Lead 6–9 IST and Supplemental Lesson Materials

Lead Facilitating Successful Implementation of Eureka Math2TM 6–9 A Story of Ratios® A Story of Functions® Supplemental Materials

Supplemental Materials

Lead: Facilitating Successful Implementation of Eureka Math2, 6–9

Contents Exploring the Strategic Curriculum Design .................................................................................................. 3 Module, Topic: Grade 6 Module 1 Topic A .............................................................................................. 3 Lesson: Grade 6 Module 1 Lesson 2....................................................................................................... 16 Lesson: Algebra I Module 4 Lesson 1 34 Supporting Teacher Preparation................................................................................................................ 57 Lesson: Grade 7 Module 3 Lesson 7....................................................................................................... 57 Topic: Grade 7 Module 3 Topic B ........................................................................................................... 75 Module: Grade 7 Module 3 .................................................................................................................... 77 Credits ........................................................................................................................................................ 81 Works Cited ............................................................................................................................................... 81 Appendix A ................................................................................................................................................. 82 Implementation Support Tool (IST) ....................................................................................................... 82 Appendix B ................................................................................................................................................. 90 Scenarios ................................................................................................................................................ 90 Appendix C ................................................................................................................................................. 92 Implementation Benchmarks................................................................................................................. 92

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A Story of Ratios®

Ratios and Rates ▸ 6 TEACH Module

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Ratios, Rates, and Percents

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Operations with Fractions and Multi-Digit Numbers

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Rational Numbers

4

Expressions and One-Step Equations

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Area, Surface Area, and Volume

6

Statistics

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Before This Module

Overview

Grade 4 Module 2

Ratios, Rates, and Percents

Grade 5 Module 6 In grade 4, students solve problems involving multiplicative comparisons, such as Blake has 4 times as many stickers as Adesh. This prior work provides a foundation for students’ understanding of ratios as multiplicative comparisons of two numbers.

In grade 5, students work with the first quadrant of the coordinate plane as they plot points to represent ordered pairs of numbers.

Topic A Ratios This topic introduces students to ratios and ratio notation. Students use tape diagrams to model ratios and solve problems. They explore different ways to group and compare objects to develop an understanding of equivalent ratios by the end of the topic.

Number of Roses Number of Daisies

Topic B Collections of Equivalent Ratios Topic B defines sets of all ratios that are equivalent ratios as ratio relationships. Students represent ratio relationships by using ratio tables, double number lines, and points in the coordinate plane. They use these models and the addition and multiplication patterns in the ratio relationship to solve for unknown quantities.

0

1

2

3

4

5

0

4

8

12

16

20

Number of Packets of Sugar Number of Grams of Sugar

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Topic C

After This Module

Comparing Ratio Relationships In this topic, students compare ratio relationships in context by using ratios to answer questions such as Which lemonade should have a stronger lemon flavor? Students use a variety of strategies to compare ratio relationships, including making direct comparisons by using a ratio table, by creating equivalent ratios, and by calculating the value of the ratio.

Topic D Rates In topic D, students develop an understanding of the rates associated with ratio relationships. They calculate unit rates and use them to solve problems involving speed, unit pricing, measurement conversions, and other real-world rate applications.

Grade 7 Modules 1 and 5 In grade 7 module 1, students extend their understanding of ratios and rates to proportional relationships. They recognize the constant of proportionality as the unit rate of a relationship. They identify, compare, and solve problems involving proportional relationships represented in graphs, tables, equations, and verbal descriptions.

In grade 7 module 5, students apply their foundational understanding of percents to a variety of other real-world contexts, including percent increase and decrease, percent error, discounts, tax, and commission.

Topic E Percents This topic introduces percents. Students understand a percent as a fraction with a denominator of 100, and they apply their ratio and rate reasoning from previous topics to solve percent problems. Students use double number lines, mental math, and other computational strategies to solve for the unknown percent, part, or whole.

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Contents Ratios, Rates, and Percents Why. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Achievement Descriptors: Overview. . . . . . . . . . . . . . . . . . . . . 10 Topic A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Ratios Lesson 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Jars of Jelly Beans • Use multiplicative reasoning to estimate the solution to a real-world problem.

Lesson 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Introduction to Ratios

Topic B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Collections of Equivalent Ratios Lesson 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Ratio Tables and Double Number Lines • Represent equivalent ratios by using ratio tables and double number lines.

• Use representations of ratio relationships to solve problems.

Lesson 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Graphs of Ratio Relationships • Plot points in the coordinate plane that each represent a ratio.

• Identify characteristics of graphs, tables, and double number lines representing ratio relationships.

• Write ratios that relate two quantities as an ordered pair of numbers.

• Use ratio language to compare two quantities.

Lesson 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Lesson 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

• Use addition patterns in tables and graphs of equivalent ratios to describe ratio relationships and find unknown quantities.

Ratios and Tape Diagrams

Addition Patterns in Ratio Relationships

• Write multiple ratios to describe the same situation.

• Represent ratios with tape diagrams.

Lesson 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Lesson 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

• Use graphs and tables to explore multiplication patterns in ratio relationships.

• Use multiplication to complete ratio tables.

Exploring Ratios by Making Batches • Create ratios by making batches of different quantities.

• Use tape diagrams to determine unknown quantities in ratios.

Multiplication Patterns in Ratio Relationships

Lesson 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Multiplicative Reasoning in Ratio Relationships

Lesson 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

• Write and use equivalent ratios when one of the numbers in the ratio is 1.

Equivalent Ratios

Lesson 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

• Find equivalent ratios by multiplying both numbers in a given ratio by the same nonzero number.

• Use equivalent ratios to find unknown quantities.

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Applications of Ratio Reasoning • Solve multi-step ratio problems by reasoning about equivalent ratios.

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Topic C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 Comparing Ratio Relationships Lesson 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Multiple Ratio Relationships

Lesson 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 Solving Rate Problems • Apply rate reasoning to solve real-world ratio problems involving speed, unit pricing, and unit conversions.

• Find an unknown quantity when given a rate and a known quantity.

• Compare ratio relationships by using graphs, tables, and double number lines.

Lesson 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

Lesson 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

Solving Multi-Step Rate Problems

Comparing Ratio Relationships, Part 1 • Compare ratio relationships by using ratio tables.

Lesson 14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Comparing Ratio Relationships, Part 2 • Compare ratio relationships by creating equivalent ratios.

Lesson 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 The Value of the Ratio • Compare ratio relationships by using the value of the ratio.

Topic D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 Rates Lesson 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 Speed • Find distance and time corresponding to a given speed.

• Identify real-world examples of rates and interpret their meanings in context.

Lesson 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 Rates • Identify rates and unit rates.

• Calculate one quantity when given another quantity and a constant rate.

Lesson 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Comparing Rates • Compare rates with like units of measurement by using unit rate.

• Solve problems involving multiple constant rates.

Topic E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 Percents Lesson 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 Introduction to Percents • Relate percents to a part-to-whole relationship where the whole is 100.

• Model percents and write percents in fraction and decimal forms.

Lesson 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 Finding the Percent • Calculate a percent when given a part and the whole.

• Discover that if multiple parts make a whole, then the percents representing the parts should total 100%.

Lesson 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 Finding a Part • Calculate a part when given the whole and a percent.

Lesson 25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 Finding the Whole • Calculate the whole when given a part and a percent.

Lesson 26. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 Solving Percent Problems • Solve multi-step percent problems.

Lesson 19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 Using Rates to Convert Units • Convert units of measurement by applying rate reasoning.

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6 ▸ M1

EUREKA MATH2

Resources Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 Achievement Descriptors: Proficiency Indicators. . . . . . . . . . . . . . . 496 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Math Past . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 Fluency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 Sample Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 Works Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 Credits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

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Why Ratios, Rates, and Percents Ratios are shown as A : B and A to B. Why not BA ?

In this module, a ratio is defined as an ordered pair of numbers that are not both zero.

Because a ratio is a pair of numbers, not a single quantity, it should only be written by using the notation A : B or A to B, which indicates this pair of numbers.

In addition, writing a ratio as a single rational number BA may give students the misconception that they can do arithmetic with ratios. However, computation with ratios and rational numbers is not the same. For example, if the ratio of the number of parts red paint to the number of parts blue paint in a mixture is 1 : 1, and it is combined with a different mixture in which the corresponding quantities are in a ratio of 1 : 2, what is the ratio of the new mixture? It is potentially 2 : 3, but it could be 3 : 4, 4 : 5, or another ratio, depending on the quantities used in each initial mixture. If students write the original ratios as 1 and 12 , however, they may be tempted to find the sum and assume that the new ratio 1

Consider the collection of shapes. a. There are 2 times as many blue circles as red circles. b. A ratio that relates the number of blue circles to the number of red circles is 4 : 2 . c. For every 4 blue circles, there are 2 red circles.

is 23 , or 3 : 2. Writing ratios as ordered pairs of numbers in the forms A : B and A to B makes it less likely that students will be confused about the properties of ratios.

The fraction BA can be used to denote the value of the ratio, which is the first number in the ratio when the second number is 1.

Why are there so many different representations in this module?

The pictorial representations in this module help students visualize the relationships between quantities in ratios, rates, and percents. These representations include tape diagrams, double number lines, tables, and graphs—all familiar tools that students continue to work with in future grades. By becoming proficient with these representations, students have multiple strategies to approach and represent a problem.

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Number of Peaches Number of Kiwis

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• Lesson 2 introduces tape diagrams as tools for representing ratios with quantities that have the same units and showing the multiplicative relationship between those quantities. The diagrams have two tapes with equal-size units.

They serve as an important tool for writing and understanding equivalent ratios, determining unknown quantities in a ratio relationship, and solving problems involving changing ratio relationships.

Number of Inches of Water

Number of Inches of Snow

1

10

2

20

3

30

• Lesson 6 introduces double number lines as tools for representing ratio relationships. These diagrams are useful for modeling situations involving different units such as distance and time.

• Lesson 6 also introduces ratio tables as tools to represent sets of equivalent ratios. This builds on prior work with tables such as measurement conversion tables in elementary grades. Students use the addition and multiplication patterns from ratio tables to write equivalent ratios, solve problems, and compare ratio relationships in context.

Number of Slices of Cheese

• Lesson 7 introduces graphs in the coordinate plane as tools to represent ratio relationships. Students make the informal observation that points representing a ratio relationship lie on the same line that passes through the origin. This learning provides a useful review of graphing in the coordinate plane before students encounter it again in module 4.

Grilled Cheese Sandwiches

y 11 10 9 8 7 6 5 4 3 2 1 0

1

2

3

4

5

6

7

8

9 10 11

x

Number of Slices of Bread

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Why aren’t students writing equations in this module?

Because module 1 is the first time that students are introduced to ratios and rates, the lessons focus on the relationships between quantities in a set of equivalent ratios. Although students do create tables and graphs, they are not asked to abstract these relationships by identifying the variables or writing an equation. This choice is intentional to preserve the focus on ratio and rate reasoning and intuitive observation of patterns rather than on the mechanics of writing an equation and substituting values into a formula.

In addition, because of this module’s focus on relationships between quantities, the rate lessons in topic D avoid expressing rates by using derived units, such as 60 miles , or as hour fractions, such as

60 miles .

1 hour

Rather, grade 6 students write this rate as 60 miles per hour and

×

Number of Bracelets

Total Cost (dollars)

4

64

1

16

7

112

1 4

×1 4

× 16

interpret it to mean that an object travels 60 miles in 1 hour. Over several examples, students observe that the units of rates are composed of two different types of quantities, such as miles and hours, and the learning focuses on understanding the meaning of rates and their units. Derived units are not necessary for students to demonstrate mastery of the grade 6 standards involving rates. Derived units are introduced in high school, when students are better prepared to understand and compute with rates as single quantities.

In module 4, after students work with single-variable equations and their solutions, students revisit ratio relationships in tables and in the coordinate plane. They define independent variable and dependent variable and explore how to write equations to model ratio relationships when given a table or a graph. This sequencing allows for the major work of grade 6—ratios and equations—to be thoughtfully spaced out and revisited throughout the school year.

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Achievement Descriptors: Overview Ratios, Rates, and Percents Achievement Descriptors (ADs) are standards-aligned descriptions that detail what students should know and be able to do based on the instruction. ADs are written by using portions of various standards to form a clear, concise description of the work covered in each module.

Each module has its own set of ADs, and the number of ADs varies by module. Taken together, the sets of module-level ADs describe what students should accomplish by the end of the year.

ADs and their proficiency indicators support teachers with interpreting student work on • informal classroom observations, • data from other lesson-embedded formative assessments, • Exit Tickets, • Topic Quizzes, and • Module Assessments.

This module contains the nine ADs listed.

6.Mod1.AD1 Write and explain ratios that describe

relationships between two quantities.

6.RP.A.1

6.Mod1.AD2 Write and explain the unit rate that describes a

relationship between two quantities.

6.RP.A.2

6.Mod1.AD3 Solve real-world and mathematical

problems by using ratio reasoning.

6.RP.A.3

6.Mod1.AD4 Represent ratio relationships by using

tables and the coordinate plane.

6.RP.A.3.a

6.Mod1.AD5 Compare ratio relationships by using

various representations.

6.RP.A.3.a

6.Mod1.AD6 Solve real-world problems by using

unit rates.

6.RP.A.3.b

6.Mod1.AD7 Model and explain percents and

problems involving percents.

6.RP.A.3.c

6.Mod1.AD8 Solve problems that involve finding the

part, whole, or percent.

6.RP.A.3.c

6.Mod1.AD9 Convert among units by using ratio

reasoning to solve problems.

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6.RP.A.3.d

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The first page of each lesson identifies the ADs aligned with that lesson. Each AD may have up to three indicators, each aligned to a proficiency category (i.e., Partially Proficient, Proficient, Highly Proficient). While every AD has an indicator to describe Proficient performance, only select ADs have an indicator for Partially 2 Proficient and/or Highly Proficient performance.

EUREKA MATH

ADs have the following parts: • AD Code: The code indicates the grade level and the module number and then lists the ADs in no particular order. For example, the first AD for grade 6 module 1 is coded as 6.Mod1.AD1.

• AD Language: The language is crafted from standards and 6 ▸ M1 concisely describes what will be assessed.

An example of one of these ADs, along with its proficiency • AD Indicators: The indicators describe the precise expectations indicators, is shown here for reference.

The complete set whole, of this or percent.

6.Mod1.AD8 Solve problems that involve finding the part, of the AD for the given proficiency category. module’s ADs with proficiency indicators can be found in the RELATED CCSSM • Related Standard: This identifies the standard or parts of standards Achievement Descriptors: Proficiency Indicators resource.

30 times the quantity); solve problems involving finding the whole, given a part 6.RP.A.3.c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 100 from the Common Core State Standards that the AD addresses. and the percent.

Partially Proficient

AD Code: Grade.Mod#.AD#

Proficient Solve problems that involve finding the part, whole, or percent.

Sana has 7 mystery novels. She says that 35% of her novels are mystery novels. What is the total number of novels Sana has?

Highly Proficient

AD Language

6.Mod1.AD9 Convert among units by using ratio reasoning to solve problems.

Related Standard

RELATED CCSSM

6.RP.A.3.d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Partially Proficient

Proficient

Highly Proficient

Convert among nonmixed units by using ratio reasoning.

Convert among units, including mixed units such as 4 feet 3 inches, to solve problems.

Solve problems involving conversion of units within ratios and rates.

Convert.

Lisa’s height is 5 feet 8 inches. What is Lisa’s height in centimeters?

Yuna runs 750 meters in 5 minutes. What is Yuna’s speed in kilometers per hour?

