So, welcome. This is the introductory lesson for the Level 5-6 webinar series for Math Kangaroo. And what you'll find today is that we are going to use a whole bunch of sample problems of different types of Math Kangaroo problems to give you a flavor of what types of things you might be asked to do. In subsequent weeks, we will focus on a particular type of problem and we'll delve more in-depth into each one of those topics so that by the time you finish the 10-week session, you'll gain experience in how to solve a wide variety of our Math Kangaroo problems and hopefully that will help you learn and score better if that's your goal. I always think learning is the goal, but I guess I'm old-fashioned. So, during the webinar, your camera and your microphone will be off. This is recorded. We do not want students having any issues with privacy getting recorded when they shouldn't be. You should have a handout with any figures. If you don't have it, all the figures will be displayed. It's just easier sometimes if you can draw on your own copies. If you didn't find that, check your class, the email link that you got with the registration, you should be able to find the link there that will help you get to the figures each week. We do have a teaching assistant who's a high school student. He'll introduce himself. He and I are both available to you on the chat. Usually when one of us is explaining a problem, it's best to ask the other one questions, but I will ask you to go ahead and put your answers in the chat for us so that we can see if we've given you enough time or if you have any problems, any questions doing these. And we do have polls for about half of the problems today, so we'll be able to launch polls and you'll be able to compare, see how many students are getting the correct answers, what wrong answers different students are getting. We'll be able to do that, okay? So there's a question about the recordings. The recordings are available after the class session. Probably tomorrow, what happens is we are being recorded in the cloud and then an employee of Math Kangaroo goes through and gets all of those and edits out the beginning and the endings or whatever. And then they'll send you a link that you can access through the registration that you've done on Math Kangaroo's website. All right. I am Sarah Segee. I am a biomedical scientist. I'm a biologist. I live in San Diego, California. I've been tutoring math and science for a very long time. I have coached an elementary school math club. I've helped with the middle school math club. I have been teaching Math Kangaroo lessons since they began them. I was teaching some of the first webinars that they had for Math Kangaroo. I've helped write a lot of these lessons. You'll notice we use old Math Kangaroo contest questions, so they're very authentic questions. I teach karate. I teach swimming. I do Zumba several times a week. That's my way of turning my brain off and doing something completely different. I like to crochet and make soap. It's a chemistry acid-base problem. Lots of calculations involved there. And if you're wondering, you know, oh, she coached elementary school, how she qualified for this, I went to engineering school. And I have children who are in engineering school. My oldest daughter has graduated from college, so I definitely can do this level and much, much beyond. Okay, we're going to let our teaching assistant introduce himself. Everyone, welcome Jacob. So I'm Jacob. I'm a 10th grader from Miami, Florida. I've been participating in Math Kangaroo since second grade, and I like to compete in other contests as well. And I volunteer to tutor students with my school math club. And I also play soccer when I have time. I've also played before when I was in middle school. So yeah. So one of the things I like to let you know is you can use math in just about any aspect of your life. So if it's trying to figure out what angle or spin to give your soccer ball, that's a math and physics problem. If you're doing some soap making or counting rows and trying to make a pattern in crochet, that's also mathematical thinking. It comes up on us everywhere. So today we will be preparing you for the Level 5-6 Math Kangaroo contest. That's a 30-question contest. You have 75 minutes to complete it. And you'll notice that Math Kangaroo contests, not every question has the same point value. There will be 10 questions worth 30 points, 10 worth, 10 worth, yeah, I guess 30 points altogether, then 10 worth 40 points, and 10 worth 50 points altogether. We expect that the first 10 problems you should be able to solve more quickly than the last 10 problems. It's all subjective. Of course, your particular experience might make a 5-point question very easy for you. But in general, we believe that they're arranged in order of difficulty. Okay. So the first questions you might find, they're kind of like exercises. You know how to do it. You know you know how to do it. You just have to go through and