So, this week, we are going to talk about algebraic thinking. We'll start with a warm-up exercise. Here you go. We have this question number five from Math Kangaroo 2018 exam. Alice subtracted two two-digit numbers, then she painted two cells. What is the sum of the two digits in the painted cell? I will be launching a poll soon, and the ABCD choice will be written on the slides here. Just want to quickly remind our four-step problem-solving strategy. Step number one, knowing what is asked here. What is the sum of the two digits in the painted cell? OK, we got all correct answers so far. So how to solve this question? Here is a strategy. Because the painted cell, we don't really know the numbers. We can actually move minus 20-something from left to right. So that becomes some unknown number. I use a square to represent that. Something 3 equals 20-something plus 25. So if you look at the single digits here, what number plus 5 becomes 3? And that will be 8. So you fill this 8 into this square, 28. So now 28 plus 25, that easily gives you 53. So now you have known these two painted cells. One is 5. The other is 8. 5 plus 8 is 13. OK, I hope this kind of gives you an easy feeling to start with. And here, let's just review what we have learned last week. Last week, we talked about patterns. Today, we'll be continuing talking about algebraic thinking. In the following weeks, we'll talk about the following topics, ratio, proportion, multiples, factors, and divisibility, time clocks, and calendars, 2D and 3D geometry problems. And finally, hands-on practice questions. So what are we learning from algebraic thinking is the ability to recognize different patterns, the ability to represent relationships using numeric expressions, the ability to make generalizations from some observed patterns or unknown relationship, and finally, analyze how things might change given the known information. It will help us to train our problem-solving skills, representation skills, and cognitive reasoning skills. The key ideas in algebraic thinking involve the following, such as equality or equivalence, inequality, as I noted on the slides. You may see the different notations, positive and negative numbers, problem-solving and critical thinking, making generalizations, patterns, variables, parameters, and relational thinking, and the ability to make generalizations in relational thinking, and also, finally, use symbols to represent the mathematical ideas behind the question. Some typical vocabulary involved here includes all of this listed here. I won't talk about each of them, but I will pick some of them to explain. I believe some of these are more familiar with, you are more familiar with, such as inequality, variables. If you don't really know something, you can assume that to be a variable. You use different symbols to represent it. You can use different expressions to represent a relationship between or among those unknown or known variables. Something here listed, vertices. So if you are using a 2D dimensional thing, for example, a triangle, when these two lines meet here, we call that vertex. You have multiple vertices here and point. And you can have 3D dimensional space as well, like this. You have different things. You can also draw a coordinate plane, for example. Coordinate plane, x and y. This is a two-dimensional space. You can use a point located on this x and y plane. And this is x0, y0. Or you can say, for example, 2, 2. This is an ordered pair, 2 and 2, that represents this point, who is located at x-axis 2 and y-axis 2. That's ordered pairs. And you can see the difference between parentheses and brackets. You use the parentheses within and brackets outside. So for example, this equation gives you some ideas how you should take operations. If you have a parentheses within the parentheses, you want to do the calculations first. So although there is a plus sign, you want to calculate a plus b first and times 3. Based upon these kind of rules, you can also say a plus b within the parentheses times 3. That can be the same or equivalent as 3 times a plus 3 times b. So all of these expressions, symbols, variables, rules, they are all combined together. On what occasions do we use which one? Depends on the problems. So next, I will show you a set of problems involved with some of the concepts here. First, I would like to start with a fun question. I would like you to think of a number. This number can be anything. I would like you to write that number down on the paper. And let's do a fun trick with it. Think of a number. Add 3 to your number. Double that, which means multiply by 2 after you add 3 to your original number. Next step, subtract 4 from it. Next step, cut that in half, which means divide that by 2. Finally, subtract your original number. Can you tell me your final answer? Can you tell me your final result? What's your number? I don't need you to tell me your original number. I just want you to tell me the final answer. OK, let me see. 1, 2, 1. I think a majority of you got it correct. 