Welcome, this is our algebraic thinking lesson or webinar for level 5-6. So welcome to Math Kangaroo. We have some new students joining us so you will notice that your videos are off and you should remain muted. This is being recorded so that is for your privacy. Anyone who misses anything or missed some previous lessons or you want to see a problem again there are recordings being made of this webinar and you will have access to watch them to catch up to see something again. Okay, we do have polls for several of our questions today. If you have some questions or want to submit some answers and get some feedback we have a TA with us, Jacob, can help you. I also take a look at my chats as I'm going. Can't always reply to everybody but we do want you to feel welcome in that you are getting some help. Just remember it can't be too personalized because we are trying to serve everybody in the group. Alright so algebraic thinking. If you are a young 5th grader and you haven't been taught algebra before and that seems like a big word I bet you actually have been taught algebra. So anybody who gave you a worksheet from kindergarten that said 3 plus blank equals 5 and you filled in a 2 you were basically already starting algebra way back then. So don't let the word algebra be anything intimidating to you and if you've done a lot of algebra then I think you might really enjoy today's lesson. Okay, here is our warm up problem. Alice subtracted two two-digit numbers. Actually she subtracted one from the other. Then she painted two cells. What is the sum of the two digits in the painted cells? Remember we don't want the digits we want the sum of the two digits in the painted cells. And I do have a poll. I'm just going to wait a minute to launch it so that you can see that picture. If you have some scratch paper you can write blank 3 minus 2 blank equals 25 so that I can launch the poll. When you answer the webinar it is anonymous when you answer a poll We do not know which students have answered and who has not or which answer you have Selected so never feel embarrassed to make a guess in Math kangaroo contests we do not subtract any points for wrong answers So if you have no idea whatsoever just do an eenie meenie miney moe If you can eliminate a few choices, you're even more likely to guess correctly and get the points anyway Okay, I'll end it. It looks like we have quite a few of you are saying you believe that the answer is 13. So let's take a look. See, it was anonymous. We don't know who said which answer, so there's nothing to worry about. I like to do my subtraction vertically. I don't know about the rest of you. So I'm going to make it some blank 3 minus 2 blank equals 25. Okay, 3 minus something equals 5. Well, it would have to be a 13, right? Because 3 minus something would be a negative number. So 13 minus 8 is 5. Okay, now in order for us to have a 2 in the tens digit in the answer, I have to have at least a 4 up here, but I know I also borrowed one, so it's actually going to be a 5 because I had to borrow or regroup to get the 13. So 53 minus 28 is 25. We can always check it by addition. So some people might have solved this problem by using addition, which is perfectly a very nice way to do it. So you can do 25 plus 2 something equals something with a 3. So 5 plus 8 is 13. Carry a 1. That's also 53. But we have to be very careful because the question asks us for the sum of the digits in the painted cells. So it's 5 plus 8 equals 13d. So there's your first exposure to what Math Kangaroo is calling algebraic thinking. But, you know, fill in the blanks is something that you've been doing for a long time, most of you, so don't worry about it. So algebraic thinking involves the ability to recognize patterns, to generalize things. So we noticed that we had to use regrouping in that previous problem, for example. Quantitative reasoning skills, all right? Subtraction, the opposite of subtraction is addition. Sometimes we might be reversing our operations in order to figure out an unknown. So you might be using variables. I'm going to encourage you, when you use variables, use an easy letter. So sometimes we use X, but if I'm talking about some sort of toys or time, I might use T. If I'm talking about apples, I might use A. I try to avoid using the letter O because O looks like a 0, and it confuses me, and no one wants to be confused when you're doing a problem, right? So algebra is frequently these variables, and we use a symbol that gives us a mathematical unknown. So I think you'll like these. Positive and negative numbers are frequently used in algebra, although in this level of the Math Kangaroo contest, I don't think I've ever seen negative numbers, but certainly by 7th and 8th grade, there will be negative numbers. So here's a little think of a number fun trick. Jacob, do you mind doing this with us? So I'm going to ask Jacob to think of a number. Jacob, don't tell us. Don't tell us. Just think of your number, okay? Now add 3 to your number. Okay. And double that. Subtract 4. Cut that in half. Subtract your original number, and you better get 1. Yeah. Did you get 1? Sounds right. Yes, sounds right. Yeah. So we can think of this like this. What was your number, Jacob? My number was 12. So if we add 3, we get 15. If we double that, we get 30. If we subtract 4, we get 26. We cut that in half, we get 13. If we subtract