Okay, so welcome to our webinar today and our topic today is drawing a picture. All right, so for today, we have two webinar instructors. My name is Diana Holland. I am a teacher, and I have been teaching for about 30 years. I have also been a mat kangaroo manager for six years, and I've been leading the weekly sessions that we have once a week for levels one and two. The other thing that I really enjoy doing is conducting math clubs that prepare my students for various math competitions. Today also, we have Noah, who is going to be helping us with the webinar, and Noah will give you a short intro about himself. Hello, everyone. My name is Noah. I'm in 11th grade right now. I've participated in Math Kangaroo since third grade, and in the past few years, I've gotten pretty good scores. All right. Thank you, Noah. He's being humble. He has got perfect scores. All right, so let's move on to some logistics about the webinar, because I saw some questions on chat. Okay, so you will not be able to communicate through audio during the webinar. You are muted, and you will not be able to turn it on, so there is nothing wrong with your computer if you cannot respond to us verbally. If you have any questions, you can always put it on chat, and Noah will answer your questions to his best ability. Please keep your questions related to today's topic, but not so much about the Math Kangaroo contest in general. We also want our participants to participate today and not just sit and listen, so you will be answering questions through polls. So as we go through the questions, when the poll pops up, we would really like you to submit your answers. The polls are anonymous. Your names are not there, and it is just on this webinar. So please make sure that you have enabled pop-ups on your browser, because otherwise you won't be able to see the polls pop up. However, if you have a different or an older version of Zoom, you might not be able to see the poll pop up. That's okay. So if the polls do not pop up, then you can just put your answer on chat also. All right? So that's not a problem. You can still participate that way. All right. So we're going to do a poll practice to make sure that it works on most of your devices. So we are going to release a poll. It's just a simple poll to ask what grade you're in, and is this your first time doing Math Kangaroo? So let's see. All right. So there are two questions, and we would like you to submit your answer. All right. It's good to see so many of you participating. So it's just a simple question. And you will notice that when the poll pops up, it's right in the center of your screen. You can just click on the poll, hold it down, and move it to the side, especially when you're trying to look at the questions later. So you have the ability to move it around so it won't block your view. So we'll see a couple more right now. We have about 92 of you participating. Yeah, I see on chat that you cannot see the poll. Like I said, there are two reasons why. One could be that you have not enabled pop-ups. The other one also, it could be it's an older version of Zoom that does not allow you to see pop-ups. So this is pretty good. We have about 73% of you participating in the poll. Again, like I said, if you cannot see the poll pop-up, that's okay. You can put your answer on chat. All right? So don't worry about it. So we're going to end the poll. It's just a trial run. 100 over a few. That's awesome. All right. So we will end the pop-up and move on to an intro of what. Oh, the pop-up was just a trial to see what grade are you in and is this your first time participating in the Math Kangaroo competition? It was just for fun. A trial run. All right. So we're going to end the poll and we are going to get into what is Math Kangaroo. All right. So what is Math Kangaroo? For some of you, I saw that quite a few of you, this is your first time doing Math Kangaroo. So Math Kangaroo is an international math competition and is held on the third Thursday in March every year. It first started in 1998, and this is the 25th year of Math Kangaroo. As a Math Kangaroo manager, I have received the materials for this year for my students. And it has a t-shirt with the 25 years on it and a little Math Kangaroo toy and all the other goodies. So we are celebrating the 25th year of Math Kangaroo this year. So for levels one and two, there are 24 multiple choice questions. And it takes, you have about, you have 75 minutes to do the questions. And the questions are what we like to call unusual and interesting problems because it is not just math facts. Now, what is 7 plus 4 plus 3 minus 2? But it is questions that, now they are questions that require logical reasoning, that requires you to think a little bit more deeply to answer the questions. That's why at Math Kangaroo, we always encourage you to use the four-step method. Step one is understand the problem. So when you read the question, I always tell my students to read it not just once, but twice. The first time, you can just read through the whole question. Then the second time, you read it a little slower and you break it down. What information am I given that I need to use? Are there information and distractors? So maybe you can circle it, underline it. What is the question asking about? Do I have to look for pairs? Do I have to look for inches? Because sometimes the questions could be given in inches and they ask you for the answer in feet. So we always take the time to understand the problem. Do not rush through. Step two is plan how to solve the