PA Coalition for World Class Math

                                                 PDE Exemplary Instruction

 This is a written transcript of the four high school math videos provided by PDE to illustrate effective instruction in high school mathematics.
 HS Math Video 1

Ms. Canady:  Number three's you are going to take out of your team folder the laminated coordinate plane sheet.  Number ones you are going to remove from the folder the overhead marker.  Number two's you will be the recorders for you teams.  You have two minutes to complete this introductory activity. You may begin.

Student:  Within your team label the blank coordinate plain below.

Student:  So which one's this one? What are these?

Student:  This is the Y and this is the X.

Student:  Then we have no, we know where our positive numbers go. So guess where our negative numbers go?  Awesome, good. Nine, eight, negative seven, negative six, negative five, negative four, negative three, negative two, negative one. Now in correspondence where do, where do our next set of positive numbers go? Up or bottom?

Student:  Up.

Student:  Up, very good.  One, two, three.

Ms. Canady:  Mathematicians, number one, tell us about the X axis. Is it horizontal or vertical?

Student:  It's horizontal, and the Y axis is vertical.

Ms. Canady:  Math genius's, number one, tell us what and where is the origin.

Student:  The origin is in the middle, and it's the perpendicular lines that intersect to the Y and the X axis.

Ms. Canady:  Ok, you did very well on the introductory activity and that activity helps you to build upon skills that you learned earlier in your schooling and those skills will also help you with today's objective, which is graphing linear equations. Linear equations, where would we be able to use linear equations in the real world? Let's do a Think Pair Share.  Think.  Talk to your partner.

[Conversation]

Does anyone else have anything to share, what you talked to your partner about.  Alexandria.

Student:  We all concluded that we can relate to linear equations by the, a roof on a house, like, it's just like the point of it, like on the y and the x axis.

Ms. Canady:  Would anybody else like to discuss what they talked about with their partner?  Jung Li.

Student:  My partner and I had talked about using graph and linear and mapping, how you search for people on a map, like when you open it there is going to be coordinate and numbers and there's way to get up there, turning left from right [fades out]...

Ms. Canady:  Ok, you guys did great with that Think Pair Share activity, but let's just review for a couple of minutes, a couple of words.  Let's review what a linear equation is. A linear equation is an equation whose solution is a straight line.  When we talk about slope, we're talking about the steepness of that line, and a couple of you when you discussed what you talked about with your partners, you talked about things like rooftops and you talked about things like pyramids and that goes with talking about slope.

 HS Math Video 2
Ms. Canady:  I'm going to do a think aloud, and we're going to do a graph of this equation.   One of the ways that we can do this is by finding the X and Y values that satisfy this equation, plotting the points and then drawing a line through those points. We can use any values that we like for X. Ok, I'm going to give X a value of one, and then I'm going to work the equation using my X value of one to get my value for Y.  

Now there's another way that we can graph this equation this time you will help me.  The way that we're going to graph this time is we're going to use your knowledge of slope and Y intercept.  Ok, and this is the formula that we're going to be working with.  What you should know about this formula is that in this formula, M equals your slope and B equals your Y intercept.  You all were talking earlier about carpenters and rooftops.  What do we mean by the Y intercept? Let's do a Think Pair Share.  Think.  Pair.

[Conversation]

Ok, Share. Ok, somebody from the difference team is going to tell us what we mean by Y intercept. Difference team who is going to talk to us. Taiji.

Student:  The answer is when the linear equation crosses the Y axis.

Ms. Canady:  Very good.  Well let's do a Think Pair Share.  Think.  What is it that we already know?  Pair.

[Conversation]

Ok, and let's share.  Somebody on the polygon team.  Polygons, who wants to talk to us?  Tell us what we already know.

Student:  The slope and the Y are the same.

Ms. Canady:  Ok, let's deal with the Y intercept first. We already know the Y intercept. How do we know that?

Student:  Because that, they line up, it line up under B.

Ms. Canady:  Ok, so we've plotted our ordered pair on the Y axis for this equation. What else is it that we need to know according to the formula?

