Discussion Sketch, Side 2* 
Time 
Teacher's question or statement 
Student's statement or question 
Teacher's response 
Before the discussion students worked independently but with support from small groups to solve the problem in their journals. 

___ Groups
of 12 = 10 groups of 6
Okay, who has something to say about A? Richard? 
R: I think that (it) is twentytwo. Groups. 
Okay, so
twentytwo groups of twelve equals ten groups of six.
(Writes 22 in blank on board.) 

Can you explain your reasoning about that, Richard? 
R: Because, I timesed twelve and ten. Twelve times ten equals twentytwo. 
Ten times twelve...
Or tw .., I’m sorry,
twelve times ten is like this. (Writes 10x12 in a column on board.)
Is that how you did it?
Okay now I want to remind you that this means twelve groups of ten. This means ten groups of twelve (writes 12 x10 in column). 
The teacher then explains that she encouraged people to draw circles in their journals for each group and write the number inside. She draws 10 circles and writes 6 inside of each. She has the class count by sixes and they get sixty. 

So I have ten groups of six here. Now, Richard, what do you think about this twentytwo groups of twelve thing?
What if I had just ten groups of twelve? How many would that be? 
R: I don’t know. 
Okay, let’s do the top row. (Points to five circles with 12 written inside.)
Twelve plus twelve is twentyfour, plus twelve is thirtysix, fortyeight, sixty. 

Now if the top row is sixty, how many am I going to have all together? Richard? 
R: 120 


120. Now this is ten groups of twelve. Richard what do you think about your idea of twentytwo groups of twelve. 
R: It’s wrong. 


Is it too big or too small? 
R: Too big. 
Okay. So we have to make this lower. 

Could it be ten? 
Class: No. 


Why not? Who can explain why not? Karim? 
K: Because it won’t be like, if you times ten and twelve then you times ten and six, they won’t be both the same answer. 
Okay, so whatever I have on this side has to be the same as what I have on this side. So it seems like it said ten times twelve is going to be a little bit, or maybe even a lot, too big. 

What do you think? Charlotte? 
You could find out that if you tried ten times twelve, that it would have to be five times twelve, because what we did was we counted by twelve the top row and then we added sixty, and sixty and it came out to sixty. Then you know that only the half the ten would be sixty. Half of the ten groups of twelve. 
So you’re basing what you’re saying there on what we just did. 

When we counted all these, we got up to, Donna Ruth, what did we get up to on that top there? 
DR: We got up to sixty. 
Okay, and that is really exactly what we want because we have ten groups of six over here and that’s sixty. 

So what number should go here? What do you think? Yasu? 
Y: Five 
Five groups of twelve. 

Can somebody please explain why it should be five groups of twelve?
Letitia? 
L: Because ten times six is sixty. Five times twelve equals sixty. 
So ten groups of six. The arithmetic we can do for that is ten times six, and the arithmetic we can do for twelve is five times twelve. 

But if you don’t happen to know what five times twelve is you can do it another way. 



Tyrone, can you explain another way if you don’t happen to know five times twelve? What else could I do? 
T: Add twelve, five times. 
Add twelve, five times.
Teacher writes the addition problem and talks through it. 

Shahroukh? 
I think ... The reason it’s correct is because twelve groups of five is sixty. And ten groups of six is sixty too, so they’re both equivalent. 
Okay, that’s good. 

That’s what we’re trying to do here. Equivalent and equals mean the same thing. Another way you could think about this problem and use a drawing to help you think about it is to look at your ten groups of six. How could I make a group of six into a group of twelve? What sort of regrouping could I do here that would help me to think about groups of twelve? 



Donna Ruth, do you have some ideas about that? 
DR: You get two sixes and um 
Okay two sixes. 

Now what do I have inside the brown circle there? 
DR: I don’t know. 


Ileana, what do I have inside the brown circle? 
I: Twelve. 
Twelve. Six and six. 

How many times can I do that here? Altogether? Connie? 
C: Five. 
Five times.
So I make groups of twelve out of my groups of six. Drawing a picture,
it’s not really a picture. It’s sort of a diagram and
I heard you talking about diagrams in science yesterday, helps you
to think through the problem. So when I see you being stuck and
not being able to help make any progress or if I see your thinking
being confused, I’m going to ask you to draw a picture because
that is something that is very important to do in mathematics. 
The class goes on to talk about another problem. 
*Quoted from: Lampert, M. (2001) Teaching problems and the problems of teaching. New Haven: Yale University Press. 