problem. So now that you have identified the key information in the question, like what information, what data you are going to use, what the question asks for, you are going to figure out how am I going to solve this problem? What strategy do I need to use to solve this problem? Do I draw a picture? Do I make a table? Do I work backwards? Do I make an organized list? These are the skills that we covered during our weekly Math Kangaroo sessions. What do I need to solve it? Again, you go back to the question. Look at the information that you have circled or underlined. Which information will I use first? So once you have planned all this, sometimes you will find that, oh, I actually need more than one strategy. And that is fine too. Because if you do the Math Kangaroo test, you will find that the questions start off really easy, and then it progresses in difficulty. So the more difficult questions, you will need to use more than one strategy to solve it. Once you have planned all that, then number three, you carry it out. You solve the problem. A lot of times students are tempted to figure it all out in their head. Please write it down. No matter how simple it is, it's always good to write it down. Because when you write it down first, it takes a little bit more time, and you can also see what you have written down. And you will say, oh, actually it's not this. It helps you catch yourself also. So then you solve the problem step by step. Number four, you look back. Once you are done, you look back to check and reflect. So you look at your answer. Does it make sense? For example, if it says that I have a certain amount of money, and I bought three apples and two oranges and now five pears, will I end up having more or less money? So this is how you figure out, does it make sense? You can also do, at school you learn how to do rounding. You can do a quick rounding and see whether your answer matches what it should be if you do a quick count. You also check, can there be a different answer? Am I looking at this answer question correctly, or am I just rushing through? So always remember to follow the four-step method in solving all the problems. So for today, our session covers draw a picture. So mad kangaroo problems have illustrations. And sometimes you have to look at the illustrations and you interpret it. What is it that we see? What is this figure or drawing trying to tell me? After you have read the problem, when we say read the problem, what information can I add to the figures that will help me solve the problem to make it easier for me to understand what the question requires? Sometimes it doesn't come with pictures, but it will help you to draw your own pictures. It's easier to use symbols. I love art, so I love to draw beautiful pictures. And sometimes my students like to do that too. And you will find on my slides I like to use cute pictures. But when you are doing the test, it's always good to just use circles, use shapes, because it saves a lot of time. So don't waste your time drawing beautiful pictures. No, you can use symbols. Use different symbols to represent different data that you have got. Or sometimes you can have graph paper ready, because it might help you, because there are already squares in it to help you count. So as we go through the questions today, we are going to use the four-step method in solving the problem. So same as the math kangaroo test, we're going to start off with simple questions, and then we're going to move on in difficulty. So the first question. Katie is in a boat on a lake. Draw the picture that she sees in the lake. So we are going to draw. So you are going to draw on the handout that you have. If you don't have a handout, it's fine. Just draw it on a piece of paper. What do you think she will see in the lake? Will she see the tree as it is, the crescent as it is, or how would it look like? And you also want to take note of where the position of the crescent would be in the lake. And it's okay if your tree doesn't look like that tree. Close it up. All right, so we are going to release the poll so that you can look at some of the answer choices of what you think the reflection in the lake looks like. And again, if it's blocking your picture, you can click on the poll, hold it, and then move it to the side. It is a pretty simple question, but be careful because you are looking at a reflection in the lake. So the tree and the crescent are not going to slide into the lake, right? It's kind of like when you look in the mirror. And you are the person in the boat in the lake. Pretend you're Katie. All right, I see a lot of answers coming in. I see a lot of answers on chat as well. If you are able to access the poll, you can just use the poll. You don't have to put the answer on poll and chat. All right, so we're going to close the poll in about five seconds. All right, and we're going to share the results with you. So as you can see, when you look at the reflection, it doesn't always look exactly the same. Like your right hand is your left hand, right? So same thing, the crescent is now on the other side of the tree. Okay, so let's move on to our next question. Together, Chip and Dale have eight nuts. They divided them equally between the two of them. Chip ate one nut. How many nuts does he have left? So now let's look at the question. What are some of the important information that we have? We have a total of eight nuts. Also very important is they divided them equally. That means they both have the same number of nuts. That's what it means. Chip ate one nut. Chip