we will begin our lesson today. The topic is patterns. So last week we gave you an overview of different types of math kangaroo problems and we are now going to start building a toolkit that you can use to solve those problems. We're going to focus more on recognizing commonalities and figuring out how to solve similar types of problems and then as we do that each week you'll see that we'll cover the various types of problems you might encounter on our contests. So just to recap, if you were not here last time, my name is Dr. Sarah Segee. I've been teaching math kangaroo for many years. I am in San Diego, California and we have a teaching assistant, Jacob, and he will introduce himself when he leads a problem. If you need to communicate with one of us, if you have more questions, you want to share your answers with us, you can do that in the chat. So you'll see Jacob and you'll see myself as hosts and you can send us messages. There are polls for some of our problems today. I don't poll every problem because the polls do take a little bit of time just to launch and give you time to fill them in and share the results and I want to make sure that we cover all of the examples we planned on using. So let me go ahead and open up the first problem today. The presentation is in a slightly different format today so please bear with me if I don't press the right button. Draw the piece that completes the pattern. There is no poll so you can just put your answer in the chat, either send it to Jacob or I so we know that you can do these problems or if you need any help. Yeah, I'm getting a lot of correct answers. I think the easiest thing to do on this one, the lesson I just taught the younger students today was about drawing and I think that's the easiest approach here is you just, you know, start filling in your little piece. So I'm going to have a quarter of a circle here. I'm going to have a pointy piece over here. I'm going to have another pointy bit of a star here. And then this is more like a little piece that kind of bends in. And the piece that matches that is indeed E. So not a very difficult one, but hopefully you like that problem. Just to warm up, this is a number three problem on our Math Kangaroo Contest, so this should be a quick three-point problem for you. Okay. Sorry, it takes just an extra click for me to get through on this thing, but you'll see why in a minute. So this week we're learning about patterns. There's different types of patterns that you'll see in math kangaroo problems. We'll talk about that in a minute. Next week we'll talk about algebraic thinking. Then we'll move on to some different things that you've probably worked with before in ratios and proportions, multiples, factors, and divisibility. We'll have a good review on time, clocks, and calendars. We're going to teach you some shortcuts and some fast ways to approach those problems. Then we have two geometry lessons before a final lesson that we call hands-on, which is kind of an experimental lesson. We'll actually maybe need some tools to work on those. Okay. Thank you, Jacob, for joining us. So what is a pattern? A pattern is anything where you can repeat something or you can find some way to predict what will come next. All right. So that's the important part of a pattern, is a pattern gives you a way to make a prediction. So sometimes we have a sequence, right? So two, four, six, eight. That's the sequence of odd numbers. It's the sequence of multiples of two, things that you should be very comfortable with, right? So you know what number will come next. You can also figure out what will be the 20th number in that sequence. It's a nice, straightforward arithmetic sequence. So can we find patterns in other types of events or objects? Is there a pattern to the days of the week? Of course. Is there a pattern to months of the year? Of course. Is there a pattern to the curls in my hair? Probably not. It's quite random. I don't think you can predict which way my hair is going to go. So I think some things are predictable. Some things are not. That's okay. So why care about patterns? Well, patterns are everywhere. You'll see patterns in nature. You can see some of these pictures that we have here. This is the top of a pine cone. You'll notice a spiral pattern in the pine cone and also in the succulent plant. You might see some other types of patterns. You can see this is expanding. This is an interesting type of pattern. We have some really interesting patterns that develop naturally in nature. The Fibonacci sequence. Perhaps some of you have found the Fibonacci, which is one plus one equals two. One plus two equals three. Two plus three equals five, and so on. You might find that one. There's fractal sequences. There's spirals, I just showed you those. And then there's the Voronoi, the Voronoi. So we will take a look at some examples of these in video. Which is why I'm in a different presentation mode so that I can actually show you some of these videos. So you can see that there's little seed crystals being formed here, and then they're expanding. So this would be a predictable fractal pattern. Here is another pattern. Here is a spiral. You'll notice spirals occur in hurricanes. Sorry if you are in Florida or in the Carolinas, and that's a touchy subject. But why in the