you should be here for the Math Kangaroo level 3-4 webinar. Our theme today is finding a pattern for solving a simpler problem. Okay, so you'll notice today we'll be looking at how patterns are used in Math Kangaroo problems. There are many different kinds of patterns, so they'll be interesting. We'll have a different variety of them today. And you'll also see that sometimes you can take a challenging, complicated problem, and you can break it down into simpler pieces, and then put those pieces back together to find an answer. So let's talk a little bit about what you need to actually be successful in the webinar today. All right, my name is Dr. Sarah Sagee. I'm a biomedical scientist. I live in San Diego, California, where today it's going to be over 80 degrees. It's been hot and dry. So if you see me taking some sips of water, it's because my throat gets very dry in this hot weather. I have been teaching math and science for 30 years. I coach an elementary school math club, and in my spare time, I like dancing, swimming, karate, and I love to go out for walks with my dog. We have a high school assistant, and I'm gonna let Alan introduce himself. Hello, I'm Alan. I'm a junior in high school. I've been taking Math Kangaroo for about 10 years now, and it's really helped spark my interest in math. I can't wait help. I can't wait to help introduce you to some of these new problem-solving concepts as well. I have two dogs. I like to sing. You may hear them in the background faintly at times. That's me. So I can see that some of you have tried to raise your hand. We can't address you individually, but if you go ahead and put your questions in the chat, Alan or I will do our best to try to give you answers. Mostly, it will be Alan answering your questions so that I can make sure that we're still conducting the webinar in a nice, organized manner. Okay, welcome. You've met us. Any questions you have, any answers you have, go ahead and put those in the chat as we go along. Okay, your microphone is going to stay muted, and we do not see your images. We do not see your videos. So the best way to reach us is use that chat. We are checking it. Both Alan and I are looking all the time. If you were able to print out that handout, it is a great way to work through the questions to work right on the handout. If you did not print the handout, all of the pictures we're using today will be on the screen, so you won't miss anything. But do have scratch paper so that you can work through the problems on your scratch paper. Like I said, ask your questions in the chat. We will do our best. We can't answer every single question, but especially if we see a question that we can't answer, we will do our best to answer it. We can't answer every single question, but especially if we see a question over and over again, we're going to try to get to those questions. We do want you to learn. And during our session, a few questions are multiple choice. All the Math Kangaroo contests are multiple choice, so some of our questions are still multiple choice. And you will see a poll that pops up on your screen, and we'll be able to answer the questions in the poll, and we'll see how everyone is doing that way. If your polls do not work on some systems, the pop-up windows do not work, always put your answer right in that chat so that Alan and I can see how you're doing, and we can explain what needs to be explained, or we can move on if you found a question to be too easy, okay? All right, so this is a practice for the poll. So I'm going to ask if Alan will launch that first poll, because it's going to cover up part of your screen. So did you notice that covered up part of your screen? Go ahead, grab on the top of that with your mouse and see if you can move it around. This is the practice for moving that poll out of your way, okay? We are not going to launch the polls until after you've had a chance to look at all the questions, but we just wanted you to have a practice. So hopefully you've been able to move that window around, and you've been able to enter what fruit you see on the screen. Pretty simple. Not everybody will see the poll. It depends on your particular setup for Zoom. Remember, if you do not see a pop-up window, if you do not see the poll, go ahead and put your answer in the chat. So if you do not have a window where you can click on red cherries, then just put red cherries in the chat. So it looks like the majority of you were able to see red cherries. Some of you thought they were apples. They're a little small for apples and with those little stems on them, they're more likely cherries, don't you think? Okay. Don't worry because we don't have very many poles. Just a few poles here and there. Okay, let's get into our first idea for Mapkangaroo. There is a four-step strategy that we recommend you use for solving problems. And this works for all types of problems, even non-math problems sometimes. The first is to understand the problem, number one. Determine what it's asking. What my students typically have found is that either the first sentence or the last sentence gives you a really big clue about what you need to answer. Make sure you answer the correct question. Number two is you're gonna have to have a plan. How will you solve that problem? What information do you need to solve it? Is the information in the problem? What information do you need to figure out? Number three is carry out your plan