# 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.

#### 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. #### Guiding Questions Create a free account or sign in to access the Guiding Questions for this Anchor Problem. #### 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 ? With Fishtank Plus, you can download a complete problem set and answer key for this lesson. Download Sample 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 about commission. • Include problems that involve fractional fees, such as late fees, registration fees, tax fees, etc. ## Target Task ? ### Problem 1 A bicycle shop advertised all mountain bikes priced at a ${{1\over3}}$ discount. 1. What is the amount of the discount if the bicycle originally costs327?
2. What is the discount price of the bicycle?
3. Explain how you found your solution to part (b).

#### References

EngageNY Mathematics Grade 7 Mathematics > Module 1 > Topic C > Lesson 14Exit Ticket

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 2

A music player was marked down by ${{1\over4}}$ of the original price.

1. If the sales price is \$128, what is the original price?
2. If the music player was marked up by ${{1\over2}}$ before it was placed on the sales floor, what was the price that the store paid for the player?
3. What is the difference between the discount price and the price that the store paid for the music player?

#### References

EngageNY Mathematics Grade 7 Mathematics > Module 1 > Topic C > Lesson 14Exit Ticket

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..

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