1 should be the correct answer. Let's see how the trick is played here. So whatever number you think of, I don't really know it. I'm going to cover it to use a picture to represent that. And I'm going to add 3 to that number. So on the right-hand side, you can see this table. Add 3. 3 dots represent 3 numbers. Double that. And then subtract 4. So I'm subtracting 4 numbers. And then cut that in half. Now I have my original number plus 1. Subtract my original number. I have 1 remaining. So did you see the pattern here? No matter what number you were originally thinking at the beginning, you will end up with the same answer. Is that cool? Why is that? Let's see this a little bit. Let's just focus on this original number right here. No matter what operations you are going to do next, let's just focus on what's happening to this original number. Remains after adding additional number. Doubled after doubling. Subtraction, but you are subtracting 4. So original number remain to be doubled. Cut that in half now becomes the original number. Twice of the original number becomes half. Finally, you subtract the original number. So that's gone. So you can see, no matter what you started with originally, it's gone. You only have 1 left. The result is 1. So this hopefully gives you some ideas of how alphabet thinking leads you to some fascinating spot. Next, let's look at a question from 2009, question 7. A bridge is built across a river. The river is 120 meters wide. One quarter of the bridge is overland on the left bank, and one quarter of the bridge is overland on the right bank of the river. The question is, how long is the bridge? I just wrote A, B, C, D, four choices for you on the screen. Later, I will be launching a poll. But please take your time to follow the four-step problem-solving strategy to solve this question. Just a hint here, how long is the bridge? That's the question. Half of you submitted answers, let's just give us a little bit more time. The last one is 480 120, 240, 60 and 480. Okay let's see the results. Most of you got it correct. The correct answer is B, 240. Let's see how. So to help you to visualize this relationship, what are we asking here? How long is the bridge? So if you use a picture to represent the bridge. A bridge is built across a river. The river is 120 meters wide. The river, one quarter of the bridge is over the land and on the left bank and one quarter of the bridge is over land on the right bank. So I'm going to use this bank. This is one-fourth. Okay this is one-fourth on the left on the right and here we know the river is a hundred and twenty but we know the river is one-half. Is that right? Because on the left on the right we have one quarter each. So one minus one quarter, another one quarter, that's one half. We also know the river is 120 meters wide. So we know 120 is just half of it. We want to know the whole length, how long is the bridge, the total, which is the whole thing. So the answer should be double the amount of 120 which is 240. Okay so I saw someone submitted answer 60 and you may wonder, I mean I may wonder, you were thinking something wrong maybe. Maybe step number one, what is being asked is misunderstood. How long is the bridge? Okay how long is the bridge? That's what is asked here. Okay so next question. Let me check if TA is here. It seems like TA is not here today so I will be reading the question. Question number two was from 2004, question eight. Three members of the rabbit family eat 73 carrots altogether during break. The father ate five carrots more than the mother. Their son ate 12 carrots. How many carrots did the mother eat that week? I put A, B, C, D on the screen. Just to remind how many carrots did the mother eat that week? There are three people here, father, mother, and son. OK, all the people who submitted the answer got it correctly. Let's see how you got it. OK, so I would recommend using a table to organize your thoughts here. Suppose the total is 73. I just write that down as 73 in the second column here. The father, mother, and son are listed here. I need to know how many did they eat, respectively. So because this question asks us to figure out how many carrots did the mother eat, so I'm going to assume that unknown number to be 8. And then the father ate 5 carrots more, so 8 plus 5. I don't really know what that is, but 8 plus 5, because 5 more than the mother. The son, 12. I need to solve for 8. Is that the problem? Step number one, we need to figure that out. So now we know the relationship. The total, the total, father, mother, son, that adds up to be 73. We also know son, 12. So now just solve for x. Is that easy? Oops, x. So how to add it up? We just need to solve for x. 2x plus 17 is 73. 