So again, this is the fifth in our series of 10 webinars. Today we're using finding a pattern to simplify a problem. If you can find the pattern, sometimes you can continue it to get an answer. Here's a warmup problem you can work on. You can put the answers into the chat or we will have a poll in just a minute. Now, remember, we've also had lessons on drawing. This might be a problem where you could try to draw in the missing piece and then match it up to one of the five choices. If you can't move the poll around and it's in your way, feel free to close it up. For most students, if you click on the top of the poll, you should be able to drag it around your screen and get it out of the way. Okay, I'll end the poll here. Everyone who answered the poll has said they think the answer is E. Let's take a look at the problem. We'll be able to figure that out. So, one of the ways that I like to do this problem is to just draw in the rest. So, if I look at the problem, I can see that there's different shapes around the different corners of the tiles. So, I see a nice rounded kind of a bit of an ovally shape here and then I see a more pointed shape off of this corner. Again, this is that same kind of pointed shape. Then finally, this is the reverse. This is like a little cave where it goes in. So, I need to find a shape that has the round almost like a flower petal, two points and then a cave and that does match. Picture E, which is what people were telling me in the chat in the poll. So, good job. All right. Now, remember today's lesson is about finding patterns. We've done a bunch of different types of, a bunch of different types of problems already. We've done a lesson that was introductory to teach us about all the different types and then we had drawing. Don't forget, we can still use drawing. I just used drawing. We were doing lists and tables for the past two weeks to look at different options, splitting problems up into different alternatives sometimes and today we're going to be using the patterns to try to simplify a pattern and that way when you try to figure out, oh, what happens in between each step and then, okay, now I can jump ahead 10 steps because I know the pattern. So, we'll be doing that today. Okay. So, extending a pattern is a frequently used strategy. If there's a pattern in data, then you can just apply that and continue to solve the problem. So, we had a problem last week where we were doubling the cups of berries that we were giving to mom every day. Doubling is a type of pattern, right? So, this strategy can be used and make a problem more simple. So, if I had asked you how many berries do we have on the fifth day, you could use multiplication to find the fifth day following the pattern. So, we're going to find patterns today and we'll see if sometimes a problem is very complicated if we make it simpler, relying on the patterns that we see, if that will help us solve it. So, don't forget, sometimes we can make a problem simpler if it looks very complicated by trying to find a pattern. This is our first problem. Well, we had the warm-up problem. This is our first numbered problem today. It comes from 2012. Father hangs the laundry outside on a clothesline. He wants to use as few pins as possible. For three towels, he needs four pins as shown. How many pins does he need for nine towels? Give you some time to think about it. You can put your answers in the chat or we'll have a poll in a couple moments. Liv, did you launch the poll? Because I'm having troubles with it. Yeah, no, the other poll was still, I think the other poll was still running. It might have been. Yeah, on mine it says ended and closed, so I don't know. Yeah, that was the warm-up poll, I believe. Do you want me to? Right. Here. Thank you. I think I just launched the main poll. Thank you. For mine, it didn't want to launch. All right, nice work. Most of you have answered that poll. We can share the results. It looks like most of you think that it's 10, option B. Let's take a look. There are a few other answers. I see some students think maybe 12. Let's take a look. So let's see if we can find a pattern here. If we add another towel onto the clothesline, you'll notice that we're only adding one pin on the top. I'm going to put them in red so they're easy to see because we're putting these pins so that each of these pins is actually holding two towels except for the pins at the ends. Those only hold one towel. So every time I add a towel, it doesn't matter what color towel or how large the towel, you'll notice I only have to add one pin. All right, so with each towel I'm putting in one pin. So I have one starter pin and then one pin is added with each towel. So if I want to hang ten towels for nine towels, I'll have my one starter pin plus one pin times those nine towels and that's going to equal one plus nine or ten pins. So all the students who said ten are correct. So if you aren't sure, try drawing out a couple more. It'll make a pattern and then you'll be able to figure it out. Question number two. Sophia is drawing kangaroos. The first one is blue, the next one is green, the one after it red, the fourth one yellow, and then again blue, green, red, yellow, and so on. In the same order. It's important. She's repeating, right? Same order. That's a pattern. What color will the 17th kangaroo be? The answers in the chat are coming out really good. I