All right, welcome back. This is the eighth webinar in our series. If you've missed any of the webinars one through seven, remember that you can go into the course where you registered for this course on your Math Kangaroo website. You log into your account and when you go to courses, you will see this webinar course. And if you go to the tab for contents, you will find the links to videos for any webinar one through seven. And then it takes about a day or two before you'll see this webinar in that place, okay? So hopefully that helps everyone to find it. Let's get working today. Our topic is working backwards. So in all of today's problems, you will have basically the answer, but we will need to go back to the beginning. I do have a lot of polls today. So I know you like to answer polls. I have quite a few. I have lots of polls. Okay, so let's try a warm-up problem. You left home to visit a friend. You went three blocks north, one block west, six blocks south, and finally four blocks east to get there. Where is your home? You'll notice this is showing you the friend's house. And we did give you, this is called a compass rose. If you forget which way the directions are, you're not sure which way they are on this figure, use this compass rose. I'll give you a few moments and you can put your answers in the chat. I will explain one thing, which is that these coordinates, I'm gonna have to click one button. These coordinates are going to be listed, X, Y, X comma Y. X is always the horizontal or the side to side, and Y is the vertical, or up and down. So this point right here is not where you started, but this point would be one across and two up. So it would be one comma two, not two comma one. The order does matter. Remember, if I get busy, you can chat with Shoria as well. I see some very good correct answers. I'm impressed. So our topic today is working backwards. So how do we work this problem backwards? We need to start with the last move that you made and we need to take that move backwards. So the last move that you made was four blocks east. So backwards of east is west. So from your friend, you need to go one, two, three, four blocks to the west, this way. Continuing backwards, it says six blocks south. Opposite of south is north. So we're gonna take six blocks to the north. One, two, three, four, five, six blocks to the north. One block west, remember opposite is east. So one block east to the right, right? And then three blocks north, the opposite will be south. So three, one, two, three, this is our home. And then we can look for the coordinates of our home. Can see our X coordinate is four and our Y coordinate is five, four comma five is answer C. So a couple of different things to think about. You're gonna do the movements backwards in the reverse order and the reverse directions. And then this problem asks you to know a little bit about how to express X, Y coordinates. If that's new and you're just learning how to do that today, then congratulations on learning something new, right? You know how I feel about that. I like learning new things and I like it when my students learn new things. All right, we are getting toward the end of our lessons. So that's good. In a problem where there is a series of actions and the end result is known, you can work backwards to find the initial state. We will have some flow charts. Sometimes you might wanna set up flow charts. Remember when we were doing our drawing lesson and we said simple drawings are always best. So this isn't a place where on a contest you need to do a fancy flow chart. A couple of arrows with a few instructions on them is more than enough usually. Undo the last step first and the first step last. And then number four is a really important one because in Math Kangaroo, we always tell you the last step of a problem is always to make sure your answer is reasonable. Check your answer, right? We use that. In the fourth step of these problems, you can always work forward. If you say, I think this is where I began, you can do the steps forward, read the problem and try and see if you get the same answer that the Math Kangaroo problem writers got. And that's a really good way to know if you are correct before you ever turn in your test. Okay, I do have a poll for this one. So I will read it and then I will launch the poll. What number needs to be written in the shaded cloud? This one is shaded. In order to get the number in the last cloud as a result of the operations shown in the picture. So this is an example of a flow chart. I'll launch the poll because the poll, the question is still there in the poll. Depending on what type of screen you have, you might have to scroll to see the answers, that's fine. Remember, when we do our polls, all the answers are anonymous, so I cannot tell, no one else can tell how you've done. So I do encourage everyone to participate in the polls, just try your best. Very good. I'm gonna share the results of the poll. So, there it is. You should be able to see those results of the poll. Three quarters of you, 75%, think the answer is five, the choice D. That is correct. Let's take a look at that. So remember, we start at the end and we work in the opposite direction with the opposite operations. So we're going to subtract four, five minus four is one. We're going to multiply by three, one times three is three. We're going to add two, three plus two is five. Now I can check myself