# Proportional Relationships

## Objective

Write equations for proportional relationships presented in tables.

## Common Core Standards

### Core Standards

?

• 7.RP.A.2.B — Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

• 7.RP.A.2.C — 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.

?

• 6.EE.B.7

• 6.RP.A.3

## Criteria for Success

?

1. Determine the constant of proportionality from a table.
2. Write an equation for a proportional relationship in the form $y=kx$ where $k$ represents the constant of proportionality.
3. Explain what the constant of proportionality means in context of a situation.
4. Explain the role of the constant of proportionality in an equation.
5. Use an equation to solve problems and determine additional values for the situation.
6. Decontextualize situations to represent them as equations and re-contextualize equations to explain their meanings as they relate to situations (MP.2).

## Tips for Teachers

?

Lessons 4 and 5 focus on representing proportional relationships as equations. Equations are abstract and can be challenging for some students to grasp. Encourage students to return to the table to show the relationship between the two quantities, either adding a column to show the constant of proportionality or drawing an arrow across rows and indicating the multiplication. Ensure that students know what the variables in the equation represent to keep the context connected to the abstract form.

#### Remote Learning Guidance

If you need to adapt or shorten this lesson for remote learning, we suggest prioritizing Anchor Problem 1 (benefits from discussion) and Anchor Problem 3 (can be done independently). Find more guidance on adapting our math curriculum for remote learning here.

#### Fishtank Plus

• Problem Set
• Student Handout Editor
• Vocabulary Package

## Anchor Problems

?

### Problem 1

Felix worked at a music shop during the summer. The table below shows some of Felix’s hours and earnings. The amount of money he earned is proportional to the number of hours he worked.

 # hours worked Amount earned ($) 1 2 18 3 5 45 10 15 135 20 180 1. Fill in the rest of the table and then write an equation that represents the relationship. 2. If Felix worked 35 hours, then how much money did he earn? 3. If Felix earned$198, then how many hours did he work?

### Problem 2

The table below shows measurement conversions between cups and ounces.

 Cups Ounces 3 24 5 40 8 64

Let $x$ represent the number of cups and $y$ represent the number of ounces. Write an equation that represents this relationship.

### Problem 3

The students in Ms. Baca’s art class were mixing yellow and blue paint. She told them that two mixtures will be the same shade of green if the blue and yellow paint are in the same ratio.

The table below shows the different mixtures of paint that the students made.

 A B C D E F Yellow 1 part 2 parts 3 parts 4 parts 5 parts 6 parts Blue 2 parts 3 parts 6 parts 6 parts 8 parts 9 parts
1. How many different shades of paint did the students make?
2. Write an equation that relates $y$, the number of parts of yellow paint, and $b$, the number of parts of blue paint for each of the different shades of paint the students made.

#### References

Illustrative Mathematics Art Class, Variation 2

Art Class, Variation 2, accessed on Aug. 1, 2017, 3:07 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

## Problem Set

? 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 examples where students are given a table and must determine the equation that represents the proportional relationship. Some example contexts include:
• Batches of recipe ingredients
• Distance, time, speed
• Miles per gallon
• Unit prices
• Unit conversions
• Paint or other mixtures
• Include problems where students explain the relationship between the table, constant of proportionality, and equation in context.

?

A lemonade is made by mixing flavored powder, $p$, with water, $w$. The chart below shows some measurements that can be used to make different amounts of lemonade.

 Amount of powder (tsp) Amount of water (cups) $\frac{1}{2}$ $2$ $2$ $8$ $3\frac{1}{2}$ $14$ $4$ $16$
1. Which equation represents this relationship?
1.   $p=4w$
2.   $w=4p$
3.   $w=4+p$
4.   $p=4\div w$
2. What is the constant of proportionality, and what does it mean in this example?

?