Unit 5 Parent Guide: Expressions and Equations
Lesson 5-1 Expressions and Order of Operations
Expressions- Combination of one or more numbers, variables, or both. Has one or more operation.
Does not include equal sign.
Example: x – 9
Term- Part of an expression that is added or subtracted.
Example: 4, 3b and b² are terms in the expression
4 + 3b - b²
Variable- A letter or symbol used to represent an unknown number or quantity that varies.
Example: y = 2x
Numerical Expression- An expression that does not include variables.
Example: 20 – 8 ÷ 2
Algebraic expressions- An expression that includes one or more variables Example: 10 (b + a)
Order of Operations- Rule that states the order the expression must be solved
Please → perform operations in Parentheses
Excuse → simplify powers (Exponents)
My Dear → Multiply and Divide left to right
Aunt Sally → Addition and Subtraction left to right
Evaluate an expression: Substitute given values for variables, then simplify using order of operations
Example:
x · (5 – y)
for x = 8 and y = 3
= 8 · (5 – 3)
= 8·2
= 16
Lesson 5-2 Expressions with exponents
Base- the ‘big’ number
Exponent- tells how many times the number is used as a factor.
Power- an expression that includes an exponent and represents repeated multiplication.
Base ← 43 → 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 = 4 · 4 · 4 = 64
Repeated multiplication-
5³= 5 · 5 · 5
Exponential Form6 · 6 · 6 · 6 = 64
 Can be used with variables: 𝑎4 · 𝑏 2 = a · a · a · a · b · b
Lesson 5-3 Translate between algebraic expressions & words:
7 · d → 7 times d or multiply 7 and d, or product of 7 and d
Diagram Expressions: Place a line through + or – symbols to separate into terms. Following order of
operations circle remaining expressions:
7 · (a – 5) - a²
Lesson 5-4 Make dot diagrams to match algebraic expressions:
3 · 5 + 3 + 5 · 2
····· ···
··
·····
··
·····
··
··
··
Simplify: find the value of the expression.
3· 5+3+5· 2
=
15 + 3 + 10
=
18
+ 10
=
28 → should be same as number of dots
Lesson 5-6
Equivalent expressions- expressions that always have the same values
Example: b + b + b and 3b are equivalent expressions
The equivalent expressions to 4b – 4 are circled.
4·b–4
(4 · b) – 4
b
b+b+b+b–4
Write the expression for the given model:
x
2
x
2
x
(4 + b) – 4
3x + 4
and x + 2 + x + 2 + x
Lesson 5-7 Commutative and Associative Properties of Addition and Multiplication:
Commutative Property of Addition
Associative Property of Addition
a+b=b+a
(a + b) + c = a + (b + c)
5+2=2+5
(3 + 4) + 1 = 3 + (4 + 1)
Commutative Property of Multiplication
Associative Property of Multiplication
a·b=b·a
(a · b) · c = a · (b · c)
4·7=7·4
(6 · 2) · 5 = 6 · (2 · 5)
Coefficient- the number part of a term when the term is a number times a variable or a number times a
product of variables.
Example:
5x + 2xy
coefficient
coefficient
Understood coefficient- if the variable is alone, it is understood that the coefficient is 1.
Example: x = 1x
Like terms- terms with the same variables, raised to the same powers.
Example:
6 + 2x + 1 + x → 6 and 1 are like terms, and 2x and x are like terms
Lesson 5-8 Distributive Property
Distributive Property- gives the opportunity to transform expressions into equivalent expressions two
different ways.
 Distributing a factor to the terms of a sum or difference:
Example: (5 + 2)c = 5c + 2c
→ c is distributed to 5 and 2
 Pulling out a common factor from the terms of a sum or difference:
Example: 5c + 2c = (5 + 2)c → c is pulled out of 5c and 2c
Equivalent expressions – yes or no
3y + 5y = 8y → yes
2 + 2m + 3 = 4m + 3 → no
m + m + 5 = 2m + 5 → yes
Lesson 5-9 Working with expressions
 Real world situation: Joe made m batches of 12 muffins. His children ate 5 of the muffins.
o Write an expression for the number of muffins that were left.
12m – 5
o Evaluate the expression when m = 4
12m – 5 =
12(4) – 5=
48 – 5= 43 muffins

Combine like terms with expressions:
-Do all computations you can
-Write each term with the coefficient in front
-Combine like terms
Example 1:
6 + 4x + 2(7x) + 1
6 + 1 + 4x + 14x
= 7 + 18x

