Okay, welcome to our Math Kangaroo class level 5 to 6. This is a webinar. So for the webinar, we'll be recording every session. If you have to miss any future session, we will be able to send you the link for you to review later. So do not worry. But we definitely would like you to participate in this session lively. Okay, so here we go. Today we are going to have the first class is introduction. This is what we are going to do for today. We have the brief introduction about what Math Kangaroo is and a strategy and a roadmap. We introduce our four step strategy. We'll go through 10 problems, like 10 sample problems, and then we'll wrap up. So just some ground rules here. Actually this is a webinar. So actually, I would like you to keep your camera off and this is the setting I set up. I will have your microphone muted. If you have some questions, you can type in the chat, our TA will be monitoring it. But we cannot promise to go over it unless we have extra time, because we have a large audience here in the class. You should have your handout ready, pen, paper and pencil during the class time to take notes and work throughout the problems. So before I start to introduce Math Kangaroo, can I have a quick survey here? Have you ever participated in Math Kangaroo before? If you have done this before, please type one in the chat. I'm just trying to count how many people have done this. Type one in the chat if you have participated in the Math Kangaroo before. If you haven't done this, type zero, please. Okay, I see. So we do have some people participated. Let me go over this part then. So as you know, this is an annual competition globally. In the U.S., we always have it on the Thursday, third Thursday of the March month. And let me take a look at the slide here. So I think for those of you who haven't really participated in Math Kangaroo competition, it's actually more like a box of chocolate. It's very tasty, right? So math questions are not really like that crazy. It's more fun. So like the box of chocolate you are seeing here, they are unusual, they are interesting, and you may discover a lot of things that you don't really expect. There are 30 problems in total. Usually the first 10 questions are often quick. The next 20 questions are trickier. This will require you to think harder and think carefully. To solve these problems, you might need to do different things, such as drawing a picture on your sketch paper, make a table or church, organize your thoughts to find a pattern described from the problem, act out the problem in your imagination, especially for those 2D or 3D geometry problems. Sometimes we require you to work backwards, figure out a lot, a list of possibilities. Also have an educated guess, started with guessing an answer and then check if it is true by putting it back into the statement. So you may be used to getting problems right on the very first try, but that may not happen with Math Kangaroo. This is a funny picture just to tell you what happened, right? So sometimes when you have an initial guess, it is right, of course, it's lucky for you to do that correctly at first time. But many times your answer is not there. So what happens? Here actually is great for you because we would like you to utilize the pencil we gave you in the test, right? The Math Kangaroo slogan on that, we will help you to utilize that pencil smartly. So sometimes it's okay, you know, whenever you make a first mistake, it's totally okay. You may need to work a problem more than once. You just need to try things to see if it works and continue trying that. Just be patient, be kind to yourself. Whenever you find, oh, my first guess didn't work, take a deep breath so that you can focus on solving the problems rather than worrying about the problems. Keep trying until you get there. You can convince yourself you will definitely get it right. So you can do this. This is the whole philosophy that we are promoting here by working on Math Kangaroo questions, we would like you to really enjoy the fun of solving math problems. Okay. So typically, we have 24 questions for level one through four, but now you are in level five and six, you will have 30 questions. No matter what grade you are in, you will have 75 minutes in total to solve all the problems. We are dividing them into three categories, easy, three points, medium, four points, hard, five points. So the problems are different from the exercises. In an exercise, we are practicing technique or a skill, you will know how to approach it and you just need to go through the steps to solve it. In a problem, instead, we don't really know at first how to approach it and demands much thought and resourcefulness. Problems often involve trying different strategies, making mistakes, try and check again and again and starting over until a solution is found. So this is a place I would like to introduce the four-step problem solving strategies. Just to have a quick sense about our audience here, can you type three if you have heard of this before in the chat? Great. So we have four-step problem solving strategies for math kangaroo questions. Step number one, we would like you to understand the problem. By reading the question carefully, you will