So this is set B, the second set of Math Kangaroo webinars. This is our level five, six. So most of you will be students going to in grades five and six, and you're preparing for the Math Kangaroo Contest coming up in March. Welcome, we're gonna do slight introduction. I will introduce myself, we'll introduce Jacob, we'll talk a bit about Math Kangaroo. In today's lesson, we do sample problems of kind of each major type of Math Kangaroo problem. Math Kangaroo problems can cover all different types of mathematical and logical material. And we're gonna give you a sample of what that might look like today. So during these webinars, you'll notice they are webinar format. So your cameras are going to be off and you will be muted during the class. But you should definitely have paper and pencil and be ready to work all of the problems. We also have handouts available almost every week where you can have the figures. That way, if we have complicated figures up on the screen, you don't have to draw them yourselves. You can just use the handouts and take your notes directly on the handouts. If you have questions as we go along, you can submit them in the chat. I also encourage you to put your answers in the chat. It lets me and Jacob know how you're doing and we try to encourage you or help you if you need it. And we have polls for about half of our problems. I don't do them for every question because they disturb the flow. They take a little too much time, but I know a lot of students love to put their answers in the poll and show us how you're doing. Those polls are anonymous. Your name will not appear anywhere. So don't worry about taking a guess, do your best. And I will say that for the Math Kangaroo Contest as well, we do not subtract any points for a wrong answer, for a wrong guess. So don't leave a blank, always give it a try. This is being recorded. So if you need to see another problem or if you missed a week, oh, I didn't quite understand what she did. I wanna see it again. Go ahead after class. It might take a few hours for the administration to post it, but in your registration for this class, there's a tab for the material and you will see a recording link there. So try to focus. If you need to take a bathroom break or whatever, try to do that, come back and you can always take a look at the recording if you missed any part of the class. Okay. My name is Dr. Sarah Sagee. I'm a biologist, a biomedical scientist. I live in San Diego, California, and I've been tutoring math and science for a very, very long time. I'm one of those students like Jacob who my friends and neighbors asked me to help with their children and things from a pretty young age. And then when my children were growing up, I was teaching at their school math clubs. That's how I started teaching math club and contest math. I've been teaching Math Kangaroo, their small group lessons, webinars, and private lessons, private one-on-one tutoring. I've been doing that for about four years. I've also taught karate and swimming. And some of my hobbies include Zumba and dancing. I like to crochet. I like to make soap from scratch. And I do have a new puppy, so I like to walk my dog a lot. And I'll let Jacob introduce himself. Yeah, thanks. So I'm a 10th grader from Florida. I go to school a little bit more North, though, so yeah. I like to participate in a lot of contests and I've been, and Math Kangaroo's one of them, and I've participated since second grade. And I also like to teach a lot of other students at my school, so yeah. Any special hobbies or interests the students might find that can relate you? Oh, yeah. I like to play soccer, so maybe some of you guys like to play soccer. Thank you, Jacob. So we're real people, so don't be afraid to reach out to us in the chat. We're here to help you. So Math Kangaroo is gonna be unlike what you've done in school. So if you've been through a Math Kangaroo class or webinar or contest before, you might know this, but if you're new to Math Kangaroo and you're taking this webinar, so you get kind of a headstart and you know what to expect come March, I'm gonna tell you, Math Kangaroo is kind of, sometimes you'll get into a question and when you really dig into it, you'll find, oh my gosh, I'm surprised I didn't know that's how I was supposed to do it. Kind of like that box of chocolates where you don't know what the flavor is inside. Okay, I want to let you know that the way you do Math Kangaroo problems may be different than the way I do them and the way that Jacob does them. The way this class is organized, we are going to have kind of a theme for each week. So we will be solving it based on the theme or the tool that's suggested for the week, but you might have something else that you like to do as well. And sometimes Jacob and I will come up with some alternate suggestions for solving problems and it's perfectly fine to use a different method if that's how you like to do it for any problem you see. So at your level, you are going to have 75 minutes to solve 30 problems. They're broken down into 10 of them are three points, 10 of them are four points and 10 of them are worth five points. So the writers