Hello, everyone. Welcome to Mexican group webinar, lesson three. Today we are going to learn the special thinking. And last week, Miss Diana taught a drawing picture. That's a very useful strategy to help you to solve the problem. And today in this lesson, we also would use a drawing picture sometimes to help you to solve some problems. Before we start the lesson, we have some new students joining. Elita, could you please quickly go through the ground rules? Yeah, so the first rule is that the student should join the Zoom meeting with the video off and voice muted for the whole session because we are recording and sending these recordings out to everybody. All the lessons will be recorded and the link will be shared after the session. So if you guys want to go back and look at how to do a problem or anything like that, you can find it in the recording. This is a webinar, so it's a large group class with less interaction. But to increase engagement, we'll have polls so you guys can vote on what you think the answer is. You guys can't speak during the session, but you can chat. You can use the chat function to ask me any questions and I'll do my best to answer as many as possible. And then lastly, the student handouts are all in the shared folders. So they're really helpful because they have diagrams and the questions on them. So please bring them and be prepared with pencils and paper. Thank you, Elita. So we always want to mention the four-step method. So this one is for the questions. You know the question, we have three sections. We have a three-point section. We have four-point and a five-point question. So for the three-point question, usually it's a one-step solution. So we do not recommend you to use this four-step method. But once you try to solve the four-step or four-point or five-point questions, so that would involve a multiple-step solution. So we strongly recommend you to use this method in addition to all the strategies. So the step one, you need to understand the question. So you need to read the question carefully, check the picture carefully, and ask yourself what the question is asking for. A lot of the time, the question, if you do not really get the question, what the question is asking for, you will answer the wrong question. Like the question gives you the five options, four right statements, one wrong statement. They didn't ask you for the right statements. They ask you for the wrong statements. If you do not read the question carefully, you might answer wrong. You might just select any one of the right statements as your answer. So that happens a lot. The second step, you have to plan how to solve the problem. You identify the problem, and you know what type of problem it is. It might be patterns, or you can use joint pictures and special thinking today and later, like work backwards, guess and check, make a list, organize a list. So those are all the strategies you could use, one or more, to help you solve the problem. So this step, you would make your plan, what you should use to help you to solve the problem. The third step, you need to carry out the plan. So you need to write down the solution step by step. You have your pencil paper. You can write down when we solve some problems in the practice test or in the real competition. We strongly recommend you to write down the solution. Because when you write down, that would help you to reduce some minor mistakes. Because when you write down something, you would have better thinking. And later, if you want to go back to check the answer, so you have something already written down, you do not need to do it again. So that's why the third step is very important. Once you select the answer, we recommend you to review the answer choice quickly. You can put the answer choice back to the question, and then to see if it applies to all the conditions. So you can do a quick check. Do not start on one question. You need to manage your time to finish all the rest of questions. At your level, there are 24 questions in 75 minutes. So here, that's from last lesson, drawing a picture. This is a very popular strategy. Like patterns, you know pattern is everywhere. And drawing a picture, that's also very helpful. Some questions, they give you the picture. But some, they didn't. So the picture helps you to see what happened here. And you also can add the actual details to the existing picture. And help you to better understand the question. So sometimes, with the picture, it's very easy for you to list the equations. And also, please remember, you always, during the picture, use the simple shapes. Do not draw a very complicated picture. Take too much time. Also sometimes, you can use different colors. Or the lines with a different weight. Some lines thick, some lines light. And also remember, you do not need to draw something very perfect. So you can just draw something you understand. Help you solve the problem. That's good enough. And draw to skill. So you also need to pay attention sometimes. Because in the question, they give you some of the items. So in the real time, the items, you have the ratio. You draw adult or you draw a kid. If you make it in the picture, you really would draw adult. It's taller. The kid is shorter. So that gives you an example. But sometimes, maybe the kid is bigger than adult. But mostly, if they give you the question, you know the size. You know which is bigger, which is smaller. When you draw the picture, you have to make it skill. So that's drawing picture. And today, we're going to learn the special thinking. Let's take a look at the first question. Elita. So you can see there's a tower on the right. And the question is asking you, what does the tower look like from