6
G R A D E
1
GRADE 6 • MODULE 1
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
1 Each lesson is ONE day, and ONE day is considered a 45-minute period.
Module 1: Ratios and Unit Rates
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NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 6•1
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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NYS COMMON CORE MATHEMATICS CURRICULUM
Module Overview 6•1
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 𝑎: 𝑏 , where 𝑏 ≠ 0 as the value 𝑎
, applying previous understanding of fraction as division (5.NF.B.3). They can then formalize their 𝑏 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
𝑁 per 100.
Students understand that 𝑁 percent of a quantity has the same value as
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.
Module 1: Ratios and Unit Rates
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NYS COMMON CORE MATHEMATICS CURRICULUM
Module Overview 6•1
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 𝑎/𝑏 associated with a ratio 𝑎: 𝑏 with 𝑏 ≠ 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.
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
2 Expectations for unit rates in this grade are limited to non-complex fractions.
3 See Glossary, Table 2.
Module 1: Ratios and Unit Rates
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NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 6•1
5.NF.B.3
Interpret a fraction as division of the numerator by the denominator ( 𝑎/𝑏 = 𝑎 ÷ 𝑏 ). 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?
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.
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., 𝑥 -axis and 𝑥 -coordinate, 𝑦 -axis and 𝑦 -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.
MP.1
MP.2
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.
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 .
Module 1: Ratios and Unit Rates
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NYS COMMON CORE MATHEMATICS CURRICULUM
MP.5
Module Overview 6•1
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 .
 Equivalent Ratios (Two ratios 𝐴: 𝐵 and 𝐶: 𝐷 are equivalent ratios if there is a nonzero number 𝑐 such that 𝐶 = 𝑐𝐴 and 𝐷 = 𝑐𝐵 . 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
30 example, 30% of a quantity means
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.)
Module 1: Ratios and Unit Rates
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NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 6•1
 Ratio (A ratio is an ordered pair of numbers which are not both zero. A ratio is denoted 𝐴: 𝐵 to indicate the order of the numbers—the number 𝐴 is first and the number 𝐵 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 cm
3
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 𝐴: 𝐵 is the quotient
𝐴
𝐵
as long as 𝐵 is not zero.)
4
 Convert
 Coordinate Plane
 Equation
 Tape Diagram
 Tape Diagrams (See example below.)
 Double Number Line Diagrams (See example below.)
 Ratio Tables (See example below.)
 Coordinate Plane (See example below.)
4 These are terms and symbols students have seen previously.
Module 1: Ratios and Unit Rates
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NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview
Representing Equivalent Ratios for a Cake Recipe
That Uses 2 Cups of Sugar for Every 3 Cups of Flour
2
3
Tape Diagram
Ratio Table
Sugar
2
4
6
Flour
3
6
9
Sugar
2: 3 , 4: 6 , 6: 9
Flour
Sugar
Flour
Double Number Line
0 2 4 6
0 3 6 9
Coordinate Plane
6•1
Module 1: Ratios and Unit Rates
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NYS COMMON CORE MATHEMATICS CURRICULUM
Module Overview 6•1
Preparation of lessons will be more effective and efficient if there has been an adequate analysis of the module first. Each module in A Story of Ratios can be compared to a chapter in a book. How is the module moving the plot, the mathematics, forward? What new learning is taking place? How are the topics and objectives building on one another? The following is a suggested process for preparing to teach a module.
Step 1: Get a preview of the plot.
A: Read the Table of Contents. At a high level, what is the plot of the module? How does the story develop across the topics?
B: Preview the module’s Exit Tickets to see the trajectory of the module’s mathematics and the nature of the work students are expected to be able to do.
Note: When studying a PDF file, enter “Exit Ticket” into the search feature to navigate from one Exit
Ticket to the next.
Step 2: Dig into the details.
A: Dig into a careful reading of the Module Overview. While reading the narrative, liberally reference the lessons and Topic Overviews to clarify the meaning of the text—the lessons demonstrate the strategies, show how to use the models, clarify vocabulary, and build understanding of concepts.
B: Having thoroughly investigated the Module Overview, read through the Student Outcomes of each lesson (in order) to further discern the plot of the module. How do the topics flow and tell a coherent story? How do the outcomes move students to new understandings?
Step 3: Summarize the story.
