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.

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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?

City names

We were driving home recently. Griffin was in the back seat reading his book about U.S. Presidents.

Griffin (9 years old): Who was the capital of Wisconsin named for?

Me: Is this a quiz or a question?

This is an important distinction with Griffin. He has both in his repertoire. If it’s a quiz, he knows (or thinks he knows) the answer and he does not tolerate speculation or conversation well. If it’s a question, then we have something to talk about.

G: Quiz.

Me: Well, then…based on your reading material, you must want James Madison as the answer.

G: Yeah.

Here I make my move. Turn it from quiz to conversation. It’s not an easy move with this guy.

Me: Does it say that in the book? That Madison is named for James Madison?

G: No. But what are the chances it would be anyone else? I mean, Wisconsin wasn’t even a state until 18XX [he knows the date, I do not recall it].

Me: What are the chances? I don’t know. Maybe not 100%, though.

G: And Jackson, Mississippi! That’s totally after Andrew Jackson!

Me: Maybe.

G: Really, what are the chances?

Since he has raised the what are the chances? question, I figure I will follow up on that. We are approaching our street at this point, which is Sherwood Avenue. Almost home.

You also need to know that the children have become fans of The Brady Bunch over the summer.

Like I said. Long story.

Like I said. Long story.

Long story about that, and I’m not sure how I feel about all of the messages it sends, but I suppose they had to outgrow Dinosaur TrainArthur and Wild Kratts at some point.

Me: Well, what are the chances that our street was named after Sherwood Schwartz, creator of The Brady Bunch?

G: Not very good! That show is from the 60s or 70s, but our street is from a lot longer ago than that!

So what do we learn?

Probability language is important to use around children. Maybeprobably, unlikely, etc. These are the words that express uncertainty. In middle school and high school, students will build formal ideas about probability (that we express chance using numbers between zero and one; that there are sophisticated rules for computing these numbers, etc.)

The more experience thinking about likelihood kids have before that time, the better off they will be.

As a society, we have a too-strong reliance on right/wrong and black/white. When behavior A makes medical condition B more likely, we behave as though A makes B certain. This makes informed decision making very difficult, and it makes educating children about probability hard, too.

So notice and express how likely things are with your children—especially when you and they are not certain about something.

Starting the conversation

There are many times when a fact or outcome is uncertain. Make predictions with your child, discuss the evidence on which you and they base your predictions. Discuss how likely you are to be right. Then notice whether your prediction came true.

You can talk about this when you roll dice in a board game, when you talk about tomorrow’s weather, when you are looking for your shoes, when you wonder who will be at a birthday party, and when you flip a coin to make a decision.

Wikipedia suggests that Jackson, MS was in fact named for Andrew Jackson, and that Madison was named for James Madison. Griffin was also right about our street. Our house was built on the street in 1928.

Summer project (3 of 3)

We went to the State Fair last week.

Photo Aug 25, 10 54 32 AMHaving wondered about the height of the Giant Slide, and having developed a technique for measuring things, we needed to collect the information required to answer our original question.

The only problem? No one but me wanted to ride the slide. This is a big change from previous years.

I went up alone in the name of mathematics.

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There are 104 steps to the top.

I asked a young woman employee how many steps she thought there were. She said 108. I told her my count and she was ready to believe it. I asked a young man employee how many steps he thought there were. He had no idea.

How can you work at the top of that thing and not be curious how many steps there are?

In any case, should someone wish to check my work next year, I got 1 set of 20 and 3 sets of 28.

I also took a kebab stick from Griffin’s dinner along with me. I broke it off at the height of one step partway up. I checked it against another step further up. Then I taped the stick into my notebook when we got home.

May not be actual size on your screen.

May not be actual size on your screen.

So then Griffin and I sat down one morning to finish this off. Recall our guesses of 40 and 45 feet.

It was a fairly conventional conversation, so I’ll just list the bullet points instead of trying to reconstruct our exact words.

  • I asked him to estimate the length of the stick, which he did—4 inches.
  • He measured the stick with a ruler—4\frac{1}{2} inches.
  • He suggested a calculator was in order.
  • I suggested that this would not be happening.
  • We sought to find 104\cdot4\frac{1}{2}.
  • He naturally subdivided this into 104\cdot4+104\cdot\frac{1}{2}.
  • His first answer to 104\cdot4 was 408; on further reflection he got 416.
  • I was a useful resource for remembering intermediate results (such as the 116).
  • Half of 104 was easy for him.
  • We ended up with 468 inches.
  • He knew we needed to divide this by 12.
  • I modeled an intelligent guess-and-check strategy for doing this by asking him to guess. I did the multiplication. You can see the results below.

For the record, I spoke aloud while doing these. E.g.

For the record, I spoke aloud while doing these. E.g. “Two 45s is 90; ten 45s is 450, so 540” Et cetera

Upon completion of our analysis, Griffin wanted to know how high the Sky Ride is.

Success.