IMG_3362

 

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

  • 1.1, then crossed out with a “2” written over it
  • 0.01
  • 0.11

When asked “What’s 1 added to 0.1?” they responded

  • 0.2
  • 0.2
  • 1 0.1 (which looks like a mixed decimal to me which is pretty cool)

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…

Three fifths triangle

 

Shared by Tracy on twitter, and a great conversation ensued.

 

Dividing Fractions using Repeated Subtration 3-4-2014

 

Thanks for this, Graham!

What’s interesting about this to me is the mental connection between division and subtraction. I doubt that this kid has anything like an explicit model of division that involves “taking away,” but it makes sense to me that the ideas of subtraction/division would be associated much in the way that addition/multiplication are.

All the more reason to make sure that there’s a robust understanding of multiplication that goes beyond “repeated addition,” no?

Part of what makes learning fractions tricky is that there at least three unnatural things to learn:

  1. The written language of fractions
  2. The spoken language of fractions
  3. The math of fractions

I work in a third grade classroom right now and I’ve heard a bunch of kids say something like the following:

a third

 

“This is a third.”

Why? There’s an enormous mushing that goes around with “fourth” and “four,” with “third” and “three.”

Related(?) mistake: 4/6 is equivalent to 1/3

Maybe that isn’t related, but I heard it out of a kid who thought a third was 3/4, so it’s probably connected somehow. Maybe you guys can figure out how.

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?)

Adding4

 

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?