do it. There's not too many steps to it and you want to get through it quickly. The others you might think, okay, this is a little bit trickier. There might be multiple steps. You might have to figure out, okay, what do I need to do first? What information do I have? What information is missing? Make sure on those problems you're working very carefully. You check your work. So we always make sure that we use a four-step process in Math Kangaroo. We like to emphasize it that the first is to understand the problem. Figure out what it is asking. We like to be a little bit tricky. Sometimes we might ask not for that one answer you think, for something different. The sum of things or something that is not possible instead of things that are possible. So be careful in your reading. There's a bit of reading comprehension. Come up with a plan. There are multiple ways to solve many of our problems. So the way you solve it might not be the way that I solve it. The way Jacob solves it might be different than the way that you solve it. It doesn't mean that anyone is doing it incorrectly. You're just using your own life experience, your mathematical education so far to do what is working for you. Sometimes we'll expose you to multiple ways because that's how you learn to try it another way and learn something new. Be very careful when working through your stuff. When you're doing a timed math contest, although to make mistakes is perfectly normal, I make them all the time, they do slow you down a little bit. So if you work carefully the first time, you'll probably get more problems done than if you go racing through and have made a lot of mistakes. Okay, so look back and check your answers. Does your answer make sense? Did you answer the correct question? Did you really read it carefully? All of that is really important. Today, like I said, we're going to do a variety of different types of problems and you'll notice that we come back to these each week. We do one type of problem. So here is our first problem. And I do have a poll for this. We have very few students with us right now, so I don't know if I'll launch the polls. I might just have you put the answers in the chat to save time and then we'll do some more problems, bonus problems at the end instead. There are 10 ducks. Five of these ducks lay an egg every day. The other five lay an egg every other day. How many eggs do the 10 ducks lay in a period of 10 days? Yeah, so some students are going ahead and sending me the answer in the chat. That's exactly the best way to communicate with me. Okay, so if we have 10 ducks total, five of them lay one egg per day. The other five lay an egg basically every other day. So they kind of lay half an egg per day. That's one way of writing every other day, right? And we're gonna multiply that by 10 days. So here we're gonna get 50 eggs. Here we're gonna get 25 eggs. So our grand total will be 75 eggs in 10 days. If you solved it with a slightly different calculation, remember I said, you have a different mathematical experience. It's no problem, okay? As long as you can follow your work, it's logical. You've come to a correct conclusion. You've done a great job. Okay, Jacob, you wanted to lead number two. Okay, so Vivian and Mike were given some apples and pears by the grandmother. Altogether, they had 25 pieces of fruit in the basket. On the way home, Vivian ate one apple and three pears and Mike ate three apples and two pears. At home, they found out that they brought home the same number of pears as apples. How many pears were they given by their grandmother? So yeah, I think I'll give you guys some time to think about this problem. If you've recently joined, please work the problem on the screen and you can put your answer in the chat to either of the hosts, to Sarah Sagi or to Jacob. Okay, so hopefully it's been enough time, um, but we're going to go over it. So we know that Vivian and Mike, they were given some apples and pears, right? And they had 25 pieces of fruit in total in the basket. And on the way home, they ate one apple and three pears. And then they ate three apples and two pears. And once they got home, they realized that they had the same number of pears as apples. So one way we could think about this is by trying to set variables. Um, so we can say that there are A, apples, and P, pears, right? And we know that the total number of fruit in the basket is 25 pieces of fruit, right? So that means that A plus P would equal 25. Now on their way home, they, um, Vivian ate one apple and three pears. So instead of having A, apples, and P, pears, it became A minus one apple and P minus three pears. Now Mike ate three apples and two pears. So now that they lost three apples, we now have A minus four apples and P minus five pears. And at home, they realized that they brought the same number of pears as apples. So that means that A minus four is equal to P minus five, right? So, but we know that A plus P equals 25. So A minus P is equal to negative