Good afternoon, Math Kangaroo. Welcome to webinar number six for level 5-6. I'm going to go ahead and put the warm-up problem on the screen. I expect in the next few minutes we'll have quite a few more students join, but you can work on that warm-up problem. Alexa is eight years older than her two sisters who are twins. The sum of the ages of the three girls is 32 years. How old is Alexa? Remember that during the meeting you can answer questions in the poll or you can put your answers in the chat for me or for Jacob if you have questions as we go along. You can send them in the chat to Jacob. I try to do my best with the answers as well. A lot of times I'm only able to give you like a thumbs up or a little smiley face but at least you'll know I'm trying. If you've just joined, you can see the warm-up problem should be displayed on your screen. Welcome to the Level 5-6 Math Kangaroo webinar. If you're joining late and you don't see the poll, you can just put the answer in the chat for me or for the co-host, Jacob. Great, most of you have answered, and I know that if you join late, you might not see the polls. That's okay. And you did really, really well. Over 90% of you say that Alexa is 16 years old. That's absolutely correct. So I'll just go over it really fast. We have Alexa is 8 years older than her two sisters who are twins. So we have Alexa plus the two twins, so we can call them Sister 1 and Sister 2. They're the same. Their age is 36 years old, and we know that Alexa is the same as her sisters plus 8. Using a little bit of algebra, we can say that there are three of those S's. Three of those S's plus 8 equals 36. Subtracting 8 from both sides, 3S equals 24, and S equals 8. Now we want to know Alexa's age, and remember, Alexa was the sister's age plus 8, so that makes her 16. It is 32. Thank you, whoever is correcting me. You're right. I do that all the time. It's just typos. Can you have typos when you're handwriting stuff? Yeah, it's 32. Excellent. Thank you for helping me out. So as I say, I'm only human, so if you see me making mistakes and you're nice and polite in the chat, just let me know. I don't mean to make mistakes, but we all have them, right? That's why we have proofreading and editing in this world. Okay. All right, so let's go back up. Today's problems are all working backwards. If you're not comfortable with working backwards, perhaps guess and check, or like the one we just did, some of them can be solved by using some algebraic equations. So if you're very comfortable with that, you have a lot of experience, you can go ahead and try the algebra. I'm not going to be doing too much of it because not many fifth graders have had formal exposure to algebra, but algebra is just a way of thinking about missing numbers or numbers you don't know yet, and you're trying to determine. So, you know, if you went through kindergarten and they said three plus blank equals seven, and you were able to undo it and say that the blank must be a four, you got introduced to algebra. So I hate it when people make algebra sound all scary. It's not. We can all do it. All right, so our four-step problem-solving method is definitely going to work today. It's going to have to understand the problem and determine what it is asking. And when we go backwards, remember, they're probably giving you the ending and asking you where did we start. When we solve a working backwards problems, we have to go the steps backwards and reverse them. So if you added something to go forward, you'll subtract to go backwards. Be very careful in completing these. And one of the great things about a working backwards problem is you can usually go forward and check your answer. So that's a really great way. We always talk about checking your answer. Does it make sense? Can you check it? You can on a lot of these. So in a working backwards problem, it's a series of events where the end is known and you need to find the initial or beginning state. Initial is a really good word for beginning. Sometimes you might use flow charts that can help you keep track of steps. Remember, if you undo the last step first, you're going to go backwards. And then if you move forward, you can check your results. So I really like these problems. We'll see how we do today. So many clicks just to go one slide. So another way to do some of these problems might be to take a reasonable guess. And then you can check and see if your guess is right. You might be able to figure out if you can revise. Do you need to have a bigger initial guess? Do you need it smaller? Maybe you can double it. So sometimes we have guess, check and revise as a strategy for figuring out some of these problems as well. My little brother has a four digit bike lock with the digits 0 to 9 on each part of the lock as shown. He started on the correct combination and turned each part the same amount in the same direction. And now the lock shows a combination of 6, 3, 4, 8. Which of the following cannot be the correct combination of my brother's lock? So this is what it shows. And remember, it says he turned them the