SFUSD Mathematics Core Curriculum Development … Mathematics Core Curriculum, Grade 7, Unit 7.2:...

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SFUSD Mathematics Core Curriculum, Grade 7, Unit 7.2: Proportional Relationships, 2014–2015

SFUSD Mathematics Core Curriculum Development Project 2014–2015

Creating meaningful transformation in mathematics education

Developing learners who are independent, assertive constructors of their own understanding

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Grade 7 Unit 7.2: Proportional Relationships

# of Days Lesson Reproducibles Number of Copies

Materials

1 Entry Task Getting Things in Proportion (2 pages) 1 per student

8 Lesson Series 1 CPM CCC2 Lesson 1.1.4 (5 pages) Resource Page 1.1.4 (optional) HW: CPM CCC2 Lesson 1.1.4 Proportion and Non-Proportion Situations (FAL):

Groupwork, pages 19-23 (5 pages) Getting Things in Proportion, Revisited (2 pages)

CPM CCC2 Lesson 4.2.2 (4 pages) Resource Page 4.2.2 HW: CPM CCC2 Lesson 4.2.2 CPM CCC2 Lesson 4.2.3 (3 pages) HW: CPM CCC2 Lesson 4.2.3 CPM CCC2 Lesson 7.2.1 (2 pages) HW: CPM CCC2 Lesson 7.2.1 CPM CCC2 Lesson 7.2.2 (3 pages) HW: CPM CCC2 Lesson 7.2.2

1 per pair 1 per pair CPM eBook 1 per pair 1 per student 1 per pair 1 per student CPM eBook 1 per pair CPM eBook 1 per pair CPM eBook 1 per pair CPM eBook

Projector and Computer with Internet access Pennies (about 15 per team) How Much is a Million? by David M. Schwartz

(optional) Rulers Mini white boards and markers Calculators

3 Apprentice Task Mid-Module Assessment Task 7-1 #3 (2 pages) Developing a Sense of Scale (FAL):

A Sense of Scale (optional, 2 pages) Sample Responses to Discuss (3 pages) A Sense of Scale, Revisited (2 pages)

1 per student 1 per student 1 per pair 1 per student

Calculators Poster paper

5 Lesson Series 2 CPM CCC2 Lesson 7.1.3 (3 pages) HW: CPM CCC2 Lesson 7.1.3 CPM CCC2 Lesson 7.1.7 (3 pages) HW: CPM CCC2 Lesson 7.1.7 CPM CCC2 Lesson 7.1.8 (3 pages) HW: CPM CCC2 Lesson 7.1.8

1 per pair CPM eBook 1 per pair CPM eBook 1 per pair CPM eBook

5 Expert Task Setting Up Shop 1 per pair Catalogs for pricing (optional) Poster or construction paper and markers Calculators

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2 Lesson Series 3 Additional Problems (4 pages) 1 per pair

1 Milestone Task Performance Task: Foundation (2 pages) Constructed Response: Chess Club

Provided by AAO Provided by AAO

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Unit Overview

Big Idea

Quantities are proportional when one quantity changes by a constant ratio in relation to the other quantity. One useful representation of the relationship is when one of the quantities is 1; this is referred to as the unit rate.

Unit Objectives

● Students understand that proportions are two or more equivalent ratios and that each quantity in a ratio must be multiplied by the same number to be equivalent, not added. (i.e., 5:2 = 10:4, because each quantity is multiplied by 2. In contrast, 5:2 is NOT = 7:4 which is derived from each quantity being added by 2). (This is review from sixth grade and not explicitly taught.)

● Students use multiple representations such as tables, diagrams, equations, and linear graphs to model AND verify proportional relationships. ● Students understand what a unit rate is and how to calculate it with whole numbers and fractions. ● Students can interpret the meaning of a point on a line, especially the origin and (1, r), where r is the unit rate. ● Students use concepts of proportional relationships and unit rate to solve for “missing measures” in proportions problems. ● Students understand why cross multiplication actually works. ● Students use concepts of proportionality and unit rate to solve multiple-step problems (percent, tax, tip, commission, simple interest, percent increase,

percent decrease, and percent error).

