So, welcome to the Mapkangaroo webinar, Drawing the Picture or Diagram, and we have a couple of things to cover today. We have quite a few questions, so I'd like to get right into it. I am Dr. Sagi, and our assistant today is Maya. Your microphone will be muted during the entire session. That does not mean you won't interact with us. You may continue to ask questions in the chat, and we will have polls where you can put the answers to questions, and we can see as a group how you're doing if you're catching on, if you're getting the ideas. So please have your chat open, make sure you're doing that, and we're going to practice with those pop-up windows for the polls, okay? All the polls are anonymous, and your chats are just coming to the hosts right now. So no one can see your questions, so there's no reason to be embarrassed about asking anything you need help with, okay? I'm Dr. Sagi. I live in San Diego, so some of you had said good afternoon, but for me, it is good morning. I'm on Pacific Time. I have been teaching for Math Kangaroo for the past several months. I've been teaching a level 3-4 class. I have been coaching a math club where I live for over six years, and just a little bit about me, I have a dog. He's in the room right now. I have four children, and I actually teach karate and swimming, and I'm going to let our TA, Maya, introduce herself. Hi, I'm Maya. I'm in 10th grade, so I'm 15 years old, and I live in Champaign, Illinois. I've been participating in Math Kangaroo for six years, I think, since fourth grade, and I'm going to be the TA today, so if you have questions, you can ask me, and a little bit about me is that I like to read, and I also like sledding. I really like the snow. So thank you. Maya is a fabulous TA. She's going to be monitoring all those chats for me, because sometimes when I'm speaking, I can't read them as fast as I would like to, but Maya is there with us, okay? All right, so on this slide, we are going to actually open up those pop-up windows. We're going to do our first poll, and we are going to see if you can move the poll around to be able to answer the question, because this has been, some students find it a little distracting. When I pop up the poll, it might cover your screen. You can just use your mouse, grab the top of the poll window, and move it around and see if you can answer, what fruits do you see? So by moving, you see the questions are in the window, and you should also be able to answer them. Just move that window around so that you can see, what is the fruit in red, and what is the fruit in green? A very mathy question, isn't it? Some of you may not see a poll. If you haven't enabled it on your web browser, or if you're using an older version of Zoom, the poll may not pop up for you. If you do not see a pop-up poll, just put your answers in the chat. That way, we know that you're still understanding the questions, okay? So it's perfectly all right to put your answers into the chat if you don't see a pop-up window. Yeah. Okay, it seems like quite a few have answered the poll, and like I said, if you can't see the poll, feel free to put your answers in the chat. That will be another way that we know that you're following along with the lesson today. You should be able to move the poll just by putting your mouse on the top of it and clicking and sliding it around. Okay, so most of you were able to answer that poll. 77 of the students were able to answer it, so that's the majority of you. Like I said, if you don't see the poll, continue to use the chat. Okay, all right. Math Kangaroo likes to teach a four-step problem-solving method, and we'll try to remember to use all of these steps on our problems. The first and most important one is to understand the problem. Determine what the problem is asking. So we will always work through what is this question asking us before we pop up a poll, okay? Then you need to have a plan on how to solve the problem, and you know that the theme today is drawing a picture, so the plan is probably going to involve some sort of picture. You might be able to do it another way, but that is our theme today. You have to carefully carry out your plan. You have to complete your calculations, organize your thoughts and steps, make sure your picture is as accurate as it can be, and then when you do have an answer, a very important step is to look back and check your answer. Did you answer the correct question? Does your answer make sense? A lot of times the questions will ask... Math Kangaroo or other math contest questions ask you for a different thing than you think they're going to ask. They might ask you for the product of numbers or the sum of numbers, or they might ask it in a strange way, so make sure you actually be very careful and answer the correct questions, okay? All right, there's some vocabulary, but I'm going to focus on those as we get to the slides that use them. So one of the unique things about Math Kangaroo contests is they like to use a lot of illustrations. There's a lot of visual and spatial questions on Math Kangaroo contests, and there's also a lot of questions that you can use drawing as a way to solve problems. So you'll see a lot of pictures on Math Kangaroo, and you'll also sometimes find that if you draw out what you're reading, it will help you solve the question, okay? When they give you a picture, remember you can modify it. Another piece of advice is when you're drawing the picture, see number three, I want you to use simple symbols. So if it says draw, it's talking about a butterfly. You're not going to want to draw a fancy butterfly. You'll run out of time. Draw something very simple. Sometimes you can just draw a simple box. You can make little circles to count things out. You don't have to do fancy pictures. There is a question later on today that will actually be easier if you draw a little bit to scale, meaning having the right dimensions. I'll show you that when we get there, okay? Most of the pictures and diagrams that you'll be asked to draw today are going to be very, very simple objects where, again, use circles, use dots, or very simple shapes, triangles, rectangles, things I'm sure you know how to draw and you're very familiar. So here is our first practice problem. So the step one is to make sure we understand the question. 