Teacher:

So for the greatest possible, you might... I don't know. What did someone write for the greatest possible common multiple? Matt?

Matt:

I just put like 6,000 and then little thingies...

Teacher:

That go on and on? Okay. Rob?

Rob:

I wrote, it's hard to tell what the highest possible number is because the numbers go on forever.

Teacher:

Yeah, that sounds like a really good answer. It's hard to tell because the numbers go on forever, right? So you can always find a bigger number. Like if I said, what's the biggest number you can think of? A billion. Well I'll say, okay, I can think of a bigger one, a billion plus one. Infinite. Well, infinite plus three. So you can always add on more numbers. So there really isn't a greatest possible common multiple, is there? It was a trick question.

Rob hit it exactly! But should I have talked so much? It was pretty much like telling the whole class it was the right answer. Maybe I should have taken more answers then polled them. I do not usually like to tell them if an answer is right or wrong, I usually like them to decide.

Teacher:

What about the smallest possible common multiple? Matt?

Matt:

60

Teacher:

60? 60 looks like it's the smallest one. We found all... a whole bunch of them, it looks like that's the smallest one. So when we talk about common multiples...

[pauses to writes two names on the chalkboard]

When we talk about common multiples...

[pauses because two students are talking]

I refuse to try to talk over students.

Teacher:

Talk about common multiples, it doesn't really make sense to talk about the biggest. You talk about about the smallest. And what we call them are least common multiples. Okay? The least common multiples. It's just... you think of all the common multiples, and the smallest one is the least common multiple. Can you see that? [adjusts overhead] Alright.

Now, let's talk about the next question. Find some common factors of 12 and 30. Alright, so let's clear off some room here. [erases overhead] Who used their factor chart to do this? Good job. Alright, so common factors of 12 and 30. Matt?

I like to encourage them to use any or all of their available resources. I think it's valuable that they know that they can use their factor chart to help them. Do they know how to find a common factor with their factor chart?

Matt:

My grandma's address, 1, 2, 3, 6.

Teacher:

1, 2, 3, and 6? Wow, now we know where your grandma lives! Alright, any other ones? Any others? Rob?

Rob:

Uh... I guess not. 15? Nope. I guess not.

I like that Rob was thinking and realized on his own that there were not anymore, but why does he raise his hand when he is not done thinking? Maybe it is a good thing that he feels comfortable enough in class that he can throw out ideas that are not fully thought out?

Teacher:

Common factors. You can look in your factor chart. Kent?

Kent:

6?

Teacher:

6? Oh, we got 6. So did we get them all, maybe? Yes, no, maybe? Yes, no, maybe? Are we missing any? Shake your head like this if we are. Shake your head no if we're not. Not missing.

Since none of the students were looking, I refused to tell them whether or not there were anymore. I want someone to prove it! Sure, it would have saved five minutes for me to just say we found them all, but I think it is important that I do not allow the students to see me as the final say.

Gil:

We're missing a lot.

Teacher:

We're missing a lot of common factors?

Gil:

Yeah.

Teacher:

So maybe we should check our factor charts. Someone have a factor chart out that they could check? [rustling] Patty?

Patty:

Does that mean it goes into both of them?

Good clarifying question. Sometimes I worry that Patty is not fully understanding what is going on in class, but her question here seems to show that she is following.

Teacher:

Yeah, it's a factor of both of them.

Matt:

That's it.

Teacher:

Okay, so maybe that's it. Raise your hand if you think that's it. Raise your hand if we're missing some. Okay, so I think this is it. What is the greatest common factor? Ann?

I cannot wait any longer, it is taking up too much class time. Only a few students are willing to state whether or not there are common factors left unsaid. I am thinking that maybe some just do not want to answer, so I take a quick poll.

Ann:

6

Teacher:

6. Shh. The greatest common factor. The biggest. Jeff?

I should have given more positive feedback to Ann here. I also should have asked her to explain how she knew.

