I'm gonna introduce myself, and then I'm going to let our TA introduce himself as well. My name is Dr. Sarah Sagee. I'm a biomedical scientist. I live in San Diego, California, where it's hot and sunny again. We did have some rain this week, which was a great surprise. I have been tutoring math and science for about 30 years, and I coach an elementary school math club here in my neighborhood, the school where my children have attended. I've been helping teach math kangaroo classes, and Alan has been my TA in the math kangaroo classes. And I like to swim. I love to walk my dog, and I've also taught karate. I'll let Alan have a moment to introduce himself. Hi, everybody. I'm Alan. I'm in high school. I really love math and physics. I've been taking math kangaroo in particular for since first grade. It's what really got me into this competition. Can people hear me? I can hear you fine. That's good. So I'm really excited to help teach you all. Thank you. Thank you, Alan. Alan, if you have a moment, if you'll go ahead and paste the link for the handout today into the chat. Some students are asking for that. So there is a handout for today's session. If you have not been able to print it yet, Alan just pasted the link to it one more time in the chat. If you do not have the opportunity to print it, all of the questions will be on the screen for you. Do have scratch paper and some pens or pencils around so that you can go ahead and sketch the drawings and work on them yourselves. It's a very big help to be able to have that scratch paper ready. Okay. You'll notice during your session that the microphone is going to be muted. We do want you to ask questions. So you will do that through the chat. We're not gonna be able to keep track of raising your hands and calling on you. We cannot see your videos, but just put it right into the chat. I have it open on my screen and Alan has it open on his screen. Alan will try to give you some feedback right in the chat. And if I see a question coming up more than once from multiple students, I'm gonna give you an answer right out loud for the entire group. The chat is anonymous. So no one knows that you asked a question or you didn't understand. I wanna help as many students understand the questions and problems as possible. So me knowing where you're having a little difficulty is going to make this webinar better for everybody. Yes, your cameras are off and your microphone is muted. There will be just a few polls during the session and that would be a pop-up window on your screen that will allow you to answer multiple choice answers. A few of our questions today are multiple choice. Most of them have been multiple choices have been removed. In those questions, just put your answer right into the chat so that Alan and I can see if you're working toward the correct answers, okay? So go ahead at any time you wanna answer a question you see on your screen, put it in the chat. If your pop-up window does not work for a poll, put that in the chat as well. So answers, questions, anything you need, your chat is the way to reach us, okay? Don't worry, we're not going to tell anybody who's asking questions. We just want to be as helpful as possible. So Math Kangaroo likes to use a four-step problem-solving method and you'll see very similar problem-solving methods from other math providers and contests as well. The first step is always to understand the problem. If you don't know what the problem is asking, you're not going to give a correct answer. I have found that typically if you check the first sentence and the last sentence of most contest problems, that will tell you what you need to have in your answer. So it might say, what is the total area? Or how much larger is one thing than another? It will ask you to find how many possible outcomes are there? It's very important that you read that part of the question so that you have the correct type of answer. Okay, next, once you understand what is the question asking, you have to figure out what information does the problem give and do I have a plan for how to solve the problem? Okay, so when you read through the problem, figure out what information is there and do I have a plan to find out the missing piece that the question's asking me to solve? Now, here's a really important part. We're all individuals and we all have different experience and different education, so our strategies may not all be exactly the same. And in today's webinar, I'm gonna do a few of the questions with multiple strategies just to show you that there are options. If the way I solve a problem today is not the way that your brain told you to solve the problem that does not mean you're wrong. It just means that we're different and that's okay. All right? One very important thing though, is when you have a plan, carry it out carefully. Don't make mistakes in adding or subtracting, right? Use your scratch paper, organize your scratch paper. Sometimes I see students who have scratch paper and they've scribbled something in this corner and they turned it around and squished it in upside down over here. I don't think that your parents are gonna limit how many pieces of paper you have. So make your scratch paper organized, make a space for each question. That way you will be able to follow your own work and you won't make needless mistakes, sloppy