Guided Notes – Writing Expressions
New Unit - Algebra
Date____________
Variable:
A ______________________ that stands for a ________________
Algebraic Expression:
Uses numbers, _________________, and operational symbols but no ___________ sign
Key Words for Operational Symbols:
Each __________ More __________ Less ___________ Share Equally ____________
Simple Expressions:
1) The product of a number and 6 ____________
2) Six increased by a number ___________
3) The quotient of a number and 2 ____________
4) 3 more than twice a number ______________
5) 5 less than the product of two different numbers __________________________
Translating Word Problems (1 step):
1) At a ballpark, team hats are sold for $15
each. Let h = the number of hats. Write an
algebraic expression for the cost of h hats.
2) Jake has four more books than Sarah. Let b = the
number of books Sarah has. Write an algebraic
expression for the number of books Jake has.
3) Mrs. Graf withdraws $160 each week to
pay the babysitter. Let w = the number of
weeks. Write an algebraic expression for the
change in her bank account after w weeks.
4) Lena is exactly 3 years older than her brother,
Peter. If p represents Peter’s age, write an algebraic
expression to represent Lena’s age.
5) Elaina and her friends went out to
dinner and decided to split the bill evenly.
The cost for dinner was $56.75. If there
were n number of people out to dinner,
how much did each person pay?
6)
Write an expression to find the total number of
yards she paddles in m minutes.
Two Step Algebraic Expressions:
Together
1) To get into the county fair, Patricia must
pay a $5 entrance fee and $2 per ride. Write
an expression to represent the total Patricia
spends at the fair.
You Try
1) Jasmine wants to rent a bike while she
is on vacation. The rental fee is $8.00 plus
an additional $2.50 for every hour the
bike is rented. If h represents the number
of hours the bike is rented, write an
expression to represent the total amount
spent.
2) Julie has $300. She wants to spend her money
on ice skating lessons. Her lessons will cost $56 per
week. Write an expression For the amount she has
after w weeks.
2) At the beginning of the day, the owner of a
restaurant opens a new case of take-out boxes.
One case holds 500 take-out boxes. He uses an
average of 35 take-out boxes each day.
Based on his average usage, write an expression to
represent the number of take-out boxes that
remain d days after the new case of boxes is
opened.
3) A group of 5 people at a restaurant
decide to split their bill evenly. If b
represents the bill and each person says
they will throw in $2 for a tip, write an
expression to represents the amount each
person spends.
3) A group of 10 people at the Village decides to
split their bill evenly. If b represents the bill and
each person says they will throw in $1 for a tip,
write an expression to represents the amount each
person spends.
4) The length of a rectangle is 13.5 cm. Write an
expression to represent the perimeter of the
rectangle, using w to represent the width.
4) The length of a rectangle is 6 cm. Write an
expression to represent the perimeter of the
rectangle, using w to represent the width.
5) Cameron had $500 in savings on January 1. Quinn had $800 in savings on January 1. Cameron deposits
$20 per week into his savings account. Quinn withdraws $15 per week from his savings account. Write
two expressions: one for the amount of money in Cameron’s savings w weeks after January 1st, and one
for the amount of money in Quinn’s savings x weeks after January 1st.
Guided Notes – Writing and Evaluating Expressions
Date_______________
To Evaluate an Expression:
1.
__________________ the equation and plug in a value for each _______________
2.
____________________________________
Examples:
1) Evaluate 3x² + 4 + y when x = -2 and
y=5
2) Evaluate 4a² + 3a + 1 when a = -2
Rewrite: ______________________
Rewrite: ______________________
Simplify: __________
Simplify: __________
3) Evaluate 2a² + 3b + c when
a = -3, b = 4 and c = 5
4) Evaluate 6(x – 2)² for x = -1.
Rewrite: ______________________
Rewrite: ______________________
Simplify: __________
Simplify: __________
5) What is the value of the expression below when n = 12 and p = -2?
𝑛
− 1 + 5𝑝 − 2𝑛 + 𝑝2
3
Rewrite: ___________________________________________________________________
Simplify: __________
Write the Expression and Evaluate:
Remember key words for writing expressions:
Each __________
More __________
Less ___________
1) To get into the county fair, Patricia must pay a
$5 entrance fee and $2 for each ride. Write an
expression to represent the total Patricia spends
at the fair.
