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4.4: Slope and Rate of Change

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

At the end of this lesson, students will be able to:

  • Find positive and negative slopes.
  • Recognize and find slopes for horizontal and vertical lines.
  • Understand rates of change.
  • Interpret graphs and compare rates of change.


Terms introduced in this lesson:

positive slope, negative slope
climbing, descending
lattice points
rate of change
undefined slope, infinite slope
interpret a graph

Teaching Strategies and Tips

Use the introduction to motivate the concept of slope. Point out:

  • Just as two points determine a unique line, a point and a slope also determine exactly \begin{align*}1\end{align*}1 line.
  • Viewing the slope as the ratio, \begin{align*}\frac{\mathrm{rise}} {\mathrm{run}}\end{align*}riserun, is useful. From one point on the line, knowing how to rise and run brings you to a second point.
  • The slope of a line is constant. That is, for any two points on the line, the ratio \begin{align*}\frac{\mathrm{rise}} {\mathrm{run}}\end{align*}riserun is the same.
  • If \begin{align*}m = \frac{2} {3}\end{align*}m=23, then \begin{align*}\mathrm{rise} = 2\end{align*}rise=2 and \begin{align*}\mathrm{run} = 3\end{align*}run=3. Since \begin{align*}\frac{2} {3} = \frac{-2} {-3}\end{align*}23=23, going down two units and then left \begin{align*}3\;\mathrm{units}\end{align*}3units will also be a point on the line.

Use Example 1 to demonstrate rise-to-run triangles for lines. The triangles are most useful when constructed on lattice points (all coordinates of the vertices are integers). This makes the slope calculation effortless. Observe that the hypotenuse runs along the line.

Use Example 2 to derive a formula for slope.

Emphasize that graphs are read from left to right.

  • Linear functions are increasing when their graphs slant up and to the right (\begin{align*}y\end{align*}y increases as \begin{align*}x\end{align*}x is increased). In this case, slope is positive since \begin{align*}\triangle y\end{align*}y and \begin{align*}\triangle x\end{align*}x are both positive (or both negative).
  • Linear functions are decreasing when their graphs slant down and to the right (\begin{align*}y\end{align*}y decreases as \begin{align*}x\end{align*}x is increased). In this case, slope is negative since either \begin{align*}\triangle y\end{align*}y or \begin{align*}\triangle x\end{align*}x is negative, but not both.

Supplement Example 4 with a skiing analogy.

  • Horizontal lines have zero slope, or no slope. This corresponds to cross-country skiing.
  • Vertical lines have undefined slope. This corresponds to falling down a cliff (undefined skiing). Vertical lines have infinite slope

Error Troubleshooting

General Tip. A common mistake is to subtract the \begin{align*}x\end{align*}x and \begin{align*}y-\end{align*}ycoordinates in different orders in the slope formula; i.e., \begin{align*}m \neq \frac{y_2 - y_1} {x_1 - x_2}\end{align*}my2y1x1x2. To avoid making this error, students can write point 1 and point 2 above the two points, and then select the coordinates in the same order. See Example 6. Of course, the choice for point 1 and point 2 is arbitrary.

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