mistakes, right? Now, the final step is always look back and reflect. Does the answer make sense? Did you answer the correct question? If you didn't answer the question that the problem is asking, then you need to go back. If you had more than one plan and you still have time, you can use a second plan and see if you get the same answer. If you get the same answer by two plans, that's good reassurance that you have a true final solution. Okay? This webinar is going to focus on geometry problems. Math Kangaroo geometry problems do require you to have at least a fifth grade level at this stage of understanding of geometry. That means that you need to know your basic shapes, right? So your basic shapes are going to include triangles, squares, rectangles, circles, right? Your basic geometric shapes that you've been learning since you were younger in elementary school. You'll also need to be able to find perimeter and area and also some of the angles in different shapes. So just a quick review, most of you know that perimeter, you can see this purple line around my rectangle. The length of that purple line is the perimeter. The perimeter is a linear one-dimensional measure. So it will be in linear units like inches or miles or meters, centimeters, just one-dimensional. The area is the amount of space that a flat shape or surface will cover. So you can think of it as these little square units and the area, the grid. If I counted up all these little squares, if these were each one unit, that would tell me how many square units were covered in the area of this rectangle. And those are two-dimensional units. So those will be in squared units. So you'll have a little superscript for squared, or you might just have the word squared or SQ sometimes also means square. So square inches, square centimeters, square meters. Okay, I'm seeing some questions about triangle congruency and similarity. You won't need to know too much about that, but you would need to know different types of triangles. For example, do you know what an equilateral triangle is or an isosceles triangle? In terms of angles, do you know what a right angle is? This rectangle contains four right angles. Each right angle is 90 degrees in measure. So those are ideas that hopefully you've seen before in school, and you would definitely need to know those for math kangaroo. Let's move on, because I like to categorize. Geometry problems are not necessarily all the same. So you have your geometric shapes, that's in common in the math kangaroo problems, but you'll notice that there are gonna be some today that we're gonna solve through qualitative means. That means you don't have to do calculating so much. You're gonna investigate them by looking at them and comparing them a lot of the times. You'll need to know, is this half of something? Is it one, which one is bigger? Why is it big? You'll need to understand things like that. And that's qualitative because you're not doing a quantity, right? Then there are gonna be some that require your basic calculations. What is the area or the perimeter or the length? Or can you measure this angle? If I split a 90 degree angle in half, what are the measures of the two angles? You'll need to know words like bisect to divide something evenly into two even parts. So some things like that will require basic calculations. And then there are gonna be problems that require you to do some intense analysis based on geometric properties. If I have an equilateral triangle, that tells me all the angles are the same, all the sides are the same, I can use that. And if I know that the area of a triangle is one half base times height, can I use that for rectangles? Can I use that for triangles? How can I solve problems knowing those properties of shapes? And I know that might sound confusing right now, but as we go through the problems and you see examples of these things, I think that you'll then understand why I've made different categories of questions. Okay, all the questions that we're doing today are from previous contests. So if you really wanna know, are these real questions? The answer is yes, they won't be this year's questions, but they were past year's questions. The contest at your level has 30 questions and they have three point values. There's three point, four point and five point questions. The contest starts out with the easier, more warmup questions. And when we go through our slides today, you'll see the problem number and the year on each of the slides. So it'll tell you the year and the problem number. And if it's a low problem number, you'll know that that was a three point question. If it's a high problem number, you'll know it's a five point question. Math Kangaroo expects that the five point questions are more challenging and will take you more time and effort to solve. So if you find that it's taking a hard question takes you more time and effort, that's the way it was designed, okay? Now remember, most, the majority of your students in your classes, if you go to public school, regular school, they would not be able to solve all the questions on a Math Kangaroo contest. They maybe solve a third to one half of them. The contests are designed to be a challenge. So when we see the tough questions today, expect to be challenged, but hopefully as we go through them and I explain them and you work a bit, it'll