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Math 7 - Module 1D - Lesson 19 - Problem Set

Added Jan 16, 2021 •

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Check your answers and work as you follow along the video where the problems from the problem set are worked out. If you still don't understand, bring it up and have questions next class or during Academic Support.

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song with this lesson summer let's read the lesson summary before we start though cuz these are all the main ideas that the work she thinks you should know given the scale factor that use the letter R whatever representing the relationship between the scale drawing length the actual length of the square of the scale factor r squared represents the relationship between the scale drawing area and the actual area all right that right there was a long way of just saying that the scale factor squared equals the area factor that was just a long way this right there it's just that all right for example if 1 inch on a scale drawing represents for in the actual length of the scale factor is 1/4 on the same drawing one square inch represents 16 squares r squared is 1/16 hairs are there Zara Square you sweat 1/4 x 1/4 this is the main idea how do you use it that's the next bit that's the subgroups let's go with this number one the Shaded rectangle shown below is a scale drawing of a rectangle whose area is 288 square feet drawing of the real life drawing which has 288 of these little squares inside what is the scale factor of the drawing o we want scale factor but what they gave us is the area so let's find the area of this one first we can't all this is one two three four four units high 1 2 3 4 5 6 7 8 8 units long so how many little rectangle I mean a little squares exist inside this 1/4 * 8/8 * 436 district has 32 32 32 little squares inside so let's write that down area equals 32 that means we got 32 little squares in here but in the real drawing it would be 288 man I ain't about to try and drop some 288 little one of these things that's annoying but that's why that's why I we we set it up no okay good just saw the height is good the overall height of this whole thing is 6 they wanted a border they put this border around here to show you that it's made up of a bunch of these little squares but we only I want this right here the Shaded rectangle yes this whole thing is 6 but we don't care about the whole thing we only care about the Shaded part so the Shaded part is just one two three four so it's there to trick you so good catch good question on the reading STAAR test will do the same thing as Lexile that little bit right there would trick you some people would turn all of it we don't need to really want that the same thing out here it's 8 not 10 now in the actual drawing the rectangle is 288 of these little squares you know what I'm too lazy to do that so I'm going to try sketch of it what is the facts Kemet so this is my drawing but the actual want some of that she put it cuz this is what came before this is the old one old and this is the new show the area it told us the area for this one is 288 288 and it wants the scale chapter of this drawing so how many times smaller how many times smaller is this drawing compared to this one I can't find the scale factor because I don't know how long this is I don't know how song this is but I can find that out let's set it up our area factor for even though area Factor is scale factor squared what it doesn't say here there's another way to find the area Factor it all they give you are the areas area factor is also the new over the edge and that's the hard part which one's the new which was the old that's where reading the Lexile comes in so I don't have a scale factor so I can't find the area factor using this way but I have both areas does it have both areas I can find the area Factor the area Factor again I am looking for what is the scale factor of the drawing this is what I want this is what else exists so what I want is from this one over that one so 32 / 288 we need simplify that cuz that'll tell us our area factor 32 and 288 what can I divide by off the top of my head I know I could / 4 but my brain cannot do Thursday fine for you sent for 32 I was going to do too I just keep on doing that way but you said for we go to 32 / 4 that is 8 288 / 4 I don't know I'm going to have to use my calculator now 288 / 4 72 all but I can divide one more time 8 and 72 can both be divided by 8 8 / 8 is 172 / 8 is 9 that my area factor is 1 nights so we found our area Factor but we just want the scale factor what is the scale factor of this drunk how many times smaller is this drawing compared to this one so look at this we have that remember there's the formula area Factor equals scale factor x scale factor but let's go the other way around area Factor equals scale factor squared which is another way to write a scale factor scale factor but let's write this for the area factor is 1/9 but instead of writing scale factor squared remember the examples we kind of did