your assistant, and I welcome you to class today. You can connect with each of us through the chat feature. If you have some questions, you can try to chat. Usually, if I'm explaining a problem, you can ask Shoria the questions. If Shoria is explaining a problem, you can ask me. I do have it open all the time, so if I see a lot of students misunderstood something or I said something incorrectly, you can go ahead and politely put it there in the chat. Let me know. I will see it. We have polls for some of the questions, not all today. You'll see because some of the answer choices are actually little figures and pictures themselves, and Zoom doesn't allow me to put all those little pictures as answer choices. So for those, you'll just put your answers in chat for Shoria and I, and we'll see how you're doing, and we'll know when we have a lot of students getting it correct or whether we need to explain it in a different way. Now, remember, Math Kangaroo Problems, we explore with a four-step problem-solving method. The first step is always to read through the problem and try to figure out what it is asking and what information you have. That also involves what information is missing. Once you know what you have, what's missing, and where you need to go, you'll make your plan. That's step two. Step three is to work very carefully through your plan. Try not to make errors. Try to double-check yourself. Step four is to look at your answer. Is your answer reasonable, and can you check it by using another method or by substituting your answer back into the problem? Sometimes there's ways to check yourself. All right, today's lesson is experimenting and acting out. So that means you'll be using your pen and paper, probably, to try a few things to see what happens. You might be using some paper. In this case, any piece of rectangular paper will help you with this problem. It can be a piece of scrap paper off your desk. I'll probably grab one in a minute. Which of the figures below couldn't be made by folding a rectangular sheet just once? Okay, so remember, we have to read carefully. Couldn't be made. And we're folding it just once. So you can try to make A, then unfold the paper, try to make B. Remember, you can only fold once. When you think you know the answer, you can put it into the chat. Yes. Do you hear me now? I have several answers from students. I'm going to do the best I can to fold my paper in front of my body so the camera picks it up. I just picked up a piece of scrap paper from my desk. So you can see it has writing on it. It doesn't matter, right? I said you could use anything. So let's see. This is a rectangular sheet of paper. Let's see if I can fold it into shape A with one fold. If I fold it, I see there's a diagonal that is one fold and I'm able to get something that looks like the shape in A. So A is definitely possible. So I'll put a little check mark next to that. It can be done. B. I still need to think about B. C. I can fold it into a square shape by just folding over part of one edge, right? That looks more square. That will work. So C is also a check mark. Now we have D. I unfolded it. Remember, unfold in between. D, I need to have a point up at the top. So if I take maybe this corner, we're going to ask the students to stay muted, please. And I fold that point up towards the top. I can actually make, you see I now have this diagonal coming down. I do have my little triangle sticking up at the top here. So I can, I think you can see it, I can make that shape in D. So that's all right. E. E looks pretty similar, only I fold a smaller piece on the sides, right? I don't go all the way over the top. So E I was also able to make. I have not been able to figure out how to make B with only one fold. So B would be the answer for this problem because it can't be made with just a single fold. So I said any paper, I hope that worked well for you. Go to the next slide. So one of the reasons we're going to use the acting out is for geometric and spatial problems. They work really well with that. So the one we were just doing was kind of a geometric shape. Could we make the shape shown in the picture with a paper? Spatial relationships may involve where things are if they're right side up, if they have backs, if they're three-dimensional, then all of a sudden we're dealing with spatial relationships. And sometimes a physical model really helps us with that. Now I have students who'll come and say but I don't have a physical model when I do the math kangaroo contest. Well to a certain extent you do, right? You do have a piece of paper and if you're quiet about it you can probably fold it during the contest, right? I mean that would work. You might be able to just use your pen for certain things. You know if you have things that are right side up versus upside down you might be able to use your pencil for that, right? And also by practicing with us in the class today you'll have more experience and your brain will be able to kind of recreate that exercise that we did today just from your memories. So the more you experiment with these things, the more you have memories of how things work and what happens when you turn different shapes. If I have a book and I turn it it's going to have a thin edge. How do I make this part be on top? What if I try to fold it up like this? What is the what is the cover going to look like? You can see I now have little flappies here. So just try experimenting with different things as you do these problems, okay? So what the