During the spring of the 2012-2013 school year Kara Imm and I were working with my 5th graders to help them visualize what it meant to divide a whole number by a rational number. The students were very quick to invert the fraction and multiply. They loved saying, “Flip and multiply.” Mind you, I had never uttered those words in the classroom. However, this class was very used to string work and representing their work through models.
We did something a little differently this time. As an assessment we had the students work on four of the problems on notecards so we could collect, assess, and think about the children who were strict calculators (calculation only no picture or model), calculators with an emerging model, or calculators with a model (perhaps not the best one but some sort of representation connected to each problem). Unlike most strings, since we were assessing to form small groups to do targeted instruction, we did not have the students talk to each other.
Here is the string:
1 divided by 1/2
2 divided by 1/2
4 divided by 1/2
3 divided by 1/2
While the students were thinking and writing, we chimed in a few times, based on what we were seeing:
- Students wanted to race to solve each problem, so we said, “I see that some of you have numbers written down. Now try to picture what you saw in your mind when you did these calculations. Why did they make sense to you? Can you sketch something for us to help us see what you saw?”
- After awhile we also added a more direct invitation, “Maybe there is a model that you have used in other ways that would make sense for this problem, too.”
- Our last push was just to get as much thinking on the card as possible,”Make sure you have used everything possible and shown us how you see this, how you solve it.”
Follow up work:
After we formed three groups (described above) we had kids form a partnership with another student. The students were given their notebooks to revise and extend their work and thinking. We acknowledged that we all had the answers to the string, but now our job was to create a context and model to help us understand why we were inverting and multiplying. We did quite a bit of talking about situations and models that would make sense. We had kids make mini-posters of their thinking and share those with the group
In the strict calculators group, we asked them to come up with a scenario (a story or a context) for any one problem in the string. When, in their own lives, would they ever need to solve the problem 3 divided by 1/2? What could the 3 and the 1/2 represent? Then, from that scenario, could they sketch something to model the situation. We coached them using language about teaching someone with a real life problem how or why you invert and multiply.
For calculators with an emerging model, we asked if the model made sense to their partner and encouraged them to justify their thinking and revise. In some cases divided by 1/2 was swapped with dividing in half (a common mistake) so we got to talk about that important difference. Similar to strict calculators, we asked the students to create a context but one that would fit their model.