Split Evenly ____________
2) Elaina and her friends went out to dinner
and decided to split the bill evenly. The cost
for dinner was $56.75. If there were n
number of people out to dinner, how much
did each person pay?
Write the expression_______________________
Write the expression _________________
If Patricia went on 10 rides, how much did she
spend?
If 8 people were out to dinner, how much
did each person pay?
Evaluate ___________________________
Evaluate ___________________________
TEST EXAMPLE:
3) A landscaper charges $30 for each job plus an additional $20 for each hour worked.
a.) Write an expression to represent the total cost of a landscape job. Explain what the variable
used in the expression represents.
b.) Explain how you identified the operation used in the expression.
c.) If the landscaper works 40 hours per week, how much does he receive on his 2-week pay
check?
Writing in Word Form:
Write the following in word form:
a.) 3x – 5
Then evaluate for x = -2 _____________________
b.) 6(x + 7)
Then evaluate for x = 3 _____________________
𝑥²
c.) −10
Then evaluate for x = -5 _____________________
d.) Write a WORD PROBLEM for the following expression:
2x + 4
Then evaluate if x = 4
Guided Notes –
Simplifying Algebraic Expressions
Date___________
Terms: A number, a __________________________________ or the product of a number and a variable.
__________________________Terms: Have the same variable or the same variable factors
______________________________: the number in front of the variable
Like Terms
Not Like Terms
*When a variable does not have
a number in front of it (like c),
there is an understood ______
in front of the variable
An expression is in ______________________________________when it has no _________________and
no _____________________.
To simplify expressions with multiple terms _____________ or ____________ the coefficients of like terms.
Examples: One Variable
Identify the terms in the expression, then combine the like terms.
1)
-13c + c
2)
2x + 3x – 2 + 4x + 5
Like Terms_________ Simplify _____________
Like Terms_________ Simplify _____________
3)
4)
0.3f – f + 10 + 0.7f + 3f – 4
Like Terms_________ Simplify _____________
m – ⅖ – 5m + ⅙
Like Terms_________ Simplify _____________
Test Example:
A store is advertising a sale where everything is 20% off. Adam and Brandi are customers discussing how
discount and tax will be calculated. Adam says he will take 0.8p to find the new price of any item. Brandi
says she will take p – 0.2p to find the new price of any item. Who is correct? Explain your answer.
Examples: Two or More Variables
Identify the terms in the expression, then combine the like terms.
1)
0.3a – b + 0.9a + 3b
Like Terms_________ Simplify _____________
2) 8f – 2t + 3f + t
Like Terms_________ Simplify _____________
Examples: Products of Two Variables
Identify the terms in the expression, then combine the like terms.
1)
3x + 2xy – 2.6x + 7xy + 7
Like Terms_________ Simplify _____________
3)
2)
3ab – a – 7 + 5ab + 5a + 4
Like Terms_________ Simplify _____________
3ab + 4b – 3.5ab + ab - 5b + b
Like Terms_______________ Simplify _________________________________
4)
3xy + y - ¼x + xy + 6x + ½y
Like Terms_______________ Simplify _________________________________
Test Examples:
1)
Which of the following are equivalent to
the following expression:
2a – 4b + (-3a) + 9b – (-4a) + 6b
Select ALL that apply.
A. 9a + 19b
B. 3a + 11b
C. 19a + 9b
D. (6a – 3a) + (15b – 4b)
E. 11b + 3a
F. (2 – 3 + 4 + 6)a + (-4 + 9 + 6)b
2) Select all that apply. Which of the
following are equivalent to:
2xy + 3x – 2x – 5 + xy
A. xy + 5x + 5
B. xy – x – 5
C. 3xy + x – 5
D. 3xy + x + 5
E. 3xy + 1x – 5
3) Adam and Shelby are shopping in a town that has a 5% tax. Adam says the final price
of any item can be found by the expression p + 0.05p, where p is the original price.
Shelby says the price of any item can be found by the expression 1.05p. Who is correct?
Explain.
Guided Notes - Simplifying Expressions with
Multiplication and Distributive Property
Date___________
An expression is in ______________________________________when it has no
_________________and no _____________________.
When you do see parenthesis, you must use:
The _______________________________Property:
a(b + c) = _____________________
*Remember when two variables are next to each other it means _____________________.
