Wheelock
Euclid
May 2001
Explorations
For the explorations portion of the workshop, you can choose your own starting point. If you are new to Excel and/or you are nervous about the mathematics, I suggest you start with the tutorial. Otherwise, feel free to jump right to the suggested activities (or to suggest your own and do it…).
Tutorial:
1) First,
do
2) Start up Excel. You will see a screen with lots of rectangular boxes. This is a worksheet. The columns are labeled with letters and the rows with numbers. Each rectangular box is called a cell. The cells are denoted A1, C17, etc. (C17 is a cell in column C, row 17 etc.). You can enter numbers, text, formulas, etc. into each cell.
3) In each row, we will represent one subtraction step in the algorithm. We don’t actually have to type very much to make the table; most of it is automatic. We will place the larger number in column A, the smaller number in column B, and the difference in column C. Start labeling each column by typing something like “larger number” in cell A1. Don’t type the quotes, and hit return when you are done. You probably will not have enough space, so you need to make column A wider. To do this, move the mouse to the gray area at the top of the worksheet, on the line between letters A and B. When you are in the right place, the cursor will change from a white cross to a black vertical line with horizontal arrows through it. Now you can drag to change the widths manually or double click to change them automatically. Label columns B and C by typing in cells B1 and C1.
4) We will
start by having Excel do
5) Now, we
want cell C2 to be 219-123. However,
instead of typing the actual number, we will use a formula that references
individual cells. In a short while,
you’ll see the power of using such a formula.
In cell C2, start by typing an equal sign (=). Formulas always start with equal signs, and
forgetting the equal sign is a common (easily corrected) mistake. Now either type A2-B2 or else click in A2
then type a minus sign and click in B2 (i.e. instead of typing cell references,
you can just click in the cell if you make a mistake doing this, type Esc and
start over). Type return. You should see 96 in cell C2. Excel does the subtraction automatically.
6) Now we’ll start on the second row of our table. This is the last row in which we’ll have to type anything! Now, we need new numbers to go into cells A3 and B3 so we can compute the next row. We know that 96 is smaller than 123, however, it is possible for the number in C2 to be larger than the number in B2 (you might wish to check your handwritten example for rows where the difference is larger than the smaller of the initial numbers). In cell A3, we wish to put the larger of B2 and C2. We don’t know which one it will be, but we don’t need to: Excel has a command called max that finds the maximum value of two numbers. In cell A3, type “=max(B2,C2).” Don’t type the quotes or the period, and you can click in the cells instead of typing them. Hit return. Now use the analogous “min” command in cell B3. You should see the appropriate numbers in each cell.
7) Now click on cell C2. Notice that in the bottom right corner of the cell there’s a little black box. This is called a fill handle. Move the cursor to this little black box. The cursor will change to a black cross. Hold down the left mouse button and drag the cursor down one cell. Let go. Cell C3 should show 27, which is A3-B3. Click on cell C3. If you look up above the worksheet, you will see the formula bar, which displays the actual formula used to calculate each cell. Note that the formula has changed automatically from A2-B2 to A3-B3 (also click in C2 to see the difference).
8) Now comes the fun part! Click in cell A3 hold down the mouse button and drag to cell C3. You should see an outline around all three cells, with the right two shaded. Note the fill handle at the bottom right corner. Drag this fill handle down as many rows as you wish (at least 10). You should see an entire table, pretty much like the one you did by hand. Click in various cells, look at the formula bar, and see how Excel has changed the formulas automatically.
9) Now
comes the other fun part. Change the
numbers in A2 and B2. The whole table
should change automatically. You might
have to drag the formulas down more cells, but otherwise, you now have a
program for executing
10) If there’s time, go on to one of the suggested activities below.
Suggested Activities (Start wherever you wish):
1. Use multiplication and division to reformulate
2. Design a spreadsheet that implements
3. Explore other things to do with Excel.
4. Bezout’s theorem is important in Number Theory. It states that if d is the greatest common
divisor of a and b, then there are integers x and y such that ax+by=d. For example, the greatest common divisor of
219 and 123 is 3 and . It is not obvious how to find x and y, but
(as you might guess),
? Can you find a way to use a spreadsheet as a
tool in solving equations such as this one?
(By the way, equations whose solutions must be integers are called Diophantine
Equations).
Euclid
May 2001
Background
The Distributive Property:
The
distributive property is key to why
3 is a factor of 18 (or we say 3 divides 18) because 18 ÷3 has no remainder.
Visually, we can represent 18 as groups of 3:
As an equation, we have .
Now, 3 is also a factor of 12, because 12 ÷3 has no remainder. Visually we have,
and as an equation we have, .
Now, the
distributive property tells us that 3 is a factor of both 18+12 and of
18-12. Visually, we see that 18+12 can be
written as 6+4=10 groups of 3:
As an equation, we have .
Similarly, 18-12 can be written as 6-4=2 groups of 3:
As an equation, we have .
For understanding
An Illustration of
We will find the greatest common factor of 72 and 30. The greatest common factor is exactly what it sounds like: it is the largest natural number (i.e. whole number) that is a factor of both 30 and 72.
By the distributive property, we know that any factor of both 72 and 30 is also a factor of 72-30=42.
Similarly, any factor of both 30 and 42 is also a factor of 42-30=12 (and thus, any factor of both 72 and 30 is also a factor of 12).
Continuing in this manner, any factor of both 30 and 12 is also a factor of
30-12=18.
Any factor of both 18 and 12 is also a factor of 18-12=6.
Any factor of both 12 and 6 is also a factor of 12-6=6.
Now we are left with 6 and 6. Following the entire chain of reasoning above, any factor of both 72 and 30 must also be a factor of 6.
Now, the
greatest factor of 6 is 6 itself. So, if
we can show that 6 must be a factor of 30 and 72, then 6 must be the greatest
common factor of 30 and 72. Of course, 6
is a factor of both 30 and 72, but to use this example to illustrate how to
prove that
Copyright 2005, Debra K. Borkovitz. You may copy or edit this material for non-profit, educational use only.
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