10 in = _____cm

AD Indicators

(1 in = 2.54 cm)

11

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Topic A Ratios Topic A introduces students to ratios. In the elementary grades, students learn to recognize number patterns, make equal groups, write multiplicative comparisons, and work with fractions. Students apply these skills in grade 6 as they recognize part-to-part and part-to-whole relationships, write ratios, and explain the meaning of ratios.

Lesson 1 presents a modeling task where students may use a variety of strategies to estimate the number of jelly beans in different-size jars.

This task prepares students for many of the ideas presented in module 1, including multiplicative relationships, ratios, rates, and measurement conversions. In the next lesson, students explore situations where multiplicative comparison language, such as “The girl has 7 times as many 5 tokens as the boy,” is not efficient or practical for describing the relationship between two quantities. Students develop an understanding of ratio language, such as “For every 7 tokens the girl has, the boy has 5 tokens,” and learn the formal definition of ratio. Students use precise ratio notation and ratio language to describe situations represented pictorially and verbally.

Students use a familiar tool, the tape diagram, to represent ratios.

As the topic progresses, students recognize that when the ratio of the number of roses to the number of daisies is 6 : 9, there are 2 roses for every 3 daisies. A digital exploration allows students to identify patterns between equivalent ratios and tape diagrams that represent equivalent ratios. At the end of the topic, the term equivalent ratios is formally defined, and students use equivalent ratio reasoning and tape diagrams to determine unknown quantities.

3

3

3

3

3

Number of Roses Number of Daisies

Students apply what they learn about ratio language and equivalent ratios in topic A to a variety of situations throughout the remainder of the module. In topic B, students represent collections of equivalent ratios in ratio tables, double number lines, and graphs and use these representations to solve problems. Later in the module, students use ratio reasoning to compare ratio relationships and solve rate and percent problems.

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EUREKA MATH2 6 ▸ M1 ▸ TA

Progression of Lessons Lesson 1

Jars of Jelly Beans

Lesson 2

Introduction to Ratios

Lesson 3

Ratios and Tape Diagrams

Lesson 4

Exploring Ratios by Making Batches

Lesson 5

Equivalent Ratios

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2

LESSON 2

Introduction to Ratios Write ratios that relate two quantities as an ordered pair of numbers.

Use ratio language to compare two quantities.

EUREKA MATH2

6 ▸ M1 ▸ TA ▸ Lesson 2

EXIT TICKET Name

Date

2

Consider the cans of blue paint and the cans of red paint.

Blue

Blue

Blue

Blue

Blue

Red

Red

Red

Red

Lesson at a Glance In this lesson, students discuss their reactions to a video, which introduces the need to use new language to compare two quantities.

After discussing a few examples, students realize that there are situations in which comparing quantities by using multiplicative comparison language is impractical or does not make sense.

Through teacher-led instruction and peer discussion, students learn how to write ratios and how to use ratio language to describe the relationship between two quantities. This lesson introduces the term ratio.

Key Questions

For parts (a)–(d), fill in the blank. a. A ratio that relates the number of cans of blue paint to the number of cans of red paint is

5:4 .

• What is a ratio?

b. A ratio that relates the number of cans of red paint to the number of cans of blue paint is

4:5 .

• When is it more practical to use ratio language instead of multiplicative comparison language?

c. There are d. For every

5 4

5

times as many cans of blue paint as cans of red paint. cans of blue paint, there are

4

cans of red paint.

Achievement Descriptor 6.Mod1.AD1 Write and explain ratios that describe relationships

between two quantities. (6.RP.A.1)

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EUREKA MATH2 6 ▸ M1 ▸ TA ▸ Lesson 2

Agenda

Materials

Fluency

Teacher

Launch Learn

5 min

30 min

• None

Students

• A New Language

• None

• From Tokens to Tea

Lesson Preparation

Land

• None

10 min

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EUREKA MATH2

6 ▸ M1 ▸ TA ▸ Lesson 2

Fluency Multiplicative Comparisons Students multiply or divide by using multiplicative comparisons to prepare for working with ratio relationships.

Directions: Answer each question.

1.

What number is twice as large as 10?

20

2.

10 is twice as large as what number?

5

3.

What number is 4 times as large as 7?

28

4.

40 is 4 times as large as what number?

10

5.

What number is 6 times as large as 11?

66

6.

600 is 6 times as large as what number?

100

7.

What number is 10 times as large as

?

5

8.

1 is 10 times as large as what number?

1 10

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1 2

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EUREKA MATH2 6 ▸ M1 ▸ TA ▸ Lesson 2

Launch

5

Students use multiplicative comparison language to describe the relationship between quantities.

Play part 1 of the Unfair Tokens video, which shows an adult giving two children different numbers of carnival tokens. Then facilitate a class discussion to elicit students’ reactions to the video.

What is the video about?

The video is about an adult giving cups of tokens to two children at a carnival. The girl gets 2 cups of tokens, and the boy gets only 1 cup of tokens. The boy is sad because the girl gets more tokens.

What might the boy be thinking?

It’s not fair that the girl gets more tokens than he gets.

It’s not fair that the girl gets twice as many tokens as he gets.

It’s not fair that he gets half as many tokens as the girl gets.

If students do not use multiplicative comparison language such as “twice as many” or “two times as many,” use the following prompt.

How can we compare the number of tokens the girl receives to the number of tokens the boy receives?

The girl receives twice as many tokens as the boy.

The boy receives half as many tokens as the girl.

Play part 2 of the Unfair Tokens video, which shows the adult giving more tokens to the boy but still results in the children receiving different numbers of tokens. Then ask the following questions.

Teacher Note In grade 5, students used multiplicative comparison language to describe relationships between numbers. For example, 2 yards is 3 twice as much as 1 yard, and 4 yards is 3 3 4 times as much as 1 yard.

3

Why is the boy still sad?

He is still sad because the girl still has more tokens than he does.

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6 ▸ M1 ▸ TA ▸ Lesson 2

EUREKA MATH2

In the first part of the video, the girl receives twice as many tokens as the boy. How can we compare the number of tokens the girl receives to the number of tokens the boy receives in the second part of the video?

The girl receives about 1 3 cups of tokens, and the boy receives about 1 1 cups of tokens.

4 4 The girl doesn’t get twice as many tokens as the boy, but she still gets more tokens than the boy.

Help students recall that after watching the first part of the video, they used multiplicative language to compare the number of tokens each child receives when the girl gets twice as many tokens as the boy. Ask students whether they can use multiplicative language to compare the numbers of tokens the girl and boy receive in the second part of the video.

Anticipate that most students will not give an answer or will find it difficult to express a comparison of the numbers by using multiplicative language.

Today, we will learn new language that we can use, instead of multiplicative comparison language, to compare quantities.

Learn A New Language Students write ratios and use multiplicative comparison language and ratio language to describe relationships between quantities.

Have students think back to part 1 of the Unfair Tokens video. Display the picture of three cups.

UDL: Action and Expression To support students in expressing learning in flexible ways, consider providing students with access to manipulatives, such as cups with marbles or coins, to help them visualize the relationship between the numbers of tokens the girl and boy get.

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EUREKA MATH2 6 ▸ M1 ▸ TA ▸ Lesson 2

Ask the following question.

Suppose each cup has 8 tokens in it. How many tokens would each child have?

The girl would have 16 tokens, and the boy would have 8 tokens.

Ask students to choose a different number of tokens that a cup might hold. Then have them determine the number of tokens each child would have. Have students write their examples on a personal whiteboard and hold it up for you to see. Write the word ratios on the board and record several students’ answers by using ratio notation as shown.

ratios 16 : 8 20 : 10 12 : 6 These pairs of numbers are called ratios. They relate the number of tokens the girl receives to the number of tokens the boy receives.

Point to the ratio 16 : 8.

We say this ratio as “16 to 8.” We can write a ratio by using a colon between the numbers or the word to.

What can we multiply the second number in each ratio by to get the first number in each ratio? Why?

Teacher Note Avoid using fraction notation such as A to B represent ratios because computations for ratios and fractions are not the same. For example, consider a batch of green paint that is 1 part yellow paint and 3 parts blue paint.

The ratio of the number of parts yellow paint to the number of parts blue paint is 1 to 3. In two batches of green paint, the ratio of the number of parts yellow paint to the number of parts blue paint is 2 to 6. Using fraction notation, students might conclude that 2 is 6 twice as large as 1 .

3

We can multiply the second number in each ratio by 2 to get the first number in each ratio because the first number in each ratio is twice as much as the second number in each ratio.

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EUREKA MATH2

6 ▸ M1 ▸ TA ▸ Lesson 2

Next, have students think back to part 2 of the Unfair Tokens video. Display the picture of a cup that has 8 tokens in it.

Language Support Have students think–pair–share about the following question. If necessary, replay part 2 of the video.

Suppose that each cup has 8 tokens in it. How many tokens would each child have if the girl receives 1 3 cups of tokens and the boy receives 1 1 cups of tokens?

4

4

The girl would have 14 tokens, and the boy would have 10 tokens.

Ask students to choose a different number of tokens that a cup might hold. Then have them determine the number of tokens each child would have if the girl still has 1 3 cups of tokens 4 and the boy still has 1 1 cups of tokens. Have students write their examples as ratios on a 4 personal whiteboard and hold it up for you to see. Write the word ratios on the board and record several students’ answers by using ratio notation as shown.

ratios 14 : 10 21 : 15 7:5

To support understanding of multiplicative comparison language, ratio language, and the word ratio, consider making an anchor chart showing a visual of sorted shapes, paired with the following color-coded sentence frames.

• There are as red circles.

times as many blue circles

• A ratio that relates the number of blue circles to the number of red circles is : .

• For every blue circles, there are red circles.

The term ratio is formally defined at the end of Learn.

What patterns do you notice about the first numbers in the ratios? What patterns do you notice about the second numbers in the ratios?

The first numbers are multiples of 7, and the second numbers are multiples of 5.

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EUREKA MATH2 6 ▸ M1 ▸ TA ▸ Lesson 2

Have students think–pair–share about the following question.

When we discussed the situation where the girl gets twice as many tokens as the boy, we said we could multiply the second number in each ratio by 2 to get the first number in each ratio. In this example, what can we multiply the second number in each ratio by to get the first number in each ratio?

We can multiply the second number in each ratio by in each ratio.

7 5

to get the first number

If students do not say that they can multiply the second number in each ratio by 7 to get 5 the first number in each ratio, suggest that they try it. Then ask them whether that worked and use the following prompts to guide the discussion.

When we multiply the second number in each ratio by 2 to get the first number in each ratio, we can compare the numbers by saying that the first number in each ratio is 2 times, or twice, as much as the second number in each ratio.

Differentiation: Support To support students with multiplying fractions, consider modeling the multiplication for one of the ratios as shown.

Have students do the multiplication for the other ratios.

10 ×

7 5

= =

10 1 70 5

×

7 5

= 14

We can multiply the second number in each ratio by 7 to get the first number in each 5 ratio. So how can we compare the numbers by using multiplicative language?

The first number in each ratio is

7 5

times as much as the second number in each ratio.

How can we use multiplicative language to compare the number of tokens the girl receives to the number of tokens the boy receives in the second part of the video?

The girl receives

7 5

times as many tokens as the boy.

Point out that it is difficult, but not impossible, to talk about fractional multiplicative comparisons. Then ask the following question.

Can we use better language to describe the relationship between the numbers of tokens the children receive? Turn and talk to your partner.

Invite students to share their suggestions. If no students use ratio language to describe the ratios, use the following prompt.

Finish this sentence: For every 7 tokens the girl receives,

.

the boy receives 5 tokens

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EUREKA MATH2

6 ▸ M1 ▸ TA ▸ Lesson 2

For every 7 tokens the girl receives, the boy receives 5 tokens. We call this ratio language. Why do you think we call it that?

Teacher Note

We call it that because it describes a ratio.

Ratio language describes the relationship between the quantities in a ratio. The phrase “7 tokens” is an example of a quantity, because it includes both a number, 7, and a unit, tokens. A ratio only includes numbers, which is why it is written as 7 : 5 and not as 7 tokens : 5 tokens.

Can you use ratio language to describe the relationship between the quantities when the girl receives twice as many tokens as the boy? How? Yes. For every 2 tokens the girl receives, the boy receives 1 token.

Direct students to problem 1. Allow them to work on the problem individually or in pairs.

1. Lisa has 9 tokens. Toby has 13 tokens. Which statements describe the relationship between the two quantities? Choose all that apply.

A. Lisa has B. Lisa has

9 13 13 9

A ratio A : B is an ordered pair of numbers that relates two quantities. A ratio, such as 3 : 2, only tells us two numbers. It does not tell us which two quantities the ratio relates. A quantity can be a discrete count of objects, such as the number of apples, or a measurement of an object, such as the measure in inches of a segment. In other words, “3” is a number, but “3 apples” or “3 inches” is a quantity. Examples of a quantity include a length, a volume, a weight, and a length of time. Model with students the precise use of the terms number and quantity.

as many tokens as Toby. times as many tokens as Toby.

C. A ratio that relates the number of tokens Lisa has to the number of tokens Toby has is 9 : 13.

D. A ratio that relates the number of tokens Lisa has to the number of tokens Toby has is 13 : 9.

E. For every 9 tokens Lisa has, Toby has 13 tokens.

F. For every 9 tokens Toby has, Lisa has 13 tokens.

Promoting the Standards for Mathematical Practice When students use ratios to accurately describe a real-world context, they are reasoning quantitatively and abstractly (MP2).

Ask the following questions to promote MP2: • What does the ratio 9 : 13 mean in the context of tokens?

• What real-world situations are modeled by ratios?

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EUREKA MATH2 6 ▸ M1 ▸ TA ▸ Lesson 2

When most students have finished, display problem 1 and its answer choices.

Discuss each answer choice one at a time. Ask students whether they think the answer choice does or does not describe the relationship between the two quantities and why. Use the following reasoning to supplement students’ understanding: • A is correct. We can multiply the number of tokens Toby has, 13, by of tokens Lisa has.

9 13

to get 9, the number

• B is incorrect. If we multiply the number of tokens Toby has, 13, by 13 , we do not get the 9 number of tokens Lisa has.

UDL: Representation

• C is correct because Lisa has 9 tokens and Toby has 13 tokens.

Consider presenting the information in another format by providing students with real objects for reference. For the tea situation, show students an actual packet of sugar and an 8-ounce cup or pictures of a packet of sugar and an 8-ounce cup.

• D is incorrect because the numbers in the ratio are written in the wrong order. Lisa does not have 13 tokens, and Toby does not have 9 tokens.

• E is correct because Lisa has 9 tokens and Toby has 13 tokens.

• F is incorrect because Toby does not have 9 tokens and Lisa does not have 13 tokens.

From Tokens to Tea Students write ratios and use ratio language to describe the relationship between two quantities.

Display the following sentence.

• Jada always puts 2 packets of sugar in her 8-ounce cup of tea.

Facilitate a discussion by using the following prompts.

What is a ratio that relates the number of ounces of tea to the number of packets of sugar?

A ratio that relates the number of ounces of tea to the number of packets of sugar is 8 : 2.

If Jada makes a 16-ounce cup of tea, how many packets of sugar do you expect her to put in her tea? Why? I expect Jada to put 4 packets of sugar in a 16-ounce cup of tea. Because 16 ounces is twice as much as 8 ounces, she will need twice as much sugar.

Pause the discussion and allow students to debate the answer to the following question.

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Differentiation: Challenge Challenge students by asking the following questions about the tea situation. Have them explain their reasoning.

• If Jada makes a 1-ounce cup of tea, how many packets of sugar do you expect her to put in her tea?