earlier one night equal scale factor x scale factor if I find one of these numbers cuz they're the same number I answer the question what's the scale factor oh no the area factor is not the same as a scale actor good question Bay excellent question the area factor is the scale factor squared and let me put this right here the scale factor x the scale factor get you the area Factor so it's a different number but it's related so if what two numbers can I multiply that are the same and gets me one nights because it's a fraction people always stuck in their head we got a fraction up top the number is 1 what numbers multiplied together get us one what same number multiplied to itself equals 1 excellent yes so 1 / 1 / 2 1 x 1 gets us 1/9 something when but here's the thing what's a number multiplied by itself gets S9 what number times what number is 9 what times what equals 9 correct 3 * 3 one-third times one third equals 1/9 and that's supposed to be the scale factor x of scale factor so we have our answer right there the scale factor equals 1 3rd what that means that this rectangle all the sides are 3 times smaller this is one third the size of this one that means this is three times bigger than this one so if this site is four times it by 3:12 if this site is 8 times it by 3 24 12 x 24 288 the area matches up it did it did not hear hear that was extra I just wanted to prove it to you we did not it didn't ask for us to find them but we could based on the scale factor it's like a whole thing that's connected that's why right here so I'm going to go back to Legos the Legos when they sell it to you you get the One Direction map that tells you how to build but then in the back of the Lego boxes if you decided not to build that you could look on this out of it different they didn't give you a map of that but you know all those pieces use the same ones in the sometimes less so all these things are connected in some way it's how do you connect it like a puzzle let's go to the next number 2 the floor plan the floor mine is a blueprint the floor plan for a home shown below 1/2 inch corresponds to 6 feet of the actual it's there to mess with you in math equals 6 ft ladder has a ruler let's look at it half an inch it was put in the white so we can see it so here's one in a half inches right here so half an inch equals 6 ft in real life but why didn't they just go all the way to 1 inch instead that's a nice one number it's there to make you think if a half an inch equals 6 ft what is weight of 1 inch pipe it in before we move on 1 inch equals what cuz I want to change it up make it easier on us if half an inch is 6 ft a whole aisle rights on my if here to hear is 6 what's your to hear good correction excellent 12 so 1 in 12 ft so we could change this here and here's why it's important let me write this down cuz you want to find a scale factor always try and find yourself after first if you go scale factor look at this we use our scale right here and to find her skin actor in real life versus the drawing 6 ft 2 1/2 6 ft in Real Life 2 1/2 inch in the dry do you want to use this that's why we'll go back to that later yes we can that's why you want to use your logic I don't want to go 6 ft 2 1/2 inch using your logic I don't need to do the extra work I know that that is 12 to 1 inch look at how much easier that is + 12/1 is another way of writing 12 Man Without drawing writing that division stuff fractions C student see student which is okay you're still passing the C stood would have done this extra work you would have done the dividing fractions thing and you would have ended up with that the person using logic I think it through before they do all this extra stuff 12 in 12 ft so let's read on bedroom two belongs to 13 year old Cassie bedroom 3 belongs to 9 year old this Cassie claims that her younger sister Alexis got the bigger room is she correct I'm going to tell you this right here and do this all the time I don't care about the bathroom I don't care about mom and dad's room I'm best I can see the kitchen that's the fun the closet and this leads out prior to the living room who know downstairs upstairs so it's just there. That's just dirty confuse you that's what you guys need to understand in B's math test you guys get what about the bathroom what about bedroom and we don't care cuz the question only asked about Cassie's room and Alexis's room so we have this let's find the area of both of these so we need the ruler on the reading STAAR test you don't yeah everyone has a different size computer monitor so everything comes different what they're going to do is there's a butt bottom of the screen does a tool button click on the tools the tools you can drag out they'll drag out a digital ruler and could shrink it make it bigger or smaller by zooming in and out on it and what you're supposed to do is you're supposed to use your mouse or turn the ruler and face it this way and then you find your measurements so bedroom 2 Cassie's room is 1.25 long 1.25 1.25 in is how tall is Cassie's room it is not a whole one inch it is .75 now let's look Alexis's