Getting Started Using the Technology at Your Disposal:
One Problem, Three Different Approaches

Mary Sue Wyss

Abstract

Pascal's Pyramid is constructed out of toothpicks and gum drops. Observations of either the number of gum drops, toothpicks, tetrahedrons, or octahedrons in each level can produce data reflective of "change". Both second and third degree polynomials are generated. The three different approaches to analyzing the data include finite differences, use of the TI-81 and/or TI-82, and Excel.

Since each approach uses the same data set, the focus is directed on the best analytic method or "tool" of investigation for the data given. In addition, through this introductory exercise, students will become acquainted with their graphing calculator (matrix key) and spreadsheet software that utilizes graphs. Exploration and analysis is geared toward junior high Algebra students but could prove to be an enjoyable introductory exercise for upper level students and technically timid teachers.

Activity A: Finite Differences

For "nice" data sets, this is usually the most direct method of deducing the function. Finite differences reinforce pattern recognition and prepare students for sequences and series.

Step 1: Make a table of x-values and y-values to represent the levels of Pascal's Pyramid and the number of gum drops per level.

Table 1.A

Step 2: Make another table that shows the differences and ratios of y to x. Since the second difference taken is constant, we know we are dealing with a quadratic function. What is it?

Table 2.A

Step 3: By taking the differences, we were offered some valuable information about the characteristic nature of our function. Naturally, our function is more involved than just x². Therefore, we may want to look at an alternative table such as ratios or multiples of y to arrive at the precise function. Let's make another table.

Table 3.A

As a result of table 3.A, we see that 2y can be generalized in terms of n. We are very close to our answer. If we divide this generalization;

2y = n ( n + 1 )

then y = n (n + 1)

2

and we are done.

Activity B: Using the TI-81/82 Graphing Calculator

While finite differences may be the most direct route to finding our function, matrices will also work. Since the function is so "nice", this may be a good way to introduce students to the use of the matrix function key on the graphing calculator.

Step 1: We need to look at our data from table 1.A of Activity A. Given the matrix multiplication of [A] x = [B] where the scalar x = [A] ^-1[B], we find that [A]^-1[B] produces the coefficients of our quadratic: y = ax²+bx+c. (We'll be using the first three points in our data set.)

To enter this into our calculator, begin by pressing . "Right arrow" over to EDIT and press . Our matrix is a 3 x 3. Press "3" and "3". Now we are ready to input our matrix given the following equations:

Step 2: We want to exit from our new [A] matrix and create a 3 x 1 matrix [B]. To exit matrix [A], press . Again, arrow over to edit and "arrow down" to 2:[B] and . You want to make a 3 x 1 matrix. Put the "y-values" into [B].

Step 3: Press , to get back to the home screen. Press if home screen is not clear. Recall that from the scalar x=[A] ^-1[B] we get our coefficients for y = ax²+ bx + c. To let the calculator multiply the matrices, press then "down arrow" to 1:[A]:. Press . Since we want the inverse, press ^. The display to screen should be [A]^-1. Now we want to go back to and "down arrow" to 2:[B]: and press . Your home screen should display the matrix [A]^-1[B]

Step 4: Press to display the scalar x.

This tells us that a = 1/2, b = 1/2, and c=0 for all practical purposes.

Thus, for y = ax²+ bx + c, the function we are looking for is y = x^{2 +} x + 0 or, as in Activity A, y = n (n + 1)

2

Activity C: Using "Excel" Spreadsheets and Graphs

Excel offers us a quick way to see the behavior of the function. Through this method, students will learn how to incorporate a little guesswork by varying the parameters. The function we are using in this exercise is nice and known; however, most other sets we will be using later are not. With this exercise, students should be able to concentrate on how to set up a template and acquire a feel for parameterizing a function.

Step 1: We'll need to set up the same tables of data as in Activity A. Each column represents the table headings as follows.

Table 1.C

Column D's heading ROC of f(x) refers to the rate of change of gum drops.

Step 2: Once you have the data in, you can select the graph icon in the top right corner. Continue by selecting the scatter plot and then press next. Select an area to place your graph and open a box for it. You should see the following display:

Graph 1.C

You can enhance the appearance of your graph by clicking twice on the graph and then accessing the axis title options. Whatever the case, it is apparent that the function is quadratic in nature for the data given. Students may put formulas into the cells B3 and C3 to create more data points and then re-graph to verify the quadratic nature of the function. The formulas appear below for cells B3 and C3.

Table 2.C

Step 3: We will be using the same intuition as in Activity A to find the function that best fits the data. We know the function is of the form y = ax²+ bx + c. Let's begin by making a column heading entitled ax² in column G. We can input the respective values for ax² in by way of a formula later.

Step 4: To vary the parameter "a", we'll need to set up a cell with the "a" value in it. We'll set that up in the "A" column as shown below.

Table 3.C

Step 5: We are now ready to input a formula into our column G that will generate the rest of the column without making the calculations ourselves. This column will be dependent on the number we have placed in cell A2. To start with we will take a=1. To put the equations into G4, use the cell display below.

Table 4.C

After highlighting down the column (however far), and entering, we have a column of new values. We would like to graph these new values against x to see how well they compare to our function. (The $ symbol between A3 and after indicates that we want that value used as a constant.)

Step 6: To get a clear visual idea of how close our guess is to the real function, we'll graph by highlighting the x column and then press and keys which will allow you to go over to the G column and highlight that.

Step 7: After both columns are highlighted, select the graph icon again from the top right corner of the screen. Select another scattergram that has two graphs showing and open a window on your spreadsheet to display the graph.

Graph 2.C

Step 8: As we can see, the graphs don't match. We can change the values of our "a" parameter to vary the graph. Try some different values of "a" to see what seems best.

Step 9: As hard as we try, the estimated data does not fit well to the actual data. We may need to consider another parameter "b" from the equation y = ax²+ bx + c. This is easy to do by setting up another cell like we did for our "a" parameter. This can be done in cell A4 and A5 as follows.

Table 5.C

Step 10: We want to change the equation in the G1 column to read ax²+ bx. After doing that, we want to go to G4 to add the "bx" to the equation. The equation should look as follows.

Table 6.C

Toy with both parameters to investigate the affect on the graph. Hold one constant and vary the other. Try some extreme points to emphasize the change before arriving at the correct solution of a = .5, and b = .5.

Enrichment Activities

Repeat the activities changing the data set by

a) counting the number of toothpicks per level,

b) finding sum of the gum drops or toothpicks per level,

c) counting the number of tetrahedrons and octahedrons per level.

Additional Resources

1. Seymour, Dale, Visual Patterns in Pascal's Triangle, Dale Seymour Publications, Box 10888, Palo Alto, CA 94303.

2. Laycock, Mary, Algebra in the Concrete, Activity Resources Co., Inc., Box 4875, Hayward, CA 94540.

3. Shimizu-Yest, Jeanne K. "Pascal's Pyramid," Master's thesis, University of Santa Clara, 1989.

Woodrow Wilson Leadership Program in Mathematics lpt@www.woodrow.org
The Woodrow Wilson National Fellowship Foundation webmaster@woodrow.org
CN 5281, Princeton NJ 08543-5281 Tel:(609)452-7007 Fax:(609)452-0066