 Hello there, Mr. Marks here, your friendly neighborhood math teacher. Remember, it's not about just getting the right answers, it's about learning and growing along the way. Today's problem is coming to us from a McDougal-Lytel geometry textbook. This problem is a great introduction into geometric concepts such as pattern recognition, so let's uncover the mysteries together. First, let's read the problem's instructions. Sketching visual patterns. Sketch the next figure in the pattern. Let's take a look at our pattern here. I first noticed that each figure in our pattern is a circle, and each of those circles is split into multiple parts. Not only this, but one of the parts in each circle is colored white while the rest are not. How would you describe how to sketch the next figure in this pattern? Oh, I know! It's just another circle broken up into more parts where we're going to color in the top left part, right? Then I say, let's test that conjecture. First, I would ask you, what do you mean by more pieces? This first circle is one, two pieces. And we can see that the second circle is one, two, three, four pieces. And this third circle is one, two, three, four, five, six pieces. Now, hold on a minute. Two pieces, four pieces, six pieces. It seems to me like the number of pieces in each circle is increasing by two each time. And that means that the fourth circle is going to have six plus two, or eight total sides. Let's make that happen. Bam! Take a quick moment to note our thinking. We had originally made the conjecture that each circle has more pieces than the next, and the top left piece is colored white. Now, we do have some new findings here. Do you see where we're going to make the change in our conjecture based on what we found? That's right. We're going to make a change right here to the word more. It's not just more pieces that are added. It's two more pieces that are added each time. Let's make that change. All right. Awesome. Now, let's review our conjecture. Each circle has two more pieces than the next, and the top left piece is colored white. Hold up. Now that we made that change, another part of our conjecture is confusing me. What about you? Each circle has two more pieces than the next, and the top left piece is colored white. That's the part that's confusing me now. What is meant by the top left piece in our new figure in the pack? An hour eight piece circle? Either of these two pieces could be considered top left. Pause for a quick moment. Which of our eight pieces here should be colored white? How would you correct our conjecture to describe which of these pieces to color white? Hey, I have an idea. Check this out. Why don't we observe our pattern using compass directions such as north, south, east, and west? Do you notice how one side of the piece that is colored white never moves from figure to figure? Let's color them in to see a little bit better. I've highlighted this left side each time. The left side never moves, which we will now call the west side. With this new information, which piece is going to be colored in in our new figure? Do you know? We're looking at the same one for each piece that's colored in. The left side is consistent or the west side. Let's make a change to our conjecture. Can you see where the change is going to be made? That's right. We're going to make the change to the bottom of our conjecture. Let's give it another read and confirm it with our sketch. Each circle has two more pieces than the next two, four, six, eight. That's correct. And the top left piece with a side to the west is colored white. Wow. It looks like we've done it. Not only have we completed our instructions, which were to sketch the next figure in the pattern, but we also created a solid conjecture to go along with that sketch. Hey, props to you for taking some time out of your day to do some math with me. I hope you followed along. And if you made mistakes, that's all good. Remember that every mistake is a step towards learning something new. This is Mr. Marks signing off. What did you think? Did you approach this problem differently? Let me know in the comments. And if you enjoyed this problem, show your support by liking and sharing this video. And don't forget to follow my page to stay up to date on more math related content. Until next time.