Thank you for joining. This is webinar number three in Math Kangaroos Level 3-4 series. This week we are going to be using a tool for solving math kangaroo problems. The tool is called Making an Organized List. That might seem a little strange, but rest assured I think by the end of the lesson you'll understand what we mean. Sometimes you have to map out all the possibilities to figure out which of those possibilities most correctly answers a problem question, right? So make sure that as we go through you're thinking, what does the question really ask? Read very, very carefully. How am I going to solve it? Today we're going to solve it by making an organized list, but this is just one approach you can use. Then we're going to work carefully and check our answer and make sure it makes sense at the end before we finally submit our responses. We do have polls for a lot of questions today, but not every single one. Sometimes it's a little hard to put figures and things into the polls. I will lead some questions and Shourya, who's a 10th grade high school student, is going to lead some questions as well. Okay? Alright, this is our warm-up problem and then we'll get into some specifics about what we mean when we say making a list. Which one of the figures on the bottom of the picture can you make using only the shapes shown in picture one? So picture one encompasses all of this. Which of the five different designs, A, B, C, D, or E, can you make if you use those same exact shapes? You're welcome to put your answers in the chat for me or for Shourya so we can see how everyone's doing. Shuri and I cannot reply to every comment in the chat, but we do our best to try to make sure we give you some feedback at some point during the lesson. I know it's a little difficult in the webinar format, so we're doing our best to make sure that everyone can learn together. I see lots and lots of correct answers. Everyone is saying that the answer is E. And since we're talking about making a list, let's go ahead and make a list of the things in picture one. I see two circles. I see one long rectangle. I see three different triangles. All right, the three triangles are different. So going through my options, choice A, I see that these two triangles are the same, and that doesn't fit my list. So I can cross that one out. I see some sort of square up here on the top of this guy. So the snowman is out. This one that looks like an ice cream cone has some small circle, and there was no small circle in the original figure. And then again, we have two same triangles, which doesn't match the three different triangles. So only E can possibly be correct. So if you did that, you did a really good job. All right, so last week, we were doing drawing a picture or a diagram. You'll notice that some of our problems this week are still gonna have pictures and diagrams, because once we've learned the tool, we don't wanna get rusty with it. We wanna keep using it. So you'll notice that sometimes we can combine things together. If you want to solve something today without making an organized list, that is okay. Just make sure that when you do another method, you still understand how to do the list. Sorry, I was looking at a message that was sent. All right, so the idea of a list is so that we can organize ourselves, right? And there's no problem with organizing. I make lists of things I need to buy at the grocery store. I make them on my phone so that I have it when I get to the store. I make lists of things I need to do before a big event. If I'm planning a party, I'll write down all the things I need to do before the party and I'll even organize that list. We'll talk about that next week when we talk about making a table. There may be different types of things I wanna list in separate lists and a table helps me do that. So I might wanna put all the food in one list and all of the clothes that I need in another list and all of the decorations in a separate list, right? Because those are gonna come from different places. One of the things we're gonna use today, not necessarily talk about is casework, okay? So casework is if I make this decision A, what options do I have? Okay, that's one case. The case is I chose A, so now I can only do things one, two, and three. But if I had chosen B, then I would be able to do things 13, 14, and 15. So sometimes you'll have to make a choice, pick a case, go through all the possibilities there, and then you might say, you know what? That was the wrong way. How many of you have ever done like just a maze puzzle or been to a corn maze? It's that time of year where we were thinking about corn mazes. So if you turn right, then after you turn right, you may have a couple of options for places you can go, right? But it might've been that you shouldn't have turned right in the first place, in which case you need to go back and turn left. Does that make sense when I use it that way? And some of you said you've been to a corn maze, yeah. So that's a very physical way of thinking about this option of casework, right? If I, oh, I turned here and these are all dead ends, so I now have to go back to where I