the 12, then we do get 1. Now, why does this work? Well, we have a representation of this. The original number is the blue mystery bag. You'll notice when we add 3 and double, now we have 2 blue mystery bags and 6. When we subtract 4, we come down to having 2 additionals. Cut it in half. Now there's only 1 additional. So no matter what number you use as your blue mystery bag, you're going to always get a 1. You can probably come up with a bunch of these little games to trick your friends. All right, here is problem number 1. A bridge built across a river. The river is 120 meters wide. One quarter of the bridges overland on the left bank and one quarter of the bridges overland on the right bank of the river. So I like to kind of even just go like this. So we have 2 things. How long is the bridge? So maybe draw a picture. Here's the river. And we have a bridge going over it. And we know that this is one quarter. Oh, not a half, a quarter. And this is one quarter. How long is the bridge? Anybody else want to put their answer into the poll? Remember, don't leave blanks on a real math kangaroo contest because blanks are definite zero and even a one in five guess is a 25% chance of getting it right. Okay, I'll end the poll here and share the results. We have a tie for answers, so let's take a look at the problems. Okay. So it tells us the width of the river is 120 meters. Notice I did not draw the scale, but that's okay. So if that's 120 meters and I have one quarter on the left and one quarter of bridge on the right, and these are both over land, that means that only half of the bridge is actually covering the river, right? This might be so that, you know, if the river bank, if the river gets a little higher, there's some space to still go over it on the bridge. There could be construction reasons why they made the bridge longer than the river is wide, but we know that 120 meters is half the length of the bridge. So if 120 is half, then 240 would be whole. So the answer is D, 240. Okay. So let's move on to our next problem. Three members of the rabbit family eat 73 carrots altogether during break. The father ate five carrots more than the mother. Their son ate 12 carrots. How many carrots did the mother eat that week? So be careful. We wanna know how many the mother ate. We have the son and we have a relationship between the father and the mother here. See if you can work that out. Do not have a poll, but you may put your answers in the chat to me or to Jacob. If you have some questions, if you need a little help or some hints, and you can, maybe Jacob will be able to help you with that. Thanks, we're getting some correct answers. We're getting some answers that are pretty close. I have a guess that some of you figured out how many carrots the father ate rather than the mother. So I'm going to use straight up algebra and we'll see how that goes. So I'll make m is the number of carrots that mother eats, carrots that mother eats. And since it tells us that the father ate five carrots more than the mother, m plus five equals the father. That's father's carrots. And then we know that the son is 12. And we know how many there are all together. So mother plus father plus the son equals 73. Okay, if you're not familiar with using algebra in this way, try to follow along, but I bet you can figure it out even without it. Okay, so when I do addition, I know that the associative property says that I can move those or even remove those parentheses because everything is added here. So I get m plus m plus I can do five plus 12 is 17 equals 73. Now, remember, m and m is two m's, but we can just leave it m and m, it's fine. When I want to figure out something and I have an addition, I can undo it by doing subtraction. For those of you who haven't done this before, you can just say, okay, m plus m can reverse it. And so I can do 73 minus 17 equals m plus m. Right? So if I subtract 73, I get 56, right? 13 minus 7 is a 6. I've regrouped, so I have 6. I have 56 equals m plus m. So half of that must be m. So m must be 56 divided by 2, which is 28 carats, that mother 8. Now, say you're not super comfortable with those variables like that. Well, you know, the son ate 12 carats, so you could do 73. Oh, sorry. Sometimes my tablet gives me a little extra line like that if I touch the screen in a funny way. 73 minus 12. So that gives me 61 carats for the mother and the father. So even if I just took the 61 carats and I divided it by 2, I would get like 30 and 31. That's the closest I can get without half a carat, right? And it could be that they ate half a carat, that's fine. So I need to have that same total, but I need there to be 6 more for father, so I could do 29 and 32. I could do 28 and 33. And when I get to 28 and 33, that's when the difference is 5. So I know that mother ate 28 carats. So that would be a slightly more indirect way to do it, but it's still involving some of your algebraic thinking ideas. Okay. For problem three, I do have a poll. The weight of three apples and two oranges is 225 grams. They're giving me another scenario. The weight of two apples and three oranges is 285 grams. Each apple weighs the same, so you can assume all the apples have the same mass. Each orange weighs the same. You can assume they have the same mass, but the apples and the oranges do not have to be the same as each other, right? So I can have one