Unit Description

Students learn to identify proportional relationships via multiple representations (e.g., tables, diagrams and graphs), and how to write, graph and interpret an equation that represents a proportional relationship. Next, students use concepts of proportional relationships and unit rate to solve for “missing measures” in proportions problems. Students also learn why cross multiplication actually works. Finally, students use concepts of proportionality and unit rate to solve multiple-step problems (percent, tax, tip, commission, simple interest, percent increase, percent decrease, and percent error).

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CCSS-M Content Standards

Ratios and Proportional Relationships Analyze proportional relationships and use them to solve real-world and mathematical problems. 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For

example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour. 7.RP.2 Recognize and represent proportional relationships between quantities.

7.RP.2a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

7.RP.2b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 7.RP.2c Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price

p, the relationship between the total cost and the number of items can be expressed as t = pn. 7.RP.2d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and

(1, r) where r is the unit rate. 7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and

commissions, fees, percent increase and decrease, percent error. The following standards support the proportions standards, but are explicitly taught in different units. Geometry Draw, construct, and describe geometrical figures and describe the relationships between them. 7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale

drawing at a different scale. Statistics and Probability Use random sampling to draw inferences about a population. 7.SP.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a

population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

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Progression of Mathematical Ideas

Prior Supporting Mathematics Current Essential Mathematics Future Mathematics

In sixth grade, students learned about ratio concepts and use ratio reasoning to solve problems. They recognized and described ratios, rates, and proportional relationships. They represented ratios, collections of equivalent ratios, rates, and proportional relationships using tape diagrams, double number lines, and function tables. Students only used whole numbers in their ratios. They also converted measurements, found unit prices, and constant speed. Additionally they found a percent of a quantity as a rate per 100 and found the whole given the part and the percent.

Students will expand their understanding of unit rate to include fractions. They will apply their understanding of ratios to decide if a proportional relationship exists or not. They will use a variety of ways to do so including equations (y = cx form only), tables, and graphs. With the graphs, they need to explain what various points on the line represent. Students also need to be able to solve a wide variety of real-world problems around percents and ratios.

Students will learn the connection between proportional relationships, lines, and linear equations. They will do this by first examining what exactly a proportion is, and understanding that a rate is in fact the slope of the graph. Students will work with two different proportional relationships that are represented in various ways (graphs, tables, equations, and diagrams) and analyze information. Students will become familiar with similar triangles and use them to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. They will use acquired knowledge regarding the slope m to derive the equation y = mx for a line through the origin (a proportional relationship) and the equation y = mx + b for a line intercepting the vertical axis at b (linear relationship).

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Unit Design

All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are both formative and summative assessments of student learning. The tasks are designed to address four central questions: Entry Task: What do you already know? Apprentice Task: What sense are you making of what you are learning? Expert Task: How can you apply what you have learned so far to a new situation? Milestone Task: Did you learn what was expected of you from this unit?

1 day 8 days 3 days 5 days 5 days 2 days 1 day Total Days: 25

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Entry Task Getting Things in Proportion

Apprentice Task Mid-Module Assessment from NY

Developing a Sense of Scale

Expert Task Setting Up Shop

Milestone Task Foundation Chess Club

CCSS-M Standards

7.RP 7.RP.1, 7.RP.2a,7.RP.2b, 7.RP.2c, 7.RP.2d

7.RP.3 7.RP.1, 7.RP.2a, 7.RP.2b, 7.RP.2c, 7.RP.2d, 7.RP.3

Brief Description of task

This task is a series of three questions adapted from the FAL that ask students to use their understanding of ratios to try to answer questions about different scenarios. Students explore proportional and non-proportional relationships.

The first part is a collection of open-response questions where students analyze a graph and its proportionality. The second part is an FAL from MAP that has students evaluating student work with their partner, then completing similar problems on their own.

This task requires students to work with one to two others to answer a variety of percentage problems through the lens of setting up their own store.

