M7N1c - Integers

M7N1. Students will understand the meaning of positive and negative rational numbers and use them in computation.
c. Add, subtract, multiply, and divide positive and negative rational numbers.
I usually don't venture into the 6-8 standards. But, since we discussed the compensation strategies recently, I thought I would discuss how two of those strategies can be used to derive the methods of calculations with integers.

Recall that the equal addition principle of subtraction states that if we add (or subtract) the same number to both the minuend and the subtrahend, the difference stays the same. Thus, 93 - 18 = (93 + 2) - (18 + 2) = 95 - 20. Another property of subtraction students encounter early on is that subtracting 0 will not change the number, that is A - 0 = A. By combining these two properties of subtracti! on, we can think about a problem like 8 - (-3) this way:
"We know subtracting 0 does not change the number. So, what can I do to change the subtrahend (-3) to 0? Add 3. But, the equal addition principle of subtraction says I have to add the same number to the minuend to keep the difference the same. So,
8 - (-3) = (8 + 3) - (-3 + 3) = (8 + 3) - 0 = 8 + 3."
Thus, you can see that subtracting a negative number is the same as adding the opposite.

We noted that there is a parallel between the compensation strategies for subtraction and division. We can actually use the equal multiplication principle of division to think about division of fraction problems, by combining it with another parallel property, dividing by 1 does not change the number. So, if you are given 3/5 ÷ 2/3, we can think like this:
"We know dividing by 1 does not change the number. So, how can we change (2/3), the divisor, into 1? Multiply by its reci! procal, of course. But the equal multiplication principle of ! division says I will have to multiply the dividend by the same number, too. So,3/5 ÷ 2/3 = (3/5 x 3/2) ÷ (2/3 x 3/2) = (3/5 x 3/2) ÷ 1 = 3/5 x 3/2."
Thus, we see that the division of fractions is the same as the multiplication by the reciprocal of the divisor.

Of course, strictly speaking, there is a minor glitch in both of these arguments. We established the four compensation strategies with whole numbers. But, we don't know if they still hold if we expand the range of numbers to integers/rational numbers. So, there is a circularity in these arguments. So, I'm not advocating these strategies to establish the computation algorithms, specially since there are other ways where students can meaningfully develop algorithms. However, I think these mathematical relationships are still interesting.

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