by going forwards. Five minus two is three, three divided by three is one. And one plus four is five. So we are a good answer. Right? So that's what I mean about being able to check yourself by going forward. Okay. Here's problem two. Problem two, Shoria, you wanted to lead this one? Yeah. Okay. So there are some pieces of candy in a bowl. Sally took half of the pieces of candy. Tom took half of the pieces left in the bowl. After that, Clara took half of the remaining pieces. In the end, there were six pieces of candy left in the bowl. We want to find out how many pieces of candy were in the bowl at the beginning. So before any of these three people took the candy. All right. So I'll just let y'all work through that. And there is a poll for this one. So once you can chat me the answer, and once many people have done that, then we can launch the poll. Okay, the poll is on there and make sure you answer it so that we can see. Okay, I think mostly everyone's answered so I'll just end the poll. So here are the results. Looks like, yeah, almost all of us picked 48, 136, 124. Okay. So in this question, the best way to work backwards would just be to start with the six, of course, starting with the last step. So if I just write down six, because this is what we're going backwards from. Let me just draw three arrows because this is the one where Sally picked first. This is the initial thing. This is what we're trying to find out. And then Sally picked half. Tom took half. Clara took half. Now we can go backwards. So the opposite of taking half of something would be multiplying by two, because taking half is the same as dividing by two, because it's multiplying by one half. So if I had six, it would be three, and then I can just multiply to get back to six. So from six, I just multiply by two, which gives me 12. I multiply by two again, because these are the candies that Tom took. That gives us 24. And then multiplying by two again gives us 48, which is the answer. Here, it's also important to note that after Sally took half of the candies in the bowl, there's less candies in the bowl. So it's not like the next people took half of the total candies, but half of the remaining ones. So you can solve this algebraically as well, but the easiest way is to just go backwards and make sort of like boxes for each person taking the candy for each step. Yay. Are you sure, yeah? Yes. So if the students are familiar with algebra, they could set it up, but we are practicing these problems in a working backwards fashion. Peter Rabbit has 20 carrots. She eats two carrots every day. She ate the 12th carrot on Wednesday. On which day did she start eating the carrots? Now, if anyone does not know the order of the days of the week, that is going to be a problem for you to solve this. We assume that you have some general knowledge for math kangaroo problems, and the days of the week and the months of the year would be part of that general knowledge. There's no poll, so you can put your answers in the chat. Thank you. A lot of you have put your answers in the chat. I'm glad you're working along with this. So in order to work backwards, I'm going to start writing on the right-hand side of the screen, and that way I can work backwards toward the left, okay? So I'm going to kind of write the days of the week backwards. Wednesday. The day before is Tuesday. The day before is Monday. The day before is Sunday. The day before is Saturday. Friday. Thursday. Wednesday. So that gets me a whole week and a day going backwards. Now, it says that Peeta eats two carrots a day, and she ate the 12th carrot on Wednesday. So I'll put the 12th carrot on Wednesday. Hopefully that makes sense to everybody. But since she eats two carrots a day, what other carrot did she eat that Wednesday? Right? She's going to eat the 11th and the 12th carrot on the same day, right? Because if we're taking them in groups of two, in pairs, it would be 11 and 12 on Wednesday. Then on Tuesday, she would eat carrot 10 and carrot 9. On Monday, it would be carrot 8 and carrot 7. Does anybody want to change their answer? Working backwards from here? Sunday is carrot 6 and carrot 5. Saturday is carrot 4 and carrot 3. And Friday is carrot 2 and carrot 1. And which day did she start eating carrots? That would be carrot number 1, right? Carrot 1 is Friday, D. Now, I know there was a little bit of a confusing piece of information right here that she eats 20 carrots. We could ask, what day does she finish eating carrots? And then you would work forward from Wednesday. But this question is asking about the first carrot, which is on Friday. So you can let Shorya know if you're still confused about that one. Hopefully, now that you see it written out kind of plain in front of you, it makes a lot of sense, right? Remember, if you didn't have the correct answer to start with, but now you understand it, the next time you're presented with a problem like this, you should do just fine. I do have a poll for this one. But this one takes some students a little extra time. So we'll see. In an ice cream shop, there was some money in a drawer. After selling six ice cream cones, there are $70 in the drawer. After selling a total of 16 ice cream cones, there are $120 in the drawer. How many dollars were there in the drawer at the start? I'll give you a few moments, and then I might try to draw a little flow chart. And I also will at some point launch a poll. We'll