Individual assessment involving proportional reasoning and percents.

Source SFUSD Teacher Created, adapted from MARS FAL from MAP

NYS Mid-Module Assessment engageny.org MARS FAL from MAP

SFUSD Teacher Created SFUSD Teacher Created, adapted from NYS End-of-Module Assessment from engageny.org and Illustrative Mathematics

Lesson Series 1

Lesson Series 2

Lesson Series 3

CCSS-M Standards

7.RP.1, 7.RP.2, 7.RP.3

7.RP.2, 7.RP.3 7.RP

Brief Description of Lessons

Students identify proportional relationships via multiple representations (e.g., tables, diagrams and graphs), and write, graph and interpret an equation that represents a proportional relationship. They will also use proportional relationships and unit rate to solve for “missing measures” in proportion problems. Students explore why cross multiplication works.

Students use concepts of proportionality and unit rate to solve multiple-step problems (percent, tax, tip, commission, simple interest, percent increase, percent decrease, and percent error).

Teachers have a chance to reteach and choose from a variety of lessons to review and reinforce concepts learned in this unit.

Sources

CPM Core Connections Course 2 Lessons 1.1.4, 4.2.2, 4.2.3, 7.2.1, and 7.2.2 MARS FAL: Proportion and Non-proportion Situations

CPM Core Connections Course 2 Lessons 7.1.3, 7.1.7, and 7.1.8

Illustrative Mathematics CPM Core Connections Course 2 Chapter 7

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Entry Task

Getting Things in Proportion

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: • Students will demonstrate their understanding of ratios and unit rate

and their ability to use those concepts in solving proportion problems. CCSS-M Standards Addressed: 7.RP Potential Misconceptions:

• Refer to the “common issue” page.

Launch: You may start with a KWL chart, and ask your students what they already know about ratios, and possibly even proportions. Tell them they will complete a series of questions individually to show you what they know already. These questions are actually from the 7th grade standards, and the later questions in this task may be challenging for many students. Explain this to students and give them a finite time to do as much of it as they can (15–20 minutes). During: Monitor and answer clarifying questions as necessary. Refer to the “common issues” page. This will give you some prompts for students stuck on different parts of the task. This would be handy to have printed as a resource for yourself while students work on the task. For ELs and other students, you may need to go over the questions in the task prior to independent time. Closure/Extension: Because the task will only take about 20 minutes, it may be wise to use the remainder of the class lesson to do a mini-lesson to review the coordinate plane and plotting (x, y) coordinates. This is a skill they will need to know for this unit. The teacher notes from the complete lesson, “Proportion and Non-Proportion Situations,” found in Lesson Series 1, gives you different teacher “moves” based on the misconceptions or mistakes. After looking at your student work, you may decide to plan a mini-lesson for the next day’s Do Now.

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Getting Things in Proportion

How will students do this?

Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 8. Look for and express regularity in repeated reasoning.

Structures for Student Learning: Academic Language Support:

Vocabulary: Proportions Sentence frames:

Differentiation Strategies: Participation Structures (group, partners, individual, other) Students will complete the task individually.

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Lesson Series #1

Lesson Series Overview: Students will identify proportional relationships via multiple representations (e.g., tables, diagrams and graphs), and write, graph and interpret an equation that represents a proportional relationship. Students use concepts of proportional relationships and unit rate to solve for “missing measures” in proportions problems. Students explore why cross multiplication actually works. CCSS-M Standards Addressed: 7.RP.1, 7.RP.2, 7.RP.3 Time: 8 days

Lesson Overview – Day 1 Resources

Description of Lesson: The objective of this lesson is for students to produce their own definition of proportional relationships. Students make predictions on how high a tower of a million pennies would be, and then use a ruler and a table to fill out how many pennies make up 1 cm, 5 cm, etc. Students discuss how they could use the height of 100 pennies to help find the height of a million pennies. In the wrap-up class discussion, point out that the relationship between number of pennies and height of tower is a proportional relationship, and ask the class to describe that relationship. Notes:

● This entire lesson is projected to be 45–60 minutes long. ● The Teacher Notes from the CPM resources provide a lot of great information on how to introduce and scaffold the lesson. ● After this lesson, students should at the very least understand that each ratio of “# of pennies” to “height of towers” are equivalent

fractions to each other, and that is how to tell that the quantities are proportional. ● At the end of class discussion, make sure students make all the points you want them to about proportions and post them

somewhere, so you can constantly add more as you continue the unit. ● This is just the introduction to proportions, and further lessons will go into different ways of verifying proportionality.

