In this Module content, you will find the Grade 6 Module on Ratios and Unit Rates from EngageNY (Eureka Mathematics). You are NOT required to review each le

6

G R A D E

New York State Common Core

Mathematics Curriculum GRADE 6 • MODULE 1

Table of Contents1

Ratios and Unit Rates Module Overview ……………………………………………………………………………………………………………………………….. 3

Topic A: Representing and Reasoning About Ratios (6.RP.A.1, 6.RP.A.3a) ……………………………………………….. 12

Lessons 1–2: Ratios ………………………………………………………………………………………………………………… 14

Lessons 3–4: Equivalent Ratios ………………………………………………………………………………………………… 28

Lessons 5–6: Solving Problems by Finding Equivalent Ratios ……………………………………………………….. 41

Lesson 7: Associated Ratios and the Value of a Ratio …………………………………………………………………. 51

Lesson 8: Equivalent Ratios Defined Through the Value of a Ratio ………………………………………………. 57

Topic B: Collections of Equivalent Ratios (6.RP.A.3a) …………………………………………………………………………….. 63

Lesson 9: Tables of Equivalent Ratios ……………………………………………………………………………………….. 65

Lesson 10: The Structure of Ratio Tables—Additive and Multiplicative ………………………………………… 71

Lesson 11: Comparing Ratios Using Ratio Tables ……………………………………………………………………….. 80

Lesson 12: From Ratio Tables to Double Number Line Diagrams …………………………………………………. 88

Lesson 13: From Ratio Tables to Equations Using the Value of a Ratio …………………………………………. 99

Lesson 14: From Ratio Tables, Equations, and Double Number Line Diagrams to Plots on the Coordinate Plane …………………………………………………………………………………………………. 109

Lesson 15: A Synthesis of Representations of Equivalent Ratio Collections …………………………………. 117

Mid-Module Assessment and Rubric …………………………………………………………………………………………………. 126 Topics A through B (assessment 1 day, return 1 day, remediation or further applications 1 day)

Topic C: Unit Rates (6.RP.A.2, 6.RP.A.3b, 6.RP.A.3d) …………………………………………………………………………… 132

Lesson 16: From Ratios to Rates …………………………………………………………………………………………….. 134

Lesson 17: From Rates to Ratios …………………………………………………………………………………………….. 139

Lesson 18: Finding a Rate by Dividing Two Quantities ………………………………………………………………. 145

Lessons 19–20: Comparison Shopping—Unit Price and Related Measurement Conversions …………. 150

Lessons 21–22: Getting the Job Done—Speed, Work, and Measurement Units …………………………… 165

Lesson 23: Problem-Solving Using Rates, Unit Rates, and Conversions ……………………………………….. 179

1Each lesson is ONE day, and ONE day is considered a 45-minute period.

Module 1: Ratios and Unit Rates

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6•1 Module Overview NYS COMMON CORE MATHEMATICS CURRICULUM

Topic D: Percent (6.RP.A.3c) ……………………………………………………………………………………………………………… 187

Lesson 24: Percent and Rates per 100 ……………………………………………………………………………………. 188

Lesson 25: A Fraction as a Percent …………………………………………………………………………………………. 197

Lesson 26: Percent of a Quantity ……………………………………………………………………………………………. 208

Lessons 27–29: Solving Percent Problems ………………………………………………………………………………. 215

End-of-Module Assessment and Rubric ……………………………………………………………………………………………… 229 Topics A through D (assessment 1 day, return 1 day, remediation or further applications 1 day)

Module 1: Ratios and Unit Rates

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6•1 Module Overview NYS COMMON CORE MATHEMATICS CURRICULUM

Grade 6 • Module 1

Ratios and Unit Rates

OVERVIEW In this module, students are introduced to the concepts of ratio and rate. Their previous experience solving problems involving multiplicative comparisons, such as Max has three times as many toy cars as Jack, (4.OA.A.2) serves as the conceptual foundation for understanding ratios as a multiplicative comparison of two or more numbers used in quantities or measurements (6.RP.A.1). Students develop fluidity in using multiple forms of ratio language and ratio notation. They construct viable arguments and communicate reasoning about ratio equivalence as they solve ratio problems in real-world contexts (6.RP.A.3). As the first topic comes to a close, students develop a precise definition of the value of a ratio 𝑎𝑎: 𝑏𝑏, where 𝑏𝑏 ≠ 0 as the value 𝑎𝑎 𝑏𝑏

