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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 = 14

2 × 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 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 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.

Download ($5.20). Also available as a printed copy.

=> Learn more and see the free samples!

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