplication symbol between the bases.

Step 4 Multiply.

So, 4 5 5 1,024.

Write as an expression using equal factors. Then fi nd the value.

1. 3 4

2. 2 6

3. 4 3

4. 5 3

5. 10 4

6. 8 5

7. 11 4

8. 15 2

9. 10 7

10. 25 4

4 3 4 3 4 3 4 3 4 5 1,024

3 3 3 3 3 3 3; 81 2 3 2 3 2 3 2 3 2 3 2; 64

4 3 4 3 4; 64 5 3 5 3 5; 125

10 3 10 3 10 3 10; 10,000 8 3 8 3 8 3 8 3 8; 32,768

11 3 11 3 11 3 11; 14,641

10 3 10 3 10 3 10 3 10 3

10 3 10; 10,000,000

15 3 15; 225

25 3 25 3 25 3 25;

390,625

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Lesson 49CC.6.EE.1

Exponents

Use one or more exponents to write the expression.

Find the value.

1. 6 3 6

____

2. 11 3 11 3 11 3 11

____

3. 9 3 9 3 9 3 9 3 7 3 7

____

4. 9 2

____

5. 6 4

____

6. 1 6

____

7. 8 3

____

8. 10 5

____

9. 23 2

____

10. Write 144 with an exponent by using 12 as the base.


12. Each day Sheila doubles the number of push-ups she did the day before. On the fifth day, she does 2 3 2 3 2 3 2 3 2 push-ups. Use an exponent to write the number of push-ups Shelia does on the fifth day.

13. The city of Beijing has a population of more than 10 7 people. Write the population of Beijing without using an exponent.

6 2 11 4 9 4 3 7 2

81

512

12 2

2 5 more than 10,000,000

7 3

100,000

1,296 1

529


Exponents

About the MathExponents can be considered as a shorthand method for representing repeated multiplication. Students will be able to relate powers of 10 to the base-ten number system. Each place value is 10 times the place to its right. Students will support their learning when they look for and make use of structure.

The LessonIntroduce Remind students that multiplication is an easier way of writing repeated addition of the same addend. Then tell them that in this lesson they will learn an easier way to represent repeated multiplication of the same factor, such as 2 3 2 3 2 3 2 3 2 3 2 3 2.

Teach Discuss the meaning of exponent and base with the students. The exponent tells how many times a number is used as a factor. The base is the number that is being multiplied repeatedly. So, the multiplication above can be written as 2 7 . Have students write a multiplication expression with a repeated factor for a partner to evaluate.

Extend the process of writing expressions with exponents to include two different repeated factors. Then have students determine the exponent for a product, given the base.


COMMON CORE STANDARDCC.6.EE.1

OBJECTIVEWrite and evaluate expressions involving exponents.

ESSENTIAL QUESTIONHow do you write and find the value of expressions involving exponents?

VOCABULARYexponent, base

MATERIALS

PREREQUISITESMultiply with more than 2 factors.

Find factors of numbers.


1152 Expressions and Equations

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Exponents








Step 4 Multiply.

So, 4 5 5 1,024.


1. 3 4

2. 2 6

3. 4 3

4. 5 3

5. 10 4

6. 8 5

7. 11 4

8. 15 2

9. 10 7

10. 25 4

4 3 4 3 4 3 4 3 4 5 1,024

3 3 3 3 3 3 3; 81 2 3 2 3 2 3 2 3 2 3 2; 64

4 3 4 3 4; 64 5 3 5 3 5; 125

10 3 10 3 10 3 10; 10,000 8 3 8 3 8 3 8 3 8; 32,768

11 3 11 3 11 3 11; 14,641

10 3 10 3 10 3 10 3 10 3

10 3 10; 10,000,000

15 3 15; 225

25 3 25 3 25 3 25;

390,625

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Lesson 49CC.6.EE.1

Exponents


Find the value.

1. 6 3 6

____

2. 11 3 11 3 11 3 11

____

3. 9 3 9 3 9 3 9 3 7 3 7

____

4. 9 2

____

5. 6 4

____

6. 1 6

____

7. 8 3

____

8. 10 5

____

9. 23 2

____





6 2 11 4 9 4 3 7 2

81

512

12 2

2 5 more than 10,000,000

7 3

100,000

1,296 1

529


Exponents

About the MathExponents can be considered as a shorthand method for representing repeated multiplication. Students will be able to relate powers of 10 to the base-ten number system. Each place value is 10 times the place to its right. Students will support their learning when they look for and make use of structure.

