I'm going to go ahead and start a little bit about the presentation. That way, those of you who have joined us on time or even a little bit early aren't sitting there doing nothing, but I'll go slowly because I do suspect we'll have more people joining us in the next few minutes. Okay. My name is Dr. Sarah Sagie. I'm going to be leading this introduction to Problem Solving with Math Kangaroo webinar level 3-4. This is our inaugural webinar for this grade level. We had one for younger students just an hour ago, and this is the one for levels 3-4. It is aimed for students primarily in third and fourth grade, although if you're a little younger or a little bit older, I bet you'll still learn a lot from this webinar. This webinar, we're going to talk about just a little bit about how the webinar is going to work. I'll introduce myself. We also have a TA who is a high school student. He will introduce himself. Hello. I'm Alan. Hello. Oh, sorry. We have an introduction slide, Alan. It's okay. Okay. Today, we are going to discuss the four-step problem solving method. This is a great method for solving all sorts of math, science, and even other types of problems. Then, we're going to go over, there's a big difference between a Math Kangaroo contest and the math that you do in school. A lot of people frequently ask me, if I get straight A's and always ace my tests in school, will I do well on a Math Kangaroo contest? The answer is yes and no. If you're the type of student who likes to think about different types of problems you can do with the skills that you're learning in school, then yes, you could do very well in a Math Kangaroo contest. If you're the type of student who just likes to do exercises, I'll explain those a little bit later, then Math Kangaroo is going to be definitely a new experience for you. So, stay tuned. We're going to talk about what kinds of different math problems you'll see in Math Kangaroo that you probably have not seen in school, but with practice and experience, you will learn how to do them. Today, I'm going to give you a few of each type of question so that you can see what that's all about. Okay, a little bit of housekeeping. Your microphone is going to be muted throughout this presentation. Right now, I have 213 attendees in the meeting. I'm expecting over 300. If you were all able to unmute and speak, we would never hear you. It would just be too many voices at once. The chat is always open. I'm not always going to be able to read the chat, but Mr. Alan Kapler will be following in the chat, and he will answer the questions that he can. We also are going to be having polls. When I have a question on the screen, I will give you a few moments to work it out. So, I hope you have some scratch paper and pencil with you so that you can work through the questions, and you'll be able to put your answers into the chat or into the poll. Like I said, if poll questions are not working for you, if you cannot see the pop-up window for polls, you can go ahead and type your answer into the chat. That way, Alan and I will know if you're following along. I hope that makes sense. So, both the chat and the polls are anonymous. No one knows who's answering the question, so you don't have to worry about feeling bad if you make a mistake, and unfortunately, you're not going to get a big high-five if you get it right, but you can give yourself that high-five. Your video is off, and no one will see you do that either, okay? My name is Sarah Segee. I'm a biomedical scientist. That means I went to college, and then I went to graduate school, and I studied actually cancer and Alzheimer's disease. I've had to use a lot of math, writing, and speaking, public speaking, in my career, so that's how I became a math teacher. I've been tutoring and working with math for 30 years. My children go to Deer Canyon Elementary School in San Diego, California, and I've been coaching their math club for six years now. I have a dog. He's actually in the room with me, but you probably won't even notice he's here. I take him walking every day, and I also can teach karate and swim lessons. I'm going to let Mr. Alan Kapler introduce himself. All right. Hello, everybody. I'm a junior in high school. Math Kangaroo was the first math contest I ever did. I've been doing it since first grade. I've finished pretty high overall in the contests, and I really like helping other people love math as well. I have two dogs. I'm in a choir in my spare time. That's me, basically. We just wanted you to know that we love math. We love to study, but we also have other interests too, and we're sure you're the same way. You don't have to make your whole life about the math contest, but that doesn't mean you can't enjoy it as well. Okay. Now that we've done our introductions, let's get onto a little bit about the lesson. We're going to talk today about problem solving coming in four steps. The first step is to always understand what is the problem. You have to determine what exactly it is you need to answer. One way to do this, you may have seen in schools, your teachers will sometimes ask you to underline important parts of a question, and I may be doing some of that today. It's important to answer the question that's being asked. Math contests are famous for trying to ask a question in a tricky way. We'll see if there are any of those today. Once you understand what the question asks, then you need to figure out how will you solve it. Do you have the information that you need? Do you have a plan? Do you know what to do with that information? If they gave you some numbers and data, do you know how to calculate it? Do you know what to do? Once you have a plan, then you actually have to do the plan, of