figure out what the timing is like as I see how you're doing in the chat. I don't know if the poll is in your way or if you can see your screens. It probably depends on your device. I'm going to start a little flowchart. So I have a drawer with some amount of money that I'm starting with. And then I sell six cones, ice cream cones. And then I know that there are $70 in the drawer. And then it says I'm selling a total of 16. And then I have $120 in the drawer. Sorry, this is a messy six. OK. So I must have sold 10 more over here, right? Because this is going to be 16 minus 6 equals 10. That should be a big clue for a lot of you if you haven't answered yet. All right, I'll share the poll and we'll discuss it, get the final answer. So you did really well. Over half of you said 40, but we had several other answers. Now this is a problem where you could take these answers and test each one. You could do that, but there's a piece of information that we need to have. And the piece of information comes right here at this part. If we sell 10 ice creams and we go from $70 to $120, we can subtract and we say that we make $50 when we sell 10 ice creams. So therefore, one ice cream must be $5, right? So every ice cream is $5. Now, if every ice cream is $5, we can back up to the first step and we sold six. So six at $5 each equals $30, right? So we need to do 70 minus 30 equals $40 to find out how much money we had in the drawer when we began the day. It's kind of a real-life question. Whenever you have a business, you of course are going to start with some money so that you can make change, right? So you start with some money in your drawer at the beginning of the day. So that explains this problem a little bit. Okay, number five. David wrote a one-digit number and next to it, to the right, he wrote another digit to form a two-digit number. Then he added 19 to this number and the sum was 72. What was the first number he wrote? And I will tell you right now, most students understand this problem well, but read the beginning very carefully. There is a poll, so I will launch that poll and we'll see how carefully everyone is reading the problem. Anybody else want to answer? Most of you replied in the poll, so thank you. And almost all of you read that problem perfectly. The answer is five. You can see that most of you have said that it is five, almost 90% of you. Let's take a look. The biggest trick in this question is, it's basically saying that we add, so we know to undo it is subtraction, right? So we have the final sum is 72 and he added 19. So if we subtract 19, that's not very difficult mathematics, right? I think most of us can do that. So we take 12, we get a three, we had to regroup. So we get 53 is the two digit number. So that's, we're only at this step. We have found the two digit number is 53, right? But if we look very carefully at this problem, it says, what was the first number he wrote? And it says, David wrote a one digit number and to the right, he wrote another. So the first one digit number he wrote was B5. So this is a problem where I know all of you can subtract 19, but you had to very carefully read that the first thing he wrote was only one digit and which of those digits, the five or the three, he wrote the one on the left first, okay? Sharia, you said you wanted to lead number six. Yes, okay. So four friends, Masha, Sasha, Dasha and Pasha, I'll just call them M, M, S, D and P, were sitting on a bench. First Masha, M changed places with D, then D changed places with P. At the end, the girls sat on the bench in the following order from left to right, Masha, Sasha, Dasha, Pasha. In which order, in what order from left to right were they sitting in the beginning? So there's gonna be this one change and then the second change. And then in the end, we have this order right here. Okay, so there is gonna be a poll for this one. So you can just chat me what you think the answer is and then we'll launch the poll when I get quite a few answers. The reason we wait for launching the poll is because having the poll on your screen interferes with some students' ability to see the problems. So if you're wondering, why don't we launch it right away, that's why. Okay, I have launched the poll, so you can go ahead and put in your answers. And once we get most answers, then I'll go ahead and work through it. Anybody else want to guess in the poll? Remember, there's no negative points. We don't subtract if you make a mistake, and no one can see who's made which response. It's all anonymous. Okay, so I've ended the poll now, here are the results. So most people have picked A, but we've got a little bit of each answer, alright. So let's just go through this, let me get my annotation thingy, okay. So in the end, we're given that they were sitting M, S, D, and P, again I'll just use letters because it's much easier to use letters when we're doing this. Otherwise it takes quite a long time to write out all the names. Okay, so at the end we have this order, and we're going to go backwards two times, because we had one change and then another one. So let's just look at what both of these changes were. Alright, so first, remember we undo the last change first, because we're going backward from the last one. So we can't undo the first one, but we have to undo the last one. So we're going to undo this one first, and then whatever goes here, we're going to undo that to get our beginning