world does a storm start to cycle itself into a spiral? It has to do with the pressure gradients and the rising air. But a spiral is a way to grow stronger and stronger. As you come to the center, you actually become more and more reinforced. So you're getting a stronger structure by making a spiral. This is the Voronoi. It's where you have a seed point. So, oh, it's trying to freeze it. You get cells, and every point in a cell is closer to the seed of that cell than it is every other cell's seed point. And if any of you can figure out, can put in the chat places in nature where you think you might see this type of a pattern, I can't give you bonus points on the contest, but you can get our congratulations. I'm gonna see if anyone provides, volunteers to provide an answer to where they think they might see this pattern in nature. Okay, so on a Math Kangaroo contest, are we gonna ask you to make a spiral? Probably not. But we will be working with number patterns, color patterns, and image patterns. I got an answer about a leaf. The cells in a leaf, if you look closely, that is a good example of the Voronoi pattern. It is that way because the leaf needs to get nutrients to all the cells. So they're kind of maximizing how they pack the cells into the space so that you're closest to the center where the nutrients are coming into your own cell. Very good. There's some other examples as well. So we have colors, images, numbers. So pattern can help you simplify a complex problem. It can reduce the number of calculations you have to make if you understand that, oh, something is gonna be repeated, then I can kind of save my time. And sometimes you might have to step back and solve a little bit of a simpler problem in order to find the pattern. So don't be surprised if that happens. Here is problem number one. And I do have a poll that I'll launch with this problem. Zosia is drawing flowers of different colors. The first flower is blue, the second, white, the next one, red, the next one, yellow, and again, blue, white, red, yellow, and so on in the same order. So we know it's a pattern because it says in the same order, what's the color of the 29th flower drawn by Zosia? And I'll launch the poll because I think you'll get this one pretty quickly. Remember, you want the 29th flower. Don't be afraid to answer the poll. It's completely anonymous, so nobody will know what answer you are using. Remember math kangaroo problems are multiple choice on the contest and there's no subtractions, no points taken away for incorrect answers, which means it's always better to make a guess than to not try, okay? So here are the results. The vast majority of you say that the answer is blue. Let's see if that's correct. If your poll does not close all by itself, just go ahead and tap close or the X. So if we do it in the pattern, we know that it's blue. I don't have a, white won't show up. Blue, white, red, and yellow. So I have a repeat every four. So if I take 29 and I divide it by four, that gives me 29 times seven is 28 with a remainder of one. So I go through this pattern seven times and then I get one more flower and that last flower must be blue. So the poll and all of you did a great job, you're correct. Number two, the picture to the right shows a pattern of hexagons. We draw a new pattern by connecting all the centers of neighboring hexagons. Which of the figures below do we get? All right, we see a few different answers. Let's read the instructions very, very carefully. Step one is always to understand what the problem is asking. It says draw a new pattern by connecting all the centers of neighboring hexagons. So the center would be this blue dot. These are neighbors, correct? So if I use my straight line tool and I start connecting neighbors, I'm going to start to get a pattern like this. And if I drew very, very carefully, can you see which of the answer options I'm starting to draw in blue? We are going to be drawing C. A is the correct pattern, but it's a little too small. And it says all of the centers of neighboring. So this one doesn't include all of the other little cross lines. This one, you'll notice, is missing the hexagons that would be at the corners, because we're going to have an actual triangle with points. All right. So far, are these good problems? Here's number three. I do have a poll, but I'll give you a few moments before I launch it, because you might want to see the information on your screen. What number is at the top of the pyramid if it is formed according to the pattern shown below? So you can see here's an X, a Y, and a Z, and we have this formula where it tells you how to make a Z if you know X and Y. And if you know any two of the three numbers, you should be able to fill in the third. You might try filling in one box at a time until you get to the top of the pyramid. That's my hint. Here's the poll. Let's see if everyone is agreeing or if we have a variety of answers. It's always interesting. Helps the teacher figure out what we need to do. No, thank you. Jacob, it looks like we have a lot of correct answers. So we can share these results. Look, it's unanimous. 