Playing By the Rules
By Michael Naylor

Building relationships between numbers starts when young children learn the counting sequence and continues to the highest levels of mathematics. This month's activities focus on finding numbers according to rules and building skills that range from early number sense to advanced algebraic reasoning.

As young children build a chain of "one more, one less" dominos, they're also building number sense.

One More, One Less Dominos (Grades K-1)
Working in pairs or small groups, the children turn a set of dominos face up. One domino is selected as the starting piece, then students take turns adding dominos to build a chain.

Unlike the ordinary domino rules, in which the numbers on adjacent dominos must match exactly, in this version the numbers must differ by one – either one greater or one less than the previous number. Can players find an arrangement that uses all the dominos? When your students have become adept at the game, challenge them to try a version in which the numbers must differ by two. The students will be having fun and building number sense that will eventually help them to master mental addition and subtraction.

Guess My Rule (Grades 2-5)
This lively game builds excitement and algebraic reasoning. Call on students to give you a number. Write it on the board, then apply a secret rule and write the result on the board. Students try to determine the secret rule.

Here are some sample secret rules: double the number and add one; multiply the number by three and add 10; multiply the number by itself (square it); subtract the number from 100; multiply the number by nine; divide the number by two and round up.

If a student thinks he or she knows the rule, he or she should call out, "Rule!" rather than saying the rule, so the game can continue. That student then takes your place, applying the secret rule to other students' guesses. If the new caller gives correct answers five times in a row, you can be certain he or she really knows the rule. As other students call out "Rule!" you can give them turns being the caller until most of the class has figured out the rule.

To end a round, ask students to explain how they figured out the rule. Interesting discussions occur when multiple rules seem to work. For example, your rule might be "double the number and add four," but a student may find that the rule "add two and then double" gives the same results. Comparing two valid rules helps your students make sense of operations.

Lateral Thinking Guess My Rule (Grades 3-8)
You can add elements to your rule that go beyond simple mathematical operations. For example, you could create a rule that does different things to odd and even numbers. These additional challenges will help to build your students' lateral thinking skills along with their algebraic sense. Here are several strange and wonderful rules to get you started.

• If the guess number is even, divide by two. If the guess number is odd, add 10. Examples: If the guess is 12, the result is 6. If the guess is 13, the result is 23.
• Add the digits of the guess together and put that number at the end of the guess. Examples: If the guess is 15, the result is 156. If the guess is 39, the result is 3912. If the guess is 7, the result is 77.
• Add one to the first guess, two to the second guess, three to the third guess, etc.
• The answer is always the number of letters in the English spelling of the guess number. Examples: If the guess is "five," then the answer is "four" (because the word "five" contains four letters). If the guess is "eleven," then the answer is "six" (because the word "eleven" contains six letters).

Advanced Guess My Rule (Grades 6-8)
These rules depend on more advanced number operations and theory. Your students may want to keep calculators handy for some of these.

• The result is the number's greatest prime factor. Examples: If the guess is 20, the result is 5 because 20 = 2 x 2 x 5; 5 is the greatest prime factor. If the guess is 21, the result is 7. If the guess is 19, the result is 19.
• The result is the product of the guess and the next two integers after it, or n(n+1)(n+2). Examples: If the guess is 5, the result is 210 (5 x 6 x 7). If the guess is 20, the result is 9240.
• The result is the remainder when the guess is divided by seven.
• Multiply the guess by the next greatest integer and divide the result by two. This is the same as the sum of the integers from one to the guess. Example: If the guess is 6, the result is 21 ((6 x 7) / 2 = 21, or 1 + 2 + 3+ 4 + 5 + 6 = 21).

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

October 2003, Vol.34, No.2