Cognitive Tool Lesson

Using Excel to Aid in the

'Optimization of Zero'

The idea is that students choose four numbers at random. They then find the absolute value of the differences of the successive numbers and repeat this process until all values are zero. By using the Excel program, students will be able to focus on the exploration of this problem as opposed to the arithmetic involved.

Directions for Excel:

Problem: Place four numbers in the first row as follows:

A B C D

For each successive row, replace the entries by the absolute value of the difference of the entry just above and the entry just to the right in the previous row. In the fourth position, use the absolute value of the differnce of the fourth and the first. (i.e. cycle)

=abs(a1-b1)

=abs(b1-c1)

=abs(c1-d1)

=abs(d1-a1)

Then drag the formula cells to fill all cells horizontally and vertically.

(I will be demonstrating and providing guidance throughout the above task.)

At this point, the students will explore the following questions. Excel is performing the arithmetic allowing students to focus on the nature of the problem: optimization of zero.

Will the process lead to a zero in all four entries for some row?

What is the largest number of rows before a zero row is generated?

What if you only use three random numbers? five random numbers?

Is there a pattern depending on an even or odd number of random numbers?

Students will test these questions by plugging in several different combinations of random numbers and exploring the results. Students will only turn in their observations and conclusions. ( I will not need to see their Excel Worksheets. It is only their tool for completing the exploration.)

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