Examples:
1)
With real numbers:
2) With variables:
2(3 + 4) =
3(x + 5) =
3)
-6(c + 4) =
4) 12(4a – 6) =
5)
-3(3f – 2) =
6)
-7(9 + 3a)
To Distribute and Simplify
Step 1: Get rid of _____________________ first by ____________________
Step 2: Identify _________________________
Step 3: Combine to __________________________
1)
3(b + 9) + 10
2) -4(c + 8) + 9c + 7
3)
4y – 7 + 8(y + 5)
4)
6(b – 9 + 2b)
You Try!
1)
3)
x(4 + 5) + 3x + 2x
11b – 2(3b + 1)
5) -6(-3c – 4) – 8(c – 10)
2) 2(5x – 3) + 3x
4) 2(a – 12) – 3(4a + 9)
1
6) -2x – (12x – 8)
4
OAA Examples:
1) Which of the following is the simplified version of:
3(x + x + y)
A.
6x + 3y
B.
3x² + 3y
C.
3x²y
D.
3x + 3y
2) Adam and Shelby go to a store where all items in the store are 10% off and there
is a 5% sales tax. They each came up with a different expression to solve for the
price of the item. Whose expression is correct? Explain.
Adam’s Expression
Shelby’s Expression
Guided Notes – Simplifying Algebraic Expressions with Shapes
Date___________
An expression is in ________________________________________________when it has
_____________________ terms possible. This means it will have no ________________and
no ________________________.
Step 1: _________________________
Step 2: Identify ______________________________
Step 3: ______________________ to simplify
Together
1)
On Your Own
1)
15a – 10(a + 4)
3y – 9(y + 5)
2) 3(c – 1) – 2(2c – 6)
2) 2(a + 4) – 9(y – 5)
3)
1
4
3)
2
4𝑦 − 36 − (10 y – 15)
5
1
2
2
𝑦 − 12 − (6 y – 18)
3
Finding the Area and Perimeter of Shapes
To find the perimeter of figures: __________ the sides.
To find the area of rectangles _____________
Find the area and perimeter of the following shapes: Write your answer using fewest terms possible.
1)
2)
P = ___________
P = ___________
A = _________
A = _________
3)
4)
P = ___________
P = ___________
A = _________
A = _________
Finding the Area and Perimeter of Triangles
To find the perimeter of figures: __________ the sides.
To find the area of triangles _____________
Find the area and perimeter of the following shapes: Write your answer using fewest terms possible.
1)
2)
P = _________________
A = _________________
P = _________________
A = _________________
Finding the Area and Perimeter of Composite Shapes
To find the perimeter of figures: __________ the sides.
To find the area of figures_____________________________________________________
Find the area and perimeter of the following shapes: Write your answer using fewest terms possible.
2)
1)
P = ___________
P = ___________
A = _________
A = _________
3)
4)
P = ___________
P = ___________
A = _________
A = _________
Guided Notes – Factoring
Date___________
Factoring: The reverse of the __________________________________________. You will remove the
_______________________________________________________________________________.
Review of Greatest Common Factor with Numbers:
1) 8 + 32
2) 49 + 84
Factors of 8 ___________________
Factors of 49 ____________________________
Factors of 32 __________________
Factors of 84 ____________________________
Greatest Common Factor ________
Greatest Common Factor ________
Rewrite ________________
Rewrite ________________
Factoring Algebraic Expressions
Step 1: Find the ____________ of the ________________________________________.
Step 2: Find the ____________ of the ________________________________________.
Step 3: Factor out ( ______________________) the GCF.
Together
On Your Own
1) Factor: 24c + 16c
1) Factor: 20x + 15x
2) Factor: 12y – 16
3) Factor: 6x + 3y
4) 6x + 24y + 6
2) Factor: 21y – 12
3) Factor: 4x + 18y
4) 21x + 7y + 14
Factoring Algebraic Expressions (with multiple variables)
Together
On Your Own
1) Factor: 12xy + 15xyz – 3xyz
1) Factor: 21xyz – 15xyz – 3xz
2) Factor: -6abc – 21abc + 15ac
2) Factor: -8ab – 20abc + 14ac
3) Factor: -32de + 14def – 18d
3) Factor: -48def + 16de – 4def
Multi-Select Test Practice
1)
2) Which expressions are a factor of
-24abc – 16ac + 40abc?