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? So I hope you understand the question. These arrows are moving from left to right. They're going to pop the balloons as they hit them. So do we have a plan for how we're going to determine how many balloons get popped? Well, one option is to modify our picture and to extend these arrows. Now how many balloons are popped by that first arrow? Let's see. I'm going to give you a minute to think about it. Extend all three arrows and determine how many balloons are popped. Some of you who are using the chat are getting the correct answer. Make sure that you extend all three arrows. I've only shown you one right now. Extend the other two as well. Make sure your answer makes sense. Double-check yourself. I'm doing the step where I carry out my plan. My plan was to extend the arrows. Does my answer make sense? I think the answer makes sense. Yes, I'm seeing a lot of correct answers in the chat. Let's go ahead and end that poll. Maya, do they see the results? Do they see the short answers? I'm not sure. I can't see the different answers. Okay. All right, let's move on to the next question. So this was a relatively simple one. You just have to extend those arrows and do some simple counting. There are eight flowers on a rosebush. Some butterflies and some dragonflies are sitting on the flowers. There's no more than one insect on each flower. More than half of the flowers have an insect on them. The number of butterflies on the flowers is twice the number of dragonflies on the flowers. How many butterflies are sitting on the flowers? Now, remember when I said we have to understand the question? There was a lot of information given in this question. The first piece of the question is that there are eight flowers. So you might want to do a drawing with eight flowers. Do you have to do fancy flowers? No. If you have the handout, you'll have some flowers. I provided those for you. But circles would be just as good as flowers for this, right? Don't get crazy on your drawing. It takes too much time. Now, the next part is we have butterflies and dragonflies sitting on those flowers. But no more than one insect on each flower. What does no more than one mean? You can have one. Can you have two? Two would be more than one. So you cannot have two insects on a flower. You can have either one or zero insects. Because you can have less, but not more. More than half of the flowers have an insect on them. So this part means you have either one or zero insects. And more than half of the flowers. What's half of eight? Right? So half of eight means that at least four. At least four flowers have an insect. And then it says the number of butterflies on the flower is twice the number of dragonflies. So there's a lot of information to process in this question. I'm going to go to the next slide where I've actually drawn those flowers. You'll see if you have the handout. Okay? Maya, that's the wrong poll. So I need to have flowers on at least four. I need to have insects on at least four of these. Right? And I have two butterflies for every dragonfly. See, it says the number of butterflies is twice the number of dragonflies. So how can I make sure I have twice as many butterflies? What if I start labeling butterfly, butterfly, dragonfly? Is that twice as many butterflies as dragonflies? So I can repeat this pattern, right? Can I do butterfly, butterfly, dragonfly again? Because I have to get past this line. I have to draw an insect on at least four flowers. And I can't continue the pattern because I don't have enough flowers to continue this pattern. So now how many butterflies are sitting on the flowers? You can answer in the poll or answer in the chat. Do you see how drawing this helped you? There was so much information to process. It was quite difficult until we actually started drawing the flowers and labeling them with insects. Then it became a much simpler problem. Did I draw fancy butterflies? No, you won't have time in a contest to draw fancy butterflies. A B and a D will be good enough. Okay. I think we're going to be able to move on to our next problem. So here's our next one. If you have the handout, that'll make it easier for you to work on this question. If not, it's pretty big up on your screen. You should be able to follow along anyway. 