Good morning, everyone. Welcome to Maths Kangaroo webinars for level one and two. My name is Van Korolava and we have with us Andrew as our teaching assistant. Andrew, could you please just say hi and introduce yourself? Hi, my name is Andrew and I've been competing in Maths Kangaroo for over seven years now. And I like to learn about math concepts and read books. Okay, thank you, Andrew. So this one is in webinar format, that means there's no video and audio. The way students interact with instructors is through answering questions through polls. And if you have any additional questions, you can send a private message chat and your message only go to me and Andrew only and Andrew will help answer your questions. And everything is recorded, that's also we also don't want you to have your videos and audios on so that you can watch the recordings if you miss the class. So the way how to make the best out of the webinars, first please print the handouts beforehand. Andrew, could you please send everyone the link to the handouts? You can find it on the registration website on Maths Kangaroo. So please print the handouts beforehand. And then it's best to attend the webinars live so that you can work on your problems with your peers. You can take part in the polls. I would really love that if you could answer the questions through the polls so that I know how we do as a class and I can adjust the pace of the class based on your performance. And at home, you can review the handouts and any additional notes with your parents. You can watch the video recordings if needed. And by all means, do additional practice problems. We have additional practice problems on Maths Kangaroo's website or can get your favorite books or whatever problems that you can find, do set additional times to do math apart from the webinars. And then we hope that the problems that we present also give you and your parents ideas to invent new problems based on that. You can change the numbers and change some of the settings. You can make the questions harder and more interesting and more challenging for additional practice. So what is Maths Kangaroo? Some of you might have participated. And for some of you, it could be your first time. It is a competition. It's a test that could, you know, around 6 million students worldwide every year participate in the competition. It's a 75-minute test with 24 questions. And what is so special about Maths Kangaroo? The questions are quite unusual. We hope they're interesting to you. Think of it as a box of chocolates. You could be surprised by what you discover inside of them. And many students say that the test is a blast for them. They had a blast doing Maths Kangaroo test. So we hope that it is something that you love, something that you look forward to do every year rather than a test that you have to do. So this is the timeline of what we cover in this test. There are, in the first 10 lessons, we go through different topics and different problem-solving strategies that help you prepare for the Maths Kangaroo test as well as to build a good foundation and hopefully make you, get you excited about math in general. And in the last question, we will do a full practice Maths Kangaroo. So the first class today is Patterns. So what are patterns? First one in the picture here, as you can guess, it's a pattern of seasons, right? Every year we go through winter, then spring, then summer, then the fall, and then winter again. This pattern just repeats and repeats every year. It's a repeating pattern. And you could think of other repeating patterns in your day-to-day life, right? Like day and night and the days of the week, Monday, Sunday, Tuesday, Wednesday, and just repeats every seven days. And what are activities that you do that's repeating every day? That is an example of repeating patterns. And what's next we have here? Anyone knows what this one is? It's a Romanesco broccoli. It's very beautiful. If you look at it here, we have a big broccoli and we have a smaller one here that looks very much the same as a big one. And then for each smaller one, we have something even smaller, but that looks very much the same as a bigger one. And it just go in here as the scale gets smaller and smaller. This pattern has a fancy name called fractals that you will learn more if you continue with math and science. And then this one, it is a, what is it? It's a Nautilus shell, something that lives in the ocean. It's a spiral pattern, right? Have patterns on land pattern. In water, we also have a pattern in the air we look at later. And this one is a honeycomb. What pattern do you see here? Each of this is a, if we sort of trace out the shape here, you have a regular hexagon, right? A perfect hexagon with six sides and they're all the same. They would call it regular hexagon. So on its own, it's a beautiful pattern with lots of symmetry, but it's another pattern that you, if you sort of notice carefully that we can basically like tie the whole plane with this hexagon without any gaps inside. And you can look up for this pattern, it's called tessellation, right? When you can tie the floor, the plane with geometric shape without any gaps inside, right? And that is what you would do if you continue with that. It's very beautiful, a lot of structures inside. And now we are in wintertime and