be a challenge that you can learn to work through. Okay? That's the idea, is to learn by challenging yourself. I can't make all the questions easy because that won't represent a real contest. All right, it's time to start our first question. So this is a number six question from several years ago. Which of the following geometric figures is not in the design to the right? So this is just asking, do you know the names of geometric shapes? I'll give you a moment to think about it. And there is a poll that Alan can pop up for this question. Oh, and I'm seeing some, there we go. Excuse the typo. Sorry, I was trying to make these quickly. If you do not see a pop-up poll, or if it's in your way, if you don't see it, put your answer right into the chat. If it's in your way, you should be able to grab onto the blue bar at the top of your mouse and just scooch it over. Yeah, we're getting lots and lots of correct responses. This question is a little different than some because it actually shows you the shape. So if you don't know, the dodecagon is going to have 12 sides. An octagon has nine sides. I would like to discuss the word regular. Regular means you have a polygon, just a closed shape. So you'll notice it's closed. We don't have any gaps. We don't have any openings. And what you'll notice about a regular polygon is that all the sides and angles are the same. People are pointing out an octagon has eight sides rather than nine. Did I misspeak? Apparently. Okay. I'm human and I misspeak. I apologize if I said nine. The octagon is definitely an eight-sided figure. So when we talk about a regular polygon, is a square a regular polygon? Give me a moment for that. I'm getting good answers in the chat. Yes, a square is a regular polygon because all the angles are the same measure and all the sides are the same length. And what kind of a triangle is a regular polygon? There's a specific triangle, which is regular. Very good. The equilateral triangle is the regular polygon. Good. Because these are concepts that you'll need to know for some other questions. I'm going to end the poll right here and show you how well you did. Most of you decided that it was the octagon. The easiest way to show you this is to show that we can actually, we can draw most of these shapes, right? So I can trace over a triangle. There's definitely a triangle. I can find squares in this shape, right? They're a little bit on an angle, but there's definitely a lot of squares in the shape. So you should be able to do this if you have the handout. Can we find hexagons in this figure? I can find hexagons in this figure. And then finally, we do have the orange dodecagons. What I do not see is any of the regular octagons. So hopefully, you were able to find those shapes in the picture. Okay? Might it be worth explaining, should it be worth explaining why this dodecagon is regular? How we know that? Well, if you'd like, I just want to make sure we have time to get through all the other questions. Okay. Just note that a regular polygon is one with all the sides have the same length and all the angles have the same length. Note that it's just the side lengths of all the other polygons, which all have the same length. And the angles is the angle from a square plus the angle from a triangle is the angle from the dodecagon. And that's the same for all of the for all of the angles of this shape. So that means it's regular. Right. Okay. I'm going to go ahead and we'll move on to the next question, which is also a question I hope that you'll be able to answer relatively quickly. Kathy draws a square with a side length of 10 centimeters. She joins the midpoints of the sides to make a smaller square. What is the area of the smaller square? So when understanding the question, remember, I like to use first and last sentences. So what is the area of the smaller square? In this case, we know that that's the green square, right? Area is the space it's covering. And what information has the question given us? It's given us this 10 centimeters, and it's actually drawn it on the diagram for you. So you should be able to figure out that you need to know the area of the square. The area of a square equals side length times the side length. Or some of you might know that it's side to the power of two. People a moment to think about it. I'm getting quite a few correct answers in the chat. Remember, you want to go ahead and put those answers right in the chat so that we can see how you're doing. Remember to give us the area. This is a multiple choice contest. So make sure that you have the correct type of units when you're answering on the multiple choice. If you had 50 centimeters as a choice, or 50 centimeters squared, you'd want to make sure that you use centimeters squared. A lot of students have gotten 50, but some have not. So let's figure out how we know it's 50. The area of the large black square, sorry, I just lost my place there for a moment. The area of large black square is going to be 10 centimeters times 10 centimeters. That is going to be 100 centimeters squared. How do we know what the green is? Well, some of you may have calculated triangles and other things. But one other way to do it, which is I think a very clever and elegant way, is to draw through the diagram. When I draw through the diagram, I know that these corners of the green triangle