other thing you'll notice is some of our problems today will have step-by-step instructions about this happens first then this then this. So make sure you're keeping the steps in order. That could be very helpful. Okay, Anne has four gray pieces as shown in picture two. So this is kind of weird because we started with picture two. Here's four gray pieces in picture two. She needs to completely cover the shape shown in picture three. If you have the handout you could actually cut these pieces out and try to put them on top of picture three. Where does she need to place the t-shape picture shown in picture one so that she can use the other three pieces to cover the rest of the shape? So this they're calling the t-shape piece and you'll see it's also over here. You only need to use it one time in picture three. I'll give you a few minutes to think about it. There's multiple choice answers you could test them or if you cut out the pieces you can try to fit them into the puzzle shape. Thank you to those students for putting responses in the chat. I can see from the responses that most students think the answer is either A or C. Let's take a look. I think that once we put in the T-shaped piece, that the next hardest piece to fit can be that zigzaggy piece. Give me a second. When we have the little zigzaggy piece, this is the one I mean that's zigzaggy. I don't know what to call that other than a zigzag. If I look at A, I can put that zigzag piece maybe here, one, maybe here, one, two, three, like this. Then where can I put the other pieces? I have one piece that's here. Then I could try to put this L-shaped piece over here, but then I cannot put in this straight piece. If I try some other arrangements, let's see where I could put in the long straight piece with four blocks. I could put it in here, so one, two, three, four across. Then where do I put in that zigzag piece? It might fit here, one, two, like this, that might fit right here, but then I have a space right here that isn't filled. So that's not going to work. So I can't find a good way to get them to fit into A. Let's try C. On C, I think that zigzag piece fits real nicely right here. That will work for me. Then the long straight piece of four will fit here, one, two, three, four will fit there. Then the L-shaped piece fits in very nicely right here, one, two, three, four. So I do like C as the best choice in this particular problem. Okay. Problem number 2 is a problem that Shurya would like to share with you. Do I need to share my screen? No, I think you should be able to work off of my screen. I think we're okay. Oh, I am. Okay. Yeah. All right. Which three of the five jigsaw pieces shown below can be joined together to form a square? So we're given these pieces and we need to form some kind of square. It can be small, it could be big. We just need to find out which three of these five pieces. So we need to use three of these five pieces. All right. Cool. So I'll give you all a minute to think about that. And there is a poll for this one. So you can type the answer in the chat when you think you've got it. Okay, I'm going to be launching the poll. Hopefully you've had a good chance. The pictures are, the main figure is in the poll, so you should be able to see it. If you do not see all the answer choices on the poll, try touching or scrolling down on the right-hand side of the poll box. That should take you to all the multiple choice answers. Okay, we have most of you have responded, but not everyone has responded to the poll. And I just want to give a piece of advice for the Math Kangaroo contest. It will be multiple choice. You will have 30 questions to answer in 75 minutes. The test is generally organized with easier questions at the beginning and harder questions at the end. So you will want to work through quickly, but if you are running out of time, do not leave blank questions. We do not deduct for a wrong answer. So it's always better to guess than to have a blank. Alright so I'll end the poll. So here are the results. We mostly everyone was picking answer D. Well we do have some C's, some E's. Alright so let's work through this. So I'm gonna start by thinking about like which piece is the hardest to blend. Oh actually before that let's just look at all the answer choices. All of them have piece one in it. So instead of having to just do a bunch of checking and guessing we can just look at which pieces work with piece one. So first I'm just gonna draw like piece one over here a little larger and now I just need to find out like which of these four pieces, which two of them fit with piece one. So the first thing I see is that like piece two and piece four are gonna be hard to fit in because they're squares and rectangles. So if we have for example piece four and then we're trying to make like a square out of it then there's really no way we're gonna be able to do it with only triangular pieces. Like even if we have four and then we have let's say two over here we'll still need two more pieces one over here and then one over here. So we can't really form a square if we already have a smaller rectangle inside of it and you can see that if you place the cutouts of if you place the cutout of a four and then just draw a box around it there's not enough space for us to form a full square. So you know it can't be two and four together so it's not gonna be B. But we still have to look at whether it can be something else with the two and four. So let's say we draw like right over let's say we draw