2 Operations with Fractions and Multi-Digit Numbers
3 Rational Numbers
4 Expressions and One-Step Equations
5 Area, Surface Area, and Volume
6 Statistics
Before This Module
Grade 4 Module 2
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.
Overview
Ratios, Rates, and Percents
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.
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.
Number of Packets of Sugar
Number of Roses
Number of Da isies
Topic C
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.
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.
After This Module
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.
Ratios, Rates, and Percents
Why
Achievement Descriptors: Overview
Topic A
Ratios
Lesson 1
Jars of Jelly Beans
• Use multiplicative reasoning to estimate the solution to a real-world problem.
Lesson 2
Introduction to Ratios
• Write ratios that relate two quantities as an ordered pair of numbers.
• Use ratio language to compare two quantities.
Lesson 3
Ratios and Tape Diagrams
• Write multiple ratios to describe the same situation.
• Represent ratios with tape diagrams.
Lesson 4 .
Exploring Ratios by Making Batches
• Create ratios by making batches of different quantities.
• Use tape diagrams to determine unknown quantities in ratios.
Lesson 5
Equivalent Ratios
• Find equivalent ratios by multiplying both numbers in a given ratio by the same nonzero number.
• Use equivalent ratios to find unknown quantities.
Topic B
Collections of Equivalent Ratios
Lesson 6
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
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.
Lesson 8
Addition Patterns in Ratio Relationships
44
74
• Use addition patterns in tables and graphs of equivalent ratios to describe ratio relationships and find unknown quantities.
Lesson 9
Multiplication Patterns in Ratio Relationships
• Use graphs and tables to explore multiplication patterns in ratio relationships.
• Use multiplication to complete ratio tables.
Lesson 10
Multiplicative Reasoning in Ratio Relationships
• Write and use equivalent ratios when one of the numbers in the ratio is 1.
Lesson 11
Applications of Ratio Reasoning
• Solve multi-step ratio problems by reasoning about equivalent ratios.
Topic C
Comparing Ratio Relationships
Lesson 12
Multiple Ratio Relationships
• Compare ratio relationships by using graphs, tables, and double number lines.
Lesson 13
Comparing Ratio Relationships, Part 1
• Compare ratio relationships by using ratio tables.
Lesson 14
Comparing Ratio Relationships, Part 2
• Compare ratio relationships by creating equivalent ratios.
Lesson 15
The Value of the Ratio
• Compare ratio relationships by using the value of the ratio.
Topic D
Rates
Lesson 16
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.
Lesson 19
Using Rates to Convert Units
• Convert units of measurement by applying rate reasoning.
354
Lesson 20 .
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.
• 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
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
Finding a Part
• Calculate a part when given the whole and a percent.
Lesson 25
Finding the Whole
• Calculate the whole when given a part and a percent.
Lesson 26
Solving Percent Problems
• Solve multi-step percent problems.
Resources
Standards
Achievement Descriptors: Proficiency Indicators
Terminology
Math Past
Sample Solutions
Works Cited
Acknowledgments
Why
Ratios, Rates, and Percents
Ratios are shown as A : B and A to B. Why not A B ?
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 A B 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 1 and 1 2 , however, they may be tempted to find the sum and assume that the new ratio is 3 2 , 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 A B 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.
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.
Number of Peaches
Number of Kiwis
• 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.
• 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.
• 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
0123456871 1901
Number of Slices of Bread
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 hour , or as fractions, such as 60 miles 1hour . Rather, grade 6 students write this rate as 60 miles per hour and 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.
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.Mod1.AD2 Write and explain the unit rate that describes a relationship between two quantities.
6.Mod1.AD3 Solve real-world and mathematical problems by using ratio reasoning.
6.Mod1.AD4 Represent ratio relationships by using tables and the coordinate plane.
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.Mod1.AD7 Model and explain percents and problems involving percents.
6.Mod1.AD8 Solve problems that involve finding the part, whole, or percent.
6.Mod1.AD9 Convert among units by using ratio reasoning to solve problems.
6.RP.A.3.b
6.RP.A.3.c
6.RP.A.3.c
6.RP.A.3.d
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 Proficient and/or Highly Proficient performance.
An example of one of these ADs, along with its proficiency indicators, is shown here for reference. The complete set of this module’s ADs with proficiency indicators can be found in the Achievement Descriptors: Proficiency Indicators resource.
