Good afternoon, Math Kangaroo. Welcome to the sixth webinar in our Level 3-4 series. So welcome back or welcome if this is the first time you're joining us. I'm Dr. Sarah Segee, and we have a teaching assistant who is Shura Vyas. And he's waving to everyone. You can communicate with us in the chat. That is the best way if you have some questions. We'll also ask you to submit some responses to problems in the chat. And we do have polls for some of our questions. And so when you fill out a poll, it's completely anonymous. No one knows what your answer is or who answered which choice. So please give it a try on a Math Kangaroo contest. You do not get penalized if you guess wrong. So it's always better to try than to leave a blank. All right. And that kind of goes with today's topic, which is guessing and checking. But Math Kangaroo doesn't want you to make just wild, crazy guesses. My kids always ask me questions and I always tell them the answer is seven. And they're like, mom, it's not seven. So don't just guess seven, right? Actually look at the problem, look at the information you have. I bet you'll see that you don't have to guess as crazy as you think. You'll probably be able to come up with some really good ideas to try. And every time you try an answer, you get information on whether you were too high or too low and you'll be able to try again. All right, let's try our first warm up problem. Aline writes a correct calculation. Then she covers two digits that are the same with stickers. Which digit is under the sticker? You'll notice I emphasized a few words. It's correct, this calculation is correct. And the digits under the stickers are the same. Okay, if you haven't written that problem down on your scratch paper, you might want to write it down when I launch the poll. It should display the problem, but in some systems you'll notice that the polls kind of cover things up or whatever, so just do what you need to. Okay, just about everybody has put an answer in the poll, so I'll share the results with you. Looks like almost everyone said seven. There's a little bit of a response for two, and that makes a little bit of sense because we see that the last digit is a four and two plus two is four, but let's see why seven is correct. Sorry, I have to click on the correct things. So I like, I don't know, I'm just a little old fashioned, maybe I like to do my addition vertically. You don't have to. So I noticed that I have a four plus a five and I'm getting a 10. So I am gonna have to do what we call carrying or regrouping. I am gonna have to make a group of 10. So in this position, I'm actually gonna need to get 14. And of course, seven plus seven is 14. And that makes the correct answer E. Some of you might've said, okay, how can I get, you know, 40 plus 50 is only equal to 90. So you may have said, okay, 104 minus 90 is 14. So you might've found a difference that way and taken 14 divided by two is seven. That's another very good method for doing this problem. So there's frequently more than one method for math kangaroo problems. Don't worry if the way Shurya or I solve it is a little bit different than yours or maybe even completely different. That's okay. If you have questions about it, you can certainly put that in the chat. Shurya would be happy to see other methods to doing the problems and help you out, okay? So you can see we've been doing several different types of methods for solving problems. This week we're on our guessing and checking lesson. We had an intro lesson at the beginning. And the last one is we're gonna use a combination of all the different tools. So you'll see how math kangaroo makes you use your tools together to solve even more complicated problems. Okay, so I have a favorite game I like to play. It's a little hard to play in this type of a format, but we're gonna use the chat and see if we can make it happen, okay? This is a game I play with my children when we go hiking or when we have long car rides. It's called Guess My Animal. And it's an example of the guess, check and revise method that math kangaroo would like you to be practicing. So you can see here, it says, guess, check and revise. So I have a statement and it says, I'm thinking of an animal that begins with the letter H. And then anyone playing the game gets to ask me yes or no questions to try to guess the animal that begins with the letter H. So if you have a yes or no question, go ahead and put it in the chat. Let's see if we can do it this way. Sharia, you can play too. Anyone have a question for how to guess my animal that starts with an H? Oh my gosh, I have somebody already guess it. That is correct, but let's try yes or no questions so we can practice the process. Okay, the question is, does it have wings? No, it does not have wings. So that limits it. Is it a mammal? Yes, it is a mammal. Any other yes, no questions? Does it swim? The answer is no, it does not swim. Well, it could swim, it just doesn't. I think that's, you don't normally think of it. It's not extinct, it does live. So several of you have guessed horse and that is correct. So you can see how the yes, no questions got us closer to it. Horses can swim. Some of you may know that horses do come in and swim. Some of you may know that horses do cross water. They do know how to swim, probably not something they prefer to do