think that's most of you for the poll. So half of you, more than half of you, think that the correct answer is blue. Then there's a little bit of green or red as options. So let's see what we can figure out. Remember, we can do this in a few ways. One is we can draw it out. You could draw out, I'm not going to draw kangaroos because it takes too long to draw kangaroos, but you could use a letter for each color and you could on this one actually probably make it out to 17. If you started with a B, a G, an R, and a Y for blue, green, red, yellow, you could repeat that blue, green, red, yellow until you got to the 17th letter. That would work fine. However, if I asked you what would be the 117th kangaroos color, you might not want to be doing that, right? So let's see if we can find a pattern. The pattern is that this repeats every four kangaroos. All right, so as long as I'm dealing with a multiple of four, I know that the pattern repeated evenly. So 17 is not a multiple of four, but 16 is. Four times four equals 16. So I know that the 16th kangaroo will be a yellow kangaroo, and then the 17th kangaroo, I'll be starting the pattern all over again, so it will be blue. And if you went ahead and said 17 divided by four is four remainder one, that is also a correct way to do this. Some students are ready to do that and look at the remainders, others not quite. So we can do it in multiple ways. It's okay. All right, we are going to have Liev lead the next question. I'll just clear these drawings so he has a clean screen. Yep. All right, so here's the problem. Monica writes numbers in the diagram in the picture so that each number is the product of the two numbers below it. What number does she write in the gray cell? So the picture here is like a pyramid of cells, I guess, where every cell is the product of the cells below it. So here's an example. So you take a cell here. Poor drawing, but cell here, cell here, cell here. Say the number here is one. The number here is four. The number at the top would be one times four, which is four. So following the same logic, if you have two and the number below it is one, then the other number would be something times one is equal to two, which would be two. So knowing that, guys, take a crack at it on your own and try and figure out which number would be in this cell. All right, so we're getting a bunch of good answers in the chat, so I'm going to go ahead and launch the poll for question three. Here it is. Once again, if the poll is in your way, just pause it. But yeah. All right. So if anyone else wants to put their answer in the poll, go ahead right now, but almost everyone has participated. Give a few more moments. All right. So I'm going to go ahead and close the poll, share the results. So as you can see, a lot of people put eight or E as the answer. So let's go through and try and solve this together. So as we saw before, this number two times one is two in that bottom right part of the pyramid, which means in the bottom left part, you follow the same logic where one times what makes two, here you would get two again. So now then you can take these two and get the product here in the middle, two times two is equal to four. And then you could go out and fill out the entire pyramid, but that's not what the problem is asking for. All the problem wants is what number is in the gray cell. And now that we have the two cells below it, you just need four times two, which is eight. And that's the number in the gray cell, which have everyone. Thank you. Good work, everybody. There were a lot of great answers. I saw in the chat and the poll, you also notice that this pyramid, although it's a pretty small one is symmetric, right? It's kind of the left and the right are the same. So that's another way you could look at the pattern here. Okay, we'll move on to the next problem. We're doing great today. We're getting through a lot of problems. All right. No one ever knows how to pronounce this name, including me. So don't worry about it. If you get to a contest and there's a name or an unfamiliar word, you can't exactly pronounce. Math Kangaroo isn't asking about your pronunciation. It's asking about your math problem solving skills. So don't worry. I'm going to say Lanaki. Lanaki builds a fence using one meter long poles. The picture shows a four meter long fence. How many poles does Lanaki need to build a 10 meter long fence? I'll give you a couple minutes to do this one. Students will have different methods. It might take a few minutes. I'm seeing fantastic work in the chat. I'd like to take a moment to be able to reply to some of you. If I don't reply on every question, it's because I can't quite keep up. Don't take it personally. I'm doing my best. All right, we'll share the results. The vast majority of the students think the answer is D, 42 poles. And that is a correct answer. So let's see what we can do to work that out together. I'm seeing a few different methods that students are using in the chat. And that is perfectly normal. It's fine. I always tell the students, you are you. Your brain has learned how to do things from your teachers and in your own way. Even the way your teacher in school does things may not be your preferred method. And here at Math Kangaroo, you can use whichever method you like. No one is going to tell you on a contest that that was the wrong way. Your way is the correct way