Student:  We need to find the slope, which is, our Y intercept is zero to two. So our slope is equal to one half. So we already have zero to two.

Ms. Canady:  Then we're going to plot our point here. Yes?

Class:  Yes.

Ms. Canady:  Ok, very good.  Then we are going to draw our line through the points.  Yes?

Class:  Yes.

Ms. Canady:  Ok, and then if you notice that these lines are both the same. Yes?

Class:  Yes.

Ms. Canady:  Ok, so that shows you that there's more than on way to graph a linear equation.
 HS Math Video 3
Ms. Canady:  Now you're going to graph linear equations as a team.  You have ten minutes to complete this activity.

[Indiscernible]

Student 1:  So where, so how will we graph it?

Student 2:  Like the B, we have the B, the B[Indiscernible] the Y equals side.  So I think you should start it from right here, the three. [Indicernible]

Student 3:  What about another, the other point?[ Conversation]

Ms. Canady:  Ok, and what about this one?  Explain to me what you did here.

Student:  This one, the Y intercept was negative three.  So we went down three spaces and then we did rise over run and we went over one and up four.

Ms. Canady:  Ok, look at your rise over run.  Your denominator, your bottom number that's your run, right?  So you're going to run over four, right? And then your numerator, your rise number is one.  So you're going to rise one, right?

Ok, now I want you guys to go back and check using the method that I showed you before, alright?  Ok, on the back of your paper, there is a problem that you are going to solve on your own, on your own. If you get stuck, in your team folder there is a pink page.  On that pink page is the answer to this problem.  Use it if you really need to, but work as much as you can without using the pink page.  Ok? Any questions?  You can begin. Ok, now what you're going to do is take a few minutes, talk to your teammates about what you did on your Team Mastery exercise.

Student:  Ok, now I need to find another coordinate.  So I will substitute it for zero.  So it will be zero, four. Start at zero, went up to four, went back to the half, went over to the two, and rose for one.  That’s the way I got it, but I'm sure there's another way she taught us, [Indiscernible].

[Indiscernible]

Now is everybody clear, is everybody clear on both ways to [Indiscernible].

Group:  Yeah.

Student:  Ok. Ok, so you see our lines [Indiscernible].
 HS Math Video 4

Ms. Canady:  Let's have someone come and show us how you solved the problem.  Ok, let's use random reporter, and let's talk to the polygons, number one.

Student:  Alright, so we know that our, to find the coordinate plain below, our formula is first, is Y is equal M X plus B, and to find our slope, M is our slope.  To find out slope is rise over run.

Ms. Canady:  Thank you Jung Li, good job.  Does everybody agree with Jung Li's answer?  Give us a thumbs up if you agree.  Ok, Katheryn.  I see that you're giving us a thumbs down, tell us what's wrong.

Student:  Why is there a zero in front of the four if the four already crosses the Y axis?

Ms. Canady:  Actually you're right, we really don't need the zero, if we know where the line is going across the Y axis.  Ok, two questions.  The first question is this, which method to solve the equation was easiest for you? Was it the first method where we found the values of X and Y and put them into the equation,   or was it the second method where we worked with slope and Y intercept?

Which method was easiest for you?  The second question is this, would we be able to determine how steep this line would be from the information that we have here?  Ok, let's do a Think Pair Share.  Think
[Conversation]

Student:  Y, M X plus B.  [Indiscernible].  All you have to do is plug in the number. It has the data, the data that you're given, and then all you have to do is put this on top and then you know [Indiscernible].

Ms. Canady:  Ok, let's have someone from the polygons.  Share your responses with us please.

Student:  Y equals M X plus B is easier, and you can determine how steep the slope is by looking at the numerator because that's the rise over run, and rise is on top.

Ms. Canady:  Ok, very good.  Thank you.

Student:  You're welcome.

