7 Beware of Patterns!
The “Look for Patterns” strategy can be particularly appealing, but you have to be careful! Do not forget the “and Explain” part of the strategy. Not all patterns are obvious, and not all of them will continue.
Problem 5 (Dots on a Circle)
Start with a circle.
If I put two dots on the circle and connect them, the line divides the circle into two pieces.
If I put three dots on the circle and connect each pair of dots, the lines divides the circle into four pieces.
Suppose you put one hundred dots on a circle and connect each pair of dots, meaning every dot is connected to 99 other dots. How many pieces will you get? Lines may cross each other, but assume the points are chosen so that three or more lines never meet at a single point.
Think / Pair / Share
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What strategies did you try? What did you figure out? What questions do you still have?
The natural way to work on this problem is to use smaller numbers of dots and look for a pattern, right? If you have not already, try it. How many pieces when you have four dots? Five dots? How would you describe the pattern?
Now try six dots. Then carefully count up how many pieces you get. It is probably a good idea to work with a partner so you can check each other’s work. Make sure you count every piece once and do not count any piece twice. How can you be sure that you do that?
Were you surprised? For the first several steps, it seems to be the case that when you add a dot you double the number of pieces. But that would mean that for six dots, you should get 32 pieces, and you only get 30 or 31, depending on how the dots are arranged. No matter what you do, you cannot get 32 pieces. The pattern simply does not hold up.
Mathematicians love looking for patterns and finding them. We get excited by patterns. But we are also very skeptical of patterns! If we cannot explain why a pattern would occur, then we are not willing to just believe it.
For example, if my number pattern starts out: 2, 4, 8, … I can find lots of ways to continue the pattern, each of which makes sense in some contexts. Here are some possibilities:
- 2, 4, 8, 2, 4, 8, 2, 4, 8, 2, 4, 8, …
This is a a repeating pattern, cycling through the numbers 2, 4, 8 and then starting over with 2.
- 2, 4, 8, 32, 256, 8192, …
To get the next number, multiply the previous two numbers together.
- 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, …
- 2, 4, 8, 14, 22, 32, 44, 58, 74 …
Think / Pair / Share
- For the last two patterns above, describe in words how the number sequence is being created.
- Find at least two other ways to continue the sequence 2, 4, 8, . . . that looks different from all the ones you have seen so far. Write your rule in words, and write the next five terms of the number sequence.
So how can you be sure your pattern fits the problem? You have to tie them together! Remember the “Squares on