was hungry. And we want to find out how many nuts does he have? That means how many nuts does Chip have? So now that we have identified all this important information, we are going to solve the problem. So we are going to look at, okay, there are eight nuts. We divided them equally. Chip ate one nut. How many does Chip have left? Okay. We will let you think for five more seconds, and then we'll release the poll for you. And you can start putting your answers on the poll. Okay, we'll have the poll out for you. Almost everybody is united in their answer. And the answers came in fast, right? Five, since most of you have already put in your answer, we'll give you five more seconds for those who have not put in the answer. And then we'll end the poll and share the answer with you. Okay, so let's see your answer, whether it's right or not. So we have Chip and Dale each have four nuts. And then we have Chip who got hungry and ate one nut. So we cross out the one nut and he has three nuts left. Good job. Let's go to our next question. How many of the 12 pairs of dancers did not dance the waltz? So there are some important information that we're looking for again. So first, there are 12 pairs. What does pair mean? That means two, right? Two makes a pair. And Johnny saw 18 people. He did not see 18 pairs, but he saw 18 people dance the waltz. So that is where it gets tricky. Now question, how many of the 12 pairs? So it's asking you how many pairs left who did not dance the waltz? So we can always underline, circle the information and try to cross them out. So very important is how many pairs did not dance the waltz? So you have to think about, okay, 18 people. How many pairs is 18 people? All right. We're going to release the poll for you to submit your answers. 12 pairs of dancers, 18 people, which is equal how many pairs dance the waltz? So 18 people will be nine pairs dance the waltz. Now you have 18 people, nine pairs out of the 12. Okay. Good. We'll give you maybe 10 more seconds to enter your answer. And then we will close the poll. All right, we'll share the answer with you. So now we see a lot of different answers here. So let's go back. So we have 12 pairs. I drew 12 pairs. I have to draw that. 18 people equals nine pairs. So if we have 12 pairs altogether and nine pairs dance, so we are left with three pairs, not six, because six are six people. So this is what we meant by being very careful when you read the problem. So if you had drawn 12 pairs, you would have X'd out nine of them and you will be left with three pairs. Okay, let's try another problem. A frog jumps on every rock, starting on rock number three and finishing on rock number nine. How many jumps does it take? So you are going to use your handout and you are going to help Mr. Frog jump on each rock. Remember, he starts on rock number three and he finished on rock number nine. Then you count the jumps as how many jumps, not how many rocks. This is a pretty easy question. So we will release the poll in maybe three seconds. Right, you can submit your answers on the poll. I love seeing all the answers come in and almost everybody is in unison. All right, we'll end the poll in five seconds since the answers are coming in so quickly. And we will share your answers with you. So if you had drawn on the picture, you would have seen that he jumped from 3 to 4, 4 to 5, 5 to 6, 6 to 7, 7 to 8, 8 to 9. And that is six jumps. Okay, let's go to a different kind of problem. So you have this sign that shows the distance. You see that when you're traveling. And it shows the distance between the villages. So there are four villages on the map. The question asks, how far is it from Bahaisty to Momaisty? Those are kind of interesting names of places, huh? So that's the most important information. So you are going to look at. So the pole, this sign is right in the middle. If you go to Bahaisty, it is four kilometers. If you go to Momaisty, it's five kilometers. So when you draw the picture, keep in mind that these two villages are in different directions. Okay. All right, we will release the poll in five seconds for you to enter your answer. You should be able to enter your answer now. How far is it from Bahaisty to Momaisty? Oh my goodness, everybody got the same answer. Almost. So it would help if you draw a line below the sign and then you mark the distance. All right, so we're going to share the results with you now. Okay, so if you had drawn the line that would have helped you because you would have seen that there is the middle point. Bahaisty is four kilometers to the left and Momaisty is five kilometers to the right. So there is a total of four plus five kilometers. And that gives you a total of nine kilometers. And if you had got nine kilometers, good job. You're not going to get lost when you're driving with your parents. Okay, we're going to go to our next question. The clowns at the circus are standing one behind another. Vivi said to Bibi, who is standing behind him, Behind me, four coworkers are standing. Bibi answered, in front of me, only three coworkers are standing. How many clowns are standing in line? So what are some of the key words? There are two clowns, Vivi and Bibi. And Vivi said, behind me are four coworkers. Bibi said, in front of me are three. And you want to know how many clowns are standing in a line. Remember, they're all standing in one straight line. And this is one of the problems that you really want to draw. So you can draw circles for the clowns or Xs for the clowns. Remember, they are in line. Make sure you mark which one is Vivi and which one is Bibi. Something like that. That was a quick peek. So you want to know altogether how many clowns are there, including Vivi and Bibi. And we're going to release the poll in five seconds for you to put your answer. 