The Function Game

The function game is a game for exploring functions with children in grades K-12. This game works well at all ages and grade levels, with only the natural changes in complexity of the functions (so, for example with a K-1 class you might use addition or subtraction of 1, 2, or 3, and might set up the game in a manipulative context, and with a middle school class, you might encourage students to use linear functions, squares, fractions or division depending on what the class was studying).

A class or group will usually play the function game by students requesting the leader to tell them outputs for the inputs they give. When a student thinks they know the function rule, the leader will give that student an input, and see if they get the right output. When a large fraction of the class thinks they know the rule, the leader or teacher will ask students to explain the rule they found. One important learning component of the function game is to discuss how rules that seem very different can actually be the same. A significant amount of algebra instruction deals with questions of when different seeming expressions are really the same.

Other important learning components of the function game from an upper elementary level and up are for students to gain experience with writing algebraic expressions to show their thinking, for students to become aware of the different characteristics of different kinds of functions, and for students to make connections between a table of values as a representation of a function and an algebraic expression as a representation of a function.

Some of the patterns students have noticed and found useful are:

Pattern #1: If the outputs in an ordered table go up by the same amount every time (+2 for example), then the formula usually has a multiplication by that amount (2n)

Example1:

formula: 2n-2 or 2(n-1)

Example 2:

formula: .5n+1 or (n/2) +1

Pattern #2: If you can figure out what the function would give at 0, that is often a good guess for the number that is added or subtracted.. In example 2, the output was 1 when the input was 0, and the formula was .5n+1. In example 1, if the input were 0, the output would be -2, and the formula is 2n-2.

Pattern #3: If the outputs start to go up by a lot, that's a clue that you have a square or a cube in there somewhere:

Example 3:

formula: 2n² or 2(2n)

Other things students found useful were: looking at the values in order, and looking at the values at 0, 1, and 10.