I did this exercise a couple of months ago, but I thought I would discuss it here in light of another math moment. Above is a standard 10 X 10 multiplication table. Most of us were required to memorize this at one point or another in grade school. The line on the diagonal shows the squares of each number. You probably noticed in grade school that the table is identical when mirrored across that line, i.e. 3*4 is the same as 4*3.
But now look at the group of boxes. If we look at the number in the middle we can refer to the numbers around it by location: so the red number is North, Green - East, Purple - South, Blue - West. What I want to point out is the relationship between these for numbers. So when I say NW that means 24 and 21.
Now that I've set this up, take a look at the NW and SE numbers. The difference between them is the same, i.e. 3. Now look at NE and SW. Again, the difference is the same, i.e. 11. It doesn't matter what number you pick, the differences between NE:SW and NW:SE will be the same. Perhaps I missed the boat on this back in grade school, but I think that is kind of cool. Also consider this, if you shift the center number down and over (say to 40), the relationship stays the same for the NW:SE numbers, and if you moved it down and over the other way (to 24) the NE:SW relationship is the same.
Now what happens if you just shift the number down (to 32). The difference for NE:SW becomes increases to 12 and NW:SE increases to 4. But if you shift it right (to 35) the NW:SE difference drops to 2 while the NE:SW difference stays at 12. How does this work, you ask?
LIKE THIS!
This was the original excel sheet I set up when I looked into this. This is the way you read it.
1 - Locate a blue line (e.g. in the upper left corner), look at the numbers that are on either side of that line (2 and 6). The difference between those numbers is equal to the axis number the line passes through (3) plus one.
2 - For the red lines, it is the axis number minus one. The easiest example is the longest that runs through the squares. all of the numbers mirrored across it are the same.
So, go back to our original example - 28. The blue line (the NE:SW difference) crosses the axis at 10, giving us a difference between numbers of 11. the red (NW:SE) line crosses at 4 resulting in a difference of 3.
Any movement along any single line will result in the same relationship between the numbers. Now, what happens if you don't know where the line will cross the axis? For instances, we can see where the blue line would cross on the number 336 and we will pretend we don't see the red line either. The key is to find the factors with the least difference. In this case it's 16 and 21. To figure out what the blue (NE:SW) difference will be take the absolute value of the difference of the two factors (i.e. 16-21 = 5) for the red, you take the sum of the factors (i.w. 16+21=37).
Now this is probably way more than you wanted to know about the multiplication table (or what I do in my free time), but there is an even simpler way to do this. It's called finding the slope! Finding the slope of a line has been done for centuries. And I just discovered it!
Not really. the multiplication table is basically one quadrant of the coordinate plane. When viewed like this (with the factors on the top and left) we are looking at the lower right quadrant (+X, -Y). If you remember from those grafting equations classes the slope is rise (the difference in Y) over run (the difference of X). Technically all of these lines have a slope of 1 because they are straight and at a 45 degree angle, but this is how it works.
As we treat this as a coordinate plane all of the factors at the top (the X axis) will be positive while the numbers at the left (the Y axis) will be negative. So we take our original box:
The differences (or slopes) between our coordinates are:
S and E = -4 (rise) plus +7 (run) = +3
S and W = -4 (rise) plus -7 (run) = -11
N and W = 4 (rise) plus -7 (run) = -3
N and E = 4 (rise) plus +7 (run) = +11
Since we only care about the distance (not the direction) we can eventually just use the absolute values which give us 3 and 11.
There you have it. I have proven that slope exists within the multiplication table. What does this mean for the world of mathematics? Probably very little. But It was a fun adventure to figure this out. As I work in education, I have been asked by many students, "When will I use math?" My 10th pre-calculus teacher, Mrs. Mead, use to answer many of those questions with, "You won't. This is to get you to think."
As this post has gone on long enough, I will simply close by agreeing with Mead. The math you see in a textbook is not the math you are doing in your head when you look at a clock, try to figure out a sale price, or even the advanced equations of speed and distance when you drive. However, the basic nature of algebra is such that it forces your brain to think on higher levels and puzzle out solutions. That is the nature of math. And that is why it is needed.
Muse on that.
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