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 one of our vocabulary words was this word indented. You notice this island has a very strange shape. It's not just like some sort of rectangle or rounded up circle. This island has a lot of little ins and outs. Those are called indentations. So let's come up with a plan to figure out how many frogs are on the island. Is there anything that tells us which is on the island and which is off the island? Do you see something in this picture that tells us? I do. It's this tree here. The tree is on the island. So if that palm tree is on the island, how can we determine what else is on the island with it? How about if we do some shading? If we shade in the island. So these parts are in the island. I'm coloring in basically the land. Can you do that too? So my plan is to color in the land and see how many frogs I encounter of coloring in the land. So this is how I would do my plan. Your plan might be slightly different. But how many frogs have I colored in? Yeah, some of you are saying by following the path. That's exactly what I've done when I've colored the island. Now how many frogs are there? I'm going to circle in those frogs that I've colored in. See how many I've colored. By now we should have lots of correct answers, right? There we go. We're doing well now. Okay, we're going to have to get to some more fun questions. So we'll share these results. Most of you have correctly counted six frogs are sitting on the island. So great job. Here's our next question. 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? Now let's understand the question. We have 20 stars total and the gold rocket has six more than the silver one. So what does six more mean? It means, for example, I could have eight and two. I could have nine and three. And you can solve this question by making a table until we get a question by making a table until we get a total of 20. So this total is 10. This total is 12. So you can make a table and you can solve the problem with a table. That would be fine. But we're going to try to use a different type of model that we're going to draw. I am going to draw something called a tape model. So if I want to imagine that this is the number of gold stars and this is the number of silver stars, this question has told me that there are six more gold stars than silver stars. Does that help if we draw a picture this way? These two tapes are the same length, aren't they? There's the same, but then this one has six more, right? So if I was going to write this out, if this is, if this is some number of stars, S, and this is the same number of stars, S, I'm gonna be able to make a little algebra equation, which is S plus six, that's the gold stars, plus S for the silver stars is gonna equal 20. We're seeing some correct answers in the chat. If I subtract six from both sides of this equation, I have S plus S equals 14. And I bet with S plus S equals 14, everyone can tell me what S must equal. That's right, now the answers are coming back pretty much correct. Double check your answer. Remember, that was a really important part. S is gonna equal seven. So that's seven silver stars. What does the question ask us? The question asks us for the gold stars. So I'm gonna need seven, oops, seven plus six in order to get the gold stars. Sorry, I'm having a little mouse problem today. So make sure you're double checking the last step, right? We had a plan, we came up with a tape diagram or making a table. Those are two plans, both work. Then we had to do our plan carefully, work it out carefully, and then we have to double check our answer. Did we answer how many gold stars there are going to be? Did we take our seven and add the six onto it to get 13? I hope you did. Here's another question. Again, I'm going to ask if you can draw on your piece of paper the scenario that I'm going to describe right now. We can fill a certain barrel with water. If we use water from six small pitchers, three medium pitchers, and one large pitcher, there's a really important word here, or from two small pitchers, one medium pitcher, and three large pitchers. And then here's the question. If we use only large pitchers of water, how many of them do we need to fill the barrel? So what do you think? What is the question asking? We're going to use only large water pitchers, and we're going to fill a barrel. But we need to come up with some way to describe the other two situations that they're talking about. I think five is hard, too. Some of you are telling me you think five is hard. Let's draw it out, okay? So we are going to have three situations, right? I'm going to divide my paper into three. In the first one, they're going to use six small pitchers, three medium pitchers, and one large pitcher. I'm just going to use letters. I'm a big fan of just writing things easy. One, two, three, four, five, six small pitchers, right? You all can do that. And I bet you can do three medium pitchers. You probably can write neater on your paper than I can on the screen. And one large pitcher. So that's this. Six small, three medium, and one large. I'm going to switch colors just to make it a little easier to see. We have an or. So this is another situation. I can use two small pitchers, one medium pitcher, and three large pitchers. So that's this. Two small, one medium, and three large. So now my picture represents what the question says, right? So that's very good that we have a good representation of the problem. We need to know what it's asking. Now it's asking, if we only use large pitchers, how many do we need? Well, I know I need at least three because there were three over here. So my number has to be at least three. How do I figure out how many additional large pitchers I'm going to use? I want to give you some time to think about it. Is there a way you can figure out how many small and medium pitchers there are in each large pitcher? We know we need more than three pitchers because when I used three pitchers, I still needed two small and a medium. Is there a way to figure out how the large pitchers and the small and the mediums are related to each other? Some of you are getting it, some of you are not, but that's okay. That's why I'm here. I'm here to teach, right? So if you noticed, in the second situation, we have two additional large pitchers compared to in the first situation. One of the neat things I like to do is I like to use all the tools