Within the first few days of school, I play a game called “What’s My Rule?” that I learned from Richard G. Brown at my first NCTM conference in Philadelphia (1989).  This game can be adapted to any level of math (I do it with piecewise functions in precalculus and really catch the students off-guard) and it helps me begin to learn their names and gets the students comfortable with speaking up in class.  I also can tell who my strong students will be and I like to give them fractions, decimals, or negatives to evaluate mentally.  I still do my version low-tech (I think it’s great for the students to see their teacher do some fast calculations on the spot), but it can definitely be adapted to use with technology.  What follows is an abbreviated class transcript.

Me:  I am thinking of a rule.  Your job is to guess it.  Everyone can play because all you have to is raise your hand, tell me your name, and give me a number.  I will then apply the rule to your number and give you the result.  If you think you know the rule, DO NOT SHOUT IT OUT!  Raise your hand, give me you name, and say, “I would like to take a quiz.”  Then I will give you the number and you will apply the rule and give me the result.  Once enough people know the rule, we will reveal the rule and I will pick a new rule.  Let’s get started.  [NOTE:  I write the ordered pairs we generate, as we generate them, in a table on the board.]

Student Alpha:  5

Me: No.  You need to say my name is _____ and my number is 5.

Alpha: My name is Alpha and my number is 5.

Me:  Thank you, Alpha; 5 yields -9.

Beta:  My name is Beta and I want to take a quiz.

Me:  OK Beta.  But you can’t guess my rule with just one ordered pair because I can change my rule on the spot.  What happens to 6?

Beta:  -8

Me:  Nope.  6 yields -11.  Give me a new number, Beta.

Beta: 3

Me:  3 yields -5.

Gamma:  My name is Gamma and my number is 10.

Me:  Clever number choice, Gamma.  10 yields -19.

Omega:  My name is Omega and I think I know the rule.  I want to take a quiz.

Me:  Ok.  What about zero?

Omega: (pause) One.

Me:  Good.  Anyone else?  Give me a number, you don’t have to take a quiz.

Delta: My name is Delta and my number is 100.

Me:  Thank you, Delta.  Another clever number choice.  100 yields -199.

Iota:  My name is Iota and I want to take a quiz.

Me:  What happens with one half?

Iota:  (pause)  Zero?

Me:  Very nice!  Yes.  One half yields zero.  Now what about you?  What’s your name?

Lambda:  Lambda.

Me:  Can you give me a number, Lambda?

Lambda:  Uh. 7.

Me:  Ok.  Raise your hand if you know the result for 7.  Yes?  What’s your name?

Epsilon:  Epsilon.  -13.

Me:  Yes. That is correct.  Can you tell me the rule, Epsilon?

Epsilon:  Yes.  You take the number, like 7.  Make it negative.  So -7.  Then add it to itself, -14.  Then add 1.  So the result is -13 because -14 plus 1 is -13.

Iota:  I was just multiplying the number by -2 and then adding 1.  Will that work too?

Me:  What do you think?

Omega:  The rule is  -2x + 1 = y.

Me:  Yes.  That would probably be the most efficient way to write it, but we could also write it -x + -x + 1 = y, which is equivalent.  Let’s try a new rule.  I have a new rule in my head and I need to call on someone who hasn’t spoken yet.  How about you?  Give me your name and a number.

Sigma:  My name is Sigma and my number is 10.

Me:  Thank you Sigma, the result for 10 is 81.

Several students mock fall out of their desks with such a big number.

Pi:  I’m Pi and my number is 6.

Me.  For 6 the result is 25.

Omega:  I want to take a quiz.

Me:  Tell me your name again.

Omega:  My name is Omega and I want to take a quiz.

Me:  What happens to 11?

Omega:  100.

Me:  Correct.  Anyone else?

Tau:  I want to take a quiz.

Me:  My name is _____ and I want to take a quiz.

Tau:  My name is Tau and I want to take a quiz.

Me:  Ok Tau.  What happens to -4?

Tau:  (thinks for a moment) 25?

Me:  Excellent work! Yes, Beta?

Beta: I know the rule.  I want to take a quiz.

Me:  Ok.  Let’s find someone to give you a number:

Kappa:  My name is Kappa.  What about 3?

Beta:  3 becomes 4.

Me:  Very nice.  What happens to zero?  (pause) Tau?

Tau:  It becomes 1.

Me:  Excellent.  Can you say what the rule is, Tau?

Tau:  Subtract 1 off of your number and then multiply the result by itself.  So zero minus 1 is -1 and the -1 times -1 is 1.  Same with 3.  3 minus 1 is 2 and then 2 time 2 is 4.

Me:  How could we write that algebraically?

Beta:  I was just doing y equals x – 1 quantity squared.

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So these two rules were:

Know your students; these two rules might be too hard or too easy for them.  Pre-algebra students might get a lot out of these two rules (especially when you give them negative and fraction input values):

Here are a couple of good piecewise functions for pre-calculus:

I also make sure to tell my students that we will not be guessing all year long, and if they were uncomfortable with what was happening not to worry!  Our job will be to learn how to write these rules without guessing, and it will take us all year to get good at it.

In just a few weeks I will be playing What’s My Rule? with all of my classes!  Except for AP Calculus… with AP Calculus I get them going in a different way with the first part of J.T. Sutcliffe’s Seeing is Believing, which I had the joy to watch her present at an AP Calculus reading in 2004.