Transcript MULTIPLYING DECIMALS - Florida Virtual School

Algebraic
Properties
Lesson 6.06
After completing this lesson, you will be able to say:
• I can generate equivalent expressions using the
algebraic properties.
Algebraic properties
Algebraic properties are operational rules
you can use to create equivalent
expressions.
Equivalent expressions have the same
value when simplified.
Commutative Property
The Commutative Property of Addition says that the order of the addends
in a sum does not matter.
The Commutative Property of Multiplication says that the order of the
factors in a product does not matter.
Commutative Property – Examples
Sarah drove 30 miles to the beach and m extra miles to the store.
The total distance can be represented as 30 + m or m + 30.
The sum is the same either way because of the commutative property
The total amount of gear needed for the beach day can be represented as
5 + (x + 2).
You can see it as the sum of 5 and another quantity x + 2.
According to the commutative property of addition, you can change the order
around of the two addends.
5 + (x + 2) = (x + 2) + 5
The sum is the same either way because of the commutative property of
addition.
Associative Property
The Associative Property of Addition says that the grouping of the addends
in a sum does not matter.
The Associative Property of Multiplication says that the grouping of the
factors in a product does not matter.
Associative Property - examples
Sarah swam 30 minutes in the morning, 45 minutes at lunch, and x
minutes in the afternoon.
The total amount she swam can be represented with the expression
30 + (45 + x) or (30 + 45) + x because of the associative property.
Sarah bought 2 cans of sunscreen for some friends, f,
and each can cost $4.
Sarah can represent the total cost of the sunscreen as
(f ⋅ 2) ⋅ 4 or f ⋅ (2 ⋅ 4) because of the associative property.
Identity Property
The Identity Property of Addition states that adding 0 to a number does not
change the identity (or value) of the number.
The Identity Property of Multiplication states that multiplying a number by 1
does not change the identity (or value) of the number
Zero is the identity element for addition because zero has no effect on the
value in a sum.
One is the identity element for multiplication because it has no effect on the
value in a product.
Because of the identity properties, you can manipulate the expression to suit
your needs while maintaining equality between two expressions.
Caution with the properties
The commutative property does not work for subtraction or division. This
means you cannot reverse the order of a subtraction or division expression and
keep the same value.
The associative property does not work if the expression contains more than
one operation
The associative property does not work if the expression contains subtraction
or division. This means you cannot move the parentheses around on a
subtraction or division expression and always keep the same value
Distributive Property
Rule
Distributive Property (with a sum): a(b + c) = a ⋅ b + a ⋅ c
Distributive Property (with a difference): a(b − c) = a ⋅ b − a ⋅ c
The distributive property says that any number multiplied to a sum or
difference of two or more numbers is equal to the sum or difference of the
products. The property allows you to rewrite a product as a sum or
difference to suit your needs without changing the value of the expression
Distributive Property
Caution
In order to apply the distributive property, a factor must be multiplied by a sum or
difference.
You cannot use the distributive property on algebraic expressions that contain
only one operation.
3(4 ⋅ 5 ⋅ 6) ≠ 3(4) ⋅ 3(5) ⋅ 3(6)
Also, the distributive property does not work when dividing a quantity by a sum
or difference
Distributive Property - example
Apply the distributive property to generate an equivalent expression for 4(x + 10)
Step 1: Multiply the first term in the parentheses by the factor. In this example,
remember there is an invisible 1 as the coefficient for the x variable.
Step 2: Bring over the mathematical operation in the parentheses.
Step 3: Multiply the second term in the parentheses by the factor, and simplify.
Therefore, 4(x + 10) is
equivalent to 4x + 40 because
of the distributive property.
Try it
Apply the distributive property to generate an equivalent
expression for 6(3n + 5 + 2c).
Check your work
Multiply the first term in the parentheses by the factor, and bring over the first
mathematical operation in the parentheses.
Multiply the second term in the parentheses by the factor, and bring over the
second mathematical symbol.
Multiply the third term in the parentheses by the factor, and simplify.
Therefore, 6(3n + 5 + 2c) is
equivalent to 18n + 30 + 12c
because of the distributive
property
Visual Models
Using models can help you understand how the distributive property works and also
help you simplify expressions more easily
Sarah and her friends laid down two beach blankets. The blankets are the
same length but of different widths. The distributive property can be used to find
the total area of the two blankets together.
The length of both blankets is 75 inches. The width of the orange blanket is
unknown, and the width of the green blanket is 52 inches.
Lets see how we can simplify this problem using both a visual model and the
distributive property
Visual Models
Sarah and her friends laid down two beach blankets. The blankets are the
same length but of different widths. The distributive property can be used to find
the total area of the two blankets together.
The length of both blankets is 75 inches. The width of the orange blanket is
unknown, and the width of the green blanket is 52 inches.
Remember the area of a rectangle = l ⋅ w.
Together, the blankets form a rectangle
that has a length of 75 inches. The width of
the rectangle is the sum of each blanket’s
width, which is x + 52
Using the area formula, the area of the rectangle can be found by
multiplying the two sides, which is represented as the expression
75(x + 52)
Using visual models to create equivalent expressions
Using the area formula, the area of the rectangle can be found by multiplying
the two sides, which is represented as the expression 75(x + 52)
Can you create an equivalent expression to represent the area of the two
blankets?
Therefore, the expression 75(x + 52) is equivalent to the expression 75x + 3,900
because of the distributive property
Try it
Rewrite the equivalent expression for
5(3x – 7) using the distributive property
Check your work
This is a product of a number and a difference.
You know the equivalent expression will be the
difference of products, according to the
distributive property.
5(3x − 7) = 5 ⋅ 3x – 5 ⋅ 7
When simplified, the expression becomes
15x − 35
Now that you completed this lesson, you should
be able to say:
• I can generate equivalent expressions using
the algebraic properties.