Patterns in hundreds chart

So what we have in this chart is all the numbers from 1 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 1 to 10, then 11 to 20, then 21 to 30, all the way to 100.

And what we're going to look at is interesting patterns that might emerge from this. 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. 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 2 in the ones place, or you're going to have a 4, or you're going to have a 6, or you're going to have an 8, or you're going to have a 0. So that one's 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: 5, 10, 15, 20, 25, 30, 35, 40, so on and so forth. And so these form these two columns on this chart. Here, we can see very clearly that multiples of five are either going to have a 5 in the ones place, like we have right over here, or they're going to have a 0 in the ones place. You might have realized that before, but 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 start at one and then we have this diagonal, and 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 to go from 1 to 10, we add 9. To go from 10 to 19, we add 9; to go from 19 to 28, we added 9. So each number is 9 plus the previous one, and if you go all the way to 91, 91 plus 9 is, of course, 100.

Now, it's important to realize these are not multiples of 9 because we started at 1, not at zero. If you started at zero, you go zero, nine, eighteen, so forth and so on. And so you go nine, eighteen, twenty-seven, thirty-six, forty-five, fifty-four, sixty-three, seventy-two, eighty-one. Those would have been the multiples of nine, but everything got shifted because we started at one, not at zero.

So we go from 1 and then 10, 19, 28, all the way down this diagonal, and then we go back to 100. So this is a really interesting thing to think about. These are all the multiples of 9 plus 1 is another way to think about it, or this is if we started at 1 and we keep adding 9s, these are all the numbers that we would highlight.

But you can see a pattern: whenever you add 9 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 9, 8, 7, 6, 5, 4, 3, 2, 1.

And even up here, we started at 0, but you don't have a lower digit than 0, so it just goes back to 9. So it goes 0, then goes up to 9, then it keeps going down, down, down, down, all the way until it gets to 0 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.