Okay, it's time to begin. I know we'll have several more students joining us because we don't have everyone here yet, so I'm going to let you have the warm-up problem. I'll go ahead and I'll put that on the screen. You can work on this problem while we wait for students to arrive. I'll give it a quick read. The picture shows three arrows that are flying and nine balloons that can't move. When an arrow hits a balloon, the balloon pops and the arrow keeps flying in the same direction. How many balloons will be hit by the flying arrows? And if you can hear me, go ahead and just send me a quick little chat that you can hear me okay. I just want to check my mic. Thank you. I appreciate that helps that. I know you can all hear me. Remember, we're going to keep our videos off. Because the meetings recorded, and we want to protect your privacy. We'll have several correct answers so we'll go ahead and we'll explain this problem and then we'll go back to the intro and I'll start right at the beginning, okay? So today's theme is drawing. Drawing pictures to help solve problems. So as you can see there's a picture here and we can extend it by doing some of our own drawing. So what I'm going to do, I don't know what color to use, I guess I'll choose purple. So it says that map kangaroo questions are very frequently word reading comprehension problems. So some important things that we have here is that the arrows are going to fly in the same direction. They don't ever like turn or twist or veer. And even if an arrow hits a balloon, the arrow keeps going. So then all we have to do is say this arrow goes through up, it goes through one, two balloons. This arrow also goes through two balloons so that makes a total of four. And this arrow will go through another two balloons which is a total of six. So everyone who put that number six or D in the chat, you are absolutely correct. Give me just a second. Those of you who are here early know that I'm dealing with a puppy in the room. And I'm sorry it's not professional but that's what happens sometimes. Okay. So let's go one slide back. See if it lets me do this. It should. Okay. So last week was our first webinar and I introduced myself. I'm Dr. Sarah Segee. I'm coming from San Diego, California. So welcome to San Diego. Welcome wherever you are. If you were here last week, thank you for coming back. Our TA is Shourya Vyas but he is not here today. So you get me and if I stop to take a little water break, please excuse me, an hour is a long time to be talking. So last week we did a brief survey. We talked about different types of math kangaroo problems and we talked a lot about the fact that there might be some visual and spatial problems. We'll see some of those. But we're going to spend our weeks looking at how you can solve various types of math kangaroo problems. We're going to use different strategies. We call it building a toolkit. And this week we're going to focus on drawing. That does not mean that drawing is the only way to solve these problems. Very frequently there's even more than one way to draw them. So I might show you one or two methods and I might say that there's other ways to do it and we might say those work very, very well for some students and this is just one option. So please keep that in mind. I'm not saying you need to draw for every problem you come to. You're going to have 24 problems on your math kangaroo contest and I'm not expecting you to draw all 24 of them. This is just one way you might try. I find this is particularly helpful when I don't know exactly what the problem is saying. Sometimes when I draw a little bit of a picture or a diagram, it helps me organize the word problem into what I'm actually trying to examine a little more closely. So you see this week is drawing a picture. Next week we'll talk about making lists. We'll go on to having a table. We'll use patterns. We will use guess and check during week six. Then we will move on to experimenting and actually acting out some steps of problems. Working backwards, that means starting at the end and going back to the beginning to find an initial state where you start. And then logic. Those are your favorite, at least one of my favorite types of problems, the logic problems. The math kangaroo contest is 24 problems at your level. So sometimes and frequently, more frequently on math kangaroo contests than on other contests, you will find that a problem is illustrated. But even if it is not illustrated, it can be helpful for you to draw your own picture or diagram. That's what I was saying. I need to read it and I need to make a picture of it. Some of us are very visual learners, and this can help you organize all the information that's presented to you. A lot of times if you have a multi-step problem, it can be helpful if you draw the steps. First I need to figure out how long this side is, and then I need to figure out what the perimeter of this problem is, and then I need to figure out what the area of the shape is. And so there might be several steps, and having a picture will help you keep it all in the correct place and not lose which step you're on, or maybe forget a number. It's very, very helpful. So I hope that you all have pen and paper ready, because we're going to be drawing along. If you are drawing something, my advice is to always use the simplest symbols that you can. Okay, if it's a rectangle, you might want to draw