In my last post, I wrote about my big-picture thoughts on Number Talks. Here's a post about the nitty-gritty.
Today in Pre-Algebra we are reviewing how to convert from fractions to decimals. I have this worksheet for all the fractions from halves to tenths. To solve this whole sheet, students will need to do some long division, so I chose a number talk that might bring up long division naturally as a strategy.
My Number Talk for today was:
Find half of 36
At the beginning of the day, I anticipate the following strategies:
- Use long division (the traditional algorithm)
- Break 36 into 30 and 6, take half of each and add the halves back together to get 18
- 6*6 is 36, so 6*3 is 18. (Not certain, but possibly
- Who knows? I'm excited to see what they come up with
Here's how it went:
Billy started us off with long division, the traditional algorithm. I'm glad he brought it up and glad to get it out of the way!
Theo said "I added a 0 and got 360. I know 360 degrees is a full circle, so 180 is a half circle. Then I took away the 0 and got 18." This was unexpected but awesome! I asked for clarification from the class as to what we are doing mathematically when we "add or take away a zero" and talked about how we are multiplying or dividing by ten.
Elsa broke up 36 into 30 and 6 and found half of each, just as I anticipated. I reminded my students of the name for this strategy: decomposing 36. I like that term because it sounds gross and is memorable for that reason.
Billy said you could count by 2's until you got to 36. I wrote it down but kind of brushed it off. In retrospect, I should have dug deeper. Why does this strategy work? Why don't we count be 3s or 4s? If we did count by 3s, what would that find for us? What a missed opportunity!
Instead, I asked everyone to try Elsa's strategy on the number 74. I got a kid to explain it, and we realized that he not only decomposed 74 into 70+4, he then decomposed 70 into 60+10! So we had a double decomposition.
Ok, these guys rocked it. I had a total of 8 strategies for solving this problem!
Karina said that 3/2 = 1.5 but she knew it was really 15, so I made another student explain that part. Anna Marie used a building-up strategy to find what added to itself to equal 30 and then 6. Above, you can see where I pointed out that both of them had decomposed 36 in the process of solving their problem.
Elizabeth Rae used a similar building-up strategy to Anna Marie, but with a twist. She said "I know 15+15 is 30, and I know that 8+8=16, so I figured out that 18+18=36" I asked her to expand on that, and she said she was using the units place to figure out that 8 and 8 make a number that ends in 6, and 18 is close to 15, so it made the most sense as an answer.
Landon started with 15 as well, but multiplied by 2 instead of adding it to itself. Then he guessed and checked until he found 18.
Jackson decomposed 36 into 20 and 16 instead of 30 and 6! What a great moment! I asked the question "So can we just decompose 36 into any pair of numbers?" and a kid said "As long as they add to 36, it should work." We will definitely need to revisit this in the future.
Reggie found a pattern with 6. 6*2 is 12, and when he added 6 to the first factor and tried 12*2 he got 24, so he added another 6 and tried 18*2 to get 36. I'm still not sure how he stumbled on this idea, but it's cool! I hope he brings it up again in the future.
Jeremy decided to divide 36 by 4, since he knew that math fact. The he multiplied than answer, 9, by 2 to get 18. I asked another student to explain why this work, and I used a visual model (at the top of the picture) to solidify the explanation.
Kayla used the traditional algorithm! Finally!
This period we had our first controversy! One student got 13.5 and another got 18. All last week, we had controversy every day, so I was actualy surprised that we hadn't had disagreement in A or B period. I always get excited when a student voices a wrong answer because they might be brave enough to share their thought process and explain their mistake.
Anna Lane decomposed the problem in the way I anticipated, into 30 and 6. Haley used the traditional algorithm.
Grant did the strategy I was anticipating but unsure of, where he knew that 6*6 is 36, so 3*6 is 18. I asked him "Why not cut both numbers in half?" And he said "that wouldn't work." And I said "why?" and it got a little confused but eventually someone said "because 6*6 means you are counting six 6s, so Grant is only counting three 6s. So that's half"
After Haley did the traditional algorithm, Macon said "I see what I did wrong. I did long division, and I knew that 2 goes into 3 once and 2 goes into 6 three times. So I got 13. THen I knew I had a remainder of 1 from the 3/2, so I brought down a zero and did 10/2 and got 5, which I added at the end as 0.5. But I needed to bring down the 6 and do 16/2"
I then pointed out that his idea was almost right! He divided 30 by 2, not 3, so that 0.5 should be in the units place, where it would add with 3 to get 8. Then he would have gotten 18. This is where I wish I had my Great Mistakes board up so I could give the class a point for Macon's great mistake.
What I Learned Today
- I can anticipate the most common strategies, but some (like Theo's or Reggie's or Jeremy's) will be totally out of left field and surprise me
- Until this year, I can't think of a single time I've had 8 students share 8 different ways to solve a single problem. B Period was on fire. And we built another bit of classroom culture that the more ideas, the better.
- Macon is brave.
- I need to pull back my participation. I did a couple of moves where I said "someone else explain that to me" but in writing this blog, I realize how many times I added something that I didn't need to add.
- I like the idea of "Try Elsa's strategy on a new problem" as a good way to extend the conversation if there aren't a ton of different strategies.
- The same kids keep sharing. I think I am going to grab a couple of shy kids on Monday and say "I would love for you to share one method in our Number Talks this week. Raise your hand on a day when you feel comfortable sharing"
- In a month or so, when I introduce the distributive property, these kids are going to roll their eyes at how obvious it is. I think I will slowly formalize my notation until I write these problems in a way that looks more and more like the distributive property.