Patterns in arithmetic
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- [Instructor] So what we have in this chart is all the numbers from one to 100 organized in a fairly neat way. It's a somewhat intuitive way to organize it where each row you have 10, so you go from one to 10, then 11 to 20, then 21 to 30, all the way to 100. And what we're gonna look at is interesting patterns that might emerge from this. So if you look at what's highlighted here in this purplish color, what numbers are highlighted there? Pause this video and think about it. Well, what's highlighted are all of the even numbers. And you can see the even numbers form these nice, neat pillars or columns on this chart. And we can look at that and immediately start to see some patterns. For example, what numbers are always going to be in the ones place for an even number looking at this chart? Well, you can see in the ones place you're either always going to have a two in the ones place or you're going to have a four or you're going to have a six or you're going to have an eight or you're going to have a zero. So that ones place digit is always going to be an even number. Let's do another example. Here we've highlighted different numbers. So pause this video and think about what's true about all of the numbers that we've highlighted? Well, you might notice that these are all multiples of five. Five, 10, 15, 20, 25, 30, 35, 40, so on and so forth. And so these form these two columns on this chart. And here we can see very clearly that multiples of five are either going to have a five in the ones place like we have right over here. So they're either gonna have a five in the ones place or they're gonna have a zero in the ones place. You might've realized that before but you see it very clearly in these two, you see it very clearly in these two columns. Let's do one other example. This one is really interesting because it's not just one of those clean column-type patterns. It looks like we started one, and then we have this diagonal, then we go to 100. What's a pattern that could describe how we go from one number to the next, or another way of saying it, what's a rule for why we highlighted these numbers in purple? Pause the video and think about that. All right, well, one thing is if we go from one number to the next, you go from one to 10, we add nine. To go from 10 to 19, we add nine. To go from 19 to 28, we added nine. So each number is nine plus the previous one. And if you go all the way to 91, 91 plus nine is of course 100. Now, it's important to realize these are not multiples of nine because we started at one, not at zero. If you started at zero, you go zero, nine, 18, so forth and so on until you go nine, 18, 27, 36, 45, 54, 63, 72, 81. Those would've been the multiples of nine. But everything got shifted because we started at one, not at zero. So we go from one and then 10, 19, 28, all the way down this diagonal and then we go back to 100. And so this is a really interesting thing to think about it. These are all the multiples of nine plus one is another way to think about it or this is if we started at one and we keep adding nines, these are all the numbers that we would highlight. But you can see a pattern. Whenever you add nine to a number, the value of the ones place decreases by one. You see that here as you go down these diagonals. You go from nine, eight, seven, six, five, four, three, two, one. And even up here, we started at zero but you don't have a lower digit than zero so it just goes back to nine. So it goes zero, then goes up to nine, then it keeps going down, down, down, down, all the way until it gets to zero again and then it starts going down from there again. So once again, interesting patterns to look at. I encourage you to look at a chart like this and think about what patterns can you find?