an actual rectangle. But if we're talking about something else, you can even just use a letter. If you're talking about a flower, you can use a circle for a flower. That's okay. If you're talking about a star, you might put a star. But you might be using, oh, we might know that we have things that are blue and things that are red. So I might use a B and an R for blue and red, okay? So just keep it in mind. Keep your symbols and things as simple as possible. You don't want to spend a lot of time drawing, but you want to be careful. Okay, so here is number one. In the picture, we see an island with a highly indented coastline and several frogs. How many of these frogs are sitting on the island? So indented means it has a part that goes in. So if I was to draw this shape, we might call that an indent. So we can see that there are some indents in this island. One of the interesting things is it doesn't tell us what is on the island, what is off the island. Let's see if you can figure that out. There's a hint in the picture, and if you see the hint in the picture, you can put that in the chat. You can also put your correct answer into the chat. And for those of you who do chat with me, you'll see I can get back to some of you, not all of you, and I'll try to make it so that I reach different students for different problems. So if you don't hear from me, it doesn't mean I'm ignoring you. It's just a matter of time. All right, so what I like to do with this problem, and I did see somebody correctly answer, what is the clue to tell you where is the island? It's the tree, right? A tree will not grow, a palm tree will not grow just in water. It has to grow on the island. Yes, and I've had a couple of students tell me that. So that's great. So then once you know that the tree is on the island, you can kind of color in the island parts, right? So I can see that this is continuous part of the island and that makes one frog. Here's two frogs. Notice this island is skinny, but that's still an island. And I can still color around these skinny parts. Oh, there's a frog that's in the water. That's okay, frogs can swim. Oh, and here's another frog. And then I have this island protruding up here as well for another frog. Island comes around and we see here's another. And then there's my final frog on the island. There are frogs off the island and that's okay, right? So let's count my frogs on the island. One, two, three, four, five, and six. Five and six. There were six frogs on the island. Tricky problem. Sometimes the coloring gets a little tricky. This is a problem where we have absolutely given you the picture, right? All you have to do is maybe do a little coloring with the picture or tracing around with your finger or whatever you need to do, okay? So hope you like that problem. This is one of our three point problems, right? So there are three points, four points, and five points. This was a three point problem. Here's another three point problem. And I do have a poll for this question. So you can put your answers in the chat or put them in the poll, but I'll give you a moment to solve it before I launch it so everyone can see. Denise fired a silver rocket and a gold rocket at the same time. The rockets exploded into 20 stars in total. The gold rocket exploded into six more stars than the silver one. How many stars did the gold rocket explode into? Why am I underlining? It's part of the critical reading of the problem. Step one is to read and understand your problem. And it's important to know that I need to know how many stars there are that are gold, gold stars. Now's your quiet working time. Seeing a lot of you answering, so I will launch the poll now. It will show the same question. Remember, if that poll gets in your way or you can't read the whole question on it, you can close it. But it does help when we can see how everyone does together. And they are anonymous, so no one's going to know which answer you put down. Or clicked, I guess. Anybody else want to put a response in the poll? Most of you have answered, but a few have not yet. Okay, we'll end it here. I'll share the results. We do have a clear majority answer. 71% of you have think that the answer is 13. I'll tell you now that is correct, but we have some other responses. Maybe the response is for three. Those are how many silver stars there are, but we wanted specifically gold stars. And I can understand why some of you have said 16. Let's take a look. If that poll does not close for you all by itself, you might have to click on the little spot that says close or the little X so you can return to your main screen. Right, so there are several ways to draw this. The simplest, well, I guess the basic level of drawing it is to just say, okay, you know what? It says that there are six more gold stars than silver ones. So if I draw six Gs for gold, that's the six extra gold stars I have. And then I know the rest are gonna be even gold and silver, right? So I can do a gold and a silver. This gives me six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. Cause I have to make sure that I draw 20 stars, right? So there, I've drawn 20 stars. I gave six more gold and then I can just count. So then I have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 13 gold. And I have seven silver. And the answer asks for the gold. So that is a very correct way to solve the problem. Is it the only way? Absolutely