one. This is what this equation gives us when we rearrange it. And we also have that A plus P is equal to 25. When we add, uh, A and P together, uh, when we add these two equations together, we get that 2A is equal to 24. So that means that A equals 12. But we have to be careful here because we want to find the number of pears that they were given by the grandmother. So since A is 12, that means that P is equal to 12 minus, uh, 25 minus 12, which is 13. So our answer would be B. Um, I also have another, um, solution that could be a little bit more intuitive. Um, here, let me just erase all this. Um, another solution path could be, um, noticing that, um, we can figure out that the number of apples and pears that they brought home is just 12, uh, 25 minus, um, so this is the number of fruit that they brought home or that they had at the beginning. So 25 is number, the number of fruit that they started at the beginning. And then Vivian ate one apple and three pears. So Vivian ate four apples, uh, four, four pieces of fruit. So that means that, um, they, they lost four pieces of fruit. And then Aiden or Mike ate, Mike ate three apples and two pears. So that means that they lost five pieces of fruit there. So in total, they now had 16 pieces of fruit when they got home. And since they had the same number of pears and apples, that means that they had eight apples and eight pears when they got home. So now we can, um, go backwards and notice that if Mike ate two pears and Vivian ate three pears, then that means that at the beginning, when they, the number of pears that they were given by their grandmother is eight plus two plus three, which is 13 pears. So that's another way of thinking about it. So, yeah. Jacob, and I like that you solved it by more than one method because that way a student who's looked at it a different way than you did the first time, there's even probably a third and a fourth way that you could look at this and still solve it algebraically. There's always, when you have a system of two equations and two variables, there's more than one way to find that. So some students may have used other options and it's all great. So I appreciate that you're showing them different ways of thinking all come out logically and will work. Okay, I do have a poll for this one. It's not a lot to read. Find the distance which Mara covers to get to her friend Bunnika. So we know that Mara, we have this little distance measure here that we have a 100 meters, corresponds to one eighth of the distance between where Mara starts and where she needs to end up at Bunnika's house. I will launch that poll very soon because I think this can be done quickly. The poll is completely anonymous, so I do not know your answers. No one else will know your answers. If you're ever not 100% sure, but you still want to put your answer in the poll, that would be great because it helps us to know what other answers you've gotten, problems you have. If it's super easy, that's fine for us, too. I'll be sharing that poll in just a few seconds. Thank you for trying. Okay, so you can see that most of you have said the answer is 800 meters. Let's see if that makes sense. If the poll is still appearing on your screen, you may need to touch a little button that says close it. I do on one of my screens, but not the other. It goes away by itself. So I know you guys are very familiar with how to use your tech and you can make it work for yourselves. Okay, so I'm going to just use a proportion. If 100 meters equals one-eighth of the trip and I want to get eight-eighths of the trip, then I'm going to have to do 800 meters, right? So I get 100 equals one-eighth and I want to know eight-eighths, so I'd have to multiply by eight. The answer is C. There are a couple of different ways to set it up. You can try to draw it out. So here's another eighth. You can do some drawing. Okay, between here, here's two more eighths. So I could start drawing things in like that. All very nice methods of doing it. Do what method is working well for you, but be accepting of learning some new methods. Right, multiples, factors, and divisibility. We'll have a whole lesson doing much more complicated ones. The least positive integer which is divisible by two, three, and four is. So we want the least and it should be positive integer. If you have any questions about any of those vocabulary words, let us know. You can ask Jacob or ask me. I think we're going to have a lot of correct answers. This is a number four. This should be one of those quick three-point exercise problems. You should have a lot of experience using multiples. So you can see everyone is answered the same, that the answer should be C, 12. So you can do this in a couple of ways, I think the first way they teach you is to do the multiples. So to write out 2, 4, 6, 8, 10, 12, and to write out 3, 6, 9, 12, and to write out 4, 8, 12. And you'll notice that the first place that all of these overlap is over here at 12. The other thing is you can use divisibility. Which number is divisible by all of these? Well, 1 is not. 6 is not divisible by 4, you cannot divide it by 4. 