The Properties of Addition
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Transcript The Properties of Addition
The Properties of
Addition &
Multiplication
As shown in different types of algebraic
equations. Yes yes.
•Closure Property
•Commutative Property
•Identity Property
•Associative Property
•Inverse Property
What is a property?
What do we mean by property?
Something that a person owns
Ex. This is Chuck’s
cow. It is his property.
What is a property?
The other definition of property
A distinctive attribute or quality of
something
Ex. What are the properties of water?
The Properties of Addition
An
addition property is
“A distinctive quality unique to problems involving addition.”
Commutative Property
Commutative Property
The order in which two numbers are added
does not change the sum
Ex. 5 + 6 = 6 + 5
Commutative Property
How will I remember?
Commute = move from one place to another
(like when you commute to school every
morning)
With addition, numbers can commute (move
around), and we still get the same answer.
Associative Property
Associative Property
The way three numbers are grouped when
adding does not change the sum
Ex. (4 + 3) + 9 = 4 + (3 + 9)
Associative Property
How will I remember?
Associate = to hang out
with someone
(like when you associate
with your homies)
With addition, numbers
can associate (be
grouped with other
numbers), and we still
get the same answer.
Identity Property
Identity Property
The sum of a number and zero (0) is the
number.
Ex. 14 + 0 = 14
How will I remember?
0 has no identity (poor guy). So there is no way that
this nothing can change a number. He is worthless.
Inverse Property
Inverse Property
The sum of a number and its opposite is zero
(0).
Ex. 24 + -24 = 0
How will I remember?
What’s the opposite of…
Let’s Practice
Which property is represented in the
equation below?
8
+ (-8) = 0
Inverse
Property
Which property is represented in the
equation below?
(12
+ 4) +13 = 12 + (4 +13)
Associative
Property
Which property is represented in the
equation below?
1234
+ 4321 = 4321 + 1234
Commutative
Property
Which property is represented in the
equation below?
98765
+ 0 = 98765
Identity
Property
This
addition
stuff is
boring.
Can’t we
do
multiplicatio
n now?
Yes yes young
grasshopper. We
can embark on the
wonderful journey
of multiplication
properties!!!
•Closure Property
•Distributive Property
•Commutative Property
•Identity Property
•Associative Property
•Inverse Property
Commutative Property
Commutative Property
The order in which two numbers are multiplied
does not change the product
Ex. 5 x 6 = 6 x 5
Commutative Property
How will I remember?
Commute = move from one place to another
(like when you commute to work)
With addition, numbers can commute (move
around), and we still get the same answer.
Associative Property
Associative Property
The way three numbers are grouped when
multiplying does not change the product
Ex. (4 x 3) x 9 = 4 x (3 x 9)
Ex. (ab)c = a(bc)
Associative Property
How will I remember?
Associate = to hang out
with someone
(like when you associate
with your homies)
With mutliplication,
numbers can associate
(be grouped with other
numbers), and we still
get the same product.
Identity Property
Identity Property
The product of a number and 1 is the number.
15 1 15
Ex.
How will I remember?
1
I am
worthles
s
Just like 0 in addition, 1 really serves no purpose in
multiplication. It is worthless and has no effect on
the number. It is the 0 of multiplication.
Inverse Property
Inverse Property
The product of a number and its reciprocal is
1.
1
15 1
15
Ex.
How will I remember?
When you are inverted, you are flipped over.
Let’s Practice
Which property is represented in the
equation below?
54
x 11 = 594
Closure
Property
Which property is represented in the
equation below?
31(5) = 5(31)
Commutative
Property
Which property is represented in the
equation below?
1
39
1
39
Inverse
Property
Which property is represented in the
equation below?
56342
x 1 = 56342
Identity
Property
Which property is represented in the
equation below?
(983
x 3)17 = 983(3 x 17)
Associative
Property
LET’S RECAP
Associative Property
States that changing the grouping in an
addition or multiplication expression does
not change the sum or product.
Associative Property
ONLY works with addition and
multiplication!
a + (b + c) is the same as (a + b) + c
a x (b x c) is the same as (a x b) x c
Associative Property – Example 1
a = 4, b = 8 , c = 3
4 + (8 + 3) = ???
