Cookies under constraints

A propos of nothing one day, I ask Griffin (9 years old at the time, finishing up fourth grade) a question.

Me: Griff, imagine you are baking cookies and you need \frac{3}{4} cup of sugar, but you only have a \frac{1}{2} cup measure. How would you get \frac{3}{4} cup?

He thinks about this for a moment.

Griffin (9 years old): You put \frac{1}{2} cup of whatever you’re measuring.

Me: Sugar.

G: Does it matter?

Me: No. I suppose not.

The conversation could end here and I would be delighted. But it does not end here.

G: You put that into the bowl, then you fill the cup halfway and put that in.

Me: And that’s \frac{3}{4} cup?

G: Yes.

Me: How do you know?

G: Because \frac{3}{4} is a half, and then half of a half.

Me: Yeah. That is what you just described. How do you know that that’s right?

G: Like a square. If you shade in half of it, and then half of what’s left, that’s the same as shading \frac{3}{4} of it.

squares

So What Do We Learn?

One question division helps answer is how many of this are in that? My question of Griffin asked how many halves are in three-fourths? This is a division question.

Griffin may not know that it is a division question. That is fine. He is thinking about a specific example of how many of this are in that? This will lead to good things further down the line.

That he sees “sugar” as a non-essential detail of the story is lovely. This will serve him well.

Griffin’s mental image for this task is a common one. He can see three fourths of a square in his mind, and he can see that this is the same as one-and-a-half halves of a square.

Finally, we learn (because I am about to tell you) that this scenario could never really happen when baking in our home. I have an awesome set of measuring cups (pictured below): \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, 1 and 1 \frac{1}{2}. (A friend—and friend of the project—has pledged to donate her \frac{1}{5} cup measure to the Talking Math with Your Kids cause.)

Stack of measuring cups

Starting the Conversation

There are so many ways to raise the question how many of this are in that? Measure each other in inches, wonder how many feet tall that is. Count your quarters, wonder how many dollars that is. Repeat with nickels, or dimes. Bake a batch of cookies using only the \frac{1}{2} cup measure.

And you can read through previous division posts for more ideas.

Milk by the gallon

Milk has been on sale at our local gas station/convenience store. Griffin and I walked up there the other day to buy some milk. Two percent milk for the kids and me; skim for Mommy.

milk.by.the.gallon

Me: Griff, the milk we just bought was $5.50 for two gallons. How much was each gallon?

Griffin (9 years old): With tax included? Or not included? I don’t do tax problems.

Note: I weep for the loss of 4% and 5% sales tax rates. They were so easy to compute mentally, and such a nice introduction to the financial world for elementary age children. Minnesota’s sales tax rate is presently 6\frac{5}{8}%. The city of Saint Paul tacks on another half percentage point. I don’t even bother computing sales tax mentally any more.

Me: No worries about taxes. There is no tax on milk.

G: OK. Two twenty-five. Er…no that’d be $4.50.

So…

$2.75!

Me: How did you do that?

G: Well, I thought it would be $2.25, but that’s half for $4.50, so there’s an extra dollar. So I split that dollar in half, which is 50 cents, put that with the $2.25, which is $2.75.

Me: Nice. I could see that thinking in your first answer; when you said $2.25. I was curious whether you used that first wrong answer or started over from scratch.

When I thought about it, I did it differently. I thought that half of $5.00 is $2.50, then I need to add half of 50 cents. Same answer, though. $2.75.

So what do we learn?

I called in to a Minnesota Public Radio program on math education last week. One of the pervasive questions in such conversations is about how kids are learning to do arithmetic in modern American schools, and it arose in this program.

The thinking Griffin is doing here is lovely, and modern math curriculum is trying to encourage more of it than in the past. He is splitting 5 \frac{1}{2} in half, and he is doing it mentally by thinking about the related multiplication facts.

This thinking is not closely related to the standard long division algorithm. One of the big challenges in school curriculum is relating mental math strategies such as Griffin’s to efficient algorithms that are more useful for complicated computations. I have a few resources parents may find helpful over at Sophia.org.

Starting the conversation

Anytime you find yourself wondering about such things, ask your child to think along with you. I wanted to know whether the gas station price for a gallon of milk was a good one. This required me knowing what the price was for each gallon. Not a hard problem for me, but I had to think for a moment. So then I asked Griffin. Do the same at the grocery store, the convenience store, the hardware store; anyplace where things are priced in groups.

If your kid needs a challenge, ask about gasoline. I paid x for y gallons yesterday. How much per gallon? This one will likely require estimation skills!

The meanings of division

I was talking with Griffin one day when he was in third grade.

Me: Do you know what 12\div 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 \div 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 \div 2?

G: Well, 15 plus 15 is 30…so…19

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

Our conversation continued.

