Lesson 12Using Graphs to Compare Relationships

Learning Goal

Let’s graph more than one relationship on the same grid.

Learning Targets

I can compare two related proportional relationships based on their graphs.
I know that the steeper graph of two proportional relationships has a larger constant of proportionality.

Lesson Terms

coordinate plane
origin

Warm Up: Number Talk: Fraction Multiplication and Division

Find each product or quotient mentally.

$\frac{2}{3} \cdot \frac{1}{2}$
$\frac{4}{3} \cdot \frac{1}{4}$
$4 \div \frac{1}{5}$
$\frac{9}{6} \div \frac{1}{2}$

Activity 1: Race to the Bumper Cars

Diego, Lin, and Mai went from the ticket booth to the bumper cars. Descriptions and tables representing their journeys are below.

Use each description to complete the table representing that person’s journey.
- Diego left the ticket booth at the same time as Tyler. Diego jogged ahead at a steady pace and reached the bumper cars in 30 seconds.
- Lin left the ticket booth at the same time as Tyler. She ran at a steady pace and arrived at the bumper cars in 20 seconds.
- Mai left the booth 10 seconds later than Tyler. Her steady jog enabled her to catch up with Tyler just as he arrived at the bumper cars.
Diego’s
time
(seconds)
Diego’s
distance
(meters)
$0$
$15$
$30$
$50$
$1$
Lin’s
time
(seconds)
Lin’s
distance
(meters)
$0$
$25$
$20$
$50$
$1$
Mai’s
time
(seconds)
Mai’s
distance
(meters)
$0$
$25$
$40$
$50$
$1$
Using a different color for each person, draw a graph of all four people’s journeys (including Tyler’s from the other day).
- Drag the names to the correct lines to label them.
- If you choose to, you can use the Paint Brush tool to change the color of each line. Select the tool, click on a color in the palette below the graph, and then click on a line. Click on the Move tool (the arrow) before changing to a new paint brush color.
- You can hide any points you create with the checkbox below the graph.
open applet in presentation mode
Which person is moving the most quickly? How is that reflected in the graph?

Diego’s time (seconds)	Diego’s distance (meters)
$0$
$15$
$30$	$50$
$1$

Lin’s time (seconds)	Lin’s distance (meters)
	$0$
	$25$
$20$	$50$
$1$

Mai’s time (seconds)	Mai’s distance (meters)
	$0$
	$25$
$40$	$50$
$1$

Print Version

Diego, Lin, and Mai went from the ticket booth to the bumper cars.

Use each description to complete the table representing that person’s journey.
- Diego left the ticket booth at the same time as Tyler. Diego jogged ahead at a steady pace and reached the bumper cars in 30 seconds.
- Lin left the ticket booth at the same time as Tyler. She ran at a steady pace and arrived at the bumper cars in 20 seconds.
- Mai left the booth 10 seconds later than Tyler. Her steady jog enabled her to catch up with Tyler just as he arrived at the bumper cars.
Diego’s
time
(seconds)
Diego’s
distance
(meters)
$0$
$15$
$30$
$50$
$1$
Lin’s
time
(seconds)
Lin’s
distance
(meters)
$0$
$25$
$20$
$50$
$1$
Mai’s
time
(seconds)
Mai’s
distance
(meters)
$0$
$25$
$40$
$50$
$1$
Using a different color for each person, draw a graph of all four people’s journeys (including Tyler’s from the other day).
Which person is moving the most quickly? How is that reflected in the graph?

Diego’s time (seconds)	Diego’s distance (meters)
$0$
$15$
$30$	$50$
$1$

Lin’s time (seconds)	Lin’s distance (meters)
	$0$
	$25$
$20$	$50$
$1$

Mai’s time (seconds)	Mai’s distance (meters)
	$0$
	$25$
$40$	$50$
$1$

Are you ready for more?

Write equations to represent each person’s relationship between time and distance.

Activity 2: Space Rocks and the Price of Rope

Problem 1

Meteoroid Perseid 245 and an unknown asteroid were traveling through the solar system.

Explore the applet to learn about the distance they had each traveled after a given time.

open applet in presentation mode

Is Asteroid x traveling faster or slower than Perseid 245? Explain how you know.

Print Version

Meteoroid Perseid 245 and Asteroid x travel through the solar system. The graph shows the distance each traveled after a given point in time.

The graph of two lines in the coordinate plane with the horizontal axis labeled "time" and the vertical axis labeled "distance." One line, labeled "asteroid x," begins at the origin and moves steeply upwards and to the right. The other line, labeled "Perseid 245," also begins at the origin and moves steadily upwards and to the right.

Is Asteroid x traveling faster or slower than Perseid 245? Explain how you know.

Problem 2

The graph shows the price, $p$ , of different lengths, $L$ , of two types of rope.

Two lines graphed in the coordinate plane labeled "Cotton" and "Nylon". There is a horizontal L axis and a vertical p axis. The line labeled "Nylon" begins at the origin and moves steadily upward and to the right. The line labeled "Cotton" also begins at the origin moves steeply upward and to the right.

If you buy $1.00 of each kind of rope, which one will be longer? Explain how you know.

Lesson Summary

Here is a graph that shows the price of blueberries at two different stores. Which store has a better price?

A graph showing two lines comparing the price of blueberries. The x axis is weight in pounds going from 0 to 5. The y axis is cost in dollars going from 0 to 40.

We can compare points that have the same $x$ value or the same $y$ value. For example, the points $(2, 12)$ and $(3, 12)$ tell us that at store B you can get more pounds of blueberries for the same price.

The points $(3, 12)$ and $(3, 18)$ tell us that at store A you have to pay more for the same quantity of blueberries. This means store B has the better price.

We can also use the graphs to compare the constants of proportionality. The line representing store B goes through the point $(1, 4)$ , so the constant of proportionality is 4. This tells us that at store B the blueberries cost $4 per pound. This is cheaper than the $6 per pound unit price at store A.