above? So it's kind of showing you from the front. But you have to think about what would happen if you were on top of it. This is a question from 2015. It's a number five. It's a three-point question. All our lessons, we will use the real problem from previous tests so you would have a clear idea about what the Mexican group questions look like. Okay, I'll end the poll now. So you guys can see that most people chose B, which is the correct answer, but a lot of people also chose A. You can see that if we're looking from above, so you can see the arrows kind of pointing down on top of the thing, what you'll end up seeing is the circle, right? Because you'll only see the cylinder behind the cone. So if we were looking at it from the front, you would be able to see that the cone has a point to it, that it's kind of a triangular shape, but since we're looking from the top, all you end up seeing is the circle from the cylinder. Yeah, thank you, Elita. If you look from the side, yeah, like you look from the side. So, I mean, you look from the side, what do you could see? You could see the triangle. And also, you could see a rectangle. So that's from the side, yeah. Also, if they ask you, if you look from bottom, what do you could see? You can put your answer into the chat. You can send it to me or send it to Elita. What do you see if you look from the bottom? If you look from bottom. Good job. Yeah, very nice. So far, I received all the answers, correct? Okay. Okay, take a look. If I give you a side face, take a look at the question again. You look from the bottom, you look up, what do you want to see? You see this coin and cylinder, they stack together. So if you look from the bottom, you only can see the bottom of the cylinder. So what do you could see? Let's circle again, yeah. So look from different angles, you might have a different look, but you have to check. You have to really rotate some subject in your mind. Or sometimes you can use the real object to help you to figure it out. So that's the special thing. It's not that easy, but it's a lot of fun. Let's continue to try more questions. Before we develop more questions, we will talk about what is special thinking. So in the Mexican group computation, you always need some question. They ask you to look at the object from the different angles and imagine what happened. So if you could move them or you could change them. So why the special thinking is very important. So because it could help you to understand how the shape and objects fit together. It could improve your ability to solve the puzzles. You see, we will have some puzzle questions later. And also, they will help you to see the patterns or some relationships between the items. So that's very helpful. So you can imagine how you turn a cube in your mind. Yeah, you could see six pieces. Yeah, top, bottom, left, right, front, back. How you turn the ball in your mind. How you turn a cylinder in your mind. So from different angles, different direction, you might see different shape. So next we talk about the strategies for the special thinking. Elita, would you like to go through this? Yeah, the first strategy is to draw the object as you imagine it. So sometimes if it gives you a word problem or if it tells you some things about the object, then you'll have a mental image of what it looks like. And so you can kind of put that onto paper so it's easier to work with. Then you can add details like shapes, lines, or arrows to show how it moves or changes. Although you guys might want to make it perfect, I think it's unnecessary. It's just for you to help like to help you understand it better. Another way is to use different colors. So if you use colors to highlight different parts of an object or to show how it changes, then it can be a lot easier to look at. So you can use red for one side of the shape and blue for another side. The third way is to act out in your mind. So you can put the object in your head and move it or rotate it to look at it from different angles. And if it's helpful, you can even like use your hands to show that it's rotating or moving around. The fourth way is to take notes while imagining. So if you write down what you think or see as you imagine the changes, you can use words like first, next, and on top to describe what happens. And if you label parts of your drawing to help you understand better, that can be really helpful too. And then the last way is to practice with real objects. This is probably the easiest way to learn spatial thinking better because you can use like blocks, papers, or other things to recreate the problem. And then you can actually see how they change as you move or rotate them. Thank you, Yilita. So using a drawing picture would be very helpful. Even if you do the spatial thinking, you are not solely just using your mind. You also can draw it out to help you to figure out what is the correct answer. So let's see the next question. Yeah, this one looks like a puzzle. With a real puzzle piece, it would be very, very easy for you to pick the right one, because it's only four pieces of the puzzle. But if you just imagine your mind, yeah, so how you solve this one, which one should be the correct answer? Correct answer. Which piece fits in the empty place in the puzzle? Any more answers? We are going to close the poll. The three-point question, usually we can take one to two minutes to solve it. OK, so here is the answer choice. So most of you pick the B. That's correct answer. Good job. So let's see how to solve this one. We have five pieces. And you know you miss a piece on the upper left corner. So for the piece, they didn't put the exact position. So you might need to rotate the pieces in your mind. So