Common Core Algebra I.Unit 1.Lesson 11.Algebraic Puzzles
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Learning Common Core Algebra I. Unit 1. Lesson 11. Algebraic Puzzles by EmathInstruction
Hello and welcome to another common core algebra one lesson by email instruction. My name is Kirk weiler, and today we're going to be doing unit one lesson 11 algebraic puzzles. This is the last lesson in unit one, but before we get started, let me just remind you that the worksheet and a homework for this lesson can be found by clicking on the video's description. As well, don't forget about the great QR code at the top of each worksheet. That will allow you to take a smartphone or a tablet and get right to this video anytime you need to.
All right, let's begin. So what we're going to be doing today is we're going to be exploring kind of cool patterns that arise with numbers. And seeing if we can't figure out or prove that those patterns, all this hold, by using what we've learned about expressions, variables, equivalent expressions, things like the properties like the associative property and the commutative and distributive properties. So let's play around with these. I love them. Exercise one, choose any number. Create the sum of two more than three times the number with two less than two times the number. What pattern is true of the result? All right, so wow, two more than three times the number of two less than two times the number. What's the pattern? All right. It says, let's explore the pattern with numbers we know before working with algebraic expressions. Even using numbers, the English is a bit tricky to decipher.
So let's do it together using a single number. Let's let the number we choose B three. And we're going to show the calculation as described in the problem. Wow. Two more than three times the number and two less than two times the number. So two more than three times the number. So three times the number, right? And then we want two more than it. All right, and we want to add the sum, right? Of two less than two times the number. So there's two times the number. And then there's two less than it. Okay. So let's really try to understand here. Here's the sum, right? That's this then. Of two more than three times the number. With two less than two times the number. All right, that's the value of this expression.
Let's do what's in parentheses. First here, we've got 9 plus two. All right. And then we've got 6 minus two. And then that's going to be 11 plus four, and that gives me 15. All right. So when the original number is three, my final result is 15. Okay, great. So is there a pattern? Well, maybe the pattern is that the number will always the result wall was. I don't know, 12 more than the original number. Three plus 12 is 15. I don't know. So let's try a few numbers and see what happens. Let's go for the number. I don't know four. Let's try that. So the same calculation. We're going to do two more than three times the number. And two less, then two times the number. All right, let's see what we have. Three times four is 12 plus two. Two times four is 8 minus two. 12 plus two is 14. Plus 6 gives me a result of 20. Um. All right, so if we start with four, we end up with 20. Let me start with three, we ended up with 15. Let's just be systematic about it.
Let's go with some nice integers. Let's see what happens. All right, so two more than three times the number. And two less than two times the number. All right, let's see what we've got three times 5, which is 15 plus two. Plus ten minus two, and that'll be 17 plus 8, and that's 25. So if we start with 5, we end with 25. Start with four, we end with 20. It will just do one more. Because we've already got three there. I know we got another row, but let's kind of move it along. So here's going to be two more than three times the number. And here's going to be two less than two times the number. So three times 6 is 18 plus two. And 12 minus two, well, that be that'll be 20 plus ten, gives me 30. So let me put this three end and the 15. So when we started with four, we got 20.
When we started with 5, we got 25 when we started with 6, we got 30, and when we started with three, we got 15. So what's the pattern? Pause the video for a second if you don't know what the pattern is immediately and think about it. So what's the pattern? It looks like the result. Is always equal. 5 times the original. 5 times the original. All right. It seems like that's the case. Now the question is, can we use properties of equality properties of equivalency to actually prove that that will always happen? All right, I'm going to clear out the text so pause the video now if you need to. All right, here it goes. All right. So now let's prove that the result that you see in the table will always be true. Let the number now be called X write an expression that translates the verbal description given in the problem for our calculation.
All right, so let's do it. Two more than three times the number. Well, if the number is X, then two more than three times the number will be three X plus two. But we want to create the sum of that with a number that is two less than two times the number. All right. Now we can use the associative property of addition to say that those parentheses don't really matter. They don't. We don't have to add the three X in the two first, and the two X and the negative two second. Then of course, we can use the commutative property of addition to flip flop that two plus two X into a two X plus two. And then now you can probably see it. Three X plus two X is 5 X two minus two is a big fat zero. So we get 5 X so if we start with X, the result is always 5 times X kind of neat. Kinda neat, right? So that's how we can prove a general result by starting with just a variable instead of a specific number.
So we're going to keep playing with these algebraic puzzles today and see where we can go with them. All right, pause the video now if you need to, write down whatever you have to, and then let's move on. Here we go. All right. Let's take a look at a fluency problem. Okay? So here's a nice multiple choice example of this. If N represents a number, which of the following expressions represents the sum of one more than twice the number and three less than 5 times the number. Pause the video now, take a moment and think about this and see if you can translate all of this into a simple binomial expression. Pause the video now. All right, let's go through it. Okay, we're summing. So we're adding. What are we adding? One more than twice the number. One more than twice the number. With three less than 5 times the number.
