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## “A third” = 3/4

Part of what makes learning fractions tricky is that there at least three unnatural things to learn:

1. The written language of fractions
2. The spoken language of fractions
3. The math of fractions

I work in a third grade classroom right now and I’ve heard a bunch of kids say something like the following:

“This is a third.”

Why? There’s an enormous mushing that goes around with “fourth” and “four,” with “third” and “three.”

Related(?) mistake: 4/6 is equivalent to 1/3

Maybe that isn’t related, but I heard it out of a kid who thought a third was 3/4, so it’s probably connected somehow. Maybe you guys can figure out how.

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## Fractions in a Factoring Tree

This mistake seems to say something significant about the way this student sees fractions, but I’m not able to piece it all together. Maybe you can?

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## “How do you write a quarter?”

A lovely third grader pointed to her page.

Her: Is this how you write a quarter?

Me: How many quarters are there in a whole?

Her: Four. How do you write a quarter?

Isn’t that interesting? What do you think is going on here?

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## Summer’s over, and we’re back with some fractions

Summer’s over. Let’s get back to work here.

A sprinkling of thoughts:

• Nicora, on a  recent post, came out against using “x out of y” to introduce fractions. She writes: “One of the most common ways to introduce fractions to young students is to talk about ¾ as “3 out of 4. And it’s part of the reason why students have so much trouble with fractions later on.” I wonder whether this is the sort of mistake that you’d see more of if you use this sort of language in class.
• I bet that this student would have trouble if asked to place these fractions on the number line.
• I don’t really get the task. Are these supposed to add up to 1? What’s the unit here?
• I don’t think that the student believes that 3/5 is more than 2/3. I’d bet that this student just isn’t paying care to the relative sizes of boxes. This aspect of the representation just doesn’t strike her as significant.
• I also notice, though, that within each activity there’s an attempt to make the boxes the same size.

I think that this last observation might be a way into a line of questioning that could help. I’d point to a shaded in box (maybe the kickball one) and ask, “What does this mean?” And then I’d point to another box and ask the same question. This would force us to bring out the unit, and the comparisons between the shaded boxes would force us to have a conversation about the relative amount of time spent at each activity. This would naturally bring us into ranking, which I think would be a good follow-up activity.

There are a bunch of interesting things here — please comment on them — but one moral I’ll take out of this is that learning math often involves becoming sensitive to nuances that would otherwise seem irrelevant.

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## “3 out of 2 doctors recommend…”

What does the student know? What does the student think is going on with fractions? How would you help?

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## How big is three-eighths?

In the student’s work below, what is “it” and why does the student think that matters?