probably something that all of us treasure is the snowflake, right? It's a beautiful dendritic patterns here. The sort of a tree trunk with branches over here. And as you notice, not all snowflake had this beautiful symmetric structure, right? It really depends on, as you sort of do more research, you know that depends on the temperatures, humidity, how the snowflake was formed in the end, how it depends on the condition when it fell to the ground. And I got this picture from an article that named the science and poetry of snowflakes that say it all, right? The complexity and the beauty of it. And hopefully that's what we would understand if we continue with math and science. So in order to understand all this fancy pattern, we start from somewhere, right? We start from the very basic. That's what the subject of today's lessons. So in a nutshell, a pattern is a way objects are arranged according to a rule, right? You can create patterns with shapes, with numbers and colors. So for example, here is a repeating patterns. We start with two hearts and a star, then two hearts and a star. So the next one, we know it has to be a heart, a heart and a star just to continue this pattern, right? And then over here, we have an, if you look at this pattern, right? Some numbers are missing. Can I figure it out? Yes, I can. Because I look at it and I see some patterns here. I look at the first three numbers, eight, 10, 12, that's counting by two. And then I know that it should be 14. I make a guess, 14, 16 and 18. And I check, well, it matches one. So this is the correct pattern. And similarly, we have 14, 12 and 10. So this is also counting by two, skip counting, but in decreasing, right? The numbers get smaller. So I make a guess it should be eight. And I'm glad because that the next one is six that fits this pattern. And then the next one should be four. So some of the main types of patterns in math kangaroo questions are addition, subtraction with picture, color, or even patterns. So what we'll be doing in today's class is we solve a bunch of questions. And why are we solving them? First, we try to figure out what the pattern is. We try to describe the pattern. And then we sort of make a summary. We identify what kind of pattern is that. Is that shape? Is that numbers? Is that colors? Okay, so let's jump right in. Okay, first question. A magician is pulling toys out of his top hat. He always pull out the toys in the same order shown in the picture here. The pattern in the picture repeats every five toys. Which two toys does he pull out next? So you will have maybe 30 seconds or so. You will probably have a handouts. And then I will launch the poll. And then you can enter your choice in the poll. If you can enter in the poll, you are also welcome to send a chat message. I launched the post, hopefully you can see it. Great, I got almost everyone. So let's get started. So this says, it looks the pattern repeats every five twice, and I usually underline important information right in the question itself. So 1, 2, 3, 4, 5, and yes, it repeats. So we make a cut right here. And it's five animals. So the next one, it starts from the mouse here. So we know that the next two should be a snail and a bird, right? And that is answer E. So great, if you got it. If you don't get it, I hope this drawing method right in the question itself will help you find out the answer in the shortest amount of time. Just draw right in here and circle the answer. And then only after that, you look at the answer choices. In some questions, it might be good to look at the answer choices first. In some questions, it might be confusing to look at the answer choice. So you solve it first and then find the correct answer. Next question, after 1 comes 3, after 3 comes 5, after 5 comes 7, what are the next three numbers? So just like a math kangaroo, we start with some warm-ups and gradually we get to more complex questions. That's great, everyone is doing great. So I can also, this is a word problem, right? But you can always, so just write out the numbers. And when we write out the numbers next to each other, it's easier for us to see the pattern. So one, and then have a three, and then have five, and have seven. If we write out this way, it decides that, oh, what is the pattern over here? This pattern is a number pattern. It's, you can call it odd consecutive odd numbers or counting by two, right? We want to be able to describe the patterns in words. And then naturally after that, we know that the next three has to be nine, and 11, and 13, right? Skip counting by two. So we have some picture patterns, some number patterns. Let's see what we have next here. Sophia arranges balls on a stairway in a pattern as shown on the picture. How will the balls be arranged on the step with the question mark here? So remember the first step is to apply the pattern. And it's best if you can describe it in words, or later on, you can have equations or things, but if you can describe in words, that would be great. So these are one of the questions that if you have the printed handouts that would that would help you you know write things right there because sorry the picture is a bit small I made it as big as I can. Okay, almost everyone got it. So let's do it together. One way you can do that, we can continue the pattern, right? You