What’s this kids next multiplication strategy? How would you help him get there?
Here’s the breakdown of student thinking about double-digit multiplication that I’m seeing as we begin our unit in my 4th Grade class.
Direct Modeling:
Direct Modeling With Composition Into Groups:
Breaking The Numbers Apart With Addition:
Breaking The Numbers Apart With Arrays:
Use of Standard Algorithm:
No Real Strategy, But Knowledge Of Multiplication by Multiples of 10:
Write down 0.1, and add a tenth to it. Write that number down. Then add 0.2 to 0.1. Write that down. Then add one to 0.1.
In case it’s hard to see, in response to “What’s one tenth added to 0.1?” these students responded
When asked “What’s 1 added to 0.1?” they responded
I’m trying to think through what class looks like tomorrow. It seems that I’ve got kids who certainly need time to work with 0.1, 3/10, 0.4 and other tenthy ideas. I also have students who don’t really have much of a grasp on how to use the hundredths place.
I’m going to take a page out of “Extending Children’s Mathematics” and give kids a version of this problem tomorrow:
Francine is making chili. She adds .1 grams of her secret ingredient to each liter of chili that she makes. If she has 5 grams of her secret ingredient, how many liters of chili can she make?
But what numbers would be most helpful to use in this problem? I’m struggling with that question right now. It seems like anything involving 0.1 or any tenths would be good, I guess. I think it’s probably most important for these students to relate decimals to whole numbers.
…and then the more practical concerns arise. What do I do for the quick finishers? They’ve done a lot of problems like this — will this problem feel tedious to them? Should I retrench with some of the part/whole stuff that we worked on yesterday? Maybe spend the first half of class solving chili problems and plan for a discussion, and then try this shading in activity again during the second half? Ooh, we could structure the second half of class around comparing 0.25 and 0.3, like my textbook says, or maybe the kids won’t be ready to discuss decimals that go into the hundredths…
I put “0.1” on the board and asked students what they’d call this. A kid said “one tenth,” but that quickly became controversial.
Question: how do you think these kids are seeing 0.1?
Shared by Tracy on twitter, and a great conversation ensued.
@TracyZager @mpershan My first thought on this to see where students are making the mistake. pic.twitter.com/7X7spt0BMI
— Bryan Anderson (@And02B) April 6, 2014
@mpershan @Anderson02B I think I would go concrete right away, and link it to sharing. If this were a brownie, which piece would you want?
— Tracy Johnston Zager (@TracyZager) April 6, 2014
@mpershan @Anderson02B @TracyZager This one is a bit different, tho. Kids seem to cue in on the equal widths here.
— Christopher Danielson (@Trianglemancsd) April 6, 2014
https://twitter.com/BHS_Doyle/status/452804242672476160
@mpershan @TracyZager I'd bet that most 7th grade students have never seen the problem with a shape other than a circle or rectangle
— Bryan Anderson (@And02B) April 6, 2014
@BHS_Doyle @mpershan @Anderson02B @TracyZager So this task needs revision and should only ask about 1/5 or 4/5.
— Christopher Danielson (@Trianglemancsd) April 6, 2014
I’m a big fan of Stadel’s Black Box. I think what makes it fun is that there’s something small to figure out (What does the black box do?) before figuring out the big thing (What’s the sum of those two fractions?)
I recently did this with my fourth graders, and it was a ton of fun. Here were some of their answers to 1/2 + 1/3:
3/4
3 1/2 / 4 (three and a half fourths)
7/8
2/3
5/6
10/12
Can you figure out how kids got each of these answers?