– Boys and girls, today’s lesson,
we’re going to start off with quick images so you need to have your
math notebooks out. Let’s date it for September 14th. We started quick images
just in the beginning of this week. Remember I’m going to show it once,
five or ten seconds, and I’m going to hide it again,
so you need to really visualize what you see and draw what you see.
I’ll give you a few seconds. I’ll show it again,
I’ll hide it again. Again, you’re going to add
to your visualization of what you see. Then I’ll show it a third time
get it to relate to math, math equations. And some people think,
we’re going to probably have different ways of viewing it,
– Okay, here we go. First time. Let’s look at it again. Are your eyes ready?
I don’t want anybody to miss it. Can you think of an
equation that might go with this? How did you know how
many there were? Can you circle your three groups
of six so I can see it even better? Okay,
let’s see what you got. Thumbs up if you got it exactly right. Thumbs up if you were really, really,
really close and you are on your way. So I know sometimes it takes us
a little bit longer than the time I can give you. Folks, let’s talk about how many this is
and how you visualize that with your math. Jose, what did you do? – I put 3×6=18. – You got 3×6=18. Can you come up here and circle
how you visualize 3×6=18, please? Thumbs up if that’s what you saw. So we got some thinkers
that are all thinking the same. Now he’s showing us,
what’s he showing us? What’s he showing us?
Ethan? – Groups of three? – Say it again, please. – Groups of three. – Groups of three.
And how many groups of three did he get? – Six.
– Six, good job. Thanks.
Anybody see it another way? Anybody see it another way?
What do you, how do you see it, Tiandra? – Two, 9×2? – 9×2.
And what did you get for 9×2? – Eighteen. – 9×2=18.
Can you come up and show it, please? Using a different color
marker so you could see it. Did anybody think like Tiandra? Oh, okay.
I see how you connected it. Nine groups of two. Did anybody do the
commutative property of that one? Because I didn’t really think
nine groups of two. I thought of it as,
what’s the commutative of 9×2? – 2×9. – Say it again.
– 2×9. – 2×9 and I’ll show you what I did.
Did anybody think of it like that? 2×9, two groups of nine? Come up, Jacob,
and show us how you saw it that way. We know that these mean the same thing. We’ve already talked about that,
but circling them might show it in a different way. Show me your two groups of nine. Good job, good job. Another way?
We’ve got another marker. We can do one more way
and then we’re going to go on with our math lesson for the day.
Another way. What did you get?
Kiera is going to show us what she’s doing. Good. So you and Jose were
doing the commutative of each other. The flip flops is what
kindergarteners call them, right? But not in fourth grade.
So a lot of us saw it as looking at groups of. As groups of. If I was going to put,
and this is just going to extend a little bit, if I was going to put all of these together,
where’s that bigger one? Well, got the little eraser out. Does that look big?
Aha. – That’s big.
– That’s big. It’s a lot quicker. If I was going to put all of these together
and make, what am I making if I put all of my rows and columns
and I fit them together, what am I making? Trent.
– An array? – What am I making?
– Array. – An array.
What would the dimensions be? If I was going to slide them all together? – Six by three. – Good job.
Come and show us. And then label where your six would go.
Good job. Cool. Cool beans,
that’s the new word. Cool beans. Good job.
Good job on quick images. We’re going to do those.
Why are we doing them? We’re going to keep doing them.
Why? What’s it going to help us do? Especially us that still are
trying to memorize some things. What’s it going to help us do?
Matthew. – Sometimes it can help us count better,
count faster. – And what’s counting
have to do with arrays? – It can help you learn times. – Exactly.
Very good. Exactly.
All right. You can keep your notebooks open
because you’re probably going to need them, but I need your pencils down
and all eyes up here on our SMART Board. We’re going to do a think-pair-share. We’re going to do a lesson today.
It’s called multiplicative comparisons. I want you to think
what does that mean? And I’m going to even
give you a little bitty hint. You all know back here on our cafe
menu how to figure out what words are. You’ve given me these ideas. How to figure out what they mean.