2x is equal to what? 58, is that correct? No, sorry, 56. Erase, eraser. See, easily make a mistake here. Hope you can avoid this. Eraser should be 6, 56. x is simply 56 divided by, oops, run out of battery here. Why is it not working? 56 divided by 2, that's 28. So that's 28. It seems like, OK, I got it right, I got the right answer. But let's check. Remember step number 4 of problem solving strategy. Reflect and check. I have 28 here. If I have 28 here, I should have 5 more. That's 33. 28 plus 5. And then do they add up to be 73? 33 plus 28, 61. Is that right? This 61. 61 plus 12. Does that give you 73? Yes, you got 73. OK, see, I just made a mistake. Suppose I have that mistake. What's happening here in step number 4 when I'm checking? I was having 58. 58. So instead of 56, I have 58. Instead of 28, I have 29. So if I ended up with 29, what's happening? I will have, I'm going to write here, 29 here, 5 more. Is that right? 34. So if I do this, 34 plus 29, that's 63. If I add 63 to 12, I have 75. Oh, oops. That's not correct number. So if I make a mistake here, if I follow strictly to the four-step problem solving, I will reflect and check and I will still find, oh, that's wrong, that's wrong. I will go back where I messed up. I will be able to correct it. OK, so just remind you, step number 1, step number 4 usually are very easily neglected by many of us, especially with challenge questions. This becomes more important for you to check, to avoid any mistakes, OK? OK, next question. Question 3, the weight of three apples and two oranges is 225 grams. The weight of two apples and three oranges is 285 grams. Each apple weighs the same, and each orange weighs the same. What is the combined weight of one apple and one orange? We'll be launching the poll. Take your time. A, B, C, D are listed on the screen. Just want to give some tips. Here, you have two different ways of solving this problem. You may be able to calculate the weight of apple and the weight of orange respectively and add it up together. That's possible, number one. Another way of solving this is you don't need to calculate them respectively. You just need to know the sum of it. So how can you quickly figure out the sum of the combined weight of one apple, one orange without further calculation? Okay, so currently among the people who submitted the answer, we have 50-50 split. The current accuracy rate is 50%. Okay, let's see. Let's work on this together. Basically, the null information, I would like to use expressions, equations to represent the null information. Because I don't know the weight of apple, I'm going to use the notation A to stand for the weight of it. I don't know the weight of orange either, so I'm going to assume that to be O, unknown parameter, variable. I say, okay, O, I don't know it. So 3A plus 2O equals 225. Similarly, 2A plus 3O equals 285. Remember, I was giving you a tip. You don't need to calculate A and O respectively, although you can do this. But that may give you more time to get the right answer. If you take a look at this on the left-hand side, 3, 2, A. On the left-hand still, how many O's do we have? 2O, 3O, 3, 2, 2, 3. If you add it up together, what do you get? 2 is, does that give you 5A plus 5O equals, okay, so here, 5A plus 5O. Okay, it's notation. 5A plus 5O, does that give you the sum of this, 225 plus 285? That's 510, is that correct? So now you have 5A plus 5O is 510. You will know, oh, what's the relationship? If I want to figure out the sum of A and O, oops, that's the eraser. I just need to figure this out. That's step number one, knowing what is asked. What is the relationship between these two? I just need to simply divide that by 5, is that correct? So I can easily see 5A plus 5O, 510. 5A plus 5O is 510. I can easily see A plus O is just a five, one fifth of 510. So that's 510 divided by five. That gives me 102, that's B. So this is the quicker way, but let's check. Let's check. Remember, step number four, right? Let's check. Let's also figure out, okay, what is the real A? What is the real O? What is the weight of apple? What is the weight of orange? So because if we agree A plus O is 102, we will know twice of that amount is 102 times two. 102 times two, is that correct? So that gives you 2A plus 2O, 204. So because 2A plus 2O is 204, I also know 2A plus 3O is 285. I can simply calculate O is a difference between these two. It's 285 minus 204, which gives you 81. So now I will know the weight of apple is 102 minus 81. This is all assuming I got it