course, right? So I can say I have a problem, which is I haven't gotten enough exercise, and I need to get some more exercise. So what am I going to do? I'm going to make a plan. My plan is I'm going to walk my dog one extra mile every day, and then I actually have to do it, right? I have to carry out my plan. I have to put the dog on the leash, put on my shoes, and make the time to go walk that extra mile every day, and then I can look back, and I can say, did I actually get that walk today? And if I didn't, I can fix it. I can still go out for a walk, right? When you are in a math contest or in school or doing science questions, you need to look back. Does the answer make sense? Does it look too big? Does it look too small? If they ask for a sum, did I give the sum? If they ask for a list, did I give a list? So make sure you look back and check to see if your answer is reasonable. And this list of steps works for lots of problems, not just your math problems. So you can keep track of that. All right. We're going to talk about the different types of math questions you'll find on math kangaroo contests. If you were here at the very beginning, you heard me say that math contests asks different questions than questions at school. Questions at school, I like to call exercises, because your teacher shows you examples of everything, right? She might even give you a study guide. If you practice all these on the practice test, then I'm sure you can get 100% on the test tomorrow. Does your teacher say that? Math contests are different. Math contests, we're giving you problems. You're going to use those skills that you've learned in school, and you're going to have to solve them to a new type of problem. So once you're used to a problem, then it just becomes an exercise. You can do it over and over again without any difficulty. But when you face a new problem, you're going to have to apply your skills to that new problem. And math kangaroo problems come in different categories. These are the six ones you find the most. The first one is number problems, and those will be most like what you see in school. So in school, when you have to add, subtract, multiply, divide, put things together, use your appendix, your order of operations, those are our number of problems. Then in school, you probably also have geometry and measurement. You can use a ruler to measure things. You might be given a figure and shown how far things are on a scale. And hopefully, you've been introduced to shapes, squares, triangles, rectangles, at least basic shapes, and you know some things about them. We'll see some of those today. Then there's visual thinking, which is not common in school, but is a big part of math kangaroo questions. And it also goes with spatial thinking. So you'll be asked to look at a picture and get some information from that picture and make some very educated guesses and check yourself using a picture. A spatial problem is a picture, but then you have to be able to use it in three dimensions. So you might have to turn it, twist it, look at it from the top, look at it from the bottom. You might have to pull it. If you have something, you might have to open it up and see what would be inside. Those would be a spatial thinking problem. Clocks also goes with calendars. Hopefully, when we get to that, I'll introduce that section a little bit. Hopefully, all of you know how to tell time on a face clock. That's the round clock with numbers. Also, hopefully, you know how many days are in a week and the days of the week and the months of the year. All of those, math kangaroo is going to assume that you know already. So if you don't know those very well, you should review them before you take a math kangaroo contest. The last is our logic problems. And those are going to be your puzzles. Those are the whodunnits and the fill in the missing blanks. And they're a lot of fun, but they're not something you see in school. And there's something that you'll need to practice if you want to get a top score on math kangaroo. Plus, I find that they're really, really entertaining to do. Our first type of question we're going to do is the most common, the most like school work. So if Mr. Allen will launch our first poll that will go along with this question. I didn't write the question. It's just a coincidence that my name is in it, but it's pretty funny. All of the questions you're going to see today come from math kangaroo contests that have been in the past. And you can actually see, you can see right here on each slide, it will tell you what year this contest question appeared. So they're real questions. The higher the number, probably the more difficult the question will be. Math kangaroo contests, the questions come in three point values, four point values, or five, three, four, or five point values. So this is going to be one of your lower point value questions. Sarah has 16 blue marbles, and I made it a little easy for you. I showed you the blue marbles on the screen. She can trade marbles in two ways, three blue marbles for one red marble, and two red marbles for five green marbles. What is the maximum number of green marbles she can get? If you don't have the poll, go ahead and put your answers in the chat. It will be fine. We'll just, we're just trying to keep track of who's able to follow along. So I'm going to actually use my drawing. I asked you to have scratch paper. That's so you can do things like this. You don't have to do fancy drawings. It's your scratch paper. If I can trade, if I can trade three blue marbles for a red marble, I can circle this three in red, and I know that that's one red marble. I can do the next one. That is also one red marble. Here's three more