have to understand exactly what is described. You will be able to determine what is really being asked and that's very important because sometimes we notice when working with students, we notice students often make mistakes by not really understanding what is being asked. Step number two, make a plan. Before you start to solve the problem, we would like you to have a specific plan. How are you going to tackle this problem? What do you need to solve it? What strategy are you going to do? Is it list and table? Is it geometry problems? Is it time and clock? Is it logical thinking or educated guess? Whatever it is, put it there on the sketch paper. Step three, implement, carry out your plan. So here we would like you to carefully complete your calculations or organize your thoughts and steps, of course, ideally on the sketch paper so that you can track your thoughts to see where it messed up. Step number four, look back to check and reflect. Wherever you have an answer, don't submit the answer right away. You want to take a look, take a pause, think, does this answer make sense to you? Did you answer the correct question? If you believe your answer is correct, then check and see if it still make all the statement from the question correctly. If not, you may want to restart over again. Okay. Matthew, if you are here, let me know because people are joining and don't want to interrupt our classes. Okay. Here we go. Here's our roadmap for the 10 weeks. Today we're having introduction session. So we'll give you a sample of questions representative of all the following categories from patterns to algebraic thinking, drawing a picture, make a table or list, work backwards, logic problems, statistics and probability problems, geometry problems. In the last session, we'll have hands-on problems. Okay. Now let's get started. The patterns is the first one. So this is from 2014, question 22. So you can see this is in the later section of the test, that will be harder. I'll start by reading the question. There are five sums, sum A last three minutes, sum B two minutes and 30 seconds, sum C two minutes, sum D one minute and 30 seconds, and sum E four minutes. These five sums are playing in the order A, B, C, D, E in a loop without any breaks. Sum C was playing when Andy left home. He returned home exactly one hour later. Which sum was playing when Andy got home? I would like you to think carefully while I'm launching the poll here. You don't have to rush. Remember, I just introduced four-step problem solving strategy. Please go through this question using those four steps. Interesting, we have four different answers chosen by different people. There is only one correct answer. So please try to use step 4 if you believe your answer is true. Put it back and see. Okay, I think more than half of you submitted the answer already. Let's take a look at this question together. You can still keep submitting your answers. If you still need time, don't rush. Just finish whatever you are doing. Okay, so here we have five songs, A, B, C, D, E. They have different lengths. So there is a pattern here. I can actually draw it out like this. A, B, C, D, E. And then again, A, B, C, D, E. So remember song C was playing when Andy left home. So we'll mark here, okay? Song C, two minutes, right? How long is D? I just wrote all the lengths of the songs underneath the letters. So D is one minute and 30 seconds. 1.5 or one and a half. E is four minutes, right? A, three. B, two and a half. The same thing. So what we can observe here is for how long when it rotates or like repeats, right? C, D, E, A, B. And then again, C, D, E, A, B. So how long is the C, D, E, A, B? Five songs in total. You can sum it up and that's exactly 13 minutes. So how many 13 minutes have passed within an hour? So you will count one, two, three, four. Okay. 13 by four is 52 minutes. It's not 60 minutes yet. But if you have the whole cycle of 30 minutes past, it will be over 60 minutes. So we will stop right here and then see. How many more minutes do we get to right there, right? When Andy got home, that's 60 minutes. So let's see. When song C, D, E play, that's seven minutes and a half. So 52 plus seven and a half. That's 59 and a half minutes. Is that right? So when Andy got home, actually the next song will be A. So correct answer will be A. So just now, let me take a look at the poll. I think majority of you got it correctly. So congratulations for those of you who chose A. If you chose B, C, D, E, you did it wrong this time. That's okay. Remember, you want to be nice to yourself. You want to see where it's messed up. So maybe it's because you will just do this mentally without really writing it down. In that case, I would encourage you to be more carefully writing it out on your sketch paper. If you are doing this kind of question in the test, also do the same. Okay, so let me end the poll and move on to the next question. Question two. Six identical black beads and three identical white beads are arranged on scale as shown in the picture above. What is the total weight of these nine beads? So here we have two different colors of beads, right? We have two scales. There's a relationship suggested, implied by these two scales. You need