like a fish would. Okay, so that's the guess my animal game. And you can see how every time you make a guess, you ask a question and you get some information. It helps you to make the next choice even more intelligent. So that's the kind of technique we're gonna use today is to be more intelligent with each new try. Here's problem number one. How many white squares need to be shaded in the picture so that the number of shaded squares is equal to exactly half of the number of white squares? Some of you have heard me say that math kangaroo is a reading comprehension exercise as well as math and logic. It says how many need to be shaded? Some already are shaded so we don't need to shade them if they're already shaded, right? So you're gonna be turning white squares gray. And it says that the shaded squares is equal to exactly half. Not exactly the same, exactly half. I do have a poll, but I'll let you guys think for a moment. All right, I will launch the poll. This little picture is in the poll. And if you've printed the handout before you came to the webinar today, you should have the picture as well. Otherwise, you can probably draw a six by four rectangle on your paper. It's six by four. When I launch my poll, it's hard for me to see the answers on the bottom, but I can scroll. When I touch on the right-hand side, I can scroll down and get to the other choices. Anyone else want to add their response in the poll? I'm seeing a variety of answers. I think we can point out where some students have read the question a little bit off. We can help you so that when we do this question again, it'll be perfect. Alright, here we go. Here are the results. So you can see we've had students kind of give us a variety of answers. I understand why some students are giving us the answer eight, but the correct answer in this case is three. But it has to do with a very critical reading of a part of the problem statement. So let's take a look at it. When you read this problem, we can see that it is a six by four rectangle. So in a six by four rectangle, there are 24 squares, right? It's six by four. And that gives me a total of 24 squares. And it is telling us that the shaded squares is exactly half the number of white squares. So if I have a shaded square, if this is my shaded column, shaded, and this is my white column. If I have one shaded, I'm going to have two white, but the total there, the total would be three. Remember, we have solved problems by using a table, right? That was one of our lessons. So we could continue with this and we could say, okay, if I have, well, let's say four shaded, I would have eight white because I have to double it and that's going to equal 12. Okay. Well, 12 is not correct, but it is half the number of 24. See I'm using my educated guessing. I'm already at half. If I want 24, then I need to double everything. So that would give me eight and 16, 16 is double eight. So this will work where I have eight of them shaded. That fulfills the half, exactly half of the number of white are shaded because half of white would be eight and eight are shaded. So I currently have five that are shaded. So I need to add three more shaded, don't I, in order to get a total of eight. So I shade in any three, then I will have the correct number eight shaded. Now can you do this in another way? Of course. I said that there's multiple ways to do math kangaroo problems. You may have simply said, okay, if my ratio is one to two, I know that I need one part shaded, two parts white. That's a total of three parts, right? So my total recipe basically would be three parts. So I could take 24 divided by three and that equals eight. So then I know that my one part is eight, my two parts is 16. So I could get there a little bit faster than making a table and doing guess and check, but it gets me to exactly the same place. Now for those students who did circle eight as their answer, it makes sense because we do have that eight here, but that is not how many need to be shaded because you're not going to count the ones already shaded. See the little reading thing in here with the math kangaroo. So be careful and read exactly what the problem says. All right. We're going to let Sharia read, lead problem number two. Okay. So Steven, wait a minute. Okay. Steven wants to write each of the digits two, zero, one, and nine in one of the boxes of the sum blank, blank, blank, plus question one. He wants to get the largest possible answer. Which digit can he write instead of the question mark? So we know each one of these four digits are going to go into one of these four places. They're not going to repeat because we need all of them and we have to find out what digit can go here to get the largest possible answer. Okay. So if you have any ideas, please tell me in the chat and you can have a few minutes to think. I think there is a poll for this one also. Okay, I have launched the poll and you can scroll down to get the answers. The picture is also on the poll. Okay, so I've shared the results for the poll, you can see A was mostly the selected answer, a few people chose B, C. Okay, so yeah, that looks like quite a bit of, you know, different answers, so let's work through it. I'm gonna take this away. Okay, so I'm gonna start by like working backwards and guessing and checking at the same time. So I'll just draw