Welcome. My name is Dr. Sarah Sagee. If you were not here last week, thank you for joining us this past week. I'm glad that you'll get to do most of the lessons. Last week was an introductory lesson. This week we will start our typical pattern, which is we do one topic and we do a whole bunch of problems solving it with that tool, with that trick you might say. This week our lesson is on finding a pattern. You will stay muted and you'll keep your video off so that when you go back, if you wanted to see this recording, you wouldn't happen to see another student in the video. Let's see. What else? I will have polls for some of the questions. I won't have polls for all of the questions because I want to make sure that we have enough time to solve more problems instead of answering just polls. You'd rather see more problems. The first few problems today, I will probably rush through a little bit so that you have more time for the problems at the end of the lesson. That is typical for Math Kangaroo. When we do our contest, we have three point questions at the beginning of the contest, four point questions in the middle, and five point questions at the end. You will need to make sure you leave enough time for the ending of a Math Kangaroo contest. Let's try finding a pattern. Let's do our warm-up problem. Adam spent four days preparing for a test. On the first day, he solved two problems. On each consecutive day, consecutive meaning in a row, he solved three times as many problems as the day before. How many problems did Adam solve altogether in preparing for the test? When we see the word altogether, we know we have to add them up, right? It means a sum. I have a poll for this one. I am going to launch it right away. You'll notice that when I launch the poll, the problem is right at the top of the poll. If it covers up your screen, you can still see the problem. If it covers up your screen and you don't like that, you could grab it by the corner and move it away. You can close it. You can do what you need to do. You can go ahead and submit your answer choice when I have quite a few students answering. I have some students already doing it. Another way, if you don't want to answer, if I don't have a poll, is to use the chat. If you'll open up your chat, you could put your answers there, especially problems where I don't have a poll. That's a great way for me to know how the students are doing. Okay, anybody else want to put their answer into the poll? I'm going to end it in a few seconds and share the results with you. The poll is anonymous. I cannot tell which students have answered or who has given a correct answer or an incorrect answer, and no one else can tell either. So you might as well take a guess, right? Also, on our Math Kangaroo contest, we do not subtract anything if you get an answer wrong, so it is better to guess than to leave something blank. Because if you guess, there's a chance you could be right. If you leave it blank, there's no chance, right? So most of you have said you think the answer is 80. Let's take a look and see if that's correct. So you notice I won't poll every question just because of the amount of time that it takes. All right. Where is that? So if we take a look at this, on the first, we have four days to prepare for the test. On the first day, he solved two problems, and each consecutive day was times three, three times. So that would be six. Then I have to multiply six times three is 18. And then I have to multiply 18 times three is 54. And now I need all together, so I have to do the sum. I need the total. So two plus six is eight. Eight plus 18 is 26. 26 plus 54, the four plus the six is 10. Five plus two is 70. I get 80. So my correct answer would be D, which is what most of you had said when you answered in the poll. So very good job. Okay. I am going to back up just a minute here. When we do math kangaroo problems, we need to make sure that we are following some steps. One is to understand the problem. That might mean reading and even rereading our problem. You'll sometimes find me underlining, just like I did when I said understand the problem. Then we're going to have to use a plan on how to solve the problem. Well, today you know that our topic is patterns. So you might try to see if there's a pattern that you can find. When you carry out your plan, be very careful. Go ahead and use your notepaper. Use that notepaper. Use your pencil. Make sure you write it out and that you can follow along and that you're able to check your answers. If you write really messy and you scribble along the margins and say, oh, I wanted to squeeze something back in over here and I'm writing really messy and I get a little disorganized, I can't possibly check my work very efficiently. So try to work carefully, neatly, and you'll do really well if you're going step by step. Okay. So today we're doing the find a pattern. You'll see that there's going to be one topic each week until we are all done. So patterns occur everywhere. Not everything is a pattern, but they happen everywhere. For example, the kangaroos in my background are in a pattern. If I erased one of them and I asked you, where does the missing kangaroo belong? You could put it back there, right? If you're wearing a striped shirt today, you could know which stripe comes next, right? Maybe you have a tile floor. There's definitely a pattern in your tile floor. There are patterns in the days of the week. There are patterns if you have a music lesson one day a week. You might have a pattern for how you get ready for school. You might do things in the same order every day when you get ready for school. There might be patterns of numbers, like the one we just did where we had numbers. A