A conversation on Division by Fractions!

I have been planning to write an article on ‘Division by Fractions’ for the past few days. My thoughts were clear, the conversations with the students were well stored in my mind, but somehow, this article was still waiting in the ‘to-do list’. Recently, I got a chance to meet Professor Ron Aharoni and attend his workshop about improving teaching skills. I purchased a few of his books as well. During the meet, I realized that we think alike when it comes to Math Education. But, when I read a chapter from his book, ‘Arithmetic For Parents : A book for Grown – Ups about Children’s Mathematics’, I realized that we even write alike! The chapter was – A conversation on Division by Fractions. I didn’t feel the need to re-invent the wheel and decided to type the part of his book  here with due credits to Professor Ron Aharoni. I will not be adding/removing any content from his chapter. So, there you go, here are some excerpts from his book –

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The explanation of the rule of division by a fraction is probably hard for children. Here is a way which is more suited to being taught in class. It is given in a form of a conversation with my daughter, Geffen. When she was in fourth grade,I took her for a walk. During that walk, which lasted less than an hour, I taught her how to divide a number by a fraction.This conversation is reproduced here quite accurately, and is used also as an opportunity for illustrating a few teaching principles. The principles involved in each stage appear parenthesized and italicized.

I: We have learned how to multiply by a fraction. Let us now see how to divide by a fraction. For example, how to calculate 10 ÷ 2 ⁄ 3. How do you think we should start?

G: We should ask a simpler question. (Start  from the simplest question possible. Also, share with the students the principles of sound thinking. The rule of starting from the simple should be explained to the students.)

I: Right. Let us start then with dividing by a simpler fraction. What is the simplest fraction you can think of?

G:  1 ⁄ 2

I: Indeed, 1 ⁄ 2 is a fraction we know and understand well. What is the simplest exercise you can think of in which we divide by 1 ⁄ 2?

G: 1 ÷ 1 ⁄ 2 (Let the students invent the problems.)

I: Can you calculate that?

G: Yes, it is 2, since 1 ÷ 2 = 1 ⁄ 2. If 6 ÷ 2 = 3 then 6 ÷ 3 = 2, likewise if 1 ÷ 2 = 1 ⁄ 2 then 1 ÷ 1 ⁄ 2 = 2.

(Well, this is smart. Not every child would see that. But what follows does not necessitate such insight.)

I: Very nice! But here is another way, which I myself understand better. 6 ÷ 2 = 3 because 2 goes into 6 three times. Do you remember what we call such division?

G: Yes, containment division. ( Use precise words, and distinguish fine points of meaning.)

I: If 6 ÷ 2 means how many times 2 goes into 6, what does 1 ÷ 1 ⁄ 2 mean?

G: How many times does 1 ⁄ 2 go into 1?

I: And how many indeed?

G: 1 ⁄ 2 goes into 1 two times. So 1 ÷ 1 ⁄ 2 is 2. (Inadvertently, we followed here another teaching principle. Try to see the same thing from as many viewpoints as possible. We saw two ways of calculating 1 ÷ 1 ⁄ 2. )

I: Could you tell me now what is 3 ÷ 1 ⁄ 2? (Add one ingredient at a time.)

G: Yes. 1 ⁄ 2 goes two times into 1. Three is 3 ones, 1 ⁄ 2 goes 3 x 2 = six times into 3. So 3 ÷ 1 ⁄ 2 is 6.

I: Right. And 4 ÷ 1 ⁄ 2 ?

G: 8.

I: And 5 ÷ 1 ⁄ 2? (Stabilizing the knowledge, by exercising.)

G: 10.

I: Could you tell me the rule?

G: Yes, dividing a number by 1 ⁄ 2 multiplies it by 2.

Because each 1 in the number contains 2 halves. (After experiencing a rule, formulate it in words.)

Remark: In class,extensive exercising is needed at this point. How many halves of an apple are there in 5 apples ? What is 5 ÷ 1 ⁄ 2? How many times does 1 ⁄ 2 go into 10 ? What is 10 ÷ 1 ⁄ 2 ? 13 chocolate bars were divided between children, and each got half a bar. How many children were there? What is 13 ÷ 1 ⁄ 2?

I: Fine. Let us now divide by a third. What is 1 ÷ 1 ⁄ 3?

G: 1 ⁄ 3 goes into 1 three times, so it is 3.

I: And what about 4 ÷ 1 ⁄ 3 ?

G: 1 ⁄ 3 goes three times into 1, so it goes 4 x 3 = 12 times into 4.

I: What is the rule for dividing by 1 ⁄ 3 ?

G: Dividing a number by 1 ⁄ 3 is multiplying it by 3.

I: Great. So, we know that dividing by 1 ⁄ 2 is multiplying by 2, and dividing by 1 ⁄ 3 is done by multiplying by 3. How do you divide by 1 ⁄ 4 ?

G: By multiplying by 4.

I: Yes, you’ve got the principle. Dividing by one over a number is multiplying by the number. (This would be too hard for her to formulate.) 

Let us now divide by 2/3, which is what we started with. Let us first return for a moment to division by 1/3. If in a party there were 10 cakes, and every child got a 1/3 of a cake, how many kids were there ?

G: 10 ÷ 1 ⁄ 3 = 30; there were 30 kids.

I: And suppose now that each kids gets 2/3 of a cake, instead of 1/3, that is, two times more than before. For how many kids will the cake suffice?

G: Every kid now gets what 2 kids got before. So, there will be half of 30 kids, which is 15.

I: Right. What exercise did you do here?

G: 10 ÷ 2 ⁄ 3, because we found how many times 2 ⁄ 3 goes into 10.

I: Right, and what operations did you do?

G: I multiplied the 10 by 3, and divided by 2.

I: So, what is the rule for dividing a number by 2 ⁄ 3 ?

G: You take the number, multiply it by 3 and divide by 2.

I: And what would be the rule for dividing by 3 ⁄ 4 ?

G: Multiply by 4, and divide by 3.

I: Very nice. And what is the rule for dividing by a general fraction?

G: You multiply by the denominator, and divide by the numerator.

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When a fact comes from outside, it is information. When it comes from within, it is knowledge.