Communities of Mathematicians – Kasman – TDG 2015
Classroom Episode: Does Multiplication Make Things Bigger?
4th Grade Routines, Spring Sequence
During snack, I said to my class, “After you clean up, we’re going to talk about whether multiplication
always makes numbers bigger.” Students started to move toward putting away their lunch boxes, but
Danielle looked me in the eye and pressed her lips together, nodding knowingly. “Yes,” she said.
“Because that’s what multiplication does.”
Teacher: Oh good! We can start with you.
Danielle: Because, like, 3 × 4 is 12, and 5 × 10 is 50.
Amir (overhearing): But 65 × 1 is 65.
Danielle: Oh…right. Except for when it’s 1.
Teacher: Save this! Hold that thought. Put your snacks away and come to the rug.
Teacher: Okay, this conversation got started while some people were still moving around. The question
is “Does multiplication always make things bigger.” (She writes the question on poster paper) Danielle?
Danielle: Well, not always, but usually.
Teacher: Say what you mean.
Danielle: Well, like 2 × 12 is 24 and 3 × 5 is 15, 24 is bigger and 15 is bigger, so usually. But not when
you use 1, because then it’s the same.
Teacher: The same as what?
Connor: It’s just the number you were multiplying by.
Nate: Anything times 1 is 1, because it’s just…1!
Teacher: Okay, so we have ‘yes’ examples and ‘no’ examples. (Writes 2 × 12 = 24 and 65 × 1 = 65
on poster paper.)
Nate: It would be true if you took out the word always. It’s the word always that is messing that up.
Teacher: How would that go?
Nate: Just, “Does multiplication make things bigger?” And that would be yes, but not always. Also, it
wouldn’t work with zero.
Marilyn: I was just going to say that. Because anything times zero is zero.
Students are asked to turn and talk to each other about this for a minute.
Danielle: So, I want to change what I said. Multiplication can never lower itself and nothing can except
subtraction.
Teacher: What?
Akeasia (who had been talking with Danielle): I know what she means. Subtraction is the only one
where a number gets lower than itself, because that’s take-away.
Teacher: The only operation where numbers get lower is subtraction?
Communities of Mathematicians – Kasman – TDG 2015
I was so happy to notice that two of my more struggling students were able to talk about operations and
how they behave – even if they weren’t correct in what they were saying. Students were talking about
operations and comparing them, without me expressly directing them away from specific numbers.
1
Carlos: But what about, like, 2 × 8 = 4?
Many students give the hand signal for “same thought”, and the teacher writes this equation down.
Teacher: This is like the problems we have been doing about a runner running half a mile for 7 days.
This could be a runner running half a mile for 8 days.
Assorted Students: Oh, right!
1
Teacher: So how is that about multiplication, again? Why is 7 times 2 multiplication?
1
2
Carlos: Because it’s 7 groups of .
Teacher: And multiplication is…
Assorted Students: Groups of a number.
Nick: I have a conjecture.
Akeasia: So do I.
Teacher: Okay, Nick, you first.
Nick: When you multiply two factors, it’s impossible for the product to be lower than both factors.
1
Because, 2 has to get bigger because there are 7 groups of it. It can’t be lower than itself. Unless it was
zero. But then zero would not be lower than itself. In multiplication, the product could be the same as a
factor or more than a factor, but not less than both factors.
Teacher: Ooh! Let’s write this down!
I was enthusiastic that “conjectures” were live in the class as well – without me asking for it, this is what
students are now geared towards thinking about. I knew right away what Nick meant, and wondered if
and when a fraction times a fraction would come up. Nick’s conjecture is true to every multiplication
situation he has been exposed to. I wondered, would I be the one to suggest 1/2 × 1/2? That would help
Nick along with his idea, but would the rest of the class, just getting used to fractions, have a way to work
with this? Could I make up a story? It couldn’t be a twist on the story they were used to – running half a
mile in half a day – too unreasonable. A half of a half of an apple is a quarter of an apple. But that
doesn’t sound like multiplication. I would wait. I would think of a story if I could, and bring it up later if
the time was right.
Akeasia: I have a conjecture. Remember?
Sara: I have a conjecture too.
Teacher: Akeasia?
Akeasia: If you add two numbers together or multiply two numbers together, multiplying the two
numbers is always bigger, the product is always bigger, than adding the two numbers together.
Teacher: Does everyone get that? Akeasia is saying that the product of two numbers is always greater
than the sum of the same two numbers.
Nick: Except 2 × 2 and 2 + 2 are the same. They’re both 4.
1
1
1
Teacher: And 10 + 2 is 10 2, which is bigger than10 × 2, which is 5.
Akeasia: Oh.
Communities of Mathematicians – Kasman – TDG 2015
Teacher: You could still work on that conjecture, Akeasia. Maybe there are certain conditions where
what you are saying is true. Do you know what I mean?
Amir: If you multiply numbers bigger than 2 × 2.
Teacher: You could work on that. Sara?
Sara: My conjecture is, if you multiply a fraction by a number the product is that fraction of the number.
Teacher: What is she saying?
Amir: I get it. Multiplying a fraction by a number is the same thing as that fraction of the number. Like
1
2
× 8 is half of 8. That’s how I do multiplication with fractions.
Several students give the “same thought” hand signal.
Teacher: Do other people get that? (Mixed nods and shakes) Okay, so we have 3 conjectures and an
original question to come back to. Does multiplication always make numbers bigger?
Class: No!
Teacher: What are all of the situations we talked about? We talked about anything times 1…
Class: Is itself.
Teacher: Anything times zero…
Class: Is zero.
Teacher: And what about fractions?
Less unanimously: Less.
One student: Less than the number that’s not a fraction.
Teacher: Danielle, you’re an amazing thinker because I’ve watched you change your mind so much in
the last 10 minutes. What do you think now if multiplication always gets bigger?
Danielle: Well, it doesn’t.
Teacher: Not always. But there was something right about what you were thinking in the very
beginning that made sense. When you thought that multiplication always made numbers bigger, you were
thinking about just certain kinds of numbers, and for those numbers, you were right. Do you know what
kinds of numbers you were thinking about? Because it’s important.
Danielle: Regular numbers.
Teacher: Yes! There’s another name for those numbers…
Nate: Whole numbers?
Teacher: Yes! So if our conjecture had read…Does multiplying whole numbers…
Nick: Not counting one…
Teacher: Greater than one, always make things bigger? What would you all say to that?
Class: Yes!
Teacher: And a whole number times a fraction…
Class: Less than the number you started with.
1
Zoe: This isn’t for a number like 2 2, though, is it?
Teacher: What do you mean?
1
Zoe: Like, 2 × 2 2?
Teacher: What do other people think?
Amir: It’s not a fraction of 2 because it’s 2 × 2 plus a fraction of 2.
End of Excerpt