We’ve done nifty, thrifty, fifty. Talked about prefixes, and
multiplicative comparisons mean? You’re thinking. Pair up with your
partners sitting around you, talk about it.
What do you think it means? – Multiplication and
it talks about multiplication and then compare, when you compare the multiplication together. – Multiple,
multiply, what about comparison? Comparison, what’s the word in there? – It’s like it’s to say,
now what are you thinking? – I’m thinking kind of the same thing. – What did you say that she agreed with? – I said multiplication comparison…
– Comparison. I know it’s hard. – It’s where like, you both
get them like, steady for each one. You have to like, flip flop them out. – Steady?
multiplicative comparison? – Like 7×3 and 3×7. – Okay. So you think it kind of means
flip flops there and you said that, the commutative property. I love the way you always
try to give examples to help me understand. That helps me.
– And you could still compare them. – Yeah, and you’re
comparing those two. Okay, what do you think? – Well I’m thinking it could
probably like, mean like, it could be a different array. – A different array? So you’re going to
compare one array to a totally different array? – Yes.
– Okay. Does anybody want to share their thoughts,
their ideas? Keana. – Me and my partner said
that multiplication… – Compare,
multiplicative comparison. – Multiplicative comparison means,
we think it means that you compare what
multiplication, multiplicate? – Ethan, you can help her out
if she wants your help. – We believe that that means that you
compare one multiplication equation with another. – Okay. Okay.
That’s a good idea. Good thinking.
What do you think? – I think that you have to put
two numbers together and see what it equals. – Any two numbers?
– No. – And you said see what it equals.
Well, let me rephrase my question. What are you going to do with those
two numbers when you put them together? You got to tell me what put together means.
Because we can do a lot to numbers. – Think of your word.
– Yeah, think of the word. Do you see any words,
small words that jump out at you, in that big long word,
multiplicative comparison? – Compare. – Compare? Okay, but you were talking
about doing something to two numbers. Do you see another word up there
to those two numbers besides compare them? – Share? – Share them?
What are you thinking, Trent? You have your hand up in the back. – I was thinking that it might,
you might have to have two numbers and you have to find the multiples. – Okay, and why did you say
two numbers with multiples? Why did you use the word multiples? – Because it says multiple. – So you’re chunking,
you’re seeing a little word in that big long word? That’s one of the things
to be said about accuracy. That’s one of the things we’re learning
about nifty, thrifty, fifty. Dejaney. – Me and my partner thought
you take two different multiplication equations and see if they equal the same. – Okay. And I will have time for one more
and then we’re going to go on and we’re going to see
if what we were thinking is really what we’re going to be doing
with multiplicative comparisons. One more.
How about Olivia? – Sure. Multiply means
multiplying by numbers. – So you think multiplicative
means multiplying numbers? We’re going to compare them? – Sure. – Sure.
Okay, we’ll try. All right, let’s focus back.
We’ve done our think-pair-share, and I want to show you a problem
that is an example of multiplicative comparison,
and it sounds like this. Tiandra picked seven apples. Reese picked four times as many apples.
How many apples did he pick? Before I even start to think about it like math,
I think about it like reading. Who are the characters in our story?
Who are the characters in our story? Kayleigh.
– Tiandra and Reese. – Tiandra and Reese.
What are Tiandra and Reese doing? Shayla. – They’re counting apples. – They’re counting apples.
Do you think they might eat them? What can you infer they
might do with them besides eat them? – They might plant other apples. – Oh, okay. They might plant the seeds out of them
and make other apples. Cool. All right.
What is your problem? Because in math the story problem
becomes your problem. So what do we need to
figure out here? Alex.
– How many apples they have? – Do we need to figure out
how many apples they have? Do you need me to re-read it for you? Okay. It says Tiandra
picked seven apples, Reese picked four times as many as apples.
How many apples did he pick? So who are we looking to see? – We’re looking for Reese. – We’re looking for Reese’s apples.
Good job. Good, good, good job. All right, hands down.
I’m starting to learn how to use this SMART Board. You are going to think this is pretty cool. Your third grade teacher
may have already showed it to you. I am going to show you
the apples that Tiandra picked, okay? You can count with me.