not. Can you do it without drawing a picture at all? Definitely you can. I've had students tell me, okay, well, if there's 20 stars, if there are 20 stars, then half of that would be 10 silver and 10 gold. But I know it's not half and half. I know there's more gold. So you might do our guess and check method, which is you start moving one of them over to the gold. You might move another one over to the gold. You might move another one over to the gold until you get a difference of six. And that could be a way to solve this problem without drawing any sort of picture. You could use algebra. If you're comfortable using algebra, you can do that. You can also do something called a bar diagram. And there's a couple of ways to do that, but here's one. You can think of your bar diagram almost like a number line where this is 20 stars altogether, right? And I know that the last six of them are gold. And then I know that the other is split half and half with half being silver. And another amount that's silver and gold is the same amount, right? So I could call that like an X if you were using some variables, but I have two of the silvers, two times the silver plus six gold. So then I say, okay, two times however many silver stars I have plus six gold stars I have equals 20. If I subtract the six from both sides, then I have two times silver equals 14. So that means that, oh, I could divide by two, but I can just say two times what equals 14? Two times seven equals 14. So I would have seven silver stars and then 20 minus seven equals 13 gold stars. So that's a way to do it with a bar diagram. Just one of the ways you could solve this problem. I have some students tell me they really like polls. I have polls for the next few questions. I don't do them for every question just because they take up some time that I'd rather be explaining different methods to you. What is the smallest number of children in a family where each child has at least one brother and at least one sister? I will launch a poll right away because I think you'll be able to read the question in the poll. Well, you can't see it, but for a moment there we had a tie among two different answers. So keep putting your replies into the poll, we're going to see if we get one that's ahead. I think that's about everybody. Now, this is very interesting. We have two answers that are coming up pretty close to each other, either three children or four children in the family. And it doesn't show up on my tablet, but on my screen I have that a few of you answered that there are six children in the family. C, the most likely answer is three or four. Let's take a look and figure it out. Coincidentally, this is the number of children that are in my family, my children. So let's see. I'm going to use S for sister. Remember, I always keep it really simple and B for brother. Well, in my family, there was a girl first, a sister. And that poor sister has no sisters or brothers, right? She's an only child until a little baby brother was born. Now this sister has a brother and this brother has a sister but they're still missing something. So I'll do it in the order it happened in my family. You can do it in any order. So this brother has a sister, but does not have a brother. And the next child that was born in my house was a boy. So now these two brothers have a brother and they have a sister. So that's great, except that this sister has no sisters. Right? There's no sister. No one can get her clothes when she outgrows them or her toys or it doesn't matter. You can have brothers or sisters. It's nice to have any kinds of siblings if you have them. But in order to answer the question and for each child to have both a brother and a sister, you would need to have at least two boys and two girls. That's four children all together. So the answer is four. All right, here's one that I recommend you draw the shape. It's a geometry problem and it's a rectangle. So that's easy. One side of a rectangle is eight centimeters long while the other is half as long. A square has the same perimeter as the rectangle. What is the length of the side of the square? So is there anybody, you can put it anonymously in the chat if you'd like, who doesn't know what perimeter means? And that's fine if you don't because that gives me a job to do is explain it. So it tells me, I'm gonna start drawing and then I'll stop to give you some thinking time. It tells me that one side of a rectangle is eight centimeters and the other is half as long. So this has to be half of eight and that equals four, says their centimeters. And then what is perimeter? Yes, I have some students tell me that they know what perimeter is. I do have a poll for this question but let me draw a little bit first. So perimeter is the length along all of the sides added together. So perimeter equals the length of all the sides. It's a sum, right? You guys know what a sum is, right? And then you might also want to draw a square. A square has all of its sides the same, right? So this is whatever side length it is, I'll call it X. They're all the same X. You might also see in geometry that people will use a line like this to indicate that segments are the same length. I will stop there. Hopefully you have some paper, you've drawn this along with me and I'll launch the poll. This is a good group answering the polls and replying in chat. That lets me know that I've given you enough