Me: So 12 \div 2 is 6 because 2 \times 6 is 12. What is 12 \div 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.

Me: What is 12 \div \frac{1}{2}?

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 \div 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.

So what do we learn?

Mathematical ideas have multiple interpretations which people encounter as they live their lives. As we learn more mathematics, we become better at connecting these different ways of thinking about ideas.

In this conversation, Griffin relies on three ways of thinking about division:

  1. A division fact is a different way of saying a multiplication fact. (12 \div\ 2 is 6 because 6 \times 2 is 12).
  2. Division tells how many groups of a particular size we can make (Ms. Starr’s class has 13 pairs of students).
  3. Division tells us how many will be in each group if we make groups that are the same size. (When he was working on 34 \div 2, Griffin put 15 in each group to start off with.)

We were just talking for fun, not homework or the state test. So I wasn’t worried about his connecting those ways of thinking. I was just curious how he would apply them to some more challenging tasks, such as dividing by 1 or by a fraction.

I was surprised by how difficult 12 \div \frac{1}{2} was for Griffin. Not because it is an easy problem, but because he could have applied his how many of this are in that? idea, or his multiplication facts idea. But he did neither and reinterpreted the task as twelve divided by half-of-twelve.

I was also surprised at the length of the pause he took for 12 \div 1. It makes sense in retrospect. After all, are you really making groups if it’s just one group? I imagine he had to think that through, rather than the number relationships involved.

Starting the conversation

When the opportunity presents itself—when you and your child are not under homework stress, not rushing to get out the door or find the dog’s leash; when you happen to be talking about number anyway—ask follow up questions. Even a simple set of division problems got a lot of good thinking out of Griffin. Problems involving 1, 0 and \frac{1}{2} are especially challenging.

Vary the size of the numbers.

Don’t worry about whether the answers are right or wrong.

Keep asking How do you know? and listening to your child’s answer.

Offer a few ideas of your own.

Quit before anybody gets frustrated or bored.

The Pumpkin Patch

On a family trip to a farm from which we have bought a tremendous amount of produce this year, Griffin and I were heading to the pumpkin patch.

20130929-215825.jpg

We had already taken the wagon ride to the other pumpkin patch; where the pie pumpkins were grown. We had helped with the harvest and had chosen several to take home. Now we were on our way to the Jack-o-Lantern pumpkin patch.

Griffin [9 years old]: We have 5 pumpkins! Is that enough to make a pie?

Me: More than enough.

G: Enough to make 5 pies?

Me: Probably not.

G: How many pumpkins go into a pie, or how many pies do you get from a pumpkin?

Me: Hmmmm… I would say about 1\frac{1}{2} pumpkins make one pie.

NOTE: This was semi-truthful. I really have no idea how many typical pie pumpkins are needed to make a pumpkin pie. I was making what I felt to be a reasonable estimate. But at the same time, I was pretty pleased with the estimate and with the math that it might encourage Griffin to do.

G: Oh! So we could make … 3 … 3 plus 1\frac{1}{2}4\frac{1}{2} … three pies! And have half a pumpkin left over!

Me: Which is \frac{1}{3} of a pie.

G: Right.

NOTE: I do not trust that he got that \frac{1}{2} of a pumpkin makes \frac{1}{3} of a pie given my estimate. He may have gotten it, and he may not have. The pumpkin patch was approaching so I let it slide.

G: Will we make three pies?

Me: No. I don’t think I’ll have the patience for that. But we can make one pie for sure.

So What Do We Learn?

Griffin is thinking about division when he figures out how many pies we can make from five pumpkins. Other similar sorts of division problems include, How many feet tall are you if you are 49 inches tall? and How many groups of four can we make in our classroom of 30 students? The pumpkin pie problem is challenging because it involves fractions.

One of the hardest parts of the thinking Griffin does here is keeping track of the units. As he counts up to 4\frac{1}{2}, he is counting pumpkins. The first 3 he utters counts pumpkins. But at the same time, he is keeping track of a number of pies. That’s the final 3 he utters: 3 pies.

I play with that idea by referring to his \frac{1}{2} of a pumpkin as \frac{1}{3} of a pie. I understand that not every parent is ready to do this on the spot. Don’t worry about that. Griffin got enough thinking from the basic conversation; the rest is gravy (or maybe whipped cream?)

Starting the Conversation

This was a special opportunity. We had some pumpkins. Griffin wanted to make things with these pumpkins. I could involve fractions.

Other such opportunities could include bags of apples, cups of flour (a standard 5-pound bag of all-purpose flour has about 18 cups), et cetera. If your child doesn’t ask the how many pies (or batches, or cakes, or whatever) question, you can ask it. But don’t make it feel like a quiz. You can just say, I wonder how many pies we could make with what we have?