Ms. Canady:   Each person on the team should have an exit sheet.  You are going to complete the exercise on the sheet.  When you are finished everything goes into your team folder and you team folder is closed

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                               And Now . . . the Middle School Lesson (from PDE)

  Middle School Math Video

Ms. Gorman:  So today what we’re going to look at is, we’re going to see if you can answer this essential question by the end of the period.  And that is, how do we solve a two step equation?  And I have three goals for you. By the end of the period it is my hope that you will be able to, identify the two operations that are present, identify the inverse operations that would be needed, and then to solve for the variable showing me seven lines of work. 

So that’s what are goals are today.  Ok?  But first, my cousin’s birthday is coming up this weekend, and he’s actually going into seventh grade.  Ok.  And I thought what better gift could I give a student going into seventh grade than a scientific calculator?  Ok.  So I thought if you don’t mind I’m not very good at wrapping gifts, so if you don’t mind I’d like to get a volunteer to wrap this gift for me.

Erica why don’t you come on up?  Ok now what I’d like you to do is on the packet that you picked up when you came in, at the top at the left side, it says official operating procedure for wrapping a gift.

So in a moment when Erica starts wrapping the gift, I want you to watch her carefully, and just make bulleted notes about the order in which she did things.  Ok?  Alright Erica, go ahead.

Wonderful.   Erica let’s see how you did.  Thank you so much.  Let’s see, oh very nice.  Good edges, thank you very much.  Alright so here’s the gift I can give my cousin this weekend.  And do you think he’ll like it?  You think? 

Well, course maybe I better test it out, and see how my gift will be received.  So I need a volunteer who would be willing to pretend to be my cousin so we can see how this gift would be received.

Is there any volunteers who would like to pretend to be my cousin?  Zach you kind of look like him, come on up here.  I got you a gift for your birthday!

Student:  Thank you.

Ms. Gorman:  You’re Welcome. Do you have any guesses as to what it might be? Or… No?

Student:  I have no idea.

Ms. Gorman:  Ok.  Now class would you please take notes on the order in which he is unwrapping the gift.

Student:  Ooh a calculator keeper.  Calculator, thank you!

Ms. Gorman:  You’re welcome.  Alright so what do you notice about the official operating procedures for wrapping a gift versus the unwrapping that Zach just did for me?

What did you notice about them?  Bailey?

Student:  It’s in reverse order.

Ms. Gorman:  Very good, it’s in reverse order.   Or it is the inverse order.  OK, very good.  Ok let’s relate this gift giving experience to math.  Just like there’s an order for wrapping gifts and unwrapping gifts, there’s an order for doing operations in math. 

And what are the order of operations?  Can anybody remind me of those?  There might even be an acronym that you memorized or maybe a saying that helps you remember the order of operations.  Becca?

Student:  Parentheses, exponent, multiplication, division, addition and subtraction.

Ms. Gorman:  Excellent!  Does that look familiar to most of you?  PEMDAS right?  Ok so is that the normal order of operations that we follow.  What do you think the inverse order would be?  Chase?

Student:  Subtraction, addition, division, multiplication, exponents, parentheses

Ms. Gorman:  Right, it kind of goes backwards right?

So you might think of it as SADMEP.  Right?  Opposite of PEMDAS.  Ok.  Inverse order of operations.  We used inverse order of operations actually a couple months ago when we looked at solving equations.  And do you remember when we made our algebraic math mobiles? 

Ok.  We used inverse operations when we solved for the variable.  Can someone remind me what the requirements were for this project, and also what we did with each line of algebra?  What were some of the requirements?  Austin?

Student:  They had to have four lines of work.

Ms. Gorman:  Right, there were four distinct lines of work.  What else?  Zach?

Student:  Had to show addition of the opposite.

Ms. Gorman:  Alright, and if you remember when we solved those equations, we used this PowerPoint slide I had for you that had some silly pictures in it.  OK?  So on your paper, below the boxes for wrapping and unwrapping gifts do make sure that you have written down the order of operations that we normally follow, as well as the inverse order of operations.

(Whispering)  There you go, great.

And then down at the bottom just to refresh your memory I have two one step equations, and we are going to solve them again using my PowerPoint.