CPM CCC2 Lesson 1.1.4 Resource Page 1.1.4 (optional) Rulers Pennies (10 per group) How Much is a Million? by David M. Schwartz (optional)

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Lesson Overview – Days 2–4 Resources

Description of Lesson: The objective of this lesson is to deepen students’ understanding of proportionality and to distinguish between direct proportion and other functional relationships. Students will also start to solve some proportion problems. We now return to the Proportion and Non-Proportion Situations task. The lesson starts with a class discussion around a PowerPoint slide of a price rate ($ for each ounce). The “teacher notes” specify what key points you would want to make during this discussion, especially around further defining proportion. Students then work in teams and use different provided scenarios and sentence frames to write and answer questions about different rates and to determine whether each scenario represents a proportional relationship. The teams write two of their questions on a blank card and switch with another pair. Each pair then checks their partners’ work. This lesson ends with a class discussion showcasing various question cards from students. Conclude by showing a slide with three types of graphs and guides students to understand that only the linear graph that goes through the origin represents a proportional relationship. At this point, this lesson provides additional problems that mirror the original entry task problems. Use these for independent practice or further pair practice for students. Notes:

● Depending on the length of the class period, this lesson could definitely stretch out over two to three days. You may want to tell students this in preparation.

● The teacher notes are extremely thorough. We suggest that you do the task yourself and refer to the teacher notes as you work to prepare to use this lesson with your students.

● It is very important to maintain a vocabulary web/word wall for this unit, because there are so many vocabulary words that are related. (e.g., “unit rate” is also referred to as the “constant of proportionality” and “multiplier” and “scale factor”) and students should understand all terms.

FAL: Proportion and Non-Proportion Situations If you want to add more practice, follow up with problems from Glencoe that provide more scenarios: Glencoe Grade 7 © 2008, Lesson 4-2, pp. 195–196; choose from problems 1–8.

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Description of Lesson: The objective of this lesson is for students to write equations, make tables, and graph lines to represent proportional relationships. Students will start to explore the significance of the origin and (1, y). This lesson takes you back to CPM, and starts with students referring to the penny tower problem from earlier in the unit. You will guide students to graph the data from the table in the penny problem. Students work in pairs or teams to solve problems 4-34 through 4-38. During the wrap-up discussion, make sure students can articulate the connection among a proportional relationship, the table values, and the graph. Also, make sure students share their thinking of how they wrote the equation. Notes:

● This should be a one-day lesson, and some of the questions may be used for homework. ● Question 4-34d refers to another previous problem in CPM that students would not have done in this unit. You may want to tell

students to skip that question. ● Also, CPM treats the “writing an equation” question as a challenge problem; however treat this as just another question to answer. ● There is a resource page provided with this lesson that you can use electronically or as a hard copy. ● Make sure to model how to properly scale the axis on a coordinate plane when graphing the penny tower problem. Students will

need to know this for the next lesson.

