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## Which Mistake Most Surprises You?

Fraction comparison for 4th Graders. They’ve been working a lot with representing fractions as circles and as rectangles. They’ve done some basic addition with fractions. Most aren’t generally able to find equivalent fractions.

What mistakes do you expect to see in the class set?

Make a prediction! Mark it down somewhere. Don’t do that internet thing of just continuously scrolling through a page at half-attention. Take a moment, form a thought. Then scroll on for the full class set of 14.

Kid 1

Kid 2

Kid 3

Kid 4

Kid 5

Kid 6

Kid 7

Kid 8

Kid 9

Kid 10

Kid 11

Kid 12

Kid 13

Kid 14

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## 5 and 1/2 x 2 and 1/4 = 7 and 3/4

I find this fascinating. This student clearly knows how that multiplying the base and the height of a rectangle gives you its area. She even knows how to multiply fraction. But when it comes to part (d), she adds the numbers instead of multiplying them.

In earlier writing I hypothesized that, when put in unfamiliar situations, students often default to an “easier” operation. This idea now seems problematic to me. What, after all, is an “easier” operation any way? And what exactly would trigger this default to some other operation? And how do we explain why competent adults — like me — make similar mistakes on my own work?

It now seems more likely to me that we associate certain pairs of numbers with certain operations. Think about the numbers 100 and 1/2. I’d suggest that most people have an association of “50” with 100 and 1/2. After all, how often have you been asked to add 100 and 1/2 together? How often have you been asked to subtract 1/2 from 100? In contrast, how often have you been asked to find 1/2 of 100?

How often have you been asked to multiply 5 1/2 and 2 1/4 together? My guess is that you — and the student above — have been asked to add these sorts of mixed numbers more often than multiply them.

The idea here is that the pairs of numbers themselves come with associations.

There’s a hard version of this claim that I don’t mean to make. I don’t mean to say that, no matter the context, you’d expect a student to add 5 1/2 and 2 1/4 together. I think a division problem with mixed numbers is unlikely to trigger associations with addition. Maybe I’m moving towards a two-part model? The sorts of mistakes we make with numbers depends both on the associations with the operation and also associations with the numbers? And things get really bad when these two associations point in the same direction?

This theory feels very testable, but at the moment I’m having a hard time articulating a possible test of it. But we should be able to mess with people’s associations with numbers and see if that changes the sorts of mistakes that they make. Ideas?

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## Three Fifths of a Triangle

Shared by Tracy on twitter, and a great conversation ensued.

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## Using Subtraction to Divide Fractions

Thanks for this, Graham!

What’s interesting about this to me is the mental connection between division and subtraction. I doubt that this kid has anything like an explicit model of division that involves “taking away,” but it makes sense to me that the ideas of subtraction/division would be associated much in the way that addition/multiplication are.

All the more reason to make sure that there’s a robust understanding of multiplication that goes beyond “repeated addition,” no?

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## 3/7 = 10/14 and 1/2 = 3/14

Nathan sends along a really lovely fraction mistake.

If I’ve got this right, the kid added 3 and 7 to get the numerator, and added 1 and 2 to get the numerator? This is a way of thinking about fractions that’s new to me. Can anyone offer a better theory or some helpful context for this kids’ thinking?

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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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I’m a big fan of Stadel’s Black Box. I think what makes it fun is that there’s something small to figure out (What does the black box do?) before figuring out the big thing (What’s the sum of those two fractions?)

I recently did this with my fourth graders, and it was a ton of fun. Here were some of their answers to 1/2 + 1/3:

3/4

3 1/2 / 4  (three and a half fourths)

7/8

2/3

5/6

10/12

Can you figure out how kids got each of these answers?

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## 3 and a half fourths

Not really a mistake, but my kids have started doing this.

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## Fraction bar as an operation

Misconceptions surrounding fractions are so well-studied that I feel a bit ridiculous sharing anything about them. Anyway…

I was chatting with this kid who was having a bunch of trouble with written fraction notation. She had been correctly solving problems that involved language such as “shade in four out of seven pieces” or “divide this shape into eighths,” but got stuck when she reached a problem that asked her to “shade in 4/6 of the shape.”

Alice: Oh, so that’s 5.

Me: Can you explain why?

Alice: Because it’s not six sixths.

Me: So, not quite.

Alice: Oh, it’s 2. Because that’s 6-4.

Me:

Alice: Or it’s 10?

Me: See…

Alice: I’m really confused here. What’s the answer?

There’s no puzzles or misunderstandings here. Alice thought that the fraction symbol was an operation between the numbers 4 and 6. And of course she did. Every other time that she’s seen two numbers and a symbol before she’s been asked to produce a third number. This is new ground for her.

I’ve been taking the advice of Brilliant Commenters Fawn, Jenny and Avery and using the language of “out of” to bridge the gap for this kid.

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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?