6.Mod1.AD8 Solve problems that involve finding the part, whole, or percent.
RELATED CCSSM
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 concisely describes what will be assessed.
• AD Indicators: The indicators describe the precise expectations of the AD for the given proficiency category.
• Related Standard: This identifies the standard or parts of standards from the Common Core State Standards that the AD addresses.
6.RP.A.3.c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30 100 times the quantity); solve problems involving finding the whole, given a part and the percent.
Language
Sana has 7 mystery novels. She says that 35% of her novels are mystery novels. What is the total number of novels Sana has?
6.Mod1.AD9 Convert among units by using ratio reasoning to solve problems.
RELATED CCSSM
6.RP.A.3.d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Indicators AD Code: Grade.Mod#.AD#
Convert among nonmixed units by using ratio reasoning.
10 in = _____cm
(1 in = 2.54 cm)
Convert among units, including mixed units such as 4 feet 3 inches, to solve problems.
Lisa’s height is 5 feet 8 inches. What is Lisa’s height in centimeters?
Solve problems involving conversion of units within ratios and rates.
Yuna runs 750 meters in 5 minutes. What is Yuna’s speed in kilometers per hour? Related Standard
EUREKA MATH2
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 5 times as many 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.
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.
Number of Roses
Number of Da isies
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
Introduction to Ratios
Write ratios that relate two quantities as an ordered pair of numbers. Use ratio language to compare two quantities.
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
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
b. A ratio that relates the number of cans of red paint to the number of cans of blue paint is 4 : 5
c. There are 5 4 times as many cans of blue paint as cans of red paint.
d. For every 5 cans of blue paint, there are 4 cans of red paint.
Key Questions
• What is a ratio?
• When is it more practical to use ratio language instead of multiplicative comparison language?
Achievement Descriptor
6.Mod1.AD1 Write and explain ratios that describe relationships between two quantities. (6.RP.A.1)
Agenda Materials
Fluency
Launch 5 min
Learn 30 min
• A New Language
• From Tokens to Tea
Land 10 min
Teacher
• None Students
• None
Lesson Preparation
• None
Fluency
Multiplicative Comparisons
Students multiply or divide by using multiplicative comparisons to prepare for working with ratio relationships.
Directions: Answer each question.
Launch
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.
Why is the boy still sad?
He is still sad because the girl still has more tokens than he does. 5
Teacher Note
In grade 5, students used multiplicative comparison language to describe relationships between numbers. For example, 2 3 yards is twice as much as 1 3 yard, and 4 3 yards is 4 times as much as 1 3 yard.
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 4 cups of tokens, and the boy receives about 1 1 4 cups of tokens. 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.
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?
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.
Teacher Note
Avoid using fraction notation such as A B to 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 6 is twice as large as 1 3 .
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.
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 4 cups of tokens and the boy receives 1 1 4 cups of tokens?
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 4 cups of tokens and the boy still has 1 1 4 cups of tokens. Have students write their examples as ratios 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.
Language Support
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 times as many blue circles as red 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.
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 7 5 to get the first number in each ratio.
If students do not say that they can multiply the second number in each ratio by 7 5 to get 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.
We can multiply the second number in each ratio by 7 5 to get the first number in each 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
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.
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?
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 9 13 as many tokens as Toby.
B. Lisa has 13 9 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 ever y 9 tokens Lisa has, Toby has 13 tokens.
F. For every 9 tokens Toby has, Lisa has 13 tokens.
Teacher Note
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.
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?
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 9 13 to get 9, the number of tokens Lisa has.
• B is incorrect. If we multiply the number of tokens Toby has, 13, by 13 9 , we do not get the number of tokens Lisa has.
• C is correct because Lisa has 9 tokens and Toby has 13 tokens.
• 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.
UDL: Representation
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.
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 4 packet of sugar in 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.
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.
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 1 2 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 3 ounces of blue paint.
e. Ryan uses 3 times as much blue paint as white paint.
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.
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.
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.
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 5 times as many tokens 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.
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.
Introduction to Ratios
In this lesson, we
• wrote ratios to relate two quantities.
• used multiplicative comparison language to compare two quantities.
• used ratio language to compare two quantities.
Examples
Terminology
A ratio is an ordered pair of numbers that are not both zero.