Complete the Mid- and End-of-Module Assessments. Use the strategies and models presented in the module to explain the thinking involved. Again, liberally reference the lessons to anticipate how students who are learning with the curriculum might respond.
Module 1: Ratios and Unit Rates
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NYS COMMON CORE MATHEMATICS CURRICULUM
Module Overview 6•1
A three-step process is suggested to prepare a lesson. It is understood that at times teachers may need to make adjustments (customizations) to lessons to fit the time constraints and unique needs of their students.
The recommended planning process is outlined below. Note: The ladder of Step 2 is a metaphor for the teaching sequence. The sequence can be seen not only at the macro level in the role that this lesson plays in the overall story, but also at the lesson level, where each rung in the ladder represents the next step in understanding or the next skill needed to reach the objective. To reach the objective, or the top of the ladder, all students must be able to access the first rung and each successive rung.
Step 1: Discern the plot.
A: Briefly review the module’s Table of Contents, recalling the overall story of the module and analyzing the role of this lesson in the module.
B: Read the Topic Overview related to the lesson, and then review the Student Outcome(s) and Exit
Ticket of each lesson in the topic.
C: Review the assessment following the topic, keeping in mind that assessments can be found midway through the module and at the end of the module.
Step 2: Find the ladder .
A: Work through the lesson, answering and completing each question, example, exercise, and challenge.
B: Analyze and write notes on the new complexities or new concepts introduced with each question or problem posed; these notes on the sequence of new complexities and concepts are the rungs of the ladder.
C: Anticipate where students might struggle, and write a note about the potential cause of the struggle.
D: Answer the Closing questions, always anticipating how students will respond.
Step 3: Hone the lesson.
Lessons may need to be customized if the class period is not long enough to do all of what is presented and/or if students lack prerequisite skills and understanding to move through the entire lesson in the time allotted. A suggestion for customizing the lesson is to first decide upon and designate each question, example, exercise, or challenge as either “Must Do” or “Could Do.”
A: Select “Must Do” dialogue, questions, and problems that meet the Student Outcome(s) while still providing a coherent experience for students; reference the ladder. The expectation should be that the majority of the class will be able to complete the “Must Do” portions of the lesson within the allocated time. While choosing the “Must Do” portions of the lesson, keep in mind the need for a balance of dialogue and conceptual questioning, application problems, and abstract problems, and a balance between students using pictorial/graphical representations and abstract representations.
Highlight dialogue to be included in the delivery of instruction so that students have a chance to articulate and consolidate understanding as they move through the lesson.
Module 1: Ratios and Unit Rates
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NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview 6•1
B: “Must Do” portions might also include remedial work as necessary for the whole class, a small group, or individual students. Depending on the anticipated difficulties, the remedial work might take on different forms as suggested in the chart below.
Anticipated Difficulty “Must Do” Remedial Problem Suggestion
The first problem of the lesson is too challenging.
There is too big of a jump in complexity between two problems.
Write a short sequence of problems on the board that provides a ladder to Problem 1. Direct students to complete those first problems to empower them to begin the lesson.
Provide a problem or set of problems that bridge student understanding from one problem to the next.
Students lack fluency or foundational skills necessary for the lesson.
More work is needed at the concrete or pictorial level.
Before beginning the lesson, do a quick, engaging fluency exercise, such as a Rapid White Board Exchange or Sprint.
Before beginning any fluency activity for the first time, assess that students have conceptual understanding of the problems in the set and that they are poised for success with the easiest problem in the set.
Provide manipulatives or the opportunity to draw solution strategies.
More work is needed at the abstract level.
Add a White Board Exchange of abstract problems to be completed toward the end of the lesson.
C: “Could Do” problems are for students who work with greater fluency and understanding and can, therefore, complete more work within a given time frame.
D: At times, a particularly complex problem might be designated as a “Challenge!” problem to provide to advanced students. Consider creating the opportunity for students to share their “Challenge!” solutions with the class at a weekly session or on video.
E: If the lesson is customized, be sure to carefully select Closing questions that reflect such decisions and adjust the Exit Ticket if necessary.
Mid-Module
Assessment Task
After Topic B Constructed response with rubric
6.RP.A.1, 6.RP.A.3 (Stem
Only), 6.RP.A.3a
End-of-Module
Assessment Task
After Topic D Constructed response with rubric
6.RP.A.1, 6.RP.A.2,
6.RP.A.3
Module 1: Ratios and Unit Rates
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