of the contest, which is a whole committee of writers, believes that as you go from number one to number 30, they're probably getting more difficult. Okay, now remember you're all individuals and so you might find a particular question difficult or hard or time consuming or very fast and it might be slightly different than the person next to you in the contest, but it's a general trend. And I think you'll find that the higher numbers take a little more time to concentrate and figure out. And some of the ways you might figure them out are to draw a picture or to make a table or a chart or list things out, look for patterns, might work backwards. You might have to try things step-by-step, try them in your imagination, try to draw them on paper in your imagination, use little symbols. Sometimes you might just wanna guess and check your answers. And it is a multiple choice contest. So if all else fails and you still have time, check the five multiple choice answers, right? That's kind of the last resort, but it's better than not having any points. So don't forget that that is an option. In Math Kangaroo, you're gonna find a lot of multi-step problems, a lot of needing to highlight and underline things, draw a little picture. So don't be afraid to draw and use your pencil. I recommend that you keep your scrap work kind of organized. So maybe put the problem number and work each problem because that way you can still go back and check your work. If you write hilly nilly a little bit over here, a little bit on the side there, or I have three centimeters of space over here and try to fill in, you're gonna have a hard time checking your work later. So go ahead and use your scratch paper, use it well, stay organized. Now, if you find that these math problems, especially at the beginning when you're first trying practice contests and pass contests for just, we recommend that you do pass contests off of our website to get a feel for it. And they're kind of tricky and hard. That's okay. It's all right to be in the process of learning how to do them. We don't come to these knowing how to do them. We can learn how to do them and you will improve if you continue to practice. So don't get too down on yourself. Always try to stay positive. Every problem that you finish is a problem where you've learned. We have a four-step problem solving strategy. The first step is to understand the problem and determine what it's asking. So this might mean read and reread the problem. Sometimes there's a lot of tiny little words. There's little words that tell you what operations you might need to use. You're gonna have to kind of get used to how Math Kangaroo does their problems, okay? Trying to figure out why my annotations aren't working the way I think they should be. Okay, plan how you're gonna solve the problem. Is that showing up? It is not. Let's see if that works. Interesting, okay. I was doing this earlier today too. So you might need to read and reread your problems, okay? Read and reread. Okay, you might wanna plan how you solve your problem. Are there keywords that tell you what operations you're gonna use? Is there a piece of information that you need to determine as a partway solution, right? You might have multiple steps where you have to go partway. So how can you tackle the problem? What do you need to do to solve it? When you work your plan, try to be careful. I know we all make mistakes and you'll find that I make mistakes as I teach the webinars. Jacob tries to catch me. You can put little notes in the chat if you think I've made an error. I do my best, but we're all humans and we all make mistakes, which is why it's careful. We need to go back and check our answers. Is the answer that I got reasonable? Is it making sense? Have I made a little mistake? If my scratch work is nice and neat, my scrap paper, then I can check myself. If my scrap paper has been really messy, it becomes difficult to check myself. So keep those things in mind, okay? Four steps, make sure you go back and check your answers each time. Why isn't it clearing all the drawings? All right, hang on, I'm going to close the annotations and try to open it again. Maybe that will work. Interesting. All right, so we have several different methods that we try to use to solve our math kangaroo problems. And you'll see the headings on today's webinar will have one of these lists on them. And we're going to use that type of method to solve the problem today. And then each week, we'll cover one of these in more depth, we'll have easy, medium and hard problems with each of these techniques, so that you can get used to practicing them. So here's our first problem for this week. And I do have a poll for this problem. So after I read through it, I'll give you some time and then I'll launch the poll. Here are five songs. Song A lasts three minutes. Song B, two minutes and 30 seconds. Song C, two minutes, just exactly two minutes. Song D, one minute, 30 seconds. And song E, four minutes. These five songs are playing in the order A, B, C, D, E in a loop without any breaks. So after E, it goes right back to A. That's kind of what we mean by reading carefully, right? What