right over here like a one like this and then we have a two somewhere around you. So the first thing is if we use our two like we can tilt it but even if we do tilt it it'll turn into something like that and then if we try to form a square our square is gonna look like this. So our square is really not even gonna be possible to form because we don't have enough space to get in the one and the two. Similarly if we have one on the bottom we still need like we still need more things over here like we'll need to have something in this spot and then if we make like a little dash line we'll need something in this spot. So basically we can't get it with any of these two pieces. So let me clear this for a moment and now we figured out that we can't have B we also can't have A because we can't use two. So you know that we're not going to be using these pieces and we're just we're figuring out how to do it with the one piece. Okay so now let's just look at C D and E again like if we try to use four with the one it's probably not gonna work like even if we have a triangle we don't have enough space for a rectangle or even if we tilt the rectangle it's it's still too long like our one would be over here our five would be over here so basically we just can't do it with four or two. So that's why C is not gonna work if we try like this big one and we try to we try to put in the one like this it's not gonna work with the four because the four is gonna be like something weird like that. So that's how we know that none of these are actually going to work. So the answer we have left is gonna be D. So we can just test out D like if we draw a three over here because we know that three is the biggest piece so we'll just put in the middle and then we'll try to draw the square like around it. So this is like the boundary of our square and now we can just put in the one and the five. So if we want we look we can look at this length and if you have the cut out you can see that it's actually the same as the bottom length of three. So that means you can make this go at the top and then you could make our five piece whose side length is the same as the side length of one go right here like that. So overall in this question I'd say that it wasn't really more about like testing every all of the answer choices you could do that that's another way you could do it but I feel like it's a it's a bit quicker if you see that all of them have piece one so then you just concentrate on how to build a square around piece one and you find out that you can't use two and four so that just gives you your answer by process of elimination. All right cool. Thank You Sharia. We're gonna go to the next problem. Okay Nora is playing with three pups on the kitchen table. In each move she takes the cup on the left flips it over and puts it to the right of the other cups. The picture shows the first move. What do the cups look like after ten moves? So this here is move one. So I'll give you a few moments to think about this. We've already done some pattern work in this course so you might think about some patterns with this as well that will probably make your work a little faster. link you to the students who are sending me the answers in the chat. That helps me know that you're working through all the problems. That's really how to make the most out of these webinars, is to work the problems during that quiet time that we give you for each problem. We need to have a total of 10 moves. Like I said, this is one of those things where you could use pencils or something on your desk because you're not going to have coffee cups with you, but you do have a pen. A pen has a point up and a point down. We start with move zero, we have three objects facing up. After three moves, we're going to have all three objects facing down. We can use these multiples of three as our pattern so that this becomes easier. After a total of six moves, that's three more moves, then we're going to have all three facing up again. After three more moves or a total of nine moves, we have all three facing down, upside down cups. If the arrow is right side up and upside down. All three would be upside down after nine moves, but we need one more move. I'm going to take this guy and move it over here and turn it facing up. I would have a down, a down, and an up cup after move 10. That looks like B, a down, a down, and an up. By recognizing that there was a little bit of a shortcut I could take by doing all the multiples of three, that made this problem faster. I didn't have to do every single one. Hopefully, you did some similar strategy to save you some time on the contest. If you didn't, now you'll be able to look for that the next time you see a problem like this. That's always something to keep in mind is we're trying to do better on these problems after we have these classes. See if this one has anything similar to what the last problem had. I do have a poll. After you have some time to work on this one, I'll launch the poll. A standard die has seven as the sum of the dots on opposite faces. That sounds like it's important. Seven is the sum of the die on opposite faces. The die is put on the first square as shown and then rolls toward the right. When the die gets to the last square, what is the total number of dots on the three faces marked with question marks? Now, remember step one? At the beginning of the class, I said step one is to read and understand the problem, to understand what information you have and what information is