 entities you want me to go over now is your chance to ask which one's from the homework you're wondering about I think I should be able to fit in a homework check later on today that's the plan stay on hope it all went swimmingly really that's cool excellent that makes things way easier then let's go right to lesson six which I don't think is gonna go as swimmingly we're gonna find out to me this is kind of a sort of a reasonably kind of a tough topic okay and what I'd like you to do once you're on lesson six is I'd like you to put your pencils down but put your eyes and your brains up what we're gonna do is we're gonna look for a pattern here's the pattern okay let's suppose I have let's see insert table sure why not let's do kind of about like that sure that'll work so let's suppose we have a rectangle rectangle and what I'm going to look at is the original size the area I'm going to add a linear scale factor can I get this all on one page new size new area area scale factor hey there that'll work on one page what's the area of a rectangle how do I find it come on you remember this this has been like I think grade six or grade five how do I find the area of a rectangle marcus length times width okay area of a rectangle is length times width so suppose my original rectangle is two by eight this is meant to be obvious what's its area come on what's its area 16 yes yes yes yes yes okay I'm a little worried if Jordan was to do that you but relax right now you're gonna be all right you sure okay 16 I'm gonna add a linear scale factor and my linear scale factor is going to be one to two we're going to double everything if I double everything my new sizes are going to be four by 16 I doubled before and I doubled so far so good what's my new area 64 here's what I want you to notice when I doubled when I doubled when I doubled the length and the width did the area get twice as big how many times bigger is my area Jordan the area scale factor ended up being one to four where'd that four come from hold that thought let's suppose Jeremiah I had another rectangle I'm picking a rectangle because it's about the easiest shape out there but this actually works for any shape with area so let's suppose I had a what do I do a two by eight three by seven rectangle which has an area mat of what thank you for doing that in your head there is hope for this generation I'm going to give you a linear scale factor the scale factor this time is going to be one to three we're going to triple this length and triple the width which instead of three by seven is going to give me triple nine by uh 21 21 what's the new area I need to calculate her now's the time to get about 180 sounds good what's the area scale factor here is it three times bigger I'll tell you right now it's not how many times bigger well to figure that out to figure out the area scale factor go new area divided by original area what do you get nine times bigger or we would say this one to nine area the scale factor not three times bigger nine times bigger anybody spot a pattern yet let's see supposing supposing supposing I made up a what five you say sure five by three you say sure three rectangle which has an area of oh help me out erin whoo-hoo see this is right it's good and supposing I did a linear scale factor of one to four I made it four times bigger so that's going to make my rectangle instead of five 20 by and instead of three 12 yes what's the new area length times width 240 what do you think the scale factor is going to be you said you spotted a pattern take a guess what do you think the area scale factor is going to be okay he's not confident enough he's going to check so he's going to go new area divided by old area and what do you get I know you got 16 this is actually a one was your hypothesis correct okay so we're going to find out I'm not going to fill in all the numbers I'm only going to fill in my hang on while I answer the phone sorry for the interruption those of you watching at home you ready Marcus thinks he spotted the pattern let's suppose I told you that the scale factor was one to five the linear scale factor was one to five what do you think the area scale factor would be Martin Marcus Marcus sorry how'd you get that hang on hold on what if the scale factor was one to nine what if I made the length nine times bigger and the width nine times bigger how many times bigger would my area be Marcus what's he doing there's a fancy word for it and you can use a math word and it's also a shape not square rooting squaring square root is going backward it turns out it turns out that the relationship between scale and area is not one to one it's one to squared which is kind of easy to remember because honestly this is how I remember area squareia which is stupid but it works for me okay you got to come up with something so let's make some educated guesses here what if we had a scale factor of one to 15 how many times larger will your area be one two yes you might need to do your calculate unless you have your squares memorized hang on hang on we'll pause the video then know what 15 squared is yeah did you know that or did you know that and okay that's a fail smarter than half of them they just