made that turn and go someplace else, right? So we do that sometimes as well when we're doing our math problems. Let's see how that goes. Shura, you had said you wanted to leave number one, and there is a poll for this one. So the question asks, how many three-digit, oh, oops. How many three-digit numbers can you create using the digits three, zero, and seven and using each digit only once? So we can only use each digit one time in our numbers. So we can't have three threes or two threes, nothing like that. So we wanna find out how many numbers can we make by this? And all numbers will have three digits. So I'll give you all a moment to think about this problem and you can chat me the answer. I'll go ahead and launch the poll. The question is in the top of the poll, so you'll still see it. When you answer the poll, it is completely anonymous. Nobody knows which answer you've used, so don't be afraid to get it wrong if that happens. It just means you're learning. Our polls represent the same thing that happens with an actual math kangaroo contest. You have multiple choice answers on the contest, and on the contest, if you mark an answer incorrect, you just get zero points. No one subtracts anything from you, so there's nothing wrong with if you're not sure or you think it's this one or this one, go ahead and make that guess. Because if you do guess correctly, you will get points, and we will not subtract. So I encourage you to go ahead and put something in the poll. I'm going to close it and share it in just a minute, probably 10 seconds. Okay, Shura, do you want to discuss the results and how to do the problem? Yep. So I can see most of you all put answer C, and I can see a few Es, all right. So let's go through this. We want to use the digits 3, 0, and 7. Now we know it's a three digit number, so let's draw in three blanks. So one thing is we can choose what the first digit will be, and then we can move on to the second one, then the third one, and the third one will only have one remaining spot. So for example, if we put a 3 in the first spot, now our 3 is already gone, so we can use a 0 and a 7 next. So in the first case, let's put a 0 here. Now since we need three different digits, we're going to have to put a 7 in the last one. In another case, we could put a 7 in the second spot. That means we only have a 0 left. So we can see that for a starting digit of 3, there are two cases that work out. However, when we use the number, let's say 7, it's also the same because we can have another 3, and then we have to use 0, or we can use 0 first, but then we have to use a 3 because we need to keep all the digits different. So again, that's two cases. Now here, it's important to know that our number can't start with 0 because if we start with 0, it'll be a number such as 037, but that's not a three-digit number because that just is 37. That's just the number 37. So remember, numbers can't start with 0, so there won't be three-digit numbers anymore. So that's why both of our cases with 0 aren't going to count because they just simplify to two-digit numbers. So that's why our answer is going to be these two 7 cases plus these two 3 cases, which will give us a total of C, 4. So I think most of y'all got that right, and it's okay if you didn't write it out in that way, and if you just thought, okay, so I'm going to put a 3, and then I'll do 07 or 70. That's also okay, but just the important thing to know in this question is that you can't start off with 0. So there was making a list, and then there was a little bit of trick. Do you know about your place values? Excellent. So here's number 2. It looks like a figure. As a matter of fact, this is spatial reasoning. This is something we'll do later on, but I think you've had some practice with figures. We made a big cube using 64 small white cubes, and then we painted 5 sides of the big cube. How many of the small cubes have exactly 2 sides painted? I might have underlined some important words in the problem statement. I have that habit. I help you get a little bit of a head start. When you're reading the contest yourself, you should be looking for those keywords. I'm going to give you a little quiet time to work on this. Bye-bye. you Give a little more explanation. It says we start with white cubes, so the green color must be the paint. Right? So you're looking for cubes with two green sides. If a side is on the inside of a cube, it's not going to get painted, right? Only the things on the very outside are getting painted, so nothing that's hidden inside is painted. When we know that five sides of the big cube are painted, but the top is not, how many sides are there on a cube? So what color are all the other sides? And that's as much as I'll give for now. All right, I'm going to start by making a list. So there are a few ways you can make this list. I think I'm going to do it by layers. I'm going to consider this to be layer one, this to be layer two, this to be layer three, and this to be layer four. So I'm going to separate my cube into four layers. Now I gave you the hint that if five sides are painted, then every side that we cannot see must also