can continue drawing here, here, and here, but it might take, you know, too much time to draw. If we can find the patterns, then we may skip drawing these three steps and get to the final step, right? So one way to do that is that I probably look at the numbers. First, we have to figure out the number of balls and the color of the balls, right? One, two, three, four, five, six, seven, eight, nine, right? After I number it, I realize that the number of balls in each step is equal to the numbers of steps, right? The cardinal numbers of the step, right? So one, two, three, four, five, six, nine. So I know that in the last step here, I would need five, I would need nine balls in total. And then that's the first question, the first pattern number, and then I look at the color over here. It always starts with black, and then black, white, black, white, black, right? So it's alternating color. It's alternating color, so we would, at nine here, we start, if you can draw, it's great, but otherwise we would do black, right? And then white, and black, and white, black, and white, until we have nine of them. Just continue until you have nine of them. So we start up here, leave, start with zero, that's not good enough. I'm sorry, with that, white, that's not good. This have two of them together, so that's not good. Here we start with a with a black, right? So black, white, black, white, but we have only four and four. It has only eight balls, so that's also not good. This one has, this not good because of these two balls that's together. This one, we have a black and white, black and white, that's good. We have a five black and four white, right? And that's nine balls. So this should be our answer total. And over here, we can look at that, but actually we have an additional one, which is a ten, so that is also the wrong answer. And I would summarize here. We have two patterns. One is numbers, patterns is consecutive numbers, and the other pattern is color pattern, alternating black and white. Yes, so that's what you do. Want to describe it and take a moment to realize what kind of patterns we are dealing with. Okay, next one. Emily builds towers in the following pattern. You can look at it visually and decipher yourself what pattern it is. That's something you need to figure out. And what will the 16th tower look like? Remember, first you want to describe the patterns. To find out the pattern, describe it. It's interesting, we have A, B, C, E as the answer, A, B, D, E as the answers. As the majority of you entered the answer choice, so let's try to figure out the patterns together. What's the patterns here? We look from one to the next, right? We see that every step we move the top shape to the bottom, right? And push everything up. Here, just move the top one to the bottom. And now the circle goes to the bottom, and finally the rectangle goes to the bottom. And because there are only four, we only have four shapes. If we do this manipulation, the pattern repeats after four times, right? And that's why you notice that at five, five looks exactly like one. But we are asked about the 16th power tower, so what should we do here? Are we going to continue to draw how the 6th, the 7th, the 8th look like? We can do so, right? But that would take a lot of time. Is there a shortcut how to get to the answer quicker by making use of the pattern over here? One way is that you can do 1, 2, 3, 4, 5. I mean, one way you can do that, you notice the patterns 1, 2, 3, 4, and it repeats itself. So it's 5, 6, 7, 8. It's similar to 1, 2, 3, 4, and then 9, 10, 11, and 12, and then 13, 14, 15, 16, right? That's one way to do that. Instead of drawing all the towers, we can just use numbers to label them, just use their labels. And we know that in each row, the patterns repeat each row exactly the same as the previous one. So we know that number 16 is exactly as number 4, right? And what would be the answer? If 16 is 4, then the answer should be... The answer is E. Which is a third of the class got the answer as E. So this question is a little bit more challenging in the sense that we have shape patterns, right? We have shape patterns, but at the same time, we also have numbers, number patterns, because here we have four different shapes, right? If we have five different shapes, then the pattern would repeat after every five towers. And yeah, this is an easy way. I mean, knowing numbers would help us get to the answer quicker, and that's why we study patterns. So we were able to generalize it, right? We don't study math, so we can just solve a specific question. We study math, so we can solve a class of questions. We can solve with big numbers because we know the rule, and this is a rule. So once we know the rule, at home, you can extend questions like this. For example, what would be the 100th tower look like, right? You can do this with your parents, and then we know the 100th tower look like, and then you know, what about, I could be even more fancy, the 101st tower look like. And that's where you see why math is useful, because it helps us generalize, because we know the rule, okay? So let's go to the next one. So Andrew, could you help with this question? Sure. So for this question, we can see this black and white bead pattern, and the question asks which dots are covered. We launched some students