The LessonIntroduce Remind students that multiplication is an easier way of writing repeated addition of the same addend. Then tell them that in this lesson they will learn an easier way to represent repeated multiplication of the same factor, such as 2 3 2 3 2 3 2 3 2 3 2 3 2.

Teach Discuss the meaning of exponent and base with the students. The exponent tells how many times a number is used as a factor. The base is the number that is being multiplied repeatedly. So, the multiplication above can be written as 2 7 . Have students write a multiplication expression with a repeated factor for a partner to evaluate.

Extend the process of writing expressions with exponents to include two different repeated factors. Then have students determine the exponent for a product, given the base.



OBJECTIVEWrite and evaluate expressions involving exponents.

ESSENTIAL QUESTIONHow do you write and find the value of expressions involving exponents?

VOCABULARYexponent, base

MATERIALS

PREREQUISITESMultiply with more than 2 factors.

Find factors of numbers.


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Exponents








Step 4 Multiply.

So, 4 5 5 1,024.


1. 3 4

2. 2 6

3. 4 3

4. 5 3

5. 10 4

6. 8 5

7. 11 4

8. 15 2

9. 10 7

10. 25 4

4 3 4 3 4 3 4 3 4 5 1,024

3 3 3 3 3 3 3; 81 2 3 2 3 2 3 2 3 2 3 2; 64

4 3 4 3 4; 64 5 3 5 3 5; 125

10 3 10 3 10 3 10; 10,000 8 3 8 3 8 3 8 3 8; 32,768

11 3 11 3 11 3 11; 14,641

10 3 10 3 10 3 10 3 10 3

10 3 10; 10,000,000

15 3 15; 225

25 3 25 3 25 3 25;

390,625


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Lesson 49CC.6.EE.1

Exponents


Find the value.

1. 6 3 6

____

2. 11 3 11 3 11 3 11

____

3. 9 3 9 3 9 3 9 3 7 3 7

____

4. 9 2

____

5. 6 4

____

6. 1 6

____

7. 8 3

____

8. 10 5

____

9. 23 2

____





6 2 11 4 9 4 3 7 2

81

512

12 2

2 5 more than 10,000,000

7 3

100,000

1,296 1

529


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Exponents








Step 4 Multiply.

So, 4 5 5 1,024.


1. 3 4

2. 2 6

3. 4 3

4. 5 3

5. 10 4

6. 8 5

7. 11 4

8. 15 2

9. 10 7

10. 25 4

4 3 4 3 4 3 4 3 4 5 1,024

3 3 3 3 3 3 3; 81 2 3 2 3 2 3 2 3 2 3 2; 64

4 3 4 3 4; 64 5 3 5 3 5; 125

10 3 10 3 10 3 10; 10,000 8 3 8 3 8 3 8 3 8; 32,768

11 3 11 3 11 3 11; 14,641

10 3 10 3 10 3 10 3 10 3

10 3 10; 10,000,000

15 3 15; 225

25 3 25 3 25 3 25;

390,625


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Lesson 49CC.6.EE.1

Exponents


Find the value.

1. 6 3 6

____

2. 11 3 11 3 11 3 11

____

3. 9 3 9 3 9 3 9 3 7 3 7

____

4. 9 2

____

5. 6 4

____

6. 1 6

____

7. 8 3

____

8. 10 5

____

9. 23 2

____





6 2 11 4 9 4 3 7 2

81

512

12 2

2 5 more than 10,000,000

7 3

100,000

1,296 1

529


14

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CC.6.EE.3

Lesson 54

Problem Solving • Combine Like Terms

Read each problem and solve.

1. A box of pens costs $3 and a box of markers costs $5. The expression 3p 1 5p represents the cost in dollars to make p packages that includes 1 box of pens and 1 box of markers. Simplify the expression by combining like terms.

2. On a heating bill, a gas company charges customers two different fees based on t therms used. The service fee costs $0.25 per therm and the distribution fee costs $0.10 per therm. The expression 0.25t 1 0.10t 1 40 represents the total bill in dollars. Simplify the expression by combining like terms.

