Welcome to webinar number six. This should be level five, six for Math Kangaroo. Today we'll be working on time clocks and calendars. And I want to spend the first minute here before we actually start completely talking about the four-step method to solving problems. You can see that that's right here on our slide. It says four-step problem solving method. What does that mean? Step number one is very, very careful reading of the problems. Math Kangaroo problems tend to also be a little bit of a reading comprehension puzzle. So you might need to read a problem two or even three times. You can underline and highlight the important information, take notes, reorganize it the way you need to see it. If you need to draw yourself a picture to understand the problem, that's a really great thing to do in step one. Once you know what you need to do, then step two is to make your plan and to try doing it. Step three is to work super carefully. Keep your work neat and organized. That helps you work carefully with fewer errors and will also make it possible for number four to check your answer. Is your answer reasonable? Does it make sense? Is there a way you can go back and check it if it was an algebra problem? Can you put your answer back in and try to do it that way? Can you solve it a different way and get the same answer? So that's step four is to double check your work. So I wanted to take time to make sure we talked about the four steps to solving problems. And that works for Math Kangaroo, works for your math classes, it works for science experiments, it works for a lot of things. Always read carefully, come up with a plan, work through your plan carefully, and then check, check yourselves. Checking can also involve your partners or peers, right? That also works. All right, here's our warmup problem. How many times faster does the minute hand of any clock move than the hour hand? I do have polls for a lot of the problems today. This is one of them. I will launch the polls pretty quickly if it's a very short problem statement like this, because you should be able to read it right on the poll itself, okay? So remember throughout the webinar, you can put your answers into the chat for me or for Jacob. You may ask Jacob or I questions. Sometimes when we're talking and explaining, we may not get to every individual question, but if we see the question pop up from multiple students, it does help us to know that we need to explain something to everybody. We do actually find that really helpful. And remember the polls are anonymous. We cannot tell what you have answered. We just can see what the whole group is answering all together. And there are no penalties for wrong guesses. On Math Kangaroo, you get it right or you get a zero. So you should always guess. You can always check your answer choices as well. With my younger students today, we were doing guess and check. And if you don't know what to guess, try the answer choices. It's a really good guess, right? I'll end the poll because most of you have completed it. And we have two answers that come up the most, 12 or 10, but every answer has been answered here. So this is interesting. Let's see how we do this. Okay, how many times faster does the minute hand of a clock move than the hour hand? So if I look at my regular face clock and I have the long minute hand, it goes around in one hour, right? The minute hand, the long hand goes around every hour. It also goes around 12 times in 12 hours, right? Cause it's a 12 hour clock, right? One, two, three, four, five, six. Everyone knows what I'm talking about? Now let's think about the hour hand. The hour hand is the shorter hand. The hour hand only moves a small portion of the clock, 1 12th of the clock every hour. So it goes around one time in 12 hours. The minute hand goes 12 times every 12 hours, the hour hand one time. So therefore the minute hand must travel 12 times as fast. So 12 times as fast B. All right, if you don't have a lot of experience telling time on a face clock, this would be a good time to practice it, okay? We are gonna be using quite a few face clocks today. Let me clear my pictures here. Okay, so we've gone through several different lessons already, we were looking at patterns, we've done some algebraic thinking, ratios, we did multiples and factors and divisibility last week. This week we're on time clocks and calendars. Now we can keep in mind where we're going and what we've done. All right, some facts that you will need to know today and keep these in mind in the future. Hopefully you already know these, but just a quick review. One year is 12 months. Most years have 365 days, except for the leap year that has 366 days. A day has 24 hours. So that means on a 12 hour face clock, you have to go around twice. One hour is 60 minutes. Half an hour is of course 60 divided by two or 30 minutes and a minute is 60 seconds. Hopefully that's complete review, you know that. A digital clock in 24 hour mode shows the time 2019. What will the clock show the next time it uses the same digits? I do not have a poll because the answers here were pictures and that's not so easy to set up in Zoom. So just put your answers in the chat. Yeah, you guys are sending