100 percent of you have said the answer is seven. I guess that means I probably don't even need to explain it. So we had know that we have a five, we have a six, so we need to have, since this is the top number, the z, is the sum divided by two, the sum would be 12. So five plus seven equals 12, and that gives us a six. Now we take seven plus nine, that equals 16. We divide that by two and we get an eight. We get six plus eight, or we can use this as the average. So we're going to have 7b, right? So if you take the sum over the number of items, that's also called the average or the mean. So that was a straightforward question, it seems. All right, problem four. Again, a problem where I do have a poll. The positive integers have been colored red, green, oh sorry, red, blue, or green. One is red, two is blue, three is green, four is red, five is blue, three is green, and so on. Do you see the repeating nature there? Renata calculates the sum of a red number and a blue number. What color can the resulting number be? You need to take any notes before the poll pops up or if the poll, if you can move it around on your screen, that's also an option. I'm going to launch it now. Okay, very nice. Again, we have, I can't say that word, it's unanimous. I can't make it into a noun, Jacob, my mouth won't work. I teach two webinars in a row, so sometimes by the middle of this webinar, my mouth is getting a little dry. Yes, most of, all of you have replied that the answer is green, which is a correct, good answer. Thank you for doing the poll. And it is because if we go red, blue, green, the red starts with one, the blue two, three, four, five, six, so you'll notice that the greens are always multiples of three. The reds are always multiples of three plus one, and the blues are always multiples of three plus two. So if we then add a red and a blue, you're gonna have multiple of three plus a multiple of three, which makes a new multiple of three, plus one plus two, which is plus three. So we have a multiple of three, which must always be green. Very good job. So sometimes even when a problem seems kind of crazy, if you just try the patterns, you could have tried a couple of examples yourself of adding different colored numbers and see what you get. That's another way to do it. And it's a great method. If you don't have the exact answer, try it. Experiment with it. Okay. Peter went hiking in the mountains for five days. He started on Monday, and his last hike was on Friday. Every day, he walked two kilometers more than the day before. When the whole trip was over, his total distance was 70 kilometers. What distance did Peter walk on Thursday? There are multiple methods for solving this problem. I will maybe show one or two, but if you solve it in a way that's a little bit different, that will be okay too. Anybody else want to take a guess in the poll? Remember, you don't get any points if you don't mark something on the contest. And there's no subtractions for incorrect answers, so it's always on this one, always better to guess. And if you can eliminate wrong answers, that makes it even better. Excellent, now everyone's at least made a guess. So you'll notice this is different than the others where we had everyone giving only one correct answer. We have several different responses. So this is a great indication that we need to take a look at the problem and see a few ways to approach it. Remember, the idea is always to be learning how to solve problems, not necessarily that you knew how to solve them before you came today, but that when you leave today, you'll know how to solve them. So Peter went hiking Monday, Tuesday, Wednesday, Thursday, Friday. We know that all together he went 70 kilometers. So there's a few ways to do this. One way is to say, you know what, let's make Monday be X. Tuesday would be X plus 2 because we know he hikes 2 kilometers more each day than the day before. That makes Wednesday X plus 4, Thursday X plus 6, and Friday X plus 8. If I add all of that up, I have 5 X's, and then adding up the constants, 2 plus 4 plus 6 plus 8 is 20. So that equals 70 kilometers. Subtracting 20 from both sides, I have 5 X equals 50, and therefore dividing by 5, X has to equal 20, 10, sorry, kilometers. But I want to know Thursday. Thursday is 10, 10 plus 6 equals 16 kilometers. So the correct answer is E, 16. Now, is that the only way to solve this? Absolutely not. You could have started with having Friday be the X. But another kind of clever way to do it is we know that there are 70 miles all together. If we just divide 70 by 5 to get the average, we'll see that 70 divided by 5 is 12 kilometers average. And on which day do we do the average? Since it's a pattern with adding or subtracting the 2 from each day, we're going to do 12 on the middle day. The median and the mean are going to be the same because we have an arithmetic sequence. Those are all words you don't know. Just think about it. We added 2 each day. If we go this way, we add 2. If we go this way, we subtract 2 each day. So this one is going to be the middle value. The middle value in this case is the average. So sorry. Did anyone correct me? I don't know. Yeah. Thanks, Allie. I've got a couple of people correcting me. You're absolutely right. My division is off. It's 14. 