really common one is like 2, 4, 6, 8. Or 2, 4, 8, 16, 32. Those are both patterns based on two, but I did different things to them, right? Once I added and when I multiplied. You can do division. There's a lot of different ways. So sometimes math kangaroo problems might seem very, very complicated or hard to solve. Sometimes we have what we call complex problems. But if you can think about it and do a simpler problem, you might learn a pattern that lets you do the harder problem. So think about that. There might be a way to do that. Okay. This picture shows a sketch of a castle. Which of the lines below is the castle? This picture shows a sketch of a castle. Which of the lines below is not part of the sketch? You might say, how is this a pattern? But what this is, is this is trying to identify pieces of a larger puzzle piece. And when we identify pieces, that helps us break things down. That's kind of one of the steps in maybe finding a pattern is to break it down. So you can go ahead and put those answers in the chat. There is no poll for this one, because I don't know how to put these little pictures as choices in a poll. All right, lots and lots of you are answering. Remember, I can't get to everybody with my little thumbs up and stuff, so you might see me thumbs up on some problems and not on others. I do the best I can, right? So yes, most of you that I've seen have put that the answer is C, and we have to be careful because it says not part of the sketch. So one of the things I like to do is I like to say, okay, this little piece, I can see that that's right here. The next piece is a diagonal and a curve. I can see that diagonal and comes up to a curve there. Piece D is a down, across, and up. I see down, across, and up here. And then finally, this piece on E is two bumps and a stick, two bumps and a stick. And the one that is not there, sorry about that, guys. What happened there? Yeah, the one that is not there is C, okay? I don't know why my annotation begins out working and then quits partway and makes me click something else, so I apologize. The colors in this picture on the right are switched. What do you think switched would mean? That's part of understanding this problem. Then the picture is rotated. What does rotated mean? What does the picture look like now? Remember, we're gonna switch. So switching, I'm gonna have to guess, means black becomes yellow and yellow becomes black. And then rotated is when you turn something either counterclockwise or clockwise, you start changing it that way. Very good. I have very, very many of you giving correct answers. The correct answer is E. Sorry, I've...there we go. All right, so let's take a look at why the correct answer would be E. For me, this one here that looks like a donut is a really good kind of landmark. I can kind of anchor everything around looking at that donut. So I'm going to ask students to stop annotating. The donut would definitely look like this when I switch it, right? But what we can see is on something like D, I'm going to count up how many circles I have in my original picture. I have 1, 2, 3, 4, 5, 6, 7. I have 7 circles. So my rotation also needs 7 circles, so D does not have enough. B doesn't have that big bullseye donut. If I look at A, in my original picture, I don't have this big dot right next to my donut, do I? So that doesn't match. And let's see, on C, I do have 1, 2, 3, 4, 5, 6, 7. I do have the correct number. Things are reversed. What seems wrong with this picture is right here, I have quite a bit of space in the original picture, and I have no space in C. So C cannot be correct. So therefore, we have the correct answer is E, and that is what most of you put in the chat. Really good. All right. All right. In a certain ancient country, numbers 1, 10, and 60 were expressed with the following symbols. People were writing down other numbers using these symbols. For example, the number 22 was written as, this is the symbol for 10. I'm sorry, it's not annotating, is it? This is the symbol for 10. This is the symbol for 1, and this is the symbol for 1. So therefore, when we put those together, we get 24. Oh, 22. Right. The next one is the 4. I got ahead of myself. We want to know, how do you write the number 124 using the symbols given? Again, I do not have a poll, because I couldn't put those little symbols into the poll. But the chat is open for you. Is that enough time? Remember, if you felt rushed and you needed more time on a problem, you'll be able to have the recording of this webinar and you can look at the problems again and have as much time as you need to by using the recordings. Okay, so when I look at this problem, there are a couple of ways to do it. I could test all of these answers and I can see which one of these is correct. So I could look at it and I could say, okay, on A, this is 10, this is 60, this is 60, and this is 41. So this gives me 134 that is not correct. So that won't work. And I can test all of them that way. That is a perfectly acceptable, good, valid, effective method. The other method is to go ahead and say, how would be the most efficient way to do this? Well, I know the biggest grouping is 60. So in 124, I have a 60 plus a 60 plus four of the ones, right? So the 60 is a colored in upside down triangle. And then the four ones is the long skinny not colored triangle. And that does look like the answer choice, E. Very good. Okay, if the numbers in each of the two rows have the same sum, what does same sum mean? Same sum, what is the value of star? Here's the star. So we need to know what value to put into this box so that I have the same sum in the rows. Rows go across this way, right? So there's a sum for this row and I'm gonna think about the sum