Good afternoon. Welcome to the ninth webinar in the Level 5-6 series for fall of 2022. I've been teaching these for a few years. So for me, it's always exciting to have new students join us each time. There will be one more lesson after this next Sunday, one more webinar. And then I want to let you know that we have a webinar series starting in January with the same level, but with different sample questions and some different topics. So if you are getting some good skills, good practice with these webinars, you can sign up to continue and come back in January. OK? All right. Let me, there we go. Today is a lesson on hands-on folding and cutting and arranging sheet. I have several props. If you did the handout, there were some props that you could cut out. I have those. If you do not have them, this lesson will be a little bit more difficult. Obviously, when you get to a contest, you may not have all of these things that you can cut out and tear holes and things. You do have paper where you can draw. You can fold your scrap paper as long as you're quiet and discreet about it, not disturbing other people. So those things are still available to you. We will kind of show you how to do it with props and also without props. One of the reasons to try it at the beginning with props is it helps train your brain. What happens when you move things? It's experience, right? Every time we try something, it becomes an easier exercise to do, kind of like the first time you tried to play a certain video game. You didn't know which buttons to press, and it was very confusing. And after a while, your fingers are going to the correct buttons. Without you thinking about it, they know what to do. All right. I don't have very many polls today because of the nature of the lesson and the answer choices. I obviously can't put these different pictures into answer choices on a poll. So go ahead and share your answers with Jacob or with myself in the chat. That is how we know that you are participating and you're ready to move on to the next question or get the answers. Is by using the chat for us. Otherwise, we don't know how long to wait in between questions. And we want you to actually do the problems and work hard, but we don't want you to sit there having nothing to do. Carol is playing with two identical cards shaped like equilateral triangles, as shown in the picture to the right. She puts the cards either next to each other or partially overlaps them. Then she traces the figure made on a piece of paper. There's only one shape below that she cannot get in this way. Which one is it? So an important part here is there's one she cannot get. So always very carefully read the problem so you know what you need to answer. And I'll wait a few moments while you do that. Yeah, we're doing really great in the chat. Thank you for participating that way. So I have my two little triangle shapes I know they're small and put them right in front of me So if the camera picks them up if I put them kind of tip to tip You can see the same shape as in a if I put them with one edge together You can see the diamond shape that we get in figure B Figure C. We just overlap them a little bit this way. I guess mine is I'm in a Reverse to you, but that's okay. Yeah, I'm sure you can mirror image that on your own Figure D is a little interesting What D is is if you take the edges and you slide them down like this Then you can see put it right in front of my face. I think then the camera picks it up It's right in front of my face, right? But I cannot from two equilateral triangles Get that shape of a square in figure E Now let's say I do not have an ability to cut out triangles I didn't bring scissors to my math kangaroo contest and even if I did the Proctor is telling me you can't use scissors It's too noisy, right? So you can kind of complete drawing the triangles just kind of complete them on the paper, right? Because we can draw so we can see that this is the two of them divided this way This is the two overlapped like this, right? This we can draw a line in between and come up with two triangles here we could have two triangles But these are two right triangles not equilateral triangles. So the answer we cannot do is E So today you might see me doing a combination of drawing and using the props that I did cut out and hopefully some of you did as well All right You So we're getting toward the end of our lessons here again, remember, please join us today Please join us next Sunday And then if you have been enjoying these if you've been learning a lot and want to continue, please consider coming again in January All right. We have a little bit of vocabulary this time We haven't had too much vocabulary. I hope I have the right file So we have identical meaning exactly the same Apex is going to be the top like that. The top of a mountain is the apex. It's usually a pointy part of something Horizontally and vertically. I believe you all know an Increasing order just least to greatest or smallest to largest. We'll figure that out so Like I said today I do have the hands-on manipulatives for many of these problems If you don't go ahead and use your drawing Use your imagination Be very careful on these problems Some will say that you may flip a figure some will say that you cannot flip the figure Sometimes we might have a rotation of a figure So there's clearly