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 1d
MIA BULJAN: Okay. So Monique, you had this idea, you said what?
STUDENT: Make two 5s, two 5s make a double.
MIA BULJAN: Oh my gosh! If this was 5s: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. You see it now Daniel?
STUDENT: You got 10, 20, 30, 40, 50, 60, 70.
MIA BULJAN: So here she goes. She can put these together.
STUDENT: And then...we need to count to ten.
STUDENT: 20, 30, 40, 50, 60, 70.
STUDENT: I don't actually get that part.
MIA BULJAN: Shh. Okay. So you don't get this part? Monique, when she saw what you did...Monique, tell her again what you thought of when you saw what she did? Oh my God!
STUDENT: Because instead of making like 5, 10, you can make doubles like ...
MIA BULJAN: Luis, answer the phone.
STUDENT: And so you can get like two 5s and make it into a friendly 10.
MIA BULJAN: So she said every time I see two 5s, I'm going to count this as 10 and these two 5s are also...
STUDENT: 10, 20, 30, 40, 50, 60, 70.
MIA BULJAN: They're just going to have to wait one second sweetheart. Eli, go explain to your mama what's going on. Okay. Okay. Hold on. Okay. This is Enmy's way. Enmy's way was using the commutative property and then counting by 5, so you skip counted also. Okay. Boys and girls, the thing, the thing that we have been working on is switching over from adding, adding, adding, adding, adding, to thinking about multiplying. So instead of saying 10 and 10 is 20, and 10 more is 30, and 10 more is 40, and 10 more is 50, on your white boards can you write that down as a multiplication problem? Now listen to my question. You have some groups of 10 here. You have some groups of 10. Think about how many groups of 10 you have and how would you write that as a multiplication problem. Can you do that on your white board? Give it a try. Ramon, turn it over. Eli, everything's okay? And she said yes? Okay. Go ahead. Does everybody understand the question? We have this 10 part, and we know that the answer is 50, but instead of writing 10, 10, 10, 10 over and over again, how can we write it as a multiplication problem? So Monique, how many groups of 10 do you see here?
MIA BULJAN: And so can you write that as ten groups of 7? Okay. So I'm going to say it one more time because I feel like I'm being a little confusing. Marlene did it in 2 parts. She did it in this 10 part and she did it in this 4 part, so I want you to write the 10 part as a multiplication problem. What did she do first with the 10s?
STUDENT: She counted them.
MIA BULJAN: She counted them, but we, let's write them as multiplication instead of counting. Okay. Turn to the person next to you and share your idea real quick. Okay. Why don't you just tell me Daniel, what you're working on here. So how would you write that as multiplication?
STUDENT: 510 and wait, 510 to make it 50, and then you do one to make it 70, and then
20 + 50 = 70 and then you have the answer.
STUDENT: But fives and you can count 14 times and you get 70, the answer.
STUDENT: I didn't actually use my multiplication.
MIA BULJAN: So who can, thinks they can tell us a multiplication sentence just for the 10 part that she did? She had some groups of 10. You can say it in words or a sentence. Uh-huh. Enmy, there are a lot of ideas on that board and I like them. I like the way you're trying to write it as a sentence. Uh-huh.
MIA BULJAN: Why 5 * 10? Everybody can you read this to me in English, instead of saying 5*10, it would be 5 groups of 10. Do we see 5 groups of 10 here?
MIA BULJAN: Let's count them. 1 group, 2 groups, 3 groups, 4, groups, 5 groups. How much was that when we do 5 groups of 10?
MIA BULJAN: Let me ask you this, and let me ask you this because this is why it matters. Do I have to count by tens or do I just know what 5*10 is?
STUDENT: Just know. Know.
MIA BULJAN: Do I just know?
MIA BULJAN: Yeah. So do I have to count that or do I just know 5*10?
STUDENT: You just know.
MIA BULJAN: Okay. So if I think of this as multiplying instead of counting, I can save some time by just saying 5*10 is 50. So was she done with her problem?
MIA BULJAN: So what's the other part of her problem written as multiplication? She had some fours, she had some fours, she had some groups of fours.
STUDENT: She could just add the 10 and 4.
MIA BULJAN: She could add them but we're thinking about multiplying so what did she, how many groups of 4 did she have?
MIA BULJAN: So we're going to write it like this, 5 groups of 4, and Enmy, you have yours written down. How much is the answer to this?
MIA BULJAN: 20. Uh-huh. So you can use ... thank you sweetheart. Really good multiplication sentences. Angel's way, you can use Angel's way of using a known fact if we think of this as multiplication instead of counting by tens. Stop. Since we know these 2 things without counting we can do multiplication faster by thinking of our multiplication known facts. And if it's one we don't know, we don't know 5*14, we haven't memorized that, but we can look for the easy pieces inside there. So today we are going to go to the music assembly, and when we come back I'm going to give each of you your own problem to solve on your tray. Okay? You think that's something you can do?
MIA BULJAN: So head back to your desks and clean up.
MIA BULJAN: I had collected the answers “69,” “70,” “56,” and “65” for the answers to the problem I presented (5 x14). Monique notices, during Enmy’s explanation of how she counted by five 14 times, that you can really count every two fives as a ten. Monique doesn’t know it, but she has invented the strategy of compensation in multiplication. She doubled one factor (5 became 10) and halved the other factor (14 became 7) so she turned 5 x 14 into 10 x 7.
This is not necessarily a strategy I’d expect my students to use yet, or even be able to articulate past this “special case” to generalizing about halving and doubling, or quartering and quadrupling, etc.
However, I did want to capture the idea so we could explore it as a conjecture (does it always work? Sometimes work? Never really work?) and to practice our productive discourse.
In the first part of this segment, as we wind down the strategy sharing, I see that I’m continuing to put more of the discussion focus on the students listening carefully to each other, rather than relying on me to revoice everything. I’m still doing quite a bit of the heavy lifting here, in terms of clarifying and orienting their thinking, but I am also giving them more space to practice hearing each other. When Enmy says she doesn’t understand Monique’s way, rather than revoicing or restating, I direct her to listen to Monique explain herself again.
In the final part of this segment, I ask them to use their whiteboards to write a multiplication number sentence. I like using whiteboards for this kind of experimental work. It’s less permanent, and therefore less intimidating, than pencil and paper work. In the video, a lot of kids are just swiping away their experiments in favor of other ideas as they discuss with their partners. There is still a lot of confusion on those whiteboards at the end of the number talk, but it was great formative information on where to go next. I imagine these students would benefit from articulating in words first, before modeling with their number sentences. Putting into words “5 equal groups of 10” to describe what Marlene had done is the conceptual part that goes with the more concrete number sentence.
Being able to connect these concepts is part of our Sociomathemical Norms. The answer of 70 is important, but it’s not the only math. Being able to connect our ideas, understand and question our classmates, and push our conceptual understanding of multiplicative reasoning is as important.