In this clip, Mia Buljan works with her students on a number talk, using a mental math approach to multiplying a two-digit number and a one-digit number. She encourages her students to identify regularity among the various approaches identified, and guides them toward identifying the commutative property.
Lesson Part 1c
MIA BULJAN: Um, girl, boy, I need a girl. Enmy, what did you do?
STUDENT: I counted by fives.
MIA BULJAN: Oh, tell me a little bit about that.
STUDENT: So I counted, I counted by fives like 5...
MIA BULJAN: How many times did you count by 5?
MIA BULJAN: Fourteen times? Boys and girls if we look at her picture do we see fourteen 5s?
STUDENT: Yes. That's what I did.
MIA BULJAN: Do we see fourteen 5s or do we see five 14s?
STUDENT: Five 14s.
MIA BULJAN: So Enmy, how did you know that you could also do fourteen 5s instead?
STUDENT: Because...maybe it was easier.
MIA BULJAN: So it was easier to count by fives. So you made your problem easier, but does anybody remember the name of the thing in math that lets us switch it around like that?
STUDENT: You can add the 5 and the 5 to make it a double.
MIA BULJAN: Okay. So you're talking about a strategy, I'm talking about an idea. Hold on one second Monique I want to hear about that. Uh-huh?
STUDENT: I kind of did the same thing.
MIA BULJAN: Okay but how did you know you could do that?
STUDENT: Because you could just say like...you could count it by 5s?
MIA BULJAN: Mm hmm. I understand your strategy but I'm wondering how did you know that you could also think of it like this? This is what you did, you did 14 groups of 5.
STUDENT: All the same answer.
MIA BULJAN: It is going to be the same answer but what's that thing in math that gives us the same answer? Andrea. We have it for addition too.
STUDENT: The...that you can, you can do, you can do addition and multiplication in like different ways and switch them.
MIA BULJAN: And switch them. Do we have to do them in order when we're adding and multiplying?
MIA BULJAN: No. Can we switch them around in subtraction? Can we just switch it in subtraction?
MIA BULJAN: Okay, so subtraction does not have this but multiplication and addition do have this. Do you remember the math name for it?
MIA BULJAN: You're going to kick yourself when I write it. You want to hear it? Commu...
MIA BULJAN: Commutative, that's right. The commutative property of multiplication says if it's easier for me to get the answer, once we understand what it's saying, if it's easier for me, I can do it in any order.
MIA BULJAN: Rogelio, how many people counted by fives to solve this problem? Excellent. Enmy, what did that look like for you? Tell me what you did. Chase you're going to have to put that away sweetheart, at your desk, not in your pocket, at your desk. Please don't let me see that again, okay? Go ahead.
MIA BULJAN: Count with her.
STUDENT: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, right there.
MIA BULJAN: How did you know to stop? Who said right there and how did you know to stop?
STUDENT: Because I waited until you get to 14.
MIA BULJAN: Were you counting on your fingers like 5, 10, 15?
STUDENT: 5, 10, 15, 20, 25, 30, 45, 50.
MIA BULJAN: 35, 40... (laughs)
STUDENT: 55, 60, 65, 70.
STUDENT: 5, 10, 20, 30.
MIA BULJAN: Count it out loud so we can hear her. Listen to her count. Do it again.
STUDENT: Uh, 5, 10, 15, 20, 25, 30, 35.
MIA BULJAN: It's probably a parent. Sorry, hold on.
MIA BULJAN: Could you hear her counting?
MIA BULJAN: Okay. Enmy, nice and loud. But listen, listen. Shhh. What? No. Nice and loud.
STUDENT: 5, 10, 15, 20, 25, 30, 35, 50, 55, 60, 65, 70.
MIA BULJAN: Hold on. Did anybody hear her make a mistake in her counting?
STUDENT: Yes, yes, yes.
MIA BULJAN: And she heard it too. So everybody let's count together again.
STUDENT: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70.
MIA BULJAN: And did it work out that time?
MIA BULJAN: I had collected the answers “69,” “70,” “56,” and “65” for the answers to the problem I presented (5 x14). Enmy is the third student I call on to share her strategy. More than half my students used her strategy to solve the problem. Sometimes I feel like our education system is so obsessed with the “basic facts” that we miss opportunities to develop our students’ conceptual understandings. Just because 5 x 14 and 14 x 5 make the same product does not mean that they are the same thing. I want to focus their [the students’] discussion on the math property that lets us switch our factors around (commutative), and I want to make sure that I’m modeling the repeated addition and skip counting for counting by fives so they can visually see how the two things (5 x 14 and 14 x 5) can be compared and contrasted.
Because so many of my students used this strategy, I wanted to focus on the discussion and the practices of being precise and critiquing the arguments of others.
In order to help the students situate their understanding of the commutative property, I am using the Sociomathematical Norm of continuing to ask questions until it makes sense. I first ask how she knows she can do 14 groups of 5 instead of 5 groups of 14, and it takes the whole group to articulate the idea that “it can be switched” or turned around, which I name as commutative. We immediately connect this word to how the commutative property works with addition and how it doesn’t work with subtraction.
When the discussion turns toward why we use the commutative property, I set it in the context of CCMP 2 [Common Core State Standards for Mathematics Standard for Mathematical Practice 2, Reason abstractly and quantitatively]. In other words, once we understand the problem, and contextualize it, we can step out of the context and use any number of strategies to solve for an answer before we step back into the context and see if our answer makes sense. Even though 14 groups of 5 doesn’t match the context we set with Marlene’s visualization of five equal groups of 14, we can articulate, with precision, what properties of the mathematics let us solve it this way.
I ask Enmy about how she knows when to stop counting by fives, and while I’m recording the skip counting on the poster we discuss how she kept track of the number of fives she had used on her fingers. It may seem odd that I appear to be forcing Enmy to count by fives in front of the whole class when she is struggling with it. However, it’s a normal part of the class culture to think about our math out loud, to want to prove and understand; so it’s not so odd for my students to struggle in front of each other. We embrace and applaud that struggle as part of learning. By having her count the fives out loud, I’m asking the rest of the class to listen to Enmy and see if they understand her and if they agree with her. Everybody heard the mistake, including Enmy, when we slowed down and listened to her count. The support comes after we identify the problem, when we all count by fives chorally.