 Middle School Math Video 2

 Ms. Gorman:  Now that you’re getting ready for eighth grade we’re going to look at some more complex algebraic equations.  Okay.

And remember algebra is useful in life so often especially in any sort of engineering field or if you do any construction in your life time like if you make a dog house or if you plant a garden, algebra can be a useful tool to you.

Okay.  But first, let’s define the new kind of equation that we’re going to be dealing with.  Remember, the essential question that you need to be able to answer by the end of this period is how do we solve a two step equation?

Remember, my goal for you as I said, when you came in is to identify the operations that are present.

The definition of a two step equation is a mathematical sentence that contains an equal sign.

Okay, so here is my first equation. Two x minus five equals twelve.  How many operations are present in this equation?  Renee?

Student:  Two.

Ms. Gorman:  Two, right.  So let’s put in two in the first box.  Okay.  Now, normally what operation would you do first? Kelsey, can you tell me what operation I would normally do first?

Student:  Multiplication

Ms. Gorman:  Correct.  I would do the multiplication first.  And then I would do?

Student:  Subtraction.

Ms. Gorman:  Subtraction.  So if I am going to undo those I am going to go backwards through the order of operations. Which one do you think I would undo first?

Student:  The subtraction.

Ms. Gorman:  The subtraction.  That’s right.  And what is, Kelsey, the inverse of subtraction?

Student:  Addition.

Ms. Gorman:  Addition, great.  So let’s write addition here. Okay would you take a look now please at the bottom of page two. And I have there our first two step equation that we’re going to try to solve.

And does anybody see how many lines of work it looks like we’re going to have?  How many?  Elizabeth?
Student:  Seven.

Ms. Gorman:  Right, it looks like it’s going to be seven lines if we count that original line. Okay. So the equation is two x plus one equals 5.  Okay.

I’m going to go through this same PowerPoint but this is going to be a little bit different than the one we used for one step equations. You’re going to see some new slides in here. 

Okay so get ready with your pencil. And let’s recopy what appears on the left side of the equation which was two x plus one.  Bailey said we are going to do subtraction. And how much are we going to subtract from both sides?

How much are we subtracting from both sides? Melanie?

Student:  One.

Ms. Gorman:  Right. So we’ll subtract one from the left side and one from the right side.  Make sure your equals signs are lined up.

Okay, next question.  Do you see things that will cancel out? And here’s where I want you to get your highlighter ready.

Okay.  On which side of the equation do you see something that is going to cancel out and what special number are we going to write down to indicate that we understand that inverse operation? Ashley?

Student:  The one and it will be zero.

Ms. Gorman:  Right, because one minus one is zero.  So, let’s write down a zero.  Okay.  And even though that might seem insignificant it is important to show that we understand inverse operations.  So get your highlighter and just highlight right over top of that zero. 

Okay, next question.  Caution, do you see any subtraction that must be rewritten?   Remember that subtracting a number is the same as adding its opposite.  Macken?
Student:  Five minus one.

Ms. Gorman:  Good, how should I rewrite that?

Student:  Five plus negative one.

Ms. Gorman:  Wonderful.  Let’s write five plus negative one.  And if you remember from a couple months ago when we looked at one step equations we also took our highlighter and we boxed these together didn’t we?

We boxed them together because they are equivalent.  Five minus one is the same as saying five plus negative one.  So we’ve got those important things highlighted.

What was your question Zach?

Student:  How come we don’t rewrite the addition as subtraction?

Ms. Gorman:  Good question, yes. This does not have to be rewritten.

Alright, let’s take a look at the work that Elizabeth just wrote up here for us.  And I just want to do a little compare and contrast between this and what we did a couple months ago when we made our math mobile.

So what do you see that is different in the new two step equations versus the equations we did a couple months ago? Austin?

Student:  There’s two operations.

Ms. Gorman:  There’s two operations now, that’s right.  Before there was only one operation. Good. What else do you see that is different or maybe something you see that’s similar? Melanie?

Student:  There’s seven lines of work instead of four.Ms. Gorman:  Right, there’s seven lines of work instead of four.  How do we check our work?  Chloe?

