Wiggins questions #3

Question 3

You are told to “invert and multiply” to solve division problems with fractions. But why does it work? Prove it.

Oh dear. If anyone on the Internet has had more to say about dividing fractions than I have, I am unaware of who that is. (And, for the record, I would like to buy that person an adult beverage!)

Unlike the division by zero stuff from question 1, this question is better tackled with informal notions than with formalities. The formalities leave one feeling cold and empty, for they don’t answer the conceptual why. The formalities will invoke the associative property of multiplication, the definition of reciprocal, inverse and the multiplicative identity, et cetera.

The conceptual why—for many of us—lies in thinking about fractions as operators, and in thinking about a particular meaning of division.

1. A meaning of division

There are two meanings for division: partitive (or sharing) and quotative (or measuring). The partitive meaning is the most common one we think of when we do whole number division. I have 12 cookies to share equally among 3 people. How many cookies does each person get? We know the number of groups (3 in this example) and we need to find the size of each group.

When dividing by a fraction, partitive division means that we know the fractional part of a group we have, and we need to find the size of a whole group.

I can mow 4 lawns with of a tank of gas in my lawnmower is a partitive division problem because I know what  of a tank can do, and I want to find what a whole tank can do. So performing the division  will answer the question.

2. Fractions as operators

When I multiply by a fraction, I am making things larger (if the fraction is greater than 1), or smaller (if the fraction is less than 1, but still positive).

Scaling from (say) 5 to 4 requires multiplying 5 by . Scaling from 4 to 5 requires multiplying by . This relationship always holds—reverse the order of scaling and you need to multiply by the reciprocal.

putting it all together

Back to the lawnmower. There is some number of lawns I can mow with a full tank of gas in my lawnmower. Whatever that number is, it was scaled by to get 4 lawns. Now we need to scale back to that number (whatever it is) in order to know the number of lawns I can mow with a full tank.

So I need to scale 4 up by .

Now we have two solutions to the same problem. The first solution involved division. The second solution involved multiplication. They are both correct so they must have the same value. Therefore,

There was nothing special about the numbers chosen here, so the same argument applies to all positive values.

We have to be careful about zero. Negative numbers behave the same way as positive numbers in this case, since the associative and commutative properties of multiplication will let us isolate any values of  and treat everything else as a positive number.

More on partitive fraction division here.

Please note that you do not need to invert and multiply to solve fraction division problems. You can use common denominators, then divide just the resulting numerators. You can use common numerators, then use the reciprocal of the resulting denominators. Or you can just divide across as you do when you multiply fractions. The origins of the strong preference for invert-and-multiply are unclear.

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Division and fractions with a third grader

I found some notes on a conversation I had with Griffin last fall. I do not remember the context for it.

Me: Do you know what 12÷2 is?

Griffin (8 years old): 6

Me: How do you know that’s right?

G: 2 times 6 is 12.

Me: What about 26÷2?

G: 13

Me: How do you know that?

G: There were 26 kids in Ms. Starr’s class [in first grade],  so it was her magic number. We had 13 pairs of kids.

Me: What about 34÷2?

G: Well, 15 plus 15 is 30…so…19

Here we see the role of cognitive load on mental computation. Griffin is splitting up 34 as 30 and 4 and finding pairs to add to each. Formally, he’s using the distributive property: .

He wants to choose and so that .

But by the time he figures out that , he loses track of the fact that  and just adds 4 to 15.

At least, I consider this to be the most likely explanation of his words.

My notes on the conversation only have (back and forth), which indicates that there was some follow-up discussion in which we located and fixed the error. The details are lost to history.

Our conversation continued.

Me: So 12÷2 is 6 because 2×6 is 12. What is 12÷1?

G: [long pause; much longer than for any of the first three tasks] 12.

Me: How do you know this?

G: Because if you gave 1 person 12 things, they would have all 12.

Let’s pause for a moment.

This is what it means to learn mathematics. Mathematical ideas
have multiple interpretations which people encounter as they live their lives. It is (or should be) a major goal of mathematics instruction to help people reconcile these multiple interpretations.

Griffin has so far relied upon three interpretations of division: (1) A division statement is equivalent to a multiplication statement (the fact family interpretation, which is closely related to thinking of division as the inverse of multiplication), (2) Division tells how many groups of a particular size we can make (Ms. Starr’s class has 13 pairs of students—this is the quotative interpretation of division) and (3) Division tells us how many will be in each of a particular number of same-sized groups (Put 12 things into 1 group, and each group has 12 things).

This wasn’t a lesson on multiplication, so I wasn’t too worried about getting Griffin to reconcile these interpretations. Instead, I was curious which (if any) would survive being pushed further.

Me: What is ?

G: [pause, but not as long as for 12÷1] Two.

Me: How do you know that?

G: Half of 12 is 6, and 12÷6 is 2, so it’s 2.

Me: OK. You know what a half dollar is, right?

G: Yeah. 50 cents.

Me: How many half dollars are in a dollar?