A ratio can be written as A to B or A: B
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.
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
B. There are 2 1 2 times as many pentagons as circles.
C. For every 1 circle, there are 2 rectangles.
D. There are 1 2 as many orange rectangles as yellow circles.
A ratio that relates the number of pentagons to the number of circles is 5 : 2 That means there are 5 2 , or 2 1 2 , 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 1 2 as many.
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.
A ratio that relates the number of students who take orchestra class to the number of students who take band class is 12 : 48
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? 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
There are 1 4 as many students who take orchestra class as students who take band class.
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
For parts (a)–(d), fill in the blanks.
A. A ratio that relates the number of strawberries to the number of bananas is 7 : 2
B. There are 7 2 times as many strawberries as bananas.
C. For every 2 bananas, there are 7 strawberries.
D. There are 2 7 times as many bananas as strawberries.
2. Consider the collection of shapes shown. Which statements correctly describe the collection of shapes? Choose all that apply.
A. A ratio that relates the number of red squares to the number of blue circles is 4 : 3.
B. There are 2 times as many triangles as squares.
C. There are 1 2 as many triangles as squares.
D. For every 1 triangle, there are 2 squares.
E. A ratio that relates the number of squares to the number of circles is 3 : 4
3. Sasha says there are 1 1 2 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 1 2 circles.
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 9 dogs, there are 15 cats.
b. For every 3 dogs, there are 5 cats.
c. There are 5 3 times as many cats as dogs.
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.
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
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.
Remember For problems 6–8, multiply.
EUREKA MATH
EUREKA MATH2
1.
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 1 2 ? Choose all that apply.
A.
B.
C.
D.
E. F.
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.
Lesson at a Glance
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.
a. Explain why a linear function would not be a good model for this situation. 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.
This lesson formalizes the term quadratic function and introduces the term average rate of change.
Key Questions
• How can we represent the distance traveled by a falling object?
• Why is a quadratic function used to model the distance traveled by a falling object over time?
Achievement Descriptors
A1.Mod4.AD14 Interpret key features of graphs and tables in terms of quantities for quadratic functions that model relationships. (F.IF.B.4)
A1.Mod4.AD16 Calculate and interpret average rates of change of quadratic functions (presented symbolically or as tables) over specified intervals. (F.IF.B.6)
Agenda Materials
Fluency
Launch 10 min
Learn 25 min
• Dropping a Ball from a Building
• A Closer Look at Distance Traveled
Land 10 min Teacher
• 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.
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. Time (seconds) 0510
Distance (feet) 102540
2. Time (seconds) 024
Distance (feet) 353535
3. Time (seconds) 0816
Distance (feet) 42444
Constant speed: Yes
Speed: 3 feet per second
Teacher Note
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
Constant speed: Yes
Speed: 5 2 feet per second
4. Time (seconds) 01224
Distance (feet) 53595
5. Time (seconds) 01836
Distance (feet) 2818
6. Time (seconds) 02040
Distance (feet) 505254
Constant speed: No
Speed: N/A
Constant speed: No
Speed: N/A
Constant speed: Yes
Speed: 1 10 foot per second
Launch
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.
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.
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 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
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.
Teacher Note
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.
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.
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
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.
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. 25
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.
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)
Distance (feet)
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 32ft/s2 and 10m/s2. Lesson 1 uses functions of the form f (t)=1 2 gt2, where 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.
2. Use the data in the table to make a graph of the distance traveled by the ball over time.
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.
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?
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?
(seconds)
UDL: Representation
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 16ft, then 48ft, then 80ft, 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 16ft/s, 48ft/s, 80ft/s, 112ft/s, and 144ft/s. Each time the speed increases by 32ft/s.
7. Write an equation for a function that gives the distance traveled in feet over time in seconds.
d(t)=16t2
8. What domain makes sense for your function? Explain your reasoning.
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.
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.
Complete the Frayer model for the term quadratic function.
Definition
A quadratic function is defined by a polynomial expression of degree 2.
Picture
Rate of change is not constant.
Quadratic Function
Linear functions
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 1 2 -second interval. Consider displaying the following calculation for the average speed from 0 seconds to 0.5 seconds:
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 of the pebble every 1 2 second.
10. What patterns do you notice in the values in the third column of the table? They increase by 5 meters per second each 1 _ 2 second.