very carefully. That's why I highly recommend scratch paper and not messy scratch paper where you just write a couple of things here and there and you turn it around and you work upside down. I recommend you keep yourself organized. So if you're doing problem number one, you can go ahead with number one and work on that problem and then go to problem number two and work on that problem. Keeping yourself organized will help you get a correct answer. Number four is look back, check and reflect. Does your answer make sense? And did you answer the correct question? So what does that mean? When you first read the question, did you have an idea? Would your answer be in the hundreds? Would your answer be a single digit? Would your answer be a number? Is your answer gonna be a non-number? How are you gonna need to answer that question? Did you do that correctly? Does that make sense? And then sometimes be very careful. Sometimes the question will have a little trick in it. Usually check that first and last sentence is the most likely place. It might say, find the sum of something or it might say, how many solutions are there? So be very careful that you're answering the right questions, okay? And I'll try to use these steps for most of the problems. Sometimes one step or another might be very simple and easy to do. So we'll go through that part quickly. Today, we're working on patterns and finding simple problems. So recognizing a pattern is a very good strategy. So when we said plan, number two, what is your plan to solve the problem? Recognizing patterns is part of a plan. Sometimes a problem might seem very complicated. So you might need to break it up into little pieces. What happens between part A and part B? What is the change that happens? How can I figure that out? And we'll have a few questions towards the end of this presentation that are gonna get quite complicated. And I hope you'll be able to use that strategy of breaking them down into simpler bits, okay? We're gonna start with our first question. It's a visual pattern. So remember there are different types of patterns. Here's a visual pattern. There should be a poll for this, which we're gonna wait. So everyone gets a chance to take a look and try to figure it out. So in your mind, take a look, what do we need to do? We need to find which one of those five pieces will fit into that pattern and match the shapes around the corners, right? So now we know what to do. Our plan. I don't know, people might have different plans. You might wanna test those different possibilities. You might want to take your hand out and you might want to draw. So what would a little piece look like? I think I can complete this piece like this, right? And this one's gonna need to be pointy. So that might be my plan is to draw. Your plan might be different. We're gonna give you a second. Hopefully you're trying to figure out which piece fills in and then we can launch the poll. All right, the poll is launching. If you do not see the poll, remember put your answer right into the chat and Alan and I will see how you're doing. So for me, I like the strategy of drawing in the finishing pieces, drawing what's missing, but everyone might have a different strategy. So I'm gonna draw this. This one goes here. And then what goes here? Another pointy one. So I need a shape that has a rounded petal here, two pointy star kind of pieces and then this concave piece here. Looks like we're doing quite well. I'm gonna end the poll. Did everyone see that my picture that I was drawing looks a lot like E, right? I have this concave piece here. I have this rounded petal at the top and I have these two pointy pieces to match piece E. Okay, now we're going to do a math pattern with numbers. So here's our mathematical one with some numbers. Give you a moment to think about this. After we simplify two plus two minus two plus two I hope you saw the pattern there, the result will be. So understand how do you find the pattern? What do you need to do to solve this? And when you have it, go ahead and put your answer right into the chat. Yeah, we're getting a lot of correct answers. We have a few slightly different results. I think you found the pattern, right? The pattern is we have this plus two, minus two, plus two, minus two, plus two, minus two, plus two, minus two. I think a lot of you know what you can do with all those plus two, minus twos, right? All those plus two, minus twos, my plan is to simplify those plus two, minus twos. Each one of these is a zero, right? And I'm gonna end up with just two. Oops, sorry, that's so messy. Two plus two, right? Almost all of you have four now. That's fantastic, good job. All right, our next one is a different, slightly different type of a pattern. This one should be on your handout. It might be easier to fill it in in your handout, but if you don't, your scratch paper will work well enough. I mean, all you have to do is make some rows of numbers, right? Monica writes numbers in the diagram in the picture so that each number is the product of the two numbers below it. Which number does she write in the gray cell? So I'm gonna start out with step number one is understand the problem. There's a little bit of vocabulary here. I can find out third and fourth graders know that product means that you do what? You're gonna have to multiply, right? Okay, so we're gonna multiply. We have to do a product of the two numbers below it. So what does