3. A radio show lasts for h hours. During that time, there are 60 minutes of air time per hour and 8 minutes of commercials per hour. The expression 60h 2 8h represents the air time in minutes available for talk and music. Simplify the expression by combining like terms.

4. A publisher sends 100 books to each bookstore where its books are sold. About 3 books are sold at a discount to employees and about 40 books are sold during store supersales. The expression 100s 2 3s 2 40s represents the number of books for s stores that are sold at full price. Simplify the expression by combining like terms.

5. A sub shop sells a meal that includes an Italian sub for $6 and chips for $2. If a customer purchases more than 3 meals, he or she receives a $5 discount. The expression 6m 1 2m 2 5 shows the cost in dollars of the customer’s order for m $ 3. Simplify the expression by combining like terms.

8m 2 5

57s

52h

0.35t 1 40

3p 1 5p 5 8p

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Lesson Objective: Combine like terms by applying the strategy use a model.

COMMON CORE STANDARD CC.6.EE.3

Lesson 54

Problem Solving • Combine Like TermsUse a bar model to solve the problem.

Each hour a company assembles 10 bikes and then packages 6 of those bikes for local shipment. An employee tests 1 bike each day. The expression 10h 2 6h 2 1 represents the number of bikes produced for international shipment in h hours. Simplify the expression by combining like terms.

Read the Problem

What do I need to fi nd?

I need to simplify the

expression .

What information do I need to use?

I need to use the like terms

10h and .

How will I use the information?

I can use a bar model to fi nd the difference of

the terms.

Draw a bar model to subtract from . Each square represents h, or 1h.

Solve the Problem

The model shows that 10h 2 6h 5 . 10h 2 6h 2 1 5 2 1

So, a simplifi ed expression for the number of bikes is .

1. Bradley sells produce in boxes at the local farmer’s market. He put 6 ears of corn and 9 tomatoes in each box. The expression 6b 1 9b represents the total pieces of produce in b boxes. Simplify the expression by combining like terms.

2. Andre bought pencils in packs of 8. He gave 2 pencils to his sister and 3 pencils from each pack to his friends. The expression 8p 2 3p 2 2 represents the number of pencils Andre has left from p packs. Simplify the expression by combining like terms.

h h h h h h h

h h h h h h

h h h

10 h

6 h 4 h

10h – 6h – 1

15b

6hlike

6h

4h

10h

4h

4h 2 1

5p 2 2

About the MathVariables can be used to write expressions when solving problems. Expressions containing like terms can be simplified by combining like terms. Students will support their learning when they model with mathematics.

The LessonIntroduce Remind students that they have used variables to write expressions. In this lesson they will simplify expressions.

Teach Discuss the meaning of like terms as terms that have the same variable to the same power. Remind students that constants are like terms. The model shows that 10h in the first row and 6h in the second row can be combined, in this case subtracted. Have students cross out 6h from each row to show that there are 4h left. Point out that 1 is a constant and does not have the variable h so it cannot be combined.

Extend the process of combining like terms to include adding or subtracting the coefficients of the like terms to combine.




OBJECTIVECombine like terms by applying the strategy use a model.

ESSENTIAL QUESTIONHow can you use the strategy use a model to combine like terms?

VOCABULARYlike terms

MATERIALS

PREREQUISITESWrite and evaluate algebraic expressions.



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Lesson Objective: Combine like terms by applying the strategy use a model.

COMMON CORE STANDARD CC.6.EE.3

Lesson 54

Problem Solving • Combine Like TermsUse a bar model to solve the problem.

Each hour a company assembles 10 bikes and then packages 6 of those bikes for local shipment. An employee tests 1 bike each day. The expression 10h 2 6h 2 1 represents the number of bikes produced for international shipment in h hours. Simplify the expression by combining like terms.

Read the Problem

What do I need to fi nd?

I need to simplify the

expression .

What information do I need to use?

I need to use the like terms

10h and .

How will I use the information?

I can use a bar model to fi nd the difference of

the terms.

Draw a bar model to subtract from . Each square represents h, or 1h.

Solve the Problem

The model shows that 10h 2 6h 5 . 10h 2 6h 2 1 5 2 1

So, a simplifi ed expression for the number of bikes is .

1. Bradley sells produce in boxes at the local farmer’s market. He put 6 ears of corn and 9 tomatoes in each box. The expression 6b 1 9b represents the total pieces of produce in b boxes. Simplify the expression by combining like terms.