me some really good responses. So if we were counting up, the next one you would maybe want to try. We have, sorry, I didn't do a couple of clicks each time. So we are starting at 2019. We maybe want, if we were counting up and this was just a regular base 10 counting, we might be tempted to count 2091. But we don't have 91 minutes in an hour, so that is not gonna work. That excludes choice A. We can't have 91, we can only go up to 59 and then we get to the next hour. So let's try that. We're gonna have the next hour 21 and then the numbers that we have are, the next lowest number is zero and then nine. So the next time that's gonna use those same digits is 2109, which is answer choice C. We're back to polls for this one. One night and day on Mars is 20 minutes longer than on Earth. How much longer is a week on Mars than on Earth? It doesn't say it here, but we will assume that the week on Mars is seven days like it is on Earth. So if you don't know, just have to go with the best assumption you can get, which is that it's seven days. If I have misread it, it is 40 minutes longer. Go with what it says on the slides. I get tongue-tied and talk over myself sometimes. So if I said 20, I'm sorry, it is 40. So if you do not see the poll, if it doesn't pop up on your screen, you may have a setting set on your computer that does not allow pop-ups. And you can try to change that before you join in next week. But if you don't have the polls, or if you don't like the polls, you can always use the chat feature. So I'm gonna end the poll. Most of the students, almost 70% have said that they believe the answer is A, four hours, 40 minutes. Is the difference between the week on Mars and the week on Earth. So let's take a look. We decided that we were gonna go with, since it didn't say, we were going to make the safest assumption, which is that there were seven days in the week on Mars, just like there are seven days on Earth, so seven days. So we would have 40 minute longer days on Mars, and we have to multiply that by seven. Four times seven is, of course, 28. So 280 extra minutes in a week. Now, none of these answers correspond, they don't say 280 minutes. So we're gonna have to figure out how many hours there are. We know that there are 60 minutes per hour. Okay. So 60 times four equals 240. So if we do the 280 minus 240, that gives us 40 extra minutes, right? So this is four hours and 40 minutes. That is answer A. So it took a little bit of a conversion with the 60 minutes per hour, but otherwise it was a pretty straightforward problem. Okay. A wall clock strikes every hour. The number of strikes corresponds to the time. So for example, at 10 a.m. and 10 p.m., you will hear 10 strikes. The clock also strikes once at the half hour mark. So at 12.30, it's gonna strike one time. At 1.30, it strikes one time. How many strikes can be heard in one 24-hour period? Give you a few moments to be able to see the whole problem big, and then I'll launch the poll. Here comes your poll if you don't like having all the polls, let me know in the chat. Some students asked for lots of polls. Others have told me they don't want them. I'm trying to find the right balance. All right, I'll be closing that poll in a few seconds. So if you haven't made a guess, go ahead and guess. Okay, we have a few varieties of answers. 180 is what 40% of you said, but coming in a pretty close second was 136 chimes or strikes of the clock. So let's see how we can figure this out. I like to break this up and think about the two different types of chimes that we have. We have hourly chimes and we have half hourly chimes. So the hourly chimes, for the first 12 hours, we're gonna have one chime, two chimes, three chimes, all the way up to 12 chimes. And we're doing the sum of this. So this is for 12 hours, right? Now we notice that we're gonna do this twice because it says in a 24 hour period, we don't give 14 chimes, we just do one through 12 again. So now how can I add this up? I'm gonna put in the 11. Well, there's a trick for adding up a sequence of numbers like this. When it's just a simple arithmetic sequence, meaning we've added the same to each, you'll notice that one plus 12 is 13, two plus 11 is also 13, three plus 10 is also 13. So we're gonna have 13 six times. And remember, we are also multiplying by two to get 24 hours. So we have 13 times 12. And that's going to equal 156. And if you don't know, this is 13 times 12. If you don't know that, maybe you know 12 times 12. 