I have on my Zoom annotation. Look at this. Small, small. Does that help? Did you notice that I have two smalls and a medium, two smalls and a medium, two smalls and a medium, and also over here I have two smalls and a medium. So those are matching groups. And look what happened in between this first scenario and the second scenario. I don't want to give everyone the answer right away. I'd like you to think. I'm seeing a lot more students figuring it out now. Do you see how drawing helped us figure this out this time? I was able to actually make a visual connection. It turns out that two of these groups got turned into large pitchers. So one of these is one of these, and this one is this other one. So if I still have one more of those groups of small, medium, and large, I think I need to make another large pitcher here. Are you happy with that? Did that make sense? For every two smalls and a medium, I could fill one large. So the answer, how many large pitchers, is going to be four large pitchers. Good work. I'm seeing a lot of great, great answers. Move that up for just a moment. All right. Everyone should see a triangle. If you don't have a paper, if you don't have the handout, you can draw a triangle on your own paper. That will help you this time. So go ahead and get yourself a triangle. This is an equilateral triangle. Equilateral, I think you know, is all three sides are the same length, right? It says, joining the midpoints of the sides of the triangle in the drawing, we obtain a smaller triangle. We repeat this one more time with the smaller triangle. How many triangles of the same size as the smallest resulting triangle fit in the original drawing? That's a lot of words for a question, isn't it? So the first thing we need to understand the question, midpoints, midpoints of a side. Midpoint sounds like the word middle, doesn't it? I bet you know what the middle is. So I would say that the midpoint of this side is probably around here. This is why we get to draw. The midpoint of this one is probably around here, and the midpoint of the bottom one is going to be directly under the top point, right? So drawing helped us a lot there, right? And it says that we are going to draw joining them. So joining them means a straight line, right? So if I join them with a straight line, the question says I get another triangle. Have you been able to draw this far on your papers and get another triangle? Now the question says we repeat this one more time with the smaller triangle. So what does repeat this? Repeat this means go right back to the beginning. It says join the midpoints. So we need to draw some new midpoints. I'll switch colors to make it easier. So I have to find the middle. Here's the midpoint. Here's the midpoint. And here's the midpoint. And it says to connect them. And now it says, so we've done all of the instructions, right? We followed all the instructions joining midpoints, the smaller triangle, repeat it one more time. It says how many triangles of the same size of the smallest resulting triangle fit in the middle? Triangles of the same size of the smallest resulting triangle fit in the original drawing. So this might be the tricky part of this question. It's not asking how many are inside the red triangle. They're asking how many of these little blue triangles could I put into the black triangle? And I think that is what makes this question tricky. How many blue triangles would fit into the black triangle? It's not the Deathly Hallows sign. It would have a circle and a line going down for the sword. I know Harry Potter. Oh, I'm seeing a little bit of the correct answer. Imagine we repeat this in all of the parts. So we repeat this here. Because it's saying in the original drawing, the original drawing was the black triangle, right? So we have to keep going with this pattern of cutting the midpoints. Your drawing doesn't have to be perfect. It's not a big deal. And then we can find out how many. So if you don't want to draw them all out, you could have noticed that you can make one, two, three, four triangles inside the red one. And there are one, two, three, four of those triangles. So without labeling them all, you could have said there are four triangles in the red. And you can make four red triangles in the black. And then you can just do four times four. Or if you're one of the people who likes to draw and count, that's a perfectly fine method too. So again, there's more than one plan. We could multiply or we could label with numbers. Does this make sense? We cut things up into smaller pieces. We do need to get a bigger number of triangles than we had originally. So check your answer if it makes sense. I'm glad it makes sense. Some of you are telling me it does. Do you think you would have gotten this answer without actually drawing on the triangle? Yeah, I don't think I would have gotten it without drawing on the triangle. So when they give you an image, go ahead and experiment with it. That's what this is about, right? Okay, it's okay if you couldn't have done it without the pictures. That's why they provide the pictures on the contests. All right, here's another one. If you have the handout, your handouts might help you with this one. Otherwise, the pictures are all here on the screen. So don't worry. Okay, so one of the vocabulary words was intersection. An intersection is a place where any two line segments meet. In this case, we're talking about roads. We're using roads as a line segment. But that's okay. How many of you have driven in a car, ridden in a car? You don't drive yet, do you? But you ride in a car. How many of you have driven in a car? But you ride in cars and you get to intersections, right? And