missing and what you need to know. I'm going to go through this one because it's a complicated statement. Seven as the sum of the dots on opposite faces. Well, a sum means we're adding opposite faces on a cube. This is a little die like you play in a game board, right? What does that mean? Well, top and bottom are opposites. So if there's one dot on the top, then I'm going to have seven minus one equals six will be the number of dots on the bottom. If I have three on the right, the left is the opposite. So I'll have seven minus three equals four on the left-hand side. I'll lead you to figure out what's on the back, right? What is rolling toward the right? It means it's just tip over. So you're just going to tip it. OK, so think about tipping it. And then we need a total for the answer. The answer is a single digit. Well, like a single number total. I guess E has two digits, but a single number as the total. I'll be quiet for a little bit, and then we'll launch the poll. Thank you. I'm getting several answers from students, so I will launch the poll again. You'll see that figure in the poll. Anybody else want to add a response to the poll? Remember, leaving it blank is a surefire way to get 0 points. But guessing, you at least have some chance of getting some points. Okay, I'll share the results. You can see the majority of students have selected B7 as the sum of the dots. That is the correct answer. That's really good. Let's see. I said this question might have something in common with the question before it. I was talking about finding a pattern. Every time we flip three, we go from right side up to all upside down and vice versa. Well, let's look at this. How many times are we tipping the die? We are tipping it one, two, three, four, five times. Around the outside of any cube, well, on a cube, there are a total of six faces, right? But if you take one to be the front and one to be the back, that means there are four faces to tip along. So when I tipped it the fourth time, I'll end up with a cube right here that is in exactly the same position as the first cube. So if I've tipped it four times, I've come right back to the beginning. So that makes it much easier for me to tell what is going to be on the fifth position. So the two has stayed in the front the entire time. The only difference is that the dots rotate. So the two is going to look like this on the fifth position. The one dot is going to be here on the question mark on the side. That will be one dot. So now my question is, what will be on the top? Well, this part here would end up on the top when I flip it, right? So after the first flip, this side becomes the top. And we already talked about over here that the top is going to have four dots. So I have a sum of 2 plus 1 plus 4. And that equals 7, which is answer B. So very good. Most of you got that. And the thing that makes this a much more doable problem is to understand that because we're just rolling it, tipping it, in the fourth position, it looks exactly the same as the first position. And we just have to tip it one more time. So another way to say it is that the second position looks exactly like this position here. All right. Number five is a problem that Shurya wanted to lead, correct? Yes. OK. So the question is asking, which of the figures below cannot be made using the two dominoes shown in the picture to the right? So we want to find what figure we cannot make. OK. And then we have three figures using these two dominoes. Yeah. And remember, the dominoes can be placed horizontally. And they can also be placed vertically. So you can place them either way. And you want to try to make all these figures using the dominoes. OK. So I'll give you all some time to think. I do not think there's a poll for this one. But you can still answer in the chats. If you have the handout, this is a good one to try to use those pieces from the handout. If not, you can do this with your imagination or by drawing on paper, it'll still work for you. It'll just be a slightly different experience. Okay, so I'm getting a few correct answers also, so I'll just go through this. Okay, so first of all, one thing to note here is that the patterns aren't symmetrical. So like, if I rotate this first domino, it's gonna turn into this. Like, if I rotate it 180 degrees, it's gonna turn into this. It's gonna have like these two dots up here and this one not down here. So this isn't the same as the first one because instead of your dots pointing sort of like to the left, they're pointing upwards. So the way you turn it matters in this one. Okay, so I feel like just to do this, the best strategy would be to just go through the answer choices and see are they possible. So if we look at A, we know that we can't place both blocks horizontally because then the top would have two blocks with just one dot each. So we can't have that. So that's why we need to split it vertically and then we can try to see if that works. So if we have a dot like this and then three dots, we know that's literally just this thing placed down there and then this one is just this block placed down there. So A has to be right. Now moving on to B. B is a bit trickier because now they're turned a bit and we actually have two ways we can think of this. We can do this horizontally and we can also do this vertically. So we can just try both ways. If we do it horizontally, well, if we turn this block on its side, it's going to give us this pattern. However, in this one, our dots are pointing up to the left