are bisectors, meaning it's even. Now what you can see is that each of these green triangles is half of a red triangle. So overall, the green square is one half the area of the black outline square. So all I have to do is I can take half of the 100 centimeters squared, and that is going to give us the 50 centimeters squared. Are there other options for doing this? Absolutely, there are other options. You could have determined correctly that if this is a bisector, that this little piece here is five, and this little piece is five, right? And so now I can draw a triangle, just trying to switch colors. I could draw this triangle here, right? And the area of that blue triangle, the formula for area of a triangle is one half times the base times the height. So the area of that blue triangle would be one half of five times five, and that is going to be 12.5. I have four of those that I would subtract. So that would also give me four times 12.5 is 50. Subtract it from the large square is going to give me 50 centimeters squared. So I've now used two strategies and come up with the same solution. That gives me very high confidence that we're correct. With the two different methods, did one of those work for most of you? Something would click, I hope. The next question is a similar type of question. So let's see if we can use what we just did on the prior question to solve this one. Here's a square with line segments drawn inside it. The line segments are drawn either from the vertices or the midpoints of other line segments. We colored one eighth of the large square. Which one is our coloring? So understand the question. Remember what midpoints mean? Midpoints mean you're dividing something directly in half. So this is the midpoint of the longer bottom section. We know these are all squares. That's in the question. That's a very important piece. They're all squares. They're the same square. We have vertices, which are the points, basically. We have the midpoints. And we now want to know which one is one eighth colored. I'll give you a moment. You can put some answers into the chat. You can put answers into the poll as well. For those of you who've answered this very, very quickly, if you have a moment, you might try to think what area of each choice is colored. So what part of A is colored? It's definitely not an eighth. It's much less than an eighth. But what fraction is that? Yeah, I'm seeing some students have said that they've seen this question before in math kangaroo books, and that would make sense, because it's a past math kangaroo question. This was on last year's test, on 2021's test. We have some students trying to find those fractions. Good work. Just a little extra challenge for you. I'm going to end the poll. The students look like they've gotten this one. Okay, so I like to solve this one in the same way we solved the last one. I draw a line where I have halves, right? You know that's half, because we know it's a midpoint, right? If I draw another line, if I cut a half and a half, I get quarters, right? And if I cut a quarter into a half, I get an eighth. So the correct answer here is D. So just think, you cut a half and a half, you get a quarter, you cut a quarter and a half, you get an eighth. And this is a question you can do without having to do a lot of calculations. So you should be able to go through it quickly on the contest, just to give you an idea. All right, the next one is a little bit different, but also a question that we're going to investigate mostly visually. So this is again, geometry with the visual investigation and comparisons. Each of five neighbors owns a rectangular plot of land with the same area. The flower gardens of their land are fenced in, that's the solid line in the pictures. Who has the longest fence? So we're going to choose one of these gardens, one of the people. And what you'll notice is the rectangle all around the outside of each one, the outside rectangle is the same. Okay, so even if their fences are not the same, the outside is the same. Okay, I'm going to clear that so it's not in the way when you're looking, give you a moment to try to figure out who has the longest fence. Remember fences, the black lines. Alan, have you been able to hear me? I hear some students say they don't hear me. No, I can hear you. Okay. If you don't hear me, you might have something, a bad connection, or your speaker is not working because, yeah, most students say they can. Thank you for confirming that. Okay. This question is a little bit trickier, it seems. We have a lot of right answers, but we have some that have not quite gotten to the correct answer. One of the ways I like to do this question, and you may be slightly different, is I like to see how does the fence of each person compare to the original outline, which is the dashed line, right? So how does, for example, Mr. John's fence compare to Mr. Adam's fence? So Mr. John has a fence that goes horizontal here. That would be exactly the same length as if it was out here. Mr. John also has a vertical fence over here that would be exactly the same length as if it was out here. So Mr. John and Mr. Adam have the same length fences, right? How about Mr. Peter? We can do the same thing with Mr. Peter. There's a vertical coming here, which would be exactly the same as the vertical if it had been on the outside, and the