Student:  It shows the value of n in the original line of work.

 Middle School Math Video 3

Ms. Gorman:  What we’re going to do next is give you an opportunity to work with a friend.  You can either pair up with someone or you can work as a small group.

But I just want to go over again some of my expectations when you are in groups and what I’d like you to do while you’re in those groups. 

You will be working with a partner to continue in the rest of the packet.  Ok and you’re going to try and solve the equations that are in the packet, showing me all seven lines of work.

What I definitely want you to take with you in addition to the packet is your mini slides hand out. 

That has all the questions on it that I’m going to ask you to ask the people in your group.

But before you do anything, I want you to look at the problem with your partner or on your small groups, and I want you to make a hypothesis or form an educated guess as to what you think the value of the variable is. 

And it’s kind of just like how when Zach was unwrapping the gift he kind of took a guess what might be in it, maybe he kind of shook it a little bit.

Ok.  Same thing as what I’d like you to do with this.  Ok, so you’re going to guess as to the value of the variable, and then you will go through and see if you you’re right. 

So you make your hypothesis and I’d like one person to read the questions that are on the hand out, and I want the other person to answer the questions. 

But everybody rights the algebra on paper.  Ok so one person asks, the other person answers, everyone writes down the algebra. 

At the very end of the paper you should see a checklist there of four things you need to make sure you did.  Do you have seven lines of work?  Are your equal signs lined up? 

Does the variable remain on the side it started on?  And then when you plug in for the value of the variable, does the equation make sense?  Is it true?

Ok.  After you do one problem then I want you to switch roles.  The other person asks the questions on the PowerPoint and the other person answers.

Ok still everybody’s writing down the algebra.  Ok?  Just a reminder, help each other. 

Ok this is new material so try to help one another as you go, and as you progress through the packet keep asking yourself, does this make sense?  What I’m doing does it make sense?  And if it doesn’t ask you partner to help you.  What don’t you understand about the level that you’re on?

Student 1:  What two operations are happening right now to the variable?

Student 2:  It’s being multiplied and added.

Student 1:  What is the inverse of each operation?

Student 3:  Division and subtraction.

Student 1:  Going backwards through the order of operations which undoing should we complete first?

Student 4:  Subtraction.

Student 1:  Should I do that to one side or both sides?

Student 5:  Both sides.

Student 1:  Caution [Indiscernible] subtraction [Indiscernible].

Student:  Forty-eight minus negative three.

Student:  It would be forty-eight plus negative three.

Student:  Wait.

Student:  Forty-eight plus negative three.

Student:  Wait, plus negative three?

Student:  Yeah.

Student 1:  What is the variable?

Student 2:  G

Student 1:  What two operations are happening right now to the variable?

Student 2:  Multiplication and addition.

Student 1:  What is the inverse of each operation?

Student 2:  Subtraction and division.

Student 1:  Going backwards through the order of operations which undoing should we complete first?

Student 2:  We should subtract eleven from each side.  Nine G equals negative nine.  Then we’ll…

Student 1:  Have we gotten the variable by solving it?

Student 2:  No.

Student 1:  What is still happening right now to the variable? [Indiscernible] What is the variable?

Student 2:  X

Student 1:  What two operations are happening right now to the variable?

Student 2:  Multiplication and subtraction. What is the inverse of these operations?

Student 1: Division and addition.

Student 2:  Run backwards through the order of operations undoing

Student 1:  Subtraction. Should I do that to one side or both sides?

Ms. Gorman:  And what is it that you’re going to do to both sides?

Student 1:  We’re going to add fifteen.

Ms. Gorman:  Yes Ladies?

Student 1:  We’re missing a step so where do we go?

Ms. Gorman:  Oh okay.  Let me go back here with the questioning.  Alright so your variable is what?

Student 1:  Adding three.

Ms. Gorman:  And what is happening to the variable right now?

Student 1:  It was being divided by three.

Ms. Gorman:  And also

Student 1:  And added by three.