Jeff:

But you're missing a factor there.

Teacher:

What factor?

Jeff:

5

Teacher:

Does 5 go into 12?

Jeff:

Oh you're doing 12 right now.

Teacher:

So we're doing... it has to go into both. It has to be a factor of both. Does that help? Alright? So what is the smallest common factor? Korey?

Jeff was obviously not listening to the discussion, since we just talked about the answer to his question with Patty. Was I too tough on him with my question? I am glad he was contributing to the class discussion though, something he usually does not do.

Korey:

One.

Teacher:

One. Alright. What do we know about the smallest common factor... about any two numbers. Andrew?

Andrew:

Always one.

Teacher:

It's always one. Yeah, so if we pick any... if we pick any two positive whole numbers, the smallest common factor's always going to be one. Am I right or wrong? Prove it. Prove it. Can anyone prove it? That's a conjecture, Andrew's conjecture. Justify the conjecture. Rob?

I like when students throw out conjectures, so I usually name the conjectures after them. It allows us to use the math vocabulary. It also motivates them to want to prove the conjecture more. But did I really allow Andrew's statement to be a conjecture? I think I verified that it was correct first and then asked someone to prove it.

Rob:

Uh... you can prove it cuz one goes into everything.

Teacher:

Ah, exactly. One divides into everything, right? Everything has one as a factor. Sh... it's too noisy. One goes into everything, right? Everything... every number, every positive whole number has one as a factor, right? So one is the smallest common factor. So if one is the smallest common factor all the time, it doesn't really make sense to make it special, right? Cuz it's like, it's like, we know that already.

Again, I jumped too quickly at the right answer. I should have made sure everyone agreed with Rob first.

Teacher:

So what we usually do when we talk about common factors is we talk about the biggest one. They are called the greatest common factor... otherwise known as the GCF. Alright? Okay? Who's, who's worked with greatest common factors and lowest common multiples before? Cool. And it shouldn't be too big of a new thing, because basically what we've been doing all this time is we've been finding common multiples or common factors. But now were just gonna look at either the biggest common factor, which is the greatest common factor, or were gonna look for the lowest common multiple, the smallest multiple. Okay? It's kind of like the Venn Diagrams we had for homework yesterday. Okay? Question... Rob?

I talked too much right here. I was just trying to summarize the definition of the terms we just finished discussing, but student volunteers could have (and probably should have) done that.

Rob:

So you had us do this POD because you wanted to look at, you wanted us to learn about the greatest common factor and the lowest...

Great question Rob! He is on fire today! He made the connection to tie it all together.

Teacher:

Right. It's an intro to the greatest common factor and the lowest common multiple. That's also why we had the Venn Diagram homework. Patty?

Patty:

Uhmm... well, is zero a real number?

Teacher:

Zero? Yeah, zero's definitely a real number.

Patty:

Well, because one doesn't go into zero.

Patty comes up with a great set of questions here! But I think I got so excited by her question that I jumped too soon. Should I have immediately told the class that zero is a real number? Maybe I should have asked them. But would that have spiraled us too far off topic? Her second question, if one goes into zero, is a great one! But I should not have changed the question to the one below. I should have stuck with her question, and then asked if one is a factor of zero afterwards.

Teacher:

Ahh, is one a factor of zero?

Gil:

NO!

Teacher:

What's the definition... someone give me a definition of a factor? Definition of a factor? Yeah, Gil?

Gil:

Uhmm...

Teacher:

Uh oh, uh oh...

I was kind of kidding around with him here. I have a relationship with him where I can do that. But I have to watch that sometimes, because it may lead to embarrassing the student.

Kent:

Gil's not right?

This is funny, because Gil has actually been "off" all period. But he always throws out such great ideas, that the class is used to him coming up with great stuff. It is funny that Kent thinks that it is unusual that Gil is not right when Gil has been wrong twice already today!