for this row. Okay, so you should be able to see the picture in the poll and you should be able to put in your answer. I'll give you a hint, we are doing patterns, right? So is there something that you see that happens more than once that would make a pattern? All right, I'll end the poll and share the results. This kind of agrees with what I was seeing in the chat as well, is that there were two most common answers, 99 or 209. Okay, so let's take a look and see what is Math Kangaroo asking us to do? Well, let's see if there's a pattern. What is the difference between one and 11? Well, obviously we've added 10, right? We have done that for each one of these, we've added 10. Okay, that's pretty easy to identify. But what do we really need to do? We need to find the sum in this row and the sum in the bottom row and have them be the same. So for example, if I got one piece of candy and then two pieces of candy and three pieces of candy, but my brother got 11 and 12 and 13, and I said, mom, that's not fair. I wanna have the same amount of candy as my brother, then I would need to get a lot of extra candy at the end to make up the difference, right? But my brother wouldn't need so many pieces of candy because he's already gotten more than I did. How many more did he get? Well, for 10 days, he got 10 more pieces than I did. So 10 times 10 is 100. Okay, so when we get to this day right here, the bottom or this column, the bottom row sum is already 100 more than the top. So I need to add 100 less. I need to add 199 minus the 100. And that gives me the correct answer of A, okay? When certain cube is painted with three colors so that every side of the cube is one of the colors and opposite sides are the same color, from which of the patterns below can this kind of cube be made? I want to go one more slide just to show what this is. These are nets, N-E-T-S, of three-dimensional shape and these can be folded. Let me show you. If I took the faces of a cube and I sliced them along the edges, kind of like wrapping paper on a present and I lay it down flat, do you see how I can get this net? And I could wrap it up on top. So think about wrapping the box, okay? So that will help us understand what the nets are. So these are the five different nets that I have for a cube. I want opposite sides to be the same color. Opposite sides are like the top and the bottom or the front and the back or the left and the right. Opposite includes the top bottom. The left right or the front back. Opposite sides never touch each other. That's a big hint for this problem. Opposite sides never touch each other. All right, I'll give you a moment to put your answer into the chat. Some of you are asking about a poll. This is the same situation where I couldn't put these nets into the poll. And I thought that students would want to see them. So there's no poll. Just use the chat. Okay. I think that the notes that I took should have been helpful. The opposite sides never touch, right? The top and the bottom of the cube never touch each other. If we start to fold these like we were wrapping a box with wrapping paper, you notice that these black sides would end up getting folded right next to each other. And also this gray side would end up touching that gray side. So A is an incorrect answer. Again, when I folded this side up, it would go right along this edge. So B is incorrect. You can see that also would happen with these grays. Same on C. These grays would end up touching each other. And also when I wrap the top of the box around, these blacks will touch each other. And even one more, those whites would touch each other. So in three ways, C is incorrect. Let's take a look at D. D looks more promising because I go gray, white, gray. But when I folded this, this black, this edge will end up touching one of the sides of this. Because this, you could think about this, if I call this the bottom, then this could be the right side. This would fold up to the top. And this would be the front. Sorry, I can't annotate as neatly as before because I had students annotating on the screens. So the correct answer is E. E is the one that's going to work here. All right. Adam and Tom are walking in the same direction around a circular table and counting chairs. So circular table. They begin their count with different chairs. Tom's 12th chair is Adam's third chair. Tom's 12th is Adam's third. And Tom's fifth is Adam's 18th. How many chairs are at the table? There is a poll for this question. You don't have to choose to do it this way. But one option is to draw a circle and to at least make some notes around your circle. I'll start that way. I'll give you a moment. And then I'll launch the poll for this one too. Okay, you might not see my drawing but hopefully you've made your own drawing if that's the method you're using. I'm going to launch the poll. Okay, most of you answered the poll not everybody and I know not everyone can see the poll so we'll close it and share here. This one I know was a trickier problem for most of you because it took longer to get your responses and that is fine. Like I said this is maybe a harder question, maybe it's worth more points and number 19 is worth five points on our math kangaroo contest. So you'll see that 41% so not even half of you think the answer is 22 and every answer got some votes. So let's take a look and see how we can solve this. So there are I've seen at least two good ways to solve this or maybe three. One way is to just draw. I guess that's the first way. So if this is Tom's number five chair you could draw in Tom's number six, seven, eight, nine, ten, eleven, and twelve. I guess I'm a little off but this could be like Tom's number six, seven, eight, nine, ten, and