below it mean? Well, the two numbers below this two are gonna be this one and this missing one. Okay, so do you have a plan? Now that we understand the problem, who has a plan? Is your plan to try to fill in the missing boxes? And remember, what is the question I told you? I gave you a big hint about that, that usually it's the first sentence or the last sentence. So we need to know what does she write in the gray cell? Make sure you're looking at the gray cell. No other cells, just the gray one is the answer that I need. So Alan is helping some of you in the chat, which is fantastic. What do I need to multiply by one? So one times something is going to give me a two. It's one times two gives me two. The computer is very laggy today. I have to yell at some children upstairs. And over here, I also need to figure out one times two is another two. So what goes in the next box? Remember, this one is going to be this product of the two below it. So the two below it, two times two is going to be four. And then we need to make the product of the next levels to get this one and also this one. It's going to be the same, isn't it? Do I get the same answer on both of those? I sure do, because I get two times four is eight. And I get two times four is eight. So you should have eight for your answer. Okay, if you didn't get it correct, don't worry about it. Fill it in now. The next one is a slightly different type of pattern. It's what we call a visual pattern. Combines a little bit of a mathematical pattern, but mostly a visual pattern. You should have this on the handout. If not, it's right here. Again, this is multiple choice. So you can put your answers into the chat or in a few minutes, we'll launch the poll. Let's see if we can understand the problem. In a certain picture, you can see that the numbers one, two, three, four, and their mirror reflections. So remember, we have to understand what is mirror reflection. So it's going to say, what is the next picture in the sequence of reflections? So here's the main question. What is the next picture? Not what's the next number, but we do need to know the next number. Remember our plan? I think we have to figure out two things for this plan. We have to figure out how do they make the reflection? Or maybe it's three. How do you make the reflection? What's the next number? And what does it look like next to its reflection? So if I take a look at the number four, this is the regular number four, right? Hang on just a minute. Okay. Brian, whoever's on the internet, get off. You're driving me like crazy. Come on. I'm very sorry. I have a busy house. And if everyone's on the computers at the same time, it slows us down. So how do we do the reflection of the four? Look, they did the reflection in the left direction, right? So there's different ways you could do a reflection. I could do a reflection down here from the bottom, but that's not what we did, right? I could do a reflection on the right-hand side, in which case it would look just like this, like a cactus maybe. But these are reflections on the left-hand side. So the next number we know has to be five. And what does the reflection look like if we make it on the left-hand side? Alan, can we go ahead and, I think I can do it. I can launch the bowl. So make a five on the left of the original five, right? So kind of a backwards five. My picture isn't perfect, but I think, yeah, the majority of you have gotten it. Very good. We don't have to run the poll very long because we're giving you some good time to solve it before we pop up the poll, right? Very good. Okay, our next question is a different type of pattern. Camilla wrote all the positive integers from one to 100 in order in a chart with five columns. Everyone know what a column is? A column goes up and down this way. So these are five columns, right? A part of the chart is shown in the picture. So this is only part. Her brother cut out a part of the chart and then he erased some of the numbers from it. Which picture represents the part of the incomplete chart cut out by Camilla's brother? So this is only the beginning of the chart on the right-hand side of the slide, right? This is gonna go on all the way till we get to 100 at the bottom. So we can cut out a piece of it and one of these is the piece that is missing. And how will we determine? So we have to make our plan. How will we determine which of these five pieces would really belong in this chart? What is our pattern? I think our pattern is A, obviously, our numbers are increasing as we go down the chart. So make sure your answers are increasing as you go down. You'll also notice that in each column, like in this column, for example, we have all the multiples of five, don't we? So we have to have multiples of five in the last right-hand column. We have five plus one in the left-hand one, six, 11, you'll notice you add five each time. Can only have two terminal digits. A terminal digit is the digit in the ones place. I can only have two or a seven in the ones place. So see which of these pieces match. Go ahead and start putting your answers in the chat and see how you're doing. And I'll launch the poll in a minute. I wanna give everyone a chance to work through it. A few students are giving me the answer B. B does follow part of the pattern, which is that you're seeing the two in the second column and you're seeing a multiple of five in the last right-hand column. But take a look at the order of the numbers. Remember, they have to be smaller on the top and increasing as you go down. And in B, you'll notice that 60 is larger than 50. So 60 should actually be lower in the chart, not higher in the chart. So we'll launch our poll. So we decided B doesn't make sense, right? Check our answer. B doesn't make sense because the 52 should have been up higher in the chart, right? 