on our contests. So I will look at it a few different ways just so you can see some options. So the first option for me is I'll do it the same way that I was doing the towels on the clothesline with the pins. I'm gonna look at how many there are to start off. I see that there's one, two poles over here on the left-hand side when I begin. Then I'm gonna see every extra meter that I do, I had to add one, two, three, four poles. So you can see as I add each other meter, one, two, three, four poles. Every time I do that, there's four poles being added, right? I'm changing colors so you can see for each meter, okay? So if I use that as my pattern, that I have two at the start, and then I add four poles for each meter. Then I'm gonna have the two plus I wanna do ten meters, no, four meters. Sorry, four meter long fence. No, pardon me, it's showing four meters, I want to get to ten meters. So I was correct the first time. I want a ten meter long fence and I need four poles every time I add a meter and I'm gonna find that. So using order of operations, I'm gonna do the multiplication, the parentheses first. So I have two plus 40 equals 42 poles, and that's what most of you said. I did have some other students looking at the pattern in different ways, okay? I wanna go through just a little bit of that. Some of you looked at how many horizontal poles are there. So you were looking at, there are two horizontal poles for each meter. So you might have said there are two horizontal for each meter. And then you might have said there are two vertical for each meter. And then some of you noticed that there might have been two to start. So you would have gotten, if you said, okay, I'm gonna do ten meters. So I'm gonna do two times ten for ten meters. This is also times ten for ten meters. And then the start, I don't have to do anything. And then you would add that up and you would also get 42 poles. So it's important to know that we can use more than one method and still come out with the same answer. The pattern that you might see could be different than someone else's. And that's all right, don't worry about it. Keep true to yourself, right? Let's move on to our next problem, see what this one is all about. Camilla wrote all the positive integers from one to 100 in order, in a chart with five columns. A part of the chart is shown in the picture on the right. So this is the original chart. Her brother cut out a part of the chart and then erased some of the numbers from it. Which picture represents the part of the incomplete chart cut out by Camilla's brother? I'm going to go ahead and launch the poll. I do appreciate people who are chatting with me and Leah. I hope he's getting back to you as well. He's telling me that you're doing a great job. Okay, I think we can end the poll. We have most of our responses. This is one where we have a couple of students giving us alternate answers. Most of you think it is C. We'll take a look at a minute, but let's take a close look at choice B and choice E to see what those choices could mean, why we could use them or not use them. Okay, so we had a few students say choice B or choice E. Let's take a look. What you'll notice is the pattern, you always have the same numbers in the same columns or at least the same ones digits in the same columns, right? So 5, 10, 15, 20, they're alternating, but we have the same ones digits in each column. Remember columns are up and down, vertical. So if we take a look at B, what we do notice is we follow that pattern. So we do have either twos or sevens in this second column, and that is correct here. And we do have fives or zeros in this fifth column. But did you notice that 60 is not before 52. If we were doing consecutive numbers from top to bottom, 52 would have to be above and this one would be 62. So that's how they tried to trick you with letter B. And then in letter E, the 94 is in a good spot. But if I fill in backwards 93, 92, 91, 90, that's all looking good. But then I'm going to go from 89 to 87. This 87 should be over here and this should be 88. So that's why choice E is not correct. So choice C is correct. And what you'll notice is if you fill this chart in, I can fill it in with 70, 71. The whole rest of it when I fill it in works just right. So you can look at the patterns. If you couldn't find, figure out what the patterns are, you could try filling in the rest of the spaces. It's another option. So again, yes, we have alternating in those verticals, we have it increasing in order, but you could fill in the numbers and make sure it works. Okay, number six. Amelie wants to build a crown using 10 copies of the token shown in picture one. So these are her tokens. Okay. 