same amount in the same direction. And we want the cannot be. There is no poll just because I couldn't put these little answer pictures in a poll. But I do have polls for many questions today. So don't worry, there will be a lot of them. Yeah, I'm getting a lot of really good answers here. So you can see from A, eight is six plus two. This is plus two, plus two. And if I do eight plus two, I'm gonna get eight, nine, and then restart at zero. So this is like plus two. So this one is okay. We're looking for the one that it cannot be. If I go, the difference between the six and the three is I went down three. I go down three from the three, I get a zero. This one is all of them have gone down three. This one is okay. If I take a look at this one, it looks like these have changed. Depends on if you wanna go up or down, but let's consider that we went down five, right? So if I go down five from the six, I get to the one. From three, what happens if I go down five? I get two, one, zero, nine, eight. So that works. From four, if I go down five, I get four. Three, two, one, zero, nine. That also works. And then the three is obvious, eight minus five. You can do the same for E. So the one that you cannot do this is C because you'll see we went down two to get to the four. And how did we get from three? If we go down two, we should have a one. And four, if we go down two, we should have a two. And six, six is correct. We did minus two, but they're inconsistent between the different numbers, the different dials. So it is C that is impossible. Number two, four squirrels ate 1,999 nuts altogether and each one ate at least 100 nuts. The first squirrel ate more nuts than any other squirrel. The second and third squirrels together ate 1,265 nuts. How many nuts did the first squirrel eat? So this is a combination of a little bit of working backwards, guess and check. There's quite a bit of logic in here. We have ate at least 100 nuts. We have the second and third squirrels together. And we have that the first squirrel ate more nuts than any other. Just a few more seconds before I close the poll. Remember, a guess is better than a blank on Math Kangaroo. But you should always check the rules for your contest or your exam to find out if you get penalized for wrong answers. If there's a penalty, then you don't want to just take a random guess. All right, here's the end of the poll. 59% of you have said 634. And we have a lot say either 629 or another answer. You might've found that you couldn't determine it, so maybe it's another answer. So let's take a look and see. There are a couple of really important facts that are definitely stated. We know that the second and third together ate the 1,265. We know that it was 1,999 altogether. So let's start with that, the 199, one, nine, nine, nine, minus this, 1,265. What do we get? Because that tells us how many the first and the fourth ate. So if we subtract, we get four, three, seven. 734 between the most nut and this one. We know that it has to be a minimum of 100 nuts. If we make this 100 nuts, that gives us 634. So we could have 634 here. So that is an answer choice, but does that work? If we take the 1,265 and we divide that by two, what do we get? We get 632 and 633 because we're not gonna do a half a nut, right? So if we put them in the order of most to least, we get 633 and 632. How do we know that these numbers are correct? Because this guy had to eat the most. So if we tried to split this into any other combination that would have a sum of 1,265, we wouldn't be able to get more nuts with the first squirrel. So this is the only combination that will fit all of the requirements given by the problem. I hope that makes sense for you. Made sense to me. So it worked for you, Jacob? Yeah, Jacob's giving me thumbs up. Sometimes I make mistakes, so you guys help me out if I do. Emily took selfies with her eight cousins. Each of the eight cousins is in two or three pictures. In each picture, there are exactly five of Emily's cousins. How many selfies did Emily take? It's kind of a good logical thinking problem. There is a question about which pictures Emily is in. It says Emily took selfies. So, Emily is in every picture, but all the rest of the information is about Emily's cousins. So, I don't think we really need to worry about Emily too much other than assuming that she is in each picture one time. Does that make sense with you, the way you read it, Jacob? So, the trend is good here where we have more people saying four than any other answer, but that's still less than 50% of you. So let's take a look at this. The way I look at this problem is I say, okay, there are eight cousins. If each of the eight cousins is in two selfies, that would be like 16 faces, right? If each of the cousins is in 24, is in three pictures, this is that they're in two and this is if they're in three, then that's 24 faces. So somewhere I have to get my total between 16 and 24. If there are five in each of the pictures, then the only numbers I can have are 15, 20, or 25, right? It has to be a multiple of five because there's five