to figure out the weight of black and the weight of white beads, and then add it up to calculate the total weight of these nine beads. I will be launching the poll soon. Please make a plan. Write it down on your paper. Calculate. The poll is up. If you are ready, please submit your answer. If you are not ready, please continue your calculation. I saw some questions about the poll. It should be on the screen whenever I launch the poll. If you are having some trouble, maybe try another device. OK. I think we have three different answers submitted so far. So let's take a look together. OK, so on the screen, on scale 2, on the left and right, you can actually see both there is a black, right? So we can actually take that one out. So in that case, we can actually know two black beads plus one white beads. The weight, the total weight of them is 30 gram. Is that right? And then scale 1, it says two black, the weight of two black beads is equal to the weight of two white beads plus 6. From here, if you do this algebraically, you can do this quickly. But here, I want to show you how you can do this using algebraic thinking rather than using algebraic equations. So from this modified scale number 2 plus the original scale 1, step 2, we can actually figure it out like this because two black equals two white plus 6 gram. So we can replace those two black beads by two white plus 6 grams on the left. So you will notice three white ones plus 6 equals 30. So over there, you can know what? You will know the weight of the white is actually 8. Is that right? Because 30 minus 6 is 24. And that's three of those white beads' weight. Is that right? So you will obtain the white beads' weight is 8 gram each. Then from there, put it back. From scale 1, you will know the black beads' weight is equal to 8 times 2. Take a look at on the right, right? On the right, right here. On the right. 8 times 2 plus 6. That's two of those black beads' weight. So divided by 2, you will obtain the black beads' weight as 11. So now you know the white, 8. You know the black, 11. The total will be 9 beads. So just count how many black, how many white, add it up together. That will be 6 times 11 because you have 6 black ones. You have three white ones, so 3 times 8. Adding up together, that will be 90, 90, 90. So I think majority of you got it correctly. So again, congratulations. 90E is the correct answer. Let me end the poll and move on to the next question. Next question is on ratio and proportion. From 2006-2008, in a certain class, 1 8th of the classes received a C on the math exam. 1 6th received a B, and 2 3rds received an A. There were no Ds. How many students received an F if there were less than 30 students in the class? There is one condition here. If there were less than 30 students in the class, we would like you to know the number of students who received the F. The poll is up. Don't rush. Feel wherever you feel ready, then submit your answer. Yeah, some of the problems are not in the handout, that's correct, because we would like you to focus on during our webinar, some of the questions we would like you to, we don't want, basically we don't want you to solve the problems beforehand, we just want you to have something to draw on whenever you have some like graphs, pictures, tables, questions there. Okay, let me grab my pencil so that I can draw on the screen, I'll be right back. OK. Let's take a look at this question together. Oops, sorry. So we have different people having A, B, C, right? We would like to sum it up to see the ratio. And then 1 minus that ratio will be the proportion of people who received F, because there is no people who received D. So let's do that. 1 8 for C, 1 6 for B, 2 3rds for A. So adding up all together, you will have 3 plus 4 plus 16 over 24. And that's 23 out of 24. So what is the rest of that? Will be 1 out of 24. So 1 over 24. You will know 1 out of 24 of all the students in a class received an F. And on top of that, you know there were less than 30 students. So person, that must be an integer. So that says the number of the students in the class is less than 30. And it must be a multiple of 24. With these two constraints, you will know the class has 24 students. And only one student received an F. So correct answer should be B. And seems like most of you, for those of you who submitted the answer, 90% of you got it correct. Great. So let me end the poll, move on to the next question. Question 4. This is from 2009. Question 21. Jane multiplied the product of 18 factors, each equal to 8, by the product of 50 factors, each equal to 5. How many digits does her final products have? So I think this question really wants you to understand the question. How many digits does her final product have when you do this together? Let me launch the poll. As I said, not every question is on your handout. Okay, interesting. We have three different types of answers already. Okay, let's take a look at this question together. Jane multiplied the product of 18 factors, each equal to 8. The 8 raised to the power like this, 8 raised to the power of 18, and then keep multiplying times product, by the product of 50 factors, each equal to 5. So times 5 raised to the power of 50. So that's the