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Investigating Properties of Real

NumbersCommutative, Associative, Identity Properties

of Addition and Multiplication

Distributive Property of Multiplication over Addition

Additive and Multiplicative Inverse Properties

Multiplicative Property of Zero

Changes in the SOL

• The properties are now taught in the following order:

Commutative Property of Addition

Does the order in which we add two quantities matter?

That is, does a+b = b+a ?

Let’s use Cuisinaire Rods to investigate the property.

Is 3+5 the same as 5+3?

They are the same because both have a length of 8 units.

3+5 = 8 5+3 = 8

Commutative Property of Multiplication

Does the order in which we multiply two quantities matter?

That is, does a x b = b x a ?

Let’s use counters to investigate the property.

Is 6x2 the same as 2x6?

They are the same because both equal 12.

Commutative Property of Multiplication

We can also use grid paper to investigate.

Cut out a rectangle with 2 rows and 6 columns and another with 6 rows and 2 columns.

They have the same area of 12 square units.

Associative Property of Addition

Does the way in which we group quantities when adding matter?

That is, does a+(b+c) = (a+b)+c ?

Let’s use Cuisinaire Rods to investigate the property.

Is 2+(3+5) the same as (2+3)+5?

They are the same because both have a length of 10 units.

Associative Property of Multiplication

Does the way in which we group quantities when multiplying matter?

That is, does a(bc) = (ab)c ?

Let’s use counters to investigate the property.

Are 3x(2x6) and (3x2)x6 the same?

They are both equivalent to 36.

Distributive Property of Multiplication over

AdditionDoes the product of a number and a

sum equal the sum of the individual products?

That is, does a(b+c) = ab+ac ?

Let’s use counters to investigate the property.

Are 2(3+5) and 2x3+2x5 the same?

They are both equivalent to 16.

Identity Properties For Addition and Multiplication

Adding or Multiplying a number by an identity number retains the “identity” or original value of that number

What number can we add to 5 and not change its value?

Zero

0+5 = 5 and 5+0 = 5

What number can we multiply by 6 and not change its value?

One

1x6 = 6 (one group of six) and 6x1 = 6 (six groups of one)

Multiplicative Property of Zero

What happens when you multiply by zero?

The result is zero.

a x 0 = 0 and 0 x a = 0

Discuss how 0x6 (zero groups of six) and

6x0 (six groups of zero) both result in 0.

Inverse Property for Multiplication

The inverse property of multiplication tells us that two numbers are inverses if their product is one (the multiplicative identity).

That is, a×1a

=1 orab

×ba

=1

Let’s use pattern blocks to show and 1

4×4 =1

2

3×

32

=1

Inverse Property for Multiplication

Lay out 4 unit pieces.

1

4×4 =1

One-forth of four gives one unit piece.

Inverse Property for Multiplication

Lay out three half pieces

Two-thirds of three-halves gives two halves which is equivalent to one unit piece

2

3×

32

=1

Inverse Property for Multiplication

3 groups of what will equal 1?

Make 3 groups

Take a unit piece and divide it into three pieces. Put one piece in each group.

3×? =1

Thus 3×13

=1

Discussion

• What did you learn from this session?

• How would you apply this to your classroom?

• What is still unclear?

• Comments and/or concerns?