12 times 12 is a dozen, it's 144. So then you just have to add one more 12 onto it. But we also have the half hour chimes. At the half hour, we only did one chime. And there's gonna be 24 of those in a 24 hour period. Right, so there's a half hour for every single hour. So in 24 hours, you would think that there are 24 half hour chimes. So we have to add six plus four is zero, carry the one eight. So 180 times the clock will chime or strike. Same difference, right? Okay, so hopefully you can do that problem the next time. And this is a great trick for finding the sums if you don't wanna have to add all the numbers together. A movie starts at 1.44 p.m. And finishes at 4.18 p.m. How long is the movie? It's a pretty straightforward question, except that look, all the answers are formatted in minutes. But try to figure it out in minutes. I think I can put this up on the poll. If you can't see when the poll is launched, go ahead and write those two times down right now, 1-47 until 4-18, and then I'll launch the poll. Anybody else want to answer the poll or put a response into the chat? We have most of you participating and I do really appreciate that. Thank you for trying every problem. It's really important to practice, right? Okay, so here we go. Most of you, 70% almost, have said 151 minutes. That is exactly right. So let's take a quick look. What we can do is we can break it up into pieces. So that's how I like to do time. You might have a slightly different approach. So it's from 1.47 to two o'clock. That's a nice, I like that piece. So I can see that from 1.47 to two o'clock is gonna be 13 minutes. And then from two o'clock to four o'clock is two hours. So that's 120 minutes. And then from four o'clock to 4.18 is, of course, 18 minutes. When I add those, 120, 13, 18, I get the 11, 151 minutes. Okay, now some of you might try to subtract. That is not a terrible way to do it. You can do 4.18 and you can think about subtracting 1.47. Now this isn't going to work exactly right because of the difference in borrowing. We're not on a base 10 system, we're on this base 60 system. But the first thing you'll notice is that eight minus seven is one. So the ones digit has to be a one, which would allow you to cross out three incorrect answers. Okay, so we can kind of use even some shortcuts to cross out wrong answers and get us closer to the correct answer. And then you just have to compare, is it 151 minutes or 91 minutes? Well, this movie's more than two hours long, so it can't be 91 minutes, it must be the 151 minutes. Ian was born on January 1st, 2002 and is older than Peter by one day less than one year. What is the date of Peter's birth? If you're not sure what this all means, read it really carefully and try to solve a problem a little simpler and then add the complicating part. I do have a poll, some students are asking, I have one, we can do it, some students are asking also about the number of days in this year. It does not matter because it's just talking about one day less than a year, right? Plus 2002 is not a leap year. So we don't need to worry about any leap years during this. And I will launch the poll. Okay looks like most of you have entered your answers into the poll so let's take a look. We have a few different choices here it looks like majority of you think it's December 31st but we don't know if it's 2002 or 2003. Let's take a look at how this will work. So remember we did some patterns and one of the advice we gave you in the pattern lesson was to try a simpler problem. So the thing that makes this complicated is it's one day less than one year older. So let's just say we have Ian was born, I'm just going to use the shorthand, January 1st 2002. Let's say he's exactly one year older than Peter. So if Ian is one year older than Peter then Peter would have been born on January 1st 2003 the following year. Right the people who are born in earlier years are older that's the first thing. So if Peter was exactly a year younger than this is his birthday but we know it's one day less than that. So we've got to get rid of one day. Which way do we get rid of it? If he's born later like January 2nd 2003 that makes him even more young. Right and that's the opposite direction. We want him to be a little less young. So let's make it December 31st and we have to go back to 2002. This is a little less young. You know it's kind of a funny way to think about it but Peter's birthday is going to be December 31st 2002. So it makes it the same year but from January to December. I hope that makes sense. So the simpler problem was to go with exactly a year and then you just had to figure out what to do with that extra day which way to go. All right. Don't worry Jacob's going to lead some questions today too. He's going to get you're going to get to have him explain and maybe you like the way he explains problems. Ann rides her bicycle throughout the afternoon at a constant speed. She sees her watch at the beginning and at the end of the ride with the following result. So here's the beginning of the ride and here's the end of the ride. What time does the clock show when Ann finishes one third of the ride? You'll notice they broke this into thirds. They're trying to make it nice and simple for you. We want the mystery time here and here are the clocks and I do not have a poll because I couldn't figure out how to make each of these clocks to be an answer choice on the poll. So just put your answer in the chat for this one. Jacob, is it just me, or do these clocks, they didn't reproduce properly in this figure, did they? Yeah, we only see one hand on the clocks. Yeah, I only see one hand. Yeah, okay, so we're not going to do this one as multiple choice. I don't, I'm sorry, I didn't notice this before. Whoever put this figure on here, we don't have the correct clock. So, in the chat, tell me the correct time. Okay, and you can do it time as it would be on a digital clock. All right, but I