blues makes one red. Three more blues makes another red, and there are three more blues that makes another red. But the question does not ask me how many red marbles I can make, does it? It asks me how many green marbles can I get? And it says for every two red marbles, I can get five green marbles. So this is two red marbles. So this group here would be five green marbles. That takes me to here. This group here is another five green marbles. And I don't have two more red marbles, there's only one. So I can't possibly get another green marble. So the answer in this case is that I can only get 10 green marbles. Alan, can we go ahead and show them the results of the poll? They've done quite well on this question. We had a rush of people toward the end, so maybe we should tell them when like 10 seconds are up. Okay. So what do you think? You did quite well. Most of you, 87%, have said that there are 10. So this is an example of the type of problem that we'll be doing. Here is another number problem, but you can notice this one has pictures involved in the question. So this would also partly be a draw a picture or make a visual question. The weight of a dog toy is a whole number. Does everyone know what a whole number is? A whole number is the numbers 0, 1, 2, 3, like that. So it does not include negative numbers. It does not include any fractions. The weight of a dog toy is a whole number. How much does one dog toy weigh? This question requires that you know something about a balance. The heavier object goes down and pushes the lighter object up. So from this question, we know that one dog from this side, one dog weighs less than 12 kilograms. And from the side on the right, we know that two dogs weigh more than 20 kilograms. And if we think about it, we also know that one dog is going to weigh more than 10 because all the dogs weigh the same amount. So if two dogs weigh more than 20, one dog weighs more than 10 and one dog weighs less than 12. So you should have been able to find that the correct answer is 11 kilograms. Let's go ahead and end that poll and share the results with you. I see in the chat, we're getting a lot of correct answers as well. So thank you for participating. This is a question in geometry and measurement, so remember the geometry and measurement questions are about shapes and distances. So what you see here is this picture shows the distance in miles between certain towns A, B, C, D, E, and F. What is the distance in miles between towns C and D? So we want to know this distance C to D. I'm going to give you a few moments with the pole to try to work it out. Notice this picture is not drawn to scale. So you can't just try to estimate based on the size of the pieces. It's not really to scale. You notice the five is pretty big compared to 34 and 21, right? There is more than one way to calculate this. So I'm going to show you at least one way. The way you have calculated it may be different. That does not mean it's incorrect. The way I'm going to do it is I have noticed this. This is the whole length here, right? And if I look at this 23 and 21, I notice that their overlap is exactly the length of CD. That's how much they overlap. So if I do my 23 plus 21, that would include CD twice. And that's going to be my 44. That included CD twice. So if I take 44 and I subtract what I see at the green arrow, the whole distance 34, that only included CD once. I find that the difference there is 10. And for me, that's the fastest way to calculate this one. But there are other ways. You could actually figure out the length of every single part by doing different things. So I could find the length of this little part because I know that 21 plus 8 plus a mystery number has to equal 31. And I can do the same and find this part because I know that 5 plus 23 plus this mystery BC piece has to equal 34. So you'd be able to then use all those different distances to find the distance CD. OK, let's see how well you did. Pretty well. We have another geometry question coming up. This one is about shapes. So hopefully all of you are familiar with squares and rectangles. A square is a four sided shape where all the sides are the same length. A rectangle, the sides are opposite and parallel, but they're not always the same length. The square is a special rectangle. A large rectangle is made up a number of squares of various sizes. The small squares here have an area of one. What is the area of the large rectangle? This question assumes that you know how to find the area of a rectangle, which is the area of the length times the width. In order to find the length and the width, we're going to need to know the length of the long sides to find the length, the length and width and the area of the large triangle of the large rectangle. It also gives us a very interesting piece of information. It tells us that the small squares have an area of four. If the area of a small square is one, sorry, it's not four, it's one, then the length times width must be one. So if the area equals one, that means that length times width equals one. And the only way that works nicely is one times one equals one. So what we're going to end up with is we know that this is a length one. And so this length here is going to have to be a three, right? Because it's three of those little ones. This length is going to end up being a four because it's this three plus a one. This length here is going to be three plus four, seven. This length here is going to be seven plus four, which is eleven. So that's the same on the outsides, right? And now we know it's a square, so this is also eleven and this is seven. So the total across the top or the bottom is going to be eighteen. So now if we come back here, we