sat there drilling it also works in the other direction what if my scale factor was one now this one i'm going to have to write by hand what if my scale factor was one to one half one to point five we're getting smaller the area scale factor would be one two how do you square a fraction it's easy you square the top over you square the bottom what's one squared in your head please what's two squared it's going to be one quarter of a or one fourth besides or divide by four if i have one and i'm making everything two thirds as large i'm shrinking it it's a reduction you know what my area scale factor is going to be what's two squared what's three squared the scale factor is going to be four ninths spot the pattern pick up your pencils it wants us to walk through the same thing here and we will because it's worth writing down in our notes but writing it a second time hopefully it'll make way more sense than doing this cold turkey they're doing this kind of a little less organized i think so it says what we want to investigate is the relationship between linear scale factor and area scale factor here's our says complete the following table our original rectangle is three by five which has an area of 15 if we have a two to one you know what i just realized i wrote these scale factors in the wrong way they want the big number to come first so this example this lovely example that i just did with you here i was writing the scale factors backwards according to the way the textbook wants i should have written this as a two to one as a three to one as a four to one as a five to one but that's okay we'll do it properly in our notes not the end of the world i always get those mixed up two to one or or a scale factor of two a linear scale factor six by ten becomes 60 the new area divided by the original area is four over one scale factor of four are you ready nine by six first of all what's the area of a nine by six rectangle you guys know the nine times table trick with your hands right right so you're shaking your heads oh man gotta pause the video again so 54 we're gonna have a one to three a one third reduction a one third reduction a one third reduction multiplying by one third is the same as dividing by multiplying by one third is the same as dividing by three not dividing by a third that's very different multiplying by one third is not the same as dividing by a third it's saying take a third of your original size reduce it by a factor of one third That's the same as dividing by three. In other words, it's going to be three by two. Yes. And the new area is going to be what, Shay? I caught you zoning out. I know. The new area is going to be what? New area. First, we're not there yet. New area is going to be, what am I going to put here? Yeah, very good. And the new area, which is six divided by the old area, which is 54. And yes, Shay, you jumped to the punchline. It turns out to be a one over nine reduction. Okay, now we're getting a little trickier. Four by eight, that's 32. How do I do this? Put your pencils down, look up. This is a scale factor. The word factor comes from when you multiply things together. Two numbers are multiplied together. We say those are factors. The answer is called the product. The way we found one third, Jeremiah, was we divided by three, I said multiplying by one third is the same as dividing by three. In fact, it's the same as timesing by one and dividing by three. If they give you a fraction, the rule is multiply by the top, divide by the bottom. Can you remember that you divide by the bottom? Doesn't that horizontal line mean divided by anyways? So if I want to find my new dimension, the four is going to be four. It's going to be times by three divided by two. That's a three over two scale factor. The new width is going to be six. And following a three to two scale factor or a 1.5 scale factor as a decimal, it's going to be eight times three divided by two, 12. What's the new area, kiddo? Six times 12, yes, use your calculator or do it in your head. Did you say 72? Did you hear? You guys are good. Someone's taught you some mental arithmetic. Now, stop. I'm going to make a prediction. You guys don't have to write this down just in case we're wrong. What's our scale factor for the area going to be if this rule holds? If this is our original scale factor, what are we expecting here? I think we're expecting nine over four. Let's see. If I go new area divided by original area, does 72 over 32 reduce to nine over four? Well, if you divide the top and bottom by eight, it does. And I've talked about your fraction buttons on your calculators and you can feel free to use those, but we'll learn how to use them. Three by 12, which has an area of 36. Now we're doing a two to three reduction. By the way, how did I know that this was an enlargement and this was a reduction? It was a reduction when your scale factor was, remember, and it was an enlargement when your scale factor was, remember, smaller than one, larger than one. Three over two as a decimal is 1.5. It's bigger than one. And I don't even really need to go to my calculator, Courtney, because the three is bigger than the two. It's bigger than one. That's going to be an