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 every 1 _ 2 second. Time, t (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.
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?
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.
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.
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.
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 in the pebble problem increased by 5 meters per second each 1 _ 2 second.
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.
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.
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
In this lesson, we
• 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.
Quadratic function
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.
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
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.
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.
a. How far above the ground is the water bottle when Mason drops it? It is 144 feet above the ground.
b. What does the point (0, 0) mean in this situation? 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.
The distance is 16 feet after 1 second and 64 feet after 2 seconds.
d. What domain makes sense in this situation? Explain your choice.
The domain should represent the time interval from the time Mason drops the water bottle until it hits the ground.
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
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
EUREKA MATH
Sample Solutions
Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.
Function f represents the distance in meters traveled by a ball dropped from a building over
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
function. Then write an expression for the distance traveled after t seconds.
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.
6. Graphs A, B, and C are shown. Determine which graph best models the distance traveled as a function of time for each situation.
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.
a. A car travels at a constant speed.
Graph A
b. A stone is dropped from the banks of a river into the water below.
Graph B
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?
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.
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?
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. 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
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
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.
a. Complete the table of values and graph function f Time (seconds)
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?
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.
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?
The average rate of change values on 1-second intervals for function f are 144112, 80, 48 16, −16, −48−80, −112, and −144. The average rate of change values on 1-second intervals for function d are 16, 4880, 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
For problems 15–17, write an equation of the line.
18. A graph of a system of two inequalities is shown.
a. Write two inequalities that define the system represented by the graph. y >2 x +3
y ≤ −3 x +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.
4 x 2−20 x +25
1. Describe
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.
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?
• How does knowing about angle relationships help us find unknown angle measures?
• How can we use if–then moves to find unknown angle measures?
Achievement Descriptors
7.Mod3.AD8 Compare algebraic solutions to arithmetic solutions of word problems. (7.EE.B.4.a)
7.Mod3.AD12 Write and solve equations to find unknown angle measures by using known facts about angle relationships. (7.G.B.5)
Agenda Materials
Fluency
Launch 5 min
Learn 30 min
• Angle Measure
• If–Then Moves
• Strategies
Land 10 min
Teacher
• None Students
• None
Lesson Preparation
• None
Fluency
Solve One-Step Equations
Students solve one-step equations to prepare for finding unknown angle measures.
Directions: Solve for the variable.
Launch
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.
Sample:
Angles:
Angle relationships:
Angles on a line
Linear pair 5
E F C
Teacher Note
A B D
Before grade 7, students refer to complementary angles as “two angle measures that sum to 90°.”
BAE and ∠DAE
EAD , ∠DAF , and ∠FAC
EAD and ∠DAC
DAF and ∠FAB
DAC and ∠CAB
EAD and ∠DAC
CAF and ∠FAE
Straight angle
Angle measures that sum to 90°
Adjacent angles
Angles at a point
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.
• 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.
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.
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.
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.
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.
Copyright
Learn
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?
Arithmetic: 903060 -= mBGA∠= °60
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.
Promoting the Standards for Mathematical Practice
Students use appropriate tools strategically (MP5) when they choose among arithmetic strategies, algebraic strategies, and tape diagrams to determine unknown angle measures.
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?
Differentiation: Support
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.
Teacher Note
In grade 6, students are introduced to the symbol m∠ABC, which refers to the measure of ∠ABC.
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
? 30 90 903060 -=
mBGA∠= 60 °
Algebraic
Let x represent the number of degrees of the measure of ∠BGA. x +=3090 x +- =-
x = 60
∠= 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.
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.
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.
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°.
Language Support
Consider adding complementary angles to the angle relationship anchor chart.
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:
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
5 = 5
Teacher Note
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.
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 x + 30 - 30 = 90 - 30 x = 60
Did we use any if–then moves there? Explain your thinking.
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.
a. Describe the relationship between ∠DAC and ∠CAB.
∠DAC and ∠CAB are a linear pair, so their measures sum to 180°.
Teacher Note
In grade 6, students solve equations by using the following properties of operations:
When
• When
The if–then moves help prepare students for the properties of equality in grade 8.
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
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.
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.
5. Consider the diagram.
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°
6. In the diagram, ∠CAB is a straight angle.
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?
3(30)
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.