Ms. Gorman:  What is the inverse of each of these operations?

Student 1:  The division is multiplication and the addition is subtraction.

Ms. Gorman:  Good

 Middle School Math Video 4

Ms. Gorman:  So for the first equation there can someone tell me please what is the variable? Elizabeth?

Student:  The variable is x.

Ms. Gorman:  Good. And what is happening right now to the variable? Chloe?

Student:  It’s being subtracted by three.

Ms. Gorman:  It’s being subtracted by three, that’s right.  Very good.  Next question.  What is the inverse of that operation?  The inverse of subtracting three?  Coleman?

Student:  Addition of the opposite.

Ms. Gorman:  The inverse of subtracting is to?

Student:  Add

Ms. Gorman:  Add.  And remember we’re not going to change the number so what number are we going to add?

Student:  Three.

Ms. Gorman:  Right, very good.

Next question.  Have we gotten the variable by itself yet?  And let me just kind of and I’m going to pick at random this time.  I’m not picking the same people all the time.

Have we gotten the variable by itself yet?  This is for Bailey.

Student:  No because it’s still next to 3.

Lisa Gorman: Good.  My next question is what is still happening right now to the variable?  Lauren?

Student 1: Thirty nine plus fifteen is…

Ms. Gorman:  And can I just ask a question?  How did you get this zero?  Where did that come from?

Student 1:  From the subtracting fifteen and then adding the...

Ms. Gorman:  Okay.  And those well subtracting fifteen and then adding what?

Student 1:  Fifteen.

Ms. Gorman:  And then adding fifteen.  Okay.  And what kind of operations did you do there?

Student 2:  Addition of…

Ms. Gorman:  You did, you had subtraction and then you did addition. So those, so subtraction and addition are pairs of?

Student 3:  Integers.

Ms. Gorman:  Well they are integers, yes, but you did inverse operations right?

Student 1:  Oh, yeah.

Ms. Gorman:  Where you’re inversely subtracting then adding.  You did it right I just wanted to make sure you understand why.

And we’re going backwards through the order of operations so which one do we undo first this addition or this division?

Student 1:  The addition.

Ms. Gorman:  Right, we’re going to undo that first.  So the opposite of adding thirteen is to?

Student:  Subtract thirteen.

Ms. Gorman:  Subtract thirteen.  And do you do that to one side or both sides?

So now I just want to follow up with the discussion of what you just did in the groups.  And I asked you to make those hypotheses first before you solved for the variable.

How many of you were right most of the time with your initial guess?  Okay.  Not too bad.  Alright. 

And then when you checked your work at the end by plugging into the original equation how many of you at least had a balanced equation when you did that?  I hope, I hope.  Okay.

What was confusing for you in that activity?  Something that was confusing?  Zach?

Student:  Number twelve.

Ms. Gorman:  Yes number twelve was confusing and I wanted to see how you guys did with that one.  What was confusing about number twelve?

Student:  It was hard to divide because I think it was forty five. It was like twelve minus, what was it?

Ms. Gorman:  Let’s see if we can look at number twelve real quick.  I think the equation was twelve minus eleven a, right? Equals forty-five.

Student:  Yeah.

Ms. Gorman:  And if we go through the questioning again my first question would be what is the variable?

Student:  A.   Ms. Gorman:  It is A.

Erica, question?

Student:  Wouldn’t you want to add twelve to both sides because it’s subtracting?

Ms. Gorman:  Good point but let me clarify one thing.  Probably when you look at this you might not see immediately how I got from this line down to this line.

The thing that we need to remember is subtracting a number is the same as doing what?

To conclude our lesson I’m going to give you this handout that has eleven questions on it.  It’s going to check your understanding of everything that we did this period.

And it’s going to help me to see if you can answer that essential question of how do we solve two step equations. 

And remember I said at the beginning of class my goals for you were to see if you can identify the two operations that are present in these equations.

So let’s see how you answered this.  How is a two step equation different from a one step equation?  And I’m going to pick from here again. Brooke.