saying that we didn't give enough time before the poll. Sorry about that. We launched the poll and you don't have to enter the answer right away. Just take some time to figure it out. We wait until a few minutes pass or until the majorities of the class respond to the poll. Alright, so it looks like most of you have answered the question and while every single answer choice was picked at least once, most of you picked the answer choice C. So the question is asking us to fill in this blank. In order to do so, we have to first find the pattern behind the beads. We can see by counting the number of black and white beads in the beginning that there is one black bead and then one white bead, two black beads and two white beads, three black beads and three white beads. So we can quickly see that the pattern so we can actually find the switches between black and white beads with the number of each type increasing by one every two times it switches. Using this we can figure out that there's next to be four black and four white beads but because we can already see that there are two black and one white bead, we can subtract those three beads from the pattern leaving us with answer choice C, two black beads and three white beads. Thank you Andrew. So as we said earlier, first we need to find out the pattern and these are pattern the number of beads and then the colors and then we see the missing one here. We need four blacks and we only have two so we need two more and we need three whites so the answer is C. And we can continue the pattern and double check here actually find black gray fits perfectly well with the pattern that we figured out. So again I would classify both number of patterns because the number of beads just increase from left to right and also color. Next one. Peter drew a pattern twice as in the picture. Which point will he reach when he draws a third pattern? Please see D. And this one, please, I'm sure this one you have in your handout. So please draw right into your handout. Okay, some student say what the point he reached, okay, that's good, you have a question to send to the chat, which point he reached, that means over here, you look at it here, for example, he reached this point, right? But he doesn't reach this point, this point he doesn't reach. So this is called grid, and over here is the points on the grid, the point, you know, it's called the lattice and where the lines meet, so these are the points on the grid, so some points, when he draws his pattern, he reaches some point, this point, this point, this point, but not this point, not this point, that's what the question means. So when he continues, draws a pattern, which is the points here he would go through, and which points he doesn't go through, so this is good. So, launch the poll, I think, good, Andrews has launched the poll. That's why we need to do more practices, not just to improve our problem-solving skills, but sometimes we get used to the language, a new concept, something that will be a little bit different from our day-to-day language. So I will continue drawing this, just like Peter. So first we figure out the pattern, right? So he draws a square, and when we see a three, we go like three steps, and then two, and sometimes big and small, and smaller. So I continue this pattern over here, I would need to go one, two, three steps up, I would need to go up here, and I count to the right three, one, two, three, I go down three, right? And then now I have to go one step to the right, and two steps up, two steps to the right, two steps down, and one, and now I do this small little square. And yes, if we draw it correctly and count it correctly, then the answer is D. He only reaches point D, and he didn't pass through any of the points A, B, C, D, E. A point is just a dot on the plane. And this one, I would say we have a number pattern, right? It's a decreasing number, like three, two, one, and three, two, one, so this actually is called periodic as well, because it's sort of repeating, right? Three, two, one, three, two, one, that repeating. And then also shape pattern, because what we're doing here is that we draw patterns of a square. Sometimes you can modify it and make a rectangle, for example, and not a square, and then we have a different pattern. Let's go to the next one. What number is covered by the question mark in the last picture below? Yes. So this is one of the questions that we want to spend really quite a bit of time trying to figure out the pattern. Some of the patterns like the previous ones are visually clear to us. The moment we look at it, right? This one might not be, so I need some time. Yes, I understand the concern with some students that you don't get the question, so let's look at it. So in each circle, I guess this is like plant, right? Could be parts of plants. So in each circle here, maybe someone does landscaping, half of it is in the shade, half of it in the sun. We have certain numbers of parts in the shade and certain in the sun, and that's the same in every picture. And someone figured out a rule, and they say that, oh, in the first picture, the number is two, the second is four, the third is three, and these numbers describe something, right? Which is not obvious to us, it's more like a puzzle, and we have to figure out what does this number mean? What