same on the other side, right? So can you follow that example? And for those of you who have not gotten an answer, can you take a look and see if you can figure it out now? Just try to push their fence lines out. So if I take Mr. Jack's vertical fence, it matches up over here, and this vertical fence matches up on the outside here, but I am actually left with four extra little bits, these four extra little bits. Does anyone else have extra little bits that don't map onto the outside? No, so I think we can end and share the poll. Most of you correctly said it's Mr. Jack, and like I'm trying to show, the reason it's Mr. Jack is because he has these four bits that cannot map out to the dotted outline. And those are extra fence lengths that he has that no one else has on their garden, okay? Trying to make sure we get through some other problems. This next one is a bit similar. I think if you were able to do this one, then the next question will also be comfortable for you. It requires a little bit of a computation, meaning a little bit of number calculating. Look at the pictures. Both shapes are formed from the same five pieces. The rectangle measures five by 10, and the other parts are quarters of two different circles. What is the difference between the lengths of the perimeters of the two shapes? So for me, important things that are in this question are, both shapes are formed from the same five pieces. So I don't have to worry, is this rectangle the same as this rectangle? It is, it is the same rectangle. Okay, it also gives us a little information about some measurements. The rectangle measures five by 10. So it's the same on both of them, five by 10. It also tells us that these rounded pieces are quarters of circles, right? So quarters of circles. So that's gonna help you as well. And then remember, what does the question ask? The question asks, what is the difference? The difference, not the length of the perimeter, but the difference between the perimeters of the two shapes. Now remember, perimeter is the distance around the outside. So we only need the outside, not any lines inside, outside lines. I'll give everyone a moment to think about it because it's very important to think. One of the things I'm seeing a little bit of questions about the quarter circles with regard to the rectangles. What we can see is that the quarter circles do have the same radii as the rectangle. So if you're not real comfortable with circles, this is a good chance to review a little circle geometry. If I draw a line all the way across a circle, that's called the diameter. If I draw a line that goes from the center, if I draw a line that goes from the center out, that is a radius, and the plural is radii. Any radii, any radius that I draw, no matter where it is in the circle, has the same length, right? So this piece has the same length as this piece. This piece has the same length as this radii, radius. So maybe that helps you. All right, I'm gonna give a few more clues. We're getting many, many correct answers, so very good. But I wanna give a few more clues. I wanna help some of the students who haven't quite gotten there, okay? If I take a look, remember, we wanna find the difference. So anything that is the same, we don't need to worry about, right? There is a large quarter circle here and a large quarter circle here. There's a large quarter circle here and a large quarter circle here. Those are the same, so I don't have to worry about them in calculating the difference. I hope that makes sense. Then if I look at the small quarter circles, there's a small quarter circle on the right, two of them, and there are two on the left as well. So finding the length of a circumference or an arc involves pi and funny calculations, but in this question, we don't have to worry about them because they're the same. The only actual difference is this has one rectangle length, which corresponds to the 10, right? And this has one, two rectangle lengths, right? This is the same as this, right? So this is two and then it also has two of the shorter radii. So the figure on the left, those straight pieces are gonna be two of the five centimeter radii and two of the 10 centimeter radii, which is gonna be 10 plus 30, 10 plus 20. The figure on the right, the only one that's there is the 10 centimeter. So how do you find the difference? Now, this one has straight pieces that total up to 30 centimeters and this one only has a 10 centimeter piece. The difference, the difference is gonna be the 30 centimeters minus the 10 centimeters. Most of you, obviously it's not equal, so that's just me mistyping. Okay, keep this idea about which parts are different, right? So we had to figure out which parts overlap and which parts are different. Keep that idea in mind for the next question. Okay, does that make sense? Our questions are each kind of building upon each other. So hopefully that will make sense when you see the next question. The frame of a rectangular painting was made from wooden pieces of the same width. And they've marked actually the width on this picture here. You can see they've marked the width. What is the width of those pieces if the outer perimeter of the frame is eight decimeters longer than the inner perimeter? Okay, I'm gonna let everyone think about that for a moment. What is the question asking? It's asking for the