you have to figure out which one is the correct one. So you can see the solution here. They mark the color over there and show you which part you should find in the answer choices. So A, B, C, D, E. But for the one you missed here, you see there is a right corner, only the two straight lines. The other two lines, they're curved. So the first one, you actually can eliminate the wrong answer with the three straight lines. So that's D. So this is not correct. Elimination is very helpful. You could have less choices to solve the problem. And then you could figure out for other corner, it should be upper left corner. So for this one, like number A, you should turn right. If you imagine your mind, you should turn right for this piece. You turn 90 degrees right. And then you could put on the upper left corner. And you would see the bottom curve line, it's not match. So this is not correct answer. And B, you need to turn left in your mind. And then you could fit on the upper left right corner. And then you check both curve lines. They all look correct. But it's better you check the rest of options because sometimes there are similar answers. There have minor difference. So it's better you look through all the answers. You can check C. You also, the C, you couldn't fit, no matter how you rotate. Because this corner, it would fit upper right corner or bottom right corner. You cannot fit the upper left. Then you would see the E. The E, you could see the two curve lines is not correct. So you can take up the right answer. Finally, you confirm B is the correct answer. There is another way I also want to mention. If you take the in-person computation, you would have the booklet to work out. So if you do the special thinking, you think that's a little hard for you, you also can turn the test booklet, turn the test paper, turn around the test paper to help you to find the right position for the piece. So that's also like you work with the real object. You do the acting out. OK, let's see the next question, Elita. Yeah, the four numbered puzzle pieces below can be used to make a picture of a fish. Which of the diagrams below shows us how to correctly put this puzzle together? Yeah, I think we can share this result. Yeah, it looks like you guys really understood this one. Almost everybody got this question correct. So great job, everybody. I can see that one should go in the top left corner because the body of the fish is clearly to the right and down of one. And then two should probably be on the bottom left because the rest of the body of the fish is either above two or to the right of two. And then for three, the body of the fish is going to be to the left of three or below three. And then for four, the body is going to be to the left of four or above four. So that makes it so that one goes in the top right corner, two goes in the bottom left corner, three is in the top left corner, two is in the bottom left corner, three is in the top right corner, and four is in the bottom right corner. So that's why it's D. And you guys can also draw it out yourselves to check. Thank you, Elita. For this one, first, you have to look at the four pieces. Make sure which part of the fish. And then that would be very easy for you to find the answer. Good job, everyone. So next one, they ask you, the kangaroo is inside how many circles? If you have the handout, if this picture on the handout, you can draw the different color to separate all the circles. Good job. It looked like, yeah, this question also, most of you picked the correct answer. Yeah. Edita, can you share the result? Yeah, so most of you picked three, and that's the correct answer. So this way, like I mentioned, you can use a different color to draw all the circles and to help you to see how many circles should be there, have kangaroo inside. And you would see the black circle didn't include the kangaroo. So there are only three. The total circle, there are four. So one black one doesn't include the kangaroo, so only three left over. OK, next question. Edita. Five square cards are stacked on a table as shown. The cards are removed one by one from the top of the stack. In what order are the cards removed? So you guys can see that some of the cards are covered by the other cards, which makes them rectangles or other weird shapes. So you have to decide what cards to remove first to remove them all. Remember they told you there are five square cards, so the original shape should be square. That would help you to decide which card on the top of another card. This is a four-point question, but I feel sometimes it's worth five point. Okay, Yadita, let's share the result. Okay. So I think you guys did pretty well with this one. Most of you got it right. So let's see. So to decide what card we have to remove first, you have to make sure that the card isn't being covered by any other cards. So that means that the first card must be a square for us to remove it. So we can see the only square that's available for us is number five. So we take away number five, and then the only square left is gonna be two, right? So then two would be next, which reveals three as a square, you can see. And then once you take away three, one is covering four. You can tell because four is like kind of a skinny rectangle. So that means it's covered because it should be a square card. So then once we remove one, then we're just left with four. So that's why A is the correct answer. Yeah. And also when you try to select the answer, if you're very confirmed which card is on the top, you also can eliminate the wrong answer. So that would help you to find the answer quickly. Yeah. Good job, everyone. Next one. Four stripes are woven into a pattern as shown