7. In the diagram, ∠AFE is a straight angle. What is the measure of ∠BFA? Check your answer.
Sample:
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.
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.
Tape Diagram
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.
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.
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.
If–Then Moves
Vertical Angles and Angles at a Point
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
Before This Module
Grade 6 Module 4
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.
Overview Expressions,
Equations, and Inequalities
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
Showing Factoring with the Tabular Model
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.
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.
Topic C
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 topic by exploring a new type of equation, x p q r = , to foreshadow work with proportional reasoning in module 5.
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.
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 <.
After This Module
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.
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.
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.
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.
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.
Implementation Support Tool
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.
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
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)
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
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 and Monitoring indicators vary by routine and are included for specific fluency routines only.
Promoting Discourse indicators are excluded from the Fluency section of the IST due to the nature of the lesson component.
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)
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
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
g. Adjusts the sequence of problems (e.g., complexity, level of abstraction) based on the existence of pervasive trends
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
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
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
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 Sprints
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
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
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
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
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
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.
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, T alking Tool, materials, visuals)
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
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
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
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
Meeting Component Purpose
a. Focuses the progression of questioning, thinking, and discourse in alignment with the purpose statement of each Learn segment
b. Honors the ratio of questioning to direct instruction, the amount of discourse, and the format of the discourse so that students generate, test, share, critique, and refine their ideas
c. Releases responsibility to students while working on lesson pages, as indicated by the lesson and student performance
d. Uses tools and models accurately to develop student understanding of the lesson’s strategies and objectives
e. 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
f. Completes Learn within or near the total time indicated, allowing time for instruction, lesson pages, and Practice Problems
Using Structures, Routines, and Activities
g. States directions clearly and includes all necessary details for engagement (e.g., topic/task, thinking job, timeframe, cues, peer interaction)
h. Prepares the environment to optimize engagement, learning, and collaboration (e.g., desk arrangement, Thinking Tool, Talking Tool, materials, visuals)
i. Uses structures, routines, and activities as opportunities for students to develop their understanding of the day’s objectives and to articulate their strategies, models, and understandings
j. Incorporates Read–Represent–Solve–Summarize (RRSS) process (grades 6–8) or Modeling Cycle (Algebra I) for solving word problems
Engaging
k. Provides appropriate think time for students to process questions or displayed content and form responses
Monitoring
l. Circulates to recognize and encourage application of mathematical understanding
m. Listens to student discourse to gauge students’ ability to articulate their understanding and use precise mathematical terminology
n. Monitors student work to notice and note trends, varied models and strategies, exemplars, and misconceptions
Promoting Discourse
o. Prompts students to elaborate or clarify responses (e.g., by saying “Why?” “How do you know?” or “Tell me more”), and use precise language to advance student thinking
p. Asks questions that invite students to make connections between solutions; models; strategies; and previous questions, lessons, modules, or grade levels
Student Expectations
Customizing
q. Uses the margin notes to improve access (e.g., UDL, Differentiation: Support, Language Support, strategic pairings/groupings)
r. Creates and inserts problems that bridge or reinforce understanding to existing problems in the sequence
Responding
s. 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 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
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
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)
Engaging
Instructional Habits
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.