A.
4
B.
8
C.
3a
D.
8c
E.
2ac
F.
4ac
G.
2abc
H.
12a
Guided Notes: Factoring Word Problems
Date___________________
If an expression has parenthesis, it means two factors are ____________________________________________.
A word problem may need factored if it is evident that you must “undo” ________________________________.
Together
On Your Own
Example Type 1 - Area
1)
The area of a rectangle is found by multiplying
its length by its width. The rectangle above has
a width of 5 units. The area of the rectangle is
15x + 40 square units. What is the length of the
rectangle?
1)
The area of a rectangle is found by multiplying
its length by its width. The rectangle above has
a width of 3 units. The area of the rectangle is
12x + 9 square units. What is the length of the
rectangle?
2) A rectangle has a width of 4 units. The area of
the rectangle is 12x + 24 square units. What is the
length of the rectangle?
2) A rectangle has a width of 7 units. The area of
the rectangle is 21x + 14 square units. What is the
length of the rectangle?
3) A rectangle has a length of 3x + 4 units. The
area of the rectangle is 12x + 16 square units.
What is the width of the rectangle?
3) A rectangle has a length of 5x + 9 units. The
area of the rectangle is 30x + 54 square units.
What is the width of the rectangle?
Example Type 2 - Perimeter
1) The perimeter of a square is 4(x + 5).
What does x + 5 represent?
2) The perimeter of a square can be written by
the expressions 20x + 32. What is the
length of one side of the square?
1) The perimeter of a triangle is represented
by the expression 3(a + 9). What does this tell
us about the triangle?
2) The perimeter of an equilateral triangle can
be represented by the expression 9x – 36.
What is the length of one side of the
triangle?
Example Type 3 - Rectangular Arrays
Rectangular arrays represent the _____________________ of a rectangle. You may have to work
backwards to find the length and width of the rectangle. Do this by ________________________________.
Examples: Fill in the missing information. Then answer the questions.
1)
2)
Write the area of the rectangle:
As a sum _______________
As a product of two factors ________________
Write the area of the rectangle:
As a sum _______________
As a product of two factors ________________
3)
Write the area of the rectangle:
As a sum _______________
As a product of two factors ________________
4)
Write the area of the rectangle:
As a sum _______________
As a product of two factors ________________
Example Type 4 - Real World Problems
1) Lucy mows five lawns. The total earned is
5(x + 30). What does x + 30 represent?
3) Xander goes to the movies with his family.
Each family member buys a ticket and two
boxes of popcorn. If there are five members of
his family, let 𝒕 represent the cost of a ticket
and 𝒑 represent the cost of a box of popcorn.
Write two different expressions for total
amount spent.
2) Mariah earns x + 20 for each article she writes
in the paper. If her final payment is 6(x + 20),
how many articles did she write?
4) Mrs. Graf decides to take her son, Wesley, and his
friends to Chuck E. Cheese for his birthday. For each
child, she pays for three rides and 15 arcade tokens.
There is a group of ten kids (including Wesley)
attending the birthday party. Let r represent the cost
of a ride and let 𝒕 represent the cost of a token.
Write two different expressions that represent the
total amount Mrs. Graf spent on the party.
Factoring with Rational Numbers
Date___________________
For some problems, you may think the expression cannot be factored. In some cases, this is
true. Other times, the directions may say factor out the _____________________________
of the variable.
Examples:
Factor out the coefficient of the variable (fractions)
1)
3)
5)
1
3
c
2
2
1 3
 m
2 2
1
 x  12
3
2)
4)
6)
2
2
j
3
9
3 3
 a
5 10
1 5
  y
3 6
Examples:
Factor out the coefficient of the variable (decimals)
1) 2.8a – 16.8
3)
1.5b – 4.5
2) -1.2k + 2.4
4) 1.1k + 10.78
The directions also may TELL you what to factor out.
1)
2)
3)
4)
1
Factor 4 out of
1
Factor −
2
1
3
𝑘 −
8
4
out of
2
3
7
𝑎 − 𝑏+
5
4
8