Patterns and Rules
By Michael Naylor

These activities will not only help your students build math skills, they'll also encourage logical thinking, algebraic reasoning and pattern recognition.

Caterpillars (Grades K-2)
Draw a series of circles to create a caterpillar (don't forget the antennae!). Place a "1" in the first circle and write the rule "+2, -1" below the caterpillar. Have the children help you fill in successive circles, adding 2 to get the next circle, then subtracting 1 to get the circle after that, and so on. You can make your caterpillar as long as you like.

Ask your students what patterns they notice (every other circle contains a counting sequence, the same numbers appear in circles three spaces apart).

Now have students draw their own caterpillars (or provide them with photocopies of one that you drew) and have them choose one of the following rules: "Start with 1, then +2, +1;" "Start with 10, then -1;" "Start with 15, then -2;" "Start with 0, then +3, -1;" "Start with 1, then +3, -2;" "Start with 1, then +2, +1, +1." What patterns can the students find?

With younger children, you can work on counting forward and backward by ones and twos. With older children, you can create more sophisticated rules that generate interesting patterns.

Finally, have the kids create rules of their own and write their rule above the caterpillar with a flap of paper taped over the rule, to hide it. Hang up the completed caterpillars and invite students to guess their classmates' rules.

Mystery triangles (Grades 3-5)
On the board, draw a triangle with circles at the vertices and boxes between the circles. In each circle, draw a small number of pips, as shown below.

Tell the students that each box should contain the total number of pips in the two adjacent circles; have them help you fill in the numbers. In the example above, the numbers are 3, 4 and 6.

Next, give the children triangles in which the boxes are filled in, but the circles are empty. The children must determine how many pips go into each circle so the totals in the boxes will be correct. It's a good idea to give students counters or beans to manipulate.

Try using these sets of three numbers in the boxes: 7, 12, 11; 16, 5, 13; 3, 11, 14. Afterwards, have students share their methods, strategies and observations.

Mystery triangles extension (Grades 5-8)
There's a wonderful algebraic pattern at work in the mystery triangles. Ask your students to examine sums and differences of the numbers in the boxes to see if they can determine a rule for finding the numbers in the circles.

The rule is: To find the number in a circle, add the two adjacent numbers and subtract the opposite number, then divide by two. It's not too difficult to see why this works, using algebra.

If the three missing quantities in the circles are a, b and c, then the three boxes contain a+b, a+c and b+c.

The sum of the two upper boxes is (a+b) + (a+c), or 2a+b+c. If the lower box, b+c, is subtracted from this sum, the total is 2a+b+c-(b+c) = 2a, which is twice the number that belongs in the upper circle. This same principle applies for the other circles as well.

Using the numbers 10, 15 and 9, students will find that 10+15-9=16, which is twice what should be in the upper circle, so the upper circle contains 8. Likewise, the bottom left circle should contain (10+9-15) / 2=2, and the bottom right contains (9+15-10) / 2=7. This will work every time!

Mystery rings (Grades 3-8)
Explore other arrangements of circles and boxes, such as a square.

It turns out to be easy. There are many possible solutions when a square is used. This will be a big surprise to your students, after having been puzzled by the triangles. Older students will enjoy exploring arrangements of five, six or more circles. Which arrangements have only one solution? Which have many solutions?

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.

November/December, 2003, Vol.34, No.3