7th GRADE UNIT 2: EQUATIONS
LESSON 1: Order of Operations and the Distributive Property
LESSON 2: Exponents and the Order of Operations
LESSON 3: Evaluating and Writing Algebraic Expressions
LESSON 4: Simplify Expressions
LESSON 5: Solving One-Step Equations
LESSON 6: Exploring Two-Step Problems
LESSON 7: Solving Two-Step Equations
LESSON 8: Solving Equations Involving the Distributive Property
LESSON 9: Review
LESSON 10: Test
LESSON 1: Order of Operations and the Distributive Property
OBJECTIVE: To use the order of operations and the distributive property
ORDER OF OPERATIONS:
Work ___________ grouping symbols.
1.)
2.)
Examples: Find the value of each expression.
a.) 30 ÷ 3 + 2 ∙ 6
b.) 30 ÷ (3 + 2) ∙ 6
d.)
 40
 25
4
e.)
84
 11
6
c.) 7(-4 + 2) – 1
f.) 9 + 3 ∙ (5 – 1)
Example: You want to buy 3 cans of soup at $0.89 each and 2 boxes of crackers at $1.59 each.
What is the total cost?
Distributive Property (uses two operations)
Algebra:
Examples:
6(12) = 6(10) + 6(2)
15(1 + 9) = 15(1) + 15 (9)
6(x + 3) = 6x + 18
Examples: Use the distributive property to find the following products.
a.) 7(59)
b.) 7(52)
c.) 9(14)
Examples: Apply the distributive property.
a.) 4(x – 6)
b.) 6(3 + y)
c.) -2(2m – 3)
d.) (-5 – x)6
f.) 8(x + 4)
e.) 5(-3m – 1)
Example:
LESSON 2: Exponents and Order of Operations
OBJECTIVE: To write and simplify expressions with exponents.
You can use EXPONENTS to show _______________ multiplication.
A POWER has two parts, a BASE and an EXPONENT.
An __________ tells you how many times a number, or _______ is used as a factor, or repeated.
5  5  5  5  125
3
READ as “Five to the third” or “Five cubed”
Examples: Write each product using exponents.
a.) -2 ∙ a ∙ b ∙ a ∙ a
b.) (-2) ∙ (-2) ∙ (-2)
c.) 4 ∙ x ∙ y ∙ y ∙ y ∙ 4
d.) (-4) ∙ (-4) ∙ (-4) ∙ (-4) ∙ y ∙ y
f.) -5 ∙ n ∙ n ∙ m ∙ m ∙ n
e.) 6 ∙ 6 ∙ 6 ∙ 6
Examples: Simplify each expression.
a.) 35
b.) 109x2
c.) -24
d.) (-2)4
e.) (-3)3xy3
f.) -33
ORDER OF OPERATIONS:
1.)
2.)
3.)
4.)
Examples: Find the value of each expression.
a.) -24 + (3 – 5)4
b.) 23 ∙ (9 – 3)2
c.) (3 + 5)2 – 2
d.) 4(10 – 8)5
f.) (4 – 1)3 – 3 x 8 ÷ 6
g.)
 9  6 .2
2 .1  2 .9
e.) 2(32 + 12)
h.)
8.2  16.3
4 .5
i.)
75
63
LESSON 3: Evaluating and Writing Algebraic Expressions
OBJECTIVE: To write and evaluate algebraic expressions.
A ____________ is a symbol that represents one or more numbers. Variables are usually letters.
An _______________ _________________ is a mathematical phrase with at least one variable.
KEY WORDS IN ALGEBRA
Operation
Addition
Subtraction
Multiplication
Division
Grouping ( )
Verbal Phrase
Expression
a) The sum of five and a number m
a)
b) Four more than a number m
b)
c) A number m plus eight
c)
d) A number x increased by nine
d)
a) The difference of five and a number x
a)
b) Four less than a number y
b)
c) A number y minus eight
c)
d) A number s decreased by nine
d)
a) The product of five and a number s
a)
b) Four times a number n
b)
c) A number n multiplied by two
c)
a) The quotient of a number z and three
a)
b) Six divided by a number z
b)
a) Three more than the quantity five times a number
z
a)
b) A number x decreased by the sum of ten and the
square of a number y
c) Four times the quantity three plus a number x
b)
c)
d)
d) The quantity of three less than a number x
divided by two
THAN means to __________ positions.
Examples: Write an algebraic expression for each word phrase.
a.) Swimming m meters per minute for 3 minutes.
b.) 12 heartbeats more than x
heartbeats.
c.) Price p decreased by 16.
d.) 20 books divided among s students.
e.) The sum of s students and 9 students.
g.) 6 less than d dollars.
f.) 12 times b boxes.
h.) Dinner bill d dollars divided by 5 friends.
i.) The cost of a package of markers is d dollars.
each.
What is the total cost in dollars of 7 packages of
Markers.
Examples: Write three word phrases for each expression
a.) x + 2
b.) 2y
j.) Nine students will hang t posters
c.) m - 50
Examples: Write an algebraic expression for each word phrase.
a.)11 times the difference of 38 and a number j.
b.) 8 more than the quotient of 5 and a
number e.
c.) A third of the difference of a number a and 14.
d.) 12 plus 35 times a number n.
e.) 4 plus a number f increased by 22.
number n.
f.) 3 more than the difference of 5 and a
You can ______________ for a variable to EVALUATE an algebraic expression.
Examples: Evaluate each expression. Use the variables p = 2, n = 3, and s = 5.
a.) 2p + 7
b.) p + (n ∙ s)
Examples: Evaluate each expression. Use the variables r = 8, s = 1, and t = 3.
r
a.) 6(t – 1)
b.)
st
LESSON 4: Simplify Expressions
OBJECTIVE: To simplify algebraic expressions using properties of operations.
3
BELL RINGER: Evaluate 4b a when a = 6 and b = 3.
2ab
___________ _____________ are terms that have the same variable factors and the same
exponents
Examples:
Non-Examples:
A ______________ is a numerical factor of a term with a variable.
Examples:
Examples: Simplify the following expressions.
a.) 5x + 9 + 2x – 4
b.) 6 + 7.2y – 4.2y + 1
Examples: Simplify each expression.
2
a.) 12 y  9   4
b.) 8 – 2(6s + 1)
3
c.) k – 2 + 7 – 9k
c.) ½ (2 – 8v) + 5
d.) 9 – 4(3z + 2)
e.) 6(2x + 3) – 4
f.) 8 – 3(7 – 2a)
Use the Greatest Common Factor (GCF) to factor expressions. You can use the distributive property
to check your Factoring.
Examples: Factor each expression.
a.) 4x + 14
b.) 18 – 24m
c.) 12y – 6
d.) 9x + 15
f.) 8c - 20
e.) 36 + 24t
LESSON 5: Solving One-Step Equations
OBJECTIVE: To solve equations by adding, subtracting, multiplication and division.
Day 1: Solving Equations by Adding or Subtracting
BELL RINGER: Factor each expression.
a.) 36n – 72
b.) 30 + 65y
PROPERTIES OF EQUALITY:
ADDITION PROPERTY OF EQUALITY:
If you ______ the same value to ______ side of an equation, the two sides remain ________.
Example:
Algebra:
SUBTRACTION PROPERTY OF EQUALITY:
If you _____________ the same value to _______ side of an equation, the two sides remain ______.
Example:
Algebra:
To SOLVE an equation, you want to get the variable __________ on ____________ of the equation.
You can use _______________________, operations that undo each other, to get the variable alone.
___________ and _______________ are ______________