room Alexis's room is 1 in bye looks like a square list double check 1 inch Alexis house Square Room I don't like Square rooms they're creepy I'd rather have a rectangle Kizi Dept so if one in 1 inch represents 12 ft manlift convert that that's easy 1 in equals 12 ft equals 12 ft area on this one super easy to find 12 x 12 so Alexis's notice I'm not driving right all this extra stuff no use your logic to figure this stuff out so Alexis's room the area is 144 12 x 12 that was super easy now we just got to spend some extra work and do Cassie's all right so let's use Cassie's we got to use the formula so we got to find the area who has the bigger area so square square feet and scale factor so we want to find the new measure old measure time's the scale factor equals the new measure so let's look at the area so the old measure is 1.25 we got to multiply it by the scale factor of 12 take us a new measure but we the other measure .75 * 12 that guesses the new measure calculator 1.25 * 1215 12 9 area so her area is so in real life that is 15 in real life that is so it's her room is longer but it's narrower cuz that's 12 but what's 15 * 9 * 935 Cassie's room vastly smaller does the area equals 135 Cassie claims that her younger sister Alexis got the big room is she right yes how would you feel if your baby sibling or how do you fix your older sibling feels if you got the bigger room do they remove their stuff and you wouldn't even know it that's one way to do it that's one way to do it that is one way to do it that is another way to do it I'm going to zoom out so that's just using basic logic and plugging in stuff doing all that extra stuff another way to do it would have been this using the area Factor so that's using just the scale factor finding the measurements in just finding the area of each room method that method one method to find the area Factor first area Factor equals the scale factor squared sore scalp 12 and square that 12 squared is 144 what do I do with this 144 I can find the area of both of these so instead of using this very similar to using this one old area find the area Factor equals the new area answer the whole thing is we we are looking for the area both of these rooms it's a better by it's like I said which is the better buy which is the bigger room a comparison things we do this twice look how easy this one is what's the area of Alexis Room 1 * 1 1 * 1 is 1 are you Alexis 1 * 1 times the area factor which is here 144 equals the new area area and real-life hers is 144 let's go look at Cassie Cassie the area of this the old one is 1.25 * 2.75 / 1.25 x 175 and the area Factor 144 * 144 and watch we should get a hundred thirty-five 1.25 * 20751.25 * 75 0.9 375 x 144 equals x 144 135 so we could compare Alexis's and Ken look at this right here look at the amount of work I put in if I if you say looking for the factor with this it's just this little bit right here look at the other way finding all the different numbers that's everywhere just practice I just I am not that good I'm only good to a certain level if you guys improve every year all of you are going to be better than me your junior year of high school I do not remember anything try and do it what's the junior high school someone said Junior High School math is probably a pre-calculus and or Algebra 2 Algebra 2 I only remember probably the first third of the book after that I have to actually be like you in I don't remember that anything up all of you right now yes I'm better right now but when you get to your junior year I will say what's the filing of the first third after November after November your junior year of high school at the 11th grade you can consider yourself smarter than me and mess it's just practice all it is practice it's like a you don't want to see me play basketball I can't dribble I I I had to look at the ground when I dribble and an ni instant got to look like an idiot that's why I will never ever let anybody take out their phone when I'm playing basketball cuz I I look stupid if you ever give me a ball during basketball time do one of two things I don't care where I'm at if I'm on the back court super far away if I cannot pass that ball I'm going to shoot it across the court cuz I am not willing to travel so it's practice people I say I want you to practice cuz one there is no free basketball court that I can hide myself in practice you know dribbling cuz dribbling at home Force when you don't have a court doesn't feel right just dribble around so just practice like everything you know practice what what little things what little things can you do I keep saying it do those i-Ready games the i-Ready games work on 44 for the first four things that I ready games do they practice your adding subtracting multiplying and dividing the next group of I ready games practices your fractions so those Concepts but you have all those five you can do almost anything if you know how to do it just because I give you a hammer can you build a house no you have to know how to use the tools plus practice it's all good and what you guys don't get right