Fitting a formula to data is an important skill, especially for making predictions. If you have a formula, you can use it to predict what would happen at the 99th stage or the 1000th stage, by just plugging that number into the formula. A word of caution: predictions are based upon the function maintaining the same rate of growth all the way to that stage. Let's examine each color individually and find the formulas for the Rectangle Pattern Challenge.
Blue The terms are 6, 10, and 14. Notice that the difference between each term is a constant - 4. That means the function is growing at a constant rate; its graph is a straight line; and its formula is in the form of y = mn + b where m is the slope and b is the y intercept (where the graph crosses the y axis).
How do we find m and b? Slope is the rate at which the function is changing. Well, that's 4, so that's m. The y intercept occurs where n = 0, so go back to n = 0 by subtracting 4 from the n = 1 stage, and we get 2. So the formula for blue is y = 4n + 2. Try all 3 stages and confirm that our formula works for all stages.
Red The terms are 2, 8, and 18. The difference between the terms are 6 and 10. It's not constant, so it's not linear. But what is it?
Let's try a difference of the differences. I need to know how many subtractions it takes to get a constant difference, but I don't really have enough information to determine that. I need another stage. Go back and look at Stage 3 and count the number of reds that we'd have on Stage 4 (add a row of reds on each side). Do you agree that it would be 32? Now our terms are 2, 8, 18, and 32. Our first differences are 6, 10, 14. And our second differences are 4 and 4. Aha! We now have our constant difference after 2 subtractions. That means we have a quadratic equation. (Yes, you are right - the rule is that the number of subtractions that it takes to get a constant difference is the largest power of the polynomial formula).
If the formula's quadratic, it's in the form of y = an2 + bn + c, and we must find the values of a, b, and c. Look at the red terms again in the table below. Notice that I added the 0 row, since there are 0 reds in Stage 0.
Stage No. (n) 0 1 2 3 4 No. Red Squares (y) 0 2 8 18 32
Since y = 0, when n = 0, substitute those values into y = an2 + bn + c and get
c = 0
When n = 1, y = 2, so let's substitute those values (and remember, c = 0) and get
2 = a + b
Substituting the next pair of n and y gives
8 = 4a +2b
This gives a system of two equations with two variables that we can solve.
2 = a + b
8 = 4a +2b
So, a = 2, b = 0, and c = 0, and the formula is y = 2n2.
Use this method of Constant Differences to find the formula for the green squares.
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