Customizing
Adaptive Implementation
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
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
Meeting Component Purpose
a. Asks questions that elicit student thinking and cause them to synthesize the day’s learning (e.g., suggested questions, key questions)
b. Focuses student discussion to align with the lesson objectives, highlighting important concepts and vocabulary
c. Honors the amount of student-to-student discourse and whole-class discourse within Land
d. Completes the Debrief within or near the indicated time
Meeting Component Purpose
a. Allows an amount of time within or near the indicated time allotted for students to independently complete as much of their Exit Ticket as possible
b. Collects Exit Tickets to analyze qualitative data to identify strengths to leverage in future lessons, varied solution paths, and misconceptions to address
Engaging
e. Provides adequate think time for students to process questions or displayed content and to form responses
f. Provides students with supports (e.g., Talking Tool, mathematical terminology) to improve access to discussion
Monitoring
g. Listens to student discourse to gauge understanding and identify opportunities to push for clearer thinking
Promoting Discourse
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
Engaging
c. Makes tools (e.g., anchor charts, concrete manipulatives, Thinking Tool) available to support independence
Monitoring
d. Circulates room while students work on their Exit Tickets to ensure student 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 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:
• Attempt to use accurate mathematical language when synthesizing the lesson’s key understandings
• Engage in discourse to articulate understanding, gain the insights of others, and evaluate errors
• Reflect on their own thinking following analysis of peer work and discourse
Customizing
j. Creates questions, in response to previously collected data, that promote synthesis of the lesson or topic
Responding
k. Affirms accuracy and efficiency in student responses while clarifying any inaccuracies and deepening understanding
l. Revisits questions from earlier in the lesson to check for growth in understanding
Responding
f. Designates the order of problems for students to complete to ensure students answer the problems that highlight the most important components of the lesson objective (within the given time)
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
• Make thinking visible by using numbers, words, and models
• Complete as much of the Exit Ticket as possible
Lead: Facilitating Successful Implementation of Eureka Math2, 6–9
Appendix B
Scenarios
Scenario 1: Imagine that a grade 6 teacher comes to your office and says, “I’m used to teaching fractions and rational numbers at the beginning of the year, so I was going to adjust modules 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: Imagine a teacher says to you, “My students didn't have Eureka Math2 last year, so they don't understand the way double number lines are introduced in this lesson. And the next lesson moves too quickly. I need to add a review day before I can teach this lesson and then a review day after this lesson to prepare my students for the next lesson.”
Sample Response: “Let’s go back and revisit the Module and Topic Overviews to see how learning progresses throughout. Then we can create a plan to support students in accessing this 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 into the trajectory of learning, return to the lesson to identify the purpose and role of that lesson in the trajectory, then review the lesson to see how the learning strategically builds and creates multiple entry points for students. Identify supports for the learning that will allow students to access the current lesson. You may also want to investigate the multiple instances of double number line usage to ensure the teacher understands that students will have continued opportunities to use and become more proficient with using them.
Scenario 3: You are completing a walk-through 15 minutes into the 45-minute math block. 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 the purpose of the observation. You might want to determine the reason that the teacher is using class time to review the previous lesson's learning You could say something like, “Because of the way that 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 grade 7 team has an 80-minute math block. A teacher approaches you and says that they are unsure of how to fit two Eureka Math2 lessons into the allotted time.
Sample Response: In order to complete two Eureka Math2 lessons in the same block, it is recommended to reevaluate the schedule to allot 90 minutes, 45 minutes for each lesson. If the teacher has specific pacing questions you could say, “We studied the module- and topic-level
Supplemental Materials
Lead: Facilitating Successful Implementation of Eureka Math2, 6–9
resources to understand the trajectory of learning, and we can use this understanding to make informed adjustments to the lessons. Understanding the purpose of each of the related lessons, and how they build upon each other, will allow us to make decisions about what to prioritize, shorten, or even consolidate to ensure that the lessons fit within our time frame without interfering with the big ideas, rigor, or coherence of the lessons. We can also use our data from previous Exit Tickets to determine students’ entry points, and use that to make strategic decisions about the lessons ”
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
Engage:
Preparing to launch 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.
Notes:
Experience
Enhance
Experience:
Learning and exploring Eureka Math2
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.
Enhance:
Extending knowledge of Eureka Math2
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.
Exemplify
Exemplify:
Skillfully implementing Eureka Math2
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.
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.
Enhance Experience
Exemplify
Leaders Teachers Students
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.
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.
There are no student actions during the Engage phase.
1The timeline mentioned is a guideline. A specific implementation may move through these phases more or less quickly.
2 The Power Up session is designed for teachers with prior Eureka Math experience.
Eureka Math2® Implementation Benchmarks –
Experience Phase (Year 1)1
Engage
Experience
Learning and exploring Eureka Math2.
Exemplify Enhance
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.
Teachers
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.
1 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
Experience Engage
Exemplify Enhance
Leaders Teachers Students
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.
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.
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.
1The timeline mentioned is a guideline. A specific implementation may move through these phases more or less quickly.
Extending knowledge of Eureka Math2.
Eureka Math2® Implementation Benchmarks
– Exemplify Phase (~Years 4 and Beyond)1
Enhance Experience Engage
Skillfully implementing Eureka Math2.
Leaders Teachers Students
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.
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.
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.
1The timeline mentioned is a guideline. A specific implementation may move through these phases more or less quickly.
