 We're live. So what we're talking about today are rates and ratios. It's a bit of vocabulary. I don't think, Shay, that I'm going to specifically say to you on a test, define a rate or define a ratio, but I will be using those words in context and expecting you to know what I mean by them. So you are going to need to know this, investigating the difference between rates and ratios. And it starts out, paragraph says, new at a stats agency did a survey of some Canadians and they asked the following question, how many text messages do you send per day on average? You can actually find the survey online if you feel like typing in that huge long HTML link. I don't feel like typing in that huge long HTML link. What they found was between ages of 18 to 34, on average, that age group sent 23 text messages a day. Figure out what page we're on now? Okay. Those little things that make a difference. Page number. Right? Eyes open. Breathing. You'll find the road to success involves all three of those, I'm sure. Okay. So they found on average 23 text messages a day between the ages of 18 to 34. By the way, your age group, do you think more than 23 per day or less than 23 per day on average? Okay. What about an ages 40 to 50? My age group? Not too many. Yeah. We're still mostly emailers if we're going to communicate electronically. I do have a feeling though, I won't be surprised if this number starts to go down because everyone's getting smartphones and you're really limited with the text messages. How many characters are you limited to in a text message? Emily? 140. Why? It's related to a second question. How much does it cost a cell phone company to send a text message? Not how much does it cost you? How much does it cost a cell phone company to send a text message? This was a question that two profs from the University of Kansas asked about three years ago, just when text messages were starting to explode and cell phone companies which used to offer them for free started charging money for them. They asked, well, what's it costing a cell phone company to send a text message? Do you guys know what it costs a cell phone company to send a text message? Sorry? Nothing. And it's related to the 140 characters. Your cell phones, if they were turned on, which I'm sure they're not, if they were in my class, which I'm sure they're not, but your cell phones are constantly sending out a signal looking for the nearest cell phone tower. And when they get a signal from a cell phone tower, this little handshake procedure that goes on, the cell phone tower identifies itself and the cell phone identifies it to the cell phone tower. It's sending those signals out no matter what. But it turns out those signals have a tiny bit of extra memory left over in them. Would you care to guess how many characters of extra memory those signals have? 140. The signal that they built into cell phones when they first brought cell phones in 15 years ago, that they budgeted into your bill back then already, that signal has room for 140 extra characters. They piggyback your text message on there and it costs them nothing because they charged you for it already, except now they're charging most of you for it again, aren't they? Text message charges are the biggest ripoff on the planet. In fact, sue the cell phone companies. Please, it's a ripoff. Should be free because it doesn't cost them anything. You guys know that? Be angry with your cell phone company. Demand better service. I think we get lousy service here in North America. Having said that, end of rant, back to math. Prompted by this survey, Rachel and Sarah decide to record the number of text messages that they sent per day and they do it on Wednesday, Thursday and Friday. It's for a communications technology project. Here's what they find. Rachel sends 46 on Wednesday, 38 on Thursday, 81 on Friday. Sarah sends 23 on Wednesday, 40 on Thursday, 54 on Friday. By the way, why would Friday spike like that? Do you think? Yeah, probably getting ready for the weekend people making plans or whatever, okay? We can interpret some of this data. Here is our vocabulary. Vocabulary number one, Sierra. A comparison of one number to another number with the same units is called a ratio. I would underline the word same units and the word ratio if I was someone who had purchased the book, which most of you have for our planning. Did I say Sierra or Sienna? Did I call you the right name, Sierra? Again, phew. If you're comparing two things and they're the same units, they're the same type of object, we call that a ratio. As compared to the ratio of text Rachel sent on Friday to the number of texts Sarah sent on Friday and write it either as A colon B or A over B. Well, this is meant to be fairly fill in the blank simple, so don't be insulted. How many texts did Rachel send on Friday? Read the chart. How many, 81? So the ratio would be 81 to, how many did Sarah send on Friday? Except traditionally, we will always express a ratio in lowest terms. 