Student:  A one step equation has four steps to it and a two step equation has seven steps to it.

Ms. Gorman:  Maybe we could say lines of work, right?  Seven lines of work for two step and one step has four lines of work.  That works.  Number two says how are they alike?

So how are one step equations and two step equations alike?  Bailey?

Student:  The goal of both of the equations are twice of the variable.

Ms. Gorman:  Well said, very good.

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                                     PDE's Elementary Math Video Transcripts

 Elementary Math Video 1

Ms. Delaney: This morning, we are going to work on something that we could actually use in our classroom.  We're going to figure out a way that we could arrange chairs so that everybody could see, as if we were maybe at the movies or at a play.  And what we're going to do this morning is imagine that these twelve pieces of paper—or at your desk they are twelve little tiny blue tiles are chairs … are you thinking about those being chairs?  Okay, what I'm going to ask you to do is, I want you to think about how you could arrange those chairs in rows and equal groups, so that we could watch a movie.  Now, what I want you to do is turn to the person next to you, turn to your partner, and you're going to share those ideas.

[Conversation]

Ms. Delaney: Okay, I would like for someone to share an answer that they came up with with their partner.  Let's start with the O'Brien family, and we'll go with number one.

What did you and your partner come up with?

Student: We came up with putting them in rows of four, like down, down ...

Ms. Delaney: Okay, into rows of four?

Student: Yeah.

Ms. Delaney: How many rows would you have?

Student: Four.

Ms. Delaney: Are you sure?

Student: Yeah.

Ms. Delaney: Can you check with your partner on that?  While she is checking with her partner, did anyone come up with something different?  Sydney, can you explain how you did yours?

Student: I did two rows of six.

Ms. Delaney: Two rows of six.  Can you show us?

Student: Yes.

Ms. Delaney: What do you think?  Thumbs up, thumbs down?  Beautiful job, Sydney.  Think about if you're in the store, if you're outside … different places that you might find ways to arrange our tiles.  So take a moment, again think to yourself … so everyone should be thinking. Now, I'd like you to turn to your partner and discuss what you're thinking about where to find ways to arrange things in the real world.  Think about different places that you've been, things that you see outside; think about going to the grocery store.  Okay, talk to your partners.

[Conversation]

Ms. Delaney: I would like the number four from each team today to share, and we will start with the O'Kelly's.

Student: In the store, the shelves are in groups.

Ms. Delaney: In the real world, the shelves [fades out] ...

Ms. Delaney: Third grade, you have given me some incredible ideas; our last idea was typed pages.  How does this connect with what we're doing in math today?  Everything up here are things that come in neatly organized groups.  Those groups are called arrays: mathematical arrangements for different numbers.  So our objective for today is going to be to create arrays for twelve.  That is our focus for today.

 Elementary Math Video 2

 Ms. Delaney: We know what our goal is for the day.  Our goal is to create arrays for twelve.  By arrays—“array” is a new word for us—array just means different ways to arrange these things.  Now, as I'm looking at our tiles, I see one group here, I see a second group here.  So, I see two groups of six. There's one, two, three, four, five, six in each group.  I wonder if there is another way I could arrange that?  I'm going to try groups of four: one, two, three, four.  So, I'm going to move these away.  One, two, three, four. Now I know that I can connect this to multiplication; it looks like something, maybe, that we have done before.  Three groups of four can also be shown as three times four, ‘cause there's four, four, and four.  This is simply repeated addition.  Think to yourself right now: how would you do all three of those things?  Ok, turn to your partner, and discuss with your partner.

[Conversation]

Ms. Delaney: This team, number three, could you please describe this array using a multiplication sentence?

Ms. Delaney: Four times three equals …

Student: … equals twelve.

Ms. Delaney: I know that we've worked with four by three.  I know we worked with three by four.  I wanted to try something totally different, but I'm going to need help from you guys. Let's get rid of these dimensions and let's try something new.  What I'd like you to imagine now, I want you to just think to yourselves: what if we took our twelve chairs, desks, tiles that we have—and we are only doing this in our minds—what if we just wanted to put them in one long row?  I want you to visualize for yourself.  What would it look like?  What might the dimensions of that row be?  What kind of multiplication sentence would match that array?  Just think to yourselves.  Okay, turn to your partner and share what those ideas are.