1,001, 1,002, 1,003, 1,004, 1,005. Here comes the poll. How many clowns are there in one straight line? There is a battle between seven clowns and nine clowns. And don't forget that when Bibi and Vivi talk, don't forget they are also included. So I'm going to give you maybe 10 more seconds to think about it. All right. I think that was more than 10 seconds. Okay, so we are going to share the answer with you. And so when Vivi said that there are four coworkers behind him, so there's one, two, three, and four behind him, that includes Bibi. When Bibi said there are three in front of him, one, two, three, that includes Vivi. So don't forget that because many of you put nine and you're counting Vivi and Bibi twice. So for this question, if you do not draw it, it kind of tricked you into the wrong answer. So drawing a picture is very important. All right. Let's move on to the next question. 13 children are playing hide and seek. One of them is the seeker. After a while, nine children have been found. How many children are still hiding? So what is the information that we need to know from this question? First, we know there are a total of 13 children. Out of the 13 children, one of them is the seeker. So how many are hiding? After a while, nine children have been found. Out of those children who are hiding, nine have been found. So how many of those children are still hiding? We will release the poll in about five seconds. So you can draw 13 children, 13 circles. One of them is the seeker. So mark the seeker or X out the seeker. And nine children have been found. So you can X out the nine and count how many are left. So we will now release the poll so that you can put in your answer. I like what I'm seeing. Everybody is united on one answer only. One and the only one. Okay, we'll close the poll in five seconds. And it looks like more and more of you are able to access the poll. That is wonderful. All right, we'll close the poll and share with you the answer. So, you have 12 children, one is the seeker, nine have been found, so there are three children left, right? Okay, so now what we're going to do, we're going to take a little switch. Instead of hearing from me all the time, you're going to get to hear from Noah, who is the TA for this webinar, who has been achieving perfect scores for many, many years in Math Kangaroo, and he does participate in a lot of other math competitions, so he's kind of like a math whiz, so pay attention to his tips. Over to you, Noah, let me start sharing. So, there we go. Give me a moment. There, okay. So, I would say one of the most important tips would be to make sure you actually get through all the questions, as there are, is it around 30? 24. All right, so you've got 24 questions that you need to finish within a certain amount of time, but as you know, the questions generally get harder and harder as you go on, but sometimes they're not always in complete order of hardness, they're just generally in order. Sometimes you might find a harder question in the beginning that you're unable to solve, and there might be an easier question near the end that you could solve. Because of this, it's really important that you do not get stuck on a question and make sure you at least read through every single question so that you could solve those easier questions that don't appear in the beginning, and you won't get stuck on the hard questions and waste time in the beginning as well. Another thing is that, as stated earlier in the four-step method, the first one is to understand the problem. You've got to, for math kangaroo, you've got to read the questions very carefully, since they frequently require you to pay close attention or you'll mess up. Like, for example, they might give you units in feet, and then they would ask for an answer in inches. Like, if let's say your answer was two feet, and then they ask for your answer in inches, if you put in two, that would be wrong. You'd have to put in 24, and so on, like that. So, let's do a problem. So we've got two, Simon has two identical aquariums. There are 24 quarts in one and 42 quarts of water in the other. How many quarts does Simon need to pour from the second aquarium into the first to have the same amount of water in both? All right. So I would suggest that you would draw a picture like with your two, with your two aquariums, how much water you have in them and then see any patterns you can find that of how much water you'd need to pour in to make them equal. All right. Since this question is a bit more difficult than the previous ones, I think I'll give you all round 20 seconds, maybe. Does that sound good? Okay. So make sure you draw it and look at the amount. Now on the next slide, now we highlighted for you some of the key information that you need to look at. Yes. There are, should we release the poll first or go to the next slide? You can go to the next slide. Oh, the next slide is just highlighted information that I re-highlighted. So once you disable annotation, you clear annotation, disable it. Alright. So let's open the poll so you guys can put in your answers. Let's go to the poll. So, thank you. Now, when we see a lot of different answers coming in, then we say, okay, I would suggest that you go back to the question, take a look, because it's not just adding or