now you're just learning this stuff you're learning you can't be a master of something you just saw you can be a user just like that same thing when you first got got to think when you first got into the first phone you never got if he has had fun. Some of you guys are pants on and I use a phone you have a thing you didn't know you were hacked a slow you had to slow at texting now y'all can be like texting a whole paragraph in 3 seconds get type senses riding emojis I stashed three big looking questions we could do around this time on the floor on the mall floor plan think about this this very different on a mall floor plan 1/4 of an inch represents 3-ft in the actual store so let's save time and use logic and not deal with that fraction here is his white space so we could see 1/4 if 1/4 Sears 1/2 1/4 right here 1/4 means is from here to here 1/4 represents 3-ft in real life in this is 3 ft what does 1 in equal then hear all the way to hear if that green is 3 ft in real life what does 1 in in that helps us cuz we don't have to do all that math this is what you want a picture in your head take a second out type it I'm going to give you a second before we cuz if no one time is it in we're going to do the long way and show all those steps if it what is it okay number 319 yes I think you played me can I watch the whole video I didn't say nothing about lesson 18 Chevelle SS we're doing 19 Seasons 19 the 18 video I got to wait because it's not converting right now the system is down but I'm going to try to do it so that's why it's so right here we want to find a scale factor here's the scale to write the scale factor remember it's new over the old but in this case the new is what we want that's 3 ft over one-fourth of an inch but I don't want to simplify that and do the work that's why I want you guys to look at this ruler and imagine this a number line right here I'm going to start over right here and erase that so it says 1/4 of an inch is 3 ft in real life but let's just look at that hears 1/4 of an inch is 1/4 here to hear that's 1/4 is that represents 3-ft Andrew wife what we don't want to play with a fraction with a whole number what does 1 inch represent think about that let's cut it up that's why I highlight it like that if here to here is 3 ft think about Legos Lego logic is this whole thing me if here to hear is 1/4 ants that represents 3 ft what does one inch when I'm saying is what does 1-inch represent then without doing all that extra long work this green here is 3 what should that be what should that be what should that be and when you put it all together as a Lego how long is it we got some good we got four of them true that's four of them but what you got four of them hold that thought there are four equal sections for equal Lego pieces each Lego pieces that green and I said is 3 long if this is 3 long that's three long that's three long that's the real on I got four what is the total length that's Lego logic I got four pieces of Legos that are the same length each three four of them what is it total 4 out of 4 is equal to 1 yes good first sections equals one inch that's why we got this now is 4 out of 4 but we got four sections of three think about Legos here to hear it here to hear is 3 here is also three here to here is also three here to hear is 3 what is all together now as one giant Lego piece 1/4 equals 3 we got four fourths which is one whole this whole thing is 1 in true each Greenpeace represents three 333 now we got one whole one in is what in real life guess what we get on top 3333 Lego logic says you add all these pieces you connect all these pieces together correct in real life it said for 1 and 1/4 inches on the it's 3ft in real life so 1 inch on the drawing means it's 12 ft in real life that's Lego logic okay so I got to show it the long way because some people are still stuck remember this is 3 / 1/4 let's do this off to the side right here 3/1 / 1/4 we don't like dividing fractions so let's change it to mold find the fraction multiplying by the reciprocal 1/4 becomes 4 / 1/3 * 4 is 12 / 112 / 1 supervisor 12 so that's our scale factor it will also help recipe finder Eric if a scale factor is 12 remember the area factor is just the scale factor squared scale factor x scale factor 144 phone number over and over again alright let's go look at this right here find the actual area store number one in store number to we're comparing so let's find the actual are that means we got to measure it so we got to measure so cats eat at the same height same height so let's measure this from there down to hear that's 1 and 1/2 in 1.75 1.75 so both of these are 1.75 High the let's see how wide they are this one right here is 1 1/2 what's a 1 and 1/2 surrounding an hour and a half cuz that's to one and a half so this is 1.5 this one right here is that's half right there so it's 1.25 so this is a little bit small for the same height but start a little bit more narrow so what is the actual area in real life we don't have any measurements in real life except for we can calculate them separately but let's just do it this