3. Expanded Area Model for Multiplication

We saw previously that drawing pictures is a good way to think about multiplication.

Multiplying is really just adding a whole bunch of things together, as long as they’re the same size. The little boxes showed that, but it got a little crazy once we used bigger numbers.

So, let’s keep on with the pattern of pretending, by trying out 14 x 7. Since I don’t actually know 14 x 7, I will break it down: 14 is really just 10 and 4. So I’ll split that side length in my picture. Then I have two little rectangles…10×7 and 4×7! I can do that!
(and if you want to get really fancy/technical, we did just demonstrate the distributive law!)

unnamedWow, that was WAY easier than expected! Maybe we can apply that same logic to our original problem?

So…
867 = 800 + 60 + 7
and
39 = 30 + 9

expanded-area-model

BAM! Still annoying…but addition is better than multiplication.

Now…how does this all help with the lattice model? That ugly beast of multiplication…

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2. Area Model of Multiplication

Another way to do multiplication is to think about it as finding the area of some rectangles.

Now, if we draw pictures of area for multiplication, we can start to visualize it! Let’s pretend, over the next few posts, that I can draw squares and rectangles. And that I can take good pictures.

For example, 3 x 2 = 6 in the area model is…

unnamed

Not too bad. But when we get to bigger problems, we don’t really want to draw out every box. It gets tedious even for small numbers, and worse as they get bigger. Who’s going to count?!? Instead, we rely on our basic math facts, and pretend that there are little boxes!

Displaying IMG_1063.JPG

It seems like we’re just reviewing our basic multiplication facts, right? Well, yes.

But the interesting part is in the next post, when we see how larger numbers (bigger than 10) mix with the area model to simplify things.

1. Lattice Multiplication: Introduction

Remember the good old days of long multiplication? You multiply, bring down, and carry. There are so many new methods, and lattice is one. It makes complete sense logically once you understand it, but it is one of the least intuitive structures. Check it out:

lattice

It’s crazy looking, and definitely not natural. Let’s get there one step at a time. Let’s solve a problem using our old-school math, and then figure it out using a few different methods. Eventually we’ll do it using lattice.

Problem:

867 x 39

Traditional Solution:

Displaying IMG_1061.JPG

Not too bad…so familiar and comfortable. Next we’ll look at basic multiplication using the area model.

6b: Following Up When Students Aren’t Meeting the Expectations

We set up expectations for lining up, and most students are on top of it. But there are 1 or 2 students who are just keeping the whole class from earning stars. What do you do? Yell at them? Hold them in their spots in line?

Nope..you should follow up with them. Instead of holding back the class if there are a few students, you can adjust the requirement to 80% of students doing it. That will take the pressure off the students who are struggling. There’s nothing worse than having a hard time, and having everyone hate you for it.


Script:
T: Johnny, come have lunch with me today. We need to check in.

J: But why? I don’t want to! I hate getting in trouble!

T: We need to check in about lining up. It’s been hard, so I want to work on it. It’s not punishment…you can still go to recess.

[after whatever it takes, Johnny comes to you for lunch]

T: Hi. What’s going on in line?

J: I hate lining up. I don’t want to be quiet and just stand there. And XX keeps poking me, because she knows it makes me upset, so I just yell.

T: I want you to be able to stand in line and help earn those stars. What do we need to do?

J: You can move me next to YY! We’re best friends, and I’ll do great there!
T: I don’t know about that…you may talk there too. I can move you to another spot, next to LL.

J: I don’t like LL. I don’t think that will work.

T: Well, LL won’t bother you though. Let’s do this: if you can help earn stars 5 times near LL, then you can earn a separate reward (break, toy, candy, etc). I know you can do it – you control yourself the rest of the day.

J: Whatever.


It may not sound like it’s going to work (Johnny isn’t that excited). Will he self-sabotage? Maybe. But right before the next transition, check in with Johnny. Remind him that he’s working for his own separate reward too. If it works, keep it up! If not, try a few more times.