know that our area equals eleven times eighteen. And if you do a little bit of calculation, can we have the results of the poll? We'll give you ten seconds for that, and then we'll put up the results of the poll before I put out the answer. It does equal one ninety eight. And if you don't know the trick for multiplying elevens, one little trick for multiplying elevens is if I'm multiplying eleven by eighteen, I can put the one here and the eight here and put the sum of those two in the middle. One plus eight is nine. So that frequently works. This is the first of our visual problems. Visual problems require you to look at an image and make some opinions about the image. Three people walked across a field of snow wearing muddy shoes. In which order do they do this? Okay, we have some interesting things to think about. We have these, you should have it on your handout if you printed your handout as well. We have these pictures of muddy footprints. If I have a muddy shoe and I walk in the snow, I'm gonna leave some mud behind on the snow. The next person who walks is going to partially cover up my footprints. So that's what we need to know is that the first person, their footprints are on the bottom. So the first footprint is on the bottom. The other thing we need to figure out is they've given you these pictures, but they didn't tell you which order they go in. So we're gonna have to assume that what they mean here is they're gonna read from left to right. So that this would be the first, this would be the second, and this would be the third in order. Because they didn't tell you what those pictures really mean. But because we're taking an English language test, we'll assume that they mean left to right. So the footprints that are on the very bottom of all the others are these footprints with the circles on them. They're below the stripes and they're below the ones with the ovals. Then you can see that these striped footprints come in and they're on top of the circle footprints, but below the ovals and the ovals cover up everything else. So we can end that poll, five, four, three, two, one, right? If we end our poll, you'll be able to see most of you did get the correct answer. That it was A, we have the circle footprints on the bottom. This one is actually first. The striped footprints come next. And on top of everything else were these ovally footprints. Okay, that one, I have to tell you, it's a number six question. That's gonna indicate that it's one of the three point questions you should be able to do quickly. The next one is a bit trickier. Okay, so get ready. I hope you have it on your handouts as well. Which key would be impossible to cut into three different figures of five shaded squares? Now, remember I said the first step is to determine what the question is asking. So this question is a little bit more confusing to me anyway. Maybe you're smarter than I am. Quite fine if you are. We have to be able to cut the key into three different figures. And each of those figures has to have five shaded squares. So the first thing you need to think about is how are you gonna cut it? You're gonna cut it by making, if I want three pieces of something, if I have something and I wanna turn it into three pieces, then typically what I would do is I would cut it twice, right? Cut and cut. Now I have three pieces. So that's the idea here. All right, so we have to cut it in three pieces, but each piece has to have a total of five little squares. So I have one, two, three, four, five. I know it's a little hard to read my writing when I write this way, but it's the best I can do on annotation. So I'm gonna make a cut here. Then I need to count another five pieces. One, two, three, four, five. I could make a cut here, but the question is asking me to cut it into three different figures. And it says, which one is impossible? Now this one, I could cut it so two figures are the same, but it's asking me, can I cut it so that they're not the same? So I'll erase that line and I'll try again. What if I cut it one, two, three, four, five? What if I cut it like this? Now I have one, two, three, four, five pieces and all three of those pieces look different. So in this first case, it is possible. I didn't have to, but it was certainly possible. You need to look for the figure where it is impossible to cut it into different looking figures. It looks like this question is a little more tricky. We have fewer students answering it. I'll start cutting some of these. I like to cut them up from the bottom because you need to have the five pieces connected to each other. So I know that these bottom pieces are gonna have to be cut this way. Otherwise they wouldn't have five little squares all together. So one, two, three, I could cut this one like this and I have different pieces. One, two, three, I can cut this one like this. And I have two of the same pieces, which is what the question, they're different because this one comes from down here. One, two, three, four, five. This little piece makes it different. I can cut this one this way. But when I get to B, look what happens. One, two, three, four, five. I can cut B like this. But then if you notice, let me change my colors. This piece here and this piece here have exactly the same shape. So let's try again. Let's try erasing those lines and we'll try again. What if I cut it in a different way? What if I cut it like, oh, I don't know. One, two, three, four, five. Then you'll notice that this piece and this piece still look the same. So we're gonna end the poll. I was able to cut A, C, D, and E into different looking pieces. But for B, no matter how I cut the top of it, I cannot cut it into different pieces. So the answer here is