way right here find the actual area in store number one in store number two old are times the area Factor equals the new area so we're just doing it twice you need to see it's two separate problems so let's do area equals 1.75 * 1.5 length times width 2.625 area of this equals 1.7 5 * 1.25 area equals 2.1875 so this one you can tell it's bigger than that one it said find the actual area in real life so we got both of the old ones now to find the actual we got to use the formula so old times area factor x the new year so the old is 2.625 time old I'm the area Factor what's our area Factor wooded area factors 144 and we'll get the new actual real area same thing right here the old is 2.1875 time's the area factor which has 144 we get our new all right let's plug it in a x that's all we got to do so got my calculator x 144 Equus 315 that's the new so in real life store number when is 315 square feet this one right here 2.625 * 144 378 that's what it is in real life so that's the area that's using that using the formula we could convert all the size to real life and then multiply those two numbers that's another way to do it so two ways to do this one I'm using just the straight up area factor in the area factor to calculate my areas again I could convert this 1.75 to what it is in real life so let's look at this in real life 1 in equals 12 ft so that's 1.5 so in real life this equals 12 1 is 12/2 is 612 and 6 in real life this is 18 ft long and this right here I got one which is 12 .75 what is a h yes 8 + 12/8 + 1220 1 and 3/4 / 12 old area I'm new here yeah I got myself lost 1.7 5 time is the scale factor 21 eye man I knew I messed up 21 18 I guess so what's 21 * 18 * 18 378 Mansi I got myself lost but which way is better I don't know it's up to you when you can use the air Factor Find the Errors of the little numbers and just tons of other that you get the new one or you find the the new lengths and just do the same which one came first the chicken or the egg that's the whole thing do you want to play with the chickens and then make an egg or do you want to hatch the egg and then play with the chickens that's the metaphor all right for the greenhouse Club is purchasing seeds for the lawn in the school Courtyard the club needs to determine how much to buy unfortunate looking at someone that right there no numbers it's just a story set up unfortunately the club meets after school the students are unable to find a custodian to unlock the door Anthony suggest they just use the school math to calculate the area that will be needed that they want that will need to be covered in see he measures the rectangular area on the map and find the length to be 10 in wide by 6 in the map of notes so this is what you would do this is actual and actual usage cuz you guys are going to use the map when you trade your comes back if you guys do the outdoor thing the math notes that the scale of 1 inch on map represent 7 ft on the actual Courtyard what is the actual area in square feet show look at all that long stuff and they give us this much space to do work well that's good enough so first let's find the original area cuz with the youth we find the original area we find the area factor x these two things together we actually get the real life measurement let's go end this is enough space to do all the work so first let's find the area Factor well it's fine to these two things separately the area of the old the area of the old as they are the math set 10 in rectangle 10 x 6 10 x 6 so the original area old area equals 10 * 6 so it's 60 so we have the area the area Factory need to find the scale factor area Factor equals the scale factor squared so we got to find the scale factor first the scale is 1 inch to seven so the scale factor is 7 FT 1 in in that's just seven so are you factor is 7 squared which is 49 and then all we got to do is plug these two things in the year old area X the area Factor let's go find a new area new real life area the new real life Courtyard equals the old which is 60 old area X the area Factor 49 and let me use my truck to calculator 2940 square feet that's how big the courtyard just look at that this whole paragraph all I require was that much space to do the work and notice all I did was just follow the recipe book and that's what okay here's the first wave beginning so he another thing someone said how do you get good at math well it's like cooking unless you can throw stuff together and just see what the experiment tastes like let's let me throw some sardines ketchup and cheese together see how that taste as any spare me yeah I see some faces or you could get a cookbook and start looking at recipes I'll look sardines I could put it in a salad weight I can if I if I get a special type of a sardine and anchovy mash it up that's a Caesar salad as long as I got some parmesan sub romaine lettuce you follow the recipe so you start learning the fall the recipe and you start seeing patterns of things that use so look at this that is my recipe old the old area X the area Factor equals the new area but then sometimes you got to look