24 is divisible by all of them, but they wanted the least, and 12 is less than 24. The same logic as to why 36 is not the correct answer. It is a multiple of all of them, but not the least multiple. So very careful reading will help you a lot with math kangaroo problems. How many hours is half of a third part of a quarter of 24 hours? Because this is somewhat of an exercise problem, let's go ahead and launch a poll. I think we'll have a lot of correct answers. Number 10 is where we kind of transition from the, oh, you should be able to do these pretty easy to, okay, we're going to make you think a little more. Okay, let's end that poll and share the results. So we have either one half or one. That's the correct answer. Let's figure it out. So I think you can do this. I'm gonna call it forwards. Or we can go backwards. It doesn't, again, everyone has their own preference. So forwards, I'm just gonna read it from the beginning to the end. I wanna know how many hours, so that's x. I wanna know how many hours is one half. Mathematically, when I see the word of, I think multiplying of 1 3rd of 1 1⁄4 of 24 hours. It's gonna be 24 over one because 24 is a whole number of hours. So then x equals, 24 is the numerator. I multiply the denominator, six times four is 24. So I get one hour. That works very well. There's, works great, no problem. Working backwards, I would do one quarter. Sorry, little typo there. Yes, you can do a typo when you're writing, it's okay. One quarter of 24 hours is gonna be six hours. And then I'm gonna take 1 3rd of the six hours, that's two hours. And then I'm gonna take a half of the two hours, that's one hour. So the really good news is I got the same answer both ways. You get the same answer both ways, that's step number four, that's checking your work, right? So remember step number four is always go back and check yourself. You do it another method, get the same answer, you can be quite confident that you will get your full points on the contest or your math tests or your physics tests or whatever it is you're doing, right, Jacob? You. Okay. Among puzzle pieces below, who have the same area? Which two? Kind of a visual problem, you have to look at it to find the answer. You notice the responses are kind of weird because they're combinations of things, you have to find a matching pair. Go ahead and put your answer in the chat, that way I'll know that everyone is working along. You start to get confused you can think about they all start with the same side square, but then they have either. Concave going in, or they have a convex going out or any or outing. I don't think I'm almost a belly buttons or something right. Yeah, I have a lot of correct answers you guys are doing really well. So let's say that this one has two out and one in that's how i'm going to call these are out ease, and this isn't any. This has two ends and one out. This one has three out. One in. This one has two and two. And this one has two out. And one in so the only two that are the same in terms of the numbers of outs and ins are one in five so that gives me answer choice be. This should be a quick exercise a number three problem, so if you feel like I could race through that we think you can too. This one, however, I think takes a little bit more concentration because it's three dimensional. It's always takes a little more concentration to rearrange a flat drawing into three dimensions in your mind. I think you can do it, which of the pieces below is needed to make the solid shown to the right into a prism. What does a prism mean? All right, good work. So, prism is a three-dimensional shape with any sort of base. It does not matter what shape the base is, but the walls come up straight from that base. So, it's like straight great walls up. So, we are going to need to fill in this space that's here, right? We have to fill in this. Sometimes it helps to draw in the space you need to fill. So, you'll notice from, you can talk about going, you know, from left to right, from front to back, and then from top to bottom. It goes into three different directions here, right? So, this one is flat. It doesn't go in our three directions, so it won't work. The same with B. Let's see. These at least go into three directions. Let's see which one of them will fit. We need to have, like, a little L shape on both sides. A little L here and a little L here, which this one lacks, so D is wrong. Now, C and E are pretty similar looking shapes. A 50-50 guess is not bad, but I don't think we have to guess. I think we can try to rotate these pieces into place. I'm going to erase over here. I'm going to, I like to use color. So, this one goes, let's see. If I take these two pieces and I put them here, those two pieces there, I could still put the other three pieces over here, but then the last two pieces are going to stick out. They're