eleventh chair. Now we know that this chair is Adam's third chair and Adam has 18 chairs to here so we can draw in Adam's number four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, and eighteen. And now I can count them all up. So if I count them I'll start hopefully I think you can see it if I do it like this right. This is one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, twenty-one, twenty-two chairs. So I could correctly do that and I could get twenty-two chairs. Is there a way to do it without drawing all those little chairs? Yes. If this is Adam's third chair and this is Adam's 18th chair I could do 18 minus 3 is 15 chairs on this side. 15 chairs on this side. And I can do Tom's 5 and Tom's 12. I can do 12 minus 5 equals 7. Then there are 7 chairs on this side and 15 plus 7 equals 22. Another way to do it is to think that, I'll even change colors right, is to think that okay if this is Tom's 12th chair but Adam's third chair then Adam started counting. This would be Adam's number two chair and this would be Adam's number one chair. So Adam started counting at Tom's number ten chair. Okay so if that's the case then we know that Tom counted a certain number he counted in between 5 and 10 is 5 right. So if we take this 18 and we add up these additional 1, 2, 3, 4 chairs then we get the 18 plus 4 equals 22. And I had a student successfully show me that if you took Tom's 12 minus the 3, they started counting with a difference of 9 between their chairs. And if you take Adam's 18 minus Tom's 12, 18 minus 5, you would get 13. And then your 9 plus 13 is also 22. So I think I've shown at least four different ways to get to the correct answer. Why would I show you four different ways? Because each one of you may have done that way or even a fifth way if you knew a fifth way. When we do math kangaroo problems I want to emphasize that the way I show it may not be exactly the way you do it and that is completely appropriate and completely fine. We can check our work we can both be correct. All right sometimes it's really good to have more than one method because you can use the second method to check if your answer was correct with the first method. So keep that in mind. A certain vase contains four flowers. One red, one yellow, and one red, blue, yellow, and one white. Kaya the bee sat on every flower in the bouquet only once. She started with the red. That's important. And she did not fly directly from the yellow to the white. How many ways could Kaya sit on all of the flowers? I do have a poll and I'll be able to launch it in just a minute. Just want to make sure everyone digests the important facts here. There are four different colored flowers. You must start on red and you cannot go from yellow to white. Okay, how's it going? I see answers in the chat and the answers in the poll. Close the poll in about 10 seconds, so if you want to put in a last-minute guess go ahead and do that. All right, here's the poll for everybody. This is another one where everything got at least one answer. I think we know it's not one, that might've been a misclick, but two and three were some answers that I saw in the chat. Four got the most votes with 43% of you saying that the answer is four. Six got just a few. I can understand why six, but we can eliminate six, I'll show you why. Okay, so you can see I started, I'm doing what's called a tree diagram. From the red, I can go to any color flower I want, right? I can go to yellow, blue, or white. Obviously I couldn't use white to draw white. So from the yellow, I cannot go to the white. So if I try to draw the white here, that is not gonna work. So I cannot go down that path. Okay, but from yellow, I could definitely go to blue, and then from blue, I can go to white. So this is one possibility, so there's one. From the blue flower, I could go to the white flower, and from the white flower, I can go to the yellow flower. Right, so I can do the yellow. I can go from blue to yellow, but from yellow, I cannot go to the white, can I? That's against the rule, yellow to white, I can't do, so this one doesn't work. So, so far I have one, two choices. If I do white as the second color, I can go to blue. I'll be able to go to blue as the fourth flower here, and I can go to red as the second flower or as the fourth flower, because there were no rules about how to go from blue to red. So that's one, two, three, four options that I can use for the flowers. Now, some of you might have said six if you were just looking at not eliminating these possibilities where you go from yellow to white. So that might be something that happened to a few of you is you didn't read this part of the problem that you can't go from yellow to white. Okay, be very, very careful that you read all of the details of every problem. That's why I said sometimes you have to reread it. You might have to read my problems more than once. If you count the small white squares in the sequence of big squares shown in the pictures below, you will get the numbers listed. That's very confusing, isn't it? We'll talk about it in a minute. If we continue this pattern, how many white squares will there be in the next big square? So this sentence was confusing. If you count the small white squares in the big square, so here's a bigger square. And if you count, there's nine total squares and eight of them are white. So that's why it is listing eight here because there are eight white. Here I have more than 21 squares, but four of them are gray and 21 are white. That's what they're saying. They're saying what happens if I draw an even bigger square, the next one in the sequence, in the next one, how many white squares will there be? Not how many total squares or how many gray, but how