do they represent? What do they describe? It's not obvious, but that's what makes it interesting, what makes us think. So take your time. Look at the numbers in the shade, number in the sun and sea. Is there any rule? I launched the poll to see if anyone wants to attempt, but don't worry, if you haven't figured it out, you can take your time. No, actually, you are doing very well. You figured it out. Okay, so I will probably, what I'm doing here, probably just recapping what you have in mind, right? So we have the shape of the sun, we have the number here. We look at two, at first you say, oh, maybe two represents the number of parts in the sun, right? But then a rule has to apply, you know, in all circumstances and not a specific case. It's two, we might think that two represents this, but then four, and here has only one, so that rule is not correct. It's good to, you know, we attempt and it's not correct, then we abandon it and find something else. And here's two, this is four, maybe two times two, two plus two is four, but here's four, so it's five, and three also doesn't, you know, that rule also doesn't work, right? So we have the shape of the sun, if each of them doesn't work, then maybe we can try to find something that relates, right? That relates the shape and the sun, and that's something we probably call the relationship. And what can we do with these two numbers, right? Some number in the shape, some in the sun, we can add them, we can subtract them. So let's try it. If we have a here, we have a four, four and two. And then we could, if we add them, we have four plus two equal to six. If we subtract them, we have four minus two equal to two. Over here, we have a five and one, right? We can add them, we have five plus one equal to six, and five minus one equal to four. And it looks like, well, it looks like we are lucky here because two is four minus two, and here four is five minus one, and we continue over here. We realize that we have how many? Seven in the shade and four in the sun. So the rule applies, the rule has to go to apply to all cases and not just one. And since this rule is correct for the first three pictures, then we can infer that it should work for the last one. And then the question mark would be the number of trees in the shade is four and minus the number of trees in the sun, three, which is equal to one, right? And then for this question, the pattern is not quite clear. So I would say that the number in each picture is equal to the difference of the numbers of trees in the shade and the number of trees in the sun. Basically shade minus sun, that is a rule that you wanted to describe in words in this case. Next one. Andrew, will you help us do this question? Sophie makes a row of ten houses with matchsticks. In the picture, you can see the beginning of the row. How many matchsticks does Sophie need altogether? All right, I'm going to launch the poll again. Please put your answer in. You take your time. The numbers are big and we have time, so it takes our time. Remember she needs to count out the number of matchsticks for 10 houses. Oh, if you cannot move the post, some students that you can just use a mouse, click on the top of the post and move it away from the question, if it blocks the question. There's a panel on top of the post, just click on the word post and quizzes, keep the mouse there and move it away so that it won't block your view. Okay, so most of you can answer to polarity, so most of you chose to correct answer to choice B, 51. So in order to solve this problem, we have to first find a pattern. We can first take a look at how many matchsticks it takes to build the first house. We can count them 1, 2, 3, 4, 5, 6. So the first house takes 6 matchsticks to make. Next we can look at the second house. Second house takes 1, 2, 3, 4, 5 matchsticks to make, because it shares 1 matchstick with the first house. We can look at the third, fourth, and fourth houses, and we can find out that they also each have 5 matchsticks. So it looks like our pattern is that each house is 5 matchsticks except for the first one. Using this, we can add 5 to 5 9 times, which gives us 45. Okay, add 6 to that, so therefore we can get 51 as our answer. Thank you, Andrew. So, yes, as usual, first we have to find out the pattern, right? And then that pattern is 6 and 5, 5, 5, because these additional houses share a wall. And now, some of you could get it very quickly, but don't worry if you haven't seen it before. There are many different ways to do it. You could see that we have 9 5s here, so we could do 4 5s and 5 5s. I see students, they are very creative in the way you calculate a big sum like that. You could add 5 and 5, right, equal to 10, and you do it. You just pair them up until you get enough, until you go through the list. You have 8 of them, and 5 plus 5 equal to 10, and the last one is 5 plus 6 equal to 11, right? So we have 40 plus 1 is 51. Or you can recognize that you can break up 6 into 1, right? So we have 1 and then 5, and this 5 plus another 5 plus 9 5 would be 10 5s. And if you're familiar with multiplication, you don't have to, but it's a shorthand notation of adding 10 5s together. That gives you 50, so that's 1 plus 50 equal to 51, right? Mathematical coding is not meant to test you, you know, big numbers and complete computations