going to come kind of out over here and hang off into space with nothing underneath. So, I don't think that C works well. Or if I try E, I think these two pieces could go here, then I make the 90-degree bend and I can come along here, and finally this will turn down and fill in this space here. So, that will fill in really nicely into the crack. So, the correct answer is E. We think about these as being a left-handed figure and a right-handed figure. So, you'll see one is kind of a rotation a little bit backwards from the other. And it is important to determine which one will fit correctly. All right. So, we have more problems. Don't worry. I just want to take a moment to wrap up a little bit. The first step is to understand what the problem is asking us. Then we need a plan on how to solve it. Was our plan always the same? Will everybody have the same plan? Absolutely not. Sometimes we have to draw. Sometimes we were using straight old algebra. Sometimes we were dividing. Sometimes multiplying. Sometimes, you saw we used ratios. We used different methods. We can check our answer by trying another method by plugging the answers back in. There's a way to do it. You'll notice that we did a lot of number systems. We did ratios. Get used to solving for expressions and equations. Math Kangaroo likes to do a lot that's visual, spatial, and geometry. That's one of the unique things about our Math Kangaroo contest compared to some others is you'll see a lot of that. You'll also see rational thinking, reasoning, and logic problems which is a lot of fun. I want to make sure some people leave early. I want to make sure you know that you can watch all of these webinar series. It takes about a day. They get edited and then they're available. If you go to your webinar registration page, so your parents or you signed into mathkangaroo.org. You have a login. When you go to the courses you've signed up for, you will find your webinar registration page and there you'll have instructions on how to access these recordings. We do recommend that you practice past contests. Our questions will never repeat themselves but we use similar styles, similar ideas. It's a really good idea to get used to how we word the problems. A lot of students who are brilliant at math won't interpret the wording that we use quite the same way that we interpret them. It's a really good idea to practice that. When you register for the Math Kangaroo contest, you will get a coupon code that will allow you to watch video solutions. If you practice a contest and you want to see a teacher solving a problem, you can get that through the video solution. You're like, I didn't get it. I need to see someone solve it. I need it step by step. Go ahead and watch those videos. They're a great resource. For each of our lessons, you'll notice we have quite a few figures. That last figure, I did a lot of drawing on it. You can get all of our figures by going to the handout link in your class registration. The same place you'll find the links for the videos of the recordings. You'll find there's a whole folder for each session. You can print out the handout. That way you'll be able to draw along or draw before I explain or try over and over again if you like. If you have questions and I haven't answered them, use the info at mathkangaroo.org email to ask them. We have some additional questions. What is the largest positive integer that evenly divides each of 56, 112, and 98? I'll give you some time to work on that one. It's the same idea as the problem we did, but I think you'll find this one is a little less obvious. So, again, there's probably multiple methods for this one. I'm going to take each number and see if I can find some factors of it. I like to work in factor pairs. It's just my personal preference. So if I take 56, oh, I see I can do that. I can do 7 times 8. That might be one that a lot of us see. 7, I can't change. I can't factor it or do anything with it because it's prime. With the 8, I could. So I could do 14 times 4. 4 is also 2 times 2. So I can do 28 times 2. That seems to pretty much be it for reasonable factors of 56. Let's see what happens if I try 112. Again, there's going to be a lot of factors of 112. I could start with, let's see, is it divisible by 8? Because that would be a good one to try. It's divisible by 8. 8 goes into 12. 1 with a 32, so 14 times 8. Oh, that's an interesting one, 14 times 8. So 14 is bigger than 8 and matches. Is it divisible by 28? OK. It should be because I can do 28 times 4. All right, so now we have an even bigger number with that 28. I like that. How about the 98? I take the 98. Let's see, can I divide 98 by 28? Definitely not. All right, so the next biggest factor, can I take 98 and divide it by 14? Let's see, 14 times, oh, I'm not good with my 14 times tables, right? 