8th Grade · 8.SP.A.3

Interpreting slope and y-intercept in 8th grade

Interpreting slope and y-intercept means saying what the two numbers in a line's equation tell you about a real situation. In 8th grade, these lines are linear models: equations that fit real data, such as hours of sunlight and plant height. Your child learns to put the slope and intercept into words with units and to use the model to make predictions.

What the standard says

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

8.SP.A.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

What it means, in plain words

Some data comes in pairs, with two measurements for each item, such as hours worked and dollars earned. When the points on a graph fall close to a line, an equation $y = mx + b$ can describe them. That equation is a linear model. The slope, $m$ , is how much $y$ changes when $x$ goes up by 1. The y-intercept, $b$ , is the value of $y$ when $x$ is 0, which is where the line crosses the y-axis.

Interpreting slope and y-intercept in 8th grade means putting both numbers into a sentence with units. In a biology experiment, a slope of 1.5 cm per hour means that each extra hour of sunlight a day goes with about 1.5 cm more plant height. In a repair shop's cost model, a y-intercept of 12 dollars means a charge of 12 dollars before any item is repaired.

A linear model fits the data closely but not exactly, so its answers are predictions. Your child uses the equation in both directions: put in $x$ to predict $y$ , or put in $y$ and solve for $x$ . A negative slope means $y$ goes down as $x$ goes up, like a candle getting shorter as it burns.

Your child should be able to:

Say what the slope means in a sentence, with the units of both quantities
Say what the y-intercept means as the value of $y$ when $x$ is 0
Use a linear model to predict $y$ from $x$ , or to find $x$ from $y$
Explain what a negative slope means in a real situation

Example

A repair shop predicts the total cost $y$ , in dollars, of repairing $x$ items with $y = 4x + 12$ . What do the 4 and the 12 mean, and what cost does the model predict for 5 items?

The slope is 4: each extra item adds 4 dollars to the cost.
The y-intercept is 12: with 0 items repaired, the cost is 12 dollars, a starting charge.
For 5 items: $y = 4 \times 5 + 12 = 20 + 12 = 32$ .
The model predicts a total cost of 32 dollars for 5 items.

Example

A candle's height $y$ , in centimeters, after burning for $x$ hours follows the model $y = -0.5x + 10$ . After how many hours is the candle 6 cm tall?

The y-intercept is 10, so the candle starts 10 cm tall. The slope is $-0.5$ , so it loses 0.5 cm every hour.
Set $y = 6$ : $6 = -0.5x + 10$ .
Subtract 10 from both sides: $-4 = -0.5x$ .
Divide both sides by $-0.5$ : $x = 8$ .
The candle is 6 cm tall after 8 hours.

Where children usually get stuck

For $y = 4x + 12$ , your child says the cost goes up 12 dollars for each item.
The 12 is the bigger number and sits at the end, so it looks like the main rate. Your child has not yet linked the number next to $x$ with the change per item.
What helps. Ask what the cost is for 0 items, then for 1 item. Seeing 12 dollars, then 16 dollars, shows that 12 is the start and 4 is the step.
For a pay model $y = 3.5x + 10$ , with $x$ in hours and $y$ in dollars, your child writes "the slope is 3.5" and stops.
Slope was learned as a number on a graph. In a model, the number only means something with its units attached.
What helps. Ask your child to fill in one sentence: for each extra ___, the ___ goes up by ___. Here it reads: for each extra hour, the pay goes up by 3.5 dollars.
For the candle model $y = -0.5x + 10$ , your child says the candle grows 0.5 cm each hour.
The minus sign gets dropped when the slope is put into words.
What helps. Ask what happens to $y$ from $x = 0$ to $x = 1$ . The height goes from 10 cm to 9.5 cm, so the candle is getting shorter.