B. And this is a more difficult question. So this would be one that would take you more time typically on a contest. So those last two questions were about shapes and visual pictures, but they were flat, right? We were assuming everything was flat. Our next one is what we call a spatial image because we're not assuming that it's flat. This is as if you're weaving strips of paper or fabric. Six strips are woven into a pattern as shown to the right. So this is how they were woven. What does the pattern look like from the back? So you're gonna have to be able to imagine that you're looking at the back of this or that you flip it over and look at the other side. And I'm gonna give you a hint. You can think about, look, this blue one goes over the yellow and under two yellows. Over two yellows, under one yellow. Under two yellows, over a yellow. And if you see it from the back, it would look the opposite to that, wouldn't it? Where we see over from the back, it's gonna look like an under. I'm going to take a few notes here. I'm going to do O for over and U for under. So this is over, over, over. This is under, under, under up. All right, we'll give 10 seconds for the poll. I know we're not giving you a lot of time, but it's because we have to cover six types of questions. So we do have to move along in order to show you examples of all the questions. You're muted, or for whatever reason, nobody can hear you. Is that better? It is. OK. Sorry about that. Sometimes my microphone gives out. I have spares. So what I was saying is if I fold off a corner of a piece of paper, it's on my right-hand side. But when I flip it around, it's on my left-hand side. We're going to end the poll here, Alan, because I don't want to end up having to miss too many questions at the end of the lecture. This is C. Where I have here on the right-hand side, these two that went under and over, I now have two overs and an under. Where I have over, over, under, I have under, under, over. It's exactly the opposite. Where I have over, under, under, I'm getting under, over, over. So C is the opposite. This is our last of the spatial thinking problems. How many of you are familiar with the concept of a net? That's the way this would be introduced in school. My audio off again? Alan, do you hear me? Yes, but it's quiet apparently. We try changing microphones. Sorry, everyone. Do you hear me better when I use that microphone? Alan, that didn't help? Apparently not, no. Does this, do I sound better on this microphone? Did that one work? I'm guessing not. It did. You can hear me now? Yes, I can. Excellent. Thank you for letting me know. We will keep trying to troubleshoot. Okay, so in school, you probably have seen a net. That's what they'll call it where you have a flattened out 3D shape and you have to fold it up and see what kind of shape it will make in three dimensions. We can go ahead and give them the next, there we go, thank you, Alan. So this is that type of question. They're calling it a plan, but it's the same as what you would have as a net in school. You're gonna take this shape. If you have your handout, you could even cut it up and fold it. There is nothing wrong on a math kangaroo contest with having a piece of scratch paper and folding it up. That is not against the rules. So you're allowed to experiment. You're not allowed to get up. You're not allowed to use a calculator, but you're allowed to experiment with your pen and pencil and your paper. If I fold up this one here, you'll notice, if I just fold it and I make this the top, so I'm looking at it, where do these two little black squares go? They're gonna end up on the bottom, right? Because these four white sides, just gonna put dots on them. These four white sides end up being the sides if I make this one here the top, right? So the little squares will not be next to the black square. So this one does not work. The little squares are not next to the black square. Also, both of these little squares are gonna end up on the bottom, aren't they? So this one doesn't work because it shows the little black squares not being on the same face of the cube. You probably won't have scissors when you do the contest, but you can certainly fold up your paper. But for now, because this is a practice session, you can go ahead and cut it out because we have to get used to what things would be like, right? So whenever you're practicing, you can use the tools you have available in order to teach yourself what to do. So just think of every practice as a teaching opportunity for yourself. And then on the real test, you'll use everything you've taught yourself. When I put these on the bottom, let's see, how will they be arranged? If I think about the bottom face, this one is gonna end up folded up and it's gonna end up over here. This one's gonna end up folded up over here. So which of those three figures shows a cube with one face that's solid black and the opposite face, the bottom face, is gonna look like the checkerboard? Now we're getting it. Now the chat answers are coming in correct now. We've given enough clues. All right, Ellen, I think we're good. We can reveal the poll on this one. It is gonna be E. You notice you cannot see the solid black top. Where is the solid black top? The solid black top is hiding over there on that side where we can't see it. But we know it's none of the others. So you can choose E because we know the others are wrong and it must look like a checkerboard. We're gonna move on to clock problems. This is a particularly tricky clock problem because not only do you have to know how to tell time on a face clock, but these clocks do not have numbers on them. So hopefully you know where to place your numbers. I noticed four