I expect Jada to put 1 packet of sugar in 4 1 ounce of tea.

• If Jada makes a 20-ounce cup of tea, how many packets of sugar do you expect her to put in her tea?

I expect Jada to put 5 packets of sugar in 20 ounces of tea.

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6 ▸ M1 ▸ TA ▸ Lesson 2

EUREKA MATH2

Can you use multiplicative comparison language to describe the relationship between the number of ounces of tea and the number of packets of sugar? If so, how?

Turn and talk to your partner.

After students finish their discussion, pose the following question.

Is there 4 times as much tea as sugar? Explain.

No. The tea is measured in ounces and the sugar is measured in packets. We cannot compare tea and sugar with multiplicative language because they do not have the same units.

Have students think–pair–share about the following questions.

What makes the tea situation different from the token situation? In other words, why can we use times as many, or multiplicative language, to compare the numbers of tokens the children receive? Why can’t we use multiplicative language to compare the number of ounces of tea and the number of packets of sugar in Jada’s tea?

In the token situation, we compared the same units, tokens. In the tea situation, the units are different, ounces and packets.

The units in the tea situation, ounces and packets, are different. What could you say to describe the relationship between these quantities that would make sense?

For every 8 ounces of tea, Jada uses 2 packets of sugar.

How many packets of sugar do you expect Jada to put in a 4-ounce cup of tea?

Explain.

I expect Jada to put 1 packet of sugar in a 4-ounce cup of tea. Because 4 ounces is half as much as 8 ounces, she will need half as much sugar.

How many packets of sugar do you expect Jada to put in a 12-ounce cup of tea?

Explain.

I expect Jada to put 3 packets of sugar in a 12-ounce cup of tea. For every 4 ounces of tea, Jada uses 1 packet of sugar.

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EUREKA MATH2 6 ▸ M1 ▸ TA ▸ Lesson 2

Invite students to think of other situations where they would use ratios or for every comparison language. Use the following sequence of questions to elicit students’ ideas: • What is interesting about the comparisons made in the token videos?

• So it’s interesting when something seems out of balance or unfair. Any other times?

• Can you imagine a time when it would be important to express the relationship between quantities like we did when comparing the amounts of tea and sugar?

For each situation that students share, ask whether the quantities in the situation have common units or different units.

Then direct students to problems 2 and 3. Have them complete the problems individually or in pairs.

2. A recipe for lemonade calls for 2 lemons and 5 cups of water. Which statements describe the relationship between the two quantities? Choose all that apply.

A. A ratio that relates the number of lemons to the number of cups of water is 2 to 5.

B. A ratio that relates the number of cups of water to the number of lemons is 5 to 2.

C. A ratio that relates the number of cups of water to the number of lemons is 2 to 5.

D. For every 5 cups of water, there are 2 lemons.

E. For every 2 cups of water, there are 5 lemons.

F. There is 2 12 times as much water as lemons.

3. To make light blue paint, Ryan mixes 2 ounces of white paint with 6 ounces of blue paint.

For parts (a)–(e), fill in the blanks. a. A ratio that relates the number of ounces of white paint to the number of ounces of blue paint is 2 : 6 . b. A ratio that relates the number of ounces of blue paint to the number of ounces of white paint is 6 : 2 . c. For every

2

ounces of white paint, Ryan mixes 6 ounces of blue paint.

d. For every 1 ounce of white paint, Ryan mixes e. Ryan uses

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3

3

ounces of blue paint.

times as much blue paint as white paint.

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EUREKA MATH2

6 ▸ M1 ▸ TA ▸ Lesson 2

After a few minutes or once most students are finished, select students to share their answers and explain their reasoning. As students share, ask them whether the quantities in the situation have common units or different units. Then use the following prompts to guide discussion about the definition of a ratio.

In the tea situation, for every 8 ounces of tea, Jada uses 2 packets of sugar. We wrote the ratio 8 : 2 to relate the number of ounces of tea to the number of packets of sugar in Jada’s tea.

Teacher Note If students ask why the numbers in a ratio both cannot be zero, ask them whether it makes sense to compare quantities that have numbers in a ratio of 0 to 0. This part of the definition is revisited in lesson 5.

Suppose that we wrote a ratio to relate the number of ounces of tea to the number of packets of sugar in Jada’s tea as 2 : 8. Does that change the meaning?

Yes. That means for every 2 ounces of tea, Jada uses 8 packets of sugar.

How would you define the word ratio? Turn and talk to your partner.

A ratio is an ordered pair of numbers that are not both zero. Why do you think the definition includes the word ordered?

It includes the word ordered because the order in which the quantities are described tells us the order of the numbers in the ratio.

Have students look around the classroom. Ask them to correctly use ratios to describe objects they see.

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EUREKA MATH2 6 ▸ M1 ▸ TA ▸ Lesson 2

Land Debrief

5 min

Objectives: Write ratios that relate two quantities as an ordered pair of numbers.

Use ratio language to compare two quantities.

Use the following prompts to guide a discussion about ratios and ratio language.

Encourage students to restate or build upon one another’s responses.

When is it more practical to use ratio language instead of multiplicative comparison language? Give an example of a relationship between two quantities by using ratio language.

It is more practical to use ratio language when the two quantities have different units, like ounces of tea and packets of sugar. For every 8 ounces of tea, there are 2 packets of sugar. It is also more practical to use ratio language when the multiplicative relationship isn’t easily calculated, like when the girl had 7 times as many tokens 5 as the boy.

Describe the meaning of ratio in your own words.

A ratio is an ordered pair of numbers that describes the relationship between two quantities.

What are two quantities that you would love to have in a ratio of 5 : 2 but would not like to have in a ratio of 2 : 5?

Sample: I would love to have the ratio of the number of hours I spend hanging out with friends to the number of hours I spend doing homework be 5 : 2.

Exit Ticket

5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

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Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

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EUREKA MATH2

6 ▸ M1 ▸ TA ▸ Lesson 2

Recap

EUREKA MATH2

6 ▸ M1 ▸ TA ▸ Lesson 2

RECAP Name

Date

2

2. There are 12 students who take orchestra class. There are 4 times as many students who take band class as students who take orchestra class. a. Write a ratio that relates the number of students who take orchestra class to the number of students who take band class.

Introduction to Ratios In this lesson, we

A ratio that relates the number of students who take orchestra class to the number of students who take band class is 12 : 48.

Terminology

•

wrote ratios to relate two quantities.

•

used multiplicative comparison language to compare two quantities.

A ratio is an ordered pair of numbers that are not both zero.

•

used ratio language to compare two quantities.

A ratio can be written as A to B or A : B.

Yes. Scott is correct because there are 4 times as many students who take band class as students who take orchestra class and 1 × 4 = 4.

1. Consider the collection of shapes shown. Which statements correctly describe the collection of shapes? Choose all that apply.

The order in which the quantities are described tells us the order of the numbers in the ratio.

1

There are 4 as many students who take orchestra class as students who take band class.

So the ratio of the number of orange rectangles to the number of blue pentagons is 4 : 5, not 5 : 4.

A. A ratio that relates the number of orange rectangles to the number of blue pentagons is 5 : 4.

C. For every 1 circle, there are 2 rectangles.

D. There are 1 as many orange 2 rectangles as yellow circles.

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The number of students who take band class is 48 because 4 × 12 = 48.

b. Scott uses ratio language to describe the ratio from part (a). He says that for every 1 student who takes orchestra class, there are 4 students who take band class. Is Scott correct? Why?

Examples

B. There are 2 12 times as many pentagons as circles.

EUREKA MATH2

6 ▸ M1 ▸ TA ▸ Lesson 2

A ratio that relates the number of pentagons to the number of circles is 5 : 2. That means there are 25 , or 2 12 , times as many pentagons as circles.

For every 2 circles, there are 4 rectangles. That means there are twice as many rectangles as circles.

So for every 1 circle, there are 2 rectangles.

For every 4 orange shapes, there are 2 yellow shapes. That means there are 2 times as many orange rectangles as yellow circles, not 12 as many.

17

18

R E CA P

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Copyright © Great Minds PBC

8/27/2021 5:28:53 PM

EUREKA MATH2 6 ▸ M1 ▸ TA ▸ Lesson 2

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

6 ▸ M1 ▸ TA ▸ Lesson 2

PRACTICE Name

Date

2

EUREKA MATH2

6 ▸ M1 ▸ TA ▸ Lesson 2

3. Sasha says there are 1 12 times as many circles as triangles in the picture in problem 2. Is she correct? Explain.

Sasha is correct because there is 1 triangle for every 1 12 circles.

1. A smoothie recipe calls for bananas and strawberries in the ratio represented by the picture.

4. At an animal shelter, 9 dogs and 15 cats are ready for adoption. Fill in the blanks to make the statements true. a. For every

A. A ratio that relates the number of strawberries to the number of bananas is B. There are

7 2

C. For every

2

D. There are

2 7

9

dogs, there are 15 cats.

b. For every 3 dogs, there are

For parts (a)–(d), fill in the blanks.

7:2 .

c. There are

5

3

5

cats.

times as many cats as dogs.

times as many strawberries as bananas.

5. Students at a middle school take an elective during the last hour of the school day. There are 11 students who take an art class. There are 3 times as many students who take a music class as students who take an art class.

bananas, there are 7 strawberries. times as many bananas as strawberries.

a. What is a ratio that relates the number of students who take a music class to the number of students who take an art class?

A ratio that relates the number of students who take a music class to the number of students who take an art class is 33 : 11.

2. Consider the collection of shapes shown. Which statements correctly describe the collection of shapes? Choose all that apply.

b. Kayla uses ratio language to describe the ratio from part (a). She says that for every 3 students who take a music class, there is 1 student who takes an art class. Is Kayla correct? Why? Yes. Kayla is correct because there are 3 times as many students who take a music class as students who take an art class.

A. A ratio that relates the number of red squares to the number of blue circles is 4 : 3.

Remember

B. There are 2 times as many triangles as squares.

For problems 6–8, multiply.

C. There are

1 2

as many triangles as squares.

6. 1,312 × 3

3,936

D. For every 1 triangle, there are 2 squares.

7. 2,214 × 4

8,856

8. 5,631 × 5

28,155

E. A ratio that relates the number of squares to the number of circles is 3 : 4.

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20

P R ACT I C E

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EUREKA MATH2

6 ▸ M1 ▸ TA ▸ Lesson 2

EUREKA MATH2

6 ▸ M1 ▸ TA ▸ Lesson 2

9. Convert 5 hours to minutes.

300 minutes

10. Each model is divided into equal sections. Which models have a shaded portion that represents the fraction 12 ? Choose all that apply.

A.

B.

C.

D.

E.

F.

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P R ACT I C E

21

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1

LESSON 1

Falling Objects Represent the distance traveled by a falling object with graphs, tables, and equations.

Explain why a linear function is not a good model for the distance traveled by a falling object.

EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

EXIT TICKET Name

Date

1

Levi drops a rock off a cliff from a height of 576 feet. The table of values shows the distance the rock travels over time.

Time (seconds)

0

2

4

6

Distance (feet)

0

64

256

576

Each time the seconds increase by 2, the distance increases by a greater amount. If I made a graph of distance over time, it would not be a line. b. Draw the graph of the function that models the distance in feet the rock travels over time in seconds.

Key Questions

y

• How can we represent the distance traveled by a falling object?

550

• Why is a quadratic function used to model the distance traveled by a falling object over time?

500 450

Distance (feet)

400 350

Achievement Descriptors

300 250

A1.Mod4.AD14 Interpret key features of graphs and tables in terms of quantities for quadratic functions that model relationships. (F.IF.B.4)

200 150 100

A1.Mod4.AD16 Calculate and interpret average rates of change

50 1

2

3

4

5

6

7

of quadratic functions (presented symbolically or as tables) over specified intervals. (F.IF.B.6)

x

Time (seconds)

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Students watch a video to compare different falling objects and notice that the weight of an object does not seem to affect how fast it falls. They explore the distance traveled by a falling object and reason why it cannot be represented by a linear function of time.

Students represent distance traveled over time as a quadratic function by using tables, graphs, and equations. They use the average rate of change over equal intervals to notice patterns that show that a falling object speeds up over time.

This lesson formalizes the term quadratic function and introduces the term average rate of change.

a. Explain why a linear function would not be a good model for this situation.

0

Lesson at a Glance

13

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EUREKA MATH2 A1 ▸ M4 ▸ TA ▸ Lesson 1

Agenda

Materials

Fluency

Teacher

Launch Learn

10 min

25 min

• Dropping a Ball from a Building • A Closer Look at Distance Traveled

Land

10 min

• Computer or device* • Projection device* • Teach book*

Students • Calculator* • Learn book* • Paper or notebook* • Pencil*

Lesson Preparation • Read the Math Past resource. * These materials are only listed in lesson 1.

Ready these materials for every lesson in this module.

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EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

Fluency Compare Speeds of Different Objects Students identify the speed at which objects are traveling to prepare to represent distance traveled by a falling object as a quadratic function of time.

Directions: Each table provides some points that represent the distance traveled by an object over time. Identify if the object is traveling at a constant speed. If it is, determine the speed of the object.

1.

Teacher Note

Time (seconds)

0

Distance (feet)

10

25

40

Time (seconds)

0

2

4

Distance (feet)

35

35

35

Time (seconds)

0

8

16

Distance (feet)

4

24

44

5

10

Constant speed: Yes Speed: 3 feet per second

2.

Fluency activities are short sets of sequenced practice problems that students work on in the first 3–5 minutes of class. Administer a fluency activity as a bell ringer or adapt the activity as a teacher-led Whiteboard Exchange or choral response. Directions for these routines can be found in the Fluency resource.

Constant speed: Yes Speed: 0 feet per second

3.

20

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Constant speed: Yes

Speed: _ ​​ ​​ feet per second 5 2

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EUREKA MATH2 A1 ▸ M4 ▸ TA ▸ Lesson 1

4.

Time (seconds)

0

Distance (feet)

5

35

95

Time (seconds)

0

18

36

Distance (feet)

2

8

18

Time (seconds)

0

20

40

Distance (feet)

50

52

54

12

24

Constant speed: No Speed: N/A

5.

Constant speed: No Speed: N/A

6.

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EM2_A104TE_A_L01.indd 21

Constant speed: Yes

Speed: __ ​​ ​​ foot per second 1 10

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EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

Launch

10

Students realize that an object’s weight does not affect how fast it falls to the ground.

Play the first part of the Falling Objects video, which shows three pairs of objects. Ask students to predict which object from each pair will hit the ground first when they are dropped from the same height at the same time. Anticipate that some students may assume the heavier objects will land first, while some students may say they will land at the same time, if they are familiar with these concepts from a science class. Since this activity is intended to dispel any misconceptions about falling objects, do not confirm or deny student guesses at this point.

UDL: Representation Presenting the falling objects situations in video format supports students in understanding the problem context by removing barriers associated with written and spoken language.

Which object do you think will land first each time?

The heavier object will land first each time.

The balls and the fruit will land at about the same time because they have similar shapes.

Next, play the second part of the video. Invite students to turn and talk with a partner about whether the video supports their predictions. Facilitate a discussion about the falling objects by using the following prompts.

Describe what you saw in the video.

The feather floats down to the ground slower than the bowling ball dropped.

Teacher Note

The video shows that the balls and the fruit seem to land at the same time.

What did you notice about the two balls and the two food items?

It looked like they hit the ground at the same time, even though one is heavier than the other.

Why do you think the feather falls more slowly than the bowling ball?

The feather is more affected by air than the bowling ball.

