Tuesday, April 13, 2010

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 subtraction, 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 reciprocal, 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.


Evelyn said...

You said in "M7N1c - Integers" that it has only been shown to hold for whole numbers. Could you expand as to whether it does hold for integers/rational numbers?

I am a retired physics teacher who works with the Chicago Lesson Study Group and I find that a big hindrance to getting students to think well mathematically are the text book that teachers must use. Using the Japanese text books for lesson study lessons helps, but then students are returned to their school texts. Could you comment?

Evelyn Mazzucco

Tad said...

Hi, Evelyn. Thank you for your questions.

These properties do hold with integers and rational numbers since they are really properties of operations, not numbers. But from students' perspective, they haven't seen them with a new type of numbers, integers or rational numbers. So, we need to be careful using something students haven't really learned/verified to develop something new. That's a part of the reason that Japanese textbooks will ask students, for example, to see if the properties of operations they learned with whole numbers work with decimal numbers after they learn to multiply decimal numbers in Grade 5.

Your second question is a much harder one, and I don't claim to have an answer.

Personally, I think a major reason for teaching mathematics in schools is to help students become better thinkers, not just to accumulate facts, rules, procedures, etc. So, helping them to think mathematically ought to be a major goal. Although some textbooks are much more useful for that purpose than others, I think we can encourage students to think mathematically even if you are using different textbooks. One thing it might be helpful for teachers to think about is to identify kinds of questions that might be helpful for mathematical thinking. For example, "How is this problem similar to what we have done before?" "What's different about this problem from what we have already learned?" "What is changing in this situation?" "What's staying the same?" etc.

Then, depending on students' mathematical development, we may have to ask those questions for students initially, but the eventual goal for us is to help students to ask those questions on their own.

I know this isn't a complete answer, but maybe others can jump in.

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Elaboration of Georgia Performance Standards by Tad Watanabe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.