, applying previous understanding of fraction as division (5.NF.B.3). They can then formalize their understanding of equivalent ratios as ratios having the same value. With the concept of ratio equivalence formally defined, students explore collections of equivalent ratios in real-world contexts in Topic B. They build ratio tables and study their additive and multiplicative structure (6.RP.A.3a). Students continue to apply reasoning to solve ratio problems while they explore representations of collections of equivalent ratios and relate those representations to the ratio table (6.RP.A.3). Building on their experience with number lines, students represent collections of equivalent ratios with a double number line model. They relate ratio tables to equations using the value of a ratio defined in Topic A. Finally, students expand their experience with the coordinate plane (5.G.A.1, 5.G.A.2) as they represent collections of equivalent ratios by plotting the pairs of values on the coordinate plane. The Mid-Module Assessment follows Topic B. In Topic C, students build further on their understanding of ratios and the value of a ratio as they come to understand that a ratio of 5 miles to 2 hours corresponds to a rate of 2.5 miles per hour, where the unit rate is the numerical part of the rate, 2.5, and miles per hour is the newly formed unit of measurement of the rate (6.RP.A.2). Students solve unit rate problems involving unit pricing, constant speed, and constant rates of work (6.RP.A.3b). They apply their understanding of rates to situations in the real world. Students determine unit prices, use measurement conversions to comparison shop, and decontextualize constant speed and work situations to determine outcomes. Students combine their new understanding of rate to connect and revisit concepts of converting among different-sized standard measurement units (5.MD.A.1). They then expand upon this background as they learn to manipulate and transform units when multiplying and dividing quantities (6.RP.A.3d). Topic C culminates as students interpret and model real-world scenarios through the use of unit rates and conversions. In the final topic of the module, students are introduced to percent and find percent of a quantity as a rate

per 100. Students understand that 𝑁𝑁 percent of a quantity has the same value as 𝑁𝑁 100

of that quantity. Students express a fraction as a percent and find a percent of a quantity in real-world contexts. Students learn to express a ratio using the language of percent and to solve percent problems by selecting from familiar representations, such as tape diagrams and double number lines or a combination of both (6.RP.A.3c). The End-of-Module Assessment follows Topic D.

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6•1 Module Overview NYS COMMON CORE MATHEMATICS CURRICULUM

Focus Standards Understand ratio concepts and use ratio reasoning to solve problems.

6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2: 1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

6.RP.A.2 Understand the concept of a unit rate 𝑎𝑎/𝑏𝑏 associated with a ratio 𝑎𝑎: 𝑏𝑏 with 𝑏𝑏 ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”2

6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Foundational Standards Use the four operations with whole numbers to solve problems.

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.3

2Expectations for unit rates in this grade are limited to non-complex fractions. 3See Glossary, Table 2.

Module 1: Ratios and Unit Rates

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Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (𝑎𝑎/𝑏𝑏 = 𝑎𝑎 ÷ 𝑏𝑏). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

Convert like measurement units within a given measurement system.

5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Graph points on the coordinate plane to solve real-world and mathematical problems.

5.G.A.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., 𝑥𝑥-axis and 𝑥𝑥-coordinate, 𝑦𝑦-axis and 𝑦𝑦-coordinate).

5.G.A.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Focus Standards for Mathematical Practice MP.1 Make sense of problems and persevere in solving them. Students make sense of and solve

real-world and mathematical ratio, rate, and percent problems using representations, such as tape diagrams, ratio tables, the coordinate plane, and double number line diagrams. They identify and explain the correspondences between the verbal descriptions and their representations and articulate how the representation depicts the relationship of the quantities in the problem. Problems include ratio problems involving the comparison of three quantities, multi-step changing ratio problems, using a given ratio to find associated ratios, and constant rate problems including two or more people or machines working together.

MP.2 Reason abstractly and quantitatively. Students solve problems by analyzing and comparing ratios and unit rates given in tables, equations, and graphs. Students decontextualize a given constant speed situation, representing symbolically the quantities involved with the formula, distance = rate × time.

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MP.5 Use appropriate tools strategically. Students become proficient using a variety of representations that are useful in reasoning with rate and ratio problems, such as tape diagrams, double line diagrams, ratio tables, a coordinate plane, and equations. They then use judgment in selecting appropriate tools as they solve ratio and rate problems.

MP.6 Attend to precision. Students define and distinguish between ratio, the value of a ratio, a unit rate, a rate unit, and a rate. Students use precise language and symbols to describe ratios and rates. Students learn and apply the precise definition of percent.

