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Lesson 6.2 of Exploding Dots

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Okay, let's go back and revisit division. Here's a division problem in a 1<--10 machine. 276 divided by 12 is 23. 12 looks like 1 dot mixed with 2 dots. We found lots of 1 dots mixed with 2 dots in the picture of 276 that we saw of two in the tens level. And three at the ones level. 23. Alright, grand. But that's us humans loving 1<--10 machine so much. What I want you to do now is repeat this idea in all machines, all at once. We're just crazy. Alright, here it goes. What do I mean by that? So I'm going to play the same game now but we're doing it in a machine but we will not tell you which machine we're in. Just the mood I'm in. I might be thinking of a 1<--10 machine. I just won't tell you. Maybe it's a 1<--2 machine or a 1<--3 machine You won't know because I won't tell you. It's the mood I'm in. But you will have a machine, which is grand. So here's a start machine. And well, what machine is it? Well you do know that the dots here always worth one because I've always set the game out that way. The thing is you don't know if the 10 dots explode to become 1, or the 2 dots explode to become 1. You don't know what number machine. 1<--10, 1<--2, who knows? Now, just to give a nod to algebra in highschool curriculum, if there's a certain thing you don't know and you want to represent it with a letter, what's the favorite letter of the alphabet for the unknown Well, to that would be x. Everyone seems to be obsessed with the letter x for an unknown so I'll follow suit and call this an 1<--x machine. x dot explode become 1 dot placed to the left. You just don't know what x happens to be. Maybe it's 10. I'm not telling you. Maybe it's 2. You don't know because I'm not telling you. It's in my head, I'm just not telling you. Alright, but you do know dots here always worth one, but you now know x of these dots x 1s makes one of these guys. This must be worth x 1s. And x of these, xx it's multiplication, xx makes one of those. This must be worth x xs. Lest, we call that x squared. And x of these, x squared, next one over, x cubed. And x x cubeds makes one of those, x to the 4th and so on. Alright, just as a check. If I do reveal to you that x really is 10 in my head, then do we really have the correct thing for 1<--10 machine? 1<--10, well let's see you get the numbers 1 if x is 10, I get 10 10 squared is 100. 10 cubed is 1000. I get 1,10, 100, 1000. In fact that is correct for 1<--10 machine. Just another check. Suppose I said to you, actually no. Instead x really was 2 in my brain all along. Is this correct for 1<--2 machine? 1,2, 2 squared 4, 8, 16. Yes, this is in fact correct for 1<--2 machine. So in some sense, this does represent all machines all at once. There it is, it could be a 1<--10 machine which case got the correct numbers there. it could be 1<--2 machine which case got the correct numbers there and so on. Grand. Now, out of the blue. I know we've been doing great school arithmetic all this time I'm going take a straight to advanced high school algebra right now. Because I want us on this very spot, right now, to do this thing. 2x squared plus 7x plus 6 divided by x plus 2. Woah! Now this fancy language for this in high school algebra, they call this polynomial division or something but woah, wha tam I really doing? I'm not talking about the language. I've got something like this to be divided by something like that. And obviously this happen to be an 1<--x machine because I've seen all these x everywhere. So, here's the truth thing about doing math Step 1. There are challenging problems have an emotional reaction. Actually, math is emotional. You should have an emotional reaction, we're all human. That looks scary and horrible to me. So I'm going to have a deep breath [exhale] and just calm my nerves down. Now I'm a bit calmer. Let's look at this. 2x squared plus 7x plus 6. In an 1<--x machine, 2x squared, I could actually draw that. 7x, is, I can do that. And 6 ones. Bingo! That's what 2x squared 7x plus 6 is. It's actually some numbering this 1<--x machine. Good. Alright then, what does x+2 look like? Well, in an 1<--x machine, it will be 1x and 2 ones. 1 dot next to 2 dots. So I 'm must be doing another division problem. In some other number machine it's just that it's this machine now. Here's the picture of 2x squared plus 7x plus 6. I'm looking for groups of 1 dot next to 2 dots. Well, do I see any? Why, yes. I'm going to get my pen 1 dot next to 2 dots. 1 dot next to 2 dots. There's one at that level. But there's another one at that level. There must be some explosions going on. All of the dots are sitting right most part of the loop just like the 1<--10 machine. Anymore 1 dot next to 2 dots? Why yes. Right here. And here. Three at that level. So how do I interpret that answer? I'm seeing two x+2 at the x level and three at the ones level. The answer must be 2x+3. Woah. In fact, look what we've just done. Look at these two pictures. In fact, these pictures are identical. Woah! Double woah, infact. Alright. So what's going on? Why is this picture of an 1<--x machine exactly the same for this picture of a 1<--10 machine? Well, think about this. Suppose I did reveal to you now that x really was 10 in my head all along. What we've just done, 1, 10, 10 squared, 100, 1000 in a 10-1 machine, And what is this number here? Well, it will be two 10 squared, 200 plus 7 times 10, 70, 6 divided by 10+2, that's 12. And what is this number here? Well, it will be two 10 squared, 200 plus 7 times 10, 70, 6 divided by 10+2, that's 12. Apparently equals, 2 times 10, 20 plus 3. 