width, the width of the frame pieces. It's asking for this piece right there where the question mark is on the diagram. What information do they give you? They tell you the difference between the outer perimeter and the inner perimeter. So the outer perimeter is all the way around the outside of the frame, outside of the frame. Inner perimeter is just this brown line right around the picture itself. Some of you are correctly answering it. This question takes a little thought for some people. So I'll be quiet for a moment. Okay, I'm getting a mix of answers. So my strategy when I get a mix of answers is to try to give you a little bit of the next step, or at least my next step. Remember yours might have been a little bit different. Don't ever feel badly if you think about the question differently than I do. I was working through some of these with my son last night and he has a slightly different thinking about some of them as well. So don't worry about that. If I take the inner perimeter and I match that up with a line on the outer perimeter, and I go around doing that, I'm gonna notice there are some gaps in the outer perimeter. We'll see if that helps some of you. How large are those gaps that I do not cover in the outer perimeter? Okay, so I'm gonna go ahead and do that. If I draw that blue line here, it's exactly the same as the blue line that's missing here, is it not? So can everyone see where I am on the bottom of the right-hand bottom of the frame? This blue line is the width of the frame, and that exactly matches this blue line here. I can draw it over here as well on the bottom left. This width of the frame is exactly the same as this piece on the outside that wasn't drawn, but yet. So how many of these little pieces do I draw? How many little widths? I would have one, two, three, four, five, six, seven, eight. So the difference between the perimeters is eight decimeters, and I have eight frame widths difference between inner and outer perimeter. So if the total difference is eight decimeters and I have eight widths, then it is one decimeter is the width, right? Because eight divided by eight is one. All right, the next question is a higher level question than this one, I think. It's gonna take some good thinking. The area of the biggest square, we had squares in one of our earlier questions. So you might be able to think back to that question with the green square in the center. The area of the biggest square shown in the figure to the right is 16 centimeters squared. And the area of the smallest square is four centimeters squared. What is the area of the medium square? So what does the question want? It wants the medium square, which is the green one, right? We need to find the area of the green square. What information does it give us? It gives us areas of two other squares. And it might help you to remember what is the formula for area? Equals side times side when you have a square, right? They're the same. I have some speedy students getting correct answers and some speedy students getting incorrect answers. This is one of those questions with multiple strategies that would all work equally well. I'm gonna give some hints here. If the black square is 16 centimeters squared and that's side times side and side is gonna be four centimeters, right? Because four times four is 16. So I can draw a four on the outside here. If I go to my red, my red square is only four centimeters squared and that is also side times side and my side must equal two centimeters. It's not a Z, it's a two. You try writing in Zoom, it's not that easy. I do know that the center red square is in the middle because we see this nice green one all the way around. So we know this is in the middle. So I can figure out if this is two and the black is, red is two, black is four, this is a one and this is a one, right? One and one. So we have multiple strategies. You can, most of you I hope know the area of a triangle. Area of a triangle is one half base times height. So there are several triangles here. You can subtract this green triangle from the blacks or you could add this brown triangle to the red. Doesn't matter. Let's see if we can find the dimensions of one of these triangles. So the triangle is going to be, doesn't matter if I call it base or height, doesn't matter. It's one plus two. And then the other dimension is gonna be one. And to find the area, we need to multiply that by one half. So I have three times one times one half, which is gonna give me 1.5 or one and a half. And there are actually four of those triangles to either add or subtract. So that's gonna be six. So this is area of triangles. So either add six to the small square or subtract six from the large square. Good. I'm going to move. There's another way to solve this question. All right, remember I said we have multiple ways? Let's move along to the next slide where I have shown another way to solve this. I work with students sometimes younger than you are starting in third grade. When I presented this question to younger students, they said, what if I draw a grid? Here's the red, sorry, the red one doesn't show up too well. What if I draw a grid around the whole thing? That works, because remember, we found that the dimension of this was four, the dimension of the smaller square was two. So if I draw a four by four grid around the whole thing, and