correct, right? I'm checking. Okay, in this case, A have 21. Now let's check step number four. Do these number add up to be 102? Yes, they are. Okay, so I have demonstrated how you can calculate the weight of apple and orange, the sum of it using two different ways. One way is without knowing the details. The other way is you are actually calculating the detailed weight for apple and orange respectively, and then you add it up together as the sum. Either way, you will get this correct answer, B, 102. Okay, next question. Question four from 2020. A father kangaroo lives with his children. They decide on all matters by taking a vote, and each member of the family gets as many votes as his or her age. The father is 36 years old, and the children are 13, 6, and 4 years old. So right now the father wins. How many years would it take for the children to have the majority of the votes if they all vote the same way? I just wrote the A, B, C, D choice on the screen, and I will be pulling up the porcelain. For those of you who submitted answers, I would like you to double check again, step number one and step number four. What is being asked here, how many years would it take for the children to have the majority of the votes if they all vote the same way? Or in another word, how many years does it take for the children to catch up with their father in terms of their combined age? That's the question here. Okay, let's see. I personally would like to write down the information given by this question using numbers. So the father is 36 years old. I wrote that down. Children, the combined age, 13, six, four, 23. What's the gap between them? 36 minus 23, that's 13, right? So for every year, how much do we bridge the gap? How much are we closer? So if you write it down, year number one, next year, father became 37, children became 26. Why? Because each of them grow one year older, right? So similarly, year two, father became 38, children became 29. So keep doing this, this pattern will keep going. So you will observe that each year, father's age increased by one, while children's combined age increased by three. So what's the gap? Every single year, children can catch up by two. That's the gap between two, right? So every single year, I catch up by two, but the total is 13. So how many years does that take? You can simply use 13 divided by two. You will have 6.5 rounded up to be seven, and C, C will be correct answer, right? Round that up to be seven. I saw some of you chose B. Maybe you forgot about this additional half, like 13, okay? Like 13, okay, if you divide 13 by two, you still have some remainder as one, a number one as remainder. So you need extra year, year seven to catch up. Or simply you can say, okay, just write it down. One, two, two twos, three twos, four, five, six, seven, right? How many twos can you get to over, get the hurdle over 13? Because the gap between these two parties are 13, okay? So correct answer is C, C, seven years, okay? Okay, next question. Yeah, I saw some of you were correcting, self-correcting, great. Just to remind, always take some time at step number four, reflect and check. Okay, good. Oops, question five. One bowl contained 26 liters of water and another bowl contained seven liters of water. The same amount of water was added to each bowl. And now the second bowl contains three times less water than the first bowl. How many liters water were added to each bowl? Let me give you some hint. For the things that you want to know, because it's unknown, you can assume that you'd be some unknown parameter or variable. You can call it whatever things that you want to call. For example, X or L stands for liters, whatever way you want to call it. And then you want to figure out a relationship, maybe by writing that down as an equation. What was the relationship between these two bowls after adding this X amount? And you write it down as an equation and then figure it out. That's way of solving method one. Another way of solving this is you use a picture to draw it to help you to visualize the relationship. I'm launching the poll right now. You don't need to hurry. Just take your time. Follow through our four-step problem-solving strategy. If you are confident your answer is correct, then make a choice. I'm going to write down the A, B, C, D choice for you. The critical information is after adding this extra amount, I don't really know that is, what the number is. But the same amount added to both bowls. And then the second bowl contains three times less water than the first bowl. That's the relationship. OK. I think so now majority who submitted the answer got it correct. Let's see how you got it correctly. As I said, you may choose to use graphics, pictures to represent the relationship. So