subtracting right away, right? There are two containers and you want to make sure they have the same amount. So what do you have to do to make sure they have the same amount of water? Because you started off one with 26 quarts and the other with 42. So you can pour here a little bit, pour here a little bit. You have to use math, so drawing a picture really, really helps. And we'll have maybe five more seconds. All right. Okay. Okay, so here is the drawing. So as you can see here, it's been split into multiple parts. We have the 26-quart bottom here. And if you notice, it's also marked off over here with 26 quarts. I think you have to move on to the next slide. It's not showing. All right. So because of this, we know that we have to split this amount of water between these two equally. Since we already know the bottom section is 26 and both. That's already been split equally. But now we have this extra part. So we would need to move half of this difference from here to here in order to make them both equal. All right. So first, we find the difference. You guys don't see the picture? No, yeah. You have to go to the other slide. Sometimes it takes a little while, so you have to double-click it. Interesting, because I'm looking over here and I can see it. This is weird. Why don't I share for you? All right. Yeah. So I'm going to share instead. Maybe because I made the slide, you know. Right here, right? Yeah. All right. I think I need to open this up again. Give me a moment. It doesn't seem like I can annotate on it. Okay, I guess we'll have you take over the sharing. I'll just do it directly from here. See if this will work. All right. Is this visible? All right. So as I was saying earlier, we can split off 26 quarts from at the bottom. Oops. 26 quarts from the bottom of both of them. So we know that there's 26 quarts already here, and then we can split off 26 quarts here. And then now we have this section of water remaining. And then to evenly split this remaining water between these two aquariums, we need to put half of it in each, right? So we would put half of the difference from one aquarium into the other one. So if we were to find the difference first, we do 42 minus 26. And then we find that this section up here is 16 quarts. All right. So now the next part we'd have to do is we divide it by two. We get eight quarts. So we must move eight quarts from the one of the aquariums to the other one in order for them to have the same amount of water. And we can check it by doing... If we do 26 plus 8, we get 34. And 42 minus 8, it's also 34. So this is the equivalent of removing eight quarts from this one. And this equation is the equivalent of adding those eight quarts you just removed into this one. All right. So let's move on to the next question. Okay. Mara is putting up a tent for a family reunion. The tent is 16 feet by 5 feet. Each 4-foot section of the tent needs a post except for the sides that are 5 feet long. How many posts will she need? All right. I would suggest that you draw a picture of the bottom of the tent and then put your posts on it. All right. Label out the sides, the lengths, and put your posts on it to make sure. All right. So there's one thing I would like to bring to all of your attention. First, it's that when we have a line and we're splitting it like a fence, let's say, if we put a post like every now and then, if you notice, we've split the fence into four sections. However, we have five posts. Make sure you keep this in mind. All right, I'll give you guys five more seconds, we'll open up the poll. All right. The poll is open. All right, so we'll share the poll with you. I just opened it. Okay. Relaunch. Okay. All right, just as a general thing, it's really important that you can reason out things like this, because reasoning, if you don't, if you don't specifically remember something, if you can reason it out, then you can still use it even if you don't remember it. Like, for example, let's say I did not know that there's one more post than the number of sections we divided in. I could, like, draw out a line, and then split it into four sections, and then I'd still notice that there's one more post than the number of sections. Being able to find patterns like this is very important, since most problems, well, pretty much all problems will have some sort of pattern in them that can help you. Okay, seeing the answers come in, so we'll close the poll. We'll close the poll in, let's say, five seconds. Put in your final answers. All right. So let me go over the explanation of the problem now. All right. So as the screen text says, we have a 16 foot by five foot tent. We have one post every four foot, including one at each corner. No posts on the short side. All right. So if we were to draw it out, it would look something like this. All right. We have each four foot section has a pole. If you've noticed, there's four four foot sections, but there are five poles. For each of these 16 feet lengths. And then we also know that we won't put in any additional poles in the five foot lengths. So we've got our answer of five plus five, 10. All right. All right. Thank you, Noah. So as you can see, those are more difficult questions that will really help it if you draw it out, because if you don't draw it out, you're going to get confused. You might forget important information. Right. So we're going to go. I'm going to skip some slides and sorry about that. Let's see. And we're going to go to the next question. It says a frog is at the bottom of a 10 meter well. Each day he climbs up three