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? When you see the matches, you can see like four, it goes with four. That's what they mean by when they touch, they have to be corresponding in match. Go ahead and see what you think the answer is for the space with the X. Clarify, you cannot change the token. You can only rotate the token. You cannot change the order of the numbers on the token itself. Just rotate it and place it down next to other tokens. Okay, that was some fast poll work. Almost all of you think the answer is four. There's a little bit, one student each for one and three. We'll see if we can decide why the answer is four. So, there are a few ways to do this problem. You notice that X is kind of exactly opposite of where we already have our numbers, right? So, it really doesn't matter if I want to fill in from this direction or if I want to fill in from this direction. It'll take about the same amount of time. So, some of you may have gone from the left, some from the right. Don't worry about it. We should get to the same place no matter which way we go around. Okay, so I'm going to go and I'm going to do the draw in method. I'm going to put in the numbers. That's one way to do this problem. And then we'll talk about if there's another way. So, I'm going to draw all my numbers right side up. It's easier for me, but I'm going to do them in the same order as the token. So, I know that I have to put the three next to the three. And then if I'm going around the token, I'm going to go around my token this way. So, I'm going to have two, one, five, four. Okay, now I have to put a five next to this piece. And again, I'm going to go around the token the same way. Five, four, three, two, one. And I'm going to fill in the last piece that I need to do before I can get to the X. I get the two, one, five, four, and then three. So, the correct answer here, it's a little hard to see, is four. And I would get the same thing if I had filled it in in the same manner, going from the other direction. Is there another way to look at this? I'm going to tell you right now, yes, there is. Because otherwise, I wouldn't try to introduce another way. We can look and see if there's a pattern for what the touching surfaces are. I'm going to call this the touching surface. So, you'll see it's five, two, four, one, three. Five, two, four, one, three. So, again, you can see that that was five, two, four, one, three. So, you might notice that that followed a pattern as well. In which case, this one would get you four on that surface. So, there's just another way to look at it. Hopefully, you did one of those methods. And again, if you went around from this side and filled in over here, that would be perfectly okay. I would, if the X was closer to one side than to another side, obviously, do the quicker side. Because this is a timed contest and you don't want to waste your time, right? And if it's a timed contest, go ahead and do one method. Keep going to the next problems. And then, if you have extra time before time is called, you can go and check yourself by using an additional method. If you use two methods to solve a problem and get the same answer, both ways, you can have higher confidence that you've gotten it correct, right? That's the way it is with mathematics. All right. I'm going to give Liev a chance to do another question. We like to hear what he has to say. He's taken these contests much more recently than I have. So, he's a really great person for you to be asking questions in the chat. Ask him about his testing strategies and ask him how he looks at problems because he does really well on these contests. All right, Liev. Yeah. So, this problem is a lot more of a visual one. But in a certain picture, you can see the numbers 1, 2, 3, and 4 drawn the way you see them with their mirror reflections being those symbols there. So, like an M, a heart, an 8, and whatever you want to think that one is. But, yeah. So, they all reflect across. Oops. Do not do that. So, they all reflect across this one axis of symmetry. It's poorly drawn, but that's roughly how they reflect. And here, this one's easier. They all reflect like that. And now, the question is, what's the next picture out of all of these? So, if you take all these reflections, what's the next one? So, what's the reflection for the number 5? So, with that information, I'm going to let you guys solve it. Feel free to chat me or Coach Ziggy what you think the answer is. And, yeah. All right, well, we've gotten a few really good answers in the chat, so I'm going to go ahead and launch the poll. And do remember that this is symmetry over just putting the same thing twice. So you have to reflect it. You want to get the answer. So here we go. Poll should be launched. All right, let me give a little bit more time for anyone else who wants to answer the poll. All right, so let me end it, share the results. So as you can see, the majority, a little over half of people answered C, though there were answers for A, B, and E as well. So I guess while a lot of you did get C, not the clearest, so let's go through this and try and I guess see how it reflects. So as you can see in all the examples, let's use the two as an example, that's a good one. So you draw the two like this. So this is the number you see, but it doesn't draw. So first thing it does is it reflects it, it mirrors it, but it doesn't draw it like this. No, it draws it on the left side to where the left side is like this. So on the right side is the original thing, and on the left it is the mirror. So now if we want to go for the number five, let's do it below. You draw number five like this, and then the mirror image on the left looks something like this. The mirror is not the best, but it looks something like that, and that would be answer C. So for everyone who got C, good job. And I understand it's a visual problem, very different, many different people answered this. So yeah, good job, everyone. That was something Lev and I were noticing is we got some different students chatting with us, and we'd like all of you