[Conversation]

Ms. Delaney: I would like the number one from the O'Reilly team.  Could you please tell us what your team came up with?

Student: We came up with one long row of twelve.

Ms. Delaney: Would it look something like this?  Is this what you were talking about?

Student: Yes.

Ms. Delaney: Number one from the O'Kelly's, could you please explain using the term “dimensions,” or the word “by,” or both.

Student: The dimensions would be one by twelve.

Ms. Delaney: Okay, it would be one by twelve.

 Elementary Math Video 3

 Ms. Delaney: What we're going to do now is—you have your twelve cubes in front of you—as teams I would like you to come up with two different arrays using your twelve tiles or cubes, the ones that we're using to represent our chairs. Once you find those ways, is it important for you to explain your answers?  Absolutely.

Student: Okay, okay.

Student: Are you sure?

Student: [Indiscernible] no, no I'm not because then this side would not be even with this side, so …

Ms. Delaney: What does it have to be to be an array?

Student: It has to be always equal.  Even, I meant that.  So it would be like this, not like that.

Ms. Delaney: So, as a team, how do we describe this array?

Student: That would be three, four [fades out] ...

Student: So, twelve by one, it has to be twelve going up and down, and one going across.

Student: [Indiscernible] Michael.

Student: Yes, because … because there's twelve ones, and if you count by ones twelve times … one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve.

Ms. Delaney: I'm going to ask this team, and I would like the number four person: share the array that your team came up with.

Student: We came up with twelve by one.

Ms. Delaney: Could you explain to the rest of the class what that would look like?

Student: It would look like … it would look the same as one by twelve, but just turned around.

Ms. Delaney: Third grade, give me a thumbs up if you agree, thumbs down if you disagree.

Looks like everyone agrees.  How about a clam clap for this team?

Now we're going to move into Team Mastery.  In Team Mastery, you're going to be working independently.  In Team Mastery, today I'm going to give you three numbers to find arrays for.  Your next job is to work together as a team.  With your teammates, you're going to compare your answers.

[Conversation]

Ms. Delaney: Can you explain how you did it?

Student: I did two by nine.

[Conversation]

 Elementary Math Video 4

 Ms. Delaney: Now, I'm going to challenge you to think a little bit harder.  Think to yourself: what numbers would only have two dimensions, two ways to arrange them?  Okay, talk to your teams, turn to your partners.

[Conversation]

Ms. Delaney: What kind of numbers did you guys come up with?

Student: Two, two, seventy, and twenty-three.   But they talked to me about this.  I want to know how they came up with this.

Ms. Delaney: Okay, so, do your teammates need to explain their answers?

Student: I came up with seventeen and twenty-three because seventeen you can only get to seventeen by ones and ...

Ms. Delaney: I like the way you're thinking.  This team, number one, did you guys come up with something?

Student: Yes, we came up with nineteen.

Ms. Delaney: With nineteen; can you explain why?  Give me the dimensions for nineteen.

Student: Because nineteen only creates two ways to do it, and nineteen is an even number.

Ms. Delaney: It's not an even number.  Juliana, can you explain about the even number?

Student: If it's not an even number, you cannot split it in half equally.

Ms. Delaney: Now, our math lesson today has almost ended, but in order to get out, you have to have a ticket.  Your ticket is going to be a pink sticky note today.  On this pink sticky note, I would like you to write the number ten at the top.  I would like you to draw an array for ten chairs, and just like we've been doing all day, label the dimensions.  Write a multiplication sentence for what you're doing.  When you have your array drawn for ten, please take your pink sticky note up and place it along the bottom of the board. This is something that will help us as we go through multiplication every single day.  It can help us when we're in the store and we're trying to figure out how many of an item to purchase.  Arrays help us to see our information organized in an easy to read way.

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