Associative Example 2
Show each side is equal by simplifying and
solving
4 x (8 x 3) = (4 x 8) x 3
Simplify:
Solve:
Apply the Associative Property
3 + (4 + 5) =
(3 + 4)
+5
Apply the Associative Property
(12 + 9) +15 =
12 + (9 +
15)
Apply the Associative Property
(48 + 7) + 32 =
48 + (7 +
32)
Prove the Associative Property
Simplify each side to show equivalency
(12 + 4) + 2 = 12 + (4 +
2)
Simplify:
Solve:
16 + 2 =
12 +
186=
18
Prove the Associative Property
Simplify each side to show equivalency
30 + (5 + 15) = (30 + 5)
+ 15
Simplify:
Solve:
30 + 20 = 35 +
15 50 =
50
Check Yo’ Self
1.
b. (8 + 1) + 2
c. k + (g + m)
2.
Simplify: 13 + 5 = 2 + 16
Solve: 18 = 18
3.
a. 3
b. 4
Commutative Property
States that two or more terms can be added in any
order without changing the sum, and two more
terms can be multiplied in any order without
changing the product.
Commutative Property
ONLY works with addition and
multiplication!
a+b=b+a
axb=bxa
Commutative Property – Example 3
5 + 9 = ???
Commutative Property- Example 4
Use the commutative property to rewrite
each number sentence in two different
ways.
9 + 6 + 3 = __________________
_________________
Apply the Commutative Property
17 + 19 =
19 + 17
Apply the Commutative Property
32 + 9 +
12
32 + 12 + 9 =
12 + 9 +
32
12 + 32 +
9 + 32 +
12
9 + 12
Apply the Commutative Property
12.5 + 19
+
7 12.5 +
7+
12.5 + 7 + 1919
=
7 + 19 +
12.5
19
+ 12.5
+
7
19 + 7 +
12.5
Challenge:
Associative or Commutative Property?
(3 + 5) + 7 = 7 + (3 + 5)
Commutati
ve
Check Yo’ Self
1.
b. 3 + 8
c. d + m
B
4.
Distributive Property
Distributive Property
A number outside parenthesis must be
multiplied by everything inside
Ex.
Distributive Property
Distributive Property
A number outside parenthesis must be
multiplied by everything inside
Ex.
How will I remember?
Distribute means “to share.”
The outside number must be distributed (shared)
with every other number.
Distributive Property
Think of it like this…
Ex. 3(5 + 2) = ???
** 3 is the candy.
We must share the candy
with everyone.
That is, we must share 3
with 5 & 2.
3
3
Distributive Property
Think of it like this…
Ex. 6(2 + 7) = ???
There is too much of a
spotlight on 6. The
spotlight must be
shared with 2 and 7
6 (2 + 7)
6(2) + 6(7)
12 + 42
54
Distributive Property
Think of it like this…
Ex. -10(3 + 9) = ???
Uh oh! Don’t lose the
negative sign as you
distribute the
spotlighted number!!
-10(3 + 9)
-10(3) + -10(9)
-30 + -90
-120
Distributive Property
Think of it like this…
Ex. -5(4 – 7) = ???
Make sure you know
where your negative
signs are in this
problem!!!!!
-5 (4 – 7)
-5(4) - -5(7)
-20 – (-35)
15
And now it’s time for …
No!! It can’t be!!
Say it ain’t so!!
Yes class. It’s time
for variables!!!
Mwahahaha!!!
Distributive Property
Think of it like this…
Ex. 3(x + 4) = ???
There is too much of a
spotlight on 3. The
spotlight must be
shared with x and 4
3 (x + 4)
3(x) + 3(4)
3x + 12
Distributive Property
Think of it like this…
Ex. -11(x + 5) = ???
Uh oh! Don’t lose the
negative sign as you
distribute the
spotlighted number!!
-11 (x + 5)
-11(x) + -11(5)
-11x + -55
Distributive Property
Think of it like this…
Ex. -5(2x – y) = ???
Make sure you know
where your negative
signs are in this
problem!!!!!
-5 (2x – y)
-5(2x) - -5(y)
-10x – (-5y)
Let’s Practice
Distributive Property
1. 4(6 + 5)
2. 6(3 + 8)
3. 15(12 + t)
4. -6(x + 12)
5. -12(3 + r)
6. -15(4 + t)
7. -9(6 + p)
8. -5(7 + t)