The “If… then…” statements below are designed to encourage students to reason about mathematics critically and creatively.

The idea here is to present the “If…” to students, then leave it up to them to fill in the “then_____.” This is wide open and there is a lot of room for students to take in multiple directions. Students may not even say what you’re hoping they say, but their responses will likely tell you a lot about what and how they’re thinking. For this reason, I think these may work best at the beginning of a unit/learning a new concept.

Students are dipping their toe in the waters of conjecture here and discovering the value of proof and reasoning in mathematics – even if it shows that their “then” idea is not mathematically sound. Some of the statements students make may be useful as whole class investigations, others may be worth a brief discussion, and still others may be saved for another day – later in the unit.

These are organized under the different types of reasoning used to complete the statement. Please share your thoughts in the comments.