Teacher:

Andrew? Andrew.

Andrew:

It ain't out of my dictionary, but... added together any number of times will eventually make the number you're getting.

Teacher:

Not added together, but you do what?

Andrew:

Multiply.

Actually, Andrew's statement with addition does work, but I missed it! Since multiplication is just a faster way of adding, you could think of a factor that way. But would the class have understood if we accepted his addition version of a definition? Would that have taken us too far off the course of our original question? Did he even mean addition? Maybe it was a mistake. Or maybe he was on to something.

Teacher:

Yeah, so, like three times two equals six. So three is a factor of two, right?

Korey:

No.

Teacher:

So using that theory...

Korey:

No, three's not a factor of 2!

Korey made a great catch here. I wish I had mentioned that to her when she said it, but I was too focused on making my point to the class, trying to get the definition of factor.

Teacher:

I mean three's a factor of six. So using that theory, is one a factor of zero? Somebody prove it. The conjecture is that zero doesn't have a factor, or one isn't a factor of zero. Jeff?

Jeff:

Uhmm, I think... [blurred...] two and then zero and one, because one is bigger than the number zero.

This was actually a really great conjecture that Jeff made. How could 1 be a factor of 0 if it is bigger? I should have mentioned what a great observation that was. We had not worked with numbers whose factors were bigger. I should have brought this back up with the class after we had discovered that 1 was a factor of 0, and asked how that could be.

Teacher:

Oh, one is bigger than zero, so it can't be a factor. Debby?

Debby:

What about one times zero equals zero?

Korey:

Yeah, yeah!

Teacher:

Ah, one times zero equals zero, so one must be a factor.

Someone:

So all numbers are factors?

Teacher:

So one is a factor of everything.

I think I misunderstood the question that one of the students threw out. But even still, I should not have just told them. I should have asked the class if they agreed with the statement and had them prove it.

Another:

Every number?

Teacher:

Every number. Bryan? Sam sit down please.

Bryan:

Uhmm... did you know about that? [the video camera]

Teacher:

Let's stay on topic. Okay? Any questions about that zero? Rob?

Rob:

Can one go into minus one?

Wow! He wants to see if 1 really is a factor of any number by proving to himself that 1 is a factor of a negative number. Rob really seems into this lesson.

Teacher:

Can one go into minus one?

Rob:

Yeah.

Korey:

Negative one!

Rob:

Negative one, whatever.

Teacher:

Well what times one equals minus one?

Rob:

Oh yeah!

Teacher:

There's like a, a number there. And we'll get more, go more into that later on in the year, okay? Gil?

I really would have liked to go with this, but I did not think it was the right time to discuss the multiplication of negative integers. We should have talked about the identity property of multiplication, and realized that anything times one equals itself. Based on his "oh yeah", I think he realized the answer as soon as I set up the problem.

Gil:

Well, uhmm, isn't all numbers factors of zero because zero times that number is zero.

What a great conjecture! Are all numbers factors of zero? It is slightly different than our other finding that 1 is a factor of all numbers.

Teacher:

Oh, so is a hundred a factor of zero?

Gil:

Yeah!

Teacher:

So maybe you're right! Another conjecture! Okay! And you guys can kind of come up with those proofs on your own, couldn't you? Okay, so put your POD away, and pull out the homework, because I want to talk about that.

I sometimes like to leave a question unsolved for them, because it shows them that not everything can be solved in a class period. But I wish we came back to it the next day or put it on the assignment for homework. We never really came back to it. I need to start making better notes about these types of things in class.

I was kind of surprised at how much telling I did during class. I had a pretty specific idea of how my class ran, but it was not until I viewed this video tape that I realized that my vision and reality were not the same. I believe that the class should discover ideas on their own. That leaves my role as the facilitator of the discussion, not the confirmer of truth. While I think that I did a fair job of keeping that role, I definitely could have done better.