does a loop without any breaks really mean? Song C was playing when Andy left home. He returned home exactly one hour later. Which song was playing when Andy got home? So when he leaves, it's song C. And when he returns, what song is playing? I'm going to go ahead and launch the poll. You will see that the question reappears in the top of the poll. And while I do that, I am going to stop my share and restart it and see if that fixes some of my annotation problems. But hopefully, the poll will still stay there for you. All right, I'll finish the poll here and share the results. You can see there's a bit of every single answer, more answer A's, but it's less than 50%, so it's not even a clear majority winner, but it is the most popular answer. So let's take a look. So you'll notice that there are five songs, A, B, C, D, and E. Song A is three minutes, B is 2.30, C is two, D is 1.30, and E is four. If we add all of those times together, that should be 13 minutes, right? If we have five and a half, seven and a half, eight, nine, nine plus four, 13 minutes. And we know that Andy is gone for exactly one hour. So in 60 minutes, we're gonna go through the playlist of 13 minutes four times. So 13 times four is 12, carry a five is 52 minutes. And then there's eight more minutes that we have to account for, right? So if we know that he left during song C, if let's say it was at the beginning of song C, where does eight minutes take us to? Two, three and a half, seven and a half. So what I'm doing is I'm counting, C is two, D is 1.30, so that takes me to three and a half. E is four, so that takes me to seven and a half minutes, 7.30. So with eight minutes, if we started at the beginning of song C, eight minutes would put me in song A at eight minutes. What if it was right near the end of song C? If it was right near the end of song C, I would still have five and a half minutes for song D and E, and then all the way until eight and a half minutes in song A. So the answer, even if it was, no matter if it was the beginning or the ending of song C, I'm still gonna be in song A one hour later. So I know some of you are asking about, is it the beginning or the ending? In this case, you can try both and you'll still end up in song A one hour later. Okay, there's no poll for this one because I thought you might wanna be able to see the figures. Six identical black beads and three identical white beads are arranged on scales as shown in the picture that's below in this case. What is the total weight of these nine beads? So if you count all those beads together, there are nine of them. So that's an interesting, make sure that that's the answer you give is the total weight of nine beads, not the weight of white beads, not the weight of black beads, the total. Okay, so let's take a look at this problem. How can we compare the different scales to try to get the weight of either the black balls or the white beads, and then maybe we can figure out the others. Well, if I'm looking at the right hand scale, I noticed that there are black beads on both sides. So, if I, for example, took away one black bead from each side, the scale would still be balanced. Right? Since we know the scale would still be balanced if we take the same thing away from both sides, I can say from here that two black equal 30 grams. Oh, two black, one white, sorry. I'm trying to get annotations working. One white plus two blacks equals 30 grams. Okay. From this side, I can see that two blacks equals two whites and six grams, so I can put that all together and I can say that one white plus, instead of writing two blacks, I can say two whites plus the six grams equals 30 grams. So this is an algebraic thinking concept called substitution, so since I have two blacks here, I can take what the two blacks is equal to there. Now I can combine one white and two whites is three whites. I can subtract six grams from both sides, so I get 24 grams. Three whites is 24 grams, and since all the whites are the same, one white must be eight grams because 24 divided by three is eight. Okay, so now that I have the weight of the whites, I can try to figure out how much the blacks must be. I can figure out, let's see, I'll change colors so it's a little easier to see. This one is eight, this one is eight, and then I have to add the six. Eight plus eight plus six is going to be 20, 16, 22, my mistake, sorry guys, 22. I've got a whole bunch of silly things going on at the same time. So this is all 22, so the two blacks equals 22. So one black, therefore, must be 11. So now let's see, I know how much one black is, I know how much one white is, but how many do I have all together when I want nine beads? It looks like I have one, two, three, four, five, six black, and I have three white. If I substitute those values in, I have six times 11 plus three times eight, that's going to equal 66 plus 24. That gives me 90 grams for all nine beads together. Okay, number three. In a certain class, one eighth of the students received a C on the math exam. One sixth received a B, and two thirds received an A. There were no D's. How many students received an F if there were less than 30 students