Finding and Fixing Slope Errors | Grade 8 Math | LevelKite

A slope calculation is a short chain of small steps, and any one of them can go wrong. In this lesson you are the checker. You read slope work that someone else wrote, find the exact step that went wrong, explain why it is wrong, and fix it.

First, recall what the number means. The slope of a line that is not vertical is the change in $y$ divided by the change in $x$ between two different points on the line. A change is the second value minus the first, so it can be negative. The slope tells you how many units $y$ changes each time $x$ goes up by $1$ . It is the same for every pair of different points on that line, so there is exactly one right answer to check against.

Key concept

What correct slope work does

Take two different points $(x_1, y_1)$ and $(x_2, y_2)$ with $x_1 \neq x_2$ , and let $m$ stand for the slope of the line through them. Correct work with the slope formula does four things:

It puts the change in $y$ on top and the change in $x$ on the bottom.
It uses the two points in the same order on top and on the bottom.
It keeps the sign of every coordinate when it subtracts.
It finishes the division correctly.

A slip in any one of these can change the answer. If $x_1 = x_2$ , the points lie on a vertical line and the bottom would be $0$ . Division by zero is undefined, so a vertical line has undefined slope.

m = \frac{y_2 - y_1}{x_2 - x_1} \quad \text{for } x_1 \neq x_2

Here are five attempts to find the slope of the line through $(-2, 5)$ and $(4, -7)$ . Two are correct. For the other three, decide which item in the list above each one breaks before you read the last column.

Five attempts at the same slope

Attempt	Work	What happened
A	$\frac{-7 - 5}{4 - (-2)} = \frac{-12}{6} = -2$	Correct: change in $y$ over change in $x$ , same order, signs kept.
B	$\frac{5 - (-7)}{-2 - 4} = \frac{12}{-6} = -2$	Correct: the other order on top and on the bottom. Both changes switch sign, so the value is the same.
C	$\frac{4 - (-2)}{-7 - 5} = \frac{6}{-12} = -\frac{1}{2}$	Upside down: this divides the change in $x$ by the change in $y$ .
D	$\frac{-7 - 5}{-2 - 4} = \frac{-12}{-6} = 2$	Mixed order: the top puts $(4, -7)$ first, but the bottom puts $(-2, 5)$ first.
E	$\frac{-7 - 5}{4 - 2} = \frac{-12}{2} = -6$	Sign slip: the bottom should be $4 - (-2) = 6$ , but the negative sign was dropped.

To check slope work, do not just scan it for something that looks odd. Rebuild it and compare.

Find the true slope yourself from the exact coordinates: change in $y$ over change in $x$ , both as second point minus first point. Divide only if the change in $x$ is not $0$ . If it is $0$ , the line is vertical and the slope is undefined.
Read the given work from the start. It is a chain of expressions joined by equals signs. Work out the value of each expression.
Stop at the first expression whose value is not the true slope. Every expression in correct work is the slope written another way, so this is where the work went wrong. If it is the very first expression, the setup is wrong. If it is a later one, the step into it changed the value.
Name the error and correct the work from there.

You can also test a final answer directly, with the move you already use to find a missing coordinate. For two points with different $x$ -values, the slope times the change in $x$ equals the change in $y$ . So a number $m$ is the correct slope exactly when starting at $y_1$ and adding $m$ times the change in $x$ lands on $y_2$ :

y_1 + m\,(x_2 - x_1) = y_2 \quad \text{when } x_1 \neq x_2

Worked example

Checking Kai's work

Kai finds the slope of the line through $(2, 9)$ and $(6, 1)$ . He writes $m = \frac{9 - 1}{6 - 2} = \frac{8}{4} = 2$ . Find the error and correct it.

Find the true slope. Call $(2, 9)$ the first point and $(6, 1)$ the second. The change in $y$ is $1 - 9 = -8$ and the change in $x$ is $6 - 2 = 4$ . The change in $x$ is not $0$ , so you can divide.
$m = \frac{1 - 9}{6 - 2} = \frac{-8}{4} = -2$
Read Kai's work from the start. His first expression has the value $\frac{8}{4} = 2$ , not $-2$ , so the error is already in the setup. A second clue: from $(2, 9)$ to $(6, 1)$ , $x$ goes up while $y$ goes down, so the slope must be negative.
$\frac{9 - 1}{6 - 2} = \frac{8}{4} = 2 \neq -2$
Name the error. On top, $9 - 1$ puts $(2, 9)$ first. On the bottom, $6 - 2$ puts $(6, 1)$ first. Kai mixed the order of the points. Reversing only one of the two subtractions changes its sign, so the whole fraction is multiplied by $-1$ . That turned $-2$ into $2$ .
$9 - 1 = -(1 - 9)$
Test the corrected slope. Start at $y_1 = 9$ and add $m$ times the change in $x$ . You land on $y_2 = 1$ , so $-2$ is correct.
$9 + (-2)(6 - 2) = 9 + (-2)(4) = 9 - 8 = 1$

Answer

Kai used the points in mixed order, so he got the opposite of the true slope. The slope of the line through $(2, 9)$ and $(6, 1)$ is $-2$ .