52 should be up high. So I think A doesn't work because I have a three in the second column. And I know that the second column should only have twos or sevens, right? Should only be twos or sevens here. This one doesn't work because the ones are in the left-hand column. You've done very well. So the problem with E is that the sevens have to be here. So our 87 would have been over here, 87, 88, 89. 89, this would be 90. And then 94 is in the right place, but this 87 should have been over there. So our correct answer in this case is C. Okay, that's the end of the polls because the rest of them are not multiple choice. So for all the rest of the questions, please put your answers into the chat. That will allow Alan and I to help you, okay? I like to explain the question until I see that the majority of you are getting the correct answer. That's how I like to teach, all right? So if I'm not seeing you getting all the way to the answer, I'll give you another step and another step until just about everyone can solve it. Number six, Emily wants to build a crown using 10 copies of the token shown in picture one. She is making, there's a typo here. She's making the pattern shown in picture two. When two tokens share a side, the corresponding numbers match. Four tokens have already been placed. Which number goes in the triangle marked with X? So again, remember I said we have to, usually the first sentence or the last sentence tells us which answer we need. So we need the number where the X is. That's what we need. So how will we do it? Some of you have played dominoes before. When we play dominoes, we usually have a tile. Then we have a second tile that goes next to it. Right, and on each tile, there are two numbers. Usually they're dots. I'm gonna write numbers, it's a little easier than dots. So let's say I have four and I have three. The next one that has to match is gonna have to be three and some other number, right? So this is sort of the idea like dominoes. So you'll notice wherever I have a one, I have to have a one on the other side where it touches. These aren't gonna change. These are just determined by this. So I'm gonna put these pieces all around until I get to the X. So I'm gonna start by filling in the first. Does it matter if I start on this side or if I start on the right side, the left or the right? In this case, I don't think it doesn't matter which side I start on, but I do have to put a three next to this three. I'm gonna write just right side up. And if the three is here, I'm gonna rotate this. This will be four and this will be five, right? This would be one, this would be two. I'm gonna change my color a little bit. So on the next one, I have to put a five here. If I have a five here, oh, let's see how I have to rotate this piece around. If I have the five, then on this side, it's gonna be the one and on this side is the two. I don't need to worry about the other one so much, right? And then I have another piece of the same puzzle. I'm gonna put the two in over here. If the two is over here and the three is here and the four comes out here. Is there another way to look at the pattern? There is, if you'll notice each time, if you look at just the numbers that are touching, I go one, three, five, two, four. So this would be one, three, five, two, four, one, three, five, two, four. So you'll notice that's a repeating pattern that goes around too. If I just look at what is the number touching? So that's another way to look at it. Another way to look at it is I'm gonna move around this piece, this token two places each time. So if I started with a one, I'm gonna go two places in this direction. The next one that will touch will be a three. The next one that will touch will be a five. Next one that will touch will be a two. Next one that will touch will be a four. So there's another way of looking at it. The reason I'm giving you multiple ways is because your own method may have been slightly different than what I did. And that doesn't mean it was an incorrect method. I just want you to know there's more than one approach and don't ever feel that you did it the wrong way. If your way makes sense and you were able to check yourself and your answer makes sense, then go for it. All right. This is a die like you might play a game, right? When you roll the die, a standard die has seven as the sum of the dots on opposite faces. The die is put on the first square as shown and then rolled towards the right. When the die gets to the last square, what's the total number of dots on the three faces marked with the question marks? There's a lot in this question. So let's break it up and try to figure out what it's asking. So we have seven as the sum of the dots on opposite sides. What does that tell us? If there's three over here, what would be on this face over there? If there's two on the front, what's back here in the back? Okay. What does it mean when it says you're going to roll it towards the right? That means roll it toward the right. And then the final part, what is the question asked