CPM CCC2 Lesson 4.2.2

Resource Page 4.2.2 (optional)


Description of Lesson: The objective of this lesson is for students to calculate unit rates (especially with non-whole numbers). Students will use unit rates to solve real-world proportion problems. This lesson also cycles back and asks students to verify whether a situation is proportional or not. Students will complete questions 4-46 through 4-48. As an introduction, have students read the first question and as a team state whether it is a proportional or non-proportional situation, and defend their answer. After, students will work through the remaining questions, which ask them to fill in a table of values and graph and find the significance of (1, y). The unit rate in this example is not an integer. Students will then be asked to write an equation for the scenario and use the equation to find other values. The closing discussion should include students sharing all the different ways they came up with to find the cost per pound, or, in other words, the unit rate. Notes:

● This should be a one-day lesson, and some of the questions may be used for homework. ● The Teacher Notes for this lesson (along with others) gives ideas for different partner work strategies a teacher could use to

structure this lesson. ● This lesson does not provide a worksheet like the previous one, so you may need to provide a mini-lesson on making your own

coordinate plane and how to scale the two axis. (This could be done during the previous lesson when you model how to graph the penny problem.)


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Description of Lesson: The objective of this lesson is that students will be able to find the missing measure in a proportion. Students will compare two different methods to do so, namely writing a proportion as two equivalent ratios, and writing a proportion as y = cx. The lesson starts with you posting a proportion and having students talk about the different methods they could use to find the missing value. Students will then work on question 7-96 through 7-99. Use the “Red Light Green Light Strategy,” in which groups stop after each question to verify their answers with you before moving on. During the wrap-up discussion, students share their different approaches to solving question 7-99. Make sure students share how they solved the problems using equations in both forms. Notes:

● This is a one-day lesson. You may decide to use some of the questions for homework. ● The CPM eBook provides a link to a mathcast for some lessons. For this lesson in particular, the video link is helpful, as the teacher

notes don’t give the same level of information around teaching and scaffolding the problems.



Description of Lesson: The objective for this lesson is that students will be able to continue exploration of different methods of solving proportions and be able to explain why cross multiplication works. Start the lesson with a recap of the previous day’s lesson. Students work together to answer questions 7-106 through 7-109. The major emphasis of these problems is for students to explore how to use equivalent ratios to solve proportion problems. In the closing discussion, make sure to point out why cross multiplication works. Notes:

● This is a one-day lesson. You may decide to use some of the questions for homework. ● This lesson does provide a few missing measure proportion problems for students to solve. However, you may want to spend

another day on increasing fluency with proportion problems from Glencoe or other resources. Additional Resources for suggestions for further practice solving proportion problems. Highway Robbery - http://illuminations.nctm.org/Lesson.aspx?id=3128 Writing Proportions (Khan Academy) - https://www.khanacademy.org/math/algebra/ratio-proportion-topic/ratios_algebra/e/writing_proportions Feeding Frenzy - http://illuminations.nctm.org/Lesson.aspx?id=2854 Glencoe Correlations 4-1 Ratios and Rates 4-2 Proportional and Nonproportional Relationships


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Apprentice Task

Mid-Module Assessment Task – Engage NY Developing a Sense of Scale - Mathshell



Math Objectives: ● Students use multiple representations such as equations, and linear

graphs to model AND verify proportional relationships. ● Students understand what a unit rate is and how to calculate it with

whole numbers and fractions. ● Students can interpret the meaning of a point on a line, especially the

origin and (1, r), where r is the unit rate. ● Students use concepts of proportional relationships and unit rate to

solve for “missing measures” in proportions problems. CCSS-M Standards Addressed: 7.RP.1, 7.RP.2 Potential Misconceptions

● Students may still be determining proportionality based on additive criteria (i.e., each salary goes up by $2 so it is proportional).

● Students may not label x- and y-axes correctly, or they make other graphing errors.

● Students may find the “reverse” unit rate to the situation described (say the unit rate is 5 instead of 1/5).

● Students still fail to understand that (1, y) should be (1, 1/5), which is the unit rate.

● Refer to the Common Issue page from the Teacher Notes for the Assessment Lesson.

● Students are confused between terms “constant of proportionality,” “multiplier,” “scale factor,” and “unit rate.”

