# Sharing Brownies

The other night at dinner as we all debriefed our days, I mentioned that I’d given my college students a problem about sharing brownies. “I should give the problem to you and see how you would solve it,” I mused to my kids, and my husband immediately said, “Let’s make brownies and solve the problem in real life!”

So we did. We threw together a pan of brownies from a mix we had sitting in our cupboard, and when they had cooled I cut and plated three square brownies for each kid. Then I handed them each a plate and a dinner knife and told them to figure out how to give every member of our 4-person family the same amount of brownie.

We like to ask sharing problems about food (“If there are 8 pancakes, how many can everyone have?”), because our kids, like all kids, are highly motivated by food and by fairness. But this is a challenging problem, without a clear, immediate solution.

Still, my kindergartener dove right in, deftly cutting all three brownies in half. She then paused for a moment with the two extra pieces before cutting them in half too and stacking a quarter brownie on top of each half-brownie portion. “Everyone gets a half and a piece,” she said when I asked her about her solution.

(Surprisingly, very few of my college-age students come up with this particular method initially, although once they have seen it they tend to prefer it. This semester one student commented on how surprised he was that a kindergartener would come up with this method right away when he, a math education major, didn’t think of it on his own.)

My 7-year-old stared and stared and stared at the brownies and I could see the gears turning in his head. Finally, several minutes after my daughter had confidently offered up her half and a piece, he said, “Okay, I think this will work,” and embarked upon a complicated cutting exercise that I would call “split the brownies into smaller and smaller pieces and hope it will all work out eventually.” First he cut one brownie into thirds, cut another brownie into fourths, cut a fourth in half, and put the half-quarter together with a full quarter to make another “third”. He then cut the last brownie into fourths, and proceeded to cut and re-cut any odd pieces out until he felt confident that he had a workable number of pieces.

He explained that everyone got a third and a quarter and a half quarter and a “small quarter” (half of half of a quarter). But when he actually distributed the pieces onto each of our plates, there were a few extra bits left behind. “Hmm, I don’t know if that really worked,” he said, and then shrugged and popped the extra pieces into his own mouth.

It was obviously interesting and fun to watch how my kids approached this problem on the very same day I watched my college students approach the same problem. But it was also interesting to listen to the informal language they used to talk about their solutions. They both already had some language for talking about fractions, and they both ran up against limits. The kindergartener could talk about halves, but once she got to quarters they became “pieces”. Nevertheless, with the motivation of actually getting to eat the brownies at the end, she was remarkably accurate at splitting the brownies into equal parts. The second grader could talk about halves and thirds and quarters, but then began talking about half-quarters and “smaller quarters” when he got down to eighths and sixteenths. And he made an interesting approximation of 1/3 by combining 1/8 and 1/4. This was not precisely equal to 1/3, but it was certainly close enough to feel fair.

My college students like to ask me questions like: “When would you start teaching this to students?” My answer is often: “Much earlier than you’d think!” When can kids start understanding fractions? Much earlier than you’d think! When can kids begin making sense of probability? Much earlier than you’d think! When can you give kids multiplication problems? Much earlier than you’d think! When can kids understand that a square is a type of rectangle? Much earlier than you’d think!

This particular problem involves both division and fractions, and while I had no idea what my 5- and 7-year-old children would do with it, I knew they would be able to do something. Kids have great ideas, and they have great ideas much earlier than you’d think!

# Seeing Improper Fractions

Today I want to put in a good word for improper fractions. You know, those fractions where the numerator is larger than the denominator, like 5/3 or 289/18.

We don’t usually tell kids about improper fractions until kind of late in their fraction learning trajectory, after they’re comfortable with “normal” fractions like 1/2 and 3/4 and 2/3 and 5/8. So children naturally get used to thinking that a fraction is like a partially filled pie. 3/4, for example, means we have a pie with 4 pieces but there are only 3 left. And then once they’re really good at this idea, we spring 5/4 on them, and the kids think, “Huh? If there are only 4 pieces in the pie to begin with, how is it possible to have 5?”

Even adults can have a hard time with improper fractions.

The conceptual root of this problem (if you’ll allow me to go into math educator mode for a moment) is that when kids see only fractions less than one, they start to think that 3/4 means “3 out of 4 things” which is not quite right, because 5/4 is a totally legitimate fraction but “5 out of 4 things” doesn’t make any sense. A better way to think about 3/4 is as three 1/4’s, where it takes four 1/4’s to make a whole. 5/4 then means five 1/4’s, where it takes four 1/4’s to make a whole. When we think of a 1/4 as a unit, we can have as many of them as we want.

So moving back to parent mode, when we give our child chances to see improper fractions, like 5/4, in real life, and when we do this early in their fraction learning trajectory, we’re not only making improper fractions themselves easier, we’re helping them develop a strong and solid understanding of what a fraction is in the first place.

Here are a few ideas for seeing improper fractions in everyday situations:

Graham Crackers

Graham crackers are great for introducing fractions to young learners because they break naturally into halves and into fourths, and because those halves and fourths are an identifiable unit. A graham cracker square can be called a half. A small graham cracker rectangle can be called a fourth. Four small rectangles make a whole cracker – that’s why they’re fourths. And it’s not at all inconceivable that you could have 5 or 6 or 7 or more small rectangles: 5/4, 6/4, 7/4, and so on.

Measuring Cups

Measuring cups are also great for thinking of fractions as units. It takes three 1/3-cup measuring cups to fill up a 1-cup measuring cup; that’s why it’s called 1/3. 1/2-cup measures and 1/4-cup measures are similar. You could experiment and see how many 1/3-cup measures would fill a glass measuring cup up to the 2 cup line. That’s six thirds (6/3)! How many thirds would it take to fill it all the way to the top, above the line? 7 thirds? 8?*

Pizza

Or quesadillas, pies, mini bread loaves… The important thing is that a) you can cut the food into equal-sized servings, and b) you have more than one whole (whole pizza, whole quesadilla, whole pie, whole loaf). If a child can identify a piece of pizza as 1/8 of a pizza, and can count pieces as eights (one pieces is 1/8, three pieces are 3/8, etc.), they can also tell how many eights 10 pieces would be, or 15, or how many eights there are in 2 pizzas.

Ruler Measurements

It’s common to use fractions in measurements – a quarter inch, a half centimeter. We have to have a way of naming measurements that are in between whole number measurements. If you’re working on measurement with your child, you’re probably using mixed numbers (e.g., 2 1/2 inches) rather than improper fractions. But go ahead and try to make the leap. If something measures 2 1/2 inches, ask: “How many half inches is that?” or “How many quarter inches is it?” This is more for older children – this is more challenging than pizza where you can see and hold and count an eighth. But it never hurts to ask something that’s beyond the child, and come back to it later if you find it’s beyond the child’s current capacity.

* As an aside, a colorful set of plastic measuring cups and spoons (maybe even one with a 1/8-cup measure) is a great Christmas or birthday gift for a child. It’s not terribly expensive, and opens up opens up all sorts of opportunities for experiences, creativity, and one-on-one time with parents or older siblings and (bonus!) they have their very own tool for thinking about and reasoning with fractions in a completely natural context.