is what is the dots on the three faces marked with question marks? Do you see the question marks all the way on the right-hand side? So they want to know what dot is, how many dots are on the front, how many dots are on the right, and how many dots would be on the top. You can go ahead and start putting your answers in the chat whenever you get to it. Some of you are doing a good job telling me what some of the dots will be, but remember, always read the question. What is it asking for? Total number of dots on three faces. So make sure you finish that up. There are some people who have questions and that's okay. That's why I'm here. What happens the first time we roll the die? The three ends up on the bottom, the two stays in the front and the one goes on the right-hand side, right? And here's something funny about a die. If we roll it, see how many times are we rolling this die? We're rolling it one time, two times, wrong tool, two times, three times, four times, five times, right? It's getting rolled over five times. And how many faces are around the outside? There's one, two, three, and four. So see if that helps you with any of it. I have a question on my screen, maybe that will help some of you. We're not rolling the dye randomly like a game, right? We're just tipping it on its side, right? So this isn't random like I'm playing Monopoly, I'm just tipping it. Right, so some of you are answering the question, rolling it five times and rolling it once is gonna give me the same thing, right? So when I roll it just one time, I'm gonna get a dye over here, right? It's as good as I can do in Zoom, guys. Well, guess what's still gonna be on the front? Still gonna have those two dots on the front, the two dots aren't gonna change on the front. So the front, the front is gonna have two. Maybe I can do this. Okay, the front is gonna have two. On the right-hand side, this one dot is gonna end up rolling to the side. So I'm gonna have one dot on the right-hand side. And then I need to figure out what will end up being on the top. And this comes from this clue right here. There's a sum of seven. So if there's three on this side, how many would be over here? On this side that we can't see, there are actually gonna be four dots. And when I rotate it over, those four dots are gonna be on the top. So I have two on the front, one on the right, and four on the top for a total of seven. And I don't have to roll it and roll it and roll it because rolling it once is enough because rolling it once and rolling it five times is the same pattern. That's why it's called a pattern. So for those of you who are giving me the numbers, remember that the question asks for the total. So make sure you add two plus four plus one. Total means sum, means add them. Yeah, now we're getting a lot of it. Very good. So remember the pattern we have to remember every time we roll it four times, it's like we got right back to the beginning. That's what our pattern is. Okay. Here's another question. Lonnick builds a fence using one meter long poles. So we know that this is one meter. The second picture shows a four meter long fence. How many poles does Lonnick need to build a 10 meter long fence? So this one's four meters and we need to extend it out to 10 meters. You may be thinking of different types of patterns to do this. The way I solved it might be different than the way you want to solve it. Remember, what is the question asking? Question is asking for how many poles? So we need the number of poles. Some of you, yeah, these look more like numbers of poles. Good. And a pole is just this one piece of wood that they're using or metal. I don't know what the brown stuff is. So if I was gonna count how many are in this fence, I could count one, two, three, four, right? So count each piece. We don't want how many more poles, we want how many poles do you need for the whole 10 meter fence. This one is tricking quite a few of you. I see quite a bit of correct answers, but I also see some people who've been tricked. So I had my way of doing this and your way might be slightly different. But what I noticed is that, let's see which is the best way to do this. This first poll and this one, they start off the fence, right? If I don't have a post to start with, I can't build any fence. So I'm gonna leave those two, no matter how long my fence is, it has to start with those two. And then I'm gonna look at how many pieces did I need to go one meter? So from my first meter, I needed a top rail, a middle rail, and two more pieces to make the second post. So for one meter, I had to add four pieces. For my next meter, how many pieces did I have to add? I had to add another four pieces for the next meter. And for the next meter, how many pieces did I have to add? You see how I'm drawing it out? I had to add four for the next. And now I'm beginning to see a pattern. Okay. See with this pattern, if you can figure it out. I'm doing, I think that's helped. I'm gonna write it out as a little bit of an equation. So for every, when I build the fence, I have to start out with two pieces. And then for every meter that I add, I need four pieces. So I'm gonna need four pieces times however many meters I want. And in this case, I want a 10 meter long fence. So now I have myself a little bit of an equation. And now we have all correct answers. That's fantastic. So yes, this is going to be two plus 40, which is 42 poles. Very good. So there are alternate