All right, so I am going to go ahead and go to the warm-up question, let you see that one. It says which piece completes the pattern? So you just have to figure out if A, B, C, D, or E best fits in the space. Do you want to put your answer in the chat? Shaurya and I tried to get to the chat and give you a little thumbs up or a little good job or tell you to try again or give you a hint maybe. So if you ever need a hint, the chat is a place you can ask Shaurya or myself, okay? All right, we'll go ahead and we'll work through this one together. So I am going to have to do it this way, we'll see what happens. So I'm gonna use orange because that's the color we had. For me, the easiest way to do this is to go ahead and to fill in what would be missing. So I can see that on this top right, I would have a piece that kind of has a point like that and that way it would match the other points in this like, I guess a four pointed star. I don't know exactly what you call that. And then we have a similar shape, but it's kind of rotated. So it's a little hollow cave that comes down on the bottom corner. This bottom left corner is opposite to the top right corner. It's the same shape, but we point up in the other direction. And then I almost have like a little flower petal that comes down from the top left. And I can see that I get that same flower petal, change color so we can see it. I get that same flower petal shape here, the same points opposite each other on the corners and also that rounded part there with E. So E is your correct answer. All right, so I'll just go back to that title slide for just a moment. Go back up to the title slide for a moment. Remember we are solving a problem with a pattern. Okay, so think about patterns today. Let's have a little conversation about patterns. So one-sided conversation because only Shuri and I get to talk, right? So I'll ask Shuri. Shuri, is there an example of a pattern in the room or on the screen that you can see right now? So at the top, it has all of our names and well, mostly black screens because only our cameras are on, but it has all of those. That's one pattern. And then at the bottom, there's like a little row of tiny symbols. That's a pattern. And then in the chat, whenever you message, it turns into a pattern of you messaging, someone replying and like that. Okay, and how do you do that? And like that. Okay, and how about, I see one in my virtual background. I see a kangaroo pattern in my virtual background. Yeah, and I have backwards kangaroos in mine. And I have, I don't know if you can see it, but my shirt kind of has a pattern. We will not be annotating on the screen, students. You need to stop or I will just close all annotations and that won't be fun for anyone. Okay, so we are focusing on patterns. Patterns can be things that you see. Patterns can be things that you hear. Patterns can be numerical patterns. How many of you can count by fives? I think you all probably count by fives or count by twos, all the even numbers. That's a pattern, right? So we might be looking for patterns like that. It might be a pattern where you multiply by the same number over and over again. How about two, four, eight, 16, 32? I kept multiplying by two. That's a pattern as well. So let's think about the patterns as we do these problems. I do have polls for some of the problems. Some of the problems today you'll notice we'll have pictures as the answers and I can't do that in polls. So this one where we have number answers, I do have a poll. 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? After you get a minute to read the problem to yourself and see the picture, I will launch the poll. I want you to have a chance to make sure you've read it very carefully. A few students have not put their responses in the poll, that's fine. I will tell you it is anonymous. Neither myself nor anyone else will know what answer you've used, so you don't have to worry about that. And on math kangaroo contests, they are multiple choice and there's never points deducted for an incorrect answer. So it's always a good idea to at least guess something, because you have a one in five chance of guessing it correctly. And if you can eliminate some wrong answers, you can have even better chances. All right, I'll end the poll and share the results. So most students have said that 73 percent, 73 percent of you have said 10 pins to hang nine towels. That is correct. Let's take a look. If I draw a towel, let's draw another towel. So as you can see, each towel that is drawn shares a pin with the towel next to it. So I did not have to add a pin, because I just put it under the same pin. Because you can see these, there's two towels sharing this pin. There's only one towel on this end pin, and I'll have to put a pin here on the end, and that will also be just for this one towel until I add the fifth towel. So the pins in the middle are shared by two towels, and the pins on the end are only for one towel. But one of the ways I like to think about is every time I add a towel to the line, I have to add in one more pin. So it doesn't matter how big my towels are, they could be really long, they can be skinny, but I add one more pin for every towel. So if I have nine towels, and I've had to use one pin for each, that's nine times one, plus I had to start with one pin all the way on the left hand side. So with my order of operations, I always do the multiplication first. So I get nine plus one equals 10. And that's probably how most of you