Lesson objective: Understand that the properties of numbers can be used to prove the rules for multiplying with negative numbers.

From 6.EE.3, students bring prior knowledge of the distributive property, and from 7.NS.1, students bring prior knowledge of additive inverses. This prior knowledge is used to justify new reasoning as students work to develop proofs of the rules for multiplying with negative numbers. A conceptual challenge students may encounter is understanding what a proof is.

Working with proofs helps students understand that sign rules are not just arbitrary facts to be memorized; rather, they follow logically from prior knowledge.

This work helps students deepen their understanding of operations because they will be able to discover the patterns or "rules" for multiplying with negative numbers (i.e., a positive times a negative is a negative, a negative times a positive is negative, and a negative times a negative is positive).

Students engage in Mathematical Practice 6 (Attend to precision) as they clearly communicate their reasoning in proofs of the sign rules for multiplying with negative numbers. They name definitions and properties that support their work.

**Key vocabulary:**

- additive inverse
- distributive property
- opposite
- properties of equality

**Special materials needed:**