70 divided by 5 is 14. So when I add the 2 to that, I'm going to get the same 16 for Thursday. When you see me make mistakes, it is perfectly okay to send me a little message to say, hey, Coach Sagi, there's a little error there. I'm human. This is why when we do any sort of writing or math and we're working with a partner or working with an editing group, we can get some more corrections out of it. So thank you. I know some teachers don't say thank you, but I do. Okay. This one they're telling us is a number pattern. Let's see if we can figure it out. On Monday, Alexandra emails a picture to five friends. For several days, everybody who receives the picture emails it the next day to two friends who haven't received the picture yet. On which day does the number of people who have received the picture become greater than 100? This problem on first reading, for me personally, is a little confusing. I happen to know how Math Kangaroo wants you to solve it and what they consider the correct answer, so I'll explain what I would have needed to know. So it says Alexandra emails the picture to five friends. So that means Alexandra and five friends have it. Then for several days, everybody who receives the picture emails it on the next day to two friends. So Alexandra is done. She emails it once. One day, one day, she does five friends. Those five friends, on one day, will each email two friends. And then those people email two friends. So these five friends don't email it every day. Only if you've just received it do you email it the next day. Does that make sense? I'll leave it. Hopefully that helps. So each person is only emailing it to two friends for one day, not every day. Thank you, everyone replied again. I'm gonna guess the correct answer is Thursday or Friday, but we even had some responses for Saturday, Sunday. Let's see. In the solution key, it doesn't talk about Alexandra receiving the email, okay? So we wanna know how many have received the picture. And it says on Monday, Alexandra emails a picture to five friends. So on Monday, on Monday, five people have received it. So this is on that day, and this will be total. Because they're asking us for the total number of people who have received the message. So there are a couple of interesting things here. So we don't know what day we're talking about, if it's the total number of people, or if it's just that day. I find the wording on this problem a little bit troublesome. You can let me know in the chat if you have similar problems with this one. On Tuesday, because each of these five have sent it to a friend, 10 new people should receive that, because it says everybody who sends it mails it to two friends. So that makes our total 15. On Wednesday, these 10 email it to 20 other people. That's gonna make our total 35. Then on Thursday, these 20 are going to email it to two new people, which is 40 people for that day, which gives us a total of 75 people who have received it with that Thursday. And then on Friday, these 40 people will send it to an additional 80 people, which is gonna send our total well over the 100. So the correct answer is that happens on Friday, if you have used this interpretation of the problem. Okay, if you're wondering on which day do 100 people receive it, that would be Saturday because the 80 people would send it to 160. But the way the problem is worded very delicately, it's saying on which day does the number of people who have received, so who have received, we're assuming is the total number of people who have received the picture. So hopefully that helps a little bit. For me, there was a little bit of a issue with the wording. There's also the question of do only these 10 people send it or do all 15 people send it? So they wanted you to say that the 10 people sent it, not the 15, only the new recipients send it the following day. Okay, Jacob wanted to lead this question. So I will let him take over reading it and giving you time to try to work through it. Thank you, Jacob. Jacob, I don't hear you. Can you hear me now? Yes, thank you. Okay, sorry about that. So I'm Jacob and this is the problem. So we have several square grids with odd number, with an odd number of rows and columns. All the small squares and grid, which are either in a row or in a column with an even number are painted white and the rest of the small squares are painted black. And grids that are one by one, three by three and five by five are shown in the picture below. How many small white squares will there be in a grid that has 25 small black squares? So yeah, I'll give you time to think about this problem. Thank you. Okay, it seems like some of us are a little bit confused, so I'll give you guys a little question. What do you guys think that the next figure will look like in the sequence? Just send me a message if you have an idea. Okay, so hopefully it's been enough time, but we're going to go over it. So notice that the next grid in the sequence is going to be a 7 by 7 because, well, we have a grid with odd number of rows and columns, and we know that this is a 1 by 1, this is a 3 by 3, and this is a 5 by 5. So the next grid in the sequence will be 7 by 7. So it's going to be 7 by 7, since that's the next odd number. And so, let's notice how many black squares are in these grids. So here, there's only one black square. Here, there's four black squares. And here, there's nine black squares. Why? Because, well, notice that if we take out all the white squares from each of the grids, well, we