Building Staircases

Your ABLE program is in the middle of building a new center. Unfortunately, funding has been cut and you are now responsible for helping finish the construction. As a math instructor, you have been charged with building the back staircase. Currently, you don’t have any information on how many steps high it must be. Explore the following questions to help guide your newly assigned responsibility.

  1. How many squares are needed if only one step is required?

One square is needed to make a single step.

  1. How many squares does it take to build only the second step?

It only requires two squares to build the second step. Think of the steps as a column and you should see the two squares stacked on top of each other.

  1. How many squares are required to formonly the third step?

Just as in the prior question, look at the third column and you can see that there are three squares in the third step.

  1. How many total squares are necessary to construct this three-step staircase?

To determine the total number of squares needed to make the staircase, simply add the number of squares together. Step 1 + Step 2 + Step 3 = 1 + 2+ 3 =6. There are six squares needed to construct a three-step staircase.

  1. How many total squares are needed to build a five-step staircase?

Using either square tiles, pictures, visualization, or another technique, you should be able to deduce that there are 15 squares needed to build a five-step staircase.

  1. How many squares does it take to make only the 11th step?

If you are interested in the 11thstep then you are only concerned with the number of tiles in that 11thcolumn, which would be 11.

  1. How many squares are required to build only the 98th step?

Examine the pattern.The number of squares required to build a specific step is the same amount as the value of the step. Thus, the 98th step would require 98 squares.

  1. Determine a rule to identify the number of squares need to make only the th step.

Examine the pattern.The number of squares required to build a specific step is the same amount as the step number. Therefore, the number of squares needed to build the th step is . In function notation,.

  1. Establish a rule to find the total number of squares required to make a staircase with number of steps. Explain how you determined this rule.

Given the staircase size , the rule to find the total number of squares is . You may have been able to arrive at this by guess and check or some other heuristic. In case you didn’t, consider the following. If we were to make a copy of a staircase — let’s use a three-step staircase — and fit it in with the original staircase, a rectangle is constructed.

The total squares required to build the rectangle is 3 * 4 = 12,but the original staircase is half of that—so 12/2 = 6. (Note that you can incorporate the concept of area here.) You already know that the total number of squares required is six, but this helps you deduce a pattern. In this example (when ), we see that the pattern is or . This rule works for the three-step staircase, but you must verify that it works for the other staircases. With a two-step staircase, you have a total of three squares. Now let’s verify the rule works. Since , we have . It works. You can check the other possibilities, too.In function notation, .

  1. Here is a different set of staircases. How many total cubes are needed to make a staircase with four steps?

Using cubes, pictures, visualization, or any other technique, you would find that you need 40 cubes to build a staircase with four steps.

  1. Following this pattern, how many cubes make up onlythe top step of a staircase with five steps?

If you are interested in the top step, then you only focus on the number of cubes in that single step, which would be five.

  1. How many cubes are required to build onlythe top step of a staircase with steps?

Examine the pattern.The number of cubes required to build the top step is the same amount as the value of the number of steps in the staircase. This is similar to examining the number of squares needed to build a step in the previous example. Thus, the top step of a staircase with steps would require cubes.

  1. How many cubes are used to build the base of a staircase with steps?

If you examine the staircase with two steps, the base is two-cubes wide and two-cubes deep. For three steps, the base is three-cubes wide and three-cubes deep. If you continue the pattern you see that the base of a staircase with steps has a width of and a depth of . That is the base is by or .

  1. Determine a function that identifies the number of cubes required to build a staircase with steps. Justify your function.

Given the staircase size , the rule to find the total number of cubes is . You may have been able to arrive at this by guess and check or some other heuristic. In case you didn’t, use the method that we previously used in 9. Recall, if you were to make a copy of a staircase — let’s use a three-step staircase — and fit it with the original staircase, you get a box. You can see that the original staircase is half of the combined box. The total cubes required to build the box is 3 * 3 * 4 = 36,but the original staircase is half of that, so 36 / 2 = 18. You already know that the total number of cubes required is 18, but this helps deduce a pattern. In this example when , see that the pattern is or . This rule works for the three-step staircase, but you must verify that it works for the other staircases. With a two-step staircase, you have a total of six cubes. Now let’s verify the rule works. Since , you have . It works, so you can check the other possibilities, too. In function notation, .

  1. Compare and contrast the difference between the different staircases and components.

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You can use the following worksheet to compare the different situations. It may be intuitive that the second three-dimensional staircase increases more rapidly than the two-dimensional staircase or a column of steps,but how do you provide evidence? One option is by looking at the tables. At the initial level, all three situations result in a single square or cube. The differences begin to be more apparent at the second level. Here you see that the second step only requires two squares, the total number of squares for a two-step staircase is three, and the total number of cubes for a two-step staircase is six. Clearly the three-dimensional staircase is increasing more quickly than the other two. If you continue to examine the trends,you see that the three-dimensional staircase is increasing quite rapidly while the number of steps in a single column is increasing at a constant rate. As an instructor, you can easily incorporate discussions about slope with this activity. You can also compare the graphs:

The blue points correspond to the “single step,” the red corresponds to the two-dimensional staircase, and the green corresponds to the three-dimensional staircase. It is here that students can see just how “steep” the green points are in comparison to the blue.

Lastly, you can compare the functions. The first function is a linear function that implies that values change at a constant rate. In this case, the rate is 1, so for every increase in step, there is an equivalent rate increase in the number of squares required to build that specific step. The second function is a quadratic function while the third is a cubic function.

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A Single Step
/ 2-D Staircase
/ 3-D Staircase

Table
Size of Single Step / Number of Squares in that Step
1 / 1
2 / 2
3 / 3
11 / 11
98 / 98
/ Table
Number of Stairs / Total Number of Squares
1 / 1
2 / 3
3 / 6
4 / 10
5 / 15
/ Table
Number of Stairs / Total Number of Cubes
1 / 1
2 / 6
3 / 18
4 / 40
Graph
/ Graph
/ Graph

Function
/ Function
/ Function

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