three blanks, plus, and then our last blank for whatever we're putting in. So since we have four choices, two, zero, one, and nine, I'll just pick one of them to put in the last blank. So let's say we pick two. Now we still want the largest possible answer, so I'm just gonna choose what to put here. So if I don't have two, if I only have nine, one, and zero, then the largest possible I can go is if I have a nine over here, and then a one, and then a zero, because otherwise I'd have like a hundred and something, or we can't even start with zero, so that's not a possibility, or I have nine over one, which is smaller. So this is the largest we can get with two, and that's going to equal 912. Okay, so now let's just do the same thing with the other one. So if I have three blanks, and now let's put in zero. So now I have nine, one, and two left. So I start with nine again, because then I would have 200, so 100s, which are smaller. Then I'll have a two, and then a one, so that I can get the highest possible number, and that's going to equal 921. And then I just do the same thing for the other ones. So we'll just do one over here, so then we have nine, two, and zero. So I have to pick nine first, then a two, then a zero, and that gives me 921 again. And now lastly, I can just try nine over here, and we can easily see that here. If we take out the nine, we have really small numbers left. The highest we can get is only 210, 219, which is really small. So let me box up all of these answers. So these are the four answers we get if we use two, zero, one, or nine. And well, we can see that this is too small. 912 is smaller than 921, and the other 921, and those happen when we have a zero. So therefore, the answer would be either zero or one. The important thing here is that if you just try the fastest you go, and you just think like, okay, I'll just put in zero for the last one to get the highest value, that doesn't really work because one has the same answer. It's sort of like thinking of it as writing 920 and then one, which is the same as writing 921 and zero. Okay. Okay, we'll move to the next problem. I do not have polls for some of the problems where we have picture answers or where there's a lot of graphics. It makes it difficult to set up a poll. So feel free to put your answers in the chat to Shoria or to me, and we can let you know how you're doing. If you need any hints or anything, let us know that as well. So this one has a lot of pictures. Together, the apple and the pear cost five cents. Together, the banana and the apple cost seven cents. Together, the pear and the banana cost 10 cents. And we want to know how much it would cost if you want to buy a banana, a pear, and an apple. We know these are not realistic prices, but it still works for math practice. I'll give you a few moments. Alright, I'm seeing several of you getting the correct answer, so that's great. Several of you had to try twice, that's alright, it's no problem. So when we have multiple choice problems and you don't know what else to do, you can always try the answers and see if you can make the answers work. Yeah, I'm getting great answers in the chat now. Okay, so I've had students suggest several different ways to solve this problem. So I've had students tell me, okay, I looked at the first two pictures and I can see that an apple and a pear cost 5 cents and an apple and a banana cost 7 cents. So therefore the banana, the banana is 2 more cents than the pear, right, because this went up by 2. So swapping a pear for banana, I have to add 2. So when we try to do that, if we say that the apple and the pear together have to be 5 cents, then we can think, okay, that could be 4 plus 1, it could be 3 plus 2, it could be 2 plus 3, or it could be 1 plus 4. I'm going to try the first one, 4 plus 1. So if this is 4 and this is 1, I know the banana is 2 more, so this would be 4 and this would be 3, and that does equal 7. So so far this is a pretty good guess. Let's try putting it together here. I have the pear being 1 and the banana being 3, and that does not equal 10. So that is not going to work. So let's try it a different way. Let's see what happens if I switch it around. Let's see if I try, I'm going to clear this off, let's see if I can clear it. Let's see if I can try it another way. Let's see if I can try making the pear 4 and the apple 1. So that makes this 1, that makes this 6, right? That will work because 1 plus 4 is 5 and 6 plus 1 is 7. Now when I do this 6 plus 4, I get 10. So now it works. And if I do that here, I'm going to have 6 plus 4 plus 1 equals 11. So that is one successful way to solve it with a bit of guess and check. You could have checked the 3 and the 2 and the 2 and the 3 combinations as well. So it would have taken maybe a few guesses, but you could get here. I had a student tell me, and I thought it was quite clever, that if I put these two together, then I'm going to have two apples, a pear, and a banana. And that would cost $0.12. And I know that the pear and the banana cost $0.10. Pear and banana, or banana and pear are $0.10. So therefore, my two apples must be $0.02, and my apple is $0.01. And all I have to do is add the apple here, and I get that it's $0.11. That works too. It's another nice way to do it. Finally, another way that I like to do it, and this works for a lot of math