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
A linear model has a slope of 3. Use the signed Δx shown below to fill in the blank for Δy.
$\displaystyle \Delta x$
$\displaystyle =$
$\displaystyle -2$
$\displaystyle \Delta y$
$\displaystyle =$
$\displaystyle 3(\Delta x) =$
Show the answer and how to solve it
Fill the blank with Δy = -6 because slope 3 multiplied by the signed x-change -2 equals -6.
1. 1The signed change in x is already shown as negative 2.
2. 2Substitute into the slope equation to find the change in y.
3. 3Multiply 3 by -2 to get the signed change in y.
Common mistake. A common mistake is ignoring the negative sign on Δx and writing Δy = 6 instead of -6.
The idea underneath. For a linear model, the intercept affects the starting value, but any change in y comes only from slope × signed change in x.
2On level
Use the table and linear model below for a job. Which sentence correctly interprets the slope?
Variable What it represents Unit
x hours worked hours
y total earnings dollars
$\displaystyle y = 3.5x + 10$
A
The model predicts that total earnings increase by about 3.5 dollars each hour.
B
The model predicts that it takes about 3.5 hours to earn 1 dollar.
C
The model predicts that total earnings start at about 3.5 dollars.
D
The model predicts that total earnings decrease by about 3.5 dollars each hour.
Show the answer and how to solve it
The correct answer is that the model predicts total earnings increase by about 3.5 dollars each hour, because slope is the change in y in dollars for each 1 hour of x.
1. 1Look at the table first. It shows that x is hours worked and y is total earnings in dollars.
2. 2In the model, the number multiplied by x is 3.5. That number is the slope.
3. 3Slope tells how much y changes for each 1 unit of x. So here the model says total earnings change by 3.5 dollars for each 1 hour worked. Because 3.5 is positive, earnings increase.
4. 4So the correct sentence is the one that says total earnings increase by about 3.5 dollars each hour. Choice-b reverses the units and says 3.5 hours per dollar, which is a common mix-up.
Common mistake. A common mistake is reversing the units and reading 3.5 as hours per dollar. In a linear model, slope is y-units per x-unit, so this slope is dollars per hour.
The idea underneath. Slope in context is always read as response-variable change per 1 unit of the explanatory variable, so the units come from y over x.

Variable	What it represents	Unit
x	hours worked	hours
y	total earnings	dollars

3On level

Look at Liam's work below. Which statement best describes the first mistake, or shows that the work is correct?

Start	$\displaystyle -5$	$\displaystyle =$	$\displaystyle -4x + 3$
Step 1	$\displaystyle -8$	$\displaystyle =$	$\displaystyle -4x$
Step 2	$\displaystyle x$	$\displaystyle =$	$\displaystyle -2$

The first mistake is in Step 2 because -8 ÷ -4 = 2, not -2.

The first mistake is in Step 1 because

-5 - 3 = -2

, not -8.

The first mistake is in Step 1 because 3 should have been added to both sides, not subtracted.

There is no mistake. The work is correct.

Show the answer and how to solve it

The first mistake is in Liam's Step 2 because -8 ÷ -4 = 2, so x should be 2.

1Check Liam's work one line at a time. The first line that does not match correct algebra is the first mistake.
2Step 1 is correct. To undo the +3, subtract 3 from both sides.
3The first mistake is in Liam's Step 2. When you divide both sides by -4, the two negatives cancel and the result is positive: $\frac{-8}{-4} = 2$ , not $-2$ .
4The correct value of x is 2, not -2. The mistake is in Step 2.

Common mistake. A common mistake is forgetting that dividing two negative numbers gives a positive result. After Liam's correct Step 1 giving -8 = -4x, the solution is x = 2, not -2.

The idea underneath. When you analyze worked algebra, compare each new line to the line before it; the first line that is not equivalent is the first mistake.

4Stretch
The candle model below uses x for time in hours and y for height in centimeters. If the height is 5 cm, what is x?
$\displaystyle y = -0.4x + 8$
Time:hours
Show the answer and how to solve it
Answer. 7.5
The answer is 7.5 because when x = 7.5, the model gives y = -0.4(7.5) + 8 = 5.
1. 1Since the height is 5 cm, replace y with 5 in the model.
2. 2Subtract 8 from both sides to undo the intercept.
3. 3Divide both sides by -0.4. Both the numerator and denominator are negative, so the result is positive.
4. 4So x = 7.5, which means 7.5 hours. Check: $y = -0.4(7.5) + 8 = -3 + 8 = 5$ ✓
Common mistake. A common mistake is treating -3 ÷ -0.4 as -7.5 instead of 7.5. Dividing two negatives gives a positive result, so x = 7.5, not -7.5.
The idea underneath. In $y = mx + b$ , multiply $x$ by $m$ , then add $b$ . To find $x$ , reverse these operations: subtract $b$ , then divide by $m$ .

$\displaystyle \Delta x$	$\displaystyle =$	$\displaystyle -2$
$\displaystyle \Delta y$	$\displaystyle =$	$\displaystyle 3(\Delta x) =$