ways to do this. You could say, oh, well for four meters, I need so much fence. You can break it up and say, I need so many going horizontally, lying sideways. And I need so many lying up and down. That's another way to break up the pattern. So you could say, okay, I need for every meter, I need two that lie down and I need two that stand up. That's okay. Everyone thinks about things a little bit differently. For me, it was easy to see, I have to add four every time I go one more meter. So that was able to get me that equation. All right, if you solved it another way, you can ask Alan about it in the chat. He might be able to help you with another way. Here's another question that we have. Johnny builds a house made out of cards. In the picture, one-story, two-story, and three-story houses are shown. How many cards does Johnny need to build a four-story house? Now there may be multiple ways to solve this question. Let's look at what it asks first. How many cards? So make sure you're getting how many cards. So for example, there are two cards here, right? And here there's one, two, three, four, five. Don't forget this is a card. So this is one, two, three, four, five, six, seven cards. So we need a four-story house. Do you have a plan? I can think of a few types of plans. Let's see if any of you have a plan. Remember to be careful. If you're drawing pictures, that's a plan. Just be careful in your drawing. If you're trying to look for a numerical pattern, that's also a plan. I see a mix of answers, quite a few correct answers. Remember, the answer has to be in cards. Well, let's, let's try a very simple plan, which is let's draw some more cards. Let's turn this three story into a four story because that's a plan, right? So I would need to put one, two, three cards on the bottom, and then I'm going to need to make some triangles, and then we can count. Is that a satisfactory plan? Of course it's a satisfactory plan. I can count one, two, 16, 17, 18, 19, 20, 21, 22, I'm going to get to 26 if I count. Is there another way? We can start to look at a pattern, right? So let's try patterning it. So every time I add a row, I have to add what? I have to add, I have to add a certain number of these little up and downs, and I have to add a certain number of the crosses, right? So on the first one, if it's only one story, I just had two cards for up and down. I have one up down combo, right? Which is two cards. For two story, I have to have three of the up down combos, and I'm going to need one of the cards that goes across. When I had the three story, I wound up with one, two, three, four, five, six of the up downs, and three, so three up down combos, and three that go across. Sorry. So for a four story, what would be the pattern? Here I added, from one story to two story, I added two up downs, here I added three up downs. I'm going to have to add four up downs, that gets me up to 10. Does that make sense with my picture? One, two, three, four, five, six, seven, eight, nine, 10 of the up down combos. And I'm going to need to add, this was two across, and this is going to be three going across. So you can start to see a pattern. Here I added, I'm adding one for the acrosses, but I'm actually adding an increasing number. So let's just look at this pattern, because I think this is fun. How do we get from one, to three, to six, to 10? And what would come next? Here I've added two, right? Here I've added three. Here I've added four. So I would have to add five the next time. So if I made a five-story building, I would have 15 of the up-down combos and I would have four going across. So I would have, this would be 15 times two, right? Plus four, that would be 34, and that's a five-story. So that's a way to extend the pattern. So remember the answer was 26 for a four-story, but if I wanna even take this question a little bit more tricky, how many do I need for five-story, would be 34. If you like drawing it in, great. But if they asked you how many would there be in a seven-story building, you could be drawing for a while. Okay. there's some people who are saying that they don't understand you don't understand the pattern or you don't understand drawing them in. The pattern okay so. i'm going to erase some of my drawing a little bit so that we can work on the pattern, a little bit more okay. I have a little bit of time so we're okay so for this is one story two story three story clearly right one story two story and story. So i'm calling this an up down up down, so this is up. i'm going to call this up down. That just goes up and down that's what i'm going to call it now how many there's one up down in the first story. When I go to the second story I have one up down on the top story, but I have two up downs on the next story, so this gives me three up downs. But it also adds. But it also adds so let's see if I can, so this is one up down this is 123 up downs. Or maybe it's easier to to or write it in the original order right here's our first one, and then I added two more up downs on the bottom. But I also added this across I added an across. So that's going to be plus one for the across. Okay. Every up down is two cards now let's try this one, how many up downs, do I have in the third picture. I have the same one, I have 23 up down and now i've added 456 up downs so here on the first story there's one on the second story there's two on the third story there's three on the fourth story, there