instead of up to the right as our block does. So that's why we need to think about can we do it vertically? Yes, we can because if we rotate this again, it's going to make this happen. So that is correct. So we can just do that and then we'll just do the other side. And that's actually the exact same that is given to us just placed down vertically. So B can also work. Now moving on to C, we can again see that we have to do it horizontally. We have to split horizontally because we can't have two one blocks next to each other. If we rotate this 90 degrees, if we rotate this block 90 degrees, we'll get something like this. One dot here and then two dots here pointing up to the right. So that's right. We can have that. And then the exact same thing happens with the right one. The block on the right, we can rotate it and it'll give us something that looks like this. So C works. So now we have A, B and C working. We can kind of see a pattern. And in D, it's like it's very similar to B except the block on the top is just turned. So we can use that by saying like, OK, we only need to think about the top block. So instead of doing what we did with B, we can just use this over here. Right. So we split it horizontally. We get that on the top. And then we turn this one like 90 degrees counterclockwise. And that gives us this. So that's correct. And by process of elimination, our only one left is going to be E. But we can still confirm it because we can split it horizontally. The top block, we can't even get that right. Like for B, we weren't able to get the top block to be like that because it's only going to point to the right. These dots are going to point to the right. So you can't split it horizontally. So now we can try to split it vertically. If we draw out a box like that and then we try to do it vertically, we can see that we can get the thing on the left. We can get this if we just turn our original block like upside down, sort of. We can get this part, but then we can't get the one on the right because it's a mirror image. Right. It looks like this and we can't ever get that. If we turn it, we're going to get something like this. So that's why E would be impossible. Yeah. So on this one, instead of like if you found a faster way, that's great. But I feel like personally, this way is like the fastest and there are a few tricks you can actually look at. So one thing I kind of missed was that if you ever see a block like this one, that's like a reflection of this one. You know, it's not going to be possible because we can't reflect anything over here. We can only like turn it around. We can't reflect it along some line. So you could just look at that. You could immediately say that E is just not possible because you have a block that's being reflected. Like even if you looked at it horizontally, the block is still being reflected. So, yeah. Okay. Thank you, Sharia. I want to point out, I think I cleared it, that one thing that students might notice is that if they trace the diagonals of the two and the three, they go in the same directions for A, B, C and D. But those diagonals go separate directions on E. So that gives you a clue that E might be something different. And then if you took a look at the difference, you'll see that there is a difference there in those diagonal directions. So those would be the two that I would investigate first. Okay. Number six. All right. You can use that rectangular sheet of paper you had before. It doesn't have to be an exact square for this. Trust me. Ella folded a square piece of paper twice and so made a square with each of its sides as long as half of the original piece of paper. What does that mean? She's folding it in fourths, right? If you take a square and you fold it in fourths, you have a smaller square, don't you, with half the length? She then cut off all four corners from the square she made. Which of the pictures below shows the piece of paper after unfolding it? So fold it, cut the corners, then unfold it. And what do you have? Awesome, a lot of students are getting it now. So I will, I've already cut my piece of paper, sorry. But imagine that I had this piece of, this is just a piece of notebook paper. And if I have folded it in half and in half again, that's what it says. I now cut all four corners. So you can see this is the, mine's a rectangle because I started with a rectangle, but it'll still give you the general idea. Okay, so I've cut all four corners. Now, when I open it, I have a piece of paper with holes, with missing corners, dips along the side edges, see if I can get it to show up, dips along the side edges and a nice hole in the center. So you can look at this. One way to see it without having cut your paper is to go ahead and see like, what would it look like if I did the quarters? If I did the quarters, you can see that here, I have not cut this corner, right? I'm missing a cut. The same thing is if I do this one, if I do B, you can see that if I had folded it, I had, I've forgotten to cut this corner. I missed a cut. That's missed the cuts on the corners out here on C and on E, I haven't cut anything out of the middle. And definitely if I was to fold it into quarters, you can see that I would have to cut out the middle part. So when I cut this one into the quarters, I can see that I've then made one, two, three, four corner cuts. So the correct answer is going to be this one. Okay. Now, again, because I've done it on the paper once, the next time I see a problem like that, I'm gonna have a