## January 26, 2014

### Ten Divided by Half

What question would you ask to prompt students to think about 10 divided by 1/2?

I'm reading The Problem With Math Is English and the author says “How many times does half go into 10?” is the only question most teachers have. Which is a problem since that has very little meaning to most students. I posed the question to Twitter and offered “How many half units fit into ten whole units?” as the only context free alternative I could come up with. Luckily, my tweeps came through with plenty more!

Bridget Dunbar: how many groups of 1/2 are in l0?

Teresa Ryan: Yes. I like using the groups idea.

Tina C: but what is a group of 1/2?

Teresa Ryan: Yeah, guess I should have thought about that longer. I was picturing groups of numbers, and 1/2 doesn't quite work. I guess what I'm thinking is breaking the 10 into individual pieces, then counting how many 1/2's there are

Tina C: that’s exactly what this book is about. Falling back on our ingrained language which doesn’t always help kids.

Kathryn Freed: how many groups of 1/2 an orange could be made from 10 oranges?

Tina C: ooh An orange is interesting. This reminds me of @TriangIemancsd's post on partitioning

Teresa Ryan: I guess I’m struggling with students seeing 1/2 of something as a group

Tina C: I agree-I don’t see half of anything as a group. But we could translate into group language via an orange

Bridget Dunbar: I picture a number line and marking off the number of halves in 10 wholes.

Justin Lanier: "How many halves are there in 10?" There’s also the other interpretation: “If 10 make a half of a pile, how big is a pile’?”

Tina C: I like this question because I find it easier to answer. No one has used that interpretation yet.

Justin Lanier: People tend to avoid this interpretation with fractions. I have shed this prejudice :)

Math Minds: or you could start with 10 divided by 2 ask for multiplication prob u could use to solve?

And then there were the people who offered a context to explain the abstract question:
Paul F: You’re making bread, and your only clean measuring scoop is a half cup. How many scoops do you need to make 10 cups?

Zach Schultz: how many "half Oreos" go into 10 Oreos. ( if we had more half dollar coins laying around I'd say how many of these go into \$10)

Jen Silverman: I need to serve each GI 1/2 pound of home-style potatoes How many servings can I get from a 10-pound bag?

Jen Silverman: I had a 10 c. bowl of popcorn. When I split it evenly btwn the Ss in my class, each S got 1/2 c How many Ss?

There were a few people who agreed that "How many halves are there in 10?" would be their go to question. What question comes to your mind first? Which approaches best help students understand the computation?

1. If you think of ½ as a unit (instead of a group), then it becomes how many units of size ½ fit into 10 units of size 1? First you have to find a common unit size (½), do the conversion from size 1 to size half, giving 20 (from the 10). Not a great explanation due to my limitations with words, but I really do think Unit is the way to go.

2. This is what I did with my son a month or so ago when we were just starting to talk about dividing fractions:

3. Justin's remark makes me think of reframing the question entirely. "10 divided by 1/2 means you have to share out 10 things to half a person. So let's consider a WHOLE person by doubling everything. 10 divided by 1/2 means the same as 20 divided by 1, which is what?"

I suppose it's not really what you're looking for, as it doesn't help with conceptualizing. On the other hand, we should be able to extrapolate that. 10 divided by 3/4, let's quadruple everything so we have 40 divided by 3 - now whole groups instead of pieces.

4. I find this conversation fascinating. I think that my instinct would be to highlight the weirdness of "What's 10 divided by 1/2?" and ask kids to make sense of it. I'd have done work to establish models of division (sharing/grouping) and I'd give room for kids to express the meaninglessness of the question.

At the end of the day we want our kids to understand what "10 divided by 1/2" means, not some contextualized version of that, not "How many times does 1/2 go into 10" or "What's 10 shared with 1/2 a friend?" or anything else. The assumption seems to be that, since kids don't understand this language we need to give them a bridge language to ease them into the abstract language. I recognize this as a teacher move that I sometimes use, but I'm not sure if I would use it here.

5. "The assumption seems to be that, since kids don't understand this language we need to give them a bridge language to ease them into the abstract language" Yes, Michael, exactly! I'm excited to read the rest of this book to see how the author recommends making the new language comprehensible. I'll try to remember to keep you all updated.

6. I agree with Michael, the situation will determine the rationalization. I think a context helps them bridge that gap between coming up with a generalization for what happens when you divide a whole number by a fraction vs fraction by a whole. I think the more important part would be why is our answer getting larger than the factors in one case and opposite in another. Difficult concept to grasp.

7. We all had some methods we could pull from to create a context. How do we help students develop that ability? I'm looking for how to have a student see a division problem and think 'grouping' or 'partitioning' or 'how many times?'

8. With some of my weaker groups I've had them drawing halfs until they get 10 'wholes'. Boils down to the same thing I guess, but is a more visual, less word-driven, way of looking at it.

I just watched a Vi Hart video which explains division by 1/2 as counting up in 1/2s: http://www.youtube.com/watch?v=N-7tcTIrers

9. Tina - How about something like "You are making bows and have 10 yards of fabric. Each bow uses a 1/2 of a yard. How many bows can you make?" That is what we use in Investigations using fraction bars and the Ss made good sense of them. Then we moved into generalizations about whole divided by fraction.

10. How many quarters can I get for 10 dollars?
How many half-dollar coins?

11. How about this -- to figure out how to do some ordinary problem, like 12 divided by 2, use a model involving pouring liquids. I have 12 liters of soda, to be divided into 2-liter bottles. Clearly, I can fill 6 bottles. Now try this one: what is 10 divided by 1/2? Answer: 10 liters divided into 1/2-liter bottles will fill 20 bottles!