In my elementary school classes, I find it very helpful to ask students to do mental math in addition to written math. I’ll often put a series of related problems on the board and ask kids to figure out the answers in their heads, and then we’ll share strategies. Some people call these number talks, other people talk about number strings. Whatever you call them, these are helpful for my arithmetic teaching for three reasons:
- Working in our heads makes certain brute-force strategies harder, so we get a focus on more efficient strategies. An example: it’s harder to do 7 x 9 = 9 + 9 + 9 + 9 + 9 + 9 + 9 in your head, so it’s an opportunity for kids to practice picking a more efficient strategy for a tricky situation, like 7 x 9 = 7 x 10 – 7.
- Students are likely to come up with different strategies for these problems, and this gives students a chance to contrast other strategies with their own. I think this helps kids better understand all these strategies.
- Finally, stringing a few related problems together can allow us to be explicit about the connections between simpler arithmetic problems (e.g. 10 x 7) and more complex arithmetic problems (e.g. 9 x 7).
I think there’s a good analogy between learning to fluently and efficiently multiply and learning to fluently and efficiently solve equations. I also think I might find it useful to introduce “equation talks” or “equation strings” this year.
Now that I’ve clarified for myself what strategies kids might use for solving equations, I feel equipped to think about what equation talks might be target those strategies.
For targeting “balance thinking”:
- x + 1 = 2x
- 2x + 1 = 3x
- 4x = 1 + 3x
- 5x + 2 = 3x
For targeting “backtracking” or “unwinding”:
- 2x = 4
- 2x + 2 = 4
- 2(x + 1) + 2 = 4
- [2(x + 1) + 2]/2 = 4
For targeting “covering up” in b – ax = c problems:
- 5 – x = 3
- 5 – 2x = 3
- 10 – 2x = 3
- 100 – 2x = 3
For targeting balance reasoning with decimals:
- x + 3 = 2x
- 0.5x + 3 = 2.5x
- 3 + 1.4x = 5.4x
- 1.1x + 6 = x
And, so on. I think this will be helpful.
(Friendly wager: I bet we’ll see a book called “Algebra Talks” or something like that published in the next five years. Takers??)