G: Two.

Me: How many half dollars are in 12 dollars?

G: [long thoughtful pause] Twenty-four.

Me: How do you know that?

G: I can’t say.

Me: One more. How many quarters are in 12 dollars?

G: Oh no! [pause] Forty-eight. Because a quarter is half of a half and so there are twice as many of them as half dollars. 2 times 24=48.

Partitive fraction division

As promised, more notebook pages on fraction division. This is based on the work I did a while back on trying to write authentic partitive division problems with fractional divisors. (As I wrote that last sentence, I reminded myself what a bizarre niche market I am trying to occupy on this here blog.)

I settled on situations involving fractional values of unit rates, such as the following.

If of a lawn takes of an hour, how much can I mow in one hour?

Before we begin, remember that if the problem were about 2 lawns in 3 hours, we would easily and naturally divide by 3. Only the numbers have changed, so the mathematical structure remains the same and we need to find .

Click each image to see it full size. If you’re into this sort of thing.

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Common numerator fraction division [#algorithmchat]

My future elementary teachers explore the common denominator fraction division algorithm at the end of the semester. Reading their work got me thinking about common numerator fraction division, and about what sense I could make of the symbols that result.

I tried to keep my work neat so others could follow it. If this sort of thing amuses you (as it obviously does me), then you’ll want to take a few minutes with the larger versions of these images. If it does not amuse (and I cannot begrudge anyone this), then you’ll just want to move along; there’s nothing here for you today.

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More on fraction division (you know you love it!)

Readers came through for me when I surveyed on ratios. So I come back to the well seeking your insights.

This time on division of fractions.

The Common Core State Standards reserve division of fractions as the only operation on rational numbers at sixth grade. The other three arithmetic operations come in the earlier grades. The Connected Mathematics response to this is going to be to compactly revisit addition and subtraction, linger a bit longer on multiplication and spend some time and depth on division.

This is a topic with which I have considerable experience.

As you surely know, the standard approach is to tell students to invert-and-multiply and then move on.

In previous versions of Connected Math, we have done a single Investigation (4 to 5 days worth of study) on division of fractions, with an emphasis on:

1. Quotative (or measurement) models, leading to
2. Common denominator algorithm

For those unfamiliar, there are two standard problem types for division: quotative (measurement) and partitive (sharing), exemplified by the following.

Quotative (measurement): I have 35 apples to package in bags. I can fit 7 apples in each bag. How many bags can I fill?

Partitive (sharing): I have 35 apples to share equally among 7 people. How many apples does each person get?

A typical quotative fraction division problem goes like this:

I have 3/4 of a cup of grated cheese. Each omelette requires 1/4 cup of cheese. How many omelettes can I make?

These types of problems lend themselves to wanting the denominators to be common; then we can divide the numerators and voila! This is justified by thinking of the 3/4 cup as 3 units and the 1/4 cup as 1 unit. The units are the same, so it doesn’t matter that they are fractional units. The basic question is How many of this are in that? Just like with the apples and bags-how many sevens are in 35?

But partitive is trickier. A typical problem goes like this:

I have 3/4 of a cup of grated cheese. This is enough to make 1/4 of an omelette. How much cheese do I need to make a whole omelette?

Now the common denominator doesn’t matter so much. Instead, I want to multiply by 4, based on the reasoning that if 3/4 is 1/4 of the whole thing, then 4 times 3/4 must be the whole thing.

For most people, this has quite a different feel from the apples and people problem that exemplifies whole-number partitive division. Having a fractional number of groups complicates the partitive problems.

What we have decided to do in Connected Math is to keep the quotative investigation more or less intact (tweaking based on our experience in the field), and create from scratch an investigation on partitive division.

But the typical examples of partitive fraction division don’t resonate with me. It’s easy enough to divide a fraction by a whole number this way (1/2 pound of peanuts shared among 3 people), but fraction by a fraction is tougher (1/2 pound of peanuts is 2/3 of a share?)

Notice that partitive division problems-whether whole-number or fractional-give unit rate answers. Apples per person. Cups of cheese per omelette. Pounds of peanuts per person.

But in those contexts, it is implausible that I would know the rate for a fraction of a unit but not the whole unit. How do I know that 3/4 cup of cheese is enough for 1/4 of an omelette without knowing how much is in a whole omelette? There is no plausible scenario under which I would have this information without also knowing the unit rate.

So now we get to the question…

What are some unit rates in which it’s plausible to know the fractional rate without knowing the unit rate?

My man Sean pointed out in a related discussion that unit rates involving time fit into this category. I can plausibly know that I walked 1/2 miles in 15 minutes, notice that this is 1/2 mile per 1/4 hour and wonder what this means for my miles-per-hour unit rate.

But in an article I reread today, the authors suggested “2/3 of a cake exactly fills 1/2 of my container”. I don’t like this one nearly as well. Exactly fills 1/2? I like it better than the omelette problem, but I feel we can do better.

How about it?