81 to 54 is not in lowest terms. What number goes both into 81 and 54? The first thing I thought was nine, and so when I went divide by nine, divide by nine, that got me nine to six, and then I went, oh, actually, apparently, something else still works. In fact, 27, and if you know your 27 times table, some of you may have seen that right away, but the point is you don't need to know your 27 times table. You can't whittle away at these with twos and threes and fives until you get them in lowest terms. Yeah, the ratio is three to two. For every two text messages that Sarah sent, Rachel sent three. You could also write it as a fraction, three over two. I've done a rant about buying a good calculator in this class, yes? Some of you may want to experiment with your fraction button. Now, if you have a fraction button and a good calculator, it looks like this, probably. It looks like a button like that, maybe. Or it may just be a divided by, let me show you what I mean. On my calculator, my fancy schmancy graphing one, if I go 81 over 54, if I hit enter and then I hit the math button twice, it gives me the fraction in lowest terms automatically, which is kind of nice, actually. Let's me know that I haven't made a dumb math mistake on a test or a quiz. Some of you with your fraction button, yours may do that, too. Play around with it, read the manual, not during the lesson, but if you have a fraction button that looks like that or you're wondering later on, call me and I'll see if I can figure out how yours works again. If you paid for the technology, I have no problem if you use the technology. Marcus, 81 over 54, enter, press math and then just go enter, enter. That's the fraction to decimal button is the really quick way to do it. Really handy, actually. Pardon me? Oh, we can fill an ocean with what you did not know, believe me. You ever watch Star Trek? Those things are almost tricorders. You can plug probes and temperature sensors and all sorts of things into those. In fact, technically, those aren't called calculators. They're called handheld computers if you actually look at the specifications for them. B, B says write a ratio in reduced form, comparing the number of texts that Sarah sent on Friday to the number of texts that Rachel sent on Friday. Again, this is meant to be fairly fill in the blankish. If the ratio of Rachel to Sarah was three to two, what is the Rachel of Sarah to Rachel? Two to three. The point of that Devon is the order matters. Write what they said first, first, and what they said second, second. Don't write it the other way around or you'll get marked wrong. C says compare the ratio found on A to the one on B. Is there a difference? Well, the order matters, but otherwise it reduces to the same thing. So ratio, same units to same units, same type of object to same type of object comparison. Then we have our second vocabulary words, a comparison of quantities which cannot be expressed in the same unit is called a rate. What's the speed limit on the low heat highway between pit meadows and the pit bridge? What? Kilometers per hour. That's a rate because kilometers and hours aren't the same units, but the per is like a divided by the diffraction. We use rates all over the place. Any time that you give a speed, that's a rate by definition. Any time you're comparing something to time, you're almost always talking about a rate. So Rachel sent text at a rate of 165 texts in three days. Where the 165 come from, we added Rachel's up. Express the total number of texts Sarah sent over three days as a rate. Again, this is meant to be kind of fill in the blank ish. How many texts did Sarah send grand total? Can someone add that up, please? Well, 54 and 40 is 94 and 20 is 110 and 317, yes? So if we wanted to express that as a rate, we would express it as 117 texts in three days. These we almost always write as a fraction. In fact, in English, we'll often say the word per and that implies a fraction. And you can see Taylor, not the same units. Most people, they'll look at that and they go, well, that's stupid, 117 texts in three days. Most people would express this as a unit rate. A unit rate is how many texts per one day or how many kilometers per hour. That 80 kilometers per hour is a unit rate. You could say, well, it's 160 kilometers every two hours, which is true, but no one says it that way. We always want the unit rate. So part two says a unit rate is a rate expressed with the denominator of one, we know, ins. Express the total number of texts Sarah sent over three days as a rate per day. How would I write this as a rate per day? What would I do with that three? I want to turn it into a one. How can I turn a three into a one? And maybe the answer is too obvious. Divide it by three and same to the top. Anyhow, what's the unit rate? How many texts per day does Sarah send? Thirty-nine texts per. So the next logical question is how many does she send in a month? E. If she continues to text at the same rate, and we have to assume that she is, how many texts would she send in one month, and we'll assume a month has 30 days. I know summer 31, February 28, let's assume a month has 30 days. I want you to notice something. Remember the unit analysis thing that I did last class, where we built this fraction change and we canceled out the units? I want you to notice, it works here too. I can write down 39 texts in one day times three, times three, Mr. Dock, sorry. How many days are we talking about here? Times 30 days, Shania, take a look. Technically, the days would cancel with the days, because one's on the top and one's on the bottom. And