would be four up downs right. So, every time I have to add one more than I had before so one I added one to get the two here I added to I have to keep adding up. didn't mean to erase that there's one up down two more up downs three more up downs for more up downs and how many of the crosses, do I add. back to this color here's the original across now I had two more cards going across so here was one here is to I would end up with three here. So, by the time I got to this next row I would have had to add four of the crosses. So you can find a pattern in that way. If it's not clear, yet what I would suggest is that you keep your piece of paper and you work on it a little bit on your own and see if you can find it on my next level. How many up downs, am I going to need one here one here. One here. And one here. And one here right, those are my up downs so that's where I counted. That I need an additional four so I have 789 10 up downs. And i'm going to need to also have more of these across is I need 123 across this. Okay. All right, we have another challenging question. So let's try this one. A meal started to hang up towels using two pegs for each towel, as shown in figure one he realized that he would not have enough pegs and began to hang the towels as shown in figure two. Altogether, he hung up 35 towels and use 58 pegs how many towels did a meal hang up in the way shown in figure one so remember always check that first and last sentence, we want to know the towels in figure one so figure one is. Two pins per towel to the pins or pegs two pegs. What would be the pattern for figure two. it's no longer two pegs per towel is it. there's a different pattern here. Which is, you have one starting pay. And then one for each town. One for each town. Let me give you some time to think about how many towels are hung up in each way. Remember the important facts. Are there he hung up 35 towels used 58 pegs those are going to be really important facts that we need to find in our problem problem. He only hung up 35 towels in all. So make sure that your answer is reasonable that he only hung up 35 towels in all. I understand some of you are confused. I'm going to give you a little bit of time. Don't worry if you're confused. This question is tricking a lot of you. I can assure you that. You notice it's called the challenging problem. No, today we're not doing 24 questions. This is our last question, so we can take as much time as we need to to work out this question. I'm seeing some answers that are greater than 35. Remember, we have to use step four and check our answer. If he only hung up 35 towels, he cannot hang up more than 35. Let's start with some things. If he hung up all the towels, all 35, like in figure two, how many pegs would he use? I mean, in figure one. So if we did everything like figure one, oh, that's a terrible color. My apologies, guys. If you did everything like figure one, and he has 35 towels, it would be 35 times two. He would need 70 pegs. Anyone can get that far, right? Okay. If Emile hung all of the towels up like figure two, he hung 35 towels up like that, he would need the one to start with, and then he would only need 35 more pegs, because let me see if I can add this to the drawing. So basically there's a peg over here that you started with, and then every towel after that, you only have to use one peg, one peg, two pegs all together, three pegs all together, four pegs all together, five pegs all together, right? So there's a starting peg, and then you just needed one peg for each additional towel. So that would give us 36 pegs. But we know that Emile hung up 35 towels and used 58 pegs. So how many, this is the, the figure two is your saving your pegs method, right? So we can call this the saving pegs, right? We're going to saving pegs method. If he hung everything up with the saving pegs method, he would have used 35, 36, pardon me, just misspoke, but we know that in fact he used 58. So let's see, 58 pegs, and the least he could have used was 36. So that gives us 22 extra pegs were used. So, if we know that Emile used 22 extra pegs, then how many towels do we think he hung up each way? The reason this is a pattern problem is because we have to figure out how many pegs per towel, how many pegs per towel. Those are patterns, right? In figure one compared to figure two, how many more pegs do we use per towel? If it's one peg per towel in figure two, One peg per towel in Figure 1, one peg per towel in Figure 2, and he used 22 extra pegs. That would be 22 towels in the two-peg way. Let's double-check it. Remember, what is step four? Step four is always double-check your answer. So if we have 22 towels in Figure 1 way, that's going to equal 44 pegs. And then we have 35 minus 22 towels. That is going to equal 13 towels in the Figure 2 way, which is going to be one for the starting peg plus 13, that's 14 pegs. Does that give me the correct total, 44 pegs in the first way and 14 pegs in the second way? So that gives me a total. So that agrees with the question, right? The question says 58 pegs, 35 towels. Did I account for everything? 58 pegs, 55 towels hung up because I have 22 plus 13, right? 