cousins per picture. So 20 is a number that's between 16 and 24, 20 would represent four pictures, four selfies. I get four, if there's five of cousins per four photos, that's 20 images of cousins, 20 faces. And that would account for having them in each picture between two and three times, each cousin two or three times. Jacob, you're looking at me puzzled. Does that work with you? Yeah, that works. I think it's a great solution. Number four, Tom thought of an integer, which Robert then multiplied by either five or six. Then Dan added either five or six to Robert's new number and Adam subtracted either five or six from Dan's new number to get 73. What integer did Tom think of? This is a five point tough question. This one does require, I think, you to really kind of analyze the possibilities and see how we can get that 75, 73, pardon me, 73. Just kind of giving you a little head start. We call the number N. We start with what they did to it. Maybe this will give you a hint. Well, I'm impressed that so many of you have submitted answers in the chat and so many of you have submitted correct answers in the poll. This is amazing. So let me share the results. Almost all of you have participated and 90% of you have the same correct answer, 12. So really good work. I'm not sure how all of you did it so well. This is the way I would do it is we have our original number that Tom thinks of. It's an integer. We know it has no fractions, no decimals, right? That's what integer means. It gets multiplied by Robert by either five or six. Then Dan adds either five or six and at the end, Adam subtracts five or six. This is for me the hardest part. So if this, we have four choices by the time we're done with Dan. If I subtract five and it's 5N plus five, then we still get 5N. The other option would be 5N minus one, right? Because if we subtract six, then we get minus the 5N minus an additional one. Same here, 5N minus six just gives us 5N, 5N minus five gives us 5N plus one, okay? And then we can do the same thing here. We're going to get 6N, we're going to get 6N minus one, we're going to get 6N plus one, and then we're going to get just 6N because if you add six, subtract six, you get 6N. So our target number here is 73. 73 is not a multiple of five or six. So this eliminates some of these options. It cannot be a multiple of five or six. It is not a multiple of five plus or minus one, right? Because it's not like a 26 or 24, it would have to be 74 to be a multiple of five minus one. So let's look, we only now have two options. It is not a 6N minus one because we know that 72 is a multiple of six, right? That would be 12 times six. So it is this one, a 6N plus one. 6N plus one equals 73. That gives us that 6N equals 72, and N equals 12, which was that answer that 90% of you got. So really, really good work on this problem. Jacob's nodding his head, he thinks it's a tricky one too. I think it's a tricky problem. So really good work, everybody. And if you're doing it by another method, that's completely fine. You know, share your method with us. Number five, there were 2,013 inhabitants on the island. Some of them were knights and the others were liars. The knights always tell the truth and the liars always lie. Every day, one of the inhabitants said, after my departure, the number of knights on the island will be equal to the number of liars, and then left the island. So this means one person leaves every day, right? After 2,013 days, there was nobody on the island. How many liars were there initially? OK, we have most of you answered the poll. I've answered as many of the questions in the chat as I can. Remember, ask Jacob as well. That way, we can get you some more help because I've got to continue and try to keep the pace of the lesson. So I'm going to end the poll and share the results. Again, 43% of you, not by any means unanimous, have said 1,006. Quite a few of you think it's impossible to determine. So let's take a look at this. Why is this a working backwards problem? That might be something to think about. Why did we put it in this theme? Now, when you're really taking the contest, you won't have that. But by practicing different types of problems, you'll be able to recognize, what am I going to do? The very last person to leave the island, the 2013th person to leave the island, said, after my departure, the number of knights and liars will be equal. And there were zero people on the island. So there were zero knights and zero liars. So that was true. So the 2013th person was a knight. I'm going to use K for a knight. That person told the truth. When they left, there were zero and zero. The person before that also said that statement. So the 2012th person to leave the island made the same statement. But there was one person left. So it could not have been true that there were even number of knights and liars. So that person was a liar. The 2011th person left on the island made the same statement, said there's the same number of knights and liars after I leave. And that is true, because there was one liar and one knight. So that person was