8th Grade · 8.EE.B.5

Graphing proportional relationships in 8th grade

Graphing proportional relationships is where 8th grade math connects rates with straight lines. In a proportional relationship, one amount is always the same number times the other, like the water a hose adds at 4 gallons per minute. Your child learns to draw these relationships as lines, to read the unit rate as the slope and to compare two rates shown in different ways.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

What it means, in plain words

Two amounts are proportional when one is always the same number times the other. That number is the unit rate: the amount of $y$ for one unit of $x$ . A hose that adds 4 gallons every minute gives the equation $y = 4x$ , where $x$ is minutes and $y$ is gallons.

The graph of a proportional relationship is a straight line through $(0, 0)$ , because zero minutes means zero gallons. The slope is the steepness of the line, which is how far it goes up for each step to the right. Graphing proportional relationships in 8th grade comes down to one idea, the unit rate as slope. The hose line goes up 4 for every step right, so its slope is 4, the same number as 4 gallons per minute.

One relationship can appear as an equation, a table, a graph or a sentence. In 8th grade, children compare two relationships shown in different ways, for example a graph of one car's trip and an equation for another car. To compare them, your child finds the unit rate of each. The larger unit rate belongs to the faster car, the better-paid job or the quicker hose.

Your child should be able to:

Graph a proportional relationship from an equation, a table or a description
Find the unit rate on a graph by reading $y$ where $x = 1$ , or by dividing $y$ by $x$ at any point
Explain why the slope of the line and the unit rate are the same number
Compare two proportional relationships shown in different ways

Example

A hose fills an empty tub at 4 gallons per minute. How do you graph the gallons added against the minutes?

Let $x$ be minutes and $y$ be gallons. The equation is $y = 4x$ .
Find a few points: after 0 minutes, 0 gallons; after 1 minute, 4 gallons; after 2 minutes, 8 gallons.
Plot $(0, 0)$ , $(1, 4)$ and $(2, 8)$ and draw a straight line through them.
The line rises 4 for each step right, so the slope is 4. The graph shows a rate of 4 gallons per minute.

Example

Car A travels by the equation $y = 50x$ , with $x$ in hours and $y$ in miles. The graph for Car B is a line through $(0, 0)$ and $(3, 165)$ . Which car is faster?

Car A's unit rate is the number in front of $x$ : 50 miles per hour.
Car B goes 165 miles in 3 hours, so its unit rate is $165 \div 3 = 55$ miles per hour.
$55 > 50$ , so Car B is faster, by 5 miles per hour.

Where children usually get stuck

From the point $(2, 8)$ on a graph of pay against hours, your child writes the unit rate as $\frac{2}{8} = \frac{1}{4}$ instead of 4 dollars per hour.
Your child divides the numbers in the order they appear in the point, $x$ first, without asking which amount is per which.
What helps. Ask for the unit rate as words with units: "dollars per hour". Dollars come first in the words, so dollars go on top: $8 \div 2 = 4$ .
Your child picks the steeper-looking line as the faster rate, even though the two graphs use different scales on their axes.
How steep a line looks depends on how far apart the grid lines are. A graph that counts by 10s on one axis can make a slow rate look steep.
What helps. Cover the pictures and find one point on each line. Divide $y$ by $x$ for each one and compare the numbers, not the drawings.
Your child graphs 4 gallons per minute as a line that starts at 4 on the y-axis instead of at $(0, 0)$ .
The 3 is the first number in the problem, so it becomes the starting point. Your child has not yet linked "proportional" with "zero of one amount means zero of the other".
What helps. Ask how much water the hose has added after 0 minutes. The answer, 0 gallons, is the point the line has to start from.