weight for apple, one weight for oranges. What is the combined weight of one apple and one orange? If anyone needs a hint, make sure you send a little message to me or chat with Jacob and we can provide some hints. Anybody else want to put their answer into the poll? Only about half of you have made an answer on the poll. Yeah, that's better, you might as well guess, right? Okay, we share the results. We have a lot of different answers. So that's really wonderful. It gives us something to think about, something to discuss together. Okay, this is a very common type of problem. I've seen it on multiple contest formats. So Math Kangaroo likes this one, but so do some other contests. So this one's really worth learning how to do. So I'm gonna use A for apple and a fancy O for oranges. Cause remember I told you I don't like O's cause they look like zeros and my brain, you know, when you're writing something, it's hard to know, was that an O or was that a zero? Plus we have zeros in some of these. So on the first, in the blue, I have three apples plus two. I'm gonna use this for my oranges and that equals 225 grams. Some of you asked me for a hint and I said, what happens if you bought all the fruit, both groups? So we would have two apples plus three oranges and that was 285 grams. So if I bought it all, I guess I'm very, very hungry today. I bought it all. I would have five apples in total and I would have five oranges in total. And I will have, in terms of weight in the package, well, five plus five is zero. And 11, I will have 510 grams. The question is asking if I have one apple and one orange. All right, it also told us that every apple weighs the same and every orange weighs the same. So since they all are the same, I can take, I have five of each, but I want one of each is what I really want. I want one apple plus one orange, right? So that is 1 5th of the 510 grams, right? Because I'm taking 1 5th of the fruits, one of each is 1 5th. So if I take my 510 and divide it by five, I get one, zero, two, 102 grams. That's how much it would cost if I wanted one of each. So it makes a lot of sense if you think, oh, I buy it all, then I have five of each and it's gonna weigh 1 5th. This is very often done with the cost of things too. I very frequently see somebody ask, if you have this many and this many costs this much, how much would this cost, right? So you might see it done in that manner. A father kangaroo lives with his children. They decide on all matters by taking a vote and each member of the family gets as many votes as his or her age. The father is 36 years old and the children are three, six and four years old. So right now the father wins. How many years will it take for the children to have majority of the votes if they all vote the same way? So you have to assume that the children all vote their votes for the same thing. Maybe they're voting to buy more candy and dad is voting that we have enough candy, but Halloween is coming, so can you ever have enough candy? Anyone who doesn't know where to start, the father is 36, so the father has 36 votes right now. The children added together, 13 plus six plus four is 23. They have 23 votes. How many votes will they each have next year? See if you can figure something out that way. Just think about what happens next year and then the year after. Okay, here are those poll results. Look at that. Again, we have a lot of different answers. Take a look and see. I think we can find a pattern in this problem. Let's see what happens. So I've already discussed that in the first year, the father gets 36 votes and the children have a total of 24 votes, right? So what will happen in the next year? The next year, the father will be 37 and the children would be 14, seven, and five. So you can see that their votes go up by three each year and dad's votes only go up by one each year, right? So there's a difference. The difference is two per year, right? Because we have plus a three or plus a one. So if I take the original difference, if I take the 36 minus 24, I'm gonna have 12 years. Well, 12 votes, right? Let's call it 12 votes. That would be more accurate, right? So in order to catch up with 12 votes, the difference is two. We're gonna need six years to catch up. Sorry, my handwriting is a little messy, but we want the children to have the majority. So we actually have to go to the seventh year. Oh yeah, somebody's corrected my math there, huh? It was, thank you, I don't mind being corrected. So yes, it's 23 votes. So I have to subtract 23 votes and you'll see that the difference, now the difference is actually it's 13. So in order to catch up those 13 votes, we'll need to go seven years. I have notes with it correctly. I don't know why sometimes I make little mistakes when I teach it. All right, so it didn't change the answer, but yeah, we have to recognize that the children get three more votes each year while the father gets only one vote each year and that allows the children to catch up. All right, one bowl contained 26 liters of water and another bowl contained seven liters of water. The same amount of water was added to each bowl and now the second bowl contains three times less water than the first bowl. How many liters of water were added to each bowl? I don't wanna explain too much because I wanna see how you interpret this problem when you're taking a real math kangaroo