now suppose the blue part represent one bowl. Another bowl contains seven liters of water. So blue, that's seven. I don't really know how many liters water added. I'm going to assume that yellow part to be x amount. So seven plus x. And that's the first bowl. 26 also plus x. So 26 plus x. And after adding that yellow part to both bowls, the relationship between two parties became 1 versus 3. So I'm going to represent that in a mathematical expression, which is x amount is yellow. So I'm going to use three times, within the parentheses, seven plus x. After adding x amount, the liters, I tripled that, three times that amount. That becomes exactly equivalent to 26 plus that extra x amount. So that's the relationship. And that's the equation. Let's solve that equation. 3 times 7, 21. 3 times x, 3x. On the left, you have 21 plus 3x. On the right, you have 26 plus x. So that gives you 2x equals 5, which gives you x equals 2.5. I saw some of you got 2. Maybe calculation went wrong. When did that happen? Even when, OK, I think I got 2. You want to plug it back. So let's suppose you got 2. 7 plus 2 is 9. 26 plus 2 is 28. Is that the relation 1 versus 3? No, right? So you will see, OK, I messed up. I went back to see where I got it wrong. And I will correct it until I got a final spot. I will check and got, yes, that's correct. OK, so that's question 5. Next question, question 6. During a competition in the Math Kangaroo summer camp in Zakopane, students were given 10 problems to solve. For each correct answer, a student was given 5 points. And for each incorrect answer, a student lost 3 points. Everybody answered all the problems. Matthew got 34 points. Philip got 10 points. And John got 2 points. How many problems did they answer correctly altogether? I will write down the A, B, C, D choice here. Now you have four choices on the screen. I'll be launching the poll soon. Now I have three people submit the answer, and they got all correctly. All of them got correctly. Let's see how you got it here. Are you using the same way of strategy? Let's see. So how many problems did they answer correctly? I don't really know that number for each of them, Matthew, Philip, and John. I'm going to call that a C for correct questions, right? For C. You can use any notation you'd like. How many points? I'm going to record that as P. C versus P. P for points, C for correctly answered. So I'm going to write down the relationship. For each of the correct answer, I gained five points. So five times C. However, for each incorrect answer, I lost three points. So minus three times the number of incorrect answer. So there's a hidden information here. The number of correct questions and the number of incorrect questions, right? They should add up to be 10. So because of that, I will know, given the correct questions, C, denoted as C, I will use 10 minus C to represent the number of incorrect answers. So in that sense, I will have this equation formulated as five C times, I'm sorry, five C minus three times parentheses, 10 minus C. That gives me the total points. And this is true for three of them, Matthew, Philip, and John. So alternatively, I can rewrite this equation as C equals something represented in P, in the format of P, right? So now I know C equals 30 plus P within a parentheses divided by eight. Why is this important? Because now I can just find the information from the statement, 34, 10, and 2, the points. I can simply plug them back into the P here and figure out the C, respectively, for Matthew, Philip, and John. So if you do this math, you will have C equals 30 plus 34 together, divided by eight. That gives you eight. Similarly, you will get five for Philip, four for John. So now you have eight, five, four. Simple math. Add all three together. Eight plus five plus four. That's 17, not 18. So C is the correct answer here. So this problem also demonstrates how you can use algebraic thinking to write down the relationship between C and P, or anything that you want to call, right? Once you figure it out, you just need to follow through. And remember, don't make any mathematical mistake. Even though you made some mistakes, you want to pause and reflect and check in step number four, trying to figure out, okay, do we get the same answer? Can we plug it back into the statement and make it still hold true, make all the information coming from the question still hold true, given your final answer? If that's the case, you can conclude, yes, I think I got the right answer. Okay? So C is correct for this one. Let's move on to the next question. A fairy has six bottles. Their volumes are 16 ounces, 18, 22, 24, 32, and 34. Some are filled with orange juice. Some are filled with cherry juice. And one is empty. There is twice as much orange