clocks on the wall. Only one of them shows the correct time. One of them is 20 minutes ahead, another 20 minutes late. And yet another is broken. What time is it now? So I think it's pretty obvious what the question wants us to answer, right? What time is it now? Remember, we have to look at the problems, look at the information that it gives us and come up with our plan. One clock is correct. One's fast, one's slow, and one is broken. So there should be three clocks that show us times that are related to each other. And one clock that shows us the time that we can't relate by being either 20 minutes slow or 20 minutes fast. And we want a clock that shows the middle time, not the fast time or the slow time. So one approach to this is to determine what time each clock shows. And one common little mistake is to think that the one goes at the top of the clock. But actually 12 always goes on the top of the clock. There are 12 hours, and so this would be one, two, three, four, five, six, seven, eight, nine, 10, 11, and 12 at the top. I hope that most of you know that this hand is the hour hand. And the long hand always shows the minutes. And when we go around the clock, this is zero minutes, five minutes, 10 minutes. You're counting by fives going around the clock for the minutes. You can be putting your answers in the chat and in the poll as I'm explaining. So given how to tell time on a face clock, I can start determining what time each of these clocks say. The first clock, the hour hand is between four and five. So I know the hour is four. The minute hand is over at the nine. So this clock here shows four, 45. I'm gonna switch colors because this clock already has some colors. In this clock, the hour hand is past the five. The minute hand is at five minutes. So I know that it's 5.05. This clock over here, the hour hand is past the five. The minute hand is at five, 10, 15, 20, 25. So five, 25. And there's one more clock. Again, it's showing after five o'clock between five and six o'clock. And this turns out to be 5.40. Now, having seen all the times on the clock, can we tell which one is the correct time? Which one is 20 minutes ahead and which one is 20 minutes behind? It requires you know something else about telling time, right? It requires that you know that there are 60 minutes in an hour. And once we get to 60, we have to restart the minutes at zero again. So the difference between 45 and 5.05 is actually five, 10, 15, 20 minutes. So I can spotlight that. This is, if this hand moves, there should be five more minutes, 10 minutes, 15 minutes, 20 minutes. So between these two clocks, from this one to this one, we have elapsed 20 minutes. And that's good because that's what we were looking for in the question, right? And then between 5.05 and 5.25, we have elapsed another 20 minutes. And then this clock here, it just doesn't even, that's the broken clock. So we don't wanna know which clock is broken. That's not what it asked. It asked, what time is it now? It has to be the middle of the three other times. So it is, Alan, close up that poll. It is 5.05. Okay, so remember, you have to be able to tell time on the clock. You have to know there are 60 minutes in an hour. Remember, 12 goes on the top of the clock. Don't make the mistake and put a one on the top of a clock. Okay, our next question is again about a clock, and it's about Casper the Friendly Ghost. Now, I don't know about you, but I'm old enough to know the cartoon Casper the Friendly Ghost. And Casper the Friendly Ghost would come out at night and he would disappear in the daytime. And that's what's happening in this question. Ghosts come out at night in cartoons. So at 6.15 a.m., Casper the Friendly Ghost vanished. Because the light comes out, it's daytime, he vanishes. And the crazy clock, which had been showing the right time until then, started to run at the right speed, but backwards. A ghost appeared again at 7.30 p.m., that's 7.30 in the evening that same day. What time did the crazy clock show? At the moment when the ghost appeared. So this is what time does the crazy clock show? The ghost appears in the evening when the sun goes down, right? So we need an evening time. We need to know what the clock will say. I'm gonna start going through it while you are working as well. If the ghost is missing from 6.15 until 7.30, how many hours have actually elapsed? And we'll put in our a.m. and our p.m. because that's gonna make a difference, right? Because it's 12 hours plus some. So from 6.15 a.m. until 6.15 p.m. would have been 12 hours. So I like to break it up. So if I break it up until 6.15 would be 12 hours. But we actually have a bit more. We have until 7.30, which is another one hour, 15 minutes. So we have to add one hour, 15 minutes to get to the 7.30 time. So all together, the ghost is gone for 13 hours and 15 minutes. Okay. Now we have to think about it. Our clock started saying 6.15, but ran backwards. So what time will that get us? Right? So if our clock runs backwards, it starts at 6.15 a.m. All right? If it runs backwards for 12 hours, it would say 6.15 p.m. Right? We have to run backwards another hour and 15 minutes. So the 15 more minutes is gonna get us to 6 o'clock p.m. I think we can start counting down on that poll. And then there's one more hour. So we're actually gonna be at 5 o'clock p.m. Okay. So the answer that you should get is 5 p.m. Can we see the results of the poll? So one really great way to think about it when you have a.m. and p.m. and that gets you a little bit confusing is think about 12 hours. So you'll notice that twice, I wrote down 12 hours, right? 