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.

9 thoughts on “Cookies under constraints

  1. Great post – and I love and covet those cups!

    “This will lead to good things further down the line.” Also – don’t you think? – this understanding stands as it is, sufficient for plenty of applications. There will always be a further down the line no matter how far we go down it.

  2. I did this with my sixth graders last year. We actually went into the kitchen and I handed them a brownie recipe and a 1/4 cup measuring cup – no other sizes. It was really interesting to hear their conversations!

  3. Lovely, and it was all done with words. What screws up the kids’s brains is a far too rapid introduction of symbols, eg (3/4)/(1/2) (extra brackets(UK) needed for the single liner), and it of course has to have a formal name “division of fractions”, and with a complete disregard for the fact that a fraction is a division anyway, which cannot be written any simpler.

  4. Just found your blog. We adopted two children (now 8 and 10)…they came to live with us three years ago and had been given almost zero help academically. We’re playing catch-up something fierce. This summer was filled with math and reading. Today, on the way to a Mexican restaurant, the kids asked why Taco Tuesday is special if you can always get tacos at the restaurant. I explained that each taco is a buck on Taco Tues. Our 8 year old was quiet for a minute, then said, “If all four of us get two tacos, that’s eight dollars!” I responded, “Hey! You just did multiplication! 4 times 2 is 8.” He was ecstatic. “I’ve never done multiplication before! Let’s do more. If ten people get four tacos, that’s…40. And if twenty people get four…that’s 80 dollars!” I was floored. Really hoping we’ve crested a mountain. I’ll be reading here again for more ideas!

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