here. What do you see when you look at it from the other side? So this is the front side. They ask you to look from the back side. What do you can see? Yeah, I think we can share the result. Just wonder, does anyone not use the poll? You do not know how to use the poll. If anyone, you do not know how to use the poll, you can send us chat. So the answer would be B. Most of you pick the correct answer. Very good. Let's look what happened. So this one, first you look from the front and they ask you when you walk to the other side, that's backside. So you should remember, when you walk to the backside, you can see in the front, you could see the two red stripes and the two green stripes. The red stripes, that's vertical and the green stripes, they are horizontal. So when you go to the backside, so the green stripes, their position doesn't change. If you look at, so if you look at, you would see the top bottom, they didn't change. It still keep the top bottom. But if you check the red stripes from the front, you could see the left stripe. If you walk to the backside, so the stripe would be on your right side. It would be on your, so that would be the different, that different position. So you have to pay attention, which one. It's the same as your front left. So if you go back to the, go to the backside, so you would see in the front side, the left, the right stripe, you are above the two green stripes. When you go to the backside, you could see the red stripes, which one, both above the two green stripes. When you go to the back, it should under both. So you would find, you know, in the backside, your left, yeah, so should be, should be, actually should be above the two green stripes. But you look from the front side, they would give you the under, yeah. So you could check the other one. It's also the same. So in the front, if you look, you look at the right stripe, it's under the green. So go to the backside, it would be above the green. So that's what you can find, the B is a correct answer. So this one, you have to imagine, yeah, imagine if you go to the backside, what you would see. Okay, next question. Which of the shapes shown below will fit this shape exactly to make a rectangle? Oh, if you couldn't see the poll, you can move your mouse to check your screen. So the poll should be there now. I'm gonna end the poll now. You guys did really good on this question. Almost all of you got it right, so good job, everybody. The correct answer was E, by the way. So you guys can see that the way we get it is kind of like if you just make a box around each of the shapes and you look at the empty space in the top, that space should match the shape that we gave them, right? So you can see the first one, A, that doesn't match because there's only one peak in the middle, I guess you would call it. The second one doesn't match either. And the third one, there's four of those small things in a row, which we don't see in our original picture. In D, there's a big valley, I guess you could call it, in the middle, so it can't be that one. But E works because you can see it makes the same shape that the other one does. But I think you guys got this concept down pretty well. Yeah, thank you, Yilita. Sometimes when you see the picture, they are missing some part. It might be better you draw it out like this one, the A, B, C, D, E. They all finish the rectangle. And then you could see what's the missing part, and you compare the options they give to you. So that's very easy for you to find the correct answer. You can finish the missing part, if you could. Next question, which of the pictures below is a shadow of the girl and the tricycle? We just opened the poll. Some students might not see the poll because if you have a single screen you can move your mouse to check. Yeah, we can share the result. Okay. Good job, everyone. Now, most of you pick the correct answer. Now, for this one, the correct answer is D. So you can see the picture here. We can check all the answer options. So the A, if you see the shadow, they miss the back wheel. That's very clear. You could eliminate this option. The B, so one ponytail lost. There are only, the girl should have two ponytails, but only one left over there. And then if you look at the C, you could see the left leg is a stick out. So it's not, it's actually the girl didn't stick after that left feet, left leg. And the D, so that would be the correct one. And you also can find the E. E has the front wheel is up. So that's not the correct shadow. So the D is the correct answer. So you look through all the options. Please check all the details. Sometimes there might be the minor difference between the option. So make sure you could eliminate all the wrong answers. Confirm your selection. Yeah, next question. This one might be a little challenging. If you have never played this kind of a toys before. Elita, go ahead. So the picture shows two gears, each with a black tooth. Where will the black teeth be after the small gear has made one full turn? Have you ever played the gear toys before? I believe some playground, they might have this kind of the toy or there are some other toys sold in the store. It's, you can play the gears together. Yeah, sometimes maybe more gears, not just two. Yeah, I think in some Lego sets when they have a motor or something, then they use gears to help things. You have to think about how this two gears move. You see the pink one move and how the purple one move. So who made the foot turn? That's a small gear. This is very important. Yeah, this is a five point question, so your early five point question you might spend three to four or five minutes I'll give you guys 10 more seconds. I think you can share the