solved the correct answer. Okay, Sophia is drawing kangaroos. The first one is blue, the next one green, the one after it red, and the fourth yellow, and then again blue, green, red, yellow, and so on in the same order. What color will the 17th kangaroo be? So we have a color pattern almost like stripes. There's a few ways to try to do this. I'll give you a few moments to take a look at it. And I do have a poll. I'm starting to see some good responses in the chat. I think you've read through the problem. So here it is in a poll. All right, 10 more seconds for the poll. Most of you have answered, just a few haven't clicked. Okay, this one, most of you have shared the same answer. You have mostly shared blue, and blue is correct. So I have, since most of you got it, I think we can just recognize that the pattern is repeated every four, right? So the pattern, see when I have to annotate this way, it's harder for me. The pattern repeats by fours, and I need 17 kangaroos. So my first four are going to be in the pattern, and then I'll get up to eight, and I'll repeat the pattern. If I add four kangaroos, I'm at 12 and repeated the pattern, and then four more gives me 16, and I've repeated the pattern. And then I need to have plus one more kangaroo to get to 17. So this was blue, green, red, yellow, blue, green, red, yellow, blue, green, red, yellow. This is blue, green, red, and yellow. And then one more, I'm going to get to blue again. So the correct answer is blue. You could do 17 divided by four, knowing that the repeat is every four, and you would get four times four is 16. And then you would see you have one left over, and that new one has to start the repeating all over again. If you wanted to, you could just do symbols, you could do a drawing, and you could do 17 little symbols, blue, green, red, yellow, in order, and you would get that the and you would get that the 17th is a blue. All right, so Shorya, you wanted to lead number three. Yes. Did I share my screen so I can like draw? Yes, absolutely. I think it will allow you to just override my share. Okay, so the question is, Monica writes numbers in the diagram in the picture below so that each number is the product of the two numbers below it. Which number does she write in the gray set? So we want to find out what number she writes here, we don't know what that is, and we know that every number is the product of two numbers below, except for this bottom row because there's nothing underneath it. So I'll give you all a minute to think about that, and you can answer me in the chat. And I think there's a poll for this one, right? Yes. I'm getting a lot of correct answers in the chat. I think we can launch the poll for the students. Some of you may see that you have to scroll down on the poll to be able to get to the answer choices because now that there's a picture, the problem is a little longer. So, on my tablet, I can scroll to see the answer choices and make a selection. Okay, so it seems like mostly everyone got this right. The correct answer is 8. So to do that, the way I would do this would be through, um, wait one. So in this box, we know this box times 1 is going to have to equal the one above it, which is 2. Right, so we need to find something times 1 that equals 2, and that's going to be 2. We use the same thing on this box because it's symmetrical. So we do this 1 times whatever number goes here is 2, so that gives us 2. And now we can find this one. So that would be 2 times 2, which equals 4. And then we can find out these boxes easily because then it's just 2 times 4 for this one, which equals 8. And then this one is also 8 because there's another 2 and a 4 underneath it. And if you wanted to check your answer additionally, you could just do 8 times 8 and make sure that it's 64, which is correct. So the pattern in this one was just that all of these, if you notice, are just powers of 2 because at the bottom it starts with 1s, then you have 2s, then you're multiplying to get 4 and 8 and then 64. But the pattern is just that the two numbers on the bottom multiply to get the number above them in a pyramid. These types of questions are actually quite common on the math community, like pyramids, triangles, things like that. This one also has symmetry between the left side and the right side. Yeah. So symmetry is another type of pattern. Okay. I will share my screen. Okay. Again, I do have a poll, but let me read through it and leave it open for a little bit so everyone can make sure they get the information they need. Lanarky builds a fence using 1-meter-long poles. The picture shows a 4-meter-long fence. How big is the fence? It's a 4-meter-long fence. The picture shows a 4-meter-long fence. How many poles does Lanarky need to build a 10-meter-long fence? And I had a TA this week who noticed that since each one of... Sorry, wrong tool. Since each one of these logs is 1-meter, when I add them up along the top, you can see that that's how we know that it's going to be 4-meters-long for this section of fence because we have 1-meter in each of these. So I'll give you a few moments to take a look. How many poles do we need? Not additional poles. You notice it says how many poles, not additional poles. So count all the poles that you need to make a 10-meter-long fence. And then I do have a poll spelled P-O-L-L for you to put in your answers, okay? Okay, the poll, I believe, does have this little picture, so I'll go ahead and launch it now. If you need to make any notes, go ahead, you