The problem involves a stack of bricks (See Figure 1). The problem is to figure out how many total bricks there would be if the pyramid was 50 bricks high with 50 bricks on the bottom row. We can see by counting that there are 10 total bricks if the stack has four bricks on the bottom.

One obvious method of attacking this problem is to create a table (See Figure 2).

With a table it may be possible establish a pattern. Upon examination we can see that the problem involves adding consecutive integers. The problem can be reworded as The Total # of bricks =1+2+3+4+5+6+...+50. So we have a problem exactly the same as adding consecutive integers from 1 to 50.

An ice cream store has 31 flavors of ice cream. How many different two scoop cones are possible? If we do not allow two scoops of the same flavor, we have a standard combination problem of 31 things taken 2 at a time—C. If we do allow two scoops of the same flavor, as is more realistic, than it will soon become clear that the problem is the same as the sum of the first 31 consecutive integers (See Figure 3).

The old handshake problem—Each person in a room shakes hands with every other person. If there are "n" people in the room, how many handshakes take place?

I am of course aware of many varied strategies to solve this problem from regrouping to solving a system of matrices. My objective here is to provide a method that allows more students success than with previously taught methods. A little basic knowledge of a computer spreadsheet will provide us with a quick answer to how many bricks are needed and even generalize the problem by providing us with a function (formula). Figures 4 and 5 show how students with a little knowledge of spreadsheets can quickly arrive at a correct solution.

We are now ready to extend the problem and find a general solution. We use the first and second differences to give us a clue about the formula (See Figure 6). The constant second differences suggest a second degree polynomial. We now take advantage of the regression capabilities of the spreadsheet. A chart of the data series must first be made. In Micrsoft Excel (Microsoft, 1998) we make a scatterplot of the data, then click directly on a data point on the chart. All the data points will be selected. Next go the Chart Menu and choose Add Trendline… .

Choose a Polynomial and degree of 2. Then choose the Options Tab and check the box that says Display equation on chart.

The regression equation of y=.5x²+.5x+2E-12 is displayed on the screen (See Figure 7). The term of "2E-12" should be recognized by students as being very close to zero. Students should also be required to explain what the "x" and "y" represent in term of the original problem. In fact, I like to have students reword the equation, replacing the variables with words such as "bricks, row, and total number of bricks." Making students explain what the spreadsheet has generated does not allow them to just hand in some "equation that the computer generated." Another way to check for student understanding is to ask them to find an answer to a similar problem without using technology or use numbers so large that they are beyond the limitations of the spreadsheet.

The spreadsheet puts a powerful problem-solving tool in the hands of students. And on a single computer screen we have a numerical, symbolic and graphical representation. If you require students to label their problem correctly and write a summary, you will also have an English representation of the problem and solution.