can form a 3 by... So let's say we have this grid. If we take out all the white squares in the grids, we can form a 3 by 3. So here, here, here, those are three columns. And here, here, and here, those are also three columns. And if you take out all the white squares in each of the grids, you end up with a 3 by 3 grid that just contains black squares. So this is 3 squared, this is 2 squared, and then this is 1 squared. So we want to find the grid that has 25 small black squares. And notice that 25, that's 5 squared. So if a 1 by 1 has 1 squared black squares, a 3 by 3 has 2 squared black squares, a 5 by 5 has 3 squared black squares, then a 7 by 7 should have 4 squared, and then a 9 by 9 should have 5 squared black squares. So we want to find the number of white squares that are in a 9 by 9, since a 9 by 9 has 25 black squares. So in order to find the number of white squares that will be in the grid, well, we just have to find what 9 squared minus 5 squared is, which is 81 minus 25, which is 56. So the answer should be D. I like the way Jacob was showing you that with the squares, because it clearly shows the pattern. So you can think of the pattern as 2 by 2, 3 by 3, 4 by 4, or by using the squares, it becomes easy to see the patterns in the bases. So it's a great explanation. So the problems can get a little bit challenging. They can require you to think a little bit, I guess this is inside a box, but a little bit differently than you've probably been asked to think in your school assignments, right? Okay, we do have a bonus problem, so don't worry. I just want to make sure that if you have to leave before we get to the answer to that you've been following all along. So sometimes we will be drawing diagrams, we'll be making tables, we'll be solving simpler problems, but you'll notice today we focused on how a pattern can be in any of these, right? So we had the patterns. Pattern recognition is really important for both math and for other everyday life types of things. So if you are setting up a concert hall, you might need some sort of pattern to set up the chairs. If you are doing sums of series and sequences, you do a sum of a sequence is called a series. So when you're adding numbers that have a certain type of change going on, you can use that pattern to help you make a shortcut formula for finding sums of those. Math Kangaroo always recommends that you start doing some practice contests. Doing practice contests is, of course, a really good way to prepare you for the actual contest that's coming up in March. If you look at our past contests, you look at your contest registration. You log into your account on the Math Kangaroo website and go to where you have registered for the math contest. And you will probably see some coupon codes for how you can get access to practice materials at a discounted rate. That's a great way to do that. Okay. Here is our bonus question. And I think that Jacob, you'd wanted to leave this one as well. Sure. Yeah. So I think that's what I saw. So we want to find what number should replace x if we know that the number in the circle in the upper row is a sum of the numbers from the two numbers, two circles right below it. So I'll give you guys some time to think about this. We do have a poll for this bonus problem that we can launch in a minute, maybe give them a little more calculation time before we put that up. At least on my screen, if I want to see the whole diagram in the poll, I can stretch it out by the corner, and that allows me to see the whole triangle pyramid thing. I'm going to share the results. Not everyone has answered, but those of you who have are doing a great job. Okay, Jacob, do you want to walk them through it? Yes, so notice that if we look at the bottom right here, then we can figure out this other number should be 9, right? And then if we look at this other triangle here, then this circle here should be 12, because 12 plus 15 sums to 27. And then if you look here at this triangle here, this circle here would be 3 because 3 plus 9 is 12. Then if you look here at this triangle, 5 plus 3 is 8. And then here, 7 plus 8, that sums to 15. So this top circle would be 15. If we look here at this triangle, triangle above the 8 and 12, the circle there would be 20 because 8 plus 12 is 20. If we look here at the triangle containing 20 and 27, the circle right above it would be 47. And then if you look at the triangle right above the 15 and 12, the circle above it would be 35 because 15 and 20 sum to 35. If we look at the top triangle here, then we know that X sums to 35 and 47 because 35 and 47 are right below it. So X would be A2. Thank you. So in this case, you'll notice the numbers themselves don't make a pattern, but how you calculate them, the operations that you're using, are making the pattern. So that's another way that Math Kangaroo might expose you to the idea of solving one little piece at a time in the pattern. I hope you liked our lesson on patterns, and Jacob and I will hopefully see you again next week, next Sunday. Hope everyone has a calm, peaceful week. No more bad weather, and I'll see you then. Bye.

pattern recognition

Math Kangaroo

problem-solving

Dr. Sarah Segee

mathematical sequences

San Diego

arithmetic series

visual spatial patterns

problem types

natural phenomena