time. That's one of the big reasons I do it. All right, so I'll end the poll. Most of you have answered. If you haven't answered, you still can be working on it. It's okay. So most of you, almost 80% have said the answer is six centimeters is the length of the side of the square. Okay, so you see, most of you have said six. You can go ahead and close that poll if it doesn't close up for you. All right, so let's take a look. We discussed that the perimeter is the distance around the outside of any shape. It doesn't have to be a square, a rectangle, any shape. The perimeter is the distance around the outside. It can have curves. It can have 25 sides. It doesn't matter. All right, so in order to do this on the rectangle, the perimeter equals the sum. So it's eight plus four. And then we know the opposite sides have the same dimensions. So we have another eight and another four plus eight plus four. If you add that all together, you are going to get 24, right? Because eight plus four is 12, plus eight is 20, plus four. And you can multiply. You can do it this way if you like. You can do eight plus four times two. That's okay. No problem using our distributive property. So now we know that here, the perimeter equals X plus X plus X plus X. Or you should know that repeated addition is the same as multiplication. So it's three times some number X. And this has to equal the same 24. So, oh, my mistake. And if you caught me, good for you. I like it when people correct me politely. It's four times this mystery number X, the side length. And four times what equals 24? Four times six equals 24. So you can have typos even when you're writing something by hand. So if you solved a problem and you look at your answer and you're thinking it's not quite right, you could have done what I just did, which was to make a small mistake that you can correct. So always double check your work. That's step four in our process is to check your answers. All right, I'm gonna skip. I don't have polls for the next few questions, but you may always put your answer in the chat. Once, oh, I need the next problem. There we go. There are eight flowers on a rosebush. Some butterflies and some dragonflies are sitting on the flowers. So we have eight of them and there's no more than one insect on each flower. So no more than means it can be one or it can also be something. Can't be two. More than half of the flowers have an insect on them. The number of butterflies on the flower is twice the number of dragonflies. So butterflies is twice the dragonflies. How many butterflies are sitting on the flowers? So make sure you answer butterflies, not total bugs, not dragonflies, butterflies. I'll give you a minute. I think this is a pretty interesting question because there's a lot we have to read and put together to get the full picture. And this is, you don't need to draw it, but when you have so many different pieces of information, sometimes putting them down on the paper will help you to organize all that information. So we know that we have eight flowers, so I'm gonna, can just do circles for the flowers. Sometimes I just do little lines, so little lines would work too, but two circles for my flowers. I'm certainly not going to draw anything fancy. We know that at least half of them have an insect on them, so half is four, right? That's the halfway line, so we have the four. Okay, so we know that there are two butterflies for every dragonflower. The number of butterflies is twice the number of dragonflies, so if I have two butterflies, I only get one dragonfly. But I haven't colored, I haven't put an insect on half of the flowers yet, right? So I have to do that again. I have to do two butterflies and one dragonfly. Now if I keep going with two butterflies, I won't have a flower for the last dragonfly, will I? So then I won't be at the two to one ratio. So this is where I have to stop, because I, like I said, I could do butterfly, butterfly, but then there's no dragonfly space, and then I would have one, two, three, four, five, six butterflies, and only two dragonflies. That's three butterflies per dragonfly, it doesn't work. So I have to use the ones that are on, so you can see that I have four butterflies. Now all that wording that was very confusing ends up being pretty simple when you see a small picture of it. Okay, here's another geometry problem. It's just a little equilateral triangle, that's not so bad. Joining the midpoints of the sides of the triangle drawing, of the sides of the triangle drawing, we obtain a smaller triangle. Let's just digest that first sentence, drawing, joining the midpoints. Does anyone want to put into the chat what they think a midpoint is? No guesses? What is the middle of something? The middle is where half is on one side and half is on the other side. So if I was drawing midpoints of the sides, maybe I'll use a brighter color, if I was drawing midpoints of the sides it would be pretty much there, right? The midpoints of the sides. And now I want to join them. Joining them usually means draw a straight line. So did that work? Did I follow the instructions? Joining the midpoints of the sides of the triangle in the drawing, we obtain a smaller triangle. Now imagine I did it perfectly and all the lines are parallel and straight and it's exactly perfect, okay? When you're drawing on a contest will it be exactly perfect? No. But do you get the idea? Of course. It