have five seconds. And here's the poll. Remember, like I said, on my tablet, I have to scroll down in order to see the answer choices. All right, I usually get about the same number of people responding to the polls. There's usually a few of you who don't, and I don't know if it's just your setup where you can't see the polls, if you're not allowing pop-ups or whatever. So hopefully that's everybody who wanted to answer the poll. And you did really, really well. You did much better than my class usually does in this. Most of you have said that the answer is 42. And again, if you want, I have to scroll to see all the results sometimes. But yeah, the correct answer is 42. Let's take a look at that. So I'm gonna solve this pretty much the same way that I solved the problem with the towels and the pins, which is I'm going to build a little bit more of this fence. And to build a little bit more of this fence, I need a poll here, a poll here, a poll, and a poll. And that would get me one meter, right? So for every one meter, I need four polls, right? So that is like these pieces here. But what that does not include is it does not include these two pieces that I had to use on the left-hand side when I first started it. So I'm gonna use exactly the same thinking that I used before, where I wanna go 10 meters times four polls. So that's gonna be 40 polls. Oops, a little bit of a spelling error there. Polls plus the two from the very beginning. So that gives me 42D. Now, are there other ways to do this? Yeah, this is Map Kangaroo and everyone is different. So some other examples of ways I've seen students do this, I've seen students break it up by the row. And they say, okay, on this top, I need one, two, three, four and I also need one, two, three, four here. So probably to go 10 meters, I will need 10 and 10, okay? I'm gonna just erase this little piece right here. It doesn't let me, okay. And then some students said, okay, but I can see that I have some vertical ones too. And I needed five to go four meters. I always need one more. So I'm gonna need 11 there. I always need one more over here as well. So I'll need 11. And then that one was just a, I wrote it out. So when they add that, they say that that is 42 polls. And that's correct as well. Just another way to look at the same problem and come up with a good solution. Okay, give me a moment to read all those words. Try to pay attention as I go through and see if you can make sense of it. Well, math kangaroo problems do involve a lot of reading and understanding. So it is important to sometimes read twice or underline things. 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. 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? This has a little bit of vocabulary. It says positive integers. The word integers means numbers with no decimal and no fractional parts. So this can be negative two, zero, four, 70,000 would work as well. They can be as big as you want. They don't have to be even. They can be odd numbers, positive numbers, negative numbers. What they can't be is 2.64. That has a decimal part. So it is not an integer. And then the other thing we're telling you is it's positive integers. So zero is not positive or negative and negative numbers don't count. So we are dealing with one, two, three, and so on as you get bigger. And you can see that in this chart right there. There is no poll because I couldn't find a way to put those little chart pieces into a poll. So just put your answers in the chat for this one. You're not sure how to start, this is one of those multiple choice problems where you can start testing the options. That probably is what you have to do because you have to test the options and one of them is going to work and the others are not going to work. Okay, getting a lot of great answers in the chat. I'm so glad that you're all participating and trying on these. Because if you just sit there doing nothing, you won't learn how to solve any new problems, right? So what some students do is they look at the original and they look at the pattern. So you'll notice that the number on the right, the ones digit number, they have a pattern in the columns. Like 5, 10, 15, 20, we have a pattern of a 5, a 0, a 5, a 0. You can see there's a pattern just alternating in each vertical column. So if I look at number 1, I have a 43 in the second column, and I know that the second column has to have only twos and sevens. So this can't work. I can look at column B and I'm like, all right, this follows the rules of those columns, doesn't it? I have a 0 in the last fifth column and I have a 2 here. But hopefully, you've noticed this went from the least number at the top to the greatest number at the bottom. This is 60 and greater than 52. So I can't count that way and follow the same pattern. So this is a no. I take a look at C, I see, okay, the 9 is in the fourth column and the 2 is in the second. That's correct. Now, I can fill in the other numbers to make sure they work, 72, 73, 74, 74, 74, 74, 75. That seems to work, like that one. 81 and 86 are supposed to be in the left column, and 87 is supposed to be in the second column. If I tried to fill this one in from 87, I would get 88, 89, and they're supposed to be a multiple of 10 or 5 in the right-hand column. So this one is no good. So the correct answer here is C. Amelie wants to build a crown using 10 copies of the token shown in picture 1. So this is the token in picture 1. She's making the pattern shown in picture 2. When two