2. Andre bought pencils in packs of 8. He gave 2 pencils to his sister and 3 pencils from each pack to his friends. The expression 8p 2 3p 2 2 represents the number of pencils Andre has left from p packs. Simplify the expression by combining like terms.

h h h h h h h

h h h h h h

h h h

10 h

6 h 4 h

10h – 6h – 1

15b

6hlike

6h

4h

10h

4h

4h 2 1

5p 2 2


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CC.6.EE.3

Lesson 54


Read each problem and solve.

1. A box of pens costs $3 and a box of markers costs $5. The expression 3p 1 5p represents the cost in dollars to make p packages that includes 1 box of pens and 1 box of markers. Simplify the expression by combining like terms.

2. On a heating bill, a gas company charges customers two different fees based on t therms used. The service fee costs $0.25 per therm and the distribution fee costs $0.10 per therm. The expression 0.25t 1 0.10t 1 40 represents the total bill in dollars. Simplify the expression by combining like terms.

3. A radio show lasts for h hours. During that time, there are 60 minutes of air time per hour and 8 minutes of commercials per hour. The expression 60h 2 8h represents the air time in minutes available for talk and music. Simplify the expression by combining like terms.

4. A publisher sends 100 books to each bookstore where its books are sold. About 3 books are sold at a discount to employees and about 40 books are sold during store supersales. The expression 100s 2 3s 2 40s represents the number of books for s stores that are sold at full price. Simplify the expression by combining like terms.

5. A sub shop sells a meal that includes an Italian sub for $6 and chips for $2. If a customer purchases more than 3 meals, he or she receives a $5 discount. The expression 6m 1 2m 2 5 shows the cost in dollars of the customer’s order for m $ 3. Simplify the expression by combining like terms.

8m 2 5

57s

52h

0.35t 1 40

3p 1 5p 5 8p


18

The following pages from the On Core Assessment Guide support

the student lessons presented earlier in this sampler:

lesson 14: Find the Whole from a percent

lesson 49: exponents

Lesson 54: Problem Solving • Combine Like Terms

Assessment Guide Sample Pages

19

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Name CC.6.RP.3cLesson 14

14 Ratios and Proportional Relationships

1. 9 is 15% of what number?

A 24

B 60

C 85

D 135

2. A train is traveling from Orlando, Florida to Atlanta, Georgia. So far, it has traveled 75% of the distance, or 330 miles. How far is Orlando from Atlanta?

A 247 miles

B 255 miles

C 405 miles

D 440 miles

3. Carmen has saved 80% of the money she needs to buy a new video game. If she has saved $36, how much does the video game cost?

A $28.80 C $63.80

B $45 D $80

4. The sixth-graders at Amir’s school voted for the location of their class trip. The table shows the results.

Class Trip Votes

Location Percent

History Museum 25%

Art Museum 35%

Aquarium 40%

If 126 students voted for going to the art museum, how many sixth-graders are at Amir’s school?

A 226 C 360

B 315 D 504

5. Marcus is saving money to buy a new DVD player. So far, he has saved 60% of the money he needs, or $45. What is the cost of the DVD player? Explain how you know.

$75; I needed to answer the question, “60% of what

number is 45?” I wrote 60 ___ 100

× 45 ___ and used equivalent

ratios to fi nd that the answer is $75.

6_MNLAEAS590479_CC_L014T.indd 14 12/10/10 1:44:49 PM

20Expressions and Equations 49

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NameLesson 49CC.6.EE.1

5. James wrote 256 as 2 8 . Janet wrote 256 as 4 4 . Explain how you know both of the students are correct. Write a word problem for which Janet’s representation could be used to answer the question.

1. The bill with the greatest value ever printed in the United States had a value of 10 5 dollars. Which is another way to write that amount?

A $10,000

B $50,000

C $100,000

D $500,000

2. Carlos represented 729 with a base and an exponent. Which of the following is NOT possible?

A The base is less than the exponent.

B The base and the exponent are equal.

C The base and the exponent are multiples of 3.

D The base is an odd number and the exponent is an even number.

3. John is making a patio in his yard. He needs a total of 15 2 concrete blocks to cover the area. How many blocks does John need?

