Intent In this activity, students practice integer arithmetic and finding and using rules for In-Out tables. They also return to the focus on language and symbolic notation begun in the earlier activities Inside Out and Pulling Out Rules.

Mathematics The six questions in this activity give students additional opportunities to express the relationships between the In and the Out in an In-Out table representation of a function. Students write algebraic equations for expressing the Out as a function of the In and use their rules to find both the Out given the In and the In given the Out. Asking students, in effect, to find both y given x and x given y emphasizes the “doing and undoing” aspect of algebraic thinking. The values in these tables also offer students the chance to use their knowledge of integer arithmetic, stressed earlier in the unit in the “hot and cold cubes” activities.

Progression This activity is particularly appropriate for students to begin as a homework assignment.

Approximate Time 20 minutes for activity (at home or in class) 30 minutes for discussion

Classroom Organization Individuals, then groups, followed by whole-class discussion

Doing the Activity Tell students that in this activity, they will look for patterns in more In-Out tables. Some of the tables are tricky and will draw upon their creativity. Students will be asked to write some of their rules as algebraic equations.

Discussing and Debriefing the Activity Have students convene in groups to briefly share findings and ask each other questions. You might then have the members of various groups begin the discussion by sharing what they noticed in one of the In-Out tables. Use Questions 2, 3, and 6 to review the usage of the term function. Ask volunteers to talk about how they translated their rules from verbal form into algebraic equations. Questions 1, 4, and 5 are nonnumeric logic puzzles. The most engaging question in the activity is probably Question 5 (the one with the funny faces). If students are stuck, you might suggest making an In-Out table in which the number of eyes and the number of hairs are both Ins. In Out Number ofeyes Number ofhairs 2 2 6 2 3 11 3 4 19 2 5 ?

Finding a rule for this table could be left open. Here are two. Out = 3(number of eyes) + 5(number of hairs) – 10 Out = number of eyes + (number of hairs)² Both formulas fit all three given rows, but the first formula gives 21 for the missing entry, while the second gives 27.