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# Connecting Percentage To Proportional Relationships

## Overview

Students connect percent to proportional relationships in the context of sales tax.

# Key Concepts

- When there is a constant tax percent, the total cost for items purchase—including the price and the tax—is proportional to the price.
- To find the cost,
*c*, multiply the price of the item,*p*, by (1 +*t*), where*t*is the tax percent, written as a decimal:*c*=*p*(1 +*t*). - The constant of proportionality is (1 +
*t*) because of the structure of the situation:*c*=*p*+*tp*=*p*(1 +*t*). - Because of the distributive property, multiplying the price by (1 +
*t*) means multiplying the price by 1, then multiplying the price by*t*, and then taking the sum of these products.

# Goals and Learning Objectives

- Find the total cost in a sales tax situation.
- Understand that a proportional relationship only exists between the price of an item and the total cost of the item if the sales tax is constant.
- Find the constant of proportionality in a sales tax situation.
- Make a graph of an equation showing the relationship between the price of an item and the total amount paid.

# Look for and Use Structure

# Mathematical Practices in Action

**Mathematical Practice 7: Look for and make use of structure.**

Have students watch the video and listen to the dialogue between Karen and Sophie. This video shows students engaged in Mathematical Practice 7: Look for and make use of structure. The students solve the problem one way and then Karen finds a faster way to solve the problem using the distributive property. She uses the structure of math to turn a two-step problem into a one-step problem.

Have students discuss whether the expressions *p* + *p*(0.05) and *p*(1.05) are equivalent. In the discussion, elicit that Karen used the distributive property to show the equivalence of these two expressions.

Emphasize that looking for and using the structure within mathematics can be very helpful and can often help make finding the solution to a problem easier.

## Opening

# Look for and Use Structure

Watch the video that shows Karen and Sophie using the structure of mathematics to solve a problem.

- Is the expression
*p*+*p*(0.05) equivalent to the expression*p*(1.05)? - How did Karen use the structure of math to show that the two expressions are equivalent?

VIDEO: Mathematical Practice 7

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will explore the connection between proportional relationships and percent in a sales tax situation.

SWD: Though informal, discuss as an introduction or preview for students with disabilities. Emphasize key information and critical concepts that will be introduced. This will support students as they work to determine salient information throughout the lesson.

## Opening

Explore the connection between proportional relationships and percent in a sales tax situation.

# Sales Tax

# Lesson Guide

Have students work in pairs on all problems and the presentation.

# Mathematical Practices

**Mathematical Practice 7: Look for and make use of structure.**

- Look for students who identify the structure of the expression 20(1.05) as a variable multiplied by a constant value.
- Watch for students who identify the structure of the formula cost = price(1 + tax) as the structure of a proportional relationship.
- Look for students who identify (1 + tax) as the constant of proportionality, either in the formula, the graph, or in making sense of the situation directly.

SWD: Deliberately group students with disabilities for this Work Time. Grouping students by skill level will allow for more efficient provision of teacher support, while grouping students heterogeneously will promote cooperative teaching and learning opportunities for students with varying mathematical skills. Monitor groups to ensure all students’ progress. It is often best practice to pair students with disabilities with typically developing peers.

# Interventions

**Student converts percentages to decimals incorrectly.**

- How can you write 50% as a decimal?
- How can you write 5% as a decimal?

**Student gets confused finding the product of a dollar amount and a percent.**

- Refer to {link to concept corner or 6th grade lesson on percents}.
- Remember that “per cent” means “per 100.”

5% = $\frac{5}{100}$ = 0.05 - 5% of 20 = 20(0.05)

**Student doesn’t recognize what the two variable quantities are in the situation or is confused by incorporating “cost” into their equation.**

- There are two variable quantities in this situation, which are the same as in the video.
- The quantity “total cost” of any number of items depends on the “prices” of the items.
- Write an equation that begins with cost = ____.

**[common error] Student thinks there is not a proportional relationship between price and total cost.**

- A proportional relationship can be represented by an equation of the form
*y*=*kx*, where*k*is the constant of proportionality. Look closely at the structure of the equation you wrote for the first question. - Look at the structure of the equation you wrote for the first question. Can you write it in the form
*y*=*kx*, like the equation below?

cost = (1 + tax)(price), with*k*= (1 + tax)

If so, what is the value of*k*in your formula?