many white. Some of you are answering in the chat. I'll put up the poll and I think you should still be able to see the picture in the poll. All right, this is an interesting one. I'm going to share the results of the poll. So one of the things I'm going to say is that of the students who answered, a huge percentage of you got it correct. So 83% got it correct, but not every student replied to the poll. So I'm wondering if maybe once you see the pattern, it becomes obvious and you get it, but until you see it, it takes a minute. It's a struggle. And that's OK, because we're here to explore and to try to find them together. So what I'm going to do is I'm going to just write some notes here. I see that this is a 3 by 3 square. So that's a total of 9 squares. I'm going to just put 9 here as the total. This is 1, 2, 3, 4, 5. This is a 5 by 5 square, which gives me a total of 25. And this is a 1, 2, 3, 4, 5, 6, 7 by 7 square, which gave me a total of 49. Now, if I want to count the gray squares, this is just 1. This is a 2 by 2. There's 1, 2 here and 1, 2 here. This is a 2 by 2 if I'm counting the gray squares. So 25 minus 4 is 21, right? 25 minus 4 gives me 21. Here I have 1, 2, 3 by 1, 2, 3, a 3 by 3. I started to write the next piece already. It's a 3 by 3, which is 9. So if I take my 49, a little messy minus 9, I get the 40. So what will I have here? If I follow this pattern, I went 3 by 3, 5 by 5, 7 by 7, following the pattern, that makes this a 9 by 9, which gives me 81 squares. And if I follow the pattern of the gray squares, a 1 by 1, a 2 by 2, a 3 by 3, this will be a 4 by 4 for the gray, which equals 16. Sorry, that's a little hard to read, isn't it? I'll write it over here, 16. So if I take the 81 minus 16, I have to regroup. So I get 11 minus 6 is 5 and 7 minus 1 is 6. I do get answer choice C, 65, which is what most of you had said. So that was great. And if you didn't see the pattern, hopefully now that I've shown and drawn out the pattern, it's very, very clear. And that's all that matters, is that when we finish the problem that it's clear to you. And you say, OK, next time I'm going to look for something like that. OK. We have some bonus problems today. Sophie makes a row of 10 houses with matchsticks. In the picture, you can see the beginning of the row. How many matchsticks does Sophie need all together? So not how many more matchsticks. Sometimes we ask you how many more matchsticks. But this is how many matchsticks all together. And here it is, a row of 10 houses. So far, I have 1, 2, 3, 4 houses. I need to make 10. There is no poll for this bonus question. So just put your answer into the chat. Very nice answers in the chat, quite a few of you. So one of the ways I like to do these problems is I like to see how many, if there's a starting kind of position that you have to have to build a house, and then how many matchsticks, in this case, you need to add for each additional unit, right? So when I do that, I see this matchstick on the left-hand side I have to start with. So that is one. Then every time I want to add to that to make a house, I have to add one, two, three, four, five matchsticks. You can see for the next house, I would add again five matchsticks, one, two, three, four, five. Every time I want to add another house, I add one, two, three, four, five matchsticks. So one way to think about this is I needed one matchstick to start, and then for every other house, I need five matchsticks, and I have 10 houses that I want to draw. So this is one plus 50, and that gives me 51. Another way to do it is to say, okay, for the first house, and this is perfectly correct. Remember I said you could think about it in different ways. For the first house, I needed six matchsticks, and then for each other house, because I don't need the left-hand wall, I only need five. So you can do six for the first one plus five times nine. That is also mathematically going to get us the exact same answer of 51. Okay, I think we have two minutes. We can probably do this last question in two minutes. There are 10 ducks. Five of these ducks lay an egg every day. Five lay an egg every day. Five lay an egg every other day. Every other day. How many eggs do the 10 ducks lay in a period of 10 days? So I'm going to start, I think, solving it. Oh, I do have a poll. I can launch the poll right now. Very good. The majority of you are sharing that you think the answer is 75. That is correct. We can think about it the way I've typed it out here. If 5 ducks are going to lay 1 egg for 10 days, that 5 times 10 is 50. The other 5 ducks are going to lay it every other day, so they're only going to lay 5 times. So 5 ducks times 5 eggs is 25, and therefore when I add that, my total is indeed 75 eggs. So very nicely done. So, when do you want to use a pattern? When you see a complicated problem and you might want to break it up into something simpler. There are many, many kinds of patterns, and we've sought different ways to detect them. They can be in drawings, they can be in the numbers, they can be in days of the week. There's a lot of different ways to do patterns. You will see there's some optional homework that you might want to try. If you go into your Math Kangaroo account, try some past contest problems. There's some optional homework you can get to practice working with patterns. I hope you liked today's lesson. We had some challenging problems, and some really great participation from all of you in the chat and the polls. I hope I'll see you again next week, same time next week. Bye, everybody.

pattern recognition

Math Kangaroo

problem-solving

interactive learning

numerical patterns

visual patterns

tree diagrams

Dr. Sarah Sagee

math contests

engagement techniques