in the class? So if you're not in a school that gives grades like this, the grades are going to be A, B, C, D, and F, no E's, so you don't need to worry about E's. I see good responses in the chat so far. Keep up the good work. Excellent work so far. So let's see. We know that there's A, B, C, D, and F. So if we put these together, we know that two-thirds got A's, one-sixth got B's, one-eighth got C's, there were no D's, and we're trying to figure out the F's. If I add these portions together, let's see how much of the whole, right? Because the whole class is going to be one, and these are fractions, parts of the whole. To add these fractions, we need to have a common denominator. The common denominator, the least common multiple here, is 24. So instead of two-thirds, if I make it out of 24, I'd have to multiply by eight, so this is 16 out of 24. This would be 4 out of 24 instead of one-sixth, and this would be 3 out of 24 instead of one-eighth. So let me see how much of a whole I have. This is 23 out of 24, so I'm only missing one-twenty-fourth to get a whole class. Now it says that there are less than 30 students in the class, and we know that these fractions are going to work evenly. It doesn't really say that, but it implies you can't have a part of a student, right? You can't take fractional students, so this is going to have to be a complete, a total multiple of 24. So if it's less than 30, there are 24 students. I don't think any of you would want me to take a part of you, and we're only missing one student, so there was one student who failed the test. Okay, Jacob wanted to leave this one, correct? Yeah, so Jane multiplied the product of 18 factors, each equal to eight, by the product of 50 factors, each equal to five. How many digits does your final product have? I'll give you guys some time to think about this question, and then we'll go over it. All right, so this is a little tricky question, so I just want to give you guys a little hint that maybe you guys should try to look for like factors of 10 and try to use exponents. Maybe that'll help. Okay, so we're going to explain it now. So, what we want to do is we want to find the number of digits in 8 to the 18 times 5 to the 50. And notice that we want to try to look for powers of 10. So, and notice that 10 is 2 times 5. So, what we can do is that this 8 to the power of 18 is basically just 2 to the 54 times 5 to the 50. So, we could take out, this is 2 to the 4 times 2 to the 50 times 5 to the 50. And then the 2 to the 4 is just 16. And then the 2 to the 50 times 5 to the 50 is just 10 to the 50. So, we want to find the number of digits in this final product. But notice that this number is just 1 with like 50 zeros. So, there's 50 zeros. And we want to multiply that by 16. But when we multiply by 16, the 1 in the front gets replaced by 16. So, we have 50 zeros, and then we have 2 digits that are 1 and 6. So, in total, we have 52 digits. So, that's the final answer. And this is a problem that requires you to know about your multiples and your exponents. So, in going from the 8 to the 18 to the 2 to the 54, what Jacob used was the knowledge that 8 is 2 to the 3rd. So, it's 2 to the 3rd to the 18th. So, you would multiply your 3 times 18. I want to share the results of the poll because half of you got it correct. So, this is one of our more difficult problems, and you're well on your way to getting there. So, hopefully after the solutions that you just saw from Jacob, you're like, okay, now I get what they were trying to do. I can do that myself next time. Here's another kind of similar problem. How many zeros does the product of the consecutive natural numbers, okay, with how many zeros does the product from the consecutive natural numbers from 1 to 50 end? So, let's look at the vocabulary here. How many zeros, and we're talking about the end. So, these are what we call terminal zeros, the ending zeros, terminal zeros. The product means you're multiplying consecutive natural numbers. So, this is 1 times 2 times 3 times 4 times 5, et cetera, right, until you get to times 50. We want to know how many zeros will be at the end of that number. There is a poll. I'll launch it. You can launch it now because the problem is pretty short to read. You'll notice that the problem reappears at the top of the poll. So, I think you can read it there. Okay, this is a five point question. This is number 21, it's five points. So it should take you a little more time. I'm gonna end the poll. Not all of you were able to solve it in the amount of time I gave you, but keep in mind it is a timed contest. So if you want to work on this some more, go ahead and use the recordings after our webinar session is done and you'll be able to see. But of those who answered it, you are getting it correct in that it is 12. So let's take a look at it. So I gave you the hint. If you want a terminal zero, all you have to do is multiply by 10, right? Anytime I multiply by 10, I get a zero at the end, don't I? Right, three times 10 is 30. And just like in the problem that Jacob did with you, two times five equals 10. So let's look at how many times if I'm multiplying these together, I