says repeat this one more time. Repeat this one more time. What does repeat mean? It means do it again. So I've got to take the midpoints of the smaller circle and join those. Now I have the smaller, the smallest resulting triangle. This is the smallest resulting triangle right here. Now it's asking how many triangles that purple size will fit in the original drawing. The original meaning the black triangle. I'll give you a minute. Now that it's drawn it should go much easier. Good work. I'm seeing a lot of correct answers, so that's great. So this triangle has a lot of symmetry. There's a lot of things we can look at that are the same, right? So because we use the midpoints, the size of this top triangle, this red triangle, and these two side triangles are all the same. So the green are equal triangles, right? So if I look at the center one, where I drew the purple, I can see that I can draw other purple triangles along the sides there too, right? And so all together, that would give me one, two, three, four purple triangles in the center green triangle. Then since I have four of those green triangles, I have four times four equals sixteen of the small triangles. So the correct answer is E16. If you need to, you can keep drawing. There's no problem if you want to keep drawing triangles and then count them off. That's okay as well. So the way I might think about multiplying and realize they're all the same, even if you don't realize that, you can continue to draw because they want to know how many fit in the original shape. So you can fill the original shape with the small triangles and see how many that is. I hope you like the problem. All right, our next one, this is one of our five-point problems. I think this one's tricky. Maybe you guys are really bright and you'll catch on right away. That would be awesome. We can fill a certain barrel with water. If we use water from six small pitchers, three medium pitchers, and one large pitcher. So that goes together. I like to help students out with this one a little bit. So you can use, you can fill it with six small, three medium, and one large pitcher. Then this is a really important word, or another way to fill it. So they're giving you two ways to fill it. Two small pitchers, one medium, and three large pitchers. So that's an or. If we use only large pitchers, so here's a third way. They want you to try a third way. So this would be a third way to fill. How many of them do we need to fill the barrel? And I do have a poll for this, but I'm going to give you some working time before I launch the poll. Okay. This is a little bit of a tricky problem. I see some of you have it right, but some are asking for a little more hints. So let's start with a little bit of hints. So I'm going back to the part that I used the orange to make the parentheses around. It says that you can use six small pictures. So I'm going to draw small with S's. S for small. That's six small pictures. Three medium pictures. I'll use M for medium. And one large picture. So that will fill up my big barrel. The other way is the pink way. The pink way uses only two small pictures. One medium picture. And three large pictures. Now I drew it in a way that should help you figure this out. See if that helps. I think that helped some of you, so I'll launch the poll now. I think we'll have some correct responses. All right, does anyone else wanna add their answer in the poll? Now the poll is overwhelmingly correct. All right, let's take a look at your results. So it seems that drawing it out really helped everyone. Most of you get the correct answer is four large pictures. Let's take a look. So what we've noticed is that what I noticed, I hope you notice it too, is that this and this are the same, right? So we don't need to worry about those. Those are the same. But then what has changed? From the orange diagram, we've gone from having these smalls and mediums to the pink diagrams only having the larges. And one student correctly told me in the chat that that means that two smalls and one medium must equal one large. So two smalls and one medium must equal one large. If we want to go to having only larges, we know we need at least three because we had three in the pink picture, but we still have two smalls and one medium. And we know that two smalls and one medium is equal to one large. And counting that up, we see that it would take four large pictures to fill the barrel. So while it was, I agree with a lot of you that it's a lot of words to read it and try to understand without making some sort of sketch. And now remember, I said sketches can just be symbols, letters or symbols in this case. And that's perfectly, perfectly good. If you wanted to do little boxes, medium boxes and big boxes, you can. So any way you want to sketch it that works well for you is perfectly okay. All right, we have a bonus problem and I think we have some time to do it. It is a 20, number 23, it is a five pointer. I think you do want to get some paper out for this. There's going to be some arithmetic involved in this one, a little bit of subtraction. That's my big hint. A rectangular paper sheet measures 192 by 84 millimeters. You cut the sheet along just one straight line to get two parts, one of which is a square. Then you do the same with the non-square part of the sheet and so on. What is the length of the smallest, what is the length of the side of the smallest