enlargement. It's going to be a reduction. How will I find out what my new dimensions are? Times by the top, divide by the bottom. It's going to be three times two, six, divided by three, two, by 12 times two, 24, divided by three, eight. The new rectangle is going to be two by eight. By the way, you guys have clued in, the abbreviation for by is times sine and x comes from construction. Area 16, new area, divided by old area. I think we're expecting an answer of four over nine. Do we get an answer of over nine? Jeremiah nods his head while yawning simultaneously. Good to know you can do two things at once. So B says, compare the linear scale factors to the area scale factors, and it wants us to complete area scale factor equals linear scale factor squared. Put a little squared on the bracket there. Take your linear scale factor, square it. Remember the abbreviation for square is a little too exponent, right? Okay. Now we're going to be getting into decimals and things, and you might want your calculators handy, although maybe you guys are better at mental math than I think. You probably want to face out and face up and get them out. Yeah, that's what I was getting at. Okay. Maggie has scanned an 8 by 10 photograph to her computer. What shapes are all photographs? Rectangles. They're not going to tell us. Assume any photograph unless they say different. It's a rectangle. Okay. Well, I have a printer in a circle. No, what is rectangle? She would like to increase the size by 44%. I would underline that. If she wants to increase the area by 44%, what does she want the new area to become? What's the original area as a percentage? This is tricky, by the way. Tricky enough that for the first time this year, Jordan, I'm using cheat notes. I did this lesson ahead of time to make sure I got all the right answers. Normally I can, okay. Put your pencils down for a second. A lot of people will go like this. Don't write this down. This is incorrect. They'll go 44%. And it wants the linear scale factor. Oh, a linear scale factor is going to be 44%. That's 0.44 is a decimal. The linear scale factor is going to be the square root of 0.44. But you get this yucky ant. I don't think that's right. I have a feeling they meant this to work out evenly. The area scale factor is 144%. Write that down. Why is it 144%? Not 44%. Yeah. She wants the original plus 44%. And what's the original? 100%. Right? The original of anything is 100%. The original size plus 44%. That's a bit of a tricky concept. Okay. Now, if the area scale factor is 144%, Mr. Dewick, can we abbreviate area scale factor as ASF? Yes. Can we abbreviate linear scale factor as LSF? Yes. I'll figure it out. What's the linear scale factor? Now we're going backwards. We're saying if you know this, what's this? How'd you go backwards? What's the opposite of squaring? I heard, what? Square rooting. The linear scale factor is going to be the square root of, now 144% is 1.44 as a decimal. Remember your percent's diffractions in decimals? What's that? What do you get? What's the square root of 101.44? Ah, 1.2, which is 120%, yes? 20% increase on the length and the width. A 20% increase on the length and the width, 120% scale factor gave us a 44% increase in the area. Now that's a bit trickier because when I see the number 20, the next number that leaps to mind is not 44. It says, explain why these scale factors are different? Well, they're one's near one's area. We already noticed. I'm not going to fill that out. We just filled the whole chart noticing that scale factors for linear scale factors and scale factors for area scale factors aren't the same. They're squares of each other or going in the other direction in the square root of each other. It says, determine the dimensions of the enlarged photograph. I want to find 120% of 8 and I want to find 120% of 12 because it was 8 by 12 originally. Sorry, 10, 8 by 10 originally. I scrolled down, didn't I? 8 by 10 originally. How do I find a percent of a number? Oh, hit number one. Look up. Sydney, what word is that? Say it nice and loud. Times almost always 99% of the time in math in anywhere. If you say the word of, hit a times on your calculator and you're fine. So it's going to be times 8, except I can't do math with 120%. What is 120% as a decimal that I can do math with? It's going to be 1.2 times 8, which is okay. The new photograph, if I want the area to be 44% larger, the new photograph is going to be 9.6 by, of means times, 120% is 1.2. The new photograph is going to be 9.6 centimeters by 12 centimeters. That gives you a 44% increase. And yes, Joe, it's ramped itself up a bit in difficulty because I don't see those numbers appearing anywhere in there. I realize that. D says blah, blah, blah, blah, forget. E, Maggie must also produce a print whose area will be reduced. I would probably underline the word reduced by 25%. If you're losing 25%, how much is left? What percent? 