what's the only unit that's left kicking around? You're going to get an answer in texts. If you don't know, because the most common question kids have is, so I love the fly or do I divide with a 30? You can actually get the units to tell you. If I divided, the 30 would be on the bottom, and I'd end up with days, days, which doesn't make a days squared, which doesn't really make much sense. If you're not sure, the unit analysis approach can bail you out. Now, I think almost everyone here was saying, Mr. Dock, when you just go 39 times three, yes, 39 times three, 40 times three would be 1200. So 39 times three is going to be 30 less than that, 1,170, someone check my math, yes? Same units, rate different units, turn the page, or next page over. How many of you are planning on buying the book or have already bought it? If you are, this page right here where it says rates and ratio summary would be a good one to either dog here, the corner, or put a little post-it note there or something. When you're studying for this unit test, which is not stuff on Thursday, but when you're studying for this unit test, this would be one of those where you would say, I better read this and make sure I know all this because this is a summary. One of the things I do like about these books is they have summaries fairly frequently. So what I've just talked about for 20 minutes, it's all summarized right there. I'll let you read that on your own time. Then we're going to change gears a tiny bit. It says using proportional reasoning, unit analysis, and unit rates to solve rate problems. Many rate problems can be solved using proportional reasoning, unit analysis, or unit rates. Really a lot of them can be solved with cross-multiplying and building those unit analysis chains. For example, Kim and Cole were given the following problem to solve. Here's the problem. Tommy purchases three bottles of vitamin water for $4.95. After paying, he gets a text message from his wife informing him that some more of his friends are coming to visit. He goes back into the store and he purchases four more bottles. What's the total amount he paid for the seven bottles of vitamin water? Danielle, I think this is a stupid question. You know why I think this is a stupid question? Because any idiot would just look at their receipt and add them up and they'd know how much they paid. Right? This is somebody somewhere thinking to themselves, I need to give these students an application problem. Oh, here, this is in a store. And so that must mean it's an, I would never do math this way. I really hate these kinds of questions. I'm going to do it because it's here and it's in the notes and it's on the curriculum. But I really get frustrated the way they artificially contrive stupid applications. I wouldn't care if I bought three and I bought four and I needed to know how much I'd paid. I'd look at my receipt or I'd see how much money I had left in my wallet. There's a bunch of different ways that I'd answer this question. I guarantee I would not cross multiply and divide. Having said that, rant over, apparently they want us to use algebra to solve this. Fine. Here's what it says. Kim solved this using proportional reasoning. She said this, 4.95 for three bottles is X dollars for seven bottles. And that works. We said that was one approach that you could use last class. I have one fraction equals one fraction. How would I solve this equation when I have one fraction equals one fraction? Two words that you learned from math eight. Do you remember? Yes. Let's all write down so that we're still awake, cross, multiply. And those five of you in the back row that are slouching more and more and more, try setting up because I'm telling you the slouching won't help you pay attention. It'll make it worse. Not that I'm mentioning any names. When I cross multiply, it's this times this equals this times this. But they've actually done two steps at once. Liam, they've gone cross multiply. They knew there was going to be a 3x on the other side, and they knew to get the x by itself. I'm going to divide by the 3. So they went cross multiply and divide. Oh, they would have gone this times this. I think there's a 7 there. They want us to kind of fill in the blanks. How much money was it for seven bottles? X. How much is X? Get your calculators out. Somebody go 495 times 7 divided by 3. 5 times 7 is 35 divided by 3 is 11.6 repeating. I'm going to go with $11.60. Am I close? Alex, I'm close, $11.55. By the way, you can also see Alex. Even if you forced me to do math like that, I'd estimate and do it in my head. I wouldn't re-prick you. I hate these contrived questions. I've gone on a big rant on them in other locations. That's one way to do it. Another way, Emily, is to convert to a unit rate. So Cole said this. Well, look, it was 495 for three bottles. How much for one bottle? Can someone go 495 divided by 3, please? A buck 65, even? And if it's a buck 65 for one bottle, how much for seven bottles? What would you do with this number? And that's what they want us to write here. They went 7 times a buck 65. And you get an answer of also $11.55. Once again, I don't care which method you use, Devin. I do want you to notice here, though, for what it's worth, Aaron, that in both methods, somewhere along the way, you times a 