Multiplication by negative numbers is usually introduced in middle school, and may be tricky to teach. In particular, the bar models that were so helpful when we solve multiplication and division problems for whole numbers and fractions in elementary grades are suddenly not very useful anymore.

In our teaching, we do not use bar models for multiplication by negative numbers. Instead, we usually use patterns to help students to arrive at the “rules”.

For example, for multiplication by negative numbers, complete the pattern:

3×4=12

3×3=9

3×2=6

3×1=

3×0=

3x(-1)=

3x(-2)=

3x(-3)=

3x(-4)=

Then using commutative property,

(-4)x3=

Then proceed in the same pattern,

(-4)x2=

(-4)x1=

(-4)x0=

(-4)x(-1)=

(-4)x(-2)=

(-4)x(-3)=

(-4)x(-4)=

From the pattern, deduce the “rules” for multiplication by negative numbers:

positive number x positive number = __________ number

positive number x (negative number) = __________ number

(negative number) x positive number = __________ number

(negative number) x (negative number) = __________ number

Extend: What happens when we divide?

This type of deductive exercise lets student appreciate that they can also apply problem-solving techniques to acquire new concepts and skills.

^{Reference: “Teaching Secondary School Mathematics – A Resource Book“, Edited by Lee Peng Yee and Lee Ngan Hoe, 2nd Edition, McGraw Hill Education (Asia)}