Addition

Teacher: Who wants to show how they solved 145 + 109? Leah?

Leah: I used the number line. [Leah points to the hops as shown in Figure 2.] I started at 145, hopped 100 to 245. Then I hopped 10 to 255. Then I had to hop back one because it was 109, not 110. So my answer is 254.

Jerome: I did it in my head, not on a number line. I knew that 145 plus 100 would be 245. To add 9, I thought in my head that adding 9 is like adding 10, except you subtract 1. So 245 plus 10 is 255, and subtracting 1 makes 254.

Teacher: Leah and Jerome, do you see anything similar between your two methods?

Leah: My way was easier because it's easier when I can see it.

Teacher: I agree that it can be easier when you can see it and not keep it all in your head. What do you think, Jerome?

Jerome: I think they are the same. When Leah was telling her way, I was like, “I did the same thing!” It's just that I didn't draw the hops. But I added the same way—first 100, then 10, then subtract 1.

Teacher: That's interesting, Jerome. Leah, do you agree with Jerome that your two methods are very similar?

Leah: Yes, I guess I do. But do I need to do it in my head? I get confused when I do that.

Teacher: No, it can be very helpful to jot down a few notes when you are solving a problem. Drawing hops on a line helps keep track of your thinking. So let's write “using the number line” and we'll call that “Leah's strategy.” We can say another strategy is “mental math,” and we'll call that “Jerome's strategy.” I'll write your thinking in a thought bubble to show you didn't write it down.

Teacher: How about 275 + 125? Who can show how they solved that problem?

Nick: I used a paper-and-pencil method, but I can't remember the name.

Teacher: That's okay. Can you show us what you did?

[See Nick's solution in Figure 2.]

Teacher: That's good, Nick. That doesn't take much writing. We will call it the compact method.

Nick: When I got all the zeros, it looked funny, so I looked back at the problem. I guess I didn't have to do it like I did.

Teacher: Could you have solved it differently, Nick?

Nick: Well, now when I look at it, I see that I have 75 and 25 and I already know that's 100. So it would have been easier if I had just done 200 plus 100 plus 100, which is 400.

Teacher: That's good insight, Nick. Did anyone else notice what Nick noticed about the 75 and 25?

Ana: I did. I like to look and see if the problem is like money, you know, like quarters and fifty cents. Then it's real easy to do.

Roberto: I did, too, but it made me think of the number line, not money. I just hopped by 25s.

Teacher: Good thinking, Ana and Roberto. It's always a good idea to first look at the numbers being added, to see if there might be an easy way of adding. Let's write another strategy on the board—thinking about money—and call it “Ana's strategy.” We'll also write “Nick's strategy,” the compact “method,” and Roberto's is the same as Leah's—using a number line.