Meet Slope

Student Summary

Here is a line drawn on a grid. There are also four right triangles drawn.

<p>Four right triangles.</p>
<p>Four right triangles each with hypotenuse on the same line. First horizontal side 6, vertical side 4. Second horizontal side 3, vertical side 2. Third horizontal side 1, vertical side fraction 2 over 3. Fourth horizontal side 6, vertical side 4.</p>  

These four triangles are all examples of slope triangles. The longest side of a slope triangle is on the line, one side is vertical, and another side is horizontal. The slope of the line is the quotient of the vertical length and the horizontal length of the slope triangle. This number is the same for all slope triangles for the same line because all slope triangles for the same line are similar.

In this example, the slope of the line is 23\frac{2}{3}. Here is how the slope is calculated using the slope triangles:

  • Points AA and BB give 2÷3=232\div 3=\frac23.
  • Points DD and BB give 4÷6=234\div 6=\frac23.
  • Points AA and CC give 4÷6=234\div 6=\frac23.
  • Points AA and EE give 23÷1=23\frac23 \div 1=\frac23.

Visual / Anchor Chart

Standards

Building On
8.G.4

Know that a two-dimensional figure is similar to another if the corresponding angles are congruent and the corresponding sides are in proportion. Equivalently, two two-dimensional figures are similar if one is the image of the other after a sequence of rotations, reflections, translations, and dilations. Given two similar two-dimensional figures, describe a sequence that maps the similarity between them on the coordinate plane.

Addressing
8.EE.6

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Building Toward
8.EE.6

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.