square you can get in this way? I know that's quite confusing, but think about it. If you have a rectangle and it says you are going to make one cut, and get two parts, one of which is a square. So if I cut it maybe around here, then this part will be a square and this part will be a rectangle. There's a wide variety of answers. Let's launch the poll and that way you'll get an idea about what each other is saying and then we'll solve it together before we have to finish up for the week. Okay, you have not all put down an answer. That's okay. This does take a little bit of time. I want to be able to work it out together. So I'll give about 30 more seconds if you wanna just put a guess in the poll. One thing about Math Kangaroo, there's no penalty for a wrong answer. We don't subtract anything off your score. So you're always better off to maybe eliminate a few choices and then guess. Okay, so I'll end the poll here. Hopefully you've made at least a guess because a guess could be right and a blank could never be right. So here we go. We have several different answers. Eight is a little bit ahead of the other choices, but we've had every choice selected because this is a tricky problem. I acknowledged it before I started, but I think it just requires some very careful subtraction. So let's see if we can do it together. Everybody here should know how to subtract, right? So I went ahead and on the next slide, I tried to draw this to scale. Okay, and I'm gonna get my notes because as you know, I'm human and I make mistakes when I subtract. So, all right. So if our first cut, our first cut is going to be this line here. And then we will get an 84 by 84 square. But we'll get this other rectangle. And what is the length of the rectangle? It's still 84 I, 84 millimeters tall. But what is the width? The width is gonna be 192 minus 84. And 109, it's 100, this whole length here is 108. Okay, so it says you do the same thing. You do the same with the non-square part. The non-square part is the rectangle. So I'm gonna make another cut. I still have 108 by 84, so I'm still gonna cut it here. And this will give me again, the 84 by 84 millimeter square. And then what do I have left? I have 108 minus 84. So this length here is 24. So now I have a rectangle along the edge that's 24 by 84. And according to the instructions, I'm supposed to cut squares, right? So I will cut here and make a 24 by 24 square. And then this remaining piece, this here will be 84 minus 24 gives me 60, right? So with a 60 by 24 piece, I can cut another 24 by 24, can't I? So then I'm gonna subtract 24 again and I get 36. Ah, well, this is 36 by 24, so I will cut it again to 24 by 24. And that leaves me a little piece here at the end, which is 24 across the bottom and 12 over here. And if I take that little 12 by 24 piece, I can cut it into two 12 by 12 squares. So I get at the end, I get two 12 by 12 squares. So the answer is C, the smallest square that I can get has a side length, side length, length of the side of 12 millimeters. So the biggest issue with this question is by drawing it carefully, I'm able to do my cuts and my subtractions without losing my place or getting confused, okay? So it wasn't that subtraction is so difficult for you, it's that there's a lot of steps. And the more steps there are, the more important it is that you remain organized. For any of you who are interested, this is a method for finding the pieces of the greatest common divider of different numbers. Is to use this method, this is an algorithm for finding the greatest common factor of 92 and 84. So it would be 12 would be the greatest factor that's common to both of them, okay? So that brings us to the end of our lesson that was on drawing. Did you find any of these problems particularly challenging other than the bonus one? Think the bonus one is the most challenging, but others were tricky as well, the filling the barrel with the pictures. Would you have been able to do that without the drawing strategy? Do you have another strategy that would work well for you? And if you do, that's fine, okay? If you had another strategy and you like it better and it works for you, then use that on the contest. But if you get to a problem and you are looking at it and you're like, I really don't know, I don't know, I don't know, then draw, try it, right? We're giving you tools that you can use, okay? So challenges for drawing is that sometimes it takes a little bit of time, or you might feel you can't draw that rectangle with the correct dimensions. It might be a little lopsided or something, that's okay. Keep them simple and do your best. You can use scratch paper on math kangaroo contests, it's no problem. I do recommend when you scratch paper that you use one problem here, one problem here, you keep it nice and neat, and you're not writing all along the edges and up the sides and in between other problems, because that makes it much harder to check your work and to keep yourself organized. I hope you liked the drawing lesson today. Next week, we're gonna work on making lists, what could happen and which one of them is correct and makes sense. So you'll have another tool after next week. Thanks for joining me this time. I'll see you all next Sunday, bye.

Math Kangaroo

problem-solving

visualization

drawing techniques

geometry

math strategies

active participation

logical deductions

math competitions

Dr. Sarah Segee