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Commutative Property
The word “commutative” is derived from the word “commute” which means to ____________ around. Consider the numerical expressions, 5 + 6 and 6 + 5.
Are the two expressions of equal value? How do you know?
Did the order of the integers change the sum?
Would the expressions 2 + 3 + 4 and 3 + 4 + 2 be equal in value? Why or why not?
The commutative property of addition states that when
adding numbers, the order of the _________ does not change the sum.
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Now, let’s consider the numerical expressions, 5 ∗ 6 and 6 ∗ 5. Are the two expressions of equal value? How do you know?
Did the order of the integers change the product?
Would the expressions 4 ∗ (3 ∗ 2) and (3 ∗ 2) ∗ 4 be equal in value? Why or why not?
The commutative property of _______________ states that when multiplying numbers, the order of the terms does not change the _______________.
! ∗ ! = ! ∗ !
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Let’s consider the numerical expressions, 5 − 6 and 6 – 5. Are the two expressions of equal value? Why or why not?
Does the order of the terms change the value of the expressions?
Are the values of the expressions, 2 − (3 − 4) and (3 − 4) − 2 equal? Why or why not?
Does the commutative property hold true for division? Let’s consider the numerical expressions, 5 ÷ 6 and 6 ÷ 5. Are the two expressions of equal value? Why or why not?
Does the order of the terms change the value of the expressions?
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Are the values of the expressions, 2 ÷ (24 ÷ 4) and (24 ÷ 4) ÷ 2 the same? Why or why not?
Let’s Practice
Use the commutative property to identify the equivalent expression of 5 + (8 + 9).
A. (5 + 8) + 9 B. 8 + 9 + 5
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Associative Property
!The word “associative” is derived from the word “associate” which can be thought of as ______________.
Let’s look at the following numerical expressions to learn more about the associative property.
Simplify (5 + 8) + 9.
Simplify 5 + (8 + 9).
What do you notice about the answers to each of the above expressions?
Did it matter which grouping was performed first?
The associative property for addition states that how terms are ______________ does not change the ______ of the
!
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!
!
Does the associative property apply to multiplication? Simplify 5 ∗ 2 ∗ 3.
Simplify 5 ∗ (2 ∗ 3).
What do you notice about the answers to each of the above expressions?
Did it matter which grouping was performed first?
The associative property for multiplication states that how terms are ______________ does not change the ______ of the expression.
!
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!
!
Does the associative property apply to subtraction? Simplify 5 − 8 − 9
Simplify 5 − (8 − 9)
What do you notice about the answers to each of the above expressions?
Did it matter which grouping was performed first?
This means that the associative property (does / does not) apply to subtraction.
!
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!
!
Finally, let’s check to see if the associative property applies to division.
Simplify 20 ÷ 4 ÷ 5
Simplify (20 ÷ 4) ÷ 5
What do you notice about the answers to each of the above expressions?
Did it matter which grouping was performed first?
This means that the associative property (does / does not) apply to division.
!
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!
Let’s Practice
Which of the following expressions represent the associative property? Select all that apply.
!!3 + 5 + 8 = 8 + 5 + 3 !! 4 ÷ 2 − 1 = 4 ÷ (2 − 1)
!! 2 + 9 + 3 = 2 + 9 + 3 !!2 ∗ 5 ∗ 9 = (2 ∗ 5) ∗ 9
Distributive Property
How can you write out 2 ∗ 5 using addition?
How can you write out 4 ∗ ! using addition?
This applies to any terms that are being multiplied!
How can you write out 3(! + !) using addition?
How many !’s are added together in repeated addition?
How many !’s are added together in repeated addition?
Let’s expand 2 8! + 3! .
Rearrange the expanded form of 2 8! + 3! .
This means that 2 8! + 3! = ____ ∗ ____ + ____ ∗ ____ = ______ + ______.
The term “distributive” comes from the word______________ which means to spread out. In this case, we distributed multiplication across addition.
The distributive property states that a number on the
________________ of a parenthesis can be multiplied through all of the terms ________________ the parenthesis.
Let’s Practice
Factoring Linear Expressions
Recall that the greatest common factor (GCF) of two terms is the _________ integer that can be ______________ evenly into both terms.
What is the GCF of 15 and 25?
We can also find the GCF between terms that contain variables.
This same idea applies to algebraic expressions!