much easier time imagining it without the paper cutting. So it's just having that experience of doing it kind of trains us how to think about it. Here's number seven. We have three transparent sheets with the patterns shown. So everyone know what transparent means? They look white here, but you can actually see through them like a window. We can rotate the sheets. Rotate means you can turn them around this way, but not turn them over. That means no flipping. When we put all three sheets exactly on top of the other, what is the maximum possible number of black squares seen in the square obtained this way if we look at it from above? So imagine that those white squares are see-through. You stack these all up and maybe put it up to the window so you can see all the black. How many black can you get? We can see that there are a total of nine squares. Very good. I have most students giving me answers of D or E. Let's take a look. The idea is we want to rotate, spin, tip these and try to cover as many things with black as possible. If I call this the first, the second, and the third, just so that it's easier for me to speak about it. If I want to get more black on the first one, I'm going to stack everything up on number 1. If I take number 2 and I rotate it just 90 degrees, then I can cover this one, this one, and this one. That's pretty good. I can actually rotate it and cover three more. That's good. Now, I still have two that are not colored. Let's see if I can color these two, the top two, these two are open. Let's see if I can color that with number 3. Well, if I put it just the way it is, I can get a red one here just the way it is by using this red one. The others are going to cover squares that have already been black. But I can't seem to get that last corner. This last corner is not covered. Let's see what happens if I rotate. If I rotate maybe this one this way a little bit, then I can put this square on top of here. That would look like this. But then I would have an open square here on this side. You can try playing with this with your cutout pages. But I never seem to be able, I either am missing a corner or I'm missing one on the side. Now, if I could flip this over, it would work, but it says that you cannot turn them over. The correct answer is 8. That is the main part of the lesson. I do have a bonus problem, so don't go away. The bonus problem is really fun. I hope you like these problems of acting out or experimenting. Something like the dominoes, we maybe had to experiment. Can you turn it this way? Can you turn it that way? Acting out might be something more like those cups, where you know what you have to do, you just have to go through and do it. Not really experimenting, you're turning the cups right side up and upside down, either by drawing them or using pencils or whatever you need to do. In terms of challenges and how to overcome that, you might not have all the materials that you need to do an acting out experiment when you're doing a math contest. The ways to challenge, to overcome that is to use your drawing, to have enough experience that you can use your imagination. That will help you. Remember, the best way to prepare for math kangaroo contests in addition to coming and watching us explain problems is to try past math kangaroo contests from the math kangaroo website. We also have videos of people solving math kangaroo contests, videos of teachers solving them. You can find that if you use your math kangaroo contest link, you will find some registration codes to help you get to those resources with a discount, or even for free for some of them. Take a look at the registration for this class and at the registration for the contest where it says event info. That's where you'll find some codes to help you save money and to keep practicing. I do really like this bonus question, and I do have a poll for it. Five cards are lying on the table in the order 5, 1, 4, 3, 2. As shown in the top row of the picture, they need to be placed in the order shown in the bottom row. In each move, any two cards may be switched. What is the least number of moves that need to be made to put the cards in the top row in order like the bottom row? Okay, I have some correct answers. I'm gonna have to close the poll just for time. I'll share it with you. So the majority of you, over 60% say it takes three moves. That's the answer that I like. That's how I can do it. You can start this in a couple of different places. I'm just gonna start by trying to get one first, okay? So if I try to get one first, I can swap one and five. And then I have one, five, four, three, two. Now I'd like to swap and maybe get two where it belongs, right, so I can swap two, and that puts five in the correct place. So that's good, one, two, four, three, five. But then you can see that three and four are in the wrong place. So I would have to swap that in order to get one, two, three, four, five. And this took me one swap, two swaps, three swaps. So hopefully you like that problem. And you can do the swaps in different orders. It doesn't matter which order you do them in. But I can't do it in less than three. So thank you for being in the webinar today. We'll see everybody next week. And don't forget to do your practice during the week, okay? See everyone next week. Thank you, Sharia. See you all next week.

problem-solving

Math Kangaroo

geometric reasoning

spatial reasoning

four-step method

physical models

patterns

contest preparation

active participation

experimentation