Launch: Give the Mid-Module Assessment Task #3, and optionally “A Sense of Scale” individually before doing the rest of the Formative Assessment Lesson (FAL) “Developing a Sense of Scale.” As students have different EL levels, it is important that each student understands the actual questions before being asked to work on it individually. Go through the different questions, asking student to think-pair-share to put what the question is asking in their own words. Use the “Common issues” section of the teacher notes to get an idea of how to address student misconceptions during the task. During: During the collaborative part of the lesson, students work in groups of two or three to come up with an efficient method for solving each problem, and then critique the reasoning of sample student work. Monitor as students work on the task, taking note of and addressing student misconceptions. Encourage students to use correct math vocabulary. Closure/Extension: Choose the most common misunderstanding made, and have students explain their thinking around what they thought about the question. Most importantly students should understand that quantities are only proportional if they have a constant rate, and share the same unit rate (aka. “constant of proportionality”). Ratios in a proportional relationship are equivalent. Give the individual task “A Sense of Scale, revisited” at the end of class or for homework.

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Mid-Module Assessment Task – Engage NY Developing a Sense of Scale - Mathshell


Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively.


Vocabulary: proportional, coordinate plane, graph, constant of proportionality, equation, point (0, 0), scale Sentence frames:

Differentiation Strategies: Go through the different questions, asking student to think-pair-share to put what the question is asking in their own words. Calculators or multiplication charts for students who struggle with basic skills. Participation Structures (group, partners, individual, other): Students work in groups of two to four while doing the collaborative activity. Students should complete the end task individually if possible.

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Lesson Series #2

Lesson Series Overview: Students use concepts of proportionality and unit rate to solve multiple-step problems (percent, tax, tip, commission, simple interest, percent increase, percent decrease, and percent error). CCSS-M Standards Addressed: 7.RP.2, 7.RP.3 Time: 5 days

Lesson Overview – Day 1–2 Resources

Description of Lesson: The objective of this lesson is that students will recognize that different multipliers find different related quantities that can be used to solve discount and increase problems. (i.e., If something is 60% off, the sale price would be “original price x 40%” or the sale price could also be “original price – (original price x 60%)”. Students will be able to calculate using both methods. “The focus of this lesson is to find the solutions to percent problems in multiple ways using different scale factors or multipliers. This requires students to recognize the complementary nature of percents less than 100%, as well as learn about scale factors greater than 100%.” —From CPM Section 7.1.3 Notes:

● This is a one-day lesson. You may decide to use some of the questions for homework.

● You will most likely need to do a mini-lesson or have students review percent to decimal to fraction equivalency as needed background information to this unit.

● Review how to find a portion (percent) of a number as background information as well.

● This lesson also uses student knowledge of diagraming and scale factor that is introduced in Lesson 7.1.2. You may need to refer to this lesson to frontload or model the first question from Lesson 7.1.3.

● You may also want to pull from Glencoe or other resources to spend a day building fluency around this objective. (Have students practice mark up, discount, commission, etc. problems.)


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Lesson Overview – Day 3–4 Resources

Description of Lesson: The objective of this lesson is that “students will develop an understanding of percent increase and decrease as scaling a quantity and will identify a multiplier related to the amount of change.” —From CPM Lesson 7.1.7 Start with a discussion of scenarios in which they want to know how much something is changing. (Examples are provided in the Teacher Notes). Students will work on answering questions 7-76 through 7-78 in groups that guide them through the process of finding percent change (either increase or decrease). The closing discussion should have students articulate what percent increase and decrease is, through the discussion of problem 7-78. Notes:


● More details on the closing discussion and scaffolds are provided in the Teacher Notes for the lesson.

● Again, you can pull from other resources to provide more practice problems for fluency.



Description of Lesson: The objective of this lesson is that students will be able to use their knowledge of percents and proportions to solve simple interest problems. Students work on questions 7-86 through 7-88. Refer to the Teacher Notes for more information. Notes:


● You can pull from other resources to provide more “practice problems” for fluency.


Draw from Lesson Series 3 Additional Resources (provided electronically) as needed.

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Expert Task

Setting Up Shop



Math Objectives: ● Students use concepts of proportionality and unit rate to solve

multiple-step problems (percent, tax, tip, commission, simple interest, percent increase, percent decrease, and percent error).