telling the truth. And that person is a knight. Do you see the pattern we're going to get into, where it's a knight, a liar, a knight, a liar? All right. So let's think about the first person to leave the island. The first person to leave the island left 2,000, sorry, 2,012 people behind. And half were knights and half were liars, right? If we're following the patterns of it alternates. So there were half of these people were knights and half liars. The first person to leave was also a knight. We want to know how many liars there were. So there was one more knight. Sorry, one more knight than liar. It's hard to write with a mouse, guys. One more knight than liar. So there were 1,007 knights and 1,006 liars. B. Be careful, you want the number of liars. Yeah, so you can think about it even in odds, back up to what is going to happen and when there's only a couple of people on the island, that's a much easier situation to imagine. And you wind up with the alternating pattern. OK. We are at the point where I get to take a sip of water and Jacob gets to lead a problem. OK, so Zev has two machines. One makes one white token into four red tokens and the other makes one red token into three white tokens. But Zev starts with four white tokens and after 11 exchanges, he has 31 tokens. We want to figure out how many of these are red. So I'll give you guys some time to think about this and then we'll go over it. Okay, so hopefully you guys had some thought for this problem, but it seems like it was very difficult, so we'll try and explain it. Do I have permission to edit the whiteboard? Now you do. Okay, perfect. So I think most of us try to figure out the amount of tokens after each exchange, but obviously going through 11 exchanges and all possibilities is very challenging, so we kind of want to find a different way to think about this. So I think a good way to think about this is, I'm going to label one operation, so operation 1 is 1 white to 4 red, and then we have another operation that converts 1 red, so operation 2 goes 1 red to 3 white. And let's say there's A of these, and then there's B of these. So since we know that there's 11 exchanges, that means that A plus B is equal to 11. But we also know that we start with 4 white tokens and we end up with 31 tokens. But notice that in the first operation, the number of tokens increases by 3, so increases by 3 tokens, and operation 2 increases the total number of tokens by 2, so this one increases by 2 tokens. So if we perform A operations of operation 1 and B operations of operation 2, that means that we have this new equation where 4 plus 3A plus 2B is equal to 31. So now we have two equations that actually become much easier to solve. So this equation becomes 3A plus 2B equals 22. We can solve this with elimination, so 2A plus 2B equals, oh sorry, this should be 27, this should be 22. And so we get A equals 5 and B equals 6. So now we know how many of each operation occurs, but we still haven't figured out how many of these are red tokens. But notice that we start with 0 red tokens. In operation 1, we gain 4 red tokens, so we get plus 20 red tokens by just doing operation 1. And for operation 2, each time we do that, we lose 1 red and gain 3 white. So the number of red tokens that we lose here will be minus 6. So when we add 0, 20, and minus 6, we get 14. So in total, there should be 14 red tokens. Jacob, I have a student asking about this elimination process. How did you get to... Oh yeah, sorry, I went through that a little bit fast. I think for 5th graders, that's a pretty big jump. So if you can break that down a little. So... Here. Okay, so... I think it's pretty clear that when we subtract 4 from each side here, we get 3A plus 2B equals 27. And then when we double this equation, we get 2A plus 2B equals 22. So now we can set 2B equal to 22 minus 2A. So we just move the 2A to the right side. And then we plug in this 2B into this equation. So we get 3A plus 22 minus 2A is equal to 27. When we subtract the 3A and the minus 2A, we get A. And then when we move the 22 to the right side, we subtract 22 from each side. So we get 27 minus 22, which is 5. Okay, so we know that A is 5. But then we can substitute the A equals 5 back into the previous equation here. So we have A plus B equals 11, but we know that A equals 5, so then we get B equals 6. So yeah, that's how we solve that equation. Hopefully that cleared any confusion up. I think some students might, if we look at this equation and this equation, you can see that the difference is there's one more A here. We have 3A and 2A. The rest of it is the same, right? So 2Bs, 2Bs, 3As versus 2As. So any increase from the 22 to the 27 must be because you added one more A. So in a more intuitive fashion, you can determine that your A must be 5. So this is a very algebraic way of solving this. If this algebra is not comfortable for you because you haven't had enough experience with it yet, that is okay. What you can do is still think about the increase of three tokens and the increase in two tokens and try to make combinations of those that will