a difference if my hand is in front of my face with my thumb my palm facing out or my palm facing Toward my own face, right? So we might that that would be flipping but I can just spin my Spin my hand without flipping it. So be careful. Make sure you're following the rules that each question gives you. All right So Which of the napkins below was made from the cutout you can see on the right? So if you take a real close look at it, you can see it was a sheet of paper Or paper napkin that's been folded in fourths. This is a fold and folded edge over here And there's a folded edge along here We've made one little triangle Nick The time of year or maybe you're making some paper snowflakes gonna be doing our own bits of paper snowflakes today Okay, this one, I didn't actually cut out a piece of paper because I want to show you an approach that you might be able to use when you're on a contest and you can't cut out a shape. So I am going to work backwards and by drawing unfold the paper. So if I start out with the paper, fold it up the way it is now, and I unfold it so that I have kind of a rectangle that looks like this, where I have one sheet in front of the other. Okay, when I unfold it, what you notice is because I've done the cut out of that little triangle shape in the center, this was a fold before, now it's not, when I unfold it, I have a piece that looks like this with a diamond shaped cut out, right? Now I'm going to open up the other fold because we had two folds in this paper, pretend I, you know what, I can actually make a square, I have, I have a tablet, it makes squares. Okay, so I have now unfolded it, the two fold lines are like this in my paper, so when I unfolded it, I have those diamond shaped cutouts like this. Now none of the figures A through E shows a page with two diamond shaped cutouts that are side by side like that. However, all I have to do is imagine rotating this 90 degrees and I will get pretty close to that shape shown in B. Another way people think about this is they put the fold lines in the figures to see if they can match up the fold lines that they could see here, these are the folds, if they could match that with the figure. So I've been trying to give you a couple of ways you might be able to work it out without being able to make the cuts by drawing and imagining it on the answer choices, so hopefully that helps. Okay, out of which of the figures below can you make the box shown in the picture? The handout did have some elaborate cutouts for this, I did not cut them all out, we are going to use this based on our 3D spatial reasoning that we have practiced a little bit with our 3D geometry, I think that will work for us. Great. Thank you for sending me answers in the chat. I hope that you're working with Jacob as well because he would be really helpful to you. We had talked about nets of three-dimensional shapes in our 3D geometry lesson, so we're kind of revisiting that a little bit here, which is why I didn't cut them out out of paper. So let's describe the figure that it's giving here. I'm going to call this, you know, this is our target figure, the one we're trying to get. You notice that the cutouts, the cutouts are along edges, but they're along opposite edges, right? Not adjacent edges, opposite edges. So we need to find one that's going to fold and have the cutouts on opposite edges. You also notice that the folds are parallel to each other, the cutouts are parallel to each other. We don't have like one on the side going this way. So if we look at A, we do have them going in the same direction. That is good, but if we just look at this, what we could call the top of our square, top of our box, you notice that you have two edges that are going to be cut out on the same face. So this is two on the same face, and that does not match with our target figure. If you take B, you'll notice that we are going to have one that's cut out at the top, and then we're going to have this edge, and this edge will line up as will this one in this one. This, see if I can change the color, this bottom cut out here is going to end up actually against this square. So we would end up with the side of a cube, I'm just going to draw the relevant edge on the bottom. We're going to end up with, this is what, so if this is the bottom, the bottom is that solid orange that I just have here, and we're going to end up with just a cut out kind of halfway here. So that one is not going to work for us. Now if we look at C, C we have them oriented in the same direction, and indeed this cut piece will join with that cut piece. So that does work, and those are going to be on the opposite edges, opposite edges, edges, they're the same parallel direction, and that is correct. You can see if we look at D, we have the same problem that we did with A, where we're going to have cutouts two on the same face. It's going to be this face here that has two, and if we look at E, what we're going to find on E and if we look at E, what we're going to find on E is the cuts are on faces, not edges, and our original figure had the cutouts on the edges, not square in the center of a face. So C is our correct answer by using what we learned from unfolding and folding nets in our 3D geometry class. All right, number three. Bob folded a piece of paper, then used a hole punch to punch exactly one hole in the paper. The