Charlie started taking piano lessons. Charlie’s first lesson lasts for 25 minutes. He decides to increase his lesson time by
10 minutes each week. The total minutes that Charlie spends in lessons is modeled by the expression, 25 + 10!, where ! is the number of weeks Charlie has been taking lessons.
What is the GCF between the two terms in the algebraic expression?
When we divide 25 by the GCF, what remains?
When we divide 10! by the GCF, what remains?
Let’s write an equivalent expression using the GCF and simplified terms.
Let’s Practice
Brandon is a weight lifter and is preparing for his next
competition. He currently bench presses 300 pounds and plans to increase his weight by 12 pounds each week. The total weight, in pounds, that Brandon bench presses is
modeled by the expression, 300 + 12! where ! is the number of weeks since Brandon started preparing for the
competition.
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Identifying Coefficients
For any term in an expression, the number that is located in before letter is known as the ______________.
What is the coefficient of the following terms? 4!
! −64!
Identify the coefficient of the terms in each expression. −3! + 2
! − 6!
1
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At the local a grocery store, large apples cost $2 each and organic bananas are $5 per bunch. The model that
represents the total price of this fruit is 2! + 5!. What are the coefficients for this expression?
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Identifying Variables
Consider the term, −17!.What is the coefficient of this term?
That leaves us with the !, which is the _______________ in this monomial.
Variables are represented in algebraic expressions with___________.
Variables are placeholders for a value that will be determined in the context of the expression. They are____________ values.
Complete the table:
Expression Coefficient Variable 15!
64!! ! −!
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Jacques wants to make a kiwi and watermelon salad. Kiwis cost $4.50 a box and watermelons cost $6 each. The total cost for Jacques’ fruit salad would be illustrated by the model, ! = 4.5! + 6!.
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Writing Expressions (Combining Like Terms)
Consider the following:Sam has two five-dollar bills and one ten-dollar bill. Janie has one five-dollar bill and three ten-dollar bills.
Let’s draw a picture to visually represent how much money each person has.
Sam: Janie:
How many bills do Sam and Janie have in total?
Are these bills all the same value?
In order to find the total value of Sam’s and Janie’s bills, we must ___________ the similar bills. In mathematics, we call these ______ ________.
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Find the total value of Sam’s and Janie’s bills by combining like terms.
The concept of combining like terms applies to expressions and equations as well.
We can _______________ the similar items together by adding or subtracting ________ terms.
Like terms are terms that have the same _______________.
Consider the following expression:
−2! + 3! − 4 − 5! + 6
Let’s identify and combine all like terms. Terms with !:
Terms with !: Constant terms:
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Simplify the following expression by combining like terms. 5! − 3! − 4 + 8! − 7
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Writing Expressions (Distributive Property)
Let’s first review the distributive property by using it to simplify 4 5! − 3 .
How would you simplify −20! + 5! + 30! − 2! + 0.5?
Let’s put it all together!
Simplify the following expression:
7 3! + 4 − 2(6! − 1)
In order to simplify expressions, first __________________ any terms on the outside of the parenthesis, then combine _____________ ______________.
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Writing Expressions (Order of Operations)
Consider 5 + 6∗ 3.How can we tackle this problem? There may be different ways.
Because different approaches can lead to different answers, we need a set of rules that everyone can follow so we get the same correct answer.
We need to have an ________ of operations. This means we need a consistent order that we follow when solving a
problem with multiple operations.
To understand how order of operations works, first think back. What were the first two math operations we learned in
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Recall that addition is a shortcut for counting _____and subtraction is a shortcut for counting ________.
After addition and subtraction, we learned how to multiply and divide. Why?
Because multiplication is repeated ____________________. and division is repeated __________________________.
In mathematics, we want to use our more powerful tools first. If these four operations are in the same expression, then we first multiply or divide, since they are more powerful.
Then, we would add or subtract.
Now, let’s return to our original problem, 5 + 6 ∗ 3. Solve it using the order of operations, where we use our more powerful tools first.
Are there even more powerful math tools? Is there a shortcut to multiplication?
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Exponents are ways to write ___________________ multiplication and division.
The order of operations rules are based on using the most powerful operations first.
Finally, take a look at these two expressions:
2 ∗ 3 + 4 ! 2 ∗ 3 + 4!
How are they different?
We want to start with grouping symbols. Some examples of grouping symbols are:
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Start with the grouping symbol, which we usually call parenthesis, then use your most powerful math tools first. The order of operations is:
Let’s Practice
Simplify the following expression:
