Measuring Up: A Middle School Experience

Measuring Slopes: Staircase Data Sheet

Please measure the vertical distance and the horizontal distance of each step on the staircase. Please record your information on this sheet.

Staircase
Step Horizontal Distance Vertical Distance
     
     
     
     
     
     
     

The Algebra of Stairs and Ramps

1a. Suppose your school wants to build a staircase to an exit on the upper level of a building. From experience, the carpenter decides that each stair should be 24 cm in the horizontal direction so that it will be big enough to handle larger feet. The distance from the ground to the base of the door is 180 cm. The carpenter also estimates that 10 stairs would be sufficient based on the height of the staircase. If all the stairs will be uniform, what will be the dimensions of each stair? How far from the building must the carpenter build the staircase?

1b. Suppose we are using the same size stair that we were using in problem 1a. If we were to climb one stair, we could think of moving a horizontal distance and moving a vertical distance. If we were to climb two stairs, we could think of traveling a horizontal distance and a vertical distance from the base of the staircase. Fill in the chart below.

Number of Stairs Climbed Total Horizontal Distance Traveled Total Vertical Distance Traveled
1    
2    
3    
4    
5    

1c. Graph the horizontal distance traveled versus the vertical distance traveled on a sheet of squared paper.

2a. Suppose the carpenter decides that each stair should be 27 cm in the horizontal direction and 18 cm in the vertical direction. What will happen if the carpenter begins the staircase in the same position as in problem 1a?

2b. Suppose we are using the same size stair that we were using in problem 2a. If we were to climb one stair, we could think of moving a horizontal distance and moving a vertical distance. If we were to climb two stairs, we could think of traveling a horizontal distance and a vertical distance from the base of the staircase. Fill in the chart below.

Number of Stairs Climbed Total Horizontal Distance Traveled Total Vertical Distance Traveled
1    
2    
3    
4    
5    

2c. Graph the horizontal distance traveled versus the vertical distance traveled on the same squared paper as in 1c.

3a. Suppose the carpenter decides to begin the staircase in the same position from the building as in problem 1a but wants to experiment with the number of stairs. Suppose he tries 12 stairs, what will the dimensions of each stair be? Suppose he tries 24 stairs, what will the dimensions of each stair be?

3b. On a new sheet of squared paper, plot the total horizontal distance traveled versus the total vertical distance traveled for each of the staircases in problem 3a. What do you notice? What do you think would happen if there were 48 stairs on the staircase?

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