you know you can go left, right, or straight. And that's what these cars are going to do. They're going to go left, right, or straight. They have, just to understand this little picture a little bit, you can see these little, these are the turn signals, the lights flashing on the outside of the cars. The arrows tell you which way the cars are going to go. We want to know, how do they look after they leave the intersection? What is your plan going to be? There's more than one correct plan. What plan are you going to use? Are you going to draw what they look like? Are you going to make arrows? Are you going to, there's a couple of ways you could try to do this. I'm going to give you some thinking time because I think that's important. I think we can put up the poll because some of the students are starting to answer on their own. Well, we're not getting a unanimous answer, but we are getting the majority getting it correct. How about if we follow some of these around? So, I'm going to, there's a couple ways to do this. I'm going to number the cars, okay? You do not have to do it this way. There are other ways to do it. That's supposed to be a six. Goodness gracious, seven, eight, nine. Okay, so I'm going to just start with car number one. Where is car number one going to end up after they go through the intersection? And again, I'm not going to make fancy pictures. I'm just going to do little rectangles. Car number one ends up over here because it went straight through. Where does car number two go? Car number two also goes straight through. And car number three is going to end up turning right and it'll be on this part of the street. Maybe you can use this plan and come up with the answer. If you want to change your answer from what you did in the poll, you can go ahead and put your new answer in the chat if you've changed your mind. And I would like to show you a different way to do it because there's not always one right way. If I want to determine how many cars wound up, for example, coming down, I could say, all right, how many of them are turning this way? And when I do that, I find, okay, I have one, two, three that are going to end up down here. So I could instead draw one, two, three down here. And then I know I can cross those out because I've done those. If I want to know how many cars wind up over here on the left, I can determine how many cars go straight or turn this way. It's only this one car here. So I would just draw one over here. So it's a similar idea, but it's just a slightly different way of thinking about it. I have somebody asking about elimination. When a contest gives you multiple choice questions, you can check the options. You can go through and say, does this one, does choice A work? Can I get to choice A? And if you can't, then you can cross out choice A. And you can go through the different choices that way. Okay, I think we can end the poll. The correct answer here was B, which is what most students in the poll did say. And I think after having a couple of ways to draw it out, I think now the students would be able to find out it was B, even if they answered it differently before. It's about learning. You don't have to get the question right the very first time. The idea is to get it right as you look at it over and over and to learn from the mistake that you might have made. Okay, Henry and John started walking from the same point. Henry went one kilometer north, two kilometers west, four kilometers south, and finally one kilometer west. John went in a whole bunch of different directions. And then we want to know which of the following must be the final part of John's walk in order to reach the point where Henry ended his walk. Now, how many of you have used maps in school or in other places? Yeah, so do you know which direction north is if you were drawing a map? If you make a compass rose, these are the directions on a compass. Do you know which way things go? North is to the top, south is to the bottom, and you should be able to read the word we from left to right. West and east. Yes, a lot of you have said north on top, right. Maya, I think we still have some people who want the link. Thank you. All right, I'm going to switch to the next slide because on the next slide I show a little piece of graph paper because I think that trying to figure out these directions with graph paper is going to be much simpler. And I included a little piece of graph paper on the handout for you. Okay, so remembering that north goes up. We can draw out these directions on your little piece of graph paper. And I even gave you a little star to know where to start drawing. So I'm going to make Henry in red and I'll do John in another color. So let's see which way Henry went. He went one kilometer north. So I'm going to have to draw one kilometer north and I can just use the grid marks for the numbers of kilometers. That's called a scale. Okay, now two kilometers west. Remember west was to the left. Four kilometers south. So remember my plan is to actually draw it using the grid for one kilometer for each square. And then one kilometer to the west. So this is where Henry winds up. You can even put a little H here for Henry, right? Now let's see where John winds up. You should be doing this on your own piece of graph paper, right? Okay, John goes one kilometer to the east. So he actually heads out in the opposite direction. Goes four kilometers south. So again, use the grid to make four. Four kilometers west. One, two, three, four. So this is where John winds up. And very careful reading the question asks, who's going to move? Who has to go where? It says John is going to walk in order to reach Henry. So