One, two… – Three, four, five, six,
seven, eight. – Oh, what’s this say? – Seven. – Exactly. I’m going to make this
page just a little bit longer, all right. Seven apples. Tiandra,
did you have fun picking your apples? – I like apples. – You like apples,
cool, all right. Reese comes along and he,
the story says he picked how many more or
how many times more? – Four. – Four times.
Who wants to show me the one time? The one time.
One time as many as Reese. Jacob, all you got to do is,
your apples are going to be red, Reese. Tiandra, yours is yellow, so you’re going to
show me one times as many and this is a, well,
we’ll see what happens. – One, two… – Three, four,
five, six, seven. – Good job.
Are you done? Good job.
That for, let’s get back because we’re in the thing. This shows that one
times as many as who? – Tiandra.
– Tiandra. That’s one, okay? All right, Trent’s ready.
Jacob said he’s ready. So Trent, you’re going to show us
two times as many but you got to use
do your thing. – One, two, three,
four, five… – Just click off of it.
There you go. – Five, six, seven. – Are you finished?
He says, yes, I’m finished. Tell me why.
I said show me two times as many. – Because I put seven. – Show me two times as many. – I did seven apples. – You did seven apples?
Is that a group of seven apples? – No. – It’s not a group? – Or yeah. – You tell me.
You think it is a group. Why is it considered to be a group? – Because I’ve put seven apples. – And you put all, we can put a circle.
Did Jacob make a group of seven apples? – Yeah. – And what did I ask you to show? – Show me two groups of seven. – Did I say show two groups of seven? – No. – You are interpreting that,
and that’s exactly what I want you to do. That’s exactly what I want you to do.
But what did I really say? – Show me two times. – What did I say? – Show me two times. – I said show me two times,
but listen to their language. They keep saying,
two groups of. Are those synonyms maybe? – Yeah. – Maybe, think about that. Jose, you want to show me
three times as many? I’m going to extend
four, five, six, seven, eight. – Oh, you caught yourself.
Why did you go back? – Because I heard eight
and I wasn’t paying attention. – Heard eight
but he wasn’t paying attention. He must have been
paying attention because he stopped. He didn’t pay attention and he would’ve kept going,
so you did pay attention. So tell me what you got here,
Jose. – I got 21. – You have 21?
How do you know? – Because I know that 7×1=7,
7×2=14 and if you have seven more than 14, you’ll get 21. – Cool. Because you got,
I’m going to say do you have seven, or you have, how many groups up here?
That’s what I want to say. How many groups do you have?
– Three. – Three.
So could I say you have three groups of seven? – Yes. – Does anybody know the synonym
of what I asked him in the beginning to do? What did I tell him? I didn’t say come up here and show me
three groups of seven. What did I say? Giovanni. – Three times seven. – I did. I said show me
three times as many, didn’t I? Good job.
Jose, can you pick someone? Dejaney, I need you to show me
four times as many using already what they have. – One, two, three,
four, five, six, seven. – Are you finished?
So can you explain what you have up on the board
using all your peers’ stuff, too? – Four times seven. – You have four times seven.
Do you have four times as many as Tiandra? – Yes.
– Yes, you do. So tell me, are we finished
with the story and how do you know? Think.
Are we finished with the story? How do you know? I’m going to read it out loud
because I think some people might need that. Tiandra picked seven apples.
What color are her apples, everybody? – Yellow. – Good.
Reese picked four times as many apples. How many apples did he pick?
Are we finished with our story? Did we solve our problem?
No? Talk to me about that,
talk to us. – Because we have to
figure out what it equals. – Oh, have we done that? – No. – How many of you think
we’ve shown what it equals? What should go with our picture
to make it very clear that we have an answer? We’ve got some equations up here.
We’ve got a visualization, a picture up here.
What else would even make it better? – Total? – What is the total,
Kayleigh? – Twenty-one,
I mean… – Kayleigh,
what were you going to say? – Twenty-one. – Twenty-one. Jose, did you say 21
when you had three groups of seven? – Yes. – Okay. Now there’s
four groups of seven. So what do we need to do? – Add 10. – It’s four groups of seven.
