[br]NOW it's time to introduce the math behind the [color=#0000ff]'rise over run'[/color] concept.[br][br]Mathematically, 'rise over run' refers to the ratio of change in y to change in x. In symbols,[br] [img]data:image/png;base64,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[/img][br]where [i][color=#0000ff]m[/color] = [/i][color=#38761d]slope[/color] and [color=#0000ff](x₁, y₁)[/color] and [color=#0000ff](x₂, y₂)[/color] are the [color=#38761d]coordinates[/color] of two points.
[br]1. Plot each pair of points (x,y) in the applet below.[br][br]2. Use the applet to obtain the [color=#0000ff]'rise'[/color] and the [color=#0000ff]'run'[/color] between each pair of points and then compute the slope. The first has been done for you.[br][br]3. List these data values on a worksheet patterned after the table below.[br][br]Pair (x₁, y₁) (x₂, y₂) rise run slope[br][br] A ( 7, 8) ( 1, 5) [u]-3[/u]_ [u]-6[/u]_ __[u]1/2[/u]__[br] B ( 5, 3) (-2, 4) ___ ___ _______[br] C (-3, -5) (-1, 8) ___ ___ _______[br] D ( 0, -5) ( 7, 8) ___ ___ _______[br] E ( 2, 0) ( 2, -3) ___ ___ _______[br] F (-8, -1) (-6, 3) ___ ___ _______[br] G ( 2, 4) ( 2, -3) ___ ___ _______[br] H ( 7, -1) (-8, -1) ___ ___ _______[br] I ( 7, -4) ( 1, -4) ___ ___ _______[br] J ( 4, -3) (-2, -9) ___ ___ _______[br][br][color=#0000ff]NOTE:[/color] The signs of 'rise' and 'run' may be interchanged, but the slopes will remain the same.[br]
[color=#0000ff]QUESTION:[/color] Given two points, how do you calculate the 'rise' between them? Briefly explain.
Use the formula: rise = y₂ - y₁. This formula gives the value of the change in y.
[color=#0000ff]QUESTION:[/color] Given two points, how do you calculate the 'run' between them? Briefly explain.
Use the formula: run = x₂ - x₁. This formula gives the value of the change in x.
[br]1. On a separate sheet of paper, determine the slope of the line that passes through the given points.[br] 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[/img][br]2. Enter this slope as a ratio in the text-box provided.[br][br]3. If you enter a correct ratio, a big[color=#ff00ff] [i]"Correct!"[/i] [/color]sign will appear.[br][br]4. Click on the "New Problem" button to generate a new problem.[br][br]Repeat until you have mastered the concept.
In future lessons, you'll have more opportunities to master this concept. We hope you ENJOYED this lesson.