would be multiplying by five. Multiplying by two is not gonna be a problem because every other number is even. So there's lots and lots and lots of twos. I don't have to worry about the twos. But let's take a look at the fives. So I have five times one, five times two, five times three, five times four, five times five. That gets me up to 55. Five times six, five times seven, five times eight, five times nine. And then five times 10 is the same as five times five times two. So let's see how many fives I have all together. I have one, two, three, four, five, six. That's a tricky one. Seven, eight, nine, 10, 11, 12 fives all together. And many more than 12 twos. So if I multiply all those fives by twos, I would get 12 terminal zeros. Zeros at the end of my product. Seems so simple now that it's explained, doesn't it? It's just hard when you don't have an explanation. And that's why we practice. That's why we learn how to do them. The king and his messengers are traveling from the castle to the summer palace at a speed of five kilometers per hour. Every hour, the king sends a messenger back to the castle who travels at a speed of 10 kilometers per hour. What is the time interval between any two consecutive messengers arriving at the castle? So make sure we want the time interval. That means it's the difference, right? And we want when the messengers arrive back at the castle. Again, another five point question. You can kind of draw out a little diagram if you need to. I like to do little timeline diagrams. You may or may not like that. Okay, I've had several of you answer. Again, not all of you are answering and that's fine if you still would like more time on these problems. But by over 50%, we are getting the correct answer, which is 90 minutes. Let's take a look at it. So I've drawn out the first part of it. Let me explain what I did. If the king travels for one hour at five kilometers per hour, he's going to go five kilometers. And at one hour, they send a messenger going back to the castle. That's what this kind of backwards arrow is here. So that messenger is traveling at 10 kilometers per hour. So to go five kilometers back will take half an hour, which means that that messenger would arrive the one hour plus that half hour after the king's departure. So it would arrive an hour and a half later. Now the king is still going at that five kilometers per hour, right? So at two hours, it's got a total of 10 kilometers when they send the next messenger back. And this guy's going at 10 kilometers per hour and we've traveled 10 kilometers. So this takes one hour to travel, which means that the arrival back at the castle could be, typos, sorry, I'm trying to erase it. Come on, erase. Will be three hours after the whole trip began. We can do that again. We can go another five kilometers, which takes us to three hours after the king has left. We'll send another messenger back. He's going again, 10 kilometers an hour. So it'll take one and a half hours to go the 15 kilometers. But we have to add the three hours because that messenger didn't leave until three hours into the journey. So we're talking about four and a half hours after the king's departure. So you could follow this pattern and you will figure that a messenger will arrive back at the castle every hour and a half or every 90 minutes. Okay, so that's something where you could just kind of work it out step-by-step. Would be one way to do that. Okay. Number seven. And again, there is a pole for this one. A large cube has an edge with a length of seven centimeters. On each of his six faces, remember king's got six faces, two diagonals are drawn in red. The large cube is then cut into small cubes with edges one centimeter long. How many small cubes will have at least one red line drawn on it? Can have more than one red line, but we're talking about at least one red line. I do suggest you might want to at least draw one face of this cube to give yourself an idea of what you're looking at. But you might not need to if you're very good at visualization. We'll launch the poll. This is the last poll we have, but we do have more problems. They just don't have polls. Okay, I'll have to end the poll here so that we can get through at least a few more problems. There are several different answers that students have given. Let's take a look and solve it together. The most popular answer is 62, but only 42%, and not even two-thirds of you have finished the problem yet. So, let's take a look. I know this one takes a little bit longer. I have a way to make it faster, though. Somebody's already drawn it for me. So, sorry, just need to make one more click here. If I look at my 7x7 square, and I draw the diagonals through it, I can see that 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, I already counted that one, 11, 12, 13 squares have red lines on them. Okay? So, that would be 13. I don't need to double count the center square. So, if there are six faces, that would be 13 times 6, right? 