be painted green because it's going to be six minus one is five. So the top is not painted, but the back is painted, the side over here is painted, and the bottom are also painted. These all have green paint. Does that make sense? So I can see in the front here that this square has two green sides, right? The top is white and the other pieces that are hiding won't get any paint. So all four corners, because it's a symmetric square and a symmetric cube, all four corners are going to look the same. So there will be four cubes with exactly two painted sides. These have only one painted side, right? These have two. And if I look at layer two, it's almost exactly the same, right? Because the top won't be painted because it's hidden by layer one. The bottom won't be painted because it's hidden by layer two. So I still have these corner pieces. So I still have four. Using the same logic, layer three would be exactly like layers one and two. So basically what you can see is I have three on each corner, so I get three times four for the top layers. Now when I look at the bottom layer, it's a little bit different. If I look at the corners of my bottom layer, they're going to be painted on one, two, can label one, two, and then also on the bottom, that's three sides on the corner. So the corners do not meet the criteria of having exactly two sides. It didn't say at least two sides, it said exactly two sides. However, if I look at this square right here, the bottom of that is green, right? Because the bottom of the whole big cube must be painted green. So the bottom of this square is green and the bottom of this square is green. So that gives me two. But as I go around my cube, I'll have even more of those. So I'm going to have two times four because there's four of those edges of the bottom. If this is the bottom edge, there's four edges on the bottom. So all together, I'm going to have 12 here and I'm going to have eight here. So 12 plus 8 is 20 cubes with exactly two painted sides. So while this might not look like a list because I didn't say here's my list, by list by kind of indicating one at a time and being organized about it, I do make a somewhat sort of a list. All right, number three. Joseph has four toys. A car, an airplane, a ball, and a ship. He wants to put them all in a row on a shelf. Both the ship and the airplane have to be next to the car. In how many ways can he arrange the toys so that this condition is fulfilled? This is an important part. Both the ship and the airplane have to be next to the car. How many of you have ever had, maybe you are walking with some friends and somebody says, I want to be next to so and so and the other person says, I wanted to be next to them. So where, so let's say, let's just call the person Joe. I want to be next to Joe. I want to be next to Joe. So where does Joe have to be so that the other two friends are both next to Joe? That will help you with this problem. Thank you. I do have a poll, so let me launch the poll for you. That way, I can see how everyone is doing. I do appreciate the people who are sending me some chat messages. Because it helps me to see what we need to explain further, or we have it all down. Sorry, I guess I was muted. So if your poll does not close, go ahead and close it up. We have most of us have said that the answer is four. So let's see what we can do. Remember in our drawing lesson, I said use very easy symbol drawings. So we're going to do S for ship, A for airplane, et cetera, right? It says the ship and the airplane have to be next to the car. So I have multiple children, and when my children all need to hold my hand, I have to have one on each hand, one on each side. So I'll have the ship and the airplane next to the car, right? But we also, the car can still be in the middle if I switch the sides of the ship and the airplane, correct? And now we know there's one other toy, which is a ball. So where is the ball going to be? Well, the ball could be here, or the ball could be here. And if we had an airplane, car, ship, the ball could be on the left, or the ball could be on the right. So that gives us four possibilities. We'll move on to the next question. Shuri, you wanted to lead this one, and there is a poll. OK. So we have that the pages of a book are numbered 1, 2, 3, 4, 5, and so on, just like any regular book. The digit 5 appears exactly 16 times, exactly. Now, we want to find out what the maximum number of pages this book can have. So keep in mind, they're looking for the maximum number, so not just how many could they have, but what is the most, what is the highest possibility? And we know the digit 5 is what we're concentrating. So I'll give you all a moment to think about this, and you can chat me the answer once you have it. OK. Thank you so much for having me. Bye-bye. Are we ready to try the poll? Thank you. Thank you. Anybody else want to make a guess in the poll before we close that up? There's a lot of different answers in this poll right now so that's good. We'll have something to explain. I've shared the results with them so they can see. Do you want to go ahead and summarize that and then explain