Multiplication in Two Ways
This is a complete lesson with teaching and exercises about how multiplication can be done in two ways, or in other words, that multiplication is commutative (the lesson does not use that term though). It is meant for third grade.
Students are asked to compare multiplications done in two different orders, along with the visual models (groupings). Students then write multiplications in two ways, matching two different groupings of the same animals. The lesson also shows how this concept applies to with number line jumps, and ends with word problems.
1. Compare the two pictures:
Three rows; four dogs in each row. 3 × 4 = 12 
3 + 3 + 3 + 3 = 12
Four columns; three dogs 4 × 3 = 12 


One row; it has five giraffes. _____ giraffes 1 × 5 = 5 
Five columns; ___ + ___ + ___ + ___ + ___ giraffes. 5 × 1 = _____ 
You can do any multiplication in two different ways, but the
result is the same. (In other words, multiplication is commutative.) 
2. Group the animals in two different ways: as rows and as
columns, and
write the multiplication fact that matches the picture. In one case,
you get the same
multiplication fact either way.
a. _____ × _____ = ______ 
_____ × _____ = ______ 
_____ × _____ = ______ 
_____ × _____ = ______ 
c. _____ × _____ = ______ 
_____ × _____ = ______ 
_____ × _____ = ______ 
_____ × _____ = ______ 
3. Draw X’s and group them in two ways to illustrate the two ways to multiply.



Multiplying in two ways on the number line

4. For each number line, write the two multiplication sentences that the arrows portray.
a. 


____ × ____ = _______ ____ × ____ = _______ 
b. 


____ × ____ = _______ ____ × ____ = _______ 
c. 


____ × ____ = _______ ____ × ____ = _______ 
d. 


____ × ____ = _______ ____ × ____ = _______ 
5. Which is the easier way to multiply?
a.

b.


c.

d.

6. Skipcount to fill in the multiplication table of 3. How does the picture relate to it?

7. Write a multiplication for each problem. Drawing can help.
a. Michael put four rocks in each of his toy buckets. He had five buckets. How many rocks did he use? _____ × _____ = _______ 
b. One small booklet has 12 pages.
How many pages are in three booklets?
_____ × _____ = _______ 
c. If
you make
groups of 4 sticks, and you have 12 sticks, how many
groups can you make? (Hint: Draw the 12 sticks.) _____ × _____ = _______ 
d.
If you make
groups of 5 sticks, and you have 20 sticks, how many
groups can you make? (Hint: Draw the 20 sticks.) _____ × _____ = _______ 
This lesson is taken from Maria Miller's book Math Mammoth Multiplication 1, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.
Math Mammoth Multiplication 1
A selfteaching worktext for 3rd grade that covers multiplication concept from various angles, word problems, a guide for structural drilling, and a complete study of all 12 multiplication tables.
Download ($5.20). Also available as a printed copy.