Common numerator fraction division [#algorithmchat]

My future elementary teachers explore the common denominator fraction division algorithm at the end of the semester. Reading their work got me thinking about common numerator fraction division, and about what sense I could make of the symbols that result.

I tried to keep my work neat so others could follow it. If this sort of thing amuses you (as it obviously does me), then you’ll want to take a few minutes with the larger versions of these images. If it does not amuse (and I cannot begrudge anyone this), then you’ll just want to move along; there’s nothing here for you today.

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Zero=half

A propos of nothing the other day, Tabitha asked a strange question.

Tabitha (six years old): Why are zero and half the same?

Me: They aren’t.

T: Like seven is one more than six, but zero and half are the same. They’re both nothing.

Me: One half? if you have half of something, that’s more than nothing.

T: But half, the number, that’s the same as the number zero.

Recall that last fall, she was not convinced that one-half was a number at all.

She now accepts that one-half is a number. But she hasn’t really dealt with the idea that there are numbers between other numbers. She is doing a bit of beautiful kindergarten logic here. Her premise is that there is only one number less than 1, namely 0. She has also accepted that one-half is a number less than 1. Therefore, one-half and zero are the same.

And—rightly—she is suspicious of this conclusion. The logic is sound, but it doesn’t make sense.

I go to work on that first premise.

Me: Oh. I see. Well, one-half—the number—is between zero and one.

I draw this picture, which I feel is certain to be totally unconvincing.

I was writing upside down. Forgive the crummy 2′s. Note the complex fraction. Take that, Common Core!

But then again, we hadn’t talked about one-half being a number since October. That last conversation seems to have been fermenting all this time, so maybe this one will do the same.

To be continued, I am sure.

Griffin on thirds

This one is from the deep, deep archives.

When Griffin was five years old, he drew the picture below.

Griffin (five years old): I’ll make the boiler one-third red, one-third green and another third red.

Me: Where did you learn about thirds?

G: You told me.

Me: When?

G: I don’t remember.

Griffin seems to have absorbed a part-whole model for thirds that roughly matches how young children often think of halves.

Compare to his performance two years later. In that later conversation, Griffin was struggling to think about one-fourth of two. This pretty much matches a major argument in the book Extending Children’s Mathematics by a subset of the CGI research team. Their claim is that a part-whole fraction model is not very useful as the beginning place for thinking about fractions; that instead the extension of sharing division to complete and equal sharing involving remainders is a more natural place for beginning fraction instruction. See also Tootsie Rolls.

Chicken skewers

Saturday was a great day. I took Tabitha to dance class, which afforded me an hour to do some reading. Then it was time for lunch and a great family tradition—Free First Saturday at the Walker Art Center.

For lunch, I chose Thai.

We shared an appetizer of chicken satays and Pad Thai. I felt guilty about the lack of adventure in my choices, then remembered that I was teaching my five year old to eat Thai food.

The satays arrived in short order.

Tabitha (five): Oh! Four of them. So we each get two.

I was hungry, so I let this go for a little while. Later, though, I followed up.

Me: You said we would each get two. What if there had been six skewers?

T: I don’t know. I don’t go that high.

…

T: Four and four. Or maybe three and three.

Me: Three and three. Good. What if there had been three skewers?

T: No answer.

Me: Why ‘no answer’?

T: Because one person gets two and the other person gets one. How is that fair?

Me: Hmmm. Good point. What could we do about that?

T: Split it in half.

Me: OK. Then how much would we each get?

T: I don’t know.

Me: Well, you’d get one whole one and a half. So skewers.

T: Right.

Re-reading our campsite conversation from a few months back, I can see that she still isn’t ready to use fractions as numbers. They still don’t really answer how many? questions in her mind.

After lunch, it was off to the Walker. Which is where we had this fabulous photograph taken. Like I said; Saturday was a great day.

Best question of the semester

Quick break from prepping and grading final exams.

My future elementary teachers always struggle to name the denominator when they need to find of .

They draw the picture.

They know that the numerator needs to be 3. And then they argue about whether the denominator should be 12 or 16.

I struggle every year to get 8 on the table as an acceptable answer. I usually end up being a voice of authority for 8, and we discuss what the whole is if you use 12 or 16 as the denominator.

My students don’t like 8 because that means the answer is of one square, but the pieces come from different squares.

This year, I had an insight that helped a lot. The question was this:

What are some situations in life when you get two same-sized parts of distinct wholes?

I opened the class session following our usual denominator debate with this question and it helped us to focus on the issue at hand.

After a few false starts (i.e. examples that didn’t really exemplify what we were after), we settled on this scenario.

When you buy a 75¢ pop from a vending machine, by inserting a dollar you get back a quarter. Do it again and now you have two quarters. Each quarter came from a different dollar, but they are still quarters. Each is one-fourth of a dollar and together they are half a dollar (even though collectively, they are one-fourth of the money you started with).

Back to the squares and we had a frame of reference for eighths.

I have been teaching this course for 8 years.