contest, you're gonna have to try to decipher the meanings of this all by yourself. Read very, very carefully. So I've made a note on the slide, I do not prefer that wording three times less water than the first bowl. You can rethink that as one-third as much water as the first bowl. All right, I'm gonna start working through this one because there is no poll. So I just noted, if you don't know how to do a problem, if you can't figure out where to start, but you still have time on a contest or a math test and it's multiple choice, Jacob, what should they do? Try each of the answer choices. Yeah, I mean, you might try an answer choice and say, oh, that one was too small and then you'll try a bigger answer choice. You might not even have to try all of them. You might be able to eliminate some. Okay, so a bowl of water contains 26 liters and another bowl of water contains seven liters. This was drawn in an interesting way. So if this whole thing is 26 liters, I have a bowl with seven liters, okay? The same amount of water is added to each bowl. And now the second bowl contains three times less water than the first bowl. So I'm going to add water to this bowl. I'm adding water to the 26th bowl and I'm adding water to the seven bowl. But now the smaller bowl is going to have... Sorry, I'm trying to undo one line. The smaller bowl is going to have 1 3rd, uh-oh. Smaller bowl is going to have 1 3rd as much as the big bowl. So it's kind of a little interesting problem. So we've only added one amount of water to each bowl, but because we're multiplying this by three, this is one times, two times, three bowls, now they're even. Okay, so if I wanted to write this as an algebra problem, I'm gonna have seven plus some volume of water X. And that is going to equal 26 plus that volume X, but I have to multiply this one by three. So seven plus X times three is 26 plus X. Now this is the distributive property. I have to do seven times three is 21, and three times X is, three X is the way we write that for short, and it equals 26 plus X. If I subtract an X from both sides, because as long as I do the same thing to both sides of an equation, my equation still remains in balance. 21, I'm having my share of typos today, plus two X equals 26, because X minus X goes away, and three X minus an X is two X. Now, if I subtract 21 from both sides, I have two X equals five, and I would divide both sides by two to get X equals 2.5. The answer is A. Now, looking at the diagram, I could try these different things. I could say, okay, what happens if this little sliver here is 2.5, and this is 2.5, and this is 2.5, right? Then instead of 26, I'm gonna have 28.5. Instead of seven, I'm gonna have 9.5 times three, and that is also 28.5. So that is a way you can check your answers, or you can work from the answer choices to get the correct solution to this problem. I believe this is when you wanted to lead, Jacob. Yeah, so during competition in the Kangaroo Summer Camp in Zakopane, students were given 10 problems to solve. For each correct answer, a student was given five points. For each incorrect answer, a student lost three points, and everyone answered all the problems, but Matthew got 34 points, Philip got 10 points, and John got two points. So how many problems did they answer correctly altogether? I'll give you guys some time to think about this. Lee, if you need to take any notes on the problem, take those now. I am going to launch the poll. The problem is in the top, but if you are one of the people who it covers your screen and you can't see things, you might want to take a couple notes. I'm going to close that poll in just a few more seconds so that Jacob can start going through the problem want to try to use our time wisely before we run out of time in our webinar today. Okay, Jacob, do you want to go ahead and help them through this? I shared the results. Okay, so what I'm going to do is I'm going to add each of these point values that Matthew, Philip, and John got. So when you add them up, you get that all together, they solved the total number of points they had was 46. And notice that, um, when you add them all up, you basically have now 30 questions. But you have the same rule, you have five points if you get it right. And minus three points. If you are wrong. So let's say that I want to try to set a variable. So let's say that, um, T questions, T questions, um, They all answered correctly. And so we want to try to find T. And let's say that 30 minus T Is the amount of questions that they answered incorrectly. So we can try and set up an equation for the amount of points. So we have 5T plus three, or sorry, Minus three times 30 minus T, that is the wrong points, is equal to 46. So we get 8T is equal to 136. So we get T is equal to 136 divided by eight. And then when you calculate that out, you get 17. So the answer should be A. I noticed some people put 13 and that's what you get when you try and find 30 minus T, which is the amount of wrong answers. So, yeah, it's pretty important to try and make sure that you're solving for the right thing. So, yeah. Thank you, Jacob. I'm going to back it up one step because if that was too, too tricky, you could just look at each individual student as well as another way to do it. So you can look at like Matthew who got 34 points. The max that Matthew could have gotten on that contest was 50 points. And