Part 1: The Americans with Disabilities Act

1.)  The Americans with Disabilities Act states that all public buildings should be handicap accessible. Sometimes this requires that an access ramp be built. The law states that these ramps should have no more than one-foot change in vertical distance for each twelve feet of horizontal distance. On your paper, draw a diagram that illustrates the previous statement. Describe in your own words what the statement means.

2.)  Refer to Handout 1. How would you determine the horizontal and vertical distance of each ramp? Determine which of these ramps meet the requirements of the Americans with Disabilities Act. Explain your process for determining the legality of each ramp.

3.)  Refer to Handout 2. Find the coordinates for each vertex of Triangle 1. Which coordinate values (i.e., x or y) are the same for point A and point C? Which coordinate values are different? What is the distance from A to C? Relate the vertical distance of the triangle to the differences in the coordinates. How could the vertical distance between two general points, say (x₁,y₁) and (x₂,y₂), be found? Which coordinate values are the same for point B and point C? Which coordinate values are different? What is the distance from B to C? Relate the horizontal distance of the triangle to the differences in the coordinates. How could the horizontal distance between two general points, say (x₁, y₁) and (x₂, y₂), be found?

4.)  In your own words define slope of a line. Explain the meaning of slope of a line. What is the slope of segment AB in Triangle 1? What is a general formula for finding the slope between two points (x₁, y₁) and (x₂, y₂)? Connect the general formula for slope to your method of determining the legality of ramp1 in Handout 1.

5.)  Find the lengths of the vertical and horizontal sides for Triangle 2. Are these the same as those in Triangle 1? What is the slope of segment AB for Triangle 2? How are the slopes of segment AB different for the two triangles? Does the direction the triangle is pointing have an effect on the slope?

Part 2: Investigating slope interactively

6.)  Using your web browser go to the Slope Calculation Activity located at:

http://www.exploremath.com/activities/Activity_page.cfm?ActivityID=45

When the activity loads, it will look like the following.

7.)  Without using the interactive activity, determine the vertical distance (Dy) and the horizontal distance (Dx) between the two points. Select the show Dx and Dy box to check your answer.

8.)  Select the show line through points box. Compute the value of the slope, m, of the line through the two points. Select the compute slope box to check your conjecture.

9.)  Notice that the right hand point (purple) is above the left hand point (orange). Grab the y₁ slider. Manipulate the slider to determine the relative positions of points whose connecting line has a negative slope (i.e., The slope of the line through the points is negative when the right hand point is ____ the left hand point.). When is the slope positive? Relate your observations to your answers in tasks 4 and 5.

10.)  Manipulate the sliders. Determine when the value of m is near zero and when m is large. How does the value of m affect the steepness of the line connecting the two points?

11.)  Place the two points on the grid such that the slope of the line connecting them is m = - ½. Determine the coordinates of two more points that lie on this line. Justify your answer.

12.)  Place the two points in a horizontal relationship. What is the slope of the line through the two points?  Manipulate the x sliders. How does the slope change as the x sliders are manipulated? Explain this phenomenon in terms of the slope formula.

13.)  Place the two points in a vertical relationship. What is the slope of the line through the two points?  Manipulate the y sliders. How does the slope change as the y sliders are manipulated? Explain this phenomenon in terms of the slope formula.

Part 3: Applications of slope

14.)  When carpenters construct rooftops, they take into consideration the pitch of the roof. The term pitch is synonymous with slope. If a roof is to have a 4/12 pitch, it should rise 4 feet for every 12 feet of horizontal run. Look at the diagram below. Determine the height of the attic at its highest point. (Diagram is not drawn to scale.)

			One foot overhangs on each side

15.)  The grade (gradient) of terrain is defined as the amount of vertical change for every 100 units of horizontal change. Often signs are posted at the top of mountains to warn truck drivers of steep slopes. Interpret the meaning of the sign below.

 

16.)  Referring to the road sign in task 15, determine the amount of vertical change over the two-mile stretch of road.