contests, a lot of math contests will kind of do this, will do a problem like this for your level. What if I bought all of it? If I bought all of it, you'll notice that I'm going to have two apples, I'm going to have two pears, and I'm going to have two bananas. And if I add the 5 plus 7 plus 10, that would cost $0.22, right? What if I wanted one of each? Well, I just have to divide everything by two. I take half of everything. So $0.22 divided by two to get one of each is going to equal $0.11. So I've just showed you three ways that you could go about that problem. All three are fantastic ways. So if you have done something similar, you could have grouped them differently and done kind of a similar method, but with different groups. That's completely fine, all right? So we will clear all this and move to our next problem. Which number should be written in the circle with the question mark? There is no poll, so go ahead and use the chat for your answers. I do like communicating with you that way. I hope Shoria is giving you some encouragement and some help as well. Which number goes where the question mark? My goodness the students were fast with this one. So you could put any number into one of these circles and you could try to follow through with that number. Okay that is a one perfectly great way to do this. All right so say for example I wanted to start at the top and I just want to put the number one over here. One minus 15 is negative 14. You guys have maybe don't work with negative numbers so maybe I'll start with a different number. Maybe I'll start with 20. 20 minus 15 is 5. 5 plus 4 is 9. 9 times 0 is 0. 0 plus 6 is 6. And 6 plus 4 does not equal 24. So it does not equal 20 it equals 24. So I'll change that I'll make that a 24. Then 24 minus 15 I have to fix it right. 24 minus 15 is going to be 11. No it's not. Pardon me. Oh my goodness. Sometimes your teachers make mistakes too. Is 9. 9 plus 4 is going to be 13. Now when I go around it works. So all you had to do is put in any number and start to go around and make it so everything worked. But probably a lot of you just saw this right away. If you saw this right away then all you had to do is say I know that anything times 0 is always 0 and that's the big clue in this one. So if you just started there with the 0 and came around you would have gotten the answer 13 right away. All right. This is a longer problem to read. So I will read it nice and slowly. I encourage you to follow along as I'm reading. Anna, Beata and Jack go to the same school. One day the librarian said to them guess how many books we have in the school library. Anna said 2010. Beata said 1998 and Jack said 2015. It turned out that the number of books in the library differed from the numbers given by the children by 12, 7 and 5. These numbers are not necessarily in the order they made their guesses. How many books are there in the school library? So one option is to underline or take notes on the important information in this problem on your scratch paper. Go ahead and figure out the important numbers and see what you can do with those. If you get totally stuck then try some of the answer choices. See if one of those makes sense. All right, several of you have figured it out, so congratulations. If you haven't and you want to ignore me and keep working, that's fine. I won't know the difference, but I will start to show you how I would go about this problem. So one of the things I like to do is I like to have things in order. I'm a pretty organized person. I like things all cleaned up and put away. So I might actually put these into numerical order. Oops, sorry, I have to enable the annotations. Hang on a sec. There we go. So I might list 1998, 2010, and 2015. That's a zero. And then I know that the differences are five, seven, and 12. Okay, so when I look at this, I'm trying to figure out what are the differences between the numbers that we already have. That would be one of the ways I might do it. So I can see that the difference here is five and the difference here is 12. So that's good, except that there's no difference of seven. So maybe I have to figure out how to get a difference of seven, five, and 12. If I try this number 2000, just cause I need some place to start. If I try the number 2000, the difference here is only two. So I can pretty quickly cross that one off. And then I can say, okay, I don't like 2000. What happens if I try the 2003? I tried 2003, my difference here is five. That's pretty good. And my difference here is seven. I'm liking this one. And my difference here is 12. So I got a pretty quick answer by testing the possible multiple choice answers. And that is a really great way to do it. No problem with that. I could have also looked at these differences of 12 and five and said, okay, I like the 12 and five, but somehow I'm missing the seven. So how could I get a seven? What if I tried something in between here and here so that I broke up that 12? 