unfolded paper can be seen in the picture. Which of the following pictures shows the lines along which Bob folded this piece of paper? So you can think about that for a minute. Great, I see students trying their best on this problem. I have both a combination of a folded piece of paper with a hole punched in it, and we can do some drawing on the screen. So one of the things you might do is you might imagine, let's say I had folded this up and punched a hole. If I folded the paper along the diagonals like in A, I would end up getting a triangle that look, a triangle that looks kind of like this, and it would have a hole in it. And when I unpunched it, I would find a hole like this in each of those. So that's one way to imagine this. If I had folded into the long skinny rectangle that I see in B and punched a hole, when I opened it up, I would see a hole in each of those. And while that is similar, there's four holes in a line, it is not four holes in the diagonal line. So here is the piece of paper. If I fold it into a triangle and then fold the edge of that triangle up, this is like the folds in D. And when I punch a hole, I have a hole in here already, but when I punch a hole through it, I see my scissors going through, and unfold it, I will have the four holes. If I can get the camera to pick them up. Come on, camera. I have four holes going along this diagonal here. One, two, three, four. So this looks, with those four little speck holes, that looks exactly like it. So this gives me four holes, just like the picture I was aiming for. If I try in D and C, I'll get four holes here, and E, I would only have three holes. So the correct answer is D. So again, multiple approaches. You can certainly go ahead and fold pieces of paper and try, because it's good practice for you to see, this is what happens when I really do it, and to train yourself to think in a three-dimensional way, but be able to put it down into only two dimensions on a contest. So, you are given identical puzzle pieces, and you are not allowed to turn them over. So, remember that was like I said, my, the palm of my hand looks different from the, with the palm facing you, or the palm facing me. So do not turn them over. Which figure cannot be made out of these two pieces? So be very careful in reading. There are two negatives in this problem statement. Two things you cannot do. Nice work. A lot of people are trying in the chat. I'm going to try to lean in real close so that you can see I have two identical pieces. I'm not going to flip, where are we? I'm not going to flip them over. You can see the little black outline on them a little bit so that you can tell I'm not flipping them. I'm going to try to make each of the pictures without flipping them. For A, if I turn this one over and I try to put them together like this. I didn't flip it, I only spun it. I can rotate it. They start out the same. If I rotate this one and put them together, then I can get the picture in A. On my paper, on my contest, I might put the line right here that shows how the two pieces go together. That's the separation between the two. Now I want to try to make the shape shown in B. If I just take those square parts and put them together, then I can kind of make that hexagon. My fingers are kind of covering the bottom of it, but there's the bottom of the hexagon. If I draw that here, you can see that it would look like this. That's how I have the two pieces put together. From C, all I really need to do is stretch and slide that so that the rectangular parts are together. Then I can make the picture that's shown in C. This is just putting them end to end. Let's take a look at E, because E is possible. I'm going to put them in the other hands. If I just put the diagonal part together, you can see that I can make that shape that is shown in E. In showing that in E, take a look. From there, I can draw my line here. That shows the two shapes. In order to successfully make that shape that is shown in D, what I would actually have to do is flip one of my pieces over and then put them together. I can make the shape in D, but remember we said we can't flip. I could do this, but one of these has been flipped. This is the one I cannot make without flipping a piece. A very tricky problem when you're talking about, I think I can put it together in the shape. Let me see if I can draw this a little bit better. If I outline this shape, you can see that this is the mirror image of it. Because it's the mirror image, I must have flipped something over. The problem says I cannot flip something over. The impossible one is D. I believe that this is a question that you want to delete, Jacob. I will clear off my drawings. Hopefully, you maybe have a circle waiting for them. Anna folds a round sheet of paper along the middle line and then she folds it once more and then one last time. In the end, Anna cuts the folded paper along the marked line. You want to find what the shape of the middle part of the paper when unfolded is. I'll give you guys some time to think about this. Okay, so we're going to start going over it. There's a lot of very close answers, so yeah, hopefully you guys can learn. So what I'm going to do is I'm going to work backwards, so we're going to start from the end and we're going to work our way back to the original round sheet. So I have a, I have a, oh no, I can't