The dialogue shown provides suggested questions and sample responses. To maximize every student’s participation, facilitate discussion by using tools and strategies that encourage student-to-student discourse. For example, make flexible use of the Talking Tool, turn and talk, think–pair–share, and the Always Sometimes Never routine.

The video shows that an object’s weight does not influence how fast it falls. A force called air resistance slows the feather down. Objects with different weights dropped from the

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11-Sep-21 11:23:59 AM

EUREKA MATH2 A1 ▸ M4 ▸ TA ▸ Lesson 1

same height will land at the same time when we do not consider air resistance. With the other pairs, the effect of air resistance is not noticeable in the video. The ball-shaped objects have similar air resistance, so they landed at nearly the same time.

What else affects how fast objects like the ones in the video fall?

Gravity has some type of effect.

What type of function might model this situation? Explain your thinking.

A linear function might model the situation if the speed of the falling objects is constant.

Teacher Note In grade 8, students learned that when speed is constant, a linear function can represent the distance an object travels over time. It would be natural for them to think that a linear function is an appropriate model to represent the falling objects.

Consider sharing some of the history of the study of falling objects.

Greek philosopher and scholar Aristotle proposed a theory of gravity that was accepted and believed for over 2000 years. He proposed that heavier objects fall at faster rates than lighter ones and in a proportional way. In the 16th century, scientist Galileo Galilei conducted an experiment with balls rolling down ramps, where he demonstrated that time taken for an object to fall is independent of the mass of the object.

Today, we will learn how to represent distance traveled by a falling object over time.

Learn

25

Math Past Consider extending the discussion about the history of the study of falling objects by referring to the Math Past resource. In addition to providing a more extended description of the theories of Aristotle and Galileo, the resource references an experiment astronaut David Scott performed on the moon during the Apollo 15 space mission. The experiment demonstrated that Galileo’s ideas about gravity are correct.

Dropping a Ball from a Building Students realize that a linear function is not a good model for an object falling to the ground.

Play the first part of the Dropping a Ball video, which shows a ball being dropped and falling to the ground as a timer tracks how many seconds pass.

What do you notice about the distance traveled by the ball at the beginning compared to the end?

The ball seems to travel less distance at the beginning and more distance toward the end.

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EM2_A104TE_A_L01.indd 23

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11-Sep-21 11:24:19 AM

EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

Do you think the ball’s speed is constant? How can we test your answer?

No. We can record the distance traveled each second. Then we can make a table or a graph to see if there is a linear relationship between time and distance traveled.

Anticipate that some students may immediately realize the relationship is not linear, while others may need to examine the situation in more detail.

Play the second part of the Dropping a Ball video, which also displays the distance the ball has traveled as the seconds pass. Ask students to work in pairs on problems 1–4. If students request it, replay the second part of the video.

After watching the Dropping a Ball video, complete problems 1–4.

1. Complete the table showing the distance the ball travels over time.

Time (seconds)

0

1

2

3

4

5

Distance (feet)

0

16

64

144

256

400

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Teacher Note The context of projectile motion introduces students to quadratic functions. To help students focus on this new type of function, the first few lessons focus on simple models for projectile motion. They use friendly approximations for the acceleration due to gravity such as 32 ft/s2 and 10 m/s2. Lesson 1 uses functions of the form f (t) = ​​ 1 ​​ gt2, where 2 g represents the acceleration due to gravity, t represents time, and f (t) represents the distance traveled. Later lessons define the height above the ground as a function of time and incorporate nonzero initial heights and initial velocities.

__

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11-Sep-21 11:24:32 AM

EUREKA MATH2 A1 ▸ M4 ▸ TA ▸ Lesson 1

2. Use the data in the table to make a graph of the distance traveled by the ball over time. y 450 400

Distance (feet)

350 300 250 200 150 100 50 0

1

2

3

4

5

6

7

8

9

10

t

Time (seconds)

3. Does the ball travel at a constant speed? Explain how you know.

No. The distance traveled each second is increasing, which means the speed is increasing.

The ball is speeding up as it falls to the ground.

4. Would a linear function be a good model for the distance traveled when the ball is dropped from the top of the building? Why? No. The rate of change is not constant.

Invite a few students to share their responses. Display the table and the graph from problems 1 and 2 and use them to reinforce that a linear function would not represent this situation accurately. If not mentioned by students, emphasize that the rate of change over equal intervals of time is not constant and the graph is curved.

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Promoting the Standards for Mathematical Practice Students reason abstractly (MP2) as they create tables, graphs, and then an expression to represent the distance traveled by the ball shown in the video.

Ask the following questions to promote MP2: • How do the table, graph, and equation confirm that the distance traveled is not a linear function of time?

• How does the function defined by d(t) = 16t2 produce a curved graph?

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EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

A Closer Look at Distance Traveled Students notice patterns and create a quadratic function that models the distance traveled by a falling object over time.

Ask students to complete problems 5–8. While they work, display the table from problem 1 that shows the distance the ball travels over time. If students need additional support, ask the following questions while displaying the table provided: • How is the distance traveled recorded in this table compared to how you recorded the distance in problem 1?

• What patterns do you notice in the distance values?

• How can you write an expression to represent the distance for time t?

UDL: Representation

Time (seconds)

0

1

2

3

4

5

Distance (feet)

16 · 0

16 · 1

16 · 4

16 · 9

16 · 16

16 · 25

Consider highlighting the patterns by annotating the table to show each distance written by using exponents. For example, write 16 · 52 next to 16 · 25. Expressing each distance by using exponents will help students generalize the equation in problem 9.

5. What patterns do you notice in the table?

Each distance is a multiple of 16. The distance increases by 16 ft, then 48 ft, then 80 ft, and so on. Each distance increase is 32 feet more than the previous one.

6. Calculate the ball’s average speed over each 1-second interval. What patterns do you notice in the speeds?

The average speeds are 16 ft/s, 48 ft/s, 80 ft/s, 112 ft/s, and 144 ft/s. Each time the speed increases by 32 ft/s.

7. Write an equation for a function that gives the distance traveled in feet over time in seconds.

d(t) = 16t2

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12/2/2020 12:58:56 PM

EUREKA MATH2 A1 ▸ M4 ▸ TA ▸ Lesson 1

0 ≤ t ≤ 5 makes sense. I can start the time from 0 when the ball first drops. From the video, I can see that after 5 seconds the ball hits the ground.

8. What domain makes sense for your function? Explain your reasoning.

Display the answer to problem 7 and invite students to share their thoughts about the patterns they noticed and how they determined the equation and the domain. Then lead a discussion to introduce the term quadratic function by using the following prompts.

We determined this function was not linear. What could we call this new type of function?

A polynomial function A squared function

16t2 is a polynomial expression, and its degree is 2. Any function defined by a polynomial expression of degree 2 is called a quadratic function. The simplest quadratic function is given by the equation y = x 2.

A quadratic function is in standard form if it is written in the form f (x) = ax 2 + bx + c, for constants a, b, and c with a ≠ 0 and for any real number x. Why do you think we call this standard form?

It looks like the standard form of a polynomial expression.

The terms are written in order of descending degree.

Have students complete a Frayer model for the term quadratic function. Encourage students to include an example of a quadratic function written in standard form in their model. Students can refer to and add to this graphic organizer through this module as they discover new characteristics of quadratic functions.

Consider displaying a completed Frayer model for students to reference throughout the module.

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EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

Complete the Frayer model for the term quadratic function.

Definition

Picture y 1600

A quadratic function is defined by a polynomial expression of degree 2.

Distance (feet)

1400 1200 1000 800

Rate of change is not constant.

600 400 200 0

2

4

6

8

10

t

Time (seconds)

Quadratic Function

f(t) = 16t

y = x2 y = 10x 2 + 5x + 3 y = 5x + 10 d(t) =

Linear functions

16t 2

Example

Nonexample

Next, direct students’ attention to the sentences that precede problem 9 and read them as a class. Direct students to complete problems 9–11 with a partner.

If students need support finding the values for speed, point out that they are computing

the change in distance divided by the change in time for each _ ​​ ​​-second interval. Consider 1 2

displaying the following calculation for the average speed from 0 seconds to 0.5 seconds: p(0.5) − p(0) ______ ​​__________ ​​ = ​​1.25 − ​​0 = 2.5 0.5 0.5 − 0

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12/2/2020 12:58:57 PM

EUREKA MATH2 A1 ▸ M4 ▸ TA ▸ Lesson 1

Lucas drops a pebble off a bridge. It hits the water after 3 seconds. The function defined by p(t) = 5t2 gives the distance the pebble travels in meters t seconds after it is dropped.

9. Complete the table and make a graph of quadratic function p. Calculate the average speed 1 of the pebble every _ ​​ ​​ second.

2

p(t)

(meters)

y

Average

45

Speed Every

_​​ 1​​ Second

0

0

−

0.5

1.25

2.5

1

5

7.5

1.5

11.25

12.5

2

20

17.5

2.5

31.25

22.5

3

45

27.5

y = p(t)

40

2

Distance (meters)

Time, t (seconds)

Distance,

35 30 25 20 15 10 5 0

1

2

3

t

Time (seconds)

10. What patterns do you notice in the values in the third column of the table?

They increase by 5 meters per second each _ ​​ ​​ second.

1 2

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EM2_A104TE_A_L01.indd 29

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EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

11. On another day, Lucas rides his bicycle at a constant speed of 5 meters per second.

Function d gives the distance in meters that Lucas travels on his bicycle over time t in seconds. a. Complete the table and make a graph of this situation. Calculate the average speed 1 every _ ​​ ​​ second.

2

d(t)

(meters)

y

Average

45

Speed Every

_​​ 1​​ Second

40

2

0

0

−

0.5

2.5

2.5

1

5

2.5

1.5

7.5

2.5

2

10

2.5

2.5

12.5

2.5

3

15

2.5

Distance (meters)

Time, t (seconds)

Distance,

35 30 25 20 15

y = d(t)

10 5 0

1

2

3

t

Time (seconds)

b. Write an equation for the function d that models the situation.

d(t) = 5t c. Why is the distance Lucas traveled on his bicycle a linear function of time?

There is a constant rate of change of 5 meters every second.

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EUREKA MATH2 A1 ▸ M4 ▸ TA ▸ Lesson 1

Once students have completed the problems, use the following prompts to facilitate a discussion.

What do you notice about the average speed of the pebble Lucas drops compared to his average speed when riding his bicycle?

The average speed is increasing as the pebble drops. Every half second, the average speed increases by 5 meters per second. The bicycle riding speed stays the same.

Linear functions, such as the one that models Lucas riding his bicycle, have a constant rate of change. When we think about a rate of change for any type of function, we call it the average rate of change. Each time you find the average speed, you calculate an average rate of change.

What was the average rate of change of the pebble function from 1 to 1.5 seconds?

From 2 seconds to 2.5 seconds?

Language Support This lesson introduces average rate of change by relating it to the average speed of moving objects. The term is formally defined in a later lesson. As this discussion progresses, consider having students annotate their work to support their informal use of the new terminology. For example, display the tables showing average speed and invite students to write average rate of change next to the column showing average rate of speed.

12.5 meters per second; 22.5 meters per second How do you think the average rate of change relates to the graphs?

The average rate of change is positive when the graphs represent an increasing function.

The average rate of change increases where the pebble graph gets steeper.

When the average rate of change is constant, like in the bicycle situation, the graph represents a linear function.

Why does the average rate of change increase for the function that models the pebble being dropped from the bridge?

Over equal intervals of time, the distance traveled increases because of the effect of gravity.

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EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

Land Debrief

5 min

Objectives: Represent the distance traveled by a falling object with graphs, tables, and equations.

Explain why a linear function is not a good model for the distance traveled by a falling object.

Display the following table and graph. Tell students that one represents the distance traveled by a falling object and the other represents an object traveling with a constant speed. Ask students to think–pair–share about which one represents each situation.

Distance (feet)

0

0

2

20

4

40

6

60

8

80

10

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EM2_A104TE_A_L01.indd 32

100

y 1600 1400 Distance (feet)

Time (seconds)

1200 1000 800 600 400 200 0

2

4

6

8

10

t

Time (seconds)

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12/2/2020 12:58:58 PM

EUREKA MATH2 A1 ▸ M4 ▸ TA ▸ Lesson 1

Which one could represent a situation like Lucas riding his bicycle at a constant speed? Which one could represent a situation like Lucas dropping the pebble off a bridge?

The table shows a constant average rate of change. That means speed is constant like Lucas riding his bicycle. The curved graph could represent the pebble being dropped off the bridge because the average rate of change over equal-size intervals is not constant.

How can we represent distance traveled by a falling object?

We can use tables, graphs, or equations to represent distance traveled by a falling object.

Why is a linear function not a good model for the distance traveled by a falling object?

We saw in the videos, tables, and graphs that when an object is dropped, it speeds up.

A linear function would represent a situation where an object has a constant speed.

What did you notice about the average rate of change for the falling objects in this lesson?

When we calculated the average rate of change, there was a constant increase in those values over equal intervals of time. For example, the average speed measurements 1 ​​ ​​ second. in the pebble problem increased by 5 meters per second each _ 2

Why is a quadratic function used to model the distance traveled by a falling object over time?

The distance traveled by a falling object over time increases by the same amount over equal intervals. A quadratic function defined by a squared term like 16t2 or 5t2 models this motion.

Exit Ticket

5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

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EM2_A104TE_A_L01.indd 33

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

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EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

Recap

EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

RECAP Name

Date

1

EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

2. Mason drops his water bottle off the edge of a cliff. It hits the ground after 3 seconds. Mason uses graphing technology to graph the distance in feet the water bottle travels as a function of time t seconds after it is dropped.

Falling Objects

y

•

used tables, graphs, and equations to represent the distance traveled by a falling object.

•

learned that a quadratic function models the distance traveled by a falling object.

Examples 1. Write whether the distance should be modeled with a linear function or a quadratic function.

Then complete the table with an expression for the distance traveled after t seconds.

Distance from the Cliff Edge (feet)

160

In this lesson, we

Terminology • A quadratic function is defined by a polynomial expression of degree 2.

• A quadratic function is in standard form if it is written in the form f (x) = ax2 + bx + c, for constants a, b, and c with a ≠ 0 and for any real number x.

Distance (meters)

128

Mason drops the bottle when t is 0, and it hits the ground when t is 3. The y-coordinates of these points show the distance traveled at those times.

112 96 80 64 48 32 16 –1

0

1

2

3

4

t

Time (seconds)

a. How far above the ground is the water bottle when Mason drops it?

Quadratic function Time (seconds)

144

It is 144 feet above the ground.

0 0

1 6

2 24

3 54

4 96

b. What does the point (0, 0) mean in this situation?

t

Mason didn’t drop the bottle yet, so the distance traveled is 0 feet. c. Use the graph to find the distance the water bottle falls 1 second and 2 seconds after Mason drops it.

6t2

The distance is 16 feet after 1 second and 64 feet after

2 seconds.

The distance increases by 6 meters from 0 seconds to 1 second and by 18 meters from 1 second to 2 seconds. The rate of change is not constant, so a linear function is not a good model.

d. What domain makes sense in this situation? Explain your choice.

Each distance is 6 times a number.

We can write the distances as 6 · 02, 6 · 12, 6 · 22, and so on. Noticing this pattern helps us write the expression.

The domain 0 ≤ t ≤ 3 makes sense because the water bottle does not fall before Mason drops it, and it hits the ground after 3 seconds.

e. Write an equation for function d that gives the distance d(t) in feet the water bottle travels t seconds after Mason drops it.

d(t) = 16t2 where 0 ≤ t ≤ 3

This is a quadratic function because it is defined by a polynomial expression of degree 2.

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EM2_A104TE_A_L01.indd 34

The domain should represent the time interval from the time Mason drops the water bottle until it hits the ground.