result. Yeah. Um, yeah, this one I think was a lot harder maybe than the other ones. It looks like the results were kind of spread out, but the correct answer was C. And I think technically most of you got C, about half of you. So the first thing we want to do is we want to count how many gears are on, I mean, how many teeth are on the small gear, right? Because every time the small gear turns with the big gear, they each just rotate one tooth, right? So it looks like there are eight gears on the small one. So that means that the big gear must also rotate eight gears for the small gear to make a full turn, right? And the next thing we want to do is we want to figure out what rotation the big gear is going to be rotating in. So because the small gear is rotating this way, you can kind of think that the big gear will have to rotate in the same direction, right? So that's why it looks like the big gear is rotating counterclockwise. So then the next thing we'll do is we'll just count eight gears down the big gear, right, going counterclockwise. So you can see the first tooth to the left of the black gear is one, and then the next is two, and then so on. And you just count until we reach the eighth tooth. And that's where the black tooth should end up being, after the small gear makes a full rotation. Yeah, so you should be very clear. So when the small gear moved, so can you see that the small gear, the tooth, would push the purple gear, the tooth? Yeah, push, push, can you see? When they push, it's opposite direction. It's not clockwise, it's anticlockwise. So pay attention when the two gears, they work to each other, they will go the different direction. That's very important. So remember this. That would help you to solve the next question. So this one, it's cogwheel A, turns around completely once. You can see the A on the left, very left. At which place is X now? So where is X? It's the third one. Yeah, middle one, if we say that's A, B, C. Yeah, middle one is B. They didn't mark, but you can mark by yourself. So the B and also the C on the very right. So they have an X, they ask you, where is the X? For this question, you look at the A, B, C. So A is the first one to turn. A work on the B, and then B work on the C. Okay, I think we can share the result. Okay. Yeah. Good job, everyone. Almost half of the class picked the correct answer. This is a very hard one, I believe, compared to all other special thinking questions. Yeah, good job. A is the correct answer. Okay, so let's go back to look at the picture first before we move to the solution. So first you look at the three cogwheels. Yeah, cogwheel A work to turn the B and the B turn the C. So they ask for the X part, where the X part would be when the A move the full round. So you would see the A, they gave you the clockwise. So they show you the clockwise. If you could see the picture, they show you the clockwise, the A move. So if you want to do this way, you would see the B connect to A would move what direction? Yeah, counterclockwise, is that right? Yeah, they would move the different direction, yeah. And then C, C would connect with B, would also move the direction opposite to B. So that means the C will follow the same direction as A. So this is a very important part you need to understand. And then they ask you, they list the name of the small tooth. Yeah, A, B, C, D, E. They ask you where the X would land it on. So when you check this one, you would think about, so like the last question, how many teeth the cogwheel A have? That means the C would move how many teeth too, because only the A will decide how many teeth would move. So the A, we would say how many teeth, A would have seven teeth. Yeah, the B, it doesn't matter here. The B just in the middle part. So it doesn't decide how many teeth C would move, only A decides. So you could see the C has six teeth. So now you would see for the C, C would move one round and one more and one more tooth. So let's look at all the steps. So you can see the first 10, where the X would be. The X at the very beginning here, yeah. And after first 10, the X move to the left part. So that's the A, position A. And then after second 10, third 10 until sixth 10, because you know there are six teeth for the C on the very right. Yeah, they didn't mark C, but we can make it a C. That's easy to see. So after the sixth 10, you would see the X back to the original place. So that you need one more 10, because there are seven teeth for the A. So you need a seventh turn. After seventh turn, the X move to the A again. So that's the answer. So this is a very important. You should understand how the gears work to each other. So the neighborhood gears always move to the opposite directions. You understand this part, that would be very helpful. And also who's the first moved view? Who would decide how many teeth they will move totally? Good job. I think half of the class make the correct answer. That's very good. Yeah. Okay. What do you think, Elita? Yeah, this is a little hard question. Yeah. Yeah, I think this question is really hard because you have to think about all three gears and which way they rotate. Yeah. Okay, next one. Yeah. All right. So in a stack of three cards with holes, the top of each card is white and the bottom is gray. Basil threaded those cards on a rope. See the picture on the right. Which of the following can he obtain without untying the rope? So the trick for this one is to look at the thread and where it's going over the card and where it's going under the card. You should check very clearly. So how the thread connect all those card. This is a pretty challenging question. So you have to imagine, because when they gave you this one, they make all the white on