first step. Write it out, whatever the question, whatever the information coming from the question, write it out. And then, step number one, you need to know what is being asked. It's asking you to find out how many digits does the final product have. You don't need to multiply everything out, but we need to find how many digits. So how to solve it. You will notice, okay, there's an 8, there's a 5. If you want to know the digits, and here we can actually find how many zeros, because here we have five, we have twos. So we can decode this 8 into 2 raised to the power of 3. So that's the step two, right? And keep doing. You will notice 2 to the power of 3, and then raised to the power of 18, that's equal to 2 directly raised to the power of 54, right? And times 5 times 50. So then, you can put this 2 and 5 together, because they have something in common. Both of them raised to 50, to the power of 50, right? So you can put them like this, 2 raised to the power of 4, times 2 raised to the power of 50, and then times 5 raised to the power of 50. So then you can put this 2 and 5 together, because that makes you a 10. That's what you want to do, right here, like 2 raised to the power of 4, times 2 times 5, all together, raised to the power of 50. So that's 16 times 10 raised to the power of 50. So, apparently, 10 raised to the power of 50, you will have 50 digits. In front of that, you will have 1 and 6, and that's two digits. So the final answer will be 50 plus 2, will be 52 digits. Did you get it correct? C is the correct answer. Okay, raise, okay, let me pause this poll and move on to the next question. Next question is about make a table or list. This strategy may help you to solve this question quickly. This is from 2003, question 21. With how many zeros does the product of the consecutive natural numbers from 1 to 50 end? Again, the product of all the consecutive numbers here. You are trying to find the zeros, very similar to the previous question. OK, I saw some questions in the chat. If you are asking about the handout, as I said, the handout won't have every questions. The handout only gives you some questions where we need you to have something to depend on during the webinar. We don't want you to solve the question beforehand. If you have some diagram, table, or picture graphs, then we'll have you in the handout. We'll have those things in the handout. So please focus on the question listed on the screen and try to solve it lively. Oh, now this is interesting. We have all five choices chosen by different people. OK, seems like this is tricky, maybe. Let's take a look at this question together. So here, I listed out, oh, maybe put out too many things at the same time. So here, we would like you to find a product of all the consecutive natural number from 1 until 50. But you need to find 0. 0 is coming from 5 times 2, right? 5 times 2. So we need to find the factors of 5. Where does it appear? So it appears how many times? Let's count. 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50. Here are the multiples of 5. And I purposefully wrote it out like this so you can see, oh, 25 is actually 5 times 5. So there are two 5s appeared. So basically, 5 appears twice in this 25. And also for 50, number 50, that's 5 times 5 times 2. There are two appearances of 5. So count. The fact of 5 actually has been repeated 12 times in the prime factorization of the product of consecutive numbers from 1 to 50, so 12 times of 5. So how about 2? How about 2? You do the same thing. You will notice for 2, 4, 6, 8, and so on, it appeared many, many times, definitely more than 12 times. Is that correct? So the factor 2 is repeated more than 12 times. But we only need to pair up the 2s with the 12 5s to get 12 factors of 10, right? So from there, you will know the product of the consecutive number from 1 to 50 ends with 12 0s because you basically have 12 pairs of 5 and 2. So the correct answer should be C. And that's kind of the majority of people's answer. Good for you. OK, let me end the poll, move on to the next question. Question 6. The king and his messengers are traveling from the castle to the summer palace at a speed of 5 kilograms per hour. Every hour, the king sends a messenger back to the castle who travels at a speed of 10 kilograms per hour. What is the time interval between any two consecutive messengers arriving at the castle? So this is a time, clocks, and calendar problems, but I definitely encourage you to draw it out to help you to see what happens. The poll is up. You don't have to rush. Take your time. Okay, nearly half of you submitted answers. Let's take a look at this question together. So, as I said, I encourage you to draw it out. Can you? Yes. Okay, so maybe for the first hour. Let's see. I don't know how long this is, but let's say at the first hour. So 60 minutes, an hour later, both the king and the messenger, his messenger arrived at here. So the king keeps going. The messenger goes back. But the messenger is traveling at twice as fast as the king. So here, it took like 60 minutes for both the king and the messenger to arrive at here. But now after an hour, because the speed of this messenger is 10 kilometers per hour, it's twice as fast as the previous speed. So it will take that messenger 30 minutes to go back. And then the king and the next messenger keep going like this. It will take another 60 minutes. 