Today we began by “jigsawing” the homework problems from last night. Teachers were assigned to “expert groups” on one of the problems: partitive vs. quotitive division, changing perimeters of a rectangle with fixed area, and the handshake problem. Experts formed new groups to discuss the problems further. Andrew showed us how the graph of y = 24/x (a hyperbola) can help thinking about changing the dimensions for a fixed area. We watched a video of 3rd graders working on the handshake problem (mini Gausses!) We also heard some great ideas from colleagues about effective ways to organize students into groups and to provide extra work for groups of students who finish early and need differentiated learning opportunities.
We discussed many models to understand fractions from a variety of perspectives, not just the “pizza model.” In particular we found that many of us overlook using fractions as distances or locations on the number line, but these other models can be of great use when justifying fraction arithmetic. We used the area model to expand our ideas about equivalent fractions and fraction multiplication. We also drew on the adjective-noun theme to justify the process of finding a common denominator.
We sat in a circle and practiced one another’s names. We discussed productive classroom norms regarding engaging in mathematical problem solving, participating as productive group members, and what strategies teachers can use to help establish such norms.
After a lunch and picture-taking break, we resumed work with fractions. We discussed why division of fractions should involve multiplying by the reciprocal. In short we did it by the same way we came to understand division yesterday: translating a division problem with an unknown result into a multiplication with an unknown factor.
We ended by working on the sheet about estimating fractions, and developing fraction number sense.
We said good-bye to our colleagues who won’t be joining us next week! We’ll miss you, and we hope to see you next summer if not before!
Math and Pedagogy Homework:
No HW, but please bring as many of your curriculum materials as you can on Monday. We’ll be re-examining your textbooks together.
But…..if you’re looking for some math to do: How many names were said when we played the name game? That is: by the time we had made it around the whole circle and each of us had taken a turn, how many names had been uttered? Hmmm…….
Have a great weekend!