22 plus 13 is 35. But double-check, what is the answer that we need to have? What does it ask? How many are hung up in the Figure 1 way? And where did I put that information? You see what I just circled in red, or rung in red, pink? 22 towels are hung up in the figure one way. And why did I come up with that 22 number? I did this. If we hang it up in the figure one way, in the figure two way, we only use our 36 pegs, 36 pegs. But I know I used 58, so I used 22 extra pegs. The extra pegs are these second pegs right here. Those second pegs are gonna be the extra pegs, right? Those I don't use in figure two. Alan, do you have another way of verbalizing it? Because I still have some students who say they don't understand. Maybe having- The way I would go about doing it is that you're going to have... So looking at the difference between the number of towels and the number of pegs, that's 23. And for each towel hung up like figure one, that's an extra peg. And all the towels in figure two, though, have only one extra peg because there's only one on the side. For example, you would use 10 pegs to hold up nine towels. So one of those 23 extra pegs is going to be used to hang up the towels in figure two, meaning that the 22 other extra pegs are gonna each be used to hang up one towel in figure one, meaning that there will be 22 towels hung up like the figure one. So that's another very elegant way to say it. So what Alan says he did is he took the 58 pegs and he subtracted 35 towels to find out how many extra pegs he has. So subtract. So he did this and he got 23 extra pegs, or 23 more pegs than towels. So how can you use 23 more pegs than towels? We know in figure two, you have one extra peg, right? That's what Alan was saying. So from figure two, there's only one extra peg. So the other 22 pegs are on towels like in figure one. So there is a completely different way to do this. And for some of you, this might be easier. So I am going to stop this share. So the question was 35 towels and 58 pegs. Okay, so Alan has correctly said that we have 58 pegs and we have 35 towels. So that gives us 23 extras and we can work that out. The other way to work it out is to make a table. So we can make a table and we could say, this can be the towels hung like figure one. And on the side, we can have figure two way. And we know that the sum has to be 35 towels, right? So let's say I hang up 10 towels the first way and 20 towels the second way. The 10 towels the first way is gonna be 20 pegs and 20 towels the second way is gonna be 21 pegs. So that's gonna give me 41 pegs. Let's say I hang 20 towels up the first way, that gives me 10 towels the second way. The 10 hung up the first way is gonna be 40 pegs. The 10 the second way is gonna be 11. Okay, this should actually be here. So that's gonna give me 51 pegs. So I'm not quite there yet, right? So I'm not quite there. What if I did 20? I need to get up to 58. What if I did 23 and that would give me, 20. Sorry, this should have been 15. That gives me 56 pegs. I'm almost at my 58. So let's just try it. What happens if we do 21 and 14? I did 21, 21 times two is 42. 14 is gonna give me 15. So that gives me 57 pegs. So if I just do one more, 22 and 13, that's my 35 towels. 22 times two is 44 and 13 plus one is 14. So that's gonna give me the 58. So you could do some guess and check. Would be another way to find that answer. Does that make sense? So we have multiple ways we can find answers. Let me wrap up the class. I'm going to change back to my slide share so that we can. So we worked on the four-step problem solving method. That requires that you understand what the problem asks. So all those times I was saying, how many poles? How many towels are this way? Make sure you answer the correct question. Plan, there's more than one plan. You can make a table. You can draw pictures. You can look for patterns. There's a lot of plans on how to solve problems. Carry it out carefully. Look back and check your answer. One way to check your answer is to use another plan. If you get the same answer by two methods, you're doing pretty well. So we look at patterns, patterns in pictures, patterns in number sequences, patterns in all different ways. So be careful. Math Kangaroo has lots of patterns. There's the pattern with the die and rolling it, right? How often does your pattern repeat? I want to thank you for participating in the Math Kangaroo webinar. This one was a particularly difficult one. The pattern lesson is a difficult lesson. So if it was troubling, just practice. You can use sample questions from the Math Kangaroo website, practice using pattern questions. They are tricky. And please consider signing up for more Math Kangaroo webinars. Alan can put the link in the chat. There's some more Math Kangaroo webinars. I want to make sure I thank Alan for all his help today. And there's supposed to be, when I close the Math Kangaroo webinar, there is going to be a survey that pops up, a feedback survey to let us know how we did. I know this was a tough lesson, but give us feedback on whether we were able to explain it, whether we showed you multiple ways, whether we gave you enough time. Let us know if you like the types of questions we used, if you want more questions, less questions, okay? I hope that you were able to learn something. That's the important part. If you were able to solve questions, even after you got hints, then you were learning today. Thank you for attending the webinar. And I hope I might see you again in another webinar. Thank you, Alan. Goodbye, everybody.

Math Kangaroo

patterns

problem-solving

webinar

Dr. Sarah Sagee

visual sequences

numeric patterns

logic puzzles

geometric patterns

arithmetic patterns