7 and a 495 and divided by a 3. You times the 495, it's on top. And the 7, it's on top. And divided by a 3. It's just that here you divided by 3 first. Which way is the best way? They both are fine. Whichever you like better. Couple more we're done. Turn the page. This one I can at least say is marginally more useful. Sometimes I do like to know if I've spent more or less than the year before. So here's what this one says. Paul hosted an annual wrap-up party for his hockey team last year. This year he's hosting the party again. Woo-hoo! Hopefully not a Game 7 Stanley Cup riot. And like last year, he'll be serving prime rib roast. Last year he bought 8 kilograms for $173.12. This year he paid $10.13 per pound. Says use unit analysis to figure out which rib roast was more expensive at the price to up or down this year. At least I can say that's a bit more of a useful question. And since there's a last year end of this year, I think the first thing I would do is this. Draw a line down the middle of my page. This is going to be last year. Danielle, how much money did he spend last year? Read the question carefully, please. So last year he spent $173.12 per. And how many kilograms of roast did he get? Let's change that to a unit rate. $173.12 per eight kilograms is how much per kilogram? $21.64, round it off properly? Or did it work out evenly? Worked out evenly? $21.64 per kilogram. What did he pay this year? Matt, what's it say? This year he bought the roast for? What? See it there, kiddo? This year he bought the roast for? Nope. OK. Oh, you got it. For what? Pound. This is in kilograms. Here we're going to do a little bit of math. So I'm going to write down $10.13. Now the abbreviation for pound is still this, LB. But it's falling out of favor. You know what they're using to abbreviate pound with now? Oh, techie people who text message? Oh, you all do. What's the pound key? What's the pound key on your phones, boys and girls? Huh? There is a pound key. What? Matt, that number key is called the pound key. That's why you hear them sometimes say you guys don't drive. If there's a traffic violation called pound 11.30, and that's a free cell caller on your star net or whatever, it's slowly being replaced with the number sign, which I think is better. Anyways, I'll still use LB. How the heck LB? It means pounds. I'll never know. Here's what I'd like to do. I'd like to change this to dollars per kilogram. I need to change pounds to kilograms. And to do that, I'm going to use that dimensional analysis trick. I'm going to multiply it by a conversion factor. I want to end up in what, Adam? Where is pounds right now on the bottom or on the top? So I better put pounds on the top so it'll cancel and kilograms on the bottom. And that'll help me end up with kilograms on the bottom. And they gave me a conversion factor. Read that to me, Adam, my friend. I guess the 2.2 goes next to the LB. And the 1 goes down. And you know what? In fact, I think it's going to be $10.13 times 2.2 divided by 1 divided by 1. But am I going to bother doing the divided by 1? No, if there's a number there, I would divide by whatever number was there, but 1. OK, what's the unit rate per kilo for 10.13 per pound? Courtney, what'd you get? I thought you looked good. So last year was 21.64. I'm going to guess it's going to be close to 21.64, like in the 20s somewhere. What'd you get, kiddo? That times that, 10.13 times 2.2. Let me see what do I get. 10.13 times 2.2. Is that OK? Do you get that? And rounding off properly to, you know what? 22.29. 22.29 per kilo, which was more expensive, Joe, when you're done yawning. Last year or this year? Yep, you awake? Ah, nothing. Wait for it, wait for it. Now you've got the adrenaline rush, right? Ah, I feel good now. The last one. I like this question because it gives you all some useful information. Most of you will buy a house someday. And for almost all of you, it will be the single biggest purchase you ever make by a long shot. What many of you don't realize is buying the house is only the first part of the fund. As soon as you own it, specifically as soon as you own the land, you get charged tax, property tax. So here's how it works. Property owners are required to pay property tax on an annual basis every year. And there's various ways to calculate the tax. This is the simplest way. Most municipalities don't use a method this simple because they want to charge you more money because governments want to charge more taxes. So the property tax amount is based on something called the mill rate. And it's called the mill rate because the number is expressed in mills where one mill is 1 tenth of a percent. It's sort of like millimeters and milliliters. There's a mill rate. And the assessed value of the property, in fact, this is the basic formula that they use. Property tax equals your assessed value, what your house is worth, divided by 1,000 times the mill rate that year. That's way too much writing, way too much writing. I'm gonna rewrite this as a formula, but I'm gonna use letters. What would be a good letter to use for the phrase property tax? I'll give you a hint. Q would be a stupid letter. What would be a good letter to use for the phrase property tax? P or T? Let's use P for property tax. Ah, you know what? It's property tax, let's use