9 R1 Get another piece of paper. We re going to have fun keeping track of (inaudible). Um How much time do you have? Are you getting tired?

Transcription

1 Page: 1 of 14 1 R1 And this is tell me what this is? 2 Stephanie x times y plus x times y or hm? 3 R1 What are you thinking? 4 Stephanie I don t know. 5 R1 Tell me what you re thinking. 6 Stephanie Well. Because I can t multiply x times y. 7 R1 Okay. So right now, where we are 8 Stephanie Um hm. 9 R1 Get another piece of paper. We re going to have fun keeping track of (inaudible). Um How much time do you have? Are you getting tired? 10 Stephanie Oh. No. I m fine.

2 Page: 2 of R1 This is so much fun. [Stephanie chuckles.] 12 R1 Okay. Here we go. Um. This is what we re dealing with: x times y, right? 13 Stephanie Um hm. 14 R1 Plus y times x. Right? 15 Stephanie Yes. [pause] Oh, but I can t move them. I have to keep them Can I just cause when I tried to do do just y plus y last time it was like 16 R1 Why don t why don t you try some numbers here and write numbers here? 17 Stephanie Easy numbers. [writes ] six plus six. That s twelve. 18 R1 So you were able to do them with numbers two and three. 19 Stephanie Yeah. 20 R1 You were able to finally add them numbers. I guess that s not helping too much. So this is 21 Stephanie Well, yeah, but there were parentheses around them here.

3 Page: 3 of R1 Okay, but let s look at the two times three and three times two. 23 Stephanie Okay. 24 R1 Look at those two terms: two times three and three times two. 25 Stephanie Um hm. 26 R1 Are they always going to be the same? 27 Stephanie Two times three. 28 R1 And three times two. 29 Stephanie And three times two. Yeah. 30 R1 Suppose you use five times six and six times five? 31 Stephanie Yeah cause it s the communative- 32 R1 So xy is that always the same as yx?

4 Page: 4 of Stephanie Yeah No Wait Yeah, cause it s the same thing. 34 R1 You just you just used a big word. What was that word? You just used? 35 Stephanie Oh. Communative. 36 R1 Commun Commutative? 37 Stephanie Commu Yeah that one. 38 R1 Yeah. What s that mean? 39 Stephanie It means for addition and multiplication, it doesn t matter the order. 40 R1 Okay so what are you doing here? This is xy and this is yx? 41 Stephanie Well I just switched em around. I just switched the two numbers around. 42 R1 And what op what operation is involved with xy and yx? 43 Stephanie Multiplication?

5 Page: 5 of R1 Multiplication. So you should be able to write xy as yx, or yx as xy. 45 Stephanie Yeah. 46 R1 It s the same. 47 Stephanie Yeah. 48 R1 If that s true and you re telling me that s true whenever you use a four and a five or an eight or a nine, or a million and a ten million. 49 Stephanie Yeah now that s true. 50 R1 Okay, so if xy is the same as yx 51 Stephanie Um hm. 52 R1 Is there a way you can rewrite any one of these so you can write that as a simpler term? 53 Stephanie xy I don t I mean I don t I don t 54 R1 Okay. You wrote xy plus xy.

6 Page: 6 of Stephanie Plus xy. 56 R1 Can you write that in a simpler way? 57 Stephanie Two x. 58 R1 Two xy. 59 Stephanie Two xy? 60 R1 Sure. 61 Stephanie [Stephanie writes.] Or is it is that 62 R1 Does that make sense? 63 Stephanie -What we re doing? Cause Yes! Two x plus times two y. Right? That would be the same thing. 64 R1 Can we say that this is two x times two y? 2(x y) 65 Stephanie No. No. So would it be like (x y) squared.

7 Page: 7 of R1 But Let s go back to this and see what (inaudible). There are three interesting things here. So let s go back and think. 67 Stephanie Okay. 68 R1 Alright. You re saying xy. 69 Stephanie Um hm plus xy. 70 R1 Plus xy. You said that s a valid thing to do. 71 Stephanie Yes. 72 R1 You can rewrite yx. Okay. And you said you can write that two parentheses x dot y. 73 Stephanie Um hm. 74 R1 Right? Now. This is confusing you a little bit this expression? 75 Stephanie Yes. 76 R1 Right?