CCSS-M Standards Addressed: 7.RP.3 Potential Misconceptions

● Students may calculate markup and percent increase and forget to add it to get the selling price.

● Students may not calculate that interest is monthly and instead use T = 1 (for one year). They may also forget to add the interest to the principal to know how much they will need to pay off at the end of the year.

● For the receipts, students might not add the tax to the subtotal to get the total.

Launch: Group students in pairs or trios to work on the task, and decide how students will submit work (report, poster, etc.). You may want students to present their work on Day 3. You may want to get catalogs so students can have examples of prices. During: Walk around to check in with students and give support tools as needed (example receipts, calculators, etc.). Encourage students to review their classwork, and address student misconceptions. If you have students make posters, you might do a gallery walk (see the Math Teaching Toolkit). Students can use sticky notes to ask clarifying questions or make constructive comments. Model for students so that they understand how to be helpful and specific: “I really like your method for finding the sales tax, but I don’t see the final price after tax,” rather than, “Good job,” or, “Not finished.” After the posters and/or presentations, ask students to reflect on this project individually. Possible reflection questions:

1. What was the most challenging part of the project? What did you learn? How did you contribute?

2. Do you have any mathematical questions? 3. Is there anything else you’d like to ask or tell me?

Closure/Extension: Use this Expert Task to assess gaps and areas of weakness. You may address specific skills as they come up, or bring the class together for a re-engagement lesson if many groups are struggling. You can draw from the Additional Resources provided for Lesson Series 1, 2, and 3.

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Setting Up Shop


Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 6. Attend to precision


Vocabulary: loan, wholesale price, markup, percent increase, selling price, markdown, tax, commission Sentence frames:

Differentiation Strategies: Some students could benefit from using a calculator, as long as they show all the work that they enter into a calculator. Some students may need to see examples for wholesale versus selling prices as well as receipts in order to contextualize their work (however most should be okay with the receipts since by seventh grade most students are consumers). Participation Structures (group, partners, individual, other): Students need to be working in small groups (recommended two to three students, although some students may benefit from larger groups; it tends to be more difficult to hold each student accountable in groups larger than three).

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Lesson Series #3

Lesson Series Overview: Students will have opportunities to work in cooperative groups on problems to solidify learning. Problems will be selected based on teacher assessment during the Expert Task. CCSS-M Standards Addressed: 7.RP.2, 7.RP.3 Time: 2 days

Lesson Overview – Day 1-2 Resources

Description of Lesson: Depending on the students’ performance on the Expert Task, put together different mini-lessons from the assessments link for CPM or from the additional resources provided with this unit. Notes: Use the additional resources provided as needed throughout the unit to address students’ misconceptions.

Illustrative Mathematics Problems: Buying Bananas, Buying Coffee, Sore Throats CPM CL 7-118 – 7-123 (these provide a good review for students) You may also draw from the Additional Resources from Lesson Series 2 (provided electronically).

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Milestone Task

Performance Assessment: Foundation Constructed Response: Chess Club



Math Objectives: ● Students use multiple representations such as tables, diagrams,

equations, and linear graphs to model AND verify proportional relationships.

● Students use concepts of proportional relationships and unit rate to solve for “missing measures” in proportions problems.

● Students use concepts of proportionality and unit rate to solve multiple-step problems (percent, tax, tip, commission, simple interest, percent increase, percent decrease, and percent error).

CCSS-M Standards Addressed: 7.RP.1, 7.RP.2, 7.RP.3

Launch: Each student will do the Milestone Task independently so you can assess his or her individual learning. Students should be able to relate the given situations to mathematical models that have been used in the lessons leading up to this point. During: During the assessment, move around the room observing and clarifying as needed. Closure/Extension: After the assessment it might be necessary to re-address areas that seem to remain unclear to the students.

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Performance Assessment: Foundation Constructed Response / Chess Club Constructed Response


Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 7. Look for and make use of structure.


Vocabulary: rate, unit rate, constant of proportionality Differentiation Strategies: Participation Structures (group, partners, individual, other): This task should be done individually.

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