Division Ratios & Proportional Relationships

File This Under “Calculator Mistake”

chris robinson

Or did I get this wrong?

Division Numbers & Operations in Base 10

91 mushrooms, 7 people



Me: What’s 91 divided by 7?

Her: [Draws hands on board.]

Me: What are these for?

Her: For counting.

My move was to nail the question down on a context and ask her the question again.

Me: Hold on. Let’s make up a division story for this question. Let’s say that 7 people are equally sharing 91 crackers.

Her: Can we change it to mushrooms?

Me: Sure.

And she starts counting on the hands. She hadn’t done this for smaller numbers, like 30 divided by 3. There she articulated that 30 divided by 3 is 10, because 3 times 10 is 30. That doesn’t seem to be on her mind right now, so I try to ask a suggestive question.

Me: [Draws 7 stick figures.] Here are the 7 people. They don’t have any arms though.

Her: Can you make one super tall and one super short?

Me: Not this time. They’re all the same armless height. Anyway, how many mushrooms can we definitely give to each person?

Her: 10.

Me: Cool, and that would take care of a bunch of the mushrooms. That would take care of 70 of the mushrooms. And how many left would there be for us to take care of?

Her: 21.

Me: Nice. So, how many more mushrooms can we give to each person?

And then she goes back to her hands and does a bunch of counting. I interrupt her and ask her whether we could give them each 4. She says no, after some thought. She says that it would have to be more than 2. It takes a little bit of thinking before she tries and confirms that 3 works.

I think that this picture, and this dialogue, captures an important step in learning multiplication and division, and how awkward it all is.

I’m very new to all of this, so I’d appreciate some comments. As is our custom on this site, here are a few prompts:

  • Umm…how did that dialogue go? What worked? What could’ve gone better, in your view?
  • I feel like there’s some wisdom here about how people learn division and multiplication that I’m not able to articulate particularly well. Maybe you can?
  • How do you ween kids off of relatively slow and sloppy methods like counting?

Looking forward to your thoughts.


Decimals Division Fractions Numbers & Operations -- Fractions Numbers & Operations in Base 10

44 crackers shared equally with 8 people…



Lots of good stuff going on here. But I don’t think I entirely understand where 1/8 came from, though I get how that gets turned into 5.8.











[I never know whether to include all the mistakes from a class set or just a few. I feel as if it’s helpful to include more mistakes, but sometimes overwhelming. My solution today is to post one especially cool mistake largely, and the others smallerly. Let me know whether that works.]

Division Numbers & Operations in Base 10

Lots of correct ways to do division


Div1 Div2 Div3 Div4 Div5


No mistakes here, but I’m sort of blown away by the variety of division techniques I saw in yesterday’s problem set. Thought that it was worth sharing.

Addition Division Multiplication Numbers & Operations in Base 10 Subtraction

Some Numbers and Operations in Base 10



Explanations? What lessons are there about the way kids think in this work?