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How to compute slope Memory Aid cheat sheet - grade 6-8

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Math Grade 6-8

How to compute slope Memory Aid Cheat Sheet

A printable reference covering slope, rise over run, slope from two points, graph direction, and undefined slope for grades 6-8.

TopicsGraphsAlgebraFunctionsRatios and ProportionsMathematicsStudy Guide
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This cheat sheet helps students remember how to compute slope from a graph, a table, or two points. Slope tells how steep a line is and whether it goes up, down, or stays flat. Students need slope to understand linear relationships, graph lines, and compare rates of change.

A simple memory aid is slope equals rise over run.

Key Facts

  • Slope is the rate of change of a line and can be written as m=riserunm = \frac{\text{rise}}{\text{run}}.
  • From two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), slope is m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
  • The rise is the vertical change, so rise=y2−y1\text{rise} = y_2 - y_1.
  • The run is the horizontal change, so run=x2−x1\text{run} = x_2 - x_1.
  • A line that rises from left to right has a positive slope, such as m=32m = \frac{3}{2}.
  • A line that falls from left to right has a negative slope, such as m=−32m = -\frac{3}{2}.
  • A horizontal line has slope m=0m = 0 because the rise is 00.
  • A vertical line has an undefined slope because the run is 00 and division by 00 is not defined.

Vocabulary

Slope
Slope is a number that describes the steepness and direction of a line.
Rise
Rise is the vertical change between two points, found by subtracting the yy-values.
Run
Run is the horizontal change between two points, found by subtracting the xx-values.
Rate of Change
Rate of change describes how much one quantity changes compared with another quantity.
Positive Slope
A positive slope means a line goes upward from left to right.
Undefined Slope
An undefined slope occurs when a line is vertical and the run equals 00.

Common Mistakes to Avoid

  • Switching the rise and run is wrong because slope is m=riserunm = \frac{\text{rise}}{\text{run}}, not runrise\frac{\text{run}}{\text{rise}}.
  • Subtracting coordinates in different orders is wrong because y2−y1x2−x1\frac{y_2 - y_1}{x_2 - x_1} must use the same point order in both the numerator and denominator.
  • Forgetting the negative sign is wrong because a line that falls from left to right must have a negative slope.
  • Calling a vertical line slope 00 is wrong because a vertical line has run 00, so its slope is undefined.
  • Using only one point to find slope is wrong because slope compares the change between two points, not the location of one point.

Practice Questions

  1. 1 Find the slope between the points (2,3)(2, 3) and (6,11)(6, 11).
  2. 2 Find the slope between the points (−1,5)(-1, 5) and (3,−7)(3, -7).
  3. 3 A line has a rise of −4-4 and a run of 22. What is the slope?
  4. 4 Explain how you can tell from a graph whether a line has positive slope, negative slope, zero slope, or undefined slope.

Understanding How to compute slope Memory Aid

A reliable method matters more than memorizing a picture. When using two points, choose one point as the starting point and keep that choice for both changes. Subtract the starting vertical value from the ending vertical value.

Then subtract the starting horizontal value from the ending horizontal value. The order can be reversed, but it must be reversed in both places. For example, moving from a point with horizontal value one and vertical value two to a point with horizontal value five and vertical value eight gives a vertical change of six and a horizontal change of four.

The slope is six over four, which simplifies to three over two. If the points are taken in the opposite direction, both changes are negative, and the same positive result appears.

A slope is a comparison, not just a single change. A line with slope three over two goes up three units every time it moves right two units. It can use bigger jumps that keep the same comparison, such as up six and right four.

This is why two lines may look different on a graph but be parallel. Their step patterns match. Simplifying a slope helps reveal this match.

A slope of one half means the vertical value changes slowly compared with the horizontal value. A slope of five means the vertical value changes quickly. The size of the slope describes steepness, while its sign describes direction.

Tables show slope through repeated changes. Check the difference between nearby horizontal values, then check the matching difference between vertical values. If the horizontal value increases by two each time and the vertical value increases by six each time, the constant rate is three vertical units per one horizontal unit.

A table represents a straight line only when this rate stays constant throughout the table. Students sometimes divide the vertical change by the wrong horizontal change. Match changes from the same pair of rows.

It is useful to write the units too. In a travel table, slope might mean miles per hour. In a money table, it might mean dollars per item.

Slope appears whenever one quantity changes in response to another. A taxi fare can have a cost per mile. A recipe can have an amount of flour per batch.

A ramp has a vertical change compared with its horizontal distance. On a graph, use grid marks carefully because one square does not always equal one unit. Read the scale on each axis before counting.

Watch for a vertical line, since its horizontal change is zero and no ordinary slope value can be calculated. A horizontal line is different.

Its vertical value never changes, so its rate is zero. Keeping these two cases separate prevents a common mistake.