Slope: Intuitive Introduction

Question 1

Below is a ramp represented by a right triangle. Right now, the green (vertical) leg of the triangle has a length of 5, and the red (horizontal) leg has a length of 9. What is the displayed value of the slope of the ramp?

[math]\frac{5}{9}[/math] [math]\frac{9}{5}[/math]

Question 2

Now drag only one of the white points so that the length of the green (vertical) leg is 4. [br]What is the displayed slope now, and is the ramp steeper or less steep than when the slope was [math]\frac{5}{9}[/math]?

The slope is now [math]\frac{4}{9}[/math] and there is no change in the steepness. The slope is now [math]\frac{4}{9}[/math] and the ramp is less steep. The slope is now [math]\frac{4}{9}[/math], and the ramp is steeper.

Question 3

Now drag whichever of the white points you need to so that the length of the green (vertical) leg is 4 and the length of the red (horizontal) leg is 3. What is the displayed slope of the ramp now, and is it more or less steep than when it was [math]\frac{4}{9}[/math]?

The slope is now [math]\frac{4}{3}[/math], and the ramp is steeper. The slope is now [math]\frac{4}{3}[/math] and there is no change in the steepness. The slope is now [math]\frac{4}{3}[/math] and the ramp is less steep.

Question 4

If the displayed slope is [math]\frac{2}{5}[/math] then which statement is [b][color=#ff0000]FALSE[/color][/b]? (If you need to use the applet to decide, feel free!)

The green (vertical) leg could be 2 units and the red (horizontal) leg could be 5 units The green (vertical) leg could be 6 units and the red (horizontal) leg could be 15 units The green (vertical) leg could be 4 units and the red (horizontal) leg could be 10 units The green (vertical) leg could be 5 units and the red (horizontal) leg could be 2 units

Question 5

Move the points so that the green (vertical) leg is 4 and the red (horizontal) leg is 8. Note the displayed slope. Repeat with respective leg lengths 3 and 6, then 2 and 4, then 1 and 2. What is true?

All slopes were displayed as [math]\frac{1}{2}[/math], and all had the exact same steepness. Not all the slopes were displayed as [math]\frac{1}{2}[/math], and some were steeper than others. All slopes were displayed as [math]\frac{1}{2}[/math], and some were steeper than others. Not all the slopes were displayed as [math]\frac{1}{2}[/math], but all were the exact same steepness.

Question 6

Now move the white points whichever way you need to so that the displayed slope of the ramp is negative. What effect does this have on the ramp?

Reflection:

Write a few sentences about what you've noticed about slope. Here are some guiding questions (but you don't have to use these to write your reflections):[br][list][*]How is steepness measured?[/*][*]What caused the ramp to be more steep, or less steep?[/*][*]What does negative slope mean?[/*][*]If two ramps have the exact same steepness, what must be true about their slopes?[/*][/list]