Got a lot of great ideas. A few that struck either me or my students:
- Several students doubled, and then doubled again to make multiplying by 4 easier
- Using the distributive property to break apart 10 and the half seems like common sense, but it was a big hit in two classes. I’m beginning to think that a flexible understanding of the distributive property is a huge part of number sense
- Among students who prefer to operate with improper fractions, there is a divide between students who see the opportunity to simplify the fraction first (21/2 * 4/1 = 21/1 * 2/1) rather than multiplying (one student converted to a common denominator of 4, for 84/4, then multiplied 84 * 4 and divided by 4*1). These are the valuable shortcuts that a) make calculation easier, but b), and more importantly, show an understanding of algebraic structure, in particular the critical importance of the commutative and associative properties when working with fractions.
On this note, I’ve been thinking more about the longitudinal structure of number talks. I’m structuring them day by day pretty randomly — whatever I think will be meaningful is what we think about. I’m curious if grouping them by structures I want students to see would be valuable. On the one hand, sustained practice with a mathematical idea could help solidify that idea, especially in lower-skilled students. On the flip side, math is math, and number talks as mixed practice have a lot of value in messaging math for math’s sake. On top of that, I’m not sure students could keep track of a weekly theme for number talks, the unit we’re in for the rest of class, and then a range of topics for warm-ups and the rest of their classes in a meaningful way. But maybe it’s something to think about.