the problem? So, yeah, lots of different answers. One of the things I think may have thrown some of y'all off is that the problem asked for how many, like it had 16 times the digit five. And I got some answers that said, I got some people that said it was 80 so I think maybe you thought that it was multiples of five and not digits. But remember, digits means like in a page, the digit must have a five either in the ones place, the tens place, like in the number. So for example, 15, like if you had the page number 15 that has the digit five, if you had something like 50 that has the digit five. So the question is going to ask us, what is the maximum number of pages that this book could have? So I'm going to start by making a list of the fives. So the questions already told us 1, 2, 3, 4, 5. So that's our first five, just the number five. Now we flip 10 pages. Now we have the number 15. That's another five. Then we get 25. That's another five. So we can keep going. Let's do 35, 45. And now when we get to 50, this is important, 50 has a five in the tens place, not in the ones place, but in the tens. So that also counts. We just want to see how many times the five appears. So now let's make a new list, but this is going to be this starting from 50. So we're going to have 50, 51, 52, 53, 54, 55, 56, 57, 58, 59. And then 60 doesn't count because it doesn't have a five. So now in our new list, notice that 55 is important because these answer choices are trying to trick us. Remember, 55 has the digit five two times, not one time. And this is important because that adds to our 16. So right now we have six over here, 5, 15, 25, 35, 45. Sorry, not six. We have five over here. Oops, it disappeared. One moment. Yeah, so we have five over here. And what that means is that we... Oh, my God. Okay. So if we make a list from 50, we have 50 through 59, right? So at first glance, that looks like it has 10 numbers in it. But remember, 55 has two fives. So instead of 10, we add one. That gives us 11. Because we need to count the double fives anyways. Thank you. So on this left list, we have five fives in this whole area. So we can see that five plus 11 is equal to 16. However, we're not done yet. So our answer isn't going to be 59 because we want the maximum page number. So remember, our book can still go on until we have another five. So the next five is going to be a 65. Since we can't have that, we're going to go one to the left. And we're going to have 64. So that means that our answer is going to be 64, which I believe is B. I think one of the tricky things in this question was understanding that 55 is going to have two and that numbers 50 through 59 need to be counted, all of them, just because they have five in the tens place, while all the other numbers, they have five in the units place. So like 5, 15, so on. Thank you, Shourya. I think that explains it very well. So just be very careful. These kangaroo questions are trying to make you think in different ways than just plain school problems. Next problem is number five. I like to explain this one using our casework concept. So when I go to explain it, you'll hear me talk about in the case that we go along the top, in the case that we walk along the bottom. So that's my hint. In the botanical garden shown in the picture, visitors walk only on marked paths. In how many different ways can one go from greenhouse A to greenhouse B if you only walk on a given path once? And we have to get from A to B, not from B to A. The paths are the uncolored parts. These are the paths. If it's got this stippled color, then you can't walk on it. So hopefully that helps. If you printed out the handout before class, you should have this picture so you'll be able to mark on your own picture. Please do not draw on my screen. Thank you. Thank you very much for joining us today, and we will see you next time. Okay, I'm getting quite a few responses, some correct, some close. So there are a few ways to do this. Last week we did drawing a picture. So we can clearly do this with drawing a picture and I'll do that first. But remember I said I was going to use a little bit of casework. So let's do in the case that from A we go along the top. So I can going to use different colors. I can come up and make the very shortest path possible. That would be from A to B. Right. I took all the first turns and I did that. Now let's say I went the same way over here. I overshot a little bit and then I came back. So that could work too. Now let's say I, instead of taking this right hand path, I took the left hand path and I'm still going along the top. I'm going to take the first path and then finally I haven't used blue yet. Sorry. I haven't used blue yet so I can do the longest path along. So if I go along the top, there are four paths that I can draw. Okay. Now this is a symmetric drawing, right? I can flip it upside down. So what if I want to go along the bottom? Because it's symmetric, I can say that there's the same number of paths that would go along the bottom as go along the top. So along the bottom, I can also do four paths. And I don't even have to draw them because I can see that