order. Okay, so we know that the second change was that D changed places with P. So first we're going to make D and P change places, that gives us M, S, P, D. So we're going backwards, and we're just swapping them, because changing places is something that we can do backwards and forwards. And now we're going to swap M with D. So this is why it's important to go from the last change to the first change, because if we didn't do that, then we'd have a problem with where D would sit. So now if we change M and D's places, that gives us D, S, P, and M, because we swapped out these. And that is going to be answer A. So just remember that whenever there's a question like this, you swap them out in the right order. So make sure when you're working backwards, not only do you have to start from your last one, but you have to go exactly backwards, like consistently from the last one to the second last and third last, and so on, so that you can get the right answer. Thank you, Sharia, that was really good advice for the students. Okay, one week, Miss Florentina sold eggs at the market every day from Monday to Friday. Again, know the days of the week, please. On Wednesday, she sold 60 eggs. On Thursday, she sold 96 eggs. And noticed that every day the number of eggs she sold was equal to the sum of the number of eggs she sold the two previous days. How many eggs did Miss Florentina sell on Monday? Okay, I'm going to launch the poll. Some of you have this correct and are already working on the little second part that I put in the chat. Okay. It looks like pretty much everyone answered the poll question. Good job. And here are the results. Two-thirds of you have said 24, but we did get some of the other responses. The 24 is correct. Let's take a look at that. So I wrote down the days of the week because I like to organize myself. And so we know we can fill in that says, on Wednesday, she sold 60 eggs. And what we want to know is how many did she sell on Monday? That is our question mark, right? It says, on Thursday, she sold 96 eggs. So all I did was fill in basically the information from the problem statement. I needed the days of the week, Monday to Friday, and Wednesday is 60 eggs, Thursday is 96 eggs. And then she says she noticed that every day of the week, the number of eggs she sold was equal to the sum of the number of eggs on the previous two days. So the opposite of a sum is a difference or subtraction, right? So if sum means addition, difference means subtraction. So to get to this Tuesday, we're going to have to do 96 minus 60. That gives us 36. We're going to have to subtract again in order to get this one. So we're going to have to do 60 minus 36. We're going to turn that into a 5 to regroup. We get 4 and we get 2. So the correct answer for the question mark is 24 eggs. And then some of you also answered, I asked, just for fun, how many does she sell on Friday? Well, when we go in this direction, we're doing the additions. When we go backwards, we do subtraction. So this one would be 96 plus 60, which is 156 eggs on Friday. OK, so I don't want to do the exit ticket left. I'm having too much fun doing these problems with you. This one is very similar to one we did. Let's try this one. Liam's school begins at 8 a.m. It takes Liam 15 minutes to walk from home to his first class. He lives nice and close to school. Liam takes 40 minutes to complete his morning routine. When he gets dressed, eats breakfast and packs his backpack. Liam doesn't like getting up in the morning, so he presses the snooze on his alarm clock to get 10 extra minutes of sleep. What is the latest time Liam can set his alarm clock so he is not late for school? Does this sound like anything you might do, Shoria? I mean, I don't walk to school, but I don't use an alarm clock either because my mom wakes me up. And that's always on time. Your mom is reliable. I have to tell everyone a secret. Liam is my son. I wrote this about my son. There is no poll, so just put your answers into the, oh, there is a poll for this one. I can launch the poll. All right, most of you answered the poll. Thank you so much for doing that. I will share with you, we have some differences answers here. Most, a little over 50%. So that is a majority say 655, but I have quite a few students saying 705. That might be the 10 minutes snooze. There might be a little misunderstanding about how snooze works on an alarm clock. But this is a very useful type of problem because there's a lot of times, I have to be someplace at a certain time and it can't be changed, that time has to work. And you'll have to figure out how long do I have to give myself the traffic to drive there? How long do I need to get ready in the morning? How long? And so you'll be actually doing calculations just like this in real life. So it's a very realistic situation. So we're gonna work backwards from school starts at 8 a.m. If school starts backward, if school starts at 8 a.m. and it takes 15 minutes to walk, then we're gonna have to back up to 745. I'm gonna label the steps so we don't get lost. 745, 15 minutes to walk. And if he takes 40 minutes to complete his wake up routine, this is his wake up routine, wake routine, that gets us to 705. And then we know that he hits the alarm clock for a 10 minute snooze. So the way snooze works on most alarm clocks is you set the first time that the alarm goes beep, beep, beep, or buzz, or music plays, whatever you have on