here's my recipe but how do I find this cuz that was ingredients for a c salad but how do I make the salad dressing look at this in order for me to find the area Factor I need the scale factor first song even though I have the overall recipe I still might need something just set aside and that's what that is right there and then the same thing right here even though I say that's the old area in the recipe what I got to find the area first so I need to have my croutons how do I make my futon might need a recipe for that but when I combine all my tools together I can use my recipe and I get my answer so it's about knowing how to connect all these different recipes all these different ideas together to get 1 new thing and that's pretty much every day learning we're always going to use something old connect it with something new to make an even newer idea a new recipe and that's what this is right here let's do the last one the company installing the new in-ground pool your fancy got swimming pool in your backyard the company installing a new in-ground pool in your backyard has provided you with scale drawing shown below if the drawing uses a scale League of this one in equals 1 and 3/4 feet whatever calculate the show amount of two dimensional space needed for the pool and its surrounding patio so this right here's the trick you so there's your swimming pool but you also need to have it's around it so you can be able to walk so the swimming pool is there to trick you we don't care about the swimming pool cuz it says for the pool and its surrounding area that means how much space do we need in real life for the whole thing that little square inside is just there to trick you that's Lexile battery so all right is this is my picture what is it in real life what is the area in real life so first let's if we're going to use the formula all area times the area factor that equals our new real life area this is our pretty much a recipe but we need to separate recipes all right we're going to go do that one in a bit the old area area if I just got to find two old recipes to make this new recipe work so we can find the old area old area Eagles and what's the area what's the area of a rectangle length times width and they gave us this drying right here and here we even got to do something before before we even find the area we got to change these into improper fractions so this makes number seven so we know it's going to be 7 down at the bottom 7 1177 77 + 380 87th and strike that down this right here 77 * 22 + 2 722 x 7 x 14 carrier one 14/150 + 5450 4 + 3 157 so we're going to multiply these two things together that's the area of the old so 7 * 749 and 8 * 15780 * 157 80 * 157 huge ginormous number one 12560 because I have my trusty kalkulatori I can divide I know that's division 12560 / 49 / 49 you can't see it cuz there's a weird glare man it's a messed up looking number I got 256 so what's 250 room 256 it's going to be a mixed number dang it call funky hold on hold on 7 * 84 doesn't work all that well so 12554 4 - 60 12560 - 12544 equals so I guess 256 + 16/49 that's what that simplifies to doesn't come out even if that's all right it's not hundred 5756 you sure oops 06 that's why I messed up a fine so I messed up the whole thing then thank you for letting me know so let's review this 80 x 156 let's use a dark color to write over it so it's 12480 12,000 180 / 49 it still doesn't get us a nice number 80 x 156 let me double-check that over 7 write 777 + SOA / 4912 480 divided by 49 I get 254 and there's a fraction right here 254 times for 9 equals force 644-6446 34 / 49 the old area that's all funky so right now we got to find our area Factor now that we got that our area Factor area Factor so let's look at this scarf under scale factor for that's not fun to do scale factor scale factor is equal to 1 and 3/4 feet over one which isn't too bad so scale factor is 1 and 3/4 change that into 1 3/4 + 7/4 and so our area Factor has to be this squared 49 over 16 that makes a super easy then okay okay okay okay actually what's use this I'm going to save that one so get that work Watch What Happens sorrow now we can now we got all the pieces so we found the old area got the scale factor scale factor got at the area Factor so we old area and area Factor now become super quick let's ride it it looks hard but as much. 12480 / 49 * what's the area Factor 9 / 16 you got your thinking man that's huge huge so I'm going to do something I'm going to do one X2 extra steps so multiply fractions multiply straight across 12480 * 49 over 49 x 16 we can flip the order we multiply thing this is the extra step but instead of 49 x 16 I'm going to rewrite it as 16 * 49 I want you guys to see this part what's 9 * 449 / 4949 / 49 yeah one this is technically * 1 I asked you don't need this I could cross that So my answer is 1 12480 / 16 which I have a calculator for 12480 / 16 780 in real life what's 780 square feet you looked at all these messed up numbers would freak me out to that's why the double check my work

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