at level 1 and 2. You can always go around it by breaking it into manageable pieces like this. The most important thing is to find out the patterns. And then if you want to have additional practice at home, you could ask, like, how if you need to build 100 houses or 102 houses, how many matches you have. You can, you know, decorate the houses and with, you know, more patterns and make fences and just create your own patterns and see if you can solve it with your parents. So we have time, so we can do two more questions. This one is number questions. Which number should replace the question mark in the pyramid? So we have the pyramids of numbers here, four, and then threes, and two, and one. Notice that one number is missing here. And then we are asked about the number on top, not the missing number in these three dots here. And the hope is that with the rest of the numbers, we can find out the rule, the pattern, so that we can fill in the missing numbers. I launched the poll because some student already sent the answer to the chat, but take your time. If you haven't figured it out, take your time. One hint is that you can find the relationship between the numbers in each row and the numbers beneath it, underneath it, or above it. Let's do it together, so 1, 3, 4, if you look at each row alone, right, 1, 3, 4, 2 doesn't, you know, they don't really follow any rules. And that means we have to look in the next row, this one is missing. So maybe we can see if there's any relationship, anything relates the numbers on top here with the numbers at bottom. And yeah, we actually found something, 4 is actually equal to 1 plus 3, and 6 equal to 4 plus 2, right? So each number is equal to the sum of the two numbers beneath it, right? And if we follow that rule, we can say 3 plus 4, if I add them up, they should equal to the missing number here is actually equal to 3 plus 4, which is 7. And then once I find the rule, I have to double check that it works for every number. And I move up here, and 11 is indeed 4 plus 7, and 7 plus 6 is indeed 13, so I'm confident that the rule works. And finally, I should apply that rule, 11 plus 13, right? So you can break it out as a 10 plus 1, and then another 10 plus 3. So that's 20, 10 plus 10 is 20, 1 plus 3 is 24. So that is our answer, 24. So this question requires a little bit of quick computations, but as you can see, you can break things like this. And most importantly, look through each row, you don't see anything, then look with relationships that connect the two rows, just like in the previous question when we do with relationship between the shaded and the sunny area. Okay, we actually have only one minute left, so I will quickly go through the summary of what we did today. As you can see, we start the class with a bunch of questions of patterns that surrounded us in nature. So try to find new patterns around you, could be shapes, could be color, and could be patterns of activities that you do during the day. And then when you solve a pattern problem, remember first to find out, we try to define the pattern. You can use equations, but at this point, it's best to describe everything in words. And then take notes of the draw of the pictures, as you can imagine the different steps, you may draw additional pictures or arrows or anything that helps you solve the question right on the paper itself, and try to predict the next steps, right, to see if your rules works. And then at the end of the day, you might want to double check everything, just to make sure that your rule works all the way from the beginning to the end. And just for some of you that who came late, I just want to quickly summarize what we thought would be the best way to make the best out the webinars, please print the handouts ahead of time. Andrew, could you send it once more through the chat, the link to the handout is on the website, you can have it here, and work on the problems actively during class and practicing the pose and do additional practice with your parents. And finally, if you have time, this is our first class, if you have time from previous year papers. So additional practice problem, you can go to Maths Kangaroo website, there are a lot of free questions, there are a lot of free questions and some, and then they also have a questions and solution for previous year, some full papers and not you need to you need to buy, I'm sorry, it's not free, but you have a 50% discount if you register for for Maths Kangaroo competitions. And the bundle books is actually a very good choice, because it's a good discount, and you have many years of exam with problems and solutions. So you can work together with your parents. And finally, Andrew, if you have time, we appreciate you couldn't just give it a quick survey. So we know how we do and we can adjust it in the next class, like the pace of the class and the material. That would be very much appreciated. And we hope to see you live, I would really appreciate if you could attend the webinars live next week. But as always, the recordings are available for reviewing at your convenient times. Okay, so thank you everyone and we'll see you next week.

Maths Kangaroo

webinar

mathematical patterns

visual aids

interactive questions

number patterns

shape patterns

skip counting

numerical sequences

educational resources