I could just flip it upside down like in a mirror, right? So the answer is eight. Now, this is sort of a list because I've been drawing and making some conclusions about the top and the bottom. I definitely used casework. But if I want to use a more clear list, one of the things I can do is I can label the intersections one, two, three, and four. So now I can make a list using those labels. There's never anything wrong with adding a few more labels to a figure. Is that an approach you've used sometimes, Shreya, in solving some problems? Yeah, it's not easy, I guess. So now the short one is A, one, three, B. And then if I wanted to go through two, I can go A, two, one, three, B. And I can go A, two, one, three, four, B. And I can go A, one, three, four, B. So those would be the paths along the top. So there's still four of them. And I can map out the exact same thing going along the bottom. In this case, I would go A, two, four, B. A, one, two, four, B. A, one, two, four, three, B. And A, two, four, three, B. So that's the same four and four paths giving me the same answer, only I was able to make a list in a little more of an alphanumeric way because I added some labels. So either one of those approaches is a really good way to do this problem. Sometimes there are so many things to draw, so many different lines, they'd be crisscrossing each other and become very difficult to count up if you had made the lines. That's also where noticing the symmetry gives me half as many lines to count and half as many things covering each other up, so that's easier. Okay, so we are going to start on some bonus problems right now. So I'm not going to do the exit just yet because we have some extra problems. I'd love to try them with you. So I do have a poll for this one. In a certain trunk, there are five chests. In each chest, there are three boxes. And in each box, there are 10 gold coins. The trunk, the chest, and the boxes are locked. Everything's locked. At least how many locks need to be opened in order to take out 50 coins? Have you ever gotten one of those packages that has you open the box and there's a box inside the box and then you open that box and there's a box inside that box? That's kind of what we're talking about here. I'll be quiet and I'll let you think about this one for a little while and then I'll launch the poll. Anybody else want to put their answer into the poll? I know some of you said you wanted polls for every question, so you should definitely be putting your answers there. Thank you, that's better. Okay, so over half of you said eight, but I'm seeing 10 and 15, and I know there were some other answers that came to me through the chat as well. So let's see how it looks. All right, so let's take a look at this. This is one where you could draw. Remember last week we had the drawing, and we drew pictures of three different sizes with small, medium, and large, and we could do that too here. We could, you know, we could make a big trunk, and then we could put five chests inside the trunk. That would work, but it's a little, gets a little cumbersome to do all that drawing for this particular question, right? See if I can clear, there we go. So how can we do it with a list instead of trying to draw? There's two ways. We can start at the very first lock we have to open, or we can start from the inside and work our way out. So I'll do it both ways. First, I know that I'm going to have to open the trunk. The trunk has to be unlocked, right? And once I unlock the trunk, I'm going to unlock one chest. When I unlock one chest, there will be three boxes, and I do need to get those three boxes, because that gets me to 30 coins, right? Because there were 10 in each, so three times 10 gets me 30 coins. But the problem says, how many do I need for 50 coins? Well, that was all the boxes in that chest, so I'm going to need to have another chest, and in that chest, I'm going to have to open some boxes, and I only need 20 more coins. I only need two times 10 coins, so I'm going to only have to open two boxes. Now, if I take the sum of all of that, that is 1, 2, 5, 6, 7, 8 locks that I've had to open. Now, I said you could do it the other way. I know that I need 50 coins, so 50 equals 5 times 10. So I have to open five, excuse me, five boxes, but there are only three boxes per chest, so I'm going to have to open two of the chests, and to get to any of the chests, I'm going to have to open the trunk, so that also gives me eight. Excuse me, a tickle in my throat. All right, so either way, from the inside or from the outside, we get the same answer, so that's good. Remember, step four is check your answers and make sure it's reasonable, and if I can get it the same, excuse me, from both ways, then I know my answer is probably really good. Do you mind reading the question for me for a minute, Shurya, so I can get a sip of water? It's arrows at the target shown in the picture. When she hits the target, she gets the points in these sections that she hit. When she misses the target, she gets zero points. Paula shoots two arrows, and she adds the number of ones. How many different scores can she get? We've