8 Page: 8 of Stephanie Well it it s saying two x times two y, though. 78 R1 Well try it. Is two xy the same as two x times two y? -Try some numbers 79 Stephanie Oh. Well okay. 80 R1 Do you do that when you re not sure if something s allowed? Do you try some numbers? 81 Stephanie Sometimes. Most of the time we re just we re not dealing with like problems where we know we don t know (inaudible). 82 R1 You will. You will. That s yet to come. [Stephanie writes: ] 83 Stephanie Two times three, that s six times four. That s twenty-four. 84 R1 Hm. 85 Stephanie And the other way, it would just be be um two times two times three times three, but that s different. That s not (inaudible). That s nine times four and that s thirty-six. 86 R1 Um, what is -? 87 Stephanie Something is wrong.

9 Page: 9 of R1 Okay. What is it this way? 89 Stephanie Oh. 90 R1 You know this way is right. 91 Stephanie Well, yeah. So (inaudible) twelve. 92 R1 Okay. So this is tw- 93 Stephanie Oh. Boy. 94 R1 So what s happening? 95 Stephanie Well, neither of them are right then! cause I mean this is 96 R1 Well, this one isn t [points to 2x 2y] Right? 97 Stephanie No. 98 R1 And this one isn t [can t see which one R1 points to]. But what about this one? [Stephanie writes: 2(2 3) 4 6]

10 Page: 10 of Stephanie Twenty-four. 100 R1 Well. I m trying to see how you got twenty-four. 101 Stephanie Oh. Wait. 102 R1 I m doing the two and the three first. 103 Stephanie Oh. 104 R1 Cause it s in the parentheses. 105 Stephanie Yeah. Yeah. That works. Okay. 106 R1 Do you see what you were just doing? 107 Stephanie Yeah. 108 R1 What you were just doing? It s common lots of people 109 Stephanie I was just cause I always do it like that cause I m used to having like a variable and when you have a variable in there you can t you have to distribute first. So I m used to distributing first.

11 Page: 11 of R1 So this isn t Is this the Distributive Law? Two xy? I mean do you need that dot? - x times y. Can you write that as xy? Have you had that yet? 111 Stephanie No. 112 R1 If you have x dot y. You have dots. Have you used dots? 113 Stephanie Yes. We used to use dots. 114 R1 So like if I write two dot x you would know that s two x, right? 115 Stephanie Yeah. 116 R1 But if I write x dot y would you know that s xy? 117 Stephanie x times yeah. 118 R1 So I didn t need that dot, maybe, huh? 119 Stephanie Um. Because 120 R1 Do I need the parentheses?

12 Page: 12 of Stephanie I don t Maybe not I don t know Because 122 R1 See I always go back to basic meaning. What helps me you see you have an xy and an xy. So I think you have an xy, you have another xy. You have one of them here and one of them here. So you have 123 Stephanie That s two xy. 124 R1 That s two of them. 125 Stephanie Okay. Yeah. 126 R1 I like to think of basic meaning and simplicity. That s what helps me. Do you see what I m saying? 127 Stephanie Yeah. 128 R1 So when I m confused, I think, Now. Wait a minute. All this language is just getting too confusing. Let s try to make it simple. What does it mean? I keep saying What does it mean? Go back to basic meaning. So you see in this lovely thing you did here this thing here looks very complicated. x times y plus y times x. Right? But as soon as you recognize that s xy and that s yx as soon as you recognize commutative, (that was very nice), that you can think of that as xy and xy. One of them and another one. And that s a nice way to think about it. 129 Stephanie Yeah. 130 R1 Okay. So what did we end up What s the sim - What s another way of rewriting this in its simplest way? Can you that s what we started with.

13 Page: 13 of Stephanie Well. Okay. Now we didn t eliminate these. 132 R1 See what you believe any more. 133 Stephanie x to the second. 134 R1 You believe that still. Okay. 135 Stephanie But what we got was plus 2 xy plus y to the second. 136 R1 Okay. 137 Stephanie That s what we have now. 138 R1 You believe that? Try some numbers and test it. 139 Stephanie Okay. Two to the second plus two (inaudible) four plus twelve plus nine. It worked. 140 R1 You like that, huh? Isn t that wonderful? 141 Stephanie It looks a lot easier than this.

14 Page: 14 of R1 Um. Okay. So you believe that this is the same as this? [indicates: ( x + y) = x + 2xy + y ] 143 Stephanie Yes. 144 R1 And we could keep testing lots of numbers? 145 Stephanie Um hm. 146 R1 Actually, you ve proved it. What you ve just done is gone through a proof. What you ve done here is is your proof is based upon you know the meaning of these things. If you think about what you ve done. I d really like you to go back and think about this when you go home. What the meaning is. When you talk about your Distributive Law. What does it mean with numbers? And then what does it mean if you don t have numbers.