see. Hold it right in front of yourself. It's awkward, but then the camera picks it up. Yes. So yeah, so this is a stripper sheet of paper, and then, so we're going to fold it right here in half, so we get here, or sorry, you get right here, then we're going to fold it right here, so another half, we get right here, then we'll fold it one more time, right here, so we get this figure, and then we're going to cut it with scissors, so right about here, and so we get something like this. So in a diagram, in a diagram, this is going to be right here, so this orange line, and then when we, so then we're going to unfold it, so we get something like this, right? So when we unfold it in our diagram, it's going to go back here, and then it's going to fold here, so we get another line right there, right? And then when we fold it back along the half, we get something like this, so in our diagram, it's going to be here, and then here, and then when we fold it, it's going to be reflected like that, and then when we fold it once again, it's going to be right here, so something like this, so like that, right? So when we reflect it in our diagram, it's going to be like this. So now, we have like two answer choices, it kind of looks like, obviously, so it can't be B, it can't be C, it can't be E, so we're looking at A and D, but notice that this line looks very parallel to this line, so when we reflect it each time, each of these angles should be like roughly 90 degrees, so our answer should be D. So yeah, I think it's kind of tricky. Probably the difference between A and D is pretty tricky, you know, it depends on how they made their cut, which one it'll look more like. I think you requested this one as well, Jacob, right? Yeah, so this question is also, oh wait, so a regular octagon is folded in half exactly three times until a triangle is obtained, as shown. Then the apex is cut off at a right angle, as shown in the figure, and you want to find when it's unfolded what it will look like. This is pretty similar, so maybe you have a stronger solver, right? So I'll give you some time to think about this. All right, so hopefully that was enough time to think about the question, but we're going to go over it. So, this problem is very similar to the other problem that we just did. So, again, we have a regular octagon, right? We have a regular octagon, and then we fold it right about this diagonal, so we get something like this. And then we fold it again right here, so we get something like this. And then we fold it again on this diagonal, so we're going to cut it in half again, and we get this right here. So, very similar to what we had last time. And then what we're going to do is we're going to cut it with scissors right here at a right angle, so it ends up being like this. We want to know what happens, so we want to know what the octagon, when we unfold it, looks like. So, we're going to apply some strategy. When we work backwards, this here will look like here. We're going to have a square here and here, because we're going to reflect it across the diagonal, so it ends up like this, right? And then we're going to reflect it again across here, so this step will look like this. Another... It's looking like a rectangle right now. And then we fold, so when we look here, this is going to look something like this right now. You can see it. And then we're going to fold it again across the main diagonal. So here, it's going to look something like this. Right, so we're going to reflect it and we get something like this. So it looks like a square inside this octagon. Once again, one of the answer choices also looks... So it looks like a square, so immediately we can roll out B, D, and E. But so we still have A and C, which both look like squares. So it's a little tricky, and I saw like some of us actually got tricked by that. So yeah, but when we look at it, in A, the side length of the square is parallel to the side of the octagon. But when we unfold it, it actually doesn't come out to that. So our answer should be C. Jacob, is there a clue you can give them as to why the point of the square is going to be pointing to a point of an octagon and not a straight edge? Sorry, can you say it again? Why is the point of the square pointing to a point of the octagon rather than the point pointing to a straight edge? Yeah, so notice that in this step right here, when we reflect it, this is a vertex of a square right at this point here. And so that point lies on one of the cuts. So when you keep reflecting it, that's going to be the vertex of the square, and it points to one of the vertices of that octagon. I think they need to have some sort of trick to know which way the square will be oriented, so that helps them. Hopefully that makes it a little clearer if they have to do a problem like this again in the future contests or whatever they're doing. Okay. Charles folds a sheet of paper in half and then repeats this form. I think they mean fold four more times. Then he makes a hole in the folded paper. How many holes does the sheet have after unfolding? There is a poll for this question so I'll let you look at it for a few minutes and then I will launch the poll. Sorry, I took the poll down because, at least for me, it is not displaying properly. I'm not getting the figure and I'm not getting the right answer choices, so sorry about that. Um, I