276 divided by 12 is 23. Of course it is. It' s what we did over here Woah! So actually, I've just done a high school algebra question and all it really is just a repeat of grade 5 but now in a more general machine. In fact if it really was a 1<--10 machine. It really is exactly the same as grade 5. If this was a 1<--2 machine, the numbers are now 1,2,4,8. What if I got here? I'll have 2 times 2 squared. 2 times 4 that's 8 plus 7 times 2 that's 14 plus 6. Alright, that's 28 divided by 2+2, that's 4. So 28 divided by 4 equals apparently, 2 times 2, 4 plus 3, 7. If this was a 1<--2 machine. If x really was 2, I've got 28. divided by 4 is 7 which is correct. In fact, I love this approach. What I'm really doing now is a whole infinitude of division problems all at once. Every possible value of x gives me new division problem which I have the answer. Brilliant! I'm not locked in my humanness anymore. I've gone beyond my human 1<--10 to any machine I like. That is power. In fact, that's high school algebra and look, it's really just a repeat grade 5. So, let me clean the board. Let's do another division. Let's make this one look really scary and I bet we can do it. It's going to be just brilliant so give me a moment to clean the board. Alright, I'm back. I'm ready for an advanced algebra polynomial division problem/ Here it is. It looks ghastly but however I think it's going to be just fine. It's really an 1<--x machine problem, just know what's x actually is. Fine, I'll deal with that. So let me draw this problem. I wanna do this top numerator, This great big number here. 1x to the 4th. 2x cubes, 4x squares, 6x, and 3 ones. This is the top line is. And I'm looking for groups at the bottom line, x squared plus 3. Alright, so what does x squared plus 3 look like. Let me just do it up here actually. It's going to be 1x squared, no x and 3 ones. So I'm really doing this division problem. I've got a picture of the top line, there it is. And I'm looking for this. These groups in that picture. 1 dot, no dots, 3 dots. I'm looking for 1, blank, 3. Anywhere in this picture. So I'm going to find groups of them, can I do it? I bet the answer is yes. Alright, let's see. Can I find some x squared plus 3 in this picture? 1 dot, no dot, 3 dots? Why, yes. Here's one dot. Here, let's skip over some no dots and then let's do 3 dots stay there. There's one at that level. Remember, all the dots really are sitting there. There must be some explosion spilling all the way to the left. So I'm really at that level. Any other one, blank, threes? Well Yes. I might split my loop this time. I hope it's okay. I'll do 1, blank, 3 So one at that level. Another one at that level, 1, bank, 3. So far so good? Grand. And what am I doing? I got a dot here and I've got 3 dots there. Look at that, 1,blank,3. One at that level. There's my answer. One there, two there. One there, how do I read that, that's at the x squared level That's at the 2 at the x level and 1. I do have my lemon right don't I? Uh huh. Yep, 1x squared, 2x, and one 1. The answer must be x squared plus 2x plus 1. And here's the lovely thing. Because this really is a grade 5 division problem if I just shown you early x was actually 10 in my head all along. This is a 1<--10 machine. The funny thing is people tend to forget a proposed that x can actually be a number I can ask, what have I just done? Well if x really was 10, I'll have what? 1,2,4,6,3. I have 12463 divided by 103 apparently is 121. Turns out that's correct. So there again. Another infinitude of problems. Different values of x gives me lots of long division problems All answers, all at once in one amazing hit just by drawing pictures. I own this idea. I can see polynomial division Despite how scary that looks, I own it. I can see it, I can do it. Good stuff. Alright. Grand so let's carry on.

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Duration: 9 minutes and 59 seconds
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Language: English
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Posted by: jamestanton on Jun 9, 2018

This is the 13th lesson in the Exploding Dots story for Global Math Week 2017 (www.theglobalmathproject.org).

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