Three less than 5 times the number. Again, just like before we can now use the associative property of addition to say one of those parentheses don't really matter. We can just remove them. We can then use the commutative property of addition to flip flop one plus 5 N and make it 5 N plus one. And now we can combine like terms to make two N plus 5 N into 7 N and one minus three into negative two. Giving me a final result of 7 N minus two. For choice one, kind of cool. All right. So this is a really nice combination of what we did in the last lesson where we were translating English into algebraic expressions. Now, we're doing it sort of for more of a purpose, if you will. All right? So pause the video if you need to because I'm going to scrub out the text, okay? All right, here we go. All clear. All right, let's move on to the next page. Okay, so let's go with that. With a more complicated problem. This is a kind of a short lesson because the problems are pretty rigorous. In this problem, we will explore calculation of the difference. Subtraction always worse than addition. The difference between the product of a number and a number 5 larger than it and the product of a number and a number 5 less than it.
Will this reveal a pattern like the last one? Like before, let's explore the pattern with numbers we know before working with algebraic expressions. Fair enough. Let the number we choose to be ten. So let's do this very complicated calculation. We're going to do a difference. There's going to be some subtraction. Okay? But what is it a subtraction of? It's a subtraction of two products. The first product is the product of a number and a number 5 larger than it. So ten and a number 5 larger than it. Minus the number, the product of a number, and a number 5 less than it, right? So here's my number ten. Here's a number that's 5 more than it. Here's a number that's 5 less than it. Now this isn't so bad. Ten times 15 is 150. Ten times 5 is 50. So when we find that difference, we get a hundred. So if I start with the number ten, I end up with the number 100.
Now that's not enough to establish any kind of pattern. For all I know, maybe the answer is well. I do, I swear the number. You know, ten times ten is a hundred. So let's play around with some other numbers. Let's just pick some at random. Let's try, let's try something like 6. Okay. Now what was the calculation I have to find the product of the number with a number that's 5 larger than it, so 5 larger than 6 would be 11. Then I have to find the difference and then 6 times the number 5 less than it. Well, 6 times 6 -5 would be one. So what do I get again? 66 -6, which gives me 60. All right. Maybe in the next calculation I'll really write that out so that we really have it. Right, let's go with the number, I don't know. Let's go with the number 8. A
ll right, so remember, I'm doing 8 and then I'm going to do 8 plus 5 and then 8 times 8 -5. I just really want you to understand where these numbers are coming from. So what will I have? I'll have 8 times 13 -8. Times three. Oh, I'm going to run out of space here. 8 times 13 is 104. I'm going to have to think about a little bit. And then 8 times three is 24. One O four -24 is 80. And I'm starting to think I see a pattern here. When I started with ten, I ended up with a hundred. When I started with 6, I ended up with 60. I started with 8. I ended up with 80. Let's do one more. Let's involve some negatives just for kicks and rinse. Let's start with two. So you'll see why negative are involved. Here I've got two times the number 5 more than two. Minus two times the number 5 less than two. So that's going to be two times 7. Minus two times negative three. Now watch out here that's going to be 14 minus negative 6. So that's going to be 14 plus 6, which is 20. Oh, I hope you see that pattern, right? It appears like the result will always be ten times what we started with. So the results is ten times the number.
Oh boy, that is some convincing evidence, right? But it's not a proof. It doesn't really justify it. It's building up evidence, all right? But it doesn't justify that we will always get ten times what we start with. So now let's prove that. Okay? So I'm going to clear out the text, pause the video if you need to. And it's gone. Now, we want to be very careful as we try to prove this, okay? But we can do it. We can do it, right? So what are we doing? We're taking a number X and we're multiplying it by a number 5 larger than it. And then we're finding the difference of that with the product of X times X -5. All right, be very, very careful. Let's do some distribution. All right, X times X is X squared. X times 5 is positive 5 X now here, I'm going to think of this as a negative X times a positive effect. That's going to be negative X squared. And then a negative times a negative is a positive 5 X all right? We're going to use the commutative property of addition to actually kind of swing this around.
As negative X squared plus 5 X, where's that? There we go, plus 5 X plus 5 X so again, I just looked at this as 5 X plus negative X squared that allowed me to flip flop using the commutative property. But now X squared plus negative X squared, and that's a big fat zero. 5 X and 5 X is ten X so our result is ten X and that's right. We saw based on the table that we were going to get a result that was ten times what we started with. And there it is. It's kind of cool. Right? All right. So algebraic puzzles. Why things are calculations work out the way they work out? Pause the video now because I'm going to clear out the text. Here it goes. And let's finish up.
All right. So today, what we did was we really use the skills that we've built up pretty much throughout this unit. Skills including things like distribution, associative and commutative properties, translating English into algebraic expressions, combining like terms, all of that came to bear in this one lesson. So it was good practice. Make sure to really work hard on that homework, and that'll give you even further practice and further reinforcement. Every time you do a manipulation, try your best to think about what property or what justifies the manipulation. The more you do that, the better you'll get and the more fluent you'll be with all of these skills, ideas and topics. All right. Thank you for joining me for another common core algebra one lesson by email instruction. My name is Kurt Weil. And until next time, keep thinking and keep solving problems.