15

16

R E CA P

Each y-coordinate on the graph is 16 multiplied by the time in seconds squared. For example, when t = 2, the distance is 16 · 22, or 64.

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12/2/2020 3:26:46 PM

EUREKA MATH2 A1 ▸ M4 ▸ TA ▸ Lesson 1

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

A1 ▸ M4 ▸ TA ▸ Lesson 1

PRACTICE Name

Date

1

4. Function f represents the distance in meters traveled by a ball dropped from a building over time.

Use the graph of function f to complete the table. Some values are already filled in.

For problems 1–3, write whether the distance is modeled with a linear function or a quadratic function. Then write an expression for the distance traveled after t seconds.

1.

Quadratic function Time (seconds)

0

1

2

3

4

t

Distance (meters)

0

2

8

18

32

2t2

EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

t

f (t)

0

0

1

5

2

20

3

45

4

80

y 80 70 Distance (meters)

EUREKA MATH2

60 50 40 30 20

y = f(t)

10 0

1

2

3

4

t

Time (seconds)

2.

Linear function

5. Use function f from problem 4 to find the average speed of the ball over each interval. The first one has been completed for you. Describe any patterns you notice.

Time (seconds)

0

Distance (meters)

0

1 2

2 4

3 6

4 8

t 2t

3.

Quadratic function Time (seconds)

0

1

2

3

4

t

Distance (meters)

0

5

20

45

80

5t2

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EM2_A104TE_A_L01.indd 35

Interval

Average Speed

t = 0 to t = 1

5____ −0 =5 1−0

t = 1 to t = 2

20 −5 _____ = 15 2−1

t = 2 to t = 3

45 − 20 ______ = 25 3−2

t = 3 to t = 4

80 − 45 ______ = 35 4−3

The average speed values increase by 10 at each interval. The average rate of change values are all multiples of 5.

17

18

P R ACT I C E

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EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

6. Graphs A, B, and C are shown. Determine which graph best models the distance traveled as a function of time for each situation.

Graph A

Graph B

y

90

80

80

80

70

70

70

60 50 40 30

Distance (meters)

90

60 50 40 30

50 40

20

20

10

10

2

3

4

t

0

Time (seconds)

1

2

3

4

t

Time (seconds)

x

f (x)

−3

9

y

0

8

y = f(x)

6

30

10 1

7. f (x) = x2

60

20 0

For problems 7–10, graph the function by completing the table, plotting the points, and then connecting the points with a smooth curve. Be sure to scale the y-axis so that each point fits on the graph.

Graph C

y

90 Distance (meters)

Distance (meters)

y

EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

1

2

3

4

t

−2

4

−1

1

0

0

1

1

2

4

3

9

Time (seconds)

a. A car travels at a constant speed.

4

2

−4

−3

−2

−1

0

1

2

3

4

x

−2

Graph A b. A stone is dropped from the banks of a river into the water below.

Graph B

−4

−6

−8

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EM2_A104TE_A_L01.indd 36

P R ACT I C E

19

20

P R ACT I C E

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12/2/2020 12:59:00 PM

EUREKA MATH2 A1 ▸ M4 ▸ TA ▸ Lesson 1

EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

8. g(x) = 5x 2

EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

9. h(x) = −5x 2

x

g(x)

−3

45

−2

20

−1

5

0

0

1

5

2

20

3

45

y

y

x

h(x)

−3

−45

−2

−20

−1

−5

0

0

1

−5

2

−20

3

−45

40

40

y = g(x)

30

20

10

−4

−3

−2

−1

0

1

2

3

4

x

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EM2_A104TE_A_L01.indd 37

20

10

−4

−3

−2

−1

0

1

2

3

4

x

−10

−10

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30

−20

−30

−40

P R ACT I C E

21

22

P R ACT I C E

−20

−30

y = h(x)

−40

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EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

12. Ana drops a ball from a balcony. It hits the ground after 4 seconds. Ana uses graphing technology to graph the distance in meters the ball travels as a function of time t seconds after it is dropped.

10. k(x) = x 2 + 5

x

y

k(x)

y

12

−3

y = k(x)

14 9

−1

6

0

5

1

6

Distance (meters)

9

−2

6

3

−4

−3

−2

−1

0

1

2

3

4

x

−3

2 3

−6

9

EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0 −5

1

2

3

t

4

−9

Time (seconds) −12

14

a. If the ball hits the ground after 4 seconds, how high is the ball when Ana drops it?

It was 80 meters above the ground. b. What does the point (0, 0) mean in this situation?

11. Which expression best models the distance in feet traveled by an object t seconds after it is dropped from the top of a tall building?

A. 16t

B. 16t 2

C. −16t 2

D. −16t

Ana has not dropped the ball yet, so the distance traveled is 0 meters. c. Use the graph to find the distance the ball falls 1, 2, and 3 seconds after Ana drops it.

E. 16

The distance is 5 meters after 1 second, 20 meters after 2 seconds, and 45 meters after

3 seconds.

d. What domain makes sense in this situation? Explain your choice.

The domain 0 ≤ t ≤ 4 makes sense because the ball does not fall before Ana drops it, and it hits the ground after 4 seconds.

e. Write an equation for function d that gives the distance d(t) in meters the ball travels t seconds after Ana drops it.

d(t) = 5t2, where t ≥ 0

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EUREKA MATH2 A1 ▸ M4 ▸ TA ▸ Lesson 1

EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

b. The function f (t) = −16t 2 + 160t is a quadratic function. How do the table and the graph of f compare to a table and graph of the quadratic function described in problem 13, d(t) = 16t 2?

13. A package falls off a window ledge. The distance in feet the package travels t seconds after it falls is given by the function d(t) = 16t 2. If the window ledge is 40 feet high, estimate when the package will hit the ground.

The package hits the ground just before 1.6 seconds because d(1.6) = 40.96.

The values in the table for f increase and then decrease. The values in the table for d only increase. Both graphs start at (0, 0). Both graphs are curved. The graph of f shows two x-intercepts, and the graph of d shows only one x-intercept.

14. Function f is defined by f (t) = −16t 2 + 160t and gives the height f (t) in feet of a foam rocket shot straight up into the air t seconds after it is launched.

c. List the average rates of change on each 1-second interval for f and d. How do the average rates of change for f on each one-second interval compare to the average rates of change for d on the same intervals?

a. Complete the table of values and graph function f.

Time (seconds)

0

1

2

3

4

5

6

7

8

9

10

Distance (feet)

0

144

256

336

384

400

384

336

256

144

0

The average rate of change values on 1-second intervals for function f are 144, 112, 80, 48, 16, −16, −48, −80, −112, and −144. The average rate of change values on 1-second intervals for function d are 16, 48, 80, 112, and 144. The average rate of change values for function f decrease by 32, and the average rate of change values for function d increase by 32.

Remember

y

For problems 15–17, write an equation of the line.

400

15.

350

y = f(t)

300

Distance (feet)

EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

250

150 100

17.

y

y

5

5

4

4

3

3

3

2

2

2

1

1

1

−5 −4 −3 −2 −1 0 −1

200

16.

y 5 4

1

2

3

4

5

x

−5 −4 −3 −2 −1 0 −1

1

2

3

4

5

x

−5 −4 −3 −2 −1 0 −1

−2

−2

−2

−3

−3

−3

−4

−4

−4

−5

−5

−5

y = 2x − 5

y = −3x + 2

1

2

3

4

5

x

y = −_1 x−3 4

50

0

1

2

3

4

5

6

7

8

9

10

t

Time (seconds)

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EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

EUREKA MATH2

A1 ▸ M4 ▸ TA ▸ Lesson 1

18. A graph of a system of two inequalities is shown. y 10 9 8 7 6 5 4 3 2 1 −10 1 −99 −88 −77 −66 −55 −44 −33 −22 −11 0 −1

1

2

3

4

5

6

7

8

9

10

x

−2 −3 −4 −5 −6 −7 −8 −9 −10 −10

a. Write two inequalities that define the system represented by the graph.

y > 2x + 3

y ≤ −3x + 3

b. Explain how you know whether an ordered pair is a solution to a system.

An ordered pair is a solution if the corresponding point lies in the intersection of the two half-planes or if the point lies on the part of a solid boundary line that borders the intersection of the two half-planes.

19. Write (2x − 5)2 as a polynomial expression in standard form.

4x 2 − 20x + 25

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7

LESSON 7

Angle Relationships and Unknown Angle Measures Identify and describe angle relationships given in diagrams.

Write and solve equations that use angle relationships to find unknown angle measures.

EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

EXIT TICKET Name

Date

7

In the diagram, ∠BAD is a straight angle. Use this diagram for problems 1–4.

C

x°

B

132° A

D

1. Describe the relationship between ∠ BAC and ∠CAD. ∠BAC and ∠CAD are a linear pair, so their measures sum to 180°.

Lesson at a Glance This lesson uses familiar angle relationships as a springboard for writing and then solving equations by using if–then moves. Students use a variety of strategies to find an unknown angle measure. Then they are introduced to if–then moves through nonmathematical statements. After seeing the mathematical version of the if–then moves, students apply them to find other unknown angle measures.

Through a mix of partner and independent work, as well as a sharing of strategies, students are encouraged to make the connection between an arithmetic strategy and an algebraic strategy. This lesson formalizes the term complementary angles.

Key Questions • What does it mean for two angles to be complementary?

2. Write an equation for the angle relationship and solve for x .

x + 132 = 180 x + 132 − 132 = 180 − 132 x = 48

• How does knowing about angle relationships help us find unknown angle measures?

The solution is 48.

• How can we use if–then moves to find unknown angle measures?

3. Check your solution to the equation.

Because 48 + 132 = 180 is a true number sentence, we know 48 is a solution to the equation.

Achievement Descriptors 7.Mod3.AD8 Compare algebraic solutions to arithmetic solutions

4. What is the measure of ∠ BAC ?

of word problems. (7.EE.B.4.a)

m∠ BAC = 48°

7.Mod3.AD12 Write and solve equations to find unknown angle

measures by using known facts about angle relationships. (7.G.B.5) Copyright © Great Minds PBC

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EUREKA MATH2 7 ▸ M3 ▸ TB ▸ Lesson 7

Agenda

Materials

Fluency

Teacher

Launch Learn

5 min

30 min

• None

Students

• Angle Measure

• None

• If–Then Moves

Lesson Preparation

• Strategies

• None

Land

10 min

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

Fluency Solve One-Step Equations Students solve one-step equations to prepare for finding unknown angle measures.

Directions: Solve for the variable.

1.

x-4 = 7

11

2.

8 + x + 7 = 25

10

3.

5.2 + z = 13

7.8

4.

4 a = 20

5.

b 3

=21

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2

5 71

2

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EUREKA MATH2 7 ▸ M3 ▸ TB ▸ Lesson 7

Launch

5

Students identify angles and angle relationships.

Display problem 1. Tell students they have 2 minutes to work on problem 1 with a partner.

Do not expect students to list all the relationships as there are many angle relationships in the diagram. Some examples are given as student responses.

1. In the diagram, DB and EC intersect at point A. List all the angles you see in the diagram. Note any angle relationships that you notice.

F C

D

A

B

E

Teacher Note Before grade 7, students refer to complementary angles as “two angle measures that sum to 90°.”

Sample: Angles:

∠BAC , ∠BAF , ∠BAD , ∠BAE , ∠CAF , ∠CAD , ∠CAE , ∠FAD , ∠FAE , ∠DAE Angle relationships: Angles on a line

∠BAE and ∠DAE ∠EAD , ∠DAF , and ∠FAC ∠EAD and ∠DAC

Linear pair

∠DAF and ∠FAB ∠DAC and ∠CAB ∠EAD and ∠DAC ∠CAF and ∠FAE

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

∠DAB

Straight angle

∠EAC Angle measures that sum to 90°

∠BAC and ∠CAF

Adjacent angles

∠EAD and ∠DAF ∠BAC and ∠EAB

Angles at a point

∠BAF , ∠FAE , and ∠EAB

After about 2 minutes, ask groups to share with the class the angles and angle relationships they identified. As students share, point to the angles mentioned. Create a list of the different types of angle relationships in the diagram.

Ensure the following points are addressed during the share: • Some pairs of angles have measures that sum to 90°. Students are introduced to the term complementary angles later in the lesson.

UDL: Representation Trace the paths of the angles as the students name the points and vertices. This helps clarify the naming convention and the relationships between angles.

• There is more than one way to name an angle. For example, ∠EAD is the same angle as ∠DAE. • When an angle is named, the middle letter in the angle name is the vertex of the angle.

Display the following two angles, which highlight the difference between an angle that needs to be identified with three letters and an angle that only needs one letter. Use the prompts that follow those angles to provide reasoning for the naming conventions.

H

B A

I J K

Language Support Consider creating an anchor chart of familiar angle relationships such as angles on a line, adjacent angles, linear pair, straight angles, angles at a point, and angles that sum to 90°. Throughout this topic, other angle relationships are introduced, so leave space for them to be added.

C

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EUREKA MATH2 7 ▸ M3 ▸ TB ▸ Lesson 7

Can ∠HKI be named ∠K? No. Point K is the vertex of three angles: ∠HKJ, ∠HKI, and ∠IKJ. It would be unclear which angle was being referred to. Three points need to be in the name.

Do we always need three points to name these angles? Why? It depends on the diagram. If naming an angle with just one point is unclear as to which angle is being named, then the name needs to include three points.

Can ∠BAC be named with just one letter? Explain.

∠BAC can be named ∠A because only one angle has a vertex at point A. Display the image from problem 1.

F C

D

A

B

E If ∠BAC has a measure of 50° , can we determine the measure of any other angles?

Turn and talk to your partner.

Today, we will find unknown angle measures by writing and solving equations.

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

Learn

Promoting the Standards for Mathematical Practice

Angle Measure Students find an unknown angle measure.

Display problem 2. Have students complete problem 2 independently. Circulate and identify students who use different methods: an arithmetic strategy, an algebraic strategy, or a tape diagram. Do not prompt or specify which strategy students need to use. All strategies are discussed later in the lesson. As students finish, have them compare their strategy with a partner.

2. What is the measure of

∠BGA?

B 30°

Ask the following questions to promote MP5: • Which tools would be the most helpful in finding the unknown angle measure?

• How can you estimate the angle measure?

Does your estimate sound reasonable?

C

Differentiation: Support

x°

A

Students use appropriate tools strategically (MP5) when they choose among arithmetic strategies, algebraic strategies, and tape diagrams to determine unknown angle measures.

G

D

F

E

If students need support with identifying ∠BGA, move them forward by helping them trace the angle. Have them place their pencil on point B, trace the angle until point G, and continue to point A. Students can also highlight the interior of ∠BGA.

Arithmetic: 90 - 30 = 60

m∠BGA = 60°

Teacher Note

After they discuss strategies with a partner, have students return to the whole group.

Did you and your partner find the same measurement for ∠BGA? Did you use the same strategy?

My partner and I both found that the measure of ∠BGA is 60°. I used an equation, but my partner drew a tape diagram.

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In grade 6, students are introduced to the symbol m∠ABC, which refers to the measure of ∠ABC.

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EUREKA MATH2 7 ▸ M3 ▸ TB ▸ Lesson 7

Ask a few students to display their work and explain their thinking. If the following strategies are not displayed by students, model them for students.

Tape Diagram

Algebraic

90

Let x represent the number of degrees of the measure of ∠BGA.

x + 30 = 90

?

x + 30 - 30 = 90 - 30 x = 60

30 90 - 30 = 60

m∠BGA = 60°

m∠BGA = 60° Facilitate a class discussion by using the following prompts.

