Function machines and In/Out tables for kids who like a rule

We play a car game. I say a number, and I say the number that comes back out of my machine. Three goes in, seven comes out. Five in, eleven out. The nine-year-old in the back seat is not allowed to guess the rule until he has three pairs, and he hates that rule more than any other. Then he gets it, and we are onto the next one.

What the sheet looks like

The function machine generator puts that game on paper. Worksheet types are find the output (the rule is given, fill the Out column), find the rule (the table is complete, write the rule), find the missing input (work backward), function machines (a drawn machine with In, the rule and Out), number patterns with blanks and "the rule is ___", or mixed. Rules can be add and subtract, multiply and divide, two-step (× 2, then + 1), or mixed, with numbers to 20, 100 or 1,000. Tables have 4, 5 or 6 rows, pages have 4, 6 or 8 problems, and the answer key is a toggle.

Why it works

• Finding the rule from a finished table is the first time most kids reason from several examples to one general statement. That is what algebra asks later, in letters.
• Working backward to a missing input is the inverse operation in disguise. A child who can undo "× 3" to get the input has understood division without being told so.
• Two-step rules force a child to check their guess against every row. One pair fits lots of rules. Four pairs fit one.

Who it suits

Second graders can do find the output with add and subtract rules to 20, four rows, four problems. Third graders move to multiply and divide and the drawn function machines. Two-step rules to 100 and find the missing input are grade 4 and 5 work, and mixed to 1,000 will keep an eleven-year-old busy. A child who struggles with the missing input page usually needs the missing addends sheet first; it is the same idea with one operation.

Print one find the rule page and hide the answer key. It is the only math sheet my son has asked to check himself.