Module 6 Test
Review
Now is a chance to review all of the great
stuff you have been learning in Module 6!
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Numerical Expressions
Algebraic Expressions
Parts of an Expression
Evaluating Algebraic expressions
Algebraic Properties
Equivalent Expressions
Exponential form
Exponential form is just a simplified way of writing a multiplication
expression where a number is being multiplied by itself
Area in Exponential Form
Since the 5 is being multiplied by itself 2
times, you can use an exponent of 2. The
area 5 ft × 5 ft written in exponential form
is 52 ft2.
When the exponent is a 2, this is called
squaring the base. So you can say "five
squared."
Volume in Exponential Form
To calculate the volume of the circus cube
you would multiply 5 ft × 5 ft × 5 ft.
5 is the base, but this time it is multiplied 3
times so the exponent in this case is 3.
Therefore, the exponential form of the
volume is 53 ft3.
When an exponent is a 3, this is called
cubing the base. So you can say "five
cubed."
Simplifying exponential numbers
35 = 3 x 3 x 3 x 3 x 3 = 3 x 3 x 3 x 3 x 3
9x3x3x3
27 x 3 x 3
Caution
When simplifying an exponent,
you must remember that
73 = 7 × 7 × 7. It does not
equal 7 × 3 or 7·3 or 73
81 x 3
243
Key Terms
Verbal expression:
A mathematical expression in which all mathematical
operations are written using words; also called a
mathematical phrase; an example is the product of three
and a number.
Variable:
A letter that holds the place for some unknown value in an
algebraic expression, such as x or y.
Algebraic expression:
An expression that contains numbers, variables, and
mathematical operations like addition, subtraction, etc; an
example is x + 2
Verbal Phrases
This chart groups commonly
used phrases by their meaning
Translating Expressions
How do you translate the expression:
“Double the total number of brownies and cookies?”
Example
Sam has to pay a $5 admissions fee for each person who attends the party.
Sam’s expression now contains two operations. The verbal expression for the
cost of admissions is “the product of five and the quantity forty minus a number.”
The first operation is to determine the total number of friends.
The second operation is to multiply by five.
Adding the parentheses around the first operation will ensure that Sam
subtracts before he multiplies.
Sam can write the algebraic expression as:
Tricks of Translating
An algebraic expression uses numbers, letters, and symbols instead of words. All
you will need to do is change the words into math.
Since algebraic expressions contain variables like the letter x, you should no
longer use x to represent multiplication
Tricks of Translating
It is important to pay attention to the order of quantities when translating
When the phrases “less than” and “subtracted from” are translated to an
algebraic expression, the order of the numbers must be reversed.
Translating expressions
Sam invited 40 friends, but his mom mentioned that some might not come.
This means he would have “forty minus some number.”
First, identify the action phrase in the verbal expression to know what
operation is being used. In this expression, the action phrase is “minus,” which
means to subtract. So the verbal expression written in algebraic form is:
Example
Rewrite “the quotient of eight and an unknown value” into an
algebraic expression
In this expression, the action phrase is "quotient", meaning division.
Parts of an Expression
Example: 2x2 – 3y – z + 6
Variable:
Expressions, whether they are mathematical or verbal, can contain variables. A
variable is a letter that holds the place for some unknown value in an expression.
expressions can have two or more unknown values. In those cases, you simply
use a new letter for each unknown value.
the given mathematical expression has three unknown values because
there are three variables: x, y, and z
Term:
Expressions are made up of terms separated by a plus or minus sign. Terms
can contain variables, numbers, or products of variables and numbers.
The given expression contains four total terms: 2x2, −3y, −z, and 6. Notice
how the plus or minus sign is attached to the term immediately following it.
Parts of an Expression
Example: 2x2 – 3y – z + 6
Factor:
Factors are numbers you multiply together to produce a product.
Look at the factors that are easily seen in the given expression.
2 and x2 are factors of 2x2
−3 and y are factors of −3y
−1 and z are factors of −z
Coefficient:
When multiplying a variable and a number to write a term, the number is listed first
and is called the coefficient. It is important to remember the sign in front of the term
also goes with that term.
