# 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:

4

4

+ 4

12

Three rows; four dogs in each row.

3 × 4 = 12

3    +    3    +   3    +   3  =  12

Four columns; three dogs
in each column.

4 × 3 = 12

Five rows;
each row has two rams.

___+___+___ +___+___ rams

5 × 2 =  _____

 Two columns; each column has five rams. ____ + ____ rams 2 × 5 = _____

One row; it has five giraffes.

_____ giraffes

1 × 5 = 5

Five columns;
each column has one giraffe.

___ + ___ + ___ + ___ + ___ giraffes.

5 × 1 = _____

 You can do any multiplication in two different ways, but the result is the same. The order of the numbers does not matter in a multiplication problem. (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.      _____ × _____ = ______ _____ × _____ = ______

 b.

_____ × _____ = ______

_____ × _____ = ______

 c.           _____ × _____ = ______ _____ × _____ = ______

 d.

_____ × _____ = ______

_____ × _____ = ______

3. Draw X’s and group them in two ways to illustrate the two ways to multiply.

 a.
 _____ × _____ = ______ _____ × _____ = ______ nine groups of 2 two groups of 9

 b.
 _____ × _____ = ______ _____ × _____ = ______ five groups of 3 three groups of 5

Multiplying in two ways on the number line

 5 × 2 = 10 2 × 5 = 10 7 × 2 = 142 × 7 = 14

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.

 2 × 10 = _____ OR 10 × 2 = _____ Two groups of ten Ten groups of two

b.

 7 × 2 = _____ OR 2 × 7 = _____ Seven groups of two Two groups of seven

c.

 3 × 4 = _____ OR 4 × 3 = _____ Three groups of four Four groups of three

d.

 11 × 3 = _____ OR 3 × 11 = _____ Eleven groups of 3 Three groups of 11

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

 1 × 3 =2 × 3 = 3 × 3 = 4 × 3 =5 × 3 = 6 × 3 = 7 × 3 =8 × 3 = 9 × 3 = 10 × 3 =11 × 3 = 12 × 3 =

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 my book Math Mammoth Multiplication 1.

#### Math Mammoth Multiplication 1

A self-teaching 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.

See more topical Math Mammoth books