we know that area is the amount of space covered, if you are not able to subtract the triangles and things before, can you do it when there's a grid on the figure? Remember, we're trying to find the green. Sorry, wavy line. So it's three of them here, and it's half of three. But I have another half of three here, so this is total of three. Here's three more, so that's 6, 7, 8, 9, 10, if I'm counting on a grid. These squares we do know are to scale because of the way this was drawn with everything centered, so we do know that this grid is to scale. But you're right, if you didn't know it was to scale, you'd have trouble. But we do know that this one worked out very well when I drew the grid on it. And then I bet some of you might have another way of doing it. Does anyone else, are there a few of you who know about Pythagorean theorem? Can I have anyone in the chat tell me if they know about Pythagorean theorem? Yes, excellent. So you can solve this question using the Pythagorean theorem. If you have successfully determined that this is four, and this is two, and so then you have a triangle with a length of three, and a triangle with a base of one, this is a right triangle. So with my right triangle, if I wanted to find out the length of the green side, sorry about the color, the green side, I can use the green side is my c squared, right? If you do not know the Pythagorean theorem, don't worry about it, okay? It is typically taught in seventh or eighth grade. But I know a lot of fifth grade and sixth grade students have already been introduced to it. So I can find the length of that green side. The a, I can use either the one or the three, it doesn't matter. One squared is one, and b squared, three squared is nine. So c squared equals 10. And of course, the area of a triangle, the area, pardon me, I'm having a lot of problems with my mouth today, the area of a square is side times side, so the area is side squared. So therefore, the area of the green square is 10. If this is beyond what you've learned so far in school, don't worry about it. We've already showed you two other ways to solve the problem. So there's more than one method. I'm glad somebody's giving me a thumbs up. Yeah, you know what? I don't know how to spell the Pythagorean theorem either. Don't worry about it. It's a math contest, not a spelling contest. We're all safe. We're not expected to know ancient Greek anyway. It's probably spelled different in Greek characters than it is in English characters too. All right, the rectangle shown in the picture was divided into squares with different length sides. The areas of some of these squares are given. They want to know the area of the entire rectangle. Now, remember, just like in the question before, if you have the area of a rectangle, the length of the side, it's side times side equals the area. And I'm going to help you out here because I think this one's a little bit tricky. Hopefully that helps all of you. If you didn't know all of your square numbers, maybe that helps. Remember, let's look at the problem. We have a rectangle and we wanna know the area. So remember the area of a rectangle equals length times width. So can you find the length and the width given the pieces that we know now? Remember, from the very beginning of the question, it says it's a rectangle. And we know that it's a rectangle. Why do we know that? This, let's see. This has squares inside it. So this length and this length are the same. But we know that this is 14. Orange is 14. And purple is 15. So we know it's a rectangle. Question is, can you find out what red is? So, the area is going to be 14 plus 18 times 15 plus 18. This is one of those questions that does have some calculations. Most of them don't have tough calculations, but there'll be a few. So hopefully you can all do multiple digit multiplication and you'll get 1,056 and does it give us the, it doesn't tell us what they are, so we'll call them square units, okay? Since it doesn't give us the measurement, we'll call them square units. And you can use that as a pretty basic rule in math. If they don't give you the units, call them square units. Okay, we're going to move on. I have a question that's a bit different. We'll try this one. So how many convex angles with different measures are made by the rays with P as a starting point? A convex angle is an angle that is larger than zero and smaller than a straight line. So I wanted to move to this because some of you have asked a little bit about angles. So if we have an angle like on a on a square or a rectangle, we call that a right angle and it's 90 degrees. If we have an angle that is smaller than 90 degrees, we can call this one an acute angle. They're talking about a convex angle. So if I draw the angle on the what we would think of as inside is convex and the angle that would be on the outside is obtuse. So this on the outside is the obtuse angle and on the inside is the convex angle. Now this question is different than the ones we've done before because this combines a couple of different things that Math Kangaroo likes to do. Math Kangaroo likes you to do combinations. So this is asking, remember first or last sentence, how many angles with different measures are made by the rays? So that's important. How many of them are there that are different? So we can add angles up. So if I, for