Invert and Multiply…but WHY?

Most of my posts so far have addressed high school level math.  This one is for younger students (and their teachers).

At another blog that I follow, Kitchen Table Math, there has been recent discussion about the “invert and multiply” rule for dividing fractions and whether it is important and/or appropriate to teach the reasoning behind this rule.

You know I am going to vote “yes”.  But what kind of derivation should we use here?

The rule can be derived algebraically.  But that doesn’t really help the grade-school students who are just learning it.  It can also be derived in a algebra-ish way.  A commenter at Kitchen Table Math pointed to the explanation found here:

http://www.moveitmaththesource.com/realfractions/mrnovaksproof.html

Mr. Novak’s explanation is elegant and concise. And it certainly answers the question: why do we invert and multiply in order to divide fractions?  But it would be nice if students also had an intuitive feel for why this works.

We can build the rule step-by-step in a manner that helps that intuition to develop.

Step 1: Division is for Sharing

You have 12 cookies – each a pretty good size —  and you need to put them in individual serving bags to hand out at a party.  By deciding how many to put in each bag, you also determine how many serving bags you can make.  For example, you could put 12 cookies in each bag, but then you would only have enough for 1 bag.  On the other hand, it you put 6 cookies in each bag, then you can fill 2 bags because 12/6 = 2. That seems obvious, but it establishes a method: we are going to divide the total number of cookies by the amount in each bag to find how many bags we can fill.  You soon notice that if you put fewer cookies in each bag, you can fill more bags.  For example:

If we take 12 cookies… And put 12 in each bag… 12 ÷ 12 = 1 We can fill  1 bag
If we take 12 cookies… And put 6 in each bag… 12 ÷ 6 = 2 We can fill  2 bags
If we take 12 cookies… And put 4 in each bag… 12 ÷ 4 = 3 We can fill  3 bags
If we take 12 cookies… And put 2 in each bag… 12 ÷ 2 = 6 We can fill  6 bags
If we take 12 cookies… And put 1 in each bag… 12 ÷ 1 = 12 We can fill  12 bags

So yes, dividing by a smaller number gives a bigger answer.  And with 12 cookies, 12 bags is the most you can fill…or is it?  I mentioned that these were pretty big cookies.  Hmmm…

 Step 2: Dividing by Fractions (of the “one over” variety)

You decide to put make the serving size a ½-cookie.  Since that serving size is even smaller, you expect to fill more bags.  Though you have not been taught how to divide fractions, you can do this one by thinking:

If I am putting ½ in each bag, then each cookie is enough for 2 bags.  So 12 cookies fill 24 bags.   I guess that must mean 12 ÷ ½ = 24.

Need to fill more bags?  Use smaller portions!  Give everyone 1/3 of a cookie…

You can see where this is going.  Let’s continue the chart…

If we take 12 cookies… And put 1/2 in each bag… 12 ÷ (1/2) = 24 We can fill  24 bags
If we take 12 cookies… And put 1/3 in each bag… 12 ÷ (1/3) = 36 We can fill  36 bags
If we take 12 cookies… And put 1/4 in each bag… 12 ÷ (1/4) = 48 We can fill  48 bags
If we take 12 cookies… And put 1/5 in each bag… 12 ÷ (1/5) = 60 We can fill  60 bags

The trend has continued.  Dividing by a smaller number gives a bigger answer.  And even if you don’t know the rule for dividing by fractions, you can certainly divide by these fractions!

To divide by “one over anything”, just multiply by the denominator.

 STEP 3: Any fraction at all!

Suppose you have those 12 big cookies and you decide to give everyone (2/3) of a cookie.  How many servings can you make?

We can figure this out using the reasoning we followed above, with one extra twist.

We have already seen that if you want to give everyone just (1/3) of a cookie, you will have enough to fill 12 × 3 = 36 bags.  But now you decide to give everyone TWO thirds of a cookie.  But you have already filled all those bags – why not just give everyone two of the bags?  You will have enough for 36 ÷ 2 = 18 servings.

Now go back and look at how we got that answer.  We don’t have to know an official rule to divide the 12 cookies into servings that each have 2/3 of a cookie.  We can just multiply by 3 and then divide by 2, which is the same as multiplying by 3/2.

Let’s try another:

Suppose you have 36 pizza pies to sell and you want to give 3/5 of a pizza to each customer.  (No one cuts pizzas into fifths, but don’t worry about that.)  How many customers can you serve before running out of pizza?

Well, if we give everyone ONE fifth of a pie, we can feed 36 × 5 = 180 customers.  Giving everyone THREE fifths means that we can only serve 180 ÷ 3 = 60 customers.

So to find 36 ÷ (3/5) we multiply by 5 and then divide by 3 which is the same as multiplying by 5/3.

Which derivation is “better”?

If I could only show students one explanation, I would go with Mr. Novak’s.  But if I had time to explore and play, I would start with what you see above.  I like that it encourages number sense and intuition.  Also, it introduces an idea that students will see again many times: that existing patterns can be extended to cover new situations.

 

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