give you 11. So you could do trial and error and make a table until you have 11 transactions, basically, but you increase 31 tokens. So you would still come up with this combination by kind of making a table and using some trial and error is another way to do it. You can look for a pattern that way. So, for example, if you did 11 that increased everything by 2, that would be an increase of 22. So that is not enough, right? Because we said that we needed an increase of 27, right? So you would have to increase more by the three tokens in order to get from 22 to 27. So that can help. All right. I would ask the students to stop annotating. I'm leaving it open so that Jacob can annotate for you, but I don't want the students annotating, please. All right, Jacob, there is number seven for you. Okay, sure. Is that one that you wanted or no? You didn't sign up for it, I guess. I can do it if you'd like. Okay. Luigi started a small restaurant. His friend, Giacomo, gave him some square tables and chairs. If Luigi uses all of the tables as single tables with four chairs each, he will need six more chairs. If he uses all the tables as double tables with six chairs each, so that's six chairs on a double table set up, he will have four chairs left over. How many tables did Luigi get from Giacomo? And I'm not sure I'm pronouncing those names correctly, but I'm doing my best. Pronouncing names is not part of the contest, guys. OK. We're over time a little bit, so I'm going to have to end the poll here and go over the solution. I've tried to draw a little picture of some of these. So the trend is correct. The answer is 10. Let's take a look at how we get there. And clearly, not everyone has gotten that correct answer or answered the poll yet, so that's OK. So this is the basic situation. If you have a single table, you would put four chairs around it. If you use a double table, you would put six chairs around it. So this is one table to four chairs. This is one table to three chairs. If we do this, we're short. We need six more. If we do this, we have four extra. So the question is, how many tables do we have? So if I separate this, let's say I separate this. So if I do tables with three chairs, then I have four extra chairs. If I do tables with four chairs, I need six. So how can that possibly be? So you can do it the way Jacob would suggest, is with some algebra. So four times the number of tables equals the chairs plus six. And then three times the number of tables equals the number of chairs minus that. Pardon me. That should be a C, not a six. The number of chairs minus four, because I have four extra. And then you can do some substitutions to solve for that, and you'll get that T equals 10. Without using the algebra, I want you to think about it in some logical terms. I can't do four times this tables, and I can't do three times. I have some extra chairs and tables that I need to move around. So if I had taken tables with three chairs, like this, I'll have four extra chairs. And I start putting those chairs, those four, around more tables. If I take, I have four plus, let me take, I need six. So let me add that six there. OK. What do I get? That's 10 chairs. That is the difference. The difference between three per table and four per table is 10 extra chairs that are needed. So that would be 10 tables. I can solve this also. I can make C, the chairs, equals 4T minus 6. And I can substitute in. I can get 3T equals 4T minus 6 for 4T minus 6 minus 4. So 3T equals 4T minus 10. So that gets me T equals 10. So some complicated ways to do it, and then some more simplistic ways to do it. All right. So wrong tools. Which of the problems did you like today? This working backwards or the guess and check problems? There's some tricky strategies. Some of the challenges is that sometimes when we do guess and check, we have to do a lot of testing. One of the things I want to point out, and I think Jacob will agree with me, math kangaroo is a multiple choice contest. So most of the time, there are only five options that you would want to guess. So the max time it would take was the time to guess five options. And if nothing else is working, that is not a bad idea. OK. A lot of times, we can simplify it. Sometimes you might know that the answer has to be even, or the answer has to be odd, or the answer has to be a multiple of a certain thing, or it has to be bracketed in between a high and a low value. So that can also help you out. So example, the selfies I knew had to be bracketed in between the 16 and the 24, because eight cousins in either two pictures. So I got a bracket, and I was able to come up with something in between. So there are some good strategies here, some good problems that we did today. I hope you enjoyed that. And we'll see everybody back here next week. And whichever team you're cheering for today, I hope they have a good Super Bowl game, OK? Bye, everybody.

Math Kangaroo

logical reasoning

problem-solving

working backwards

guess-and-check

algebraic thinking

trial and error

interactive learning

pattern recognition

problem constraints