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.

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 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 … … three pies! And have half a pumpkin left over!

Me:Which is of a pie.

G: Right.

NOTE: I do not trust that he got that of a pumpkin makes 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 , 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 of a pumpkin as 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?*

This is classic real-world example of the quotative model of division. For a given total of whatzits, how many portions of whoozits can you make if it takes n whatzits per whoozit? Or another way of putting it, You’re given a total supply, t, of something and a fixed measure of that something, n, needed per unit of another thing. How many units of that other thing can you make?

The more typical division situation kids tend to understand with little or no initial instruction is the partitive or “fair-sharing” model. There, the total supply, t, is again fixed. But it’s the number of fair (equal) shares you want to make that is given and the SIZE of those shares that must be found. Since many children have shared (and most by school age, if not sooner, understand the difference between sharing and fair-sharing, to the extent that they can say, “Tommy splint the candy bar in half, and I got the bigger half,” or “I split the cupcake in two and gave my sister the smaller half” versus “We shared the jellybeans three ways and everyone got 7.”

In either case, remainders can be a tricky idea to wrap one’s head around mathematically, though probably not practically. “We shared the jellybeans three ways. Everyone got 7, and I ate the extra one” would not phase a lot of young kids. Same with “We shared the jellybeans three ways. My friends each got 7 and I got 8.” But “We shared the jellybeans three ways: my friends each got 7/22nds of the total, and I got 8/22nds is NOT going to occur to anyone not actively forced to think about it that way in school, or cleverly led to it by a parent who talks math with his/her kids a lot. Remainders aren’t hard. Rational numbers, however, are abstract, particularly when we don’t split the remainder equally, so that the above becomes, “We shared the jellybeans three ways and everyone got 7 1/3 beans.” (Go ahead: split a jellybean in thirds. I dare you).

Going back to the original situation you’ve presented, I love that you found or stumbled into a naturally-occurring quotative division situation for your son to think about (or he created it via his own curiosity and hunger). His understanding of the rational number issues may not be complete, but it’s pretty sophisticated for his age/experience. He does appear to have some grasp of the protean notion of the unit and how that affects how we think about the quotient. It will be interesting to see, as his hold of that idea solidifies, if he can avoid or at least quickly recognize and recover from the common error that people make when shifting between fractions and decimals in remainder situations. That seems to be a real difficulty for kids and adults alike, leading to a delightful area of “gotcha!” questions with which our high-stakes test question writers, and just good old everyday math teachers, may torment their students. :^)

p.s.: Love me some pumpkin pie.