your alarm clock, you set it for the first time. Then when you hit the snooze bar, depending on the settings on your alarm clock, it's usually about 10 minutes, it'll do it again. But we have to set it for the first time you want it to make noise. That would be the time that you set on the alarm clock. So that would be B, 655 AM. You notice that in this subtraction here, this subtraction here, we had to remember that there are 60 minutes in one hour. We did some time and calendar problems, right? So 60 minutes in one hour, if I subtract five minutes, I get to seven. And if I subtract five more minutes, I'm at 655. So hopefully most of you are careful about being able to subtract 60 minutes in the hour, okay? You didn't try to make this a 95 or something like that. All right, we will go and we will do a problem that was before this that I said was similar to one we already did. Okay, in a chocolate shop, there was some money in a drawer. After selling three boxes of chocolate, there are $40 in the drawer. After selling a total of 12 boxes of chocolate, there are $94 in the drawer. How many dollars were there in the drawer at the start? Okay, so does this sound familiar? Instead of selling chocolates, we did one with ice cream. Let's see if you can do it all by yourselves now. All right, thank you so much for sending me your answers in the chat. I wanna thank this whole group today is doing a really good job participating and participating in the polls. So thank you. So we are going to find out, all right, in a chocolate shop, there was some money in a drawer after selling three boxes of chocolate. There are $40 in the drawer. And after selling a total of 12, there are $94 in the drawer. So this is pretty much the same way that we did the previous problem. If we have a total of 12, that means we can do 12 minus three. So that is nine additional boxes, right? And we have to do the same thing with the money. We have to do 94 minus 40. So that's $50 was for nine, sorry, $54. Delete, delete. There we go. $54 for nine boxes. And 54 divided by nine equals $6 per box. Okay, so three boxes times $6 equals $18. And now to get this one, we're gonna have to do 40 minus 18. Remember to regroup, you get a 10. So we started with $22 be in the drawer. Now, if we work this forward, we still need to know $6 per box. We still need to have that. So in this part, you have to do this part first. There really is no way without doing that part first to work through this problem. Okay, so this is kind of a necessary work backwards problem because even if we tried to work forward, if we didn't know $6 per box, we wouldn't be able to go start at the beginning. But once you know that, you can find the correct answer. And hopefully now you practice it twice, you'll do it every time correctly. So I hope you liked the working backwards problems. Remember, working backwards might not be the only way to solve these. As Sharia kind of suggested, there might be an algebra way to solve some of these problems. You could also test your answer choices. And some of these problems, you would have been able to just test the five different answers. But it might take you longer than just working backwards did. And when you're doing a timed contest, you don't wanna do things the long way if you do not have to. If you can't think of any other way, and you have time, always test the answer choices. That would be a good strategy rather than getting no points. So when do you use the working backwards strategy? When they give you the ending and they ask you about the beginning, it's also called the initial state. And the challenges to overcome might be how long this takes you to do that. But I think now that you've practiced, you'll be very proficient at it. Now, remember, the best way to prepare for an entire Math Kangaroo contest is to try to do a practice contest or two. That will show you how many different types of questions will appear, how long it takes you to complete all 24 questions, because that's really important is to practice your pacing, your time, how long it takes. Okay, if you're registered for the Math Kangaroo contest, you can have an access to some video solutions. So you can always test yourself with those video solutions. Shoryo, do you have any last advice for them before we say goodbye and thank them for great work today? I would say like practice as much as you can. I mean, if you can't, like, you don't have to take a lot of time to sit down and do it on one day, but you can just, you know, every day if you have time or not every day, but like every other day or just something like that, just make sure each week you sit down, do a few problems. You know, it just helps exercise your brain, even if you can't do it for quite a long time. So again, just look at past Math Kangaroo papers. Like I, all I did was the past papers. That's literally how I practiced for Math Kangaroo. I just did those. So the practice questions will really help you. So just do that. Do you wish there had been a course like this so that you could have somebody explaining the different types to you? Yeah, that would have been nice. Okay. Thank you. I hope you've had a wonderful Thanksgiving. I will see everybody next week. We only have two more of these webinars, so I hope you're getting the good feel for all of our Math Kangaroo types of problems. Bye everybody. Bye.

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