A 30

B 125

C 152

D 225

4. Which is a way to write 2 3 2 3 2 3 5 3 5 with exponents and two bases?

A 2 3 3 5 2

B 3 2 3 2 5

C 2 5 3 5 5

D 2 5 3 10 3 5

2 8 means 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2. I grouped the 2s

by 2 to get 4 3 4 3 4 3 4. Then I multiplied 16 by 16, which

is 256. Since my second step was equivalent to 4 4 , I know

the two numbers are equivalent. Possible word problem:

Janet baked cookies for the bake sale. She put 4 cookies

in a bag and 4 bags in a basket. Then she put 4 baskets in

a box. Finally, she put 4 boxes in a bin. How many cookies

did she bake if she fi lled 4 bins?


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Name CC.6.EE.3Lesson 54


1. Sandwiches cost $5, French fries cost $3, and drinks cost $2. The expression 5n 1 3n 1 2n gives the total cost in dollars for buying a sandwich, French fries, and a drink for n people. Which is another way to write this expression?

A 10n

B 10 n 3

C 30n

D 30 n 3

2. Jackets cost $15 and decorative buttons cost $5. The delivery fee is $5 per order. The expression 15n 1 5n 1 5 gives the cost in dollars of buying jackets with buttons for n people. Which is another way to write this expression?

A 25n

B 25 n 2

C 20n 1 5

D 20 n 2 1 5

3. Dana has n quarters. Ivan has 2 fewer than three times the number of quarters Dana has. The expression n 1 3n 2 2 gives the number of quarters they have altogether. Which is another way to write this expression?

A 2 n 2

B 2n

C 4 n 2 2 2

D 4n 2 2

4. Scarves cost $12 and snowmen pins cost $2. Shipping is $3 per order. The expression 12n 1 2n 1 3 gives the cost in dollars of buying scarves with pins for n people. Which is another way to write this expression?

A 14 n 2 1 3

B 14n 1 3

C 17 n 2

D 17n

5. Debbie is n years old. Edna is 3 years older than Debbie, and Shawn is twice as old as Edna. The expression n 1 n 1 3 1 2 3 (n 1 3) gives the sum of their ages. Simplify the expression by combining like terms. Explain how you found your answer.

4n 1 9; First, I used the Distributive Property to rewrite

the expression as n 1 n 1 3 1 (2 3 n) 1 (2 3 3), or

n 1 n 1 3 1 2n 1 6. Then I used the Commutative

Property of Addition to switch the order of

3 and 2n. Finally, I combined the like terms:

n 1 n 1 2n 1 3 1 6 5 4n 1 9.


21

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Name CC.6.EE.3Lesson 54


1. Sandwiches cost $5, French fries cost $3, and drinks cost $2. The expression 5n 1 3n 1 2n gives the total cost in dollars for buying a sandwich, French fries, and a drink for n people. Which is another way to write this expression?

A 10n

B 10 n 3

C 30n

D 30 n 3

2. Jackets cost $15 and decorative buttons cost $5. The delivery fee is $5 per order. The expression 15n 1 5n 1 5 gives the cost in dollars of buying jackets with buttons for n people. Which is another way to write this expression?

A 25n

B 25 n 2

C 20n 1 5

D 20 n 2 1 5

3. Dana has n quarters. Ivan has 2 fewer than three times the number of quarters Dana has. The expression n 1 3n 2 2 gives the number of quarters they have altogether. Which is another way to write this expression?

A 2 n 2

B 2n

C 4 n 2 2 2

D 4n 2 2

4. Scarves cost $12 and snowmen pins cost $2. Shipping is $3 per order. The expression 12n 1 2n 1 3 gives the cost in dollars of buying scarves with pins for n people. Which is another way to write this expression?

A 14 n 2 1 3

B 14n 1 3

C 17 n 2

D 17n

5. Debbie is n years old. Edna is 3 years older than Debbie, and Shawn is twice as old as Edna. The expression n 1 n 1 3 1 2 3 (n 1 3) gives the sum of their ages. Simplify the expression by combining like terms. Explain how you found your answer.

4n 1 9; First, I used the Distributive Property to rewrite

the expression as n 1 n 1 3 1 (2 3 n) 1 (2 3 3), or

n 1 n 1 3 1 2n 1 6. Then I used the Commutative

Property of Addition to switch the order of

3 and 2n. Finally, I combined the like terms:

n 1 n 1 2n 1 3 1 6 5 4n 1 9.


NOTES

NOTES

NOTES

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