75, that's going to be our scale factor for area. Our area scale factor, our area scale factor. Is that okay if we abbreviate it that way? We're good. Is 75%, or as a decimal, 0.75. What's our linear scale factor? If we know the area scale factor, how do we go backwards? And this one won't work out evenly, by the way. How do we go backwards? What do we do? Square root. The linear scale factor is going to be the square root of 0.75, or the square root of 75%. And since they want to the nearest 100th of an inch, give it to me to three decimal places. We'll carry an extra decimal place just to be safe. What do you get when you go square root of 0.75? Sorry, 0.660? I don't think so. I think it's going to be a bigger number. I think that's wrong, too. That sounds better. I'm pretty sure square root of 0.75, square root, not squared, square root of 0.75 is 0.8660. Is that what you said? Okay, because you said 660, and I was going, no, I'm sure there was an eight in there somewhere. 0.8660, or 86.60%, whichever way you want to think about it. This wants the dimensions of the print. You guys stay on this page. I'm going to turn back one page. Once we have the linear scale factor, 1.2 as a decimal, or that, how do we find the dimensions? What did we do with that linear scale factor? Jordan, what did we do? Times it. Okay. It was originally 8 by 10. The new dimensions are going to be 8 times 0.8660, and 10 times 0.8660. What do you get? 8 times 6.93, and 8.66 centimeters. If you're finding this a bit tricky, this is a bit tricky. Maps. Last one. Yeah, last example. Example two. Marco, who is the owner of Mapit Incorporated, has produced a map of Canada with his employees. So the area of the province of Alberta is approximately 661,850 kilometers squared, or square kilometers. By the way, can you see another easy way to remember that area is the square of the linear scale factor? What do you notice, Joe? I know you were elsewhere. That's why I'm getting your attention. Staring right at me and creeping me out to be wet on it. That's okay. Ready? Here's the question. Can you spot another easy way to remember that area scale factor is the square of linear scale factor? Look up, look up, look up, look up. What do you notice on the units? Yeah, more specific. What do you notice on each of those units? Would that maybe help you remember that area is the square? Really? Very convenient. So in real life, Alberta is this. The linear scale, sorry, on Marco's map, the area is that. The linear scale factor can be written in the form of 1 to x. Calculate the value of x. Okay. The area scale factor on the map is what? Well, real life is 661,850 square kilometers. On the map, it's 264.74 square kilometers. So do we want to write map first or actual area first? Let's go actual area divided by map area. You okay there? Sure. What's the area scale factor here? Please pull up your calculators and type. Sorry, 2500. The map is 2500 times bigger. Then okay, that's the area scale factor. You know what? Let's put ASF, area scale factor. Is he right? Is it 2500? Somebody else get that too? Okay. If that's the area scale factor, what's the linear scale factor? How do we go backwards and find the linear scale factor? It's either squaring or square rooting. Which one? Square rooting. Okay. So the linear scale factor is going to be the square root of 2500. I think this will work out evenly. Will it not? Will it not? 50? Yes. The linear scale factor, the map is 50 times bigger. So if they want us to write it as 1 colon x, we would write this. 1 and on the map it was centimeters colon 50 kilometers because on real life it was kilometers. Alex, is that okay? Vaguely? Kind of? Okay. We found the area scale factor by going new area divided by old area. Got that. Found the linear by, oh, well if we know the linear, we square it to find the area. If we know the area scale factor, we square root it to find the linear. That's where that came from when I said, okay, that's the, this is, this is the map because I put the map on top, the map on top. So this is the map 50. What was the map measured in? Sorry, I said map. This is the real life. This is the real life. I put the real life on top. The real life is 50 kilometers for every on the map 1 centimeter. So on the same map, the area of the province of BC is represented as that big. Find the actual area in real life. This is area. Joe, you know how I know? Square area. Yep. It's dumb, but it works for me. And they want the actual area. I'll call that X. The actual area divided by the map area equals the actual area divided by the map area. There's our conversion factor right there, is it not? And I think everything's lined up properly, actual to actual map to map. What do you get? Oh, how do I solve this cross multiply? Have I mentioned lately that that cross multiply thing that you learned in grade eight is one of the most useful skills you'll learn? I have no idea what the answer is. Are we bigger than Alberta or smaller than Alberta? Like quite a bit. Almost a full third. 9 4 2 0 9s to the nearest what 100 square kilometers. 9 4 2 100 kilometers square, square kilometers is how you pronounce it. You need to practice this and wrap your brain around it. I don't think this lesson was constructed particularly well, but here's what I'm going to say. You should try. I think number two is totally fair game. Yeah, skip number one. I think number three is good. I think number four is good. Five is good. I'll go six A. Eight is good. There you go.