meters. Each night he slides down one meter because he wasn't climbing. On what day will he reach the top of the well and escape? So important information, the well is 10 meters high or deep. Every day he will climb up three meters, but then in the night he slides down one meter. So on which day will he reach the very top of the well so he can escape? So draw, I would draw a line or draw a rectangle. So you can draw a rectangle and you will mark it 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And then you're going to put your arrow for day one. He goes up three, but he drops down one. So he will be at two feet from the bottom, right? And then you're going to keep doing that. So drawing pictures for especially the harder questions really, really helps. Oh yeah, you're currently sharing the presentation mode of the PowerPoint and not the slides themselves. Like the slides are there, but they're like small. Oh, that's weird. Let me try again. Good catch, Noah. Let me try again. I wonder what happened to that one. I'm going to try again and see if that works better. Okay, so let me do this. Sorry about that. It's not working properly. Okay, we'll just use this because the other one is not working so well. Okay, so if you draw a picture, then you should see you have a 10 meter well and you keep drawing and you mark and Noah will share the poll with you to enter your answer. All right, that was quick how the answers came in. It asks you also, I noticed that some of you mentioned what day, but in the question, it does not say what day it is. So it's day one, day two, day three. So your answer is not what day because he could have started on a Saturday. All right, we'll end the poll and you can see your answer because most of you got the answer right. So if you draw it like I did, you will find that it is day five that he will finally escape from the well. Now I have a very interesting question for you to try. This is about a dragon. A dragon has three heads. Every time a hero cuts off one head, three new heads grow. It's hard to get rid of the dragon. The hero cuts one head off and then he cuts one more head off. So your question is, how many heads does the dragon have now? Okay, so draw a picture. You can draw a circle and then it has three lines coming from it. and then you cross out one head and then you add on three new heads. Something like that. Okay, we will release the poll so you can put in your answer. All right, five more seconds. We'll share the results with you. Okay, so he has three heads. You cut off one head, it grows. That means it's left with two heads. Now it grows three more heads. So the dragon has five heads. And then you cut off one more head and it grows two more heads. So you have a total of seven heads. So drawing will make this so much easier. All right, I have one question I wish I thought that might be interesting for you because it's like you're in the classroom. There are 20 students. Eight of them are boys. At how many desks do the girls sit if two girls sit at each desk? So important information. There are a total of 20 students. Eight of the 20 students are boys. You want to know how many desks do the girls sit, not how many girls, and two girls sit at each desk. Aha, they share desks. Okay, so what is your strategy? So first, you can't just draw first. You have to think of, okay, there are 20 students. If eight of them are boys, then there are 12 girls. And if 12 girls sit two at a desk, how many desks will there be? So let's share the poll with you to put in your answer. I thought this would be a classroom question you can relate to. So we'll end with something familiar. Remember, it asks you how many desks, not how many girls, how many desks, and there are two girls at each desk. Okay, five more seconds to put in your answer, and then Noah will share with you the results. All right, we'll end the poll, and if there are 12 girls and 2 girls sit at each desk, you have 12 divided by 2, 6 desks. So I would like to remind you again of the four-step method, where number one, you understand the problem, two, you plan how to solve the problem, then you carry out the plan, and when you are done, always check and reflect. You always need to check. And you know, to check doesn't mean you look at the answer, okay, good, I answered it. To check means you go back to the problem and you answer it again, all right? So we're at the end of our webinar today. These are some of the reminders on how to draw a picture, some of the things we can use to draw a picture, and just some reminders at the end of this webinar, you will receive an email for a survey. Please take a couple of minutes to complete the survey so we can help. It will help us improve the presentation. There are two more webinars left for Levels 1 and 2. On next Saturday, it will be how to make a table. I know there's so much data, how do you make a table to put in your data and help you solve the problem, and then sometimes we have to go backwards to solve the problem. So there are also webinars for Levels 3 and 4, and 5 and 6, if you have siblings. So at the end of this webinar, you will receive an email to put on chat. I will, yeah, okay. So thank you very much for spending this one hour with Noah and me, and we are so glad that you're able to join us, and we hope to see you again next week. Thank you very much, and have a good weekend. Bye, everyone.

Math Kangaroo

drawing method

problem-solving

logical reasoning

Diana Holland

Noah

math competition

four-step method

interactive webinar

feedback survey