tokens share a side, the corresponding numbers match. Let me show you what that means. Do you see how these share a side and they're both the number 2? These share a side and they're both the number 1. That's going to tell you how to put this coin in for the rest of the crown. Four tokens have already been placed. Which number goes in the triangle marked with X? If you were able to print out the handout before class, you should have this picture in that handout. I'm getting some correct answers and some other answers, and that's good, because we want there to be a little bit of a challenge in these webinars, right? Don't want to have only questions that everybody can do. We want you to learn new types of problems and have a bit of a challenge. So, for me, the easiest way to do this is to start filling in the tokens. And you can fill them in from the left side or the right side, that's just your personal choice. I think the X looks a little bit closer to the left side, so I'm going to do it from the left side. So, I have to put the number three here because these are adjoining sides, right? And I'm going to notice that the numbering goes in this direction around the token, so three, four, five. The one and the two are there also, but I don't need to worry about it. I don't have to fill in every single square because what I really want is the little triangle with the X. So, I know that for my next token that I place, I'm going to have to match up the five. If I match up the five and I go around the token, from five, the next one is a one and a two, so there's my two. And then if I keep going around, I'm going to have to start with a two because they match up. From two, if I go around the token, I'm going to get two, three, four. So, the correct answer would be D, four. There is another pattern that's going to happen here because of the way we rotated our token. If I start like over here where I see the five, you'll notice that the next one that is a match line is a two, five, two, four, one. So, that would give me five, two, four, one, three. Five, two, four, one, three. So, five, two, see if I can do a bright color so you can see that. Five, two, four, one, three. Five, two, four, one, three. So, that's another way to look at the pattern. Okay. I think that Shoria wants to lead this problem. Do you want to share your screen, Shoria? Yeah, let me do that. Okay, okay. There we go. Okay. Okay, so in a certain picture, here we can see the numbers one, two, three, and four. And we also have the reflection. So, we reflected the one, reflected the two, the three, and the four. Now, we want to find out what is the next reflection, this next picture going to look like in the sequence. So, remember, one, two, three, four. And so now, we're probably going to be doing the number five. So, I let y'all have a few moments to think about that. And you can answer me in the chat if you think you have an answer. Okay, I'm getting a lot of answers, so I think I'll go ahead and start working through it. So, first of all, you can see that this reflection is done through a line that's like this, right? We're going, we're reflecting it sort of horizontally in the sense that our four, for example, is going like this. So, if you were looking at yourself in a mirror and I would like, I would raise my hand like this, then it would be the other hand in the image. So, sort of like this. So, just like that for the three, we have something like this and so on. So, if we draw five and then we draw a little imaginary line like that, then we just reflect this over here. So, that would be this. This line stays the same and we make this half circle. So, when we join them together, it looks something like this, which is going to be answer C. I think most everyone got that right as well. Yeah, so if you look at the other answer choices, you can see that some of them were upside down or even sideways. So, make sure that you had that right side up five on the right hand side, right? All right. I do have a poll for this one. This one says 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? Give you a few moments. Like I said, there is a poll, but I think you might want to have a little time to count and see what you can do with this one. I want to point out that this had the same words that we had in one of the previous problems where it's saying how many total cards, not how many more cards, but how many total cards. Okay, so I'm gonna launch the poll. We haven't had a poll in a few problems. I think it's time to launch one. Anybody else want to make a guess in the poll? Remember, we do not know who's guessing, so you don't have to worry about us making any judgments or anybody else seeing your answers. And also, on the Real Math Kangaroo Contest, you do not want to leave blanks, because blanks are definitely no points. And even if you guess correct, you get points. All right, I'll end the poll, share the results. So if you can't see, go ahead and scroll down. More than half of you have said 26, which is really good work. That is correct, but not everyone got it. So let's explain it so we can all see it together. All right. So you remember a few weeks ago we did a lesson on drawing. So one of the ways that I like to do this is to just finish the drawing. They already have a drawing that has three stories. So to get a drawing with four stories won't be that difficult, right? I can just add another layer here, and I see that I have to add cards that way, horizontally. And then