for every incorrect answer, you don't get five points and you lose three points. So each incorrect answer is basically you're subtracting eight. Right? So if you do the 50 minus 34, you're going to get 16 was subtracted. So that's two wrong answers. So that means that we got eight correct answers and you can do that for each of the people who did the contest and then add that total. So that's another way to do it. If you want to do one person at a time. Because I know the algebraic equation you set up Jacob is very correct, but it might if you are new to algebra, that might be a little hard to set up. So it's possible to break it down into smaller pieces as well. I said to clear the drawings. There we go. All right. Number seven. Was that for you as well, Jacob? Sure. Yeah, I can do it. So notice that. Oh, so a fairy has six bottles. Their volumes are 16 ounces, 18 ounces, 22 ounces, 24 ounces, 32 ounces and 34 ounces. Some of them are filled with orange juice and some are filled with cherry juice, but one is empty. And there is twice as much orange juice as cherry juice. And we want to find the volume of the empty bottle. So I'll give you some time to think about this as well. I know it might be rushing for some of you, but I am going to launch the poll so that we will be finishing our webinar just about on time today. Okay, I know not all of you have answered the poll, but what I'm seeing is that we have every single answer choice. So, that to me means that Jacob is going to do a real good job explaining this and that everyone will understand it when he's done. All right, so basically what we first want to do is you want to. Find the volume, the total amount of liquids across all bottles. So, when you do that, we have 16 plus 18 plus 22 plus 24 plus 32 plus 34 and that comes out to being. 146 I think. Yes, and then notice that if we set the amount of. If we set the amount of cherry juice to x, then the amount of orange juice is going to be 2x, right? So, that means that the total amount of orange juice and cherry juice is equal to 3x. And so, that means that if we take 146 minus the volume of the empty bottle. Then, this number should be divisible by 3. So, what we can do is we can take each of the bottles or I guess like each of the answer choices and test which one makes the most sense. So, if we have 146 minus 18, this is 128 and that's not divisible by 3 because. Yeah, I mean, yeah, like, well, there's some of. Yeah, some of the digits is not divisible by 3, so and that's the rule for the visibility of number 3. If we take the number 34, we have. 112, the sum of the digits that's 4. And so, that's also not going to be divisible by 3 because 3 does not divide 4. If we have 146 minus 24, that is 122, that is also not divisible by 3. So, yeah, because the sum of the digits is 5. If we take 146 minus 32, that is 114 and the sum of digits that is 6. So, that is divisible by 3, so that is a possible answer. And then, if we take the last answer choice, this is 124, which is also not divisible by 3 because the sum of digits is 7, which is not divisible by 3. So, the answer should be 32 and we can also find a case where that actually does work. So, yeah, that should be the answer. Okay, that's exactly the method to use for this one. So, it's using a little bit of our divisibility rules. It's using a little bit of our logical thinking. We had to do some figuring out that we had to have 3x divisibility by 3 because there was twice as much and you have to have two parts of one and one part of something. You have three parts all together. So, excellent. So, Math Kangaroo can ask you interesting problems like that. So, one of the things you'll need to be able to do to answer Math Kangaroo problems that involve algebraic thinking is to recognize patterns. We had that with the voting ages, right, where the children's ages increased by 3 as a total each year and the parents only 1. You'll need to work carefully in organizing. You might need to use some variables to represent some unknowns. Read the problems more than once, right. We had that 3 times fewer. I had to read that a whole bunch of times before I figured out that they meant that there was one third as much water, right. There might be some interesting vocabulary used in some of these Math Kangaroo problems. So, get used to reading them. The best way to get used to reading Math Kangaroo problems is to do practice Math Kangaroo contests. When you registered for this webinar, you got a discount code that will allow you to access Math Kangaroo contests from past years. Those are wonderful to do to practice. They will start out harder, but as you get more experience, you should be able to go through them faster, more accurately, and you should be getting more comfortable. You can also, if you've registered for the Math Kangaroo contest, you should be able to have a code to watch some video solutions so you can see some teachers solving problems just like Jacob and I do here. Remember, you can always use the answer choices. Test those answer choices and you can always check your answers. You can fill the answer back in and make sure it makes sense. So, thank you for joining us on our webinar about algebraic thinking. Thank you, Jacob, for all your help today. And I hope to see you again next week. Bye, everybody.

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