What are some similarities and differences between the strategies?

All strategies lead to the same measurement of 60°.

All strategies use the measure of ∠BGC. All strategies use the angle relationship of ∠BGA and ∠BGC having measures that sum to 90°.

The tape diagram is another visual representation of the situation.

The algebraic strategy requires us to solve an equation.

The arithmetic strategy is straightforward and simple because it doesn’t have a variable.

How do you know that the measures of ∠BGA and ∠BGC sum to 90°?

The little square by the vertex shows us that their measurements sum to 90°.

By definition, two angles with measures that sum to 90° are called complementary angles.

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

Invite a few students to go to the board to sketch other examples of complementary angles. Ask students to explain how they know they have drawn complementary angles.

If students only draw adjacent angles, present the following pair of angles to the class.

70°

Language Support

20°

Consider adding complementary angles to the angle relationship anchor chart.

Ask students if these angles are complementary. If students answer no because the angles are not adjacent, refer them to the definition.

Does our definition say they must be adjacent?

No. It says only that the sum of the angle measures needs to be 90°.

Are these complementary angles? Why? Yes. Because 70 plus 20 is 90, they are complementary angles.

Display the following diagram.

20° 40° 30°

Are these complementary angles?

No. The definition states that complementary angles are two angles with measures that sum to 90°. The diagram shows three angles with measures that sum to 90°.

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EUREKA MATH2 7 ▸ M3 ▸ TB ▸ Lesson 7

If–Then Moves Students use if–then moves to find unknown angle measures.

Display the following statements: • If you live in Kansas, then you live in the United States.

• If it is Wednesday, then it is a weekday.

• If you are in the United States, then you are in North America.

• If it is a weekday, then it is Wednesday.

What do you notice?

All the statements are in the form of If …, then ….

Are all the if–then statements true? Explain your thinking.

The first statement is true because Kansas is in the United States.

The second statement is true because Wednesday is a weekday.

The third statement is true because the United States is in North America.

The fourth statement is false because there are other weekdays besides Wednesday.

Now let’s look at mathematical if–then statements.

Read and display the following statement to the class:

Teacher Note

If a = b, then a + c = b + c.

Assume a, b, and c are numbers. Give a thumbs-up if you agree with the statement and a thumbs-down if you disagree with the statement.

Have students discuss what the statement means with a partner. Invite a few pairs to share their reasoning.

If two quantities are equal, I can add the same amount to both quantities and they will still be equal.

For example:

4=4 4+1=4+1

If students question the limitations of c being a nonzero number in the last if–then statement, ask them the following questions.

• What is the unknown factor equation that represents 4 ÷ 0?

• What number times 0 gives us a product of 4?

Because dividing by 0 is undefined, c cannot be 0 in that statement.

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

Repeat the thumbs-up or thumbs-down process, partner discussions, and sharing for the remaining three if–then moves.

If a = b, then a - c = b - c.

If a = b, then a ⋅ c = b ⋅ c.

If a = b and c ≠ 0, then a ÷ c = b ÷ c.

These statements are known as if–then moves. Let’s take another look at the algebraic method used to solve problem 2.

Display the algebraic method from problem 2.

Let x represent the number of degrees of the measure of ∠BGA.

x + 30 = 90

Teacher Note

x + 30 - 30 = 90 - 30 x = 60 Did we use any if–then moves there? Explain your thinking.

• a+b−b=a

Yes. We subtracted 30 from both sides of the equation.

Now, let’s look at another problem that uses if–then moves.

Direct students to problem 3. Have students work in pairs to complete parts (a) and (b).

3. In the diagram, ∠DAB is a straight angle.

23°

• a−b+b=a • When b ≠ 0, a ⋅ b ÷ b = a.

• When b ≠ 0, a ÷ b ⋅ b = a.

The if–then moves help prepare students for the properties of equality in grade 8.

C x°

In grade 6, students solve equations by using the following properties of operations:

B

A D a. Describe the relationship between ∠DAC and ∠CAB.

∠DAC and ∠CAB are a linear pair, so their measures sum to 180°.

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EUREKA MATH2 7 ▸ M3 ▸ TB ▸ Lesson 7

b. Write an equation for the angle relationship and solve for x.

x + 23 = 180 x + 23 - 23 = 180 - 23 x = 157 The solution is 157.

Have students return to the whole group. Model how to check the solution to an equation by completing part (c) together. Emphasize that substituting the correct solution for the variable creates a true number sentence. If students end up with a false number sentence, then they need to determine whether they made an error when substituting the value or when solving the equation. Tell students that they are expected to check their solutions to equations from now on. c. Check your solution to the equation.

157 + 23 = 180

Teacher Note Problem 3(c) shows each step associated with checking the solution. Moving forward, only the sentence describing how the answer is a solution is provided. But it is expected that students will always complete such steps to check their solutions.

180 = 180 Because 157 + 23 = 180 is a true number sentence, we know 157 is a solution to the equation.

Have students continue to work with their partner to complete problems 3(d)–6. Circulate as students work and listen for instances where students identify if–then moves. d. What is the measure of ∠DAC?

m∠DAC = 157° 4. ∠A and ∠B are complementary angles. The measure of ∠A is 34.5°. Write and solve an equation to determine the measure of ∠B. Check your solution.

Let x represent the number of degrees of the measure of ∠B.

x + 34.5 = 90 x + 34.5 - 34.5 = 90 - 34.5 x = 55.5 The measure of ∠B is 55.5° because 55.5 + 34.5 = 90 is a true number sentence.

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

5. Consider the diagram.

E

C

D 25° x °

25°

B

A a. Describe the relationship among ∠CAE, ∠EAD, and ∠DAB.

∠CAE, ∠EAD, and ∠DAB have measures that sum to 90° b. Write an equation for the angle relationship and solve for x.

25 + x + 25 = 90 x + 50 = 90 x + 50 - 50 = 90 - 50 x = 40 The solution is 40. c. Check your solution to the equation.

Because 25 + 40 + 25 = 90 is a true number sentence, we know 40 is a solution to the equation. d. What is the measure of ∠EAD?

m∠EAD = 40°

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EUREKA MATH2 7 ▸ M3 ▸ TB ▸ Lesson 7

6. In the diagram, ∠CAB is a straight angle.

D

E C

x°

F

x° x° A

(3x)° B

a. Describe the relationship among ∠CAE, ∠EAD, ∠DAF, and ∠FAB.

∠CAE, ∠EAD, ∠DAF, and ∠FAB are angles on a line, so the sum of their measures is 180° b. Write an equation for the angle relationship and solve for x.

x + x + x + 3x = 180 6x = 180 6x ÷ 6 = 180 ÷ 6 x = 30 The solution is 30. c. Check your solution to the equation.

Because 30 + 30 + 30 + 3(30) = 180 is a true number sentence, we know 30 is a solution to the equation. d. What are the measures of ∠CAE, ∠EAD, ∠CAD, and ∠FAB?

m∠CAE

m∠EAD

m∠CAD

m∠FAB

x = 30

x = 30

x + x = 30 + 30

3x = 3(30)

= 60

= 90

m∠CAD = 60°

m∠FAB = 90°

m∠CAE = 30°

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m∠EAD = 30°

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7 ▸ M3 ▸ TB ▸ Lesson 7

EUREKA MATH2

When students are finished, confirm their answers. Then facilitate a class discussion by using the following prompts.

What angle relationships are in problems 3–6?

Problem 3 has a linear pair.

Problem 4 has complementary angles.

Problem 5 has adjacent angles with measures that sum to 90°.

Problem 6 has angles on a line.

When did you use if–then moves?

As we solved each equation, we used an if–then move. Each time, we used it directly after we wrote the equation.

How did you check your solution to each equation?

I took the solution I got and substituted it for the variable in the original equation. When it made a true number sentence, I knew the solution to the equation was correct.

Strategies Students find the value of x.

Direct students to problem 7. Consider having them work independently for about 2 minutes. If students solve arithmetically, challenge them to write and solve an algebraic equation or use a tape diagram. Allow for productive struggle. Students practice solving equations of this form throughout the topic, so it is not imperative that all students exhibit mastery in using an algebraic method now.

When the struggle is no longer productive, or when most students have finished, tell them to compare their work with a partner. Identify specific students to share their strategy with the class.

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EUREKA MATH2 7 ▸ M3 ▸ TB ▸ Lesson 7

7. In the diagram, ∠AFE is a straight angle. What is the measure of ∠BFA? Check your answer.

B

C 36° 36° x°

A

D

x° F

E

Sample:

Teacher Note Students should not assume that angles are right angles. However, students may recognize that ∠AFC and ∠EFC both have a measure of 90°. That is because they are made up of identical pairs of angles and each pair forms half of a straight angle. As long as students can justify their thinking, encourage them to use this information to help them solve the problem.

Arithmetic strategy:

180 - 36 - 36 = 108 108 ÷ 2 = 54 The measure of ∠BFA is 54°.

Because 54 + 36 + 36 + 54 = 180 is a true number sentence, we know that 54° is the measure of ∠BFA. Invite students who used varied solution strategies to share their work with the class. Ask students to look for connections among the strategies. For example, a student who used an arithmetic strategy had to divide 108 by 2 to get 54. We see this same move in the algebraic strategy as 2x = 108.

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EUREKA MATH2

7 ▸ M3 ▸ TB ▸ Lesson 7

Tape Diagram

Algebraic

2x + 36 + 36 = 180

180 x

x

36

36

2x + 72 = 180 2x + 72 - 72 = 180 - 72 2x = 108

180 x

x

2x ÷ 2 = 108 ÷ 2

36

36

x = 54 m∠BFA = 60°

72 180 x

x

36

36

180 – 72

72 180 - 72 = 108

180 x

x 108

x

m∠BFA = 60°

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72 x

108

36

36

2 units = 108 1 unit = 108 ÷ 2 1 unit = 54 x = 54

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EUREKA MATH2 7 ▸ M3 ▸ TB ▸ Lesson 7

Land Debrief

5 min

Objectives: Identify and describe angle relationships given in diagrams.

Write and solve equations that use angle relationships to find unknown angle measures.

Facilitate a class discussion by using the following prompts. Encourage students to add to their classmates’ responses.

What does it mean for two angles to be complementary?

Two angles are complementary if their measures sum to 90°.

How does knowing about angle relationships help us find unknown angle measures?

The angle relationships help us write an equation to represent the situation. For example, when we have complementary angles, we need to write an equation that shows the two angle measures sum to 90°. When we have a linear pair, we need to write an equation that shows that the two angle measures sum to 180°.

How can we use if–then moves to find unknown angle measures?

If–then moves help us solve the equation that represents the angle relationship. If we add, subtract, or multiply by any number on both sides of the equation, the sides remain equal. We can also divide both sides of the equation by any nonzero number and have the sides remain equal.

Exit Ticket

5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

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Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

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Topic B Unknown Angle Measurements In topic A, students use the distributive property to write expressions in equivalent ways.

Students build on this work in topic B when they write and solve equations to find unknown angle measurements.

Students begin the topic by recalling how to name angles and identify angle relationships from a diagram. Students use prior knowledge of angle relationships and their understanding of a new term, complementary angles, to write equations. As students look for efficient methods to solve these equations, they are introduced to the use of if–then moves. Students practice checking solutions to equations by using the original equation and the problem context.

In a digital lesson, students continue to write equations to represent angle relationships.

Students apply their understanding of a new term, supplementary angles. Students find unknown angle measures by using tape diagrams and equations. Students determine that when solving an equation, they can divide both sides of the equation by a nonzero number or multiply both sides of the equation by its multiplicative inverse and get the same result.

Students continue to write and solve equations that represent angle relationships. They are introduced to two new angle relationships: vertical angles and angles at a point. In the last lesson, students apply the new learning from the topic to solve more challenging, multi-step angle relationship problems.

In topic C, students continue working with equations, gaining fluency in solving equations containing all types of rational numbers. In grade 8, the if–then moves are formalized as the properties of equality. Those properties are used in solving more rigorous equations throughout the high school grades.

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If–Then Moves Assume a, b, and c are numbers.

If a = b, then a + c = b + c.

If a = b, then a - c = b - c.

If a = b, then a ⋅ c = b ⋅ c.

If a = b and c ≠ 0, then a ÷ c = b ÷ c.

Vertical Angles and Angles at a Point

D (5x)° E

(7x + 96)° A

35°

C

B

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Progression of Lessons Lesson 7

Angle Relationships and Unknown Angle Measures

Lesson 8

Strategies to Determine Unknown Angle Measures

Lesson 9

Solving Equations to Determine Unknown Angle Measures

Lesson 10 Problem Solving with Unknown Angle Measures

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Before This Module

Overview

Grade 6 Module 4

Expressions, Equations, and Inequalities

In grade 6, students apply properties of operations to solve one-step equations of the forms x + p = q and px = q for cases in which p, q, and x are nonnegative rational numbers. In this module, students extend this work to negative rational numbers and to the forms px + q = r and p(x + q) = r. In grade 6, students write inequalities of the form x > c or x < c to represent a constraint in a problem. Such inequalities have infinitely many solutions that can be represented on a number line. In this module, students extend that learning to include solving inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Students solve inequalities that are less than or equal to and greater than or equal to and interpret the solution set in the context of the problem.

Topic A Equivalent Expressions Topic A centers on writing expressions in equivalent forms. Students move from doing familiar work with numerical expressions to determining when algebraic expressions are equivalent. Through the application of properties of operations—namely the distributive property—as well as the use of the tabular model, students multiply and factor expressions with rational and negative numbers.

Modeling the Distributive Property with the Tabular Model

8

l

–2

8l

–16

8(l - 2) = 8l - 16

Showing Factoring with the Tabular Model

3

5x

–4

15x

–12

15x - 12 = 3(5x - 4)

Topic B Unknown Angle Measurements In topic B, students use familiar and new angle relationships to write and solve equations that help determine unknown angle measures. Students continue to use properties of operations and visual models to solve equations. They are introduced to a new strategy for solving equations: if–then moves.

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If–Then Moves Assume a, b, and c are numbers.

If a = b, then a + c = b + c.

If a = b, then a - c = b - c.

If a = b, then a ⋅ c = b ⋅ c.

If a = b and c ≠ 0, then a ÷ c = b ÷ c.

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Topic C

After This Module

Solving Equations Students continue to use if–then moves in topic C to fluently solve equations of the forms px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Students engage in historical mathematics to determine the advantages and disadvantages of presenting problems rhetorically and symbolically. Later, students use the structure of an equation to make a simpler problem. Throughout this topic, students participate in activities and puzzles to simulate play when fluently solving equations. They end the q x topic by exploring a new type of equation, p = r , to foreshadow work with proportional reasoning in module 5.

Grade 7 Modules 4 and 5 In module 4, students extend solving equations to applying formulas to determine area, circumference, and surface area. Students solve percent equations in module 5.

Grade 8 If–then moves are formalized as the properties of equality in grade 8 and are used to solve more rigorous equations throughout high school.

Topic D Inequalities In topic D, students apply what they know about solving equations to solve inequalities.

They begin by graphing boundary numbers and testing numbers to determine the correct region on the number line to shade. Students determine whether solving by using if–then moves is more efficient. Through experimentation, students notice that when both sides of an inequality are multiplied or divided by a negative number, the inequality sign must be reversed to maintain a true number sentence. This discovery necessitates additional if–then moves for inequalities.

The work done in this module supports work with linear equations in one and two variables in grade 8. Students also extend their understanding of equivalent expressions when they solve systems of linear equations in grade 8.

If–Then Statements for Inequalities Let a, b, and c represent numbers.

If a < b, then a + c < b + c.

If a < b, then a - c < b - c.