12, I can get a seven. Maybe I want it to be a seven here because seven plus five is also 12. And then I get a five. So if I do 1998 plus five, that's gonna give me my 2003. And I already tested 2003. So I know that that gives me the correct differences. Like I said, I did that by trying to turn this 12 into a five and a seven, because I know that seven plus five is going to give me 12. And that way I would have all of these differences covered. So just a couple of approaches for trying that problem. Hopefully one of those makes sense and works for you in the way that your brain would like to process this problem. Remember, we're all unique and a little bit different. Number six is a problem that Shorya wanted to lead. So I'll leave him to that. Okay. So the numbers one to nine are placed in the square shown with one number in each square. And the sums of all pairs of neighboring numbers are shown. Which number is in the shaded square? So we want to find out which number is gonna be here. We know they're from one through nine and they're all used one time because there's nine boxes. And all the pairs have the sums which are like 15, 17, 39, so on. Okay. So I guess I'll leave you all to think about that. And you can respond in the chat. And there is a poll for this one. I guess I'll end the poll now because I see most of y'all have answered. So, let's see. Most have picked 7, but there are a few people who have picked 5, 6, 8, and 9. So, okay, quite a bit of diverse answers. So, let me just, um, let me just Okay, so what I thought here is that there's a lot of squares, so if we just try to guess and check, like, all the numbers from 1 through 9 in the first square, that's gonna take way too much time. Like, even if we're able to narrow it down, it'll take too long. So the best way to do this is, um, if you look at this one, this sum is pretty important because it's a sum of 3. And we know that the only numbers that sum to 3 are 1 and 2, because we can't have 0. So that means that we figured out that there's only two choices for these two boxes. They're either gonna be 1 or 2. So then we can just use that. So we can guess and check with 1 and 2. So let's say we put 2 right here, we put 1 right here. So now all we have to do is just use this whole pattern. So if we have, you know, for example, 1 plus 8 is 9, 8 plus 7 is 15, 7 plus 4 is 11. But now we have a problem, because 4 is half of 8, so we'll need another 4. But we can't repeat numbers. So that means that putting 1 here is actually not the right choice. So then that just means we're going to have to put 4 here. So let me draw a new box down here. So that means in our third and fourth boxes, we'll need to have a 1 and a 2. And then we can just make sure that we are right. Since we've used process of elimination, we should be right, but we can just make sure. So if we have 1 here and 2 here, these two are going to be 9, that gives us 7. These two are going to be 15, that gives us 8. These two are going to be 11, that gives us 3. These two are going to be 8, so that gives us 5. And then the last one is going to be 4, because they add up to 9. And then we can just go back from here. These two have to be 7, so they're going to be 6. And then this one's going to be 9, because that's 15. So guessing and checking helps in this one, because once you find out that the sum of 3 is the one you need to look at, the one you need to focus on, because it only has one possibility, you have two ways of guessing and checking. You can do 1 and 2, and then you can do 2 and 1. So narrowing it down to two cases, sort of, makes it a whole lot easier. All right, so the correct answer here turns out to be 7. 7 is in the gray box. And this is kind of like that guess my animal game, right? If I told you that the animal is a mammal, then you wouldn't try to tell me, is your animal, oh, it was H. So you wouldn't say, is your animal a housefly? Because a housefly is not a mammal, right? So if we can limit the number of guesses we have to make, that's going to save us a lot of time. So that's why this is a really cool example with the number 3 here. All right, number 7 also has a poll. The numbers 2, 3, 5, 6, and 7 are written in the squares of the cross in this figure in such a way that the sum of the numbers in the row, rows read across, is equal to the sum of the numbers in the column, up and down. Which of the numbers can be written in the center square of the cross? And you'll see the answer choices, so you'll have to maybe test several of these to see if it can be in the center of the cross. You want sums across and down to be the same. I'll give you a few minutes, and then we'll launch the poll. Okay I'll launch the poll because I want to make sure we can get to the next question as well because it's a fun one. So you can't see all of that on your paper remember you're just making like a plus and you have the numbers 2, 3, 5, 6, and 7. Anybody else want to make a guess in the poll? All right, I'm gonna end the poll. Half of you got it correct, and that's really good. This is one of our harder problems. This is a five-point problem. Remember, we have three, four, and five points. Number 20 is going to be a five-point problem. So this is one that should take you a little more thinking time, a little more investigation. Five or seven are the correct answers for what can be in the center. Let's take a look. Something to keep in mind is when I'm adding numbers, if I am adding A plus B plus C, and I'm also adding D plus B plus E, I don't really have to worry about the B because it's in both of them, right? So I just want to make sure that the top red box and the bottom red box have the same sum as the left and right blue boxes. Does that make