example, wanted to consider the angle made by this over here, I can add 10 plus 20 and I can get 30. But that is the same measure as this angle. So they're not different. So remember different and how many of them are there? Not what are they? I don't need a list of 10 degrees, 20 degrees, 30 degrees. I need to know how many. I'm going to move to the next slide because in the next slide I've added a few labels which will help us speak about these together. I added just the letters because if I'm making the list, it's nice to be able to organize my list sometimes. So if I want the angle APB, oh goodness, that equals 10. If I want the angle APC, that equals 30. So you can make a list of all the different angles now that you have labeled them. So As far as I know, that's the list of all of the angles that you could possibly make. We want to know only the different ones, so we can cross out any that are repeated, right? So 30 is repeated. What else is repeated? 50 is repeated. And that gives me a total of eight that are different and unique. Okay, we're going to wrap up there. We had to skip a few slides, but that's okay. I'd rather give you thorough answers to the ones we did. So remember, the four-step problem-solving method is to understand what the problem is asking. Like on that last question, if we didn't pay attention to the fact that it was different measures, we would have gotten the wrong answer. We would have gotten 10 instead of 8. Plan how to solve the problem. Don't worry if there's more than one plan or somebody, your neighbor says, I did it a different way. That's okay. Be careful in carrying out your plan and check your answer. Did you answer the correct question? And geometry problems can involve understanding the properties of different shapes, knowing what a square means, knowing that a right triangle, you have the length, you have a base and a height, just when you have a right triangle. Notice that sometimes you can just do a visual examination of a problem and get an answer. And sometimes you're going to have to do careful calculations. I'm going to give Alan a minute. I know we're over time, but I'm going to give Alan a minute to give you a little bit of advice because he's taken more of these contests than I have. And I know some of you will take the contest very soon. Right. Yeah. So it's important when you go to the test. Well, first of all, you need to make sure that you don't spend all your time on one question. If you have a particular, if you have something in particular that is really stumping you, it's probably best to just go on to the next question. After all, it's not the end of the world if you miss a question in the middle of the test. Second, make sure to use your scratch paper. It's really easy to use your scratch paper and make sure to read the question thoroughly. I can't tell you how many questions I've missed because I missed a word like area instead of perimeter or something like that. It's really easy to miss questions on Math and Guru because they're often puzzle questions. So be very careful, write everything out. That's why you need scratch paper. If you need to do a diagram, there's a question, how many points will be taken off if a question is skipped? They don't deduct any points. You just don't receive the points. You don't get any deduction for skipping a question. You don't get a deduction for skipping a question. It's just you get zero points if you don't get it correct. And you get three or four or five points if you get it right, depending on where in the test it is. So that's another Yeah, there's one other question. You do not get deducted for a wrong answer. So a good testing strategy is to answer as many questions correctly and confidently as you can. Go back to harder questions if you have time left over. And then if you're running out of time completely, fill in the bubbles, right? You might get some. Yeah, like if you have five, if you have five questions left that you don't know how to do, just put in C for all of them, for example. And on average, you'll get at least one of them correct. No guarantees, but it's a way to increase your score a bit. Okay. All right. So there are still resources available. This is our last webinar, but there are resources available on the Math Kangaroo website. You can still get sample questions, sample contests, and there are video solutions available. So go ahead and try those video solutions. At the end of today's webinar, there is a feedback survey to let us know how you enjoyed today's webinar, how you thought that Alan and I did explaining questions, if you liked the questions, things like that. So when I close up the Zoom, please go ahead and fill out that feedback survey. I hope you've learned something. I hope you were challenged because the idea was to give you a challenge. And I hope that you all do really well, practice up and do really well using our advice on the actual Math Kangaroo contest and also in your math classes. I hope that math becomes something fun that you enjoy doing. All right. Thank you so much for participating in the webinar today. Don't forget to do the form as we close. Okay. Thanks a lot.

Dr. Sarah Sagee

biomedical scientist

math competition

Math Kangaroo

problem-solving

geometry principles

multiple-choice strategies

Alan

math tutoring

enjoyment of mathematics