At first, this is what I thought the student had done:

- First, the student drew six circles to represent “out of 6 books.”
- Then, they distributed, one-by-one, the 66 books into each of the 6 circles. (If they just put 11 in each, why tally them?)
- Then, the student searched for a way to represent the “5 out of” that are non-fiction.
- It follows that the remaining books are fiction. That makes six sixes, or 36 books.

But then Bridget and Julie came in with a fantastic, different interpretation. Their’s feels like an improvement on my first draft.

you don't think it's possible that they saw the 5 separate from the 6? As in 5 are fiction and 6 are non-fiction…

— Bridget Dunbar (@BridgetDunbar) June 17, 2016

Agreed. "There are 5 non=f for every 6 f." In person I'd ask, "Which do you think there are more of, non-f or f?"

— Julie Wright (@julierwright) June 17, 2016

We then got to work trying to come up with some activities to address this work. Suppose that your class of 6th Graders try this problem, and a lot of your class has struggles that are similar to the work above. You’re planning tomorrow’s lesson. What activity would you begin class with?

This is what we came up with. Which of these activities do you think would be most helpful? Are there any changes you would make to any of them? Is there a combination and sequence of these activities that you think would work particularly well? (I took a shot at sequencing them below. Some details on activity structures are here.)