the top. So you would see the top gray, top gray, top gray. That would be the correct order for all the faces because each card has two faces. Top is white, bottom is gray. So they stack them by this way, all the white on the top, gray on the bottom. So if you check the order of all the faces, that would be white, gray, white, gray, white, gray. Or you can go the other direction. It's a gray, white, gray, white, gray, white. So this is very important for you to select the answer. So you should check the answer choices. So hold the faces next to each other. If you found the two same color face next to each other, that wouldn't be your choice. That's very important. I think this one is a super challenging, yeah? Yeah, I think so. I think this one's very tricky. You have to really be able to move it around in your head, I think. Yeah, and also you need to think what's the neighborhood of faces, yeah, for the different card. Remember, if you found the face the same color next to each other, that wouldn't be your choice. You must have the different color next to each other. Yeah, I think maybe we should work together for this question because, yeah, the students get the correct answer. Yeah. You can share the pool. Okay. Yeah. Um, so I think this one was really, really hard because only three of you got it right. The correct answer was E. So let's see. So the first thing let's try and do is you see how on some of them the string is on top of the first half but covered on the second half or some of them it's covered in the first half of the card and on top of the card in the second half. Let's flip them all so that the string is, um, on top of the first part of the card and covering the second half of the card because that's how it's kind of shown to us in the picture, right? So if we can flip them all like that, then whichever option has it so that all three are the same color when the string is the same way for each card would be the correct answer, right? So let's start with A. So for A, you can see that the string is on the first half of the first card and on the first half of the second card, but not on the first half of the third card. So if we flip that third card over, then we get it so that the string is the same for all three cards, but we can see that that's not, those are, um, two different colors. There's gray and white for the cards, so it can't be A. So then we'll move on to B, right? So for B, it's the same thing. The first and second cards are good, but the third card has to be flipped. And we get gray, white, white, which is also not good because all three have to be the same color. And then for C, you can see that, again, it's the same thing. The third card has to be flipped over, right? So that the string is the same way for all three cards. So if we flip the third card over, then we get white, gray, gray, which is still not correct. And then D, um, I saw a lot of you guys chose D probably because that's how the cards are presented to you, but you can see that actually the third card is different from the first two because the string is laying on the card in a different way. So we have to flip that over. So it's actually not three white cards. It's two white cards and one gray card, which is, um, wrong. And then for E, you can see the third card has to be flipped over, which does give us three gray cards. So since all three cards are the same color, then we know that we can get to this, um, configuration without untying the rope. Thank you, Yelita. For this one, like they give you before, you know the, the thread get into the card with white and get out with gray. So this other side. And the next card also get in with white, get out with gray. And the third card, same way. If you go to the different direction, you would get in with gray, get out with white. So you would know that the faces next to each other, they would be the opposite color, not the same color. So if you also check this way, you can see that A, if you get into the card with gray, you get out with white, but you can see the next card you're getting in with white. So you have two white next to each other. So that's also help you to figure out this is not the correct answer. Yeah, so this is another way you can check which one would be the correct answer. You can check. So the string, the thread get into the card, what color in, what color out. So the next card, next card, what color in, what color out. So that's also help you to find the correct answer. So this is a pretty challenging. So I have seen a lot of students couldn't find the correct answer. So that's why every time when you see some of the questions with picture, you have to check very carefully and read the question carefully, understand what question ask for, what condition they already gave to you. Yeah. Thank you, everyone. So today we learned the special thinking strategy. This is a very helpful. So please try to do more practice. If you have some items at home, like different 3D shapes, so you can try to rotate them and try to help you to understand how to look at from different angles, different side. So please also sometimes remember you can draw the picture and you can take notes and help you also use different colors, all the different, like a thick or thin lines, help you to draw the picture. Yeah. Thank you, everyone. So we will send you the recording shortly after the class and hope to see you back next week and have a wonderful rest of the day. Thank you, Elita. Yeah. Yeah. Thank you, everybody. Yeah. Thank you, everyone. Bye bye.

spatial thinking

mathematics

problem-solving

visualization

diagrams

interactive exercises

real-world examples

competition preparation

strategies

student engagement