14, 28, 28 times 2 is going to be 56. If I do 56 times 2, then I'm going to get too big. So let's try 14 times 7. So I'm going to figure out that in doing these, 14 was the largest common factor of all of them. So the largest positive integer that evenly divides each is 14. On my way to school, I must pass by the mall, which is 3 7ths of the way home to school, the way from home to school. If the distance from my home to the mall is 240 yards, what is the distance from my home to the school? So, we can think that there are 7 parts of the trip from home to school, we can divide it up into 7 pieces. So for the first 3 pieces out of the 7, it's 240 yards, but I want to know how many there would be if I had the full 7 out of 7. So I can set it up like this. If you've solved proportions, you can cross-multiply and then divide and set it up so that you can solve for x. So this is a perfectly acceptable way to do it. The other way, which was a little bit more to my preference because I don't like to deal with big numbers, is if I know that 3 7ths equals 240, then I know that 1 7th, if I divide 24 by 3, I'm going to get 80. And then I know how to do 8 times 7, so 8 times 7 equals 506, but it's 80, so it would be 560. So the correct answer is going to be 560 yards the whole way. Now it wouldn't be beyond math kangaroo to ask what is the distance from the mall to school instead of the whole distance, so be very careful that you've read correctly and you'll answer the correct problem. We have one more of these 3D solids, and I think that Jacob will lead this one for you. So the solid is painted on all sides and on the top. How many squares are painted? I'll give you some time to think about this. I think maybe we need to clarify where it says how many squares are painted. They mean the small square faces, so they mean one of these little squares is one. Okay, since we're running out of time, I think, Jacob, do you want to show them at least one method for doing it? Um, yeah, sure. So, um, we can look at it from each of the sides. So when we look at it from here, we can see that this side and this square and this square are all seen. Now, when you look at it from this side, we can see that this square, these squares, this square, and this square here are all counted. And then when you look at it from the top, you can see that these squares, this square, and this square here are all counted. And then when you look at it from this here, you can see that this, um, 2 by 5, so this is 5 and this is 2, right? So that 2 by 5 rectangle there is counted. When you look at it from the bottom here, you can see that this 2 by 4 is also counted. We're not counting the bottom, Jacob. Oh, yeah, yeah, that's right. Yeah. So just be real careful on reading the problems. It says it's painted on all sides and top. It never mentions painting the bottom. And then when you look at it from here, you can see that this 5 by 4 rectangle is counted except this corner square. So we can compute the surface area of the 4 when it becomes like a prism, and then we can add in this square here and then this square that's right here. It's kind of hard to notice, but yeah, it's like this square here. That's another missing square that we don't have. So when we add all of them up, we have 2 by 4. So that's when you look at it from this side and this side, we can have a 4 by 5 minus 1. So that's when you look at it from this side and this side. And then when you look at it from the top, you have a 2 by 4. Or sorry. Oh, yeah, this one should be 2 by 5, I believe. Yeah, because when you look at it from this side and this side, it should be 2 by 5. And then this one should be multiplied by 2 and this one should be multiplied by 2. And then we also have these two squares here. So that should be plus 2. So when you compute all this up, it should be 20 plus 38 plus 8 plus 2, which should be 58. So these two plus 10, which is this two. And then this should be 68. So that should be the right answer. Thank you, Jacob. So if you looked at it slightly differently, that's no problem. We're all going to have our own preference for the visualization here, but Jacob's approach is working very well. And it is important to note, yeah, that when you remove those pieces, you actually expose those two red faces that you don't normally see. So I hope you enjoyed these math kangaroo questions today. Remember, next week, instead of jumping around and doing an introduction to a whole bunch of different topics, we'll be focusing on one topic and delving more in depth, starting with easier problems and getting into more complicated levels. Your places to continue to work during the week is to, after tomorrow perhaps, go ahead and re-watch this video if there are any problems that you didn't understand you want to look at again. Do practice contests. Use the video solutions that are available all through the Math Kangaroo website. We'll hope you'll come back next week. Have a wonderful week. Thank you, Jacob.

Math Kangaroo

problem-solving

webinar series

math exercises

Sarah Segee

Jacob teaching assistant

algebra

spatial reasoning

interactive learning

contest preparation