which way does John need to go to reach Henry? Not Henry meeting John, John meeting Henry. It makes a big difference that we read the question carefully, right? Did I carry out my plan very carefully using the graph paper and the scale and making sure I knew my cardinal directions? I did carry out the directions carefully. It looks like our poll is coalescing on the answer B, one kilometer north. And a lot of people in the chat are saying B as well. Would this question have been difficult without graph paper? Yes, but can you make yourself a little piece of graph paper on contest? Of course you can. Even if it's not perfect, you could certainly draw a small grid on your scratch paper. Here's another thing you can do. When you do your math work, do it on graph paper. Always do your math work on graph paper. Then you'll have a grid anytime you need it. One of the things I've found out with my own children is that working on graph paper, if you're trying to add, like let's say you're adding or multiplying, working on graph paper and putting one number on top of the other number, that's going to make it easier for you to do your math work on graph paper. Working on graph paper and putting one number in each box of the grid actually helps you stay organized. And you won't end up putting a ones in the tens place or having your things cross in different directions. So I like working math on graph paper. Here's another practice problem. A rectangular sheet of paper measures 192 by 84 millimeters. Are you going to have a size paper that big when you're doing the contest? No, you're not going to have one that big when you're doing the contest, it's okay. You can draw a rectangle on your paper that's to scale, just like we were doing the scale before. You cut the sheet along just one line, 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 side of the smallest square you can get in this way? Now we saw this with the triangle problem, right? That it's good to start with the picture, to start with the general shape. And then we have this other similar thing where it says, and so on. We're going to be repeating a process. So I have a rectangle on my next slide that's going to make this easier, right? I drew my rectangle pretty much to scale. It's never perfect, but if I look at this, I know that 84 can go into 192 more than two times, right? So I had to make the one dimension at least two times the other dimension. Now let's see if we understand the question. What does it mean? You cut the sheet along one straight line to get two parts, one of which is a square. I have to use a straight line and I have to cut out a square. If I know that this is 192 and this is 84, how will I draw a square? I'm going to have to come up with a length of 84 on this side as well. So I have an 84 by 84 square. Again, remember, I didn't draw it exactly perfectly, but let's say that this was a square. This is my 84 by 84 square. Now it says you do the same with the non-square part of the sheet. So this is a non-square. And if I have the dimensions, it's 84 on this side, right? And it's going to be 108 on this side. So I can still draw another 84 by 84 square, can't I? I can cut it off here. That would give me another 84 by 84 square. Are you following along? Are you doing this on your own papers? You should be. And then what is the length that I have left over here? I have 24 because 108 minus 84 is 24. Now when I'm going to draw another square, so remember, like these are all gone now. So I'm going to draw just in this rectangle. I'm just working on this rectangle here, which is 24 by 84. How will I draw a line to make a square there? I'm going to draw another 24. So I've got 24 and that gives me a 60 over here, right? I guess I can do another 24 then, can't I? If I have 60, I can cut off another 24. And that's going to give me 36, which means I can still cut off some more. And my scale is going off a little bit, but I should be able to come up with two squares at the bottom. What was 36 minus 24? 12, right? And this 24 gets cut into halves. So this is a more difficult problem to keep track of what you're doing. Does this make sense? If I have a rectangle that's 192 by 84, can I cut it up into 12 by 12 squares? How could you test that? Is 192 divisible into squares 12 by 12? Is 84 divisible by 12? It is 12 millimeters, that's right. If you didn't get it the first time, why don't you go ahead and write these measurements down and you can work on it again later. Sometimes it's hard to get your dimensions right when you're chopping up a rectangle like this, but you should come down to 12 by 12 squares. Could you have done this without chopping it up and making a model like this? You might have been able to do it if you knew that 192 and 84 were both divisible by 12. You might've been able to figure out that you could move 12 by 12 squares by using some factors or divisibility of 192 and 84. So just to give you another option, because I know some people like to solve things in different ways. Okay, this is our last question for this webinar. The map shows three bus stations at points A, B, and C. A tour from station A to the zoo and port and back to A is 10 kilometers. A tour from B to the park and zoo and back is 12 kilometers and a tour from C to the port and back to C is 13 kilometers long. So if you look at this picture, you can see they're talking about these different loops. All right, so if I just want to make this little loop to go from A to the zoo to the port and back to A, I would travel 10 kilometers. So each of these little numbers inside almost a triangle shape, it's basically the perimeter of that triangle. How long you would travel if you were