12 hours. Cause I know that takes me from a.m. to p.m. Our next question is about a calendar. This can be, I'm gonna try to show a little calendar here in my window. See a calendar with weeks, days of the week, just as the whole month, right? This is January. Okay. So think about that kind of a calendar with the whole month on a page as we introduce this question. One of the things you'll notice on my calendar is that some days appear four times on four weeks and just a few days appear five times, right? Cause there are typically, I'm gonna just write this down. There are 30 to 31 days in a month, right? One year in March, there were five Mondays. Which day of the week below could not appear in this month also five times? Well, maybe you don't know how many days there are in March, but you know, there are 30 or 31, right? There happened to be 31, but let's say you didn't remember cause that's okay. If there are 30 or 31 days in a month, and if I do the first 28 days, first 28 days is gonna equal exactly four weeks, right? So each day would be four times. So only two to three days can be five times. Hope that makes sense. Now, if I write out the days of the week in order, which I hope you can do, I'm gonna abbreviate them because remember it's a contest. I can't take too long, right? Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday. Can't take too long on contests. Always use your shortcuts when you're on a contest. So I know that the day of the week that we're talking about that comes five times is Monday. So a day has to be one or two away. Can't be more than three days away if it's gonna appear again. So this would be walking backwards one day, two days, three days away, or actually only three, sorry. Let me erase that because I'm gonna confuse you with that. So if Monday is the first day that appears, the other three days could be Sunday or Saturday. If I'm working in the other direction, the days that could appear could be Monday, Tuesday, and Wednesday. So which of these choices could not appear five times? Friday is not a choice, but Thursday certainly is a choice. So that's the one that you would ring. And again, if you don't remember that it's 31 days, not 30 days, this Thursday, Friday, couldn't work in either case. So look for those days. All right, we're gonna have to keep moving. This next question is a logic question. This is a visual puzzle that you're going to see. This is a question number four. So this is one you should come up with a nice quick answer. We won't spend a lot of time on it. The picture shows five screws in a block. Four of the screws are the same length. One screw is shorter. Which screw is the short one? I can see already the poll is going really well for this one. If I have a screw and it's as long as screw number two and I unscrew it a bit, you can see, oh, well, I don't know where number three ends, but it could be there. Number four could just barely be touching the block. So those, I can't say which one, if one of those is shorter or not, you're kidding. But if I compare screw number one to screw number five, the bottom of screw number five is missing from the bottom of the block, isn't it? So screw number five is definitely shorter than screw number one. And in this case, you would have to select that as your answer. Our poll shows that most of you believe that was correct too. Here's another logic puzzle. On your handouts, if you have the handout, I gave you several places for trying because one method is trial and error. You may also be able to do this without trial and error and I'll show you that in a minute. Leon wants to write the numbers from one to seven in the grid shown. Two consecutive numbers cannot be written in two neighboring cells. So is there anybody, if you don't know what the word consecutive means, that's a pretty big word. Consecutive numbers are numbers that go in order. So two and three are consecutive numbers. Five and six are consecutive numbers. 100, 101 are consecutive numbers. You cannot put consecutive numbers into neighboring cells. Oh, we have to understand the question. What is a neighboring cell? Neighboring cells are those that meet at the edge or at a corner. So if I highlight, if I highlight this square, it has neighbors. This would be a neighboring cell and this would be a neighboring cell. What number can be written in the cell marked with the question mark? That's the question. We need to fill in the question mark. Okay? So how many neighbors does the question mark have? It actually has one neighbor, two neighbors, three neighbors, four neighbors, five neighbors. The question mark rectangle is the one with the most neighbors. So that's gonna be how we know what number to put in there. I'm gonna just write out the numbers one through seven because there are seven squares and seven numbers, one through seven. So we know each number can only appear one time. Hopefully you all understood that. Remember that's part of the understand the problem. There are seven squares, seven numbers. The one in the middle, it has to be, there has to be five other numbers that are not consecutive to it. So if I pick, for example, in my little picture here, if I pick the number four and I try to find five numbers that are not consecutive to it, I'm gonna be able to use this one, this one, this one, and this one. That's only four numbers that are not consecutive. So I would not be able to put the number four in the middle. I'll end up having to put a three or a five next to it, which would make it consecutive numbers in neighboring squares. So this one is out. I know I can't do all of the even numbers because four was an even number and it didn't work. I certainly can't do all seven numbers. What if I try again and I write up my numbers? What if I decide I'm going to put number one? Then for non-consecutive