think you're muted. I'm sorry was unmuted for that whole last problem as well. So what I was saying is, when we go, when we unfold. My annotation doesn't work. When we unfold we go from one to two, we multiply by two. When we unfold we go from two to four, we multiply by two again. When we unfold, we have our four on one side and our four on the other we're multiplying by two again. And if we keep that we will end up with 32 holes punched in the unfolded paper. All right. Eva cut a paper napkin into 10 pieces. She then also cut one of the pieces into 10 pieces. She repeated this process, two more times into how many pieces. Did she cut the napkin. Be careful working through this to follow the instructions exactly. I think that will help. I think that's most of you have answered in the poll. Anybody else want to take a guess? Remember, leaving a blank on a math kangaroo is always a zero points, but you have a chance of guessing correctly. So if you're getting to the end of time on a contest or a test where we do not penalize for an incorrect guess, then you should guess and maybe try to guess it right. OK, so we have quite a few of you, over 3 4ths of you, saying the answer is 37. That's really good. Excellent work. So if she takes that paper napkin and she cuts it into 10 pieces, then she takes one of those pieces and she cuts it into 10 more. So instead of saying 10, she's going to have 9, and then she has 10 that she cut that one piece into. She repeats that again. So instead of 10, she's going to have 9 and make a 10. And here's the final repeat. She's going to have 9 and make a 10. So then she gets 27 plus 10 is 37 pieces of paper. Vadim has a square piece of paper divided into nine cells. He folds the paper as shown, overlapping horizontally and then vertically, so that the gray square ends up on top. Vadim wants to write the numbers from one to nine into the cells, so that once the paper is folded, the numbers would be in increasing order with numbers one on the top layer. What numbers should he write instead of A, B, and C? Notice this fold is a little different than the folds we've seen before. This, I'm gonna call it an accordion fold. You notice it's a zigzag type of fold, not fold in half and fold in half and fold in half. These are thirds in a zigzag fashion. Because we're running out of time, I will save the poll here. More than half of you think the answer is A. That looks like your favorite answer so far. So I have the little picture. Let me try to get nice and close to the camera and see if we can get it to work. I will fold it zigzag, just like it showed, and then zigzag again with that gray square on the top. When I do that, you will notice that this square becomes square number 1, square number 2, and square number 3 in this little stack right here. Then this one is upside down. It comes in 4, 5, and 6. And then the 6 is going to be right here on top of 7, 8, and 9. So A is 6, B is 4, and C is 8. That matches A. I know we went a little bit over time because of the muting problems, and I wanted to make sure we tried to do those bonus questions if we could. So my recommendation is, if these are difficult for you, find some more practice problems like these. We have them in our past Math Kangaroo contests. Go ahead and do the cutting out, do the drawing the shapes, do the unfolding, and try to map it out because it is a training process. You do get better at these. You get better at visualizing what happens when I unfold it. What happens if I cut it this way? What would happen if I cut it that way? Experiment with it. If it's paper snowflake season, go ahead and make paper snowflakes. And if I want to make a certain design, figure out how you have to cut the edges of your folded-up paper in order to get them to come out the way you want. This is all exercise and training. Just like if I want to be able to draw a dog, I'm going to have to find some instructions and draw dogs over and over again until I improve. And everyone's like, oh, that's the cutest dog I've seen. It might take some of us longer than others, but you do get better when you practice doing these. If these were very intuitive and amazing for you, that's wonderful. You would probably be the exception. Jacob, do you find that these took you some experience to improve and get better at these? Yeah, definitely. So it works with all sorts of new challenges that you're doing. Just keep trying at them. Don't be harsh on yourself. And every time you succeed, you know that you're doing a little bit better than you did before you even started to try to practice. One more lesson next week. And there is going to be more webinars coming up in January. If you'd like to continue to try some new topics with Math Kangaroo, it would be great to see you. Don't forget to use our past contests to try all different types of problems and to learn how to solve them in the correct amount of time. Past contests are going to be a great way to do that. All right. You do get scratch paper on your Math Kangaroo contest. So don't be afraid of trying to draw things or even fold your paper a little bit if you're nice and quiet. All right, everyone. I'll see you next week. Bye-bye.

webinar

spatial reasoning

visualization skills

paper folding

geometric shapes

problem-solving

hands-on tasks

practice exercises

interactive learning

contest preparation