60 minutes, right? So when this person arrives at here, then he sends another messenger back. Is that right? So then the messenger needs to travel how long to get here? So from here, 60 minutes later, here this person arrives, takes like 30 minutes to arrive here, then here it takes another 30 minutes. Is that right? So you will notice that one hour after leaving the castle, the king is five kilogram away from the castle, and the first messenger is sent back, right? It will take the messenger half an hour to get back to the castle, which is 90 minutes from the start of the journey. And then two hours later, two hours after leaving the castle, the king is 10 kilograms away, the second messenger is sent back, it will take this messenger one hour to get back. So that's three hours from the start of the journey. Right? So three hours is 180 minutes, and then the previous messenger is 90 minutes. Is that right? So 180 minutes minus 90 minutes, that's 90 minutes. Okay. Oops, where's my... So that's just right what I was talking about. The second messenger, the first messenger, it took like 90 minutes for him to get back since the start. And then the second messenger is two hours, sorry, three hours. So that's 180 minutes. So you do that math, 180 minus 90 will be 90 minutes. And the correct answer will be D. Okay, and that's the majority, I think. Let me check. Yes, that's the majority of your answer. Okay. So this question, I think you can do this on the picture to help you to organize the dots. Also, you don't need to do this because here you have this 10, you have this 5. You are trying to calculate two consecutive messengers arriving at a castle. If you do the math, you can do the same thing. So it doesn't matter what strategy you use, your preferred problem solving strategy, but you will get the same answer. No matter what you do, you will have the same answer. Okay. You can keep doing like this. The third, like this, if you do this way, then the third one, it will be four hours and a half. So four hours and a half will be 70 minutes. Okay. Okay. Let's move on to the next question. And stop the poll. Yes. Next question is a geometry problem. Question 7. A large cube has an edge with a length of seven centimeters. On each of the six faces, the two diagonals are drawn in red. The large cube is then cut into small cubes with edges of one centimeter long. How many small cubes will have at least one red line drawing on it? Now, I've seen four different types of answers submitted. Remember to use our step four. If you believe your answer is true, pause and check. OK, seems like we need to rush up a little bit. So I know some of you are still making your choice. Let's start taking a look at a question together. So we have this cube, a large cube. We would like to see this question from one face first. So take a look at this picture here. Here, how many small this square has red color? You will count 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3. And down here, you need to minus that. So this is 7 plus 7 minus 1, because you have 3, 3, 3, 3 on both ends. So each face, you have 13. You have 13. All has this red color. And then how many faces do we have for the cube? That's the next step, right? Because we have a cube, 1, 2, 3, 4, 5, 6, all faces of the cube. You multiply that together. That's 78. So some of you, or most of you, I think currently majority of you chose 78. However, is that the case? Is that case? If you draw a cube like this, so think about this. On this corner, because there's a large cube, is that right? Like this, like this, right? Think about this piece. This is dashed line. Think about this. I'm putting this right here. You can imagine this corner piece has 1, 2, 3, these three sides, basically this side, this side, and then the other side, which you cannot see. All of them has this color. So it has been counted how many times? Three times, so triple times. The corners have been counted three times. So how many corners do we have? We have 1, 2, 3, 4, 5, 6, 7, 8, right? So you have eight corner cubes. They have been counted three times. You need to minus those. So that's 13 times 6, but not quite there yet, minus 2 times 8, because it's been counted three times, triple, right? So minus the additional 2, so 2 times 8. So that's how you get the correct answer, 62. So we have three people got it right. Congratulations. You have been thinking this question carefully enough, and you got it right. Good job. Question eight. Question eight, work backwards. The figure in the picture consists of seven squares. Square A has the greatest area, and square B has the smallest area. The length of the sides of two squares are given, how many squares identical to square B would it take to fill in square A completely? You are given two square, the side length, two for this one, and then three for this one. You need to know how many identical squares to square B would it take to fill in square A completely. So let me launch the poll. Okay, since we don't really have too much time left, I will be quick to start. Most of