The coefficients in this expression are 2, −3, and −1
Parts of an Expression
Example: 2x2 – 3y – z + 6
Constant:
Constants are numbers that stand alone. They are called constants because
they have fixed value
The constant of this expression is 6
Example
Identify the parts of: 4(t + 3) – s
Variables: t and s
Term: There are two terms in this expression: 4(t + 3) and –s
Factor: Parentheses not only group operations together, they also mean
multiplication of two factors. 4 and t + 3 are factors of the term 4(t + 3).
As well, –1 and s are factors of the terms –s
Coefficient: The coefficients in this expression are 4, 1, and –1
Remember, if there is no number in front of a variable, it is
understood as 1 times that variable and is not written
Constant: In this expression, the second factor of the term 4(t + 3) has a constant
of 3
Evaluating Expressions
Substitution means replacing one thing with
something else
You can substitute for the variables when you know their values. Then you
follow the order of operations to simplify the expression
Evaluating Expressions
Let’s use substitution to evaluate the expression 5x + 100, if x equals 7
Since you know the value of the variable is 7, you can replace the letter x
with the number 7. That’s all there is to substitution
The algebraic expression is now a numerical expression and can be easily
simplified. To evaluate the expression, follow the order of operations.
Example 2
Evaluate the expression 2x2 + 4y, when x = 8 and y = 1.5
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 - 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.
Key Terms
Like terms:
Terms in an expression that have the same
variable part; for example, 5x and 4x are like
terms.
Unlike terms:
Terms in an expression that do not have the same
variable part; for example, 5x and y are unlike
terms.
Examples of like and unlike terms
Like Terms
7x, x, and –3x are like terms because all
the terms contain the same variable part
x.
Unlike Terms
7x and y both have a single variable, but
the terms are not alike since different
variables are used.
y2, 4y2, and –8y2 are like terms because all 4a and 4b, although they have the same
the terms contain the same variable part coefficient are unlike terms because the
y2.
terms do not have the same variable.
4ab, –8ab, and ab are like terms because
all the terms contain the same variable
part ab.
2x and –5x2, although each term contains
the same variable x, are not alike because
each x variable has a different exponent.
All constants terms are like terms because 6x and 4xy, although both terms have an x
they do not contain a variable
variable, only one term has the y variable,
so these terms are not like terms.
Combining Like Terms
Combine like terms:
A process of combining terms that have identical variable
parts.
Combining like terms means combining the coefficients
of the like terms. The purpose of combining like terms is
to help simplify an expression.
Combining Like Terms – example
In the expression 3y + 2y, notice how both terms contain the exact same
variable y. For this expression, you can combine the like terms by adding the
coefficients of the terms. Use the distributive property to factor out the y
variable, and then add the coefficients to combine the like terms.
Therefore, the expression 3y + 2y simplifies to 5y because of combining like
terms using the distributive property. This means that 3y + 2y, (3 + 2)y, and 5y
are all equivalent expressions.
Equivalent Expressions
In an algebraic expression, you can use the mathematical properties to group
and combine like terms. The properties also allow you to prove that two
expressions are equivalent.
Proving Equivalent Expressions: Method 1
Show that the expression 3x + 2 + 2x + 5 is equivalent to 5x + 7. Can you
identify which properties are used to show that the two expressions are
equivalent? Think about which property allows you to move from step to step
Proving Equivalent Expressions: Method 2
Show that the expression 3x + 2 + 2x + 5 is equivalent to 5x + 7. Can you
identify which properties are used to show that the two expressions are
equivalent? Think about which property allows you to move from step to step
You can prove these expressions are equivalent by using substitution.
Substitute 1 for x, and simplify. This is a great way to check your math
work to ensure you used the properties correctly.
Proving Equivalent Expressions: Method 3
Show that the expression 3x + 2 + 2x + 5 is equivalent to 5x + 7. Can you
identify which properties are used to show that the two expressions are
equivalent? Think about which property allows you to move from step to step
If two expressions are truly equivalent, the value of the expression should be
equal regardless of the number that is substituted. Let’s substitute 5 for x to
be sure this works for another number.
You have now had a chance to review all of the great
stuff you learned in Module 6!
• Numerical Expressions
• Algebraic Expressions
• Parts of an Expression
• Evaluating Algebraic expressions
• Algebraic Properties
• Equivalent Expressions
Have you completed all assessments in module 6? Have you completed your
Module 6 DBA?
Now you are ready to move forward and complete your module 6 test. Please
make sure you are ready to complete your test before you enter the test
session.