sense? So I'm only gonna add up pairs of numbers. So you see me adding up pairs instead of triplets because I'm trying to work a little faster, and I don't need to worry about the center one. It's in both the across and the down column row. So if I take a look at the numbers that I start with, I need to come up with pairs that have the same sum. And I can see that two and seven have the sum of nine. And if I want another sum of nine, I can use three and six. That leaves me five for the center. So I know that five is one of the options for the center. So I can cross out only three and only seven, right? Does that make sense? Let's take a look at it again. I remember I'm adding up the top and the bottom, and I don't really worry about the middle. So how about a five in the center? If I put the five in the center, I can make the sum of two and seven. Oh, I already did that one, sorry. Did the five in the center. So what if I try a different type of sum? What if I cross off the seven and I do two plus six and three plus five? So I'll put seven in the center and two plus six. Two plus six is eight. Three plus five is eight. So seven also works. So I can cross off only five. So now I'm left with, that's either choice D or E. Let's see what happens if I try to put that three in the center. If I put three in the center, that gives me choices of two, five, six, and seven. And I need to come up with two pairs with the same sum. If I do, I can't do the two low ones and the two high ones. That's clearly not gonna work. If I try to do the outsides and the insides, I have 11 and I have nine. So it's not going to match up. So three does not work, and that leaves me the choice is five or seven for the center of this cross. So I really wanna do the bonus problem. So I hope you like guessing and checking, but the really important part about guessing and checking is to use the best, most intelligent guess that you can. And if it doesn't work out, adjust, do that revision. So remember on that problem where I said, oh, you know what, I wound up with 12 squares and I needed 24, the revision was to double it, right? So make sure you're using those revisions. Elena wants to write the numbers from one to nine in the square shown below. So just like the problem that Sharia was leading, the numbers one to nine. The arrows always point from a smaller number to a larger one. She has already written five and seven. Which number should she write instead of the question mark? Now, because we're almost out of time, I'm gonna work this one out loud with you. The first thing I noticed, you notice five and seven, they're not exactly in consecutive order. There's no six here. So that tells me they don't have to be exactly like one, two, or three, four. I can skip numbers as long as I go from smaller to larger. One of the things I'm gonna look for is I looked at this box, I found all the arrows were going out. So this has to be the smallest number because all the arrows are going out. And this one has to be the largest number because all the arrows are pointing toward the largest one, right? So that helps me out. So if I list my numbers, one, two, three, four, five, six, seven, eight, nine, I already have the five, the seven, the one, and the nine done. Between seven and nine, I have to use eight. So eight is taken up too. So now I need something in between one and nine. That's not a big help. I have limited some answers out. So what can I use that is going to be less than five over here in the bottom? What if I try a four? Does that work? So four, something less than four could be a three and a two. Now you'll notice that in this particular problem, I don't know which way those could swap. They could swap and it wouldn't matter, but that doesn't affect the outcome because what I really need to know is this box, not that box down there. It's not gonna affect it. But I can cross out the two and the three. I've used them already. I've used the four. The only number that I have not used so far now is the six. So the answer here has to be that six needs to be in the box with the question mark. Now, remember I told you, guess and check doesn't mean just a wild guess. Use the information. So what did I use? I used that the arrows point from small to large to fill in some of these numbers, the one and the nine. Just like Sharia used the fact that the only way to get a sum of three was with a one and a two. So always use, and we had that problem with the circle, with the times zero. So we knew we had to put a zero there. So while we might say guess, we don't just mean guess any possible number. We mean use the best clues that you have to make a really good guess. And if you don't know where to guess, remember that problem with the books in the library, you can always use the answer choices as your first guess and see what happens if you try that. And that could give you information as well. So hope you've liked our guess and check lesson. Remember, don't random guess, intelligent guesses. And I'll see everybody next week. Sharia, you wanna say goodbye and thank everyone? Sure, yeah, thanks for coming, guessing and checking, real cool. All right, see y'all next week. Bye.

Math Kangaroo

webinar

Level 3-4

Dr. Sarah Segee

Shura Vyas

guessing and checking

problem-solving

interactive problems

strategic guessing

math contest