you got it right, for those of you who submitted already, so take a look here. This one should be simple, if you draw it out correctly like this. If you know this is 2, then you will know this is 2. Is that right? You will know this is 2. And since this is also squared, this is 2, then this is 2. And then, because you know this is 3, so the additional part here is 3 minus 2 is 1. So you will know this is 1, side length of B is 1. So now, because this is 1, and this is 2, this is 2, and this is 2, right? This is 2, because you know this is 1, this is 1, then this is 1. So once you know this is 1, then 2 plus 2 plus 1 is 5. You will know A here, the length of that is 5, the side length. So now, A is 5 times 5, 25. B is 1 times 1, is 1. So how many? 25. Good job. B is correct answer. Okay, next question, logical problems. Three pirates were asked how many coins and how many diamonds their friend Graybeard had. Each of the three answered truthfully to one question but lied answering the other. Their answers are written on the piece of paper it pictured. What is the total number of coins and diamonds that Graybeard has? Okay, let's take a look at this question together. Currently, most of you chose C. Let's see if that's correct. So take a look at statement 2 and 3, because we were told that each of the three answered truthfully to one question, but lied answering the question, the other. Because 2 and 3, they both said he has seven coins, but they're disagreeing on the number of diamonds, right? So because of that, you can actually conclude seven coins must be right, because they cannot be right on a diamond, both. So seven coins must be right. Then if seven coins are right, then how many diamonds? Then go back to statement number one, because eight coins is wrong, because we already said seven coins are right, so eight cannot be right, then six diamonds must be right. So from there, you can know seven coins and six diamonds, six plus seven is 13. Good job for those of you who chose C. Okay, so you can see this is question 26 already, but actually it's not really that hard. We can make it even harder, but this is just representative of logical problems. Whenever you see this kind of questions, don't freak out. Just try to find a strategy here. Try to find a place where you can pick on and then see which one is right. So from this place, we said coins. They have to agree on the coins, because diamonds cannot be right. So C is correct for this one. Okay, next question is our last question for today. Question 10. The faces of a cube are painted black, white, or gray, so that opposite faces are of different color. What of the following is not a possible net of this cube? Remember, step number one is saying which one is not possible. Not possible, okay? So, okay, the pole. Okay, let's take a look at this question together. So here, we would like you to mentally think how these came together as a cube. You can maybe start by, oops, start by this one. Basically, there's four faces, wrap it up. For example, A will be the white here, black here, gray here, and then white. Can you imagine this, right? So similarly, you can do this for B, C, D, E. And then you will notice for this cube, so for example, for A, you will notice if I put the white, white goes here, the black goes here, the gray goes here, the white goes underneath. And then the gray will be this side. And then the black will be this side, okay? So mentally think about this. If you go through A, B, C, D, E, you will notice for the E, you have black, black, okay? Here's the completely black, black, black. Here, you have gray on this side. This is like this side gray. And then here, the white goes to the behind thing. I can't see that, but that's white. And then here, also white. You will have white in front, white in the rear. They are opposite. So that doesn't satisfy this opposite faces are of different color. So E is not correct. Therefore, it's not a possible net of this cube, okay? So this question really wants you to utilize your mental thinking. Try to find a way to see, to visualize this cube in your head. And you will get the E as correct answer. Okay, so let's wrap up quickly here. Today, we just gave you a sample set of all different types of questions. And you will notice every question requires you to use a different strategy to solve the question. You will also see Math Kangaroo focus on problem solving and critical thinking. It's not really just math calculation and boring calculations, right? So in this introduction section, we have exposed you to 10 popular question types. In the future sessions, we'll go over each of them particular type in depth. Okay, so we hope you enjoy the webinar today. I will posting a link in the chat. So please take your time to finish the class evaluation before you go. Just give me a second here. Here we go. Okay. Oops. Everyone. So welcome to the webinar and I will see you next week the same time. Again, if you can't make it, we will always be recording this. You will be able to find it in the folder. Okay.

Math Kangaroo

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algebraic thinking

geometry

logic

statistics

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