I have to add the up and down cards to kind of make that two parts of a triangle, and that would make the four-story house, right? So now I'm going to look at it in levels. Now, again, this is just my way. Remember, you are you, and your brain might work a different way. I look at it, and I say, OK, I had to go up and down in this level, so that's two. And then over here, I have just one card going across horizontally. And then for the next story, I had to go up and down and up and down, so that was four. But I have just two cards coming across. Now, if I start to look at a pattern, I have one, two, three, four, five, six here. So this is a pattern two, four, six, and it goes eight. So that's the pattern of even numbers, right? Evens are plus two. If I look at the other pattern that I have, I have one, two, and here I have three cards that I added. So that is just a single addition each time. Now, if I add all of that up, I get... I like to add using some grouping methods. Perhaps some of you learned this in like second grade. I'm still in second grade, just, you know. So eight plus two is 10, four plus six is 20, 23, 24, 25, 26 when I add them together. I just make fewer mistakes when I do those groups and just cross them out as I go. That's my personal preference. That may not be how you add. You might add top to bottom and that's correct too. It's just a choice. Okay. So we are not going to do the exit ticket. I have a bonus bonus problem for you because I thought we would finish a little bit. Early. So I'm going to read the bonus bonus problems to you. The numbers three, seven, 11, 15 are the beginning of an arithmetic sequence. Sequences are numerical patterns. An arithmetic sequence has a common difference between each successive number. What's the common difference in this sequence? What is the 10th number in this sequence? And what would be the 20th number in this sequence? And we don't have a lot of time, so we'll work on it together, but I think it's good to recognize the beginning of the sequence. This is the simplest one where we add each time. So two, not two, right. Three, seven, 11, 15. And I think you can see that I have added four and I have added four and I have added four. So the first part, what is the common difference in this? The common difference is four. What is 10th number in this sequence? Well, that's a little bit of a trick because I didn't start with four or one or even zero, did I? I started with three. Okay. So notice how many times do I have to add four? To have two numbers, I add four once, but to have three numbers, I add it twice. To have four numbers, I added it three times. So just like that close line that we had where we added one pin. So I'm gonna need for the 10th number, I'm gonna need to add four nine times. And then I have the original number, which is a three. So if I add four nine times, nine times four is 36. And then I add the three because I started with a three. So my 10th number would be 39. And some of you might write out 10 numbers. That wouldn't be so bad. But that's gonna be a little bit harder when I ask you what is the 20th number in the sequence. For the 20th number in the sequence, I'm gonna do plus four, not 20 times, but 19 times. And then I have to add the original number, three. So 19 times four, nine times four is 36, carry three, four plus three is seven, right? So 76 plus three equals 79. So that, it was a way that you can solve these types of problems. I hope you like that. What do you think of those, Surya? Have you seen sequences before? Middle school, right? Actually, in later years of the Math Kangaroo, sequences actually pop up quite frequently. So they're also really just all about patterns, yeah. I'm giving them a bonus bonus. They get something a little bit early. Bonus for coming to Math Kangaroo webinars. Okay, so let me head back up. Let me page up. It should. It should let me page up. So the exit ticket was, I hope you found this pattern strategy to be helpful. And remember, the best way to get used to all different types of Math Kangaroo problems is to continue doing those practice contests. Make sure you are doing something to prepare, do a few problems each week. The other thing that a whole practice contest does that no webinar can actually do for you is tell you how fast you need to work. So you need to be able to work through the correct number of problems in only 90 minutes. So 24 problems in 90 minutes. You do have to do some of the questions pretty quickly. And you'll practice that if you try a whole contest. Do you have any helpful hints, Surya, before we wrap up? One of the things I'd say is whenever you're doing questions with patterns, it's always helpful to draw one or two patterns after the one the question has, like in the problem with the fence and the one with the cards. It's always helpful if you draw a little bit more just so that you can visualize the pattern and see what it is. Because sometimes it's not very clear just by looking. That's a good point. Because we did discuss that drawing was a good method for solving problems. And so just because we're now calling something else patterns doesn't mean that we've stopped using a tool that we were practicing before. All right, everybody. Have a really good November. I'll see everybody next Sunday. Bye for now. Bye.

Math Kangaroo

pattern problems

arithmetic sequences

reflective symmetry

numerical relationships

problem-solving

time management

pattern recognition

contest preparation