T for tax. T, write this down, equals. What would be a good letter to use for assessed value? V, you say? Over 1,000 to still 1,000 times. What would be a good letter to use for the mill rate? So apparently to figure out your taxes, the value of your house divided by 1,000 times whatever the mill rate is that year. Let's look at part A. Charlene paid $1,924.39 in property tax last year when the mill rate was 4.6038. What's her home's assessed value? So take a look at this equation here, this little formula. Are they asking me in part A to find T, V, or M? Read the question carefully. Are they asking me to find T, V, or M? V, that means they told me everything else. So read the question, what's the tax? Can't tell whether you're dead or confused. I think dead. Ah, nothing. Danielle didn't even twitch that time. Okay. Yeah, Danielle, I think the tax is $1,924.39 equals. We're trying to find the value divided by 1,000 times. What's M, the mill rate according to this question, Boston? What's M, the mill rate according to this question? Read, my friend. Yes, 4.6038. Let's get the V by itself. I'd like to do this in one step, so you're ready. What's the 1,000 doing to the V? Adding, subtracting, multiplying, or dividing? Dividing, so how will I move it over? What's the opposite of dividing by 1,000? Timesing. What's the 4.6038 doing to the V? Adding, subtracting, multiplying, or dividing? Timesing. What's the opposite of timesing by 4.6038? How will I move it over to the other side? There's your equation solving very, very quickly. I think you're gonna get this. I think the answer is going to be the value is $1,924.39. Alex, if I'm correct in hearing you, I would times by 1,000 and divide by 4.6038. What was this person's house? What was Cindy Charlene's, sorry, what was Charlene's house assessed at? And by the way, if you want the shock of your life, if you haven't already, if your parents are okay in telling you, ask them how much they pay in property tax, it's stunning, you think. Well, they own the home, how can they tax it? They don't need to have rules that make sense for taxing. They'll still tax you. What'd you get? It's in the hundreds of thousands, I think it's over 400,000, if I'm roughly in my head. What'd you get, Emily? Four, read me the digits, four, so three, five, because we'd go to dollars and cents. Yeah, her house was assessed at a little over $400,000, $418,000 and 35 cents. Is that a reasonable answer? Is that what houses are worth roughly give or take? Yeah, they're in the hundreds of thousands. B, the last question, no cheer. B asks, B asks, how much does her property bill change for the current year? If her property is assessed 10% lower and the mill rate increases to 5.2113, okay. First question is this. What are they asking me to find in this question? I think they're asking me to find T. In fact, I think they want me to find the increase, which means I'll find T and then subtract it from last year's tax. But they want me to find T, which is this equation right here. Let's write it down. T equals V over 1,000 times M. You gonna be okay, Boston? You can deal with the distraction? Sure, okay. Boston, once again, since you did so well last time, what's M, what's the mill rate in this question? 5.2113, and this is gonna be a 1,000 on the bottom here still. The real question is, it says her current property value is 10% lower. If I lose 10%, what do I have left as a percentage? 90, put your pencils down and look up. Don't write this next little bit down. I think what we wanna put here is 90% of, need more room to write, $418,000.35. That's what's going to go there. The real question is math eight review now, how do I find 90% of a number for a candy? Who can remember? What about 0.90? Boston, you just, hang on. You just impress me. Not a boy. First of all, you can't do math with a percent. You always have to make it a decimal, and 90% is 0.9. You learn the trick of dividing by 100 or moving the decimal place over two spots. And if you want a little trick of the trade in math, the word of means times. How much tax this year? 0.9 times that, times that, divided by 1,000, get your calculators out. I'd like to know how much tax we're paying this year. Marcus, what'd you get? Oh, I thought you raised your hand like you had an answer. I'm getting $160 and 49 cents, sorry, 1,960 dollars and 49 cents. Okay, 1,960 for, 1,960.49. And I'll add the units dollars. How much more than last year? Go that minus that. We're not gonna bother. What's the whole point of this? Rates and ratios and unit rates and cross-multiplying are your friends. They're handy. Turn the page. What's your homework? Your homework is as follows. Number one, number two, three A, sorry, not number two, three A but not B or C. Number four, number five. With me, Devin. Yeah, I know. That's why I'll come back to it and make fun of you when I do. Okay. Six A and B, seven. Number seven is asking which of these two answers is correct and why? Cause one person got nine minutes and one person got 4.5 minutes. Eight yogis yogurt and nine. You had about a half hour left. You'll get it done easily in class and no homework in theory, except what is your real homework? Make sure you're prepared for the quest on Thursday. I'm gonna hit stop. Then I'm gonna take some, hang on, stop.