Try four practice questions

Real 8th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Look at the equation below. Another relationship increases by 5 for each 1-unit increase in x. Fill in the table, then choose which relationship has the greater unit rate.
$\displaystyle y = 3x$
Item Answer
Equation unit rate
Description unit rate
Greater rate
Show the answer and how to solve it
The table should show 3 for Equation unit rate, 5 for Description unit rate, and Description for Greater rate because 5 is greater than 3.
1. 1In a proportional equation written as y = kx, the coefficient of x is the unit rate. Here the coefficient is 3, so the equation's unit rate is 3.
2. 2The verbal description says the relationship increases by 5 for each 1-unit increase in x, so its unit rate is 5.
3. 3Since 5 is greater than 3, the Description has the greater unit rate.
Common mistake. A common mistake is confusing which number belongs to which relationship. The coefficient in y = 3x is 3 (the equation's rate), and 5 comes from the verbal description.
The idea underneath. In y = kx, the coefficient k is already the unit rate, so comparing the relationships means comparing that coefficient to the rate stated in words.
2On level
The graph shows Lila's pay y in dollars after x hours of yard work. Maya says, "Lila earns 4 dollars for each 1 hour of yard work." Jayden says, "Every 1 hour, Lila earns 4 more dollars." Who is correct?
Lila's Yard Work Pay
A
Only Maya
B
Only Jayden
C
Both students
D
Neither student
Show the answer and how to solve it
Both students are correct because each statement describes the unit rate of 4 dollars per hour, just using different wording.
1. 1Look at the graph. The line passes through the origin (0, 0) and the marked point (2, 8).
2. 2Compute the unit rate: change in pay divided by change in hours.
3. 3Maya says Lila earns 4 dollars for each 1 hour. The unit rate is 4, so Maya is correct.
4. 4Jayden says every 1 hour Lila earns 4 more dollars. That is also a per-1-hour statement with the same rate of 4. Jayden is also correct.
5. 5Because both statements describe the amount of pay for each 1 hour and both give 4, both students are correct.
Common mistake. A common mistake is dismissing one student because the wording is different. Both statements express the same rate: 4 dollars for each 1 hour.
The idea underneath. On a proportional graph, slope tells how much y changes for each 1 unit of x, while the origin shows the starting amount is 0.
3On level
Look at the proportional relationship on the graph. Select all calculations that correctly find the unit rate k.
A
$\displaystyle k = {}$
$\displaystyle \frac{10}{4}$
B
$\displaystyle k = {}$
$\displaystyle \frac{4}{10}$
C
$\displaystyle k = {}$
$\displaystyle \frac{10 - 5}{4 - 2}$
D
$\displaystyle k = {}$
$\displaystyle 10$
E
$\displaystyle k = {}$
$\displaystyle \frac{10 - 5}{2 - 4}$
Show the answer and how to solve it
The valid calculations are $k = \frac{10}{4}$ and $k = \frac{10 - 5}{4 - 2}$ because both equal $\frac{5}{2}$ , the slope of the proportional line.
1. 1Look at the marked points on the line. The graph goes through the origin and also through $(2, 5)$ and $(4, 10)$ .
2. 2Because this is a proportional graph through the origin, use $y/x$ from a point on the line. Using $(4, 10)$ :
3. 3You can also use change in $y$ over change in $x$ between two points on the line. From $(2, 5)$ to $(4, 10)$ , the rise is 5 and the run is 2.
4. 4Now check the other calculations. $\frac{4}{10}$ flips run and rise, 10 is only a y-coordinate, and reversing only the x-subtraction changes the sign.
Common mistake. A common mistake is to flip rise and run and write $\frac{4}{10}$ instead of $\frac{10}{4}$ . Another common error is reversing only one subtraction, which changes the sign even though the line rises from left to right.
The idea underneath. On a proportional graph, every correct slope calculation from points on the line gives the same unit rate because $y/x$ and change in $y$ divided by change in $x$ match the same constant ratio.
4Stretch
Look at the equation below. In another relationship, y decreases by $\frac{1}{2}$ for each 1 increase in x. Ethan says the equation has the greater unit rate because $0.8 > 0.5$ . Which evaluation is best?
$\displaystyle y = -0.8x$
A
The claim is correct.
B
The claim is wrong. Absolute values were compared instead of signed rates.
C
The claim is wrong. The verbal rate is $-2$ , not $-\frac{1}{2}$ .
D
The claim is wrong. $-0.8$ and $-\frac{1}{2}$ are equal.
Show the answer and how to solve it
The best evaluation is: The claim is wrong. Absolute values were compared instead of signed rates. The verbal rate $-0.5$ is greater than the equation rate $-0.8$ .
1. 1The equation is $y = -0.8x$ , so the equation's unit rate is $-0.8$ .
2. 2The verbal description says $y$ decreases by $\frac{1}{2}$ for each 1 increase in $x$ . Decreases means negative, so the verbal unit rate is $-\frac{1}{2}$ .
3. 3Rewrite $-\frac{1}{2}$ as a decimal to compare with $-0.8$ .
4. 4Compare the signed rates: $-0.5$ is closer to zero than $-0.8$ , so $-0.5 > -0.8$ . The verbal rate is actually greater.
5. 5Ethan compared the absolute values $0.8 > 0.5$ and concluded the equation rate is greater. Dropping the negative signs reverses the true comparison for negative numbers. Absolute values were compared instead of signed rates.
Common mistake. A common mistake is comparing $0.8$ and $0.5$ as positive numbers. Since both rates are negative, $-0.5$ (closer to zero) is actually greater than $-0.8$ .
The idea underneath. To compare unit rates fairly, rewrite them in the same number form first, because equivalent fractions and decimals represent the same slope.