Worked example

Checking Maya's work

Maya finds the slope of the line through $(-3, 1)$ and $(5, 7)$ . She writes $m = \frac{7 - 1}{5 - (-3)} = \frac{6}{5 - 3} = \frac{6}{2} = 3$ . Find the error and correct it.

Find the true slope. The change in $y$ is $7 - 1 = 6$ . The change in $x$ is $5 - (-3)$ . From $-3$ you go up $3$ to reach $0$ , then $5$ more to reach $5$ , so the change is $8$ . In symbols, subtracting $-3$ is the same as adding $3$ . The change in $x$ is not $0$ , so you can divide.
$m = \frac{7 - 1}{5 - (-3)} = \frac{6}{5 + 3} = \frac{6}{8} = \frac{3}{4}$
Check Maya's first expression. It matches the correct setup and has the value $\frac{6}{8} = \frac{3}{4}$ . So far her work is correct: change in $y$ on top, change in $x$ on the bottom, same order.
Check her second expression. Its value is $\frac{6}{2} = 3$ , not $\frac{3}{4}$ , so the step into this expression changed the value. The top is right, because $7 - 1 = 6$ . On the bottom, $5 - (-3)$ became $5 - 3$ . Maya dropped the negative sign of $-3$ .
$\frac{6}{5 - 3} = \frac{6}{2} = 3 \neq \frac{3}{4}$
Test the corrected slope. Start at $y_1 = 1$ and add $m$ times the change in $x$ , which is $8$ . You land on $y_2 = 7$ , so $\frac{3}{4}$ is correct.
$1 + \frac{3}{4} \cdot 8 = 1 + \frac{24}{4} = 1 + 6 = 7$

Answer

Maya's setup is correct, but she changed $5 - (-3)$ into $5 - 3$ and lost the negative sign. The slope of the line through $(-3, 1)$ and $(5, 7)$ is $\frac{3}{4}$ .

A zero on top and a zero on the bottom mean different things. A common mistake is to treat them alike: calling $\frac{0}{6}$ undefined, or calling $\frac{7}{0}$ zero. That fails because division asks which number times the bottom gives the top.

$(-4, 6)$ and $(2, 6)$ : $\frac{6 - 6}{2 - (-4)} = \frac{0}{6} = 0$ , because $0 \cdot 6 = 0$ . The line is horizontal and its slope is $0$ .
$(3, -2)$ and $(3, 5)$ : the change in $y$ is $5 - (-2) = 7$ and the change in $x$ is $3 - 3 = 0$ . No number times $0$ gives $7$ , so division by zero is undefined. The line is vertical and its slope is undefined, not $0$ . A vertical line also does not give $y$ as a function of $x$ , because one $x$ -value is paired with more than one $y$ -value.

Three points on one line

x	y	y divided by x
$1$	$5$	$5$
$2$	$7$	$3.5$
$4$	$11$	$2.75$

Worked example

Checking Dev's work

The table above lists three points on one line: $(1, 5)$ , $(2, 7)$ and $(4, 11)$ . Dev looks at the first row and writes that the slope is $\frac{y}{x} = \frac{5}{1} = 5$ . Find the error and correct it.

Find the true slope from two rows, $(1, 5)$ and $(2, 7)$ . The change in $x$ is $2 - 1 = 1$ , which is not $0$ , so you can divide.
$m = \frac{7 - 5}{2 - 1} = \frac{2}{1} = 2$
Read Dev's work. Its value is $5$ , not $2$ , so his setup is wrong. He divided one point's $y$ -value by its $x$ -value. He never found a change in $y$ or a change in $x$ .
$\frac{5}{1} = 5 \neq 2$
See why the shortcut fails here. This line has one slope, but the last column of the table gives a different number in each row. Dividing $y$ by $x$ (with $x \neq 0$ ) gives the slope only for a proportional relationship, $y = kx$ with $k$ a constant, whose line passes through the origin. This line is straight, but the relationship is not proportional.
$\frac{5}{1} = 5, \quad \frac{7}{2} = 3.5, \quad \frac{11}{4} = 2.75$
Test the slope $2$ on a different pair of rows, $(1, 5)$ and $(4, 11)$ . This is a fair test because the slope is the same between any two different points on a line that is not vertical. Start at $5$ and add $2$ times the change in $x$ . You land on $11$ .
$5 + 2(4 - 1) = 5 + 2(3) = 5 + 6 = 11$

Answer

Dev divided a single $y$ -value by its $x$ -value instead of dividing a change in $y$ by a change in $x$ . The slope of the line is $2$ .

To check slope work, find the true slope yourself from the exact coordinates, then read the given work from the start. When the two $x$ -values are different, every expression in correct work equals the slope, so the first expression with a different value shows where the error is. Watch for an upside-down fraction, mixed point order, a lost negative sign, a single $y$ -value divided by a single $x$ -value, and a misread zero. Test your corrected slope with $y_1 + m(x_2 - x_1) = y_2$ . When the two $x$ -values are equal, the line is vertical and its slope is undefined.