In this number talk, Mia Buljan engages her third grade learners with a one-digit by two-digit multiplication problem, 5x14. She comments that her students have shown that while some numbers are easier for them to decompose (for example, tens and fives), with other numbers it is more difficult for students to mentally identify “friendly” numbers to use to help solve the problem. She invites students to share the different solutions they found, and then defend their thinking.
In a debrief with coach and colleague Erika Isomura, Buljan notes that ambiguity of language can make this kind of mental math particularly challenging for learners: do they really understand the difference between three groups OF 36 and 36 composed of three groups?
MIA BULJAN: Okay. So Monique, you had this idea, you said what?
STUDENT: Make two 5's, two 5's make a double.
MIA BULJAN: Oh my gosh! If this was 5's: 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 5's and make it into a friendly 10.
MIA BULJAN: So she said every time I see two 5's, I'm going to count this as 10 and these two 5's 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 10's?
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. Amy, 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 510? Everybody can you read this to me in English, instead of saying 510, 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.