MP.7 Look for and make use of structure. Students recognize the structure of equivalent ratios in solving word problems using tape diagrams. Students identify the structure of a ratio table and use it to find missing values in the table. Students make use of the structure of division and ratios to model 5 miles/2 hours as a quantity 2.5 mph.

Terminology New or Recently Introduced Terms

 Equivalent Ratios (Two ratios 𝐴𝐴:𝐵𝐵 and 𝐶𝐶:𝐷𝐷 are equivalent ratios if there is a nonzero number 𝑐𝑐 such that 𝐶𝐶 = 𝑐𝑐𝐴𝐴 and 𝐷𝐷 = 𝑐𝑐𝐵𝐵. For example, two ratios are equivalent if they both have values that are equal.)

 Measurement of a Quantity (A measurement of a quantity is a representation of that quantity as a multiple of a unit of measurement. The multiple is a number called the measure of the quantity. Examples include 3 inches or 5 liters or 7 boys with measures 3, 5, and 7, respectively.)

 Percent (One percent is the number 1 100

and is written 1%. Percentages can be used as rates. For

example, 30% of a quantity means 30 100

times the quantity.)

 Quantity (illustration) (Examples of a quantity include a length, an area, a volume, a mass, a weight, a length of time, or a speed. It is an instance of a type of quantity.) All quantities of the same type have the properties that (1) two quantities can be compared, (2) two quantities can be combined to get a new quantity of that same type, and (3) there .always exists a quantity that is a multiple of any given quantity. These properties help define ways to measure quantities using a standard quantity called a unit of measurement.)

 Rate (illustration) (A rate is a quantity that describes a ratio relationship between two types of quantities. For example, 1.25 miles

hour is a rate that describes a ratio relationship between hours and

miles: If an object is traveling at a constant 1.25 miles hour , then after 1 hour it has gone 1.25 miles, after

2 hours it has gone 2.50 miles, after 3 hours it has gone 3.75 miles, and so on. Rates differ from ratios in how they describe ratio relationships—rates are quantities and have the properties of quantities. For example, rates of the same type can be added together to get a new rate, as in 30 miles

hour + 20 miles hour = 50 miles

hour , whereas ratios cannot be added together.)

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 Ratio (A ratio is an ordered pair of numbers which are not both zero. A ratio is denoted 𝐴𝐴:𝐵𝐵 to indicate the order of the numbers—the number 𝐴𝐴 is first and the number 𝐵𝐵 is second.)

 Ratio Relationship (A ratio relationship is the set of all ratios that are equivalent ratios. A ratio such as 5: 4 can be used to describe the ratio relationship {1: 4

5 , 5 4 : 1, 5: 4, 10: 8, 15: 12, … }. Ratio

language such as “5 miles for every 4 hours” can also be used to describe a ratio relationship. Ratio relationships are often represented by ratio tables, double number lines diagrams, and by equations and their graphs.)

 Type of Quantity (illustration) (Examples of types of quantities include lengths, areas, volumes, masses, weights, time, and (later) speeds.)

 Unit of Measurement (A unit of measurement is a choice of a quantity for a given type of quantity. Examples include 1 cm, 1 m, or 1 in. for lengths, 1 liter or 1 cm3 for volumes, etc. But the choice could be arbitrary as well, such as the length between the vertical bars: |——————|.)

 Unit Rate (When a rate is written as a measurement (i.e., a number times a unit), the unit rate is the measure (i.e., the numerical part of the measurement). For example, when the rate of speed of an object is written as the measurement 1.25 mph, the number 1.25 is the unit rate.)

 Value of a Ratio (The value of the ratio 𝐴𝐴:𝐵𝐵 is the quotient 𝐴𝐴 𝐵𝐵

as long as 𝐵𝐵 is not zero.)

Familiar Terms and Symbols4

 Convert  Coordinate Plane  Equation  Tape Diagram

Suggested Tools and Representations  Tape Diagrams (See example below.)  Double Number Line Diagrams (See example below.)  Ratio Tables (See example below.)  Coordinate Plane (See example below.)

4These are terms and symbols students have seen previously.

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Representing Equivalent Ratios for a Cake Recipe That Uses 2 Cups of Sugar for Every 3 Cups of Flour

Coordinate Plane

Flour Sugar

2

4

6

3

6

9

Ratio Table

Tape Diagram

0 2 4 6

0 3 6 9

Sugar

Flour

Double Number Line

2

Sugar

Flour

3

2: 3, 4: 6, 6: 9

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