# Proportional Relationships

## Objective

Solve multi-step ratio and rate problems using proportional reasoning, including fractional price increase and decrease, commissions, and fees.

## Common Core Standards

### Core Standards

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• 7.RP.A.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.

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• 5.NF.B.6

• 6.RP.A.3

## Criteria for Success

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1. Find a fractional amount of a quantity and reason about what to do with the value using the context of the problem.
2. Find an original value when given either a discount or a markup and a fractional rate.
3. Use different strategies to solve price increase, decrease, commission, and fee problems, including tape diagrams, proportions, equations, and other proportional reasoning.

## Tips for Teachers

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• In terms of pacing, this lesson may be extended over more than one day depending on the amount of practice and/or exposure to different variations of problems students need. The Target Task includes 2 problems which could be split over 2 days.
• The Anchor Problems specifically cover the topics of price increase and price decrease. The other topics of commissions and fees should be included in the problem set.
• Percent problems are not included in this lesson; all problems involve fractional amounts rather than percentages. In Unit 5, students will revisit this topic, but with percentages.

#### Remote Learning Guidance

If you need to adapt or shorten this lesson for remote learning, we suggest prioritizing Anchor Problems 1 and 3 (benefits from worked example). Find more guidance on adapting our math curriculum for remote learning here.

#### Fishtank Plus

• Problem Set
• Student Handout Editor
• Vocabulary Package

## Anchor Problems

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### Problem 1

Peter’s Pants Palace advertises the following sale:

If a pair of shoes costs $40, what is the sale price? A pair of pants usually sells for$33. What is the sale price of the pants?

#### References

EngageNY Mathematics Grade 7 Mathematics > Module 1 > Topic C > Lesson 14Example 1

Grade 7 Mathematics > Module 1 > Topic C > Lesson 14 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

### Problem 4

A motorcycle dealer paid a certain price for a motorcycle and marked it up by ${{1\over5}}$ of the price he paid. Later, he sold it for \$14,000. What is the original price?

Solve this using two different strategies.

#### References

EngageNY Mathematics Grade 7 Mathematics > Module 1 > Topic C > Lesson 14Example 4

Grade 7 Mathematics > Module 1 > Topic C > Lesson 14 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

## Problem Set

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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

• Include problems similar to each anchor problem, asking students to find both the prices after the markup or markdown, and the prices before the markup or markdown.
• Include problems that involve fractional fees, such as late fees, registration fees, tax fees, etc.

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### Problem 1

A furniture store offers a deal to new customers. On your first purchase, you can use a $\frac{1}{4}$ discount on the price of any table.

Brian is a new customer at the store and wants to buy a table that originally costs $96$. What discounted price would Brian pay for the table?

### Problem 2

A washing machine is on sale, and its price is marked down by $\frac{1}{5}$ of its original price. The sale price of the washing machine is $304$. What is the original price of the washing machine before the discount?

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