If a < b and c is a positive number, then a ⋅ c < b ⋅ c.

If a < b and c is a negative number, then a ⋅ c > b ⋅ c.

If a < b and c is a positive number, then a ÷ c < b ÷ c.

If a < b and c is a negative number, then a ÷ c > b ÷ c.

Versions of these statements can be written to begin with >, ≤, or ≥ instead of <.

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Why Expressions, Equations, and Inequalities I notice that this module does not encourage the use of manipulatives. Why not?

The use of manipulatives can support student engagement and provide differentiation and equity. Manipulatives can promote student thinking and aid in communicating about the mathematics being learned. Manipulatives often bridge learning from the conceptual stage to the pictorial or abstract stages of learning. However, students may lose the chance to deepen their understanding of concepts if manipulatives are used in isolation from mathematical connection.

Algebra tiles are a widely used manipulative to engage students in understanding the “rules” for solving equations. Although the tiles can be helpful in representing variables and constants, these and other difficulties may occur when students employ these manipulatives: 1. Students must assign numbers and variables to a set of colored tiles. Students are then tasked with remembering which color and shape of tile represents a positive number, which color and shape represents a negative number, and which color and shape represents a variable.

2. Use of colored tiles connects easily to addition, but some learners cannot conceptualize subtracting a negative number or multiplying when the first factor is negative. Some interpretations of division cannot be modeled by using the tiles.

3. Tiles do not represent non-integer rational numbers and cannot be used to model arithmetic with rational numbers.

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Further, use of algebra tiles often imposes procedural directions that mask the mathematics happening when an equation is being solved. The properties of operations, and, in this module, the if–then moves, are not clear when focus is placed on the movement of the manipulative rather than on the solving of the equation.

I notice that this module includes standards for geometry. Why are these standards addressed in this module?

Students understand and apply angle relationships to determine unknown angle measures. These relationships necessitate equivalence. To determine whether angles are complementary, students understand that the two angle measures must sum to 90°. To determine whether angles are supplementary, students understand that the two angle measures must sum to 180°. A natural approach to determine unknown angle measures in these and other cases is to solve for the unknown by using an equation. Determining unknown angle measures drives the need to solve equations. Students use equations to show why angles are equal in measure.

Why is the word simplified not used in this module?

The word simplified has multiple meanings depending on the different situations in which it is used. For example, when asked to simplify a fraction, students perform a different task than they would when they simplify an expression. When a student is directed to simplify in any situation, a specific description of the term should be provided that is appropriate for that situation. If you choose to have students simplify expressions in this module, we recommend that you define simplify for each case in which it is used, and we encourage you to accept all equivalent forms of the expression as correct.

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Supplemental Materials

Lead: Facilitating Successful Implementation of Eureka Math2, 6–9

Credits Great Minds® has made every effort to obtain permission for the reprinting of all copyrighted material.

If any owner of copyrighted material is not acknowledged herein, please contact Great Minds for proper acknowledgement in all future editions and reprints of this handout.

Works Cited Great Minds. Eureka Math2TM. Washington, DC: Great Minds, 2021. https://greatminds.org/math.

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Implementation Support Tool 6–9 A Story of Ratios® A Story of Functions®

Implementation Support Tool 6–9

Overview Using Eureka Math2 to its fullest potential takes time, reflection, and continuous intentional preparation. This process is accelerated with the partnership between a teacher and an instructional coach (e.g., administrators, district level coaches, facilitators). The Implementation Support Tool (IST) supports both teachers and coaches in creating the optimal experience for every student in a Eureka Math2 classroom. It describes teacher practices that are essential to each component of a Eureka Math2 lesson.

Understanding the Implementation Support Tool The IST has the look of an evaluation rubric and observation checklist; however, it is not intended to be either. Understanding the IST and its structure helps to properly inform coaching, reflection, and preparation. The bullets below outline these important understandings.

•

There is not a linear progression across the IST. Teacher actions described in the rightmost column do not necessarily have greater instructional impact than teacher actions described in the leftmost column. Always consider the impact that the teacher actions described on the IST will have on student learning independently or in direct comparison with other teacher actions.

•

The teacher actions described on the IST are not all-or-nothing actions. Each can be performed with varying degrees of effectiveness.

•

The teacher actions described on the IST are like the keys on a piano. Just as playing a song requires playing the right notes at the right times, exemplary instruction requires using the correct teacher actions at the appropriate times for appropriate reasons. Not every teacher action is appropriate at all times.

•

Most teacher actions on the IST are strengthened by the presence of other teacher actions. Before focusing on the development of a teacher action, check if there is a prerequisite teacher action that should be developed first.

Using the Implementation Support Tool Like most tools, the IST is most effective when used as intended. This tool is not meant to be used in an evaluative fashion or as a checklist during teacher observation. Instead, it supports implementation the following ways.

•

Coaches can use the IST to develop a deeper understanding of Eureka Math2 during early implementation observations.

•

Coaches can use the IST to support their analysis of observation data following its collection. It is not recommended for use during classroom observations because it can distract coaches from collecting as much specific, objective observation data as possible.

•

Coaches can use the IST to name instructional priorities for targeted professional development, PLCs, and 1:1 coaching cycles.

•

Coaches can refer to the IST for shared language during feedback conversations with teachers following observation.

•

Coaches can use the IST with teachers to prompt goal setting and self-reflection.

•

Individual teachers or teams can use the IST during lesson preparation to look for opportunities to leverage prioritized teacher actions.

•

Individual teachers can develop their practice by using the IST to reflect on their current instruction. This reflection can be supported by using the IST to analyze recorded classroom footage.

Note: For an explanation of the structure of the IST, refer to Appendix I: Structure. © Great Minds PBC

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Implementation Support Tool 6–9

Fluency (Optional) In A Story of Ratios and A Story of Functions, Fluency is not part of the 45-minute lesson structure. If lesson time permits, a fluency activity can be administered as a bell ringer or adapted as one of the teacher-led routines below.

Fluency uses activities that solidify and build students’ ability to use mathematical procedures flexibly, accurately, efficiently, and appropriately. Students become familiar with fluency routines because of their consistent use across modules and grade levels, allowing for efficient teaching and learning. All fluency routines benefit from the general fluency indicators below. The most common routines—Whiteboard Exchange, Choral Response, Count By, and Sprints—have additional sets of indicators to support implementation.

Choral Response

Whiteboard Exchange

General Fluency Indicators

Core Implementation

Instructional Habits

Adaptive Implementation

Meeting Component Purpose a. Completes fluency activity within or near suggested time (10 minutes for Sprints, 3–5 minutes for all other routines) b. States directions clearly and includes all necessary details for engagement (e.g., topic/task, timeframe, cue) c. Prepares the environment to optimize engagement and learning (e.g., location of the activity, materials, concrete manipulatives, visuals)

Engaging and Monitoring indicators vary by routine and are included for specific fluency routines only.

Using Routines a. Uses the suggested sequence to build fluency through repeated practice with a targeted concept or skill b. Gives immediate, concise, and specific feedback to each student based on the accuracy of their response (e.g., asks questions, prompts to include the unit) c. Returns to students with incorrect initial response to validate their corrections

Engaging d. Provides appropriate work time before signaling students to show their whiteboards e. Establishes expectations for participation to maintain efficiency and ensure student accountability Monitoring f. Scans every student’s whiteboard to notice and note trends, varied solutions, exemplars, and misconceptions

Responding g. Adjusts the sequence of problems (e.g., complexity, level of abstraction) based on the existence of pervasive trends

Using Routines a. Uses the suggested sequence to build fluency through repeated practice with a targeted concept or skill b. Uses established cues (hand signals or verbal indicators) to prompt every student to respond in unison c. Displays or states correct responses

Engaging d. Provides appropriate think time before signaling students to chorally respond Monitoring e. Listens to students’ responses to confirm accuracy and recognize outliers (e.g., mistakes, lack of participation)

Responding f. Prompts students to use precise mathematical language g. Adjusts the sequence of prompts (i.e., decreased or increased complexity) to improve access or provide challenge h. Responds to errors with scaffolds such as questioning or concrete or pictorial supports to improve access

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Promoting Discourse indicators are excluded from the Fluency section of the IST due to the nature of the lesson component.

April 2022

Customizing d. Administers alternate fluency activity to increase engagement or in response to a demonstrated student need e. Scaffolds the sequence with additional problems to provide access (e.g., lower complexity initial problems, intermediate problems to ease complexity, extension problems) Responding f. Models or thinks aloud about a parallel problem or sequence to address a widespread, small-gap misconception and then re-engages students in a similar sequence (excluding Sprints)

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Implementation Support Tool 6–9

Fluency, Continued

Sprints

Count By

Core Implementation

Instructional Habits

Using Routines a.

Uses concise, clear signals to guide students to count in unison (upward, downward, and stopping) b. Repeats practice of counting up and down to help students commit sequences to memory and recognize patterns c.

States unit, starting number, and ending number prior to activity

Engaging d. Uses appropriate pace for the complexity and familiarity of the count

Using Routines a.

Directs students to complete the sample problems to ensure their understanding of the purpose of the sprint b. Frames the sprints as an opportunity for growth (e.g., encouraging and celebrating effort, success, and improvement) c.

Provides 1 minute for students to complete as many problems as possible, in order without skipping d. Reviews the correct answers to each sprint at a brisk pace and with an established verbal cue that allows for all students to follow along e. Provides an opportunity for students to complete more problems on sprint A or to analyze and discuss the patterns in Sprint A to increase success on sprint B f.

Directs students to record their performance and to calculate and celebrate growth after sprint B

Engaging g. Provides appropriate think time for students to internalize patterns before transitioning to sprint B

Monitoring e. Listens for accurate counting by all students f. Listens to key junctures or anticipated points of struggle within the sequence (e.g., crossing over ten or hundred) to assess for fluency

Monitoring h. Circulates during the sprint to provide encouragement, ensure students are completing the problems in order, and look for areas where fluency breaks down i. Listens to student responses during correction to identify where a discussion about patterns is warranted j. Listens to peer-to-peer discussions about patterns in sprint A to ensure students have named high-leverage takeaways to apply in sprint B

Adaptive Implementation

Responding g. Alters the pace of the count based on student responses h. Focuses on counting up and down at key junctures or known points of struggle within the sequence i. Responds to errors by pausing the count to ask targeted questions or to provide concrete/pictorial supports j. Gradually removes scaffolds to increase academic ownership

Responding k. Prompts students to analyze specific sequences of problems in preparation for current and future lessons l. Names or has students name the high-leverage patterns identified within peer-to-peer discussions prior to sprint B

Students Expectations

During fluency activities, students in a Eureka Math2 classroom should do the following: • Participate fully (verbally, on whiteboards, with hands when appropriate, etc.)

• Respond to established signals and prompts • Include units in responses, when appropriate • Make corrections, when appropriate

© Great Minds PBC

In addition to the indicators for all fluency activities, during sprints, students in a Eureka Math2 classroom should also do the following: • Complete problems in order, with urgency, for 60 seconds • Look for and communicate about the patterns in the sprint • Work to improve fluency by looking for patterns and applying them

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Implementation Support Tool 6–9

Launch Launch creates an accessible entry point to the day’s learning through activities that build context, create a need for new learning, or activate prior knowledge. Every Launch ends with a transition statement that sets the goal for the day’s learning.

Core Implementation

Meeting Component Purpose a.

Aligns facilitation to the purpose statement of Launch

b.

Provides opportunities indicated by lesson for students to notice; wonder; apply; and articulate strategies, choice of models, and understandings

c.

Honors the amount of teacher-to-student discourse, student-tostudent discourse, and whole-class discourse within Launch

d.

Concludes Launch by using the given transition statement to set the goal for and make an explicit connection to the day’s learning

e.

Completes the Launch within or near the time indicated

Using Structures, Routines, and Activities f.

States directions clearly and includes all necessary details for engagement (e.g., topic/task, thinking job, timeframe, cues, peer interaction) g. Prepares the environment to optimize engagement, collaboration, and learning (e.g., desk arrangement, Thinking Tool, Talking Tool, materials, visuals)

Instructional Habits

Engaging h. Provides appropriate think time for students to process questions or displayed content and form responses i.

Encourages flexible thinking (e.g., varied models and strategies) and connections to prior knowledge to access content

Monitoring j. Circulates to recognize and encourage application of mathematical understanding k. Listens to student discourse for understanding, connections to prior knowledge, and evolving reasoning l. Monitors student work to notice and note trends, varied models and strategies, exemplars, and misconceptions to leverage during class discussion Promoting Discourse m. Prompts students to elaborate (e.g., by saying “Why?” “How do you know?” or “Tell me more”) to make student thinking more visible, clear, and complete

Adaptive Implementation

Customizing p. Uses the margin notes to improve access (e.g., UDL, Differentiation: Support, Language Support, strategic pairings/groupings) Responding q.

Calls on students or selects student work strategically to highlight trends, varied solutions, connections to prior content, and misconceptions

r.

Models the use of or directs students to use specific sections of the Talking Tool and Thinking Tool to support discourse and metacognition

n. Asks questions that invite students to make connections between solutions, models, strategies, and previous lessons, modules, or grade levels o. Prompts students to restate, build on, or evaluate other students’ responses to enhance discussion and strengthen habits of discussion Student Expectations

During Launch, students in a Eureka Math2 classroom should do the following: • • • • •

Generate, test, share, critique, and refine their ideas Engage in student-to-student and whole-class discourse Process presented images and information by noticing and wondering Formulate and articulate their thinking regarding solution pathways, strategies, models/representations, and understandings Ask and answer questions when engaging in routines and activities

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Implementation Support Tool 6–9

Learn Learn presents new learning related to the lesson objective, usually through a series of instructional segments. This lesson component takes most of the instructional time. Suggested facilitation styles vary and may include explicit instruction, guided instruction, group work, partner activities, and digital elements.

Core Implementation

Instructional Habits

Engaging

Meeting Component Purpose a. b.

k.

Focuses the progression of questioning, thinking, and discourse in alignment with the purpose statement of each Learn segment Honors the ratio of questioning to direct instruction, the amount of discourse, and the format of the Releases responsibility to students while working on lesson pages, as indicated by the lesson and

l.

Uses tools and models accurately to develop student understanding of the lesson’s strategies and

m.

f.

Completes Learn within or near the total time indicated, allowing time for instruction, lesson pages,

States directions clearly and includes all necessary details for engagement (e.g., topic/task, thinking job, timeframe, cues, peer interaction)

Promoting Discourse o.

Prepares the environment to optimize engagement, learning, and collaboration (e.g., desk

you know?” or “Tell me more”), and use precise language to advance student thinking

Uses structures, routines, and activities as opportunities for students to develop their Incorporates Read–Represent–Solve–Summarize (RRSS) process (grades 6–8) or Modeling Cycle

p.

Asks questions that invite students to make connections between solutions; models; strategies; and previous questions, lessons,

(Algebra I) for solving word problems

Calls on students or selects student work strategically to highlight trends, varied solutions, misconceptions, and broader mathematical understanding

t.

Models or directs students to use specific sections of the Talking Tool and Thinking Tool to support metacognition and discourse

u.

Verifies possible solutions throughout Learn by inviting students to share work or revealing objective-aligned

Prompts students to elaborate or clarify responses (e.g., by saying “Why?” “How do

understanding of the day’s objectives and to articulate their strategies, models, and understandings

j.

s.

trends, varied models and strategies,

arrangement, Thinking Tool, Talking Tool, materials, visuals) i.

understanding to existing problems in the sequence

exemplars, and misconceptions

Using Structures, Routines, and Activities

h.