numbers, I have a choice of seven, six, five, four, and three. I can find five. The only way I can fill this in and not have consecutive numbers is if I use one of the numbers at the end of my sequence. So I can put a one here. I don't wanna put the two next to it. So I'll put the two way over here. Then I can put any of the other numbers around it. It doesn't matter. Because none of these are consecutive to one, only two is. The other way to do it is if I put the number seven in there, six is consecutive to seven. So I put six far away. Then any of the other numbers can go in here. It doesn't matter where they go as long as it's not next to it. So the only numbers that fit here are one and seven E. So it's looking pretty good in the poll. Definitely it was trickier than the question before it, wasn't it? We are going to just make this in our hour time because we have one more question to solve. And I think you'll like this one. Five sparrows sit on a branch as shown below. Each sparrow chirps the same number of times as the number of sparrows it sees. Now, you and I might think that's confusing but it gives us an example to help explain it because we need to understand the problem. For example, David chirps three times. So I'm going to look, here's David and he's going to chirp one, two, three times because David sees three other sparrows. Okay, now I get what the question says. It doesn't matter which way these sparrows face. David sees all of them. One sparrow turns to look in the opposite direction. Again, each of the sparrows chirps the same number of times as the number of sparrows it sees. This time, the total number of chirps is more than the first time. Which of the sparrows turned to look in the opposite direction? Okay, I'm understanding the question. I need to know which sparrow turned around. The one that turns around that's correct is the one that's going to see more birds because that's going to make for more chirps. So which of these birds, when turned around, will see more other birds? So we can go through one by one and we can see how many birds does each one see. Angel is already looking in the direction with all the other birds, right? So we wouldn't want to turn Angel around. So she's not the correct choice. Bertha is only looking at one bird in this direction. But if Bertha turns around, then she could see three birds. So she would chirp more times. What about Charlie? It doesn't matter if Charlie turns around, does it? Because Charlie sees two birds in either direction. So it's not Charlie. David is already facing in the direction where there are three birds. In the other direction, he'd only see one. So we don't want to turn him. That would be fewer chirps. And the same for Aglio. He's already facing all the other birds. We do not want to turn him around. So the one we do want to turn around would be Bertha so that Bertha sees three birds, not one. And how many more chirps would we get? If Bertha goes from one chirp to three chirps, that's an additional two chirps, right? Because three minus one equals two chirps. So we'd have more chirps when Bertha turns around. Okay? Yeah, sometimes reading the question, sometimes reading the question is the biggest trick. So let's go over, let's review all the things that we learned today. We have a review here. The four-step method is going to be useful for anything that's not a simple exercise. You need to understand what the problem asks. You need to make a plan how to solve your problem and carry out your plan. Carrying out your plan may require very careful work on your scratch paper. If you have a complicated question and you get an addition fact wrong, you're going to get the whole answer wrong. So careful on your carrying out the plan and then look back and check your answer. Okay? There are six types of questions and some of these are not familiar from school. So practicing for Math Kangaroo will help you. Number of questions, geometry measurement questions. We did those visual and spatial questions. Clocks and calendars. And you notice there was some interest, not just what time is it, but what happens if a clock runs backwards? What happens if a clock is fast or a clock is running slow? And then we have logic questions. Those can be the whodunits. It can be like a Sudoku. It can be fill in the blanks. And you notice that there was overlap. Our last question was both a visual question and a logic question, right? Because you had to see the picture of the birds or you would never be able to answer the question. So there is visual there. So don't be afraid that questions could have combinations of types. On Math Kangaroo website, there are a lot of practice questions. Once you register for the contest, you can access to practice contests. I would encourage you to do those and use the video solutions as well. I want to thank you for participating in the Math Kangaroo webinar. I want to thank Alan Kappler very much for his help with the chat, with the polls. Did you want to say goodbye or anything, Alan? Yeah, goodbye, everybody. I hope you enjoyed it. When you complete the Zoom today, there will be a survey that pops up. Please, parents, students, if you can take just a minute to fill out the survey, the webinar series is a new feature for Math Kangaroo and we want to make them as useful and educational for you as you can. So your feedback will really help us to continue to make them better and better. When you're doing anything new, you always want to improve. Thank you very much. I appreciate your attention. I hope you had a good hour with me. Bye.

Math Kangaroo

problem-solving

webinar

Dr. Sarah Sagie

Alan Kapler

geometry

logic puzzles

interactive session

practice exercises