think we can still solve the problem by telling the correct time. At one third of the ride. Hey, I'm sorry that the answer choices are not as they should be. But like I said, I think we can just do this and determine the time. So we're going to ignore these answer choices. And I will make sure that this slide gets corrected before it's used again. But let's take a look. Ann rides her bicycle throughout the afternoon at a constant speed. So whatever her constant speed is, if her speed is constant, then the time is going to be even between these thirds, right? This is a third, this is a third, and this is a third. So since the speed is the same, it's going to be one third the time. So let's try to figure out the times. When we tell times on a clock, we have 12 at the top, 1, and 2. And then this is the 6th. So we're between 1 o'clock and 2 o'clock on the hour. And when the hand is directly to the bottom, that's 30 minutes. So this is a 1.30 o'clock. Using the same logic, this is between 3 o'clock. Can't even read that, can you? Sorry. This is between 3 and 4 o'clock. And again, we're at the bottom. So this is at 3.30. So now we know that the bicycle ride was from 1.30 to 3.30. So it was a two-hour ride. That's a reasonable bicycle ride. I hope she had a good time. So if we have to divide that into thirds, we know that an hour is 60 minutes. Two hours makes it 120 minutes. And a third of that is 40 minutes. Right? The other way you could do it is a third of an hour is 20 minutes. And since we had two hours, it would be 20 times 2, or 40 minutes. There's always more than one way to think about these things. So now we need to do 1.30 plus 40 minutes. Well, 30 plus 40 is 70. We can't have the 70. So we have to take 60 over for the next hour. So it's 2, and then there would be 10 minutes. 70 minus 60 is 10 minutes left. You could also try to count it out with your fingers or other ways of telling time. You could do 1.30, and then you could do 40 minutes by counting. I even use my hands, right? So 1.30, 1.40, 1.50, 1.60. OK, that's 2 o'clock, and then 2.10. So that's another way to do it. So 2.10 PM is the time we are looking for. OK, let me turn the page of my notes. Mr. Kowalski was asked how old he was. Did I read that? When Mr. Kowalski was asked how old he was, he said, I have lived 44 years, 44 months, 44 weeks, 44 days, and 44 hours. How many years old is Mr. Kowalski? Remember, we're only asking about the years, right? So when you ask me how old I am, I don't tell you so many years, so many months, so many days. I tell you how many years. And even if it's past my birthday, I usually don't add that extra year on until I actually get to my birthday. Is that the way you do it too? If you say I'm 12 years old, even though you're 12 years old and five months, you still say 12 years old. So keep that in mind. There is a poll for this question, which I'll launch in a minute. I will go ahead and launch that poll in a second. If you are one of those people who don't have the polls pop up on top of the problem. Remember, it's 44 years, 44 months, weeks, days and hours. Okay, so you can make little notes and have that all everything is 44. Most of you have answered the poll, so I will go ahead and explain the problem. I know a few of you are still working, I can tell because you're sending me some questions on the chat. That's wonderful. If you have more questions while I'm explaining, I can't get to all of them, you can ask Jacob as I go along, okay? So here, half of you have said 48, there's some 47s, 49s. This problem takes a little bit of calculation, so let's take a look at that. Okay, so we know that we have 44 years to start with. It gave us that, 44 years. Then we're talking about 44 months, months. So I'm putting my years here, and remember, we count years as completed years, right? So like I was saying, if you're 12 years and 6 months, you normally, someone asks, how old are you? I'm going to say I'm 12 years old, I'm going to say I'm 12 years and 6 months old. 44 months, we have 12 months per year, 12 months per year, right? So this gives us another 3 years, but it also gives us, 3 is going to be 36, so this gives us an extra 8 months. So if we got 4 more months, we'd be adding another year. So let's see what's next. We have 44 weeks, and there's somewhere between 4 to 5 weeks per year, sorry, weeks per month. My mouth is not going the right way. So 44 weeks is going to be somewhere around 11 months. Remember I said we had 8 months and now we have 11 months, so that's definitely going to give us another year. This is 19, let's say it's 19 months, we're approximating a little bit. So if it's 19 months, we have 19 minus 12 is going to give us an extra 7. Alright, so now we have 44 days, 44 days is 1 month, because we have about 30 days in a month. So if we do 44 minus 30, we've got another 14 extra days. And then we have 44 hours, 44 hours divided by 24, this is less than 2 days. So even if we take the 14 days plus 2, we have 16 days. So we're not going to get any more years out of this. We do have extra months, but no extra years. So when we add 44 plus 3 plus 1, we get 48 years. Like I said, we do have some extra months and weeks, but that's not how we tell