Monitors student work to notice and note

Creates and inserts problems that bridge or reinforce

Responding

use precise mathematical terminology n.

and Practice Problems g.

pairings/groupings) r.

ability to articulate their understanding and

Uses the mathematical language of the lesson or related language (e.g., ratio relationship and value of the ratio) to demonstrate precise yet accessible terminology

Listens to student discourse to gauge students’

Uses the margin notes to improve access (e.g., UDL, Differentiation: Support, Language Support, strategic

Circulates to recognize and encourage application of mathematical understanding

objectives e.

q.

form responses

Monitoring

student performance d.

Customizing

to process questions or displayed content and

discourse so that students generate, test, share, critique, and refine their ideas c.

Provides appropriate think time for students

Adaptive Implementation

exemplars v.

Addresses learner variance by challenging or supporting students (e.g., modeling, targeted questioning, connecting to prior learning, adjusting complexity)

w. Adjusts pacing within Learn segments to meet the needs of students while honoring the objective

modules, or grade levels Student Expectations

During Learn, students in a Eureka Math2 classroom should do the following: •

Apply concepts, skills, models, and strategies connected to the lesson objective to solve problems

•

Make thinking visible by using numbers, words, models, and tools

•

Attempt to use accurate mathematical language in discourse and writing

•

Ask questions to clarify their thinking or understand the reasoning of others

•

Articulate understanding in whole-class, peer-to-peer, and teacher-to-student discourse

•

Make connections between concepts and skills and between current and previous content

•

Use the Read–Represent–Solve–Summarize or Modeling Cycle process to solve word problems

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Implementation Support Tool 6–9

Practice Problems Practice problems may be assigned for completion outside of class or used during additional instructional time as an opportunity for in-class independent practice. The indicators below should be used when a teacher has assigned Practice problems to be used during class time.

Core Implementation

Meeting Component Purpose a. Allots time for every student to work on the Practice problems independent of teacher guidance b. States directions clearly and includes all necessary details for engagement (e.g., topic/task, timeframe, peer interaction)

Instructional Habits

Engaging c.

Limits time with individual students to prioritize monitoring every student’s work d. Asks students questions to prompt the continuation of work while maintaining the cognitive lift Monitoring e. Monitors every student’s work to notice and note trends, varied models and strategies, exemplars, and misconceptions Promoting Discourse indicators are excluded from the Practice problem section of the IST due to the nature of the lesson component.

Adaptive Implementation

Customizing f.

Designates the order of problems to complete to ensure practice with the most important components of the lesson objective g.

Creates and assigns additional problem sequences, as needed, to provide access or advance learning

Responding h. Directs students to leverage classroom resources (e.g., anchor charts, other examples in their own work, Talking Tool, Thinking Tool, Lesson Recap) i.

Responds to pervasive misconceptions by questioning or modeling with a parallel example to provide a scaffold in real time

Student Expectations

While working on given Practice problems, students in a Eureka Math2 classroom should do the following: • • • • •

Engage in practice, independent of the teacher, for the allotted time Think about, make sense of, and represent problems before solving them Show work and explain their thinking as indicated Refer to resources (e.g., anchor charts, other examples from the lesson, Thinking Tool, Lesson Recap) for support before asking for help Adjust existing work based on teacher feedback

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Implementation Support Tool 6–9

Land Land helps facilitate a brief discussion to close the lesson and provides students with an opportunity to complete the Exit Ticket. Suggested questions, including key questions related to the objective, help students synthesize the day’s learning. The Exit Ticket provides a window into what student understand that can be used to inform instructional decisions.

Core Implementation

Instructional Habits

Meeting Component Purpose a.

Engaging

Asks questions that elicit student thinking and

e.

cause them to synthesize the day’s learning (e.g., suggested questions, key questions)

f.

Debrief

lesson objectives, highlighting important concepts and vocabulary Honors the amount of student-to-student discourse Completes the Debrief within or near the indicated time

j.

improve access to discussion g.

Creates questions, in response to previously collected data, that promote synthesis of the lesson or topic

Provides students with supports (e.g., Talking Tool, mathematical terminology) to

Responding

k. Affirms accuracy and efficiency in student responses

Monitoring

and whole-class discourse within Land d.

Customizing

Provides adequate think time for students to process questions or displayed content and to form responses

b. Focuses student discussion to align with the

c.

Adaptive Implementation

while clarifying any inaccuracies and deepening

Listens to student discourse to gauge understanding and identify opportunities to push for clearer thinking

understanding

l.

Promoting Discourse

Revisits questions from earlier in the lesson to check for growth in understanding

h. Prompts students to elaborate or clarify responses (e.g., by saying “Why?” “How do you know?” or “Tell me more”), or to use precise language to advance student thinking

i.

Prompts students to restate, build on, or evaluate other students’ responses to enhance discussion and strengthen habits of discussion

Meeting Component Purpose

Engaging

a. Allows an amount of time within or near the

c.

indicated time allotted for students to

support independence

independently complete as much of their Exit Ticket

Exit Ticket

Responding

Makes tools (e.g., anchor charts, concrete manipulatives, Thinking Tool) available to

b. Collects Exit Tickets to analyze qualitative data to identify strengths to leverage in future lessons, varied solution paths, and misconceptions to

Designates the order of problems for students to complete to ensure students answer the problems

Monitoring

that highlight the most important components of the

d. Circulates room while students work on their Exit Tickets to ensure student

as possible

f.

lesson objective (within the given time)

accountability and gather formative data around unproductive struggle

e. Monitors every student’s work to note trends and confirm the effectiveness of the lesson to inform customization of the next day’s lesson Promoting Discourse indicators are excluded from the Exit Ticket section of the IST due to the

address

nature of the lesson component.

Student Expectations

During the Debrief component of Land, students in a Eureka

Math2

classroom should refer to the day’s activities to do the

following:

During the Exit Ticket component of Land, students in a Eureka Math2 classroom should do the following: •

Solve a problem representative of the lesson objective

•

Attempt to use accurate mathematical language when synthesizing the lesson’s key understandings

•

Make thinking visible by using numbers, words, and models

•

Engage in discourse to articulate understanding, gain the insights of others, and evaluate errors

•

Complete as much of the Exit Ticket as possible

•

Reflect on their own thinking following analysis of peer work and discourse

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Supplemental Materials

Lead: Facilitating Successful Implementation of Eureka Math2, 6–9

Appendix B Scenarios Scenario 1: Imagine that a 6th grade teacher comes to your office and says, “I’m used to teaching rational numbers at the beginning of the year, so I was going to adjust Module 2 and 3 to come before Module 1.”

Sample Response: “The writers of the curriculum carefully crafted the scope and sequence, considering the placement of content down to the lesson level. There is a logical reason for the organization of the modules. Let’s go into the front matter of the modules to find out why Modules 2 and 3 come after Module 1.”

Scenario 2: “My students didn't get Eureka Math2 last year, so they don't understand the way double number lines are introduced in this lesson. And tomorrow's lesson moves too quickly. I need to add a review day before I can teach this lesson and then after this lesson to prepare my students for tomorrow.”

Sample Response: ““Let’s go back and revisit the module and topic overview to see how learning progresses throughout to create a plan to support students in accessing the day’s lesson. We only get a few days to use for responsive teaching, so we want to make sure those days are strategically placed, just in time.”

In a scenario such as this, it may be helpful to revisit the module and topic to see where the lesson fits in to the trajectory of learning and return to the lesson to identify the purpose and role of that lesson in that trajectory of learning. Then, review the lesson to see how the learning strategically builds and creates multiple entry points for students. Identify support for learning that will allow students to access the day’s lesson. You may also want to investigate the multiple instances of double number line usage to ensure the teacher sees that students will have continued opportunities to use and become more proficient with using them.

Scenario 3: You are completing a walkthrough and it is 15 minutes in to the 60-minute math block and the teacher is checking students’ homework while they are completing a math worksheet.

Sample Response: Pacing will be impacted by the decision because there are only 30 minutes left in the block and 45 minutes of content in the lesson. The response to this will depend on the specific situation and purpose of the observation. You might want to determine the reason the teacher is using class time to review the previous day’s learning. You could say something like, “The way the Eureka Math2 lessons are designed, you can be confident that there are multiple entry points embedded into the lesson to help activate prior knowledge.”

Scenario 4: The seventh-grade team has a 80-minute math block, on a block schedule. A teacher approaches you and says that they are unclear on how to fit two Eureka Math2 lessons in the allotted time. What do you do?

Sample Response: It is recommended to reevaluate the schedule to a lot 60 minutes for second through fifth grade Eureka Math2 lessons. In order to complete two Eureka Math2 lessons in the same block, it is recommended to reevaluate the schedule to allot 90 minutes. We can use our

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Supplemental Materials

Lead: Facilitating Successful Implementation of Eureka Math2, 6–9 study of the trajectory of learning through module and topic level resources to make informed adjustments to the lessons. Understanding the purpose of each of the related lessons, and how they build, will allow us to make decisions about what to prioritize, shorten or even consolidate, to ensure that they fit within our time frame without sacrificing the big ideas, rigor and coherence of the lessons.

We can also use our data from previous Exit Tickets to determine students’ entry points and help us make strategic decisions about the lessons.

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Eureka Math2® Implementation Benchmarks Overview The Eureka Math2 curriculum aims to help all students think deeply about mathematics and become critical thinkers and problem solvers. A successful implementation of Eureka Math2 takes dedication from all stakeholders and progresses through the following four phases.

Engage

Experience

Enhance

Exemplify

Prior to Year 1

Year 1

~Years 2-3

~Years 4 and Beyond

Engage: Preparing to launch Eureka Math2

Experience: Learning and exploring Eureka Math2

Enhance: Extending knowledge of Eureka Math2

Exemplify: Skillfully implementing Eureka Math2

Teachers and leaders ensure that plans and materials are in place for the implementation of Eureka Math2. All members of the learning community understand the rationale behind the adoption and are invested in its success.

Teachers and leaders gain knowledge of Eureka Math2.

They identify and explore key structures and aspects of the curriculum. As the learning community gains experience, implementation results may vary across classrooms and schools.

Teachers and leaders increase their understanding of and familiarity with Eureka Math2. They become more consistent in their pacing and lesson facilitation. Educators customize lessons to meet students’ needs while maintaining the curriculum’s rigor and intentionality.

Teachers and leaders facilitate a highly effective implementation of Eureka Math2 across classrooms and schools. Educators effectively maintain pacing and respond to student data when planning instruction.

Notes:  This resource is not intended as an evaluative tool. Instead, it should guide the progression of an implementation as each phase builds on previous phases. 

The timeline provided is a guideline. Previous experience with Eureka Math may lead to progressing through the phases at a faster pace.



In this resource, leaders may include, but are not limited to, district administrators, curriculum directors, principals, assistant principals, and instructional coaches. Teachers include, but are not limited to, general education teachers, special education teachers, intervention specialists, and paraprofessionals.

Eureka Math2® Implementation Benchmarks—Engage Phase (Prior to Year 1)1

Engage Preparing to launch Eureka Math2.

Experience

Leaders  

 





Identify individuals to lead and support implementation and define their roles.

Ensure access to all print and digital curriculum materials for leaders, teachers, and students.

Plan professional development for leaders and teachers.

Participate in Lead: Facilitating Successful Implementation professional development.

Introduce the learning community to Eureka Math2 and ensure buy-in for the implementation.

Understand how Eureka Math2 assessments guide instruction.

Enhance

Teachers 

 

Participate in Launch: Bringing the Curriculum to Life or Power Up: Transitioning to Eureka Math Squared professional development.2 Preview print and digital curriculum resources.

Organize materials such as teacher editions, student materials, and manipulatives.

Exemplify

Students 

There are no student actions during the Engage phase.

The timeline mentioned is a guideline. A specific implementation may move through these phases more or less quickly.

The Power Up session is designed for teachers with prior Eureka Math experience.

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2

Eureka Math2® Implementation Benchmarks – Experience Phase (Year 1)1

Engage

Experience

Leaders 







Set expectations for teachers regarding the implementation of Eureka Math2 (e.g., use Eureka Math2 daily).

Use observations and feedback conversations to support teachers toward optimal implementation of Eureka Math2 .

Develop a culture of curriculum study and provide supporting structures for teacher collaboration and planning.

Celebrate teachers’ progress in attempts to incorporate new practices from the curriculum.

Exemplify

Teachers 

  





1

Enhance

Learning and exploring Eureka Math2.

Participate in Teach: Effective Instruction with Eureka Math2 and Assess: Embedded Opportunities to Support Instruction professional development sessions.

Study curriculum materials to prepare for instruction.

Follow the Fluency, Launch, Learn, and Land lesson structure.

Learn the models and strategies emphasized in Eureka Math2 and use them as indicated.

Maintain pacing, with the understanding that students develop proficiency over time.

Introduce instructional routines (ie: What Doesn’t Belong, Math Chat) found in the curriculum.

Students 

   

Learn the models and strategies emphasized in Eureka Math2 and use them as indicated.

Provide written and verbal explanations of mathematical problem solving.

Ask and answer mathematical questions.

Engage in mathematical discourse.

Actively engage in instructional routines (ie: What Doesn’t Belong, Math Chat) found in the curriculum.

The timeline mentioned is a guideline. A specific implementation may move through these phases more or less quickly.

Eureka Math2® Implementation Benchmarks – Enhance Phase (~Years 2–3)1

Engage

Experience

Leaders 







Enhance

Plan professional development and additional support for new teachers or teachers new to a grade level.

Use observations and feedback conversations to ensure lesson customizations maintain fidelity to the lesson objectives and support students’ needs.

Establish structures for using assessments to effectively inform instructional decisions.

Provide feedback that praises progress and pushes practice on content and instruction.

Exemplify

Extending knowledge of Eureka Math2.

Teachers 









Participate in the Adapt: Optimizing Instruction and Inspire: Discourse, Engagement, and Identity professional development.

Effectively use the curriculum materials and student data to adapt daily instruction.

Facilitate instructional routines, highlighting the Standards for Mathematical Practice, to promote student discourse.

Encourage students to use models and strategies flexibly to solve problems and build their understanding of mathematics.

Use data from assessment reports (Topic Quizzes, Module Assessments, etc.) to inform instructional choices.

Students 





Use models and strategies flexibly to solve problems and build understanding of mathematics.

Use mathematical language in verbal and written communication about mathematics and problem solving.

Demonstrate persistence in learning math and solving problems.

The timeline mentioned is a guideline. A specific implementation may move through these phases more or less quickly.

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Eureka Math2® Implementation Benchmarks – Exemplify Phase (~Years 4 and Beyond)1

Engage









Enhance

Experience

Leaders Maintain systems for ongoing professional development, collaboration, and planning, accounting for varied experiences of teachers.

Use observations and feedback conversations to ensure lesson customizations differentiate based on students’ needs.

Analyze classroom and school data to identify inequity and implementation concerns, and make a plan to address them.

Maintain systems for teacher development to ensure all teachers receive frequent support, observation, and feedback.

Exemplify Skillfully implementing Eureka Math2.

Teachers  



 

Develop understanding of alignment with prior and successive grade levels.

Make connections to prior or upcoming content, provide scaffolds and extend learning.

Use data from assessments to inform instruction to meet whole class, small group, and individual student needs.

Promote student-initiated and studentto-student discourse.

Generously share experiences to mentor teachers new to the curriculum.

Students     

See themselves as mathematicians, expressing confidence in their ideas.

Independently notice, wonder, and make connections to prior learning.

Consistently provide clear explanations of problem solving processes.

Offer critiques of others’ mathematical work.

Demonstrate curiosity through mathematical wonderings.

The timeline mentioned is a guideline. A specific implementation may move through these phases more or less quickly.

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