# Answers

Let

*c*= the total cost*c*=*p*+ 0.05*p**c*= 20 + 0.05(20) = (1.05)(20)*c*= (1.05)(20) = 21

The total amount Lucy paid was $21.00.- Yes, there is a proportional relationship between the price of an item and the total amount paid for it.
- The number 1.05 is the constant of proportionality. It is the constant factor you multiply the price of an item by to determine the total amount paid for the item.

## Work Time

# Sales Tax

Lucy bought a blouse with a price of $20. The sales tax was 5%. What is the total amount she paid for the blouse?

- Write an equation to find the total amount Lucy paid for the blouse.
- Solve the equation.
- If every item in the store has a sales tax of 5%, is there a proportional relationship between the price of an item and the total amount paid for it?
- How does the number you used in your equation relate to the constant of proportionality?
- Make a graph that shows the relationship between the price of any item in the store and the total amount paid.

## Hint:

- What is the price of one blouse?
- What amount do you need to multiply the price by to get the total amount that Lucy paid for the blouse?

# Prepare a Presentation

# Preparing for Ways of Thinking

Look and listen for students who:

- Have a clear grasp of the proportional relationship between the price of an item and the total amount paid for it.
- Discuss and debate whether 0.05 or 1.05 is the constant of proportionality.
- Understand that 20 + 0.05(20) = 20(1 + 0.05) by the distributive property.
- Discuss the scaling on the axes for the graph they must create representing the relationship between the price of an item and the total amount paid.
- Discuss how the constant of proportionality is shown on their graph.
- Discuss and debate whether the graph is a straight line or a series of points in a line.
- Understand what each point on the graph represents.

# Challenge Problem

- Answers will vary. Ideally students will identify approaches with and without the distributive property.

## Work Time

# Prepare a Presentation

Prepare a presentation about Lucy buying a blouse in which you show and explain each part of your work.

# Challenge Problem

- Find a different method that Lucy could have used to calculate the total cost of the blouse.

# Make Connections

# Mathematics

Examine two approaches to solving the problem and connect the approaches sufficiently so that students recognize their equivalence.

- The first approach is more closely tied to the presentation of the problem. Add 5% of 20 to 20: cost = 20 + 0.05(20)
- The second approach builds on the first and involves the distributive property. 5% of 20 added to 20 has two terms, each a product of 20; 20 can be “taken out” of each term using the distributive property to write an equivalent expression, which can be simplified:

cost = 20 + 0.05(20) = 20(1 + 0.05) = 20(1.05)

SWD: If students are struggling to the point of frustration, help them move toward particular approaches. The CCSS include students’ understanding various ways to solve problems, so while it may be necessary to promote or suggest a particular approach, it is important to clarify for students the various ways to solve the problems.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates’ explanations of their equations, graphs, and solutions.

## Hint:

As your classmates present, ask questions such as:

- What are the variables in this situation?
- How does the form of your equation relate to the form of the formula for a proportional relationship?
- If you wrote a general formula, what would it be? How would it help you?
- Where is the percent sales tax in your formula? How did you decide to change the percent to that value?
- How did you figure out how to make your graph? Can you explain the steps you took?

# Proportional Relationships and Percent

# A Possible Summary

When there is a constant tax percent, the total cost for items purchased—including the price and the tax—is proportional to the price. To find the cost, multiply the price(s) of the item(s) by (1 + *t*), where *t* is the tax percent, written as a decimal. The constant of proportionality is (1 + *t*) because of the structure of the situation. Because of the distributive property, multiplying the price by (1 + *t*) means multiplying the price by 1, then multiplying the price by *t*, and then taking the sum of these products.

ELL: Make sure you write what you are saying for the summary on an anchor chart so students can follow along. It will be helpful to hang this chart in a prominent location in your room to support students when they organize their data.

# Additional Discussion Points

Review the following concepts with the class:

- How to find the total amount in a sales tax situation
- How a proportional relationship exists between the price of something and the total cost when the sales tax is constant
- How percent connects to the constant of proportionality

## Formative Assessment

# Summary of the Math: Proportional Relationships and Percent

Write a summary about solving sales tax problems and how proportional relationships and percent are connected.

## Hint:

Check your summary.

- Do you explain how to find the total cost when there is sales tax?
- Do you point out that the sales tax must be constant in order for a proportional relationship to exist between the price of each item and the total cost of the item?
- Do you explain how percent is connected to the constant of proportionality?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of the class. Review the reflections to find out if students understand how the sales tax percent connects to the constant of proportionality in a sales situation with a constant sales tax.

## Work Time

# Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**In a sales situation with a constant sales tax, the sales tax percent connects to the constant of proportionality in the following way …**