I love all the multiplication that this kid understands. I think they’re totally ready to be able to handle this sort of multiplication.

How would you build on what they know? What problem would you use to help take them to the next step?

When kids are learning to give fractions meaning, I think they often struggle to figure out how the numerator and denominator are coordinated. Here we see a middle step in understanding, maybe: it’s not that the numerator and denominator are totally disconnected. They’re just coordinated in a way that doesn’t really correspond to how they actually work together (i.e. denominator tells you the “unit” and the numerator tells you the “quantity.”)

Maybe the progression of learning looks like this:

- 2/3 means “2 and 3,” nothing to do with each other. Totally baffling notation.
- 2/3 means “2 by 3” or “2 times 3,” some more familiar situation where two numbers can be coordinated in a relation.
- 2/3 means “2 thirds,” which is a productive way to coordinate the numerator and denominator.

Thoughts? Am I overinterpreting this as a middle step in a progression, when it’s actually just a totally uncoordinated interpretation of the fraction?

From Bedtime Math:

Big kids:The record distance for a thrown boomerang to travel is 1,401 feet. If it traveled exactly 1,401 feet on the return trip too, how many feet did it travel in total?Bonus:Meanwhile, the longest Frisbee throw is 1,333 feet – about a quarter of a mile! How much farther from the thrower did the boomerang travel than the Frisbee?

From the submitter, who sends in the thinking of two of his students:

(1) first student, having doubled the boomerang distance in the earlier question, now doubles the frisbee distance and calculates (2801 – 2666) feet.(2) Second student gets an 100 board and spends a short time calculating 100 – 33 = 67. Then thinks for a long time during which I’m sure he is going to say 67 + 1 = 68, but never quite does it. I stay silent until he announces: 667. No clue where the extra 600 came from. He wasn’t willing to write down or draw anything to explain his thinking.

Interesting! I’m inclined to put the first student in the “extending the thinking you’d do in one model to a less familiar situation” category and the second student in the associational mistake (same link) category.

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?

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?

That’s weird, right?

They clearly get the visual model. Now, granted, it’s hard to apply this visual model when multiplying by “one and a half.” Still, there’s a clear attempt to work it out with pies, and then they wrote four. I mean, what’s going on?

Maybe the kid was just adding instead of multiplying. After all, 2 and a half and 1 and a half makes 4. Maybe he forgot what operation he was working on. He was confused.

But then you work your way through the stack of papers, and you see this mistake coming up a bunch.

Why do kids that clearly get that we’re multiplying end up adding?

You might say, hey, these were just guesses from students who were unable to grapple with a difficult problem. They just wrote *anything *down. You’re going to have to trust me, because I was there, that this wasn’t the case. These were kids who were, like, I’ve got this, what else you got?

If you’ve followed my work for the last year or so, you know that I’m really into exponent mistakes. I’m inclined to connect this multiplying fractions error with some of the things that I’ve shown you all about exponents. This seems, to me, to be another situation where kids default to a computationally easier operation when faced with applying an operation in difficult context. Sometimes that’s exponentiation defaulting to multiplication, but here it’s multiplication defaulting to addition.

Based on the first of these, I’d think that the student was mistakenly adding instead of subtracting. But how could that also explain the second mistake?

On the other hand, it’s hard to imagine that the student is subtracting in the second case, since they end up with a number that’s larger than what they started with.

Ideas?