What do we need to add to the three groups of seven? – Seven more.
– Seven more. Get it going. I’m waiting.
We’re waiting. – Twenty-eight? – Very good.
Twenty-eight what? – Twenty-eight apples. – Why should we label them?
Why should we label every answer in a math story? – So the person knows
what you’re talking about. – Absolutely.
What’s the story about? – Apples. – Well we don’t want to put oranges.
– No. – We don’t want to put raindrops.
– No. – We want to put apples.
So if I was going to answer the question, how many apples did he pick?
Using our visualization, the manipulatives on the SMART Board,
using the math equations that you gave me, what will we put for our answer? Oh, every hand should be up,
because good listeners should’ve heard Kayleigh. She worked hard on that. – Twenty-eight. – Ayana. – Twenty-eight. – Twenty-eight what? – Twenty-eight apples. – Very good.
Twenty-eight apples. That is an example
of multiplicative comparison. Are you changing your predictions
a little bit? Are you still going, huh? How many of you are like that, huh?
It’s okay if you’re like huh. Go ahead.
Don’t be shy. How many of you are still wondering,
I need more help, I want to see it again? Well, guess what?
You’re going to see it again, because I have a feeling
most of you are going huh? It’s brand spanking new.
I want to show you another problem, okay? It’s an example of
multiplicative comparisons. Think about what you were
predicting it was all about, and let’s see if you change your mind. All right, here’s the thing. Go back to what you were thinking
multiplicative comparisons was all about and I want you
to focus on the repeated words. I love it, because repeated words is
what we were doing in writing. Focus on the repeated words.
Aliya’s little sister is two feet tall. Her dad is three times as tall as her little sister.
How tall is her dad? Tiandra picked seven apples.
Reese picked four times as many apples. How many did he pick? Are there some repeated words
in both of those problems? Are there some repeated words
in both of those problems? Are there some repeated words? Aaron, are there some repeated words?
– Yes. – Yes.
What do you see? – Picked? – Picked?
Do you see pick? Oh, okay, I see what you’re doing.
I want to know, yeah, pick is in here twice,
but I want to know if there are some repeated words in both stories.
You’re absolutely right, so let me rephrase it.
Comparing both stories, are there some repeated words? Would you like some help or you thinking?
More time to think or help? – Help. – Help.
Tiandra? In the white striped shirt,
Reese? The gray shirt.
Oh, Ayana. – Yes.
– Where are they? – Little sister and apples. – Little sister and apples.
Are they the same words? I want the words in this one
to be in this one. You are picking
repeated words in one story. – Times. – Times.
Is that all? – And. – There’s just one little bitty word
that we should probably be doing. – How. – No, I’m not thinking that.
One that kind of goes with times. Her dad is three times
as tall as her little sister. Reese picked four times as many.
A little bitty word. – As many? – Say it again. – As many. – Times as,
times as. This one’s times as tall;
this one’s times as many, okay? So think.
Go back to your think-pair-share. Think about multiplicative comparison. – Multiplication comparison?
– What does that mean? – Still I’m thinking it’s where you can,
like the communative and try and make them like balance those,
and I guess that’s all. – I think it’s the same thing,
for you could keep them balanced. – Like a story problem, and then it can like help you
with story problems with math, about counting and multiplication. – I think it’s actually like where you can,
I’m like trying to chunk it, like chunk it so like, it’s in the word like compare it.
Like compare for, I don’t know, balancing. – Do you have anything else to say? – I think we can share, pair. – Anything?
– I don’t know. – Wait a minute, when I look at
the picture, I see two on this side and six on this side,
So what do you mean? You’re going to have
to explain balance to me. – Like, you just said
they’re commutative. – Oh, so you’re saying ‘balance’ for
another word for ‘commutative’? Gotcha. Okay, give me five. Does anybody want to share their thoughts now?
Because I know in the beginning, a lot of people were saying
we’re going compare multiplication problems. Is that really what we did?