our age. We usually tell our age by the completed number of years that have elapsed since we were born. So the correct answer for this would be 48 years old. He's not yet 49. I hope that helps. It's a kind of a complicated question, but think about your real life and how you would answer the question if somebody asks you your age. Okay, this is a question I believe that Jacob wants to lead. So let me clear all of my squiggles off of here, and we'll let Jacob. Yeah, so I am released a homing pigeon at 7.30am. The pigeon arrived at its destination at 9.10am. How many miles did the pigeon travel if it flies 4 miles in 10 minutes? I'll give you guys some time to think about this. Looks like we have most of the responses, Jacob. I'll go ahead and share those results. So about three fourths of the students are saying the answer is 40. You want to tell them how that works? Yeah, so first we want to find the total time it takes for the pigeon to arrive at its destination. So from 7.30 to eight, that's 30 minutes. From eight to nine, that's 60 minutes. And then from nine to nine, 10, that is 10 minutes. So when you sum all this up, it's 100 minutes. So we want to find the amount of miles the pigeon travels in 100 minutes, but we know that the pigeon travels four miles in 10 minutes. So four miles in 10 minutes. And if we multiply this by 10, we get that there's 40 miles in 100 minutes. So the pigeon will travel 40 miles in total. So the answer should be C. We did a lesson on ratio and proportion. So once they figured out the time, they should be able to do the ratios. Good work, everyone. Okay, so at three o'clock the minute hand and hour hand make a right angle. What will the measure of the angle between these hands be after 10 minutes? I'll give you guys some time to think about this poem as well. You may remember we've done lessons where we draw. This might be one where you want to draw the clock hands. I think because we're almost out of time, we'll skip the poll, and we'll let Jacob draw the times on these clocks and explain how to find the angles. Okay, so for three o'clock, I'll draw it on the left circle. It will look like this is the minute hand, and then the hour hand will be something like that, so 3, 2, 1, and then 12. On the other hand, so this is 3 o'clock. On the other hand, 3, 10, which is 10 minutes after 3 o'clock, it will look something like the minute hand will be pointing at the 2. And then the hour hand will be a little bit away from the 3, but it'll still be very close to 3. So we want to find this angle. Notice that the minute hand is pointing at the 2, and the angle between the 2 and the 3 is 1 12th the entire circle. So the angle between 2 and 3 should be 1 12th times 360, which is 30. And now we want to find the angle between 3 and 4, 3 and wherever this minute hand is pointed to. So notice that the distance between the 3 and the point where the minute hand is pointing to is 1 6th the distance between 3 and 4. So if this is where 3, 10 is located for the minute hand, then the distance between the 3 and the 3, 10 is, for example, we'll set the variable as x, and the distance between 3 and 4 would be 6x, because it's 10 minutes is 1 6th of 60. And we know that the angle between 3 and 4 is same as the angle between 2 and 3. So the angle between 3 and 4 is 30 degrees. So the angle between 3 and 3, 10 is 1 6th times 30. So this is 5. And we want to sum these. So you get 30 plus 5, which is 35. So the answer should be E. All right, I want to point out that because Math Kangaroo is a multiple choice contest, there were quite a few of these answers we could eliminate, right? We could see that this is a pretty small distance, so you could get rid of the 90, the 80, the 60, and then you could figure it out between the 30 and the 35. And Jacob very wisely explained that the distance, the angle between any two numbers is 30 degrees, and then we knew we went past. So just using some logical thinking in that way, trying to break up the clock and is it more or less, you could come up with a pretty good guess, which is definitely something you could do if you couldn't calculate exactly. But Jacob's calculations are correct, so thank you. So that is going to take us to the end of our lesson on clocks, calendars, time. I hope you like this lesson, and I started off with the four-step problem solving method. So read very, very carefully, plan, carry it out, check your answers. Remember some of those basics that we had at the very beginning of the lesson about how many days in a week, how many minutes in an hour, hours in a day, months in a year, days in a year. Those types of constant facts are something that you should review. It's something that you kind of need to keep in the back of your mind. This is the earth world that we live in, and how we calculate time and calendars. So thank you very much for attending our webinar today. We are more than halfway through the series, and Jacob and I look forward to seeing you next week for webinar number seven. Bye everybody.

Math Kangaroo

time calculations

problem-solving method

clock hands

Mars day conversion

arithmetic sequences

time intervals

linear calculations

circle geometry

logical reasoning