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Investigating
Properties of Real
Numbers
Commutative, 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 Addition
Does 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).
1
That is,a   1 or
a
a b
 1
b a
1
2 3
 4  1 and   1
Let’s use pattern blocks to show
4
3 2
Inverse Property for
Multiplication
1
 4 1
4
Lay out 4 unit pieces. 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
 1
3 2
Inverse Property for
Multiplication
3? 1
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.
1
Thus 3   1
3
Discussion
• What did you learn from this session?
• How would you apply this to your classroom?
• What is still unclear?
• Comments and/or concerns?