traveling in a bus. Okay, a tour from the zoo to the park and port and back is 15 kilometers. So that's the inside triangle is 15 kilometers. The question is asking how long is the shortest tour from A to B and C and back to A? So can we draw that out on our map? What is the shortest tour from A to B and C and back to A? That would follow this line, right? A to B, that's the shortest way. B to C, this is the shortest way. And C back to A, this is the shortest way. So we want to go around the outside of our map. What do you think? Now that we understand that we need to find the shortest distance around the outside of the map, do you have a plan for how to solve this? I wanna give it to you right away. I want you to think of it. I'll give you some clues as we go along, but I wanna make sure you have a little bit of thinking. How about if we look at the triangles? Pretend these are triangles and each one has three sides, right? How many sides do we need to make up the larger loop around, right? I need this segment, this segment, this segment, this segment, this segment, and this segment. So there's six little segments that I need to make the larger triangle. How can I find the lengths of those six segments? If you're getting somewhere, you can put it in the chat. So I'm going to change color here for a minute. I'm going to just look at this 12-kilometer-long loop. This 12-kilometer-long loop has three segments. One, two, three. And how many of those are in common with the orange part of the map? Two out of the three are in common, right? And I could say the same thing about this 13-kilometer-long loop, that two out of the three are shared. And also with the 10-kilometer-long loop, that two out of three sides are going to be shared with the path that I actually want to make. That means that there are some parts I do not want. Do we know how long the parts you do not want to travel are? You do not want to travel on this part in the middle, right? And fortunately for us, we know how long that red part is, don't we? The red part is 15. So what would happen if we took all of these loops, the 10, the 12, and the 13 loops, but cut away the red loop? What do you think? If I took the 10 loop on the top, and I took the 12 loop on the left, and I took the 13 loop, that is going to equal 35 kilometers. A lot of you are giving me 35 kilometers, that's correct. But I don't want to take the red part. So I have to subtract the red part. So that's going to give us a total of 20 kilometers. Maya, you had a way to explain this by labeling it with some different letters and points. Do you want to try to do it that way with the segments? Uh, sure. Okay, so Maya is going to give you an alternate way. I'm going to clear my drawings, and we'll let Maya do it another way, because everyone has their own preferred method of thinking in their heads. And Maya was doing it another way, and we'll let her explain it since it was more natural for her. Um, I don't think I can annotate, so I don't know if I can. Let me see. Can you now? Yes, thank you. Okay. So, um, the way I was thinking of doing it at first was just labeling each of the paths with a letter. So you can say A, and I'll make them lowercase so that they're not the same as the points. Okay, and so the next thing I did is I wrote down what we already know. So the first thing we know is that the length of A plus F plus G, which is this path here, um, we know that's equal to 10. And then the next thing we know is that B plus, sorry, um, next thing we know is that B plus, um, C plus H is equal to 12. Um, that, because that is this path here, which we know is 12. And then we also know that I plus E plus D is 13, and that G plus H plus I is 15. And so in the end, it's sort of very similar to the previous method that you saw, because then we can write what we're looking for, which is A plus B plus C plus D plus E plus F. And, um, what we want to do is we want to somehow add or subtract the things we know in order to get this. And so then, like in the previous method, you can see that if you add the first three and then subtract the 15, then you'd get rid of the G plus H plus I that you don't want. And so overall, you would get, um, 10 plus 12 plus 13 minus 15. But yeah, so it's very similar way of thinking about it, but a little bit different. But everyone has their own, you have to respect that everyone's mind thinks a little bit differently. And the fact that we can label things and kind of discuss them differently and come up with the same answer is reassuring that the mathematics is working. So don't worry if you solved questions in a slightly different way than I presented them here. That's perfectly normal to have your own pattern of thinking. Okay, so that's our last question for today. And I want to wrap up. Remember the four-step problem method, understand what the problem asks. There was a lot of vocabulary, a lot of multiple step things we had to go through very carefully and understand the questions. If you're leaving, there is going to be a survey. I would like you to stay and do the survey for us, plan your problem, carry out the plan and check your answers. Okay. Laugh kangaroo problems do have pictures or drawing pictures may help you. Okay. Thank you for attending today. When I close up this, um, there will be a survey so that you can help give us feedback and we can do an even better job the next time we teach a webinar. Thank you, Maya. You're fantastic. And all the students who are participating. Thank you so much. Bye.

Math Kangaroo

diagram techniques

problem-solving

visual reasoning

spatial reasoning

interactive webinar

alternative methods

visual strategies

teaching improvement