13 times 6, that gives me 18, that gives me 78. So, that might be one answer you might try is 78. However, if I think about this as being a cube, right, and I draw a red diagonal on the top of that cube, you'll notice that the corners are going to have diagonals from different directions on them, right? So, if I had continued this cube, wrong color, coming down like this, I would still have a red diagonal over here. So, each corner actually has multiple red lines, and I don't want to count each corner three times, because it would be on three faces that it touches. So, how many corners are there on this? There are eight corners on the whole thing, right? So, the eight corners, we don't want to triple count. Does that make sense, triple count? Because they would be on three different faces. So, I have to take 8 times 3, which is 24, and then I will subtract that 24 from the 78. So, 78 minus 24. Oh, I want to single count them, not triple count them. So there's the mistake. OK, I knew it. Sorry if I make mistakes, guys, but this is what happens when you're trying to do it live in front of a real audience of students. So I don't want to triple count them, but I do want to single count them. So I'm going to get rid of 16 of them that I've counted an extra two times. So I do the 78 minus 16, which is going to get me to 62. So I will have 62 pieces that have some red line on them. So don't triple count, single count, and that will get you there. All right, I think this is a problem you wanted to cover, Jacob. Yeah, so the figure in the picture consists of seven squares, and square A has the greatest area, and square B has the smallest area. The lengths of the sides of the two squares are given. How many squares identical to square B would it take to fill square A completely? So I'll give you guys some time to think about this. Okay, so I'm going to go over it now. The key thing to notice is that since this is of length 2, that means that this length here is also 2, because they're all squares. Now, notice that since this is 3, that means that this length here is also 3, so that means that the side length of B is 1. Okay, so side length. So, now we know the side length of B, but we want to solve for the side length of A. Okay, so since each of these small squares are identical, that means that they each have a side length of 2. So, this total length is 6. Okay, but B has a side length of 1. So, since this has a side length of 6, this length here, side length of 6, that means that the side length of A is 6 minus 1, which is 5. So, now we have the side length of A, and we have the side length of B, so we want to find the number of squares identical to B, which would fill the square A completely. So, let's say that there's k squares that are, let's say that the answer that we want is k, so that means that k squares are identical to square B, and that is the amount that it's needed to fill square A completely. So, then we've got k times the area of B, which is 1 squared, should be equal to 25, which is the area of A. So, when we solve this, we have, this is just 1, so it's equal to 25. So, the number of squares needed to, number of squares identical to B that is needed to fill square A is just 25, so that's our answer. Students can think of the side length, since A is a 5 by 5 square, they could also multiply 5 by 5 to know that you would need 25 1 by 1 squares to fill it. That's excellent. Okay, so hopefully you have enjoyed this exploration of different types of math kangaroo problems. So, we got to get through 8 of the questions, and I know that some of them get tricky, they take a little more time, and the polls take us a little bit of time, so there's a bit of a balance. I like to have the polls, so I can see how you're doing, and you can actually input your answers, but at the same time it takes a little time for us to do that. So, in the next few weeks, what you're going to see is we're going to take each of those types of problems, and we will explore several problems of different difficulties using those tools. So, if your favorites are geometry problems, we're going to have a whole lesson of geometry problems. If you like the factors and the exponents and different things like that, you'll notice that we have a lesson that covers that. So, I hope that today was challenging, but not too exhausting for you. It should be that you are learning, you couldn't do all of the problems when you got here, but now you feel confident that you could do them if you saw them again. And another way to do that is to go ahead and get the recordings and see if you can do them. In your registration for this class, remember there are links for handouts so that you will have the figures for the next time you're here with us on Sunday. There's also links to optional homework. You can do past contests for homework, or sometimes we have practice tests, practice problems that look a lot like the ones we've done today, so that you can go back and try those on your own. I hope you liked the webinar. Thank you, Jacob, for your help. Thank you all of you for being here. And Jacob and I will see you again next Sunday. Have a really great week.

Math Kangaroo

webinar

grades five and six

Dr. Sarah Sagee

problem-solving strategies

mathematical concepts

geometry

fractions

confidence building

supportive learning