to give us a try with the chat. We'd like to hear from all of you, and if you don't understand something, especially Lev is here to help. Sometimes I'm in the middle of presenting, but whoever is not presenting, we'd love to help you get to the right answer and learn a little bit more. This was a different type of question, right? And in my chats with some of my TAs this week, it came up that this was a problem that tricked some of the TAs when they were younger as well, because it's a different type of thinking. And then I said, I got tricked by a problem when I was younger that was a pattern, and I'll put it up right now on this one. The pattern was letters. So the question is, what's the next letter? And you think, huh, okay, I have two Ts, two Fs, maybe I have two Ss, and in that case, you'd be correct. But then the next question is, what is the next letter? If anyone knows the next letter and wants to put that in the chat, at the end of the lesson today, I think I'll have time to show you what this pattern is. It's not very mathematical, well, it's kind of mathematical. All right, we'll move to our next problem. Should be able to get to the next problem, let's see. There we go. So we have a bonus problem, but I just want you to know that was the main part of the lesson. We're going to do the bonus problem when we have time. Remember to keep practicing math kangaroo contests. You'll see some of these questions if you do practice contests, because these questions come exactly off of old exams. The best way to study for a new exam is to look at the old exams. We're going to use similar types of ideas each year, right? This is our last problem. I do want to remind you that next week is the time change. We go back to standard time. So if you don't change your clock and you come and you try to join the meeting, you'll notice I won't be here because next week is our fall back. We get an extra hour of sleep. So keep that in mind. Change the clocks next weekend. Johnny builds a house made out of cards. In the picture, such one-story, two-story and three-story houses are shown. How many cards does Johnny need to build a four-story house? Keep in mind that the question is asking how many total cards does Johnny need so not how many more cards But how many total? I'll launch the poll. If you're not ready for the poll and you don't want to do it, just close it up. I haven't gotten too many answers, but trying to get finished on time today. Okay, we've got a little of every answer, just kind of fun. I think this is the first question today where we've had every answer, right, Laev? Yeah, I think so too, it's really evenly spread out. Yeah, so I think that we need to make sure we go over this question and work it out together. Now, we've done a couple of strategies. The first strategy we worked on a few weeks ago was draw. So we can just draw this. We only want to go one more step. This is already three-story, we only need to go to four-story. So you could literally just draw it in, right? Not a big deal. As long as you're drawing the correct way, you can do that. And then you count one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, and you'll get the correct answer, right? Not exactly the strategy we're trying to learn today with patterns, but sometimes brute force is a really great method, right? Drawing is not wrong. It's not wrong to use a method that you like and makes you comfortable. But let's look for the pattern. I think the pattern here is to look for stories, right? So when I have one up and down on the first story, so it's just one times two. On the second story, I have two up and downs. So that's two times two. On the third, I have three up and downs. So that's three times two. And then I have four on the fourth story, right? Now, if I look in between, I have one for there. I have two and I have three. And I don't actually have little floors on the fourth story because you see there's no floors here. So I have to follow that pattern. Now, if I add these up, this is gonna be two and one and four and two and six and three and eight, right? And if I add all of those up, again, you know what? I like to add by pairing up to tens. You probably have learned that, right? So 10, 20, 23, 24, 25, 26. So you can look for the pattern of the ups and downs. It increases. There's one more up and down on each level. There's one more card in between each level. That was patterns. So either way, I am gonna get the 26. I don't know if I have a blank slide on the, I do have a blank slide on the end of this presentation. Excellent. So remember, I had this. And I told you that S was next. And then I asked, what is the next letter? All right, this isn't one that normally people get. I bet you know which number is next now, which letter is next now. So then of course we get E for eight. And it doesn't repeat E again, does it? It goes to N for nine. So just something silly that I added to the end. Remember the time change next week. I hope you liked our lesson on patterns. And we'll see you again next Sunday. Thanks, Lynn, for helping out today and helping all the students with the chats. Remember, ask us questions during the lessons. We don't mind. See you later. Bye!

patterns

problem-solving

interactive exercises

visualization

symmetrical patterns

sequence prediction

real-world applications

audience participation

mathematical principles