Example 2:
m + 2n + m + 3n
m + m + 2n + 3n
= 2m + 5n
Distributive Property with GCF
42 + 35
^
^
Example:
-Pull out GCF →
𝟕
𝟕
42 + 35
^
^
-Find other factor →
-Rewrite as a multiplication problem by using the
𝟕·𝟔
𝟕·𝟓
GCF as the coefficient with the remaining factors
42 + 35
^
^
combined inside parentheses.
𝟕·𝟔
𝟕·𝟓
=
7(6 + 5) or 7(11)
Lesson 5-10 Showing how tables, graphs, diagrams and equations relate
Double Number Line- a pair of number lines that show how two variables relate to each other.
Vary- to change in value
-How To Plot Values In A Double Number Line
Example: If you buy potato salad at a deli, your cost is related to the amount you buy. The more you
buy, the greater your cost. The amount and the cost vary together. In other words, as the amount of potato
salad changes, the cost changes.
p is the number of pounds of potato salad
c is the cost in dollars
One number line represents p and the other represents c.
c=0
dollars
pounds
p = 0 lb
pounds
c=1
c=2
c=3
c=4
c=5
p = ¼ lb p = ½ lb p = ¾ lb p = 1 lb p = ?
Lesson 5-11 Motion at a Constant Speed
Dependent Variable- a value that depends on another value. Example: The cost of gas depends on the
number of gallons purchased.
Independent Variable- a value that has influence on another. Example: The number of gallons purchased
affects the cost of gas.
- Plot Values in a Double Line Graph and Write Equations for Constant Speed:
Example: Suppose a student walks at a constant rate of 7 feet every 2 seconds.
-To label a double line graph you need to find how far they walk every second by dividing.
7 ÷ 2 = 3½ , then add 3½ for every second they walk on the double line graph.
3½
7
10 ½
14
Equation: d = 3.5 t
Lesson 5-12 Relating
tables,graphs,equations
#1 An online music store is offering a special #6 On the grid, plot the
on music. In order to receive the special the
customer will have to pay a one-time fee of
1. Read situation
$8. After the fee each CD will cost $5.
2. Study the completed row in the table.
CD
Cost
One
Total
3. Use the information to determine the
purchased
for
the
time
Cost
$t
unit rate row. Fill in.
c
CD
($)
fee
4. Determine the pattern to fill in
1
remaining table rows.
5
8
13
#3
values in the table for c
and t. Draw a line through
these points. Show the
unit rate triangles.
#7 Explain how the table,
the graph, and the equation
represent the cost of each
5. Use bottom row to write an equation
CD.
2
10
8
18
representing the total cost for any
-Table- $5 per CD 1:5
#2
3
15
8
23
number of CDs.
-Graph- Unit rate triangle
4
20
8
28
6. Follow directions for graph.
-Equation- 5c
7. Gives explanation for cost of each CD.
C
5c
8
5c + 8
#8 Explain how the graph
8. Gives explanation for one time special
fee.
#5 Write an equation to represent the total and the equation represent
*Solving Equations: 5c + 8 = t
cost in dollars for t and for c CDs. 5c + 8 = t the one-time special fee.
-Graph- line exits at the 8
5 (1) + 8 = t
-Equation- +8
5 + 8 = 13
Lesson 5-13: Writing Equations and Comparing Equations in ‘Real World Situations’

Company A charges $30 for 24 bracelets, plus a flat rate of $4 for shipping any
number of bracelets.
$1.25
Unit Price = 24 ) 30
Variable b = number of bracelets
Equation: 1.25b + 4 = total cost

Company B charges $13 for 10 bracelets, and does not charge for shipping.
$1.30
Unit Price = 10 ) 13
Variable b = number of bracelets
Equation: 1.30b = total cost
If 20 bracelets are purchased from each company, which company has the better total cost?
Company A
1.25b + 4
1.25 (20) + 4
25.00 + 4
$29.00 = total cost
Company B
1.30b
1.30 (20)
$26.00 = total cost
*Company B has the better total cost
Lesson 5-14: Inequalities
Inequality- a statement that compares two expressions using one of these symbols:
> greater than
> greater than or equal to
Example: 4 + 7 > 10
< less than
< less than or equal to
≠ not equal to
Solution of an Inequality- a value that can be substituted for the variable in an inequality to make a true
statement.
Example: x = 10 is a solution of x + 3 < 20 because x + 3 < 20 is true.
Statements as Inequalities


3 is greater than or equal to t
60 is less than 8 times 8
3>t
60 < 8 ∙ 8
Inequalities using Words


4 ∙ m < 16
7∙7>7+7
4 times m is less than or equal to 16
7 times 7 is greater than 7 plus 7
Real World Inequalities


Children younger than 3 years old are admitted to the amusement park for free. (age) a < 3
You must be at least 50 inches tall to ride the roller coaster. (height) h > 50
Graphing Solutions (> and <)
Solution Set- the set of all solutions of an inequality. Most inequalities have an infinite (endless) number of
solutions.
Graphing Solutions (> and <)
Lesson 5-15: Solutions of Equations and Inequalities
Solution- a value of the variable that makes the equation true.
Solving- finding the solution.
Solutions
Solving for Making Sides Equal
Lesson 5-16: Addition and Subtraction Equations
Inverse Operations: operations that undo (cancel) each other.
Addition and Subtraction are inverse operations
x + 9 = 20
-9 -9
x = 11
x – 7 = 10
+ 7 +7
x = 17
62 + d = 89
- 62
- 62
d = $27
Lesson 5-17: Multiplication and Division Equations
Multiplicative Inverse: the product of a number and its multiplicative inverse is 1.
9
1
∙
1
9
9
=9=1
3
4
4
12
∙ 3 = 12 = 1
Inverse Operations: operations that undo (cancel) each other.
Multiplication and Division are inverse operations
2x = 18
x ÷ 2 = 12
2
2
∙2 ∙2
x=9
x = 24
12p = 276
12
12
p = 23 people