made this problem especially tricky, so be very careful. She shoots two arrows, so she can score with both. She can score with only one arrow, and one arrow could miss, because it says she misses, right? So think about it. She could miss with both arrows, so zero is a possible score. I'm getting lots of responses. I'm not getting very many correct responses, so think very carefully, make a list, be careful, and I will launch the poll for you. All right, you must be working very carefully because you're taking your time and that's good. But I'm going to go ahead and start explaining because I'm afraid we'll run out of time before we finish this problem. And I like this problem. So I'll share the results of the poll so far. We have most people saying seven, but we have replies for every single answer choice. So what that means to me is this problem needs some explaining, right? We want to make sure that even if we're not getting everything correct when we start the day, that by the end of the class, we say, all right, then we understand now. And the next time I see a problem like that, I'm going to keep that in mind. I'm going to remember that they tried to trick me with that before, and I'm going to get it right. So I'm going to make different cases, right? The first case is that she misses two arrows. So then the score is zero. Now I'm going to make the case that she misses with one arrow. So the score for the other arrow could be 30, 50, or 70. Now the final category is that she hits twice, right? If she hits twice, she can have a 30 and a 30. That equals 60. She can have a 50 and a 50. That equals 100. She can have a 70 and a 70. That equals 140. Because there's enough room to hit each place with two arrows, right? So that's possible. And then she can have the combinations. So we can have 30 plus 50 and 30 plus 70. When we add, we get an 80, we get 100. If we got a 50, then you can get a 70, and that is 120. Now, when I look at that, I'm going to notice, I hope I didn't miss any. When I look at that, I noticed that the 100 appears twice. Do you see that? And they asked about different scores. So I'm not going to double count 100. Now let's see, are all the other scores unique? I think they don't repeat. So now I can count up how many different scores are here. One, two, three, four, five, six, seven, eight, nine possible different scores. So some of you might've thought that it was 10 because there were 10 ways that she could score or not score. Sometimes students forget that, oh, you might miss and have a zero. But sometimes students forget that you could get a 30 and a 30 or a 50 and a 50. So that would give you maybe answer choice A if you didn't count those as well. So hopefully, now it's crystal clear and everyone's like, of course, nine was the right answer all along. So we are going to have to stop there. We are going to go back to the slide that had the wrap up on it. So I hope you will make use of this organized list strategy. Your list doesn't have to be super neat and pretty just so that you can read it and understand where you are. If you need to count things up, you gotta make sure that you can follow your own writing. I always like to keep my problems separate when I'm doing things on notepaper so that I don't have to try to figure out what I wrote in the margins and what got squeezed in over here. So try to use the paper in a nice organized way. Which of these problems did you find tricky? Which of these problems do you think that the list really helped? For me, I think that that last question, B2, the list really helped. I kind of find the opening the chest problem tricky, but if you wanna send me a little chat message, you can tell me what you thought. Okay, so remember, you have a code in your Math Kangaroo registration for the contest that will allow you to look at solutions, video solutions. You have a discount code included with this webinar so that you can do past Math Kangaroo contests. They may still be a little difficult for you if this type of a contest is new and you're not very experienced, but as we do more sample problems together in our webinars, those contests should get better for you. You should be able to work a little faster, a little more accurately. You should be feeling more and more confident each time you do a practice test. Do you have any advice in our last 30 seconds, Shreya? What did you think of the class today? So- Especially for casework problems, things like if you have diagrams, you wanna find out which way to go, things like that. And even in places where you might not think it useless, like in the cube problem, in the chest problem. So yeah, I'd say, you know, use your paper, do it organized. Yeah. And I wanna thank Shreya for helping me when I had that tickle in my throat. This is what teamwork is about. So hopefully you sometimes work with teams as well when you're doing your own schoolwork and other projects. I'll see everybody next week. Thanks so much for coming. Bye-bye. Bye.

Math Kangaroo

Level 3-4

webinar

organized list

problem-solving

casework

math strategy

student interaction

real-time feedback

practice resources