Common Core Standards: Math
Expressions and Equations 8.EE.B.6
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.
Joe Weakleg is in a bicycle race—one of those long, grueling marathons that are on TV for hours and hours on Sunday afternoons. This part of the race is uphill and it seems to Joe that the more he pedals, the steeper the hill is getting.
After the race, he takes a step back. (Well, actually a whole lot of steps.) He stands back and looks at the road from a distance. As it turns out, the road he was on was one long straight line.
Besides learning that he needs to train better before attempting another one of those races, Joe learns that if points are collinear—or on the same straight line—then the slope between different parts of that line is the same. If it's pretty steep at the beginning, then it retains that same "steepness" until the very end.
Students should know that mathematically, this means the line's slope stays the same. They can use similar triangles to solidify their understanding of this concept.
In the diagram above, ΔABE and ΔACD are similar, which means that all their corresponding sides are in proportion. The ratio of CD to BE is the same as that of CA to BA and DA to EA. So while the distance from A to B isn't the same as from A to C, their steepness is the same.
If students need an example, use the triangle's vertices as points. For instance, we can find the slopes of the line segments if A is at (0, 0), B is at (4, 3), and C is at (12, 9). Using the slope formula, we can calculate the slope of AB, AC, and even BC.
So regardless of the distance, all three line segments have the same slope. Again, students should be aware that slope represents the steepness of a line and that negative slopes are possible.
Finally, it's time to introduce them to linear equations in standard form. Students should know that lines can be expressed in the form y = mx where m is the slope because given an x coordinate, all we need to do is multiply it by the slope to find the y coordinate. Once students are comfortable with this, make sure to tweak the formula slightly to get y = mx + b, where b is the y-intercept.
If students get confused, make sure to pedal back to where they were last on track. Once they've flexed and trained their mathematical muscles enough, they should be closer to Lance Armstrong than Joe when it comes to linear equations. Although after a marathon's worth of B.O., they probably don't want to stand too close to either one of them.
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