Curious Circles: Introduction to Algebraic Patterns

Two circles can be arranged in five ways: 

1. one circle can be inside the other (without intersecting)

2. they can be separated (again, without intersecting)

3.  they can overlap (intersecting at two points)

4. or they can coincide (intersecting at an infinite number of points)

5.  or they can touch at one point (called a tangent point)

In how many ways can a given number of circles be either separated or inside each other?

(NOTE: the situations in which the circles overlap or coincide or touch at one point are NOT counted here).

For 2 circles, there are 2 different possibilities.

For 3 circles, there are 4 different possibilities. Most of the children figured out three of these four possibilities. The hint I gave them to complete the fourth solution was to look at my face and two eyes. Almost immediately they saw the last solution. See all solutions for 2, 3, 4, and 5 circles on the last two pages of the first pdf. 

After the children solved the fourth solution to 3 circles, I had them explore 4 circle solutions. After they had solved 4-6 of these solutions, I told them that there were 9 total solutions. Using trial and error, better known by the children as "guess and check," they maybe found 6-7 of the nine solutions.

Then I gave them the ultimate hint, I drew the four solutions again for 3 circles. The children were puzzled that I only drew 3 circle solutions. Then I confused them even more by repeating these four solutions. When I asked the if those eight sketches could lead to the solutions for 4 circles, they began to realize that all they had to do was to add a single separate circle to each of the four solutions and then completely encircle the other four. Now they had eight solutions for 4 circles. The saw the 9th solutions pretty easily and then a lightbulb went off: if they wanted the solutions for five circles, they could repeat the process with each of the 9 solutions for 4 circles and easily come up with 18 solutions. Then they saw that there were two missing solutions to make a total of 20.

The older children began to see a pattern emerge and that they could continue this process indefinitely; and some tried.

During the week, the children should try to find all 20 solutions for 5 circles using the systematic approach described above. If they are really industrious, they should try to find all 44 solutions for six circles.

The older children should use the algebraic pattern to find the number of solutions to as high a number as they can. The second pdf has the solutions to 20 circles; there are 2,835,694 solutions. 

When I mentioned this to the older children, I asked them whether they would rather use the pattern to come up with the higher number of solutions or whether they would like to draw all 2+million solutions. Of course, they chose the pattern.

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