Did we compare multiplication problems? Yes or no?
– Yes. – Thumbs up if you think we did
compare multiplication problems. Okay, explain why we did that.
How we did that. We were comparing multiplication problems.
Can you explain it? Do you have evidence to explain it? Matthew. – We compared two words
that were both in them. – We compared two words in them. Did those two words
have to do with multiplication? – Yes. – Did we compare multiplication problems?
Because if you said we did, you’ve got to prove it
or I’m not going to believe it. So you people that
just raised your hands, you better be thinking
about how you’re going to prove it. If you can’t prove it you might say ugh, now I don’t know so much if
I really believe that’s what we did. I agree with you.
We compared the problems, the word, the story problems
and found a repeated word. What do you think we really did? Because when I think, and
maybe I’m thinking different than you, but when you all say we
compared multiplication problems, I’m thinking I compared seven times four
to two times three. Did I do that? – No. – No. Well, that’s what I’m thinking,
because that’s what that means to me. So who can tell me what they think
multiplicative comparison means and prove it with evidence,
just like when we infer in reading. We must prove it with evidence. Ethan, nice and loud. – I believe that multiplicative comparisons
means when you compare the two factors in a
multiplication equation. – Oh, say that again
nice and loud. – I think it is when you compare
two factors of a multiplication equation. – So, I’m going to bring up this one
and you talk to us about what you just said.
Like what would be the factor? Come on up. All eyes. Ms. Pratt’s the student,
Ethan’s the teacher. – Factors would be four and seven. – How do you know that? – I know that because 7×4
would equal 28 and… – Did our picture show that?
Did our picture show 7×4 or the commutative 4×7? Did it? – Yes. – Can you circle that so people can see
where it says four groups of seven or seven groups of four?
Everyone look at it. You’re good.
Why didn’t you circle that one? – Because that is the,
is the number we’re multiplying. – Yeah, it is.
And who’s that about? – Tiandra. – Tiandra.
And what is the rest of the story about? We’ve got to use this information,
but what is the question focusing on? – It’s focusing on
how many apples Reese picked. – Which one of these
colors shows Reese? – Red. – Yeah.
So he said, and let me restate it, and if I’m not saying it right,
you help me, because sometimes kids understand
your language better than they understand my language because they’re
used to hearing their peers speak. So you’re saying that
multiplicative comparisons are when you’re really comparing the
two factors in a story to get a product? – Yes. – Go back to your seat.
Good job. To sum it up I want to look at the different ways
we can say 4×7 equals 28? And we know the commutative property, 7×4 is 28.
Let’s look at the next one. I wonder if we can come up with a, we could say it this way.
Let’s say it. – Four groups of seven
is twenty-eight. – We could say it this way. – Seven groups of four
is twenty-eight. – We could say it this way. – Four rows of seven
is twenty-eight. – We could say it this way. – Seven rows of four
is twenty-eight. – I’m going to show you this
and I want you to read it carefully, because there’s another one after it. I’m going to show them to see if you
can guess what it is. We could say… – Twenty-eight is
four times as many as seven. – Or? What could the other way be?
No shouting out. Trent, what are you thinking?
Loud. – Seven times as many,
seven is 28 times as many as four. – Does that make sense? Is seven
twenty-eight times as big as four? If you think about the pictures,
that means we have twenty-eight fours. Is twenty-eight fours going to give us seven?
– No. – Look at how it’s written. Twenty-eight
is four times as many as seven. – Look at your factors there. – We switched out factors
so what are we going to switch out here? – Seven and twenty-eights? – No. Is 28 a factor?
– Got to think of factors. – Seven. – And…? – Four? – Yeah, because 4×7 is 28,
so we switched them out. We got to switch them out here.
We’re getting the commutative. We could say… – Twenty-eight is… – That’s what that said.
– Change your factors. – Twenty-eight is
seven times as four. – As many… – As many as four. – Say it really loud everybody. – Twenty-eight is
seven times as many as four. – These are groups.
This is what we learned in third grade. This is an array.
This is what we learned in third and fourth grade. And this is called what?