juice as cherry juice. What is the volume of the empty bottle? ABCD choices are 16, 18, 32, and 34. Just to remind you, step number four, check your answer. Reflect to see if it is correct. What is the volume of the empty bottle? If your answer is true. Let me give you a hint here. There's one bottle is empty among six bottles. One is empty. There's twice as much orange juice as cherry juice. So that means for the five left, five bottles left, somehow the combination can make it one group is twice as much as the other group. So that means what? If you add it up together, excluding that empty bottle, the sum of the volumes should be a number divisible by 3. Is that correct? So with that hint, can you think further how to represent this relationship in an equation? OK, seems like this might be a little bit of a challenge. Let's work on this together. OK, so the total volumes of all these bottles together, let's just add up together. That's to be 146 ounces. I don't really know which one is empty. I'm going to call that as v, the volume of the empty as the v. So excluding that empty bottle, I'm going to use 146 minus v. They should be a multiple of 3. Why is that? Again, there's twice as much orange juice as cherry juice. Suppose cherry juice is 1, then the orange juice should be twice of that amount. If you add the volume of orange juice and cherry juice together, that should be 3 times cherry juice. So because of that, I can apply this rule. Remember the vocabulary we reviewed earlier from the earlier slides? Divisibility rule for 3 states that a number is completely divisible by 3 if the sum of its digits is divisible by 3. So this is like a shortcut for you to remember. I want to find a number to check if it is divisible by 3. I can just do that by checking if the sum of its digits is divisible by 3. So let's just do that for each of the bottles. A, 16. I try that. What do you get? 130. Is that divisible by 3? No. I do that for each of the answers here. Is this divisible by 3? No. Still, I keep doing this. And I will find, oh, if I assume the empty bottle is 32, and I got 114. Because 1, 1, 4, they can be added to be 6. That's divisible by 3. So I will say, yes, that's correct answer. Also, another way, you can just divide this by 3 and see, did you get integers? So that's 38, right? So the empty bottle should be volume as the 30. Not 38, I'm sorry. I mean the 32, 32. So that's question 7. Question 7. So from this question, very importantly, you need to find the pattern, the hidden relationship. Because the cherry juice is half of the amount or volume of orange juice. Or orange juice is twice as much as cherry juice. So the total ended up to be some multiple of 3, multiple of 3. And you have to eliminate some empty bottle among the 6. So you can try 6 of them one by one. And you will find, OK, if I eliminate 32, then the rest, they should be a multiple of 3. So that's question 7. And finally, there's a bonus question behind this. Let me wrap up this quickly. So by algebraic thinking, we hope you have gradually developed your ability to recognize the patterns, represent different relationships given from the statement, and make generalizations from the question, and finally, analyze how things change. When solving these type of questions, you will have to read the problem more than once, identify the vocabulary used, recognize the type of skill required, and also solve and check your answer. So as today, we have seen some people who were really quick at giving me the answer. But the answer that they instantly give may not be correct. So make sure that you spend your time at step number 4 of problem-solving strategy, check, pause, and reflect and check. Finally, there's this bonus question for you. There are three friends, Adam, Tom, and Paul, went to swimming pool 15 times. Adam bought the tickets for all of them 8 times, and Tim did the same for 7 times. Paul gave $30 to his friends, which, as he calculated, he owed for the pool tickets. How should Adam and Tom split those $30 so that each boy pays the same amount for the pool tickets? So I think we are over time. I'm going to leave this question to you as homework. And our next webinar will start with the answer for this question. And before I end this webinar, I would like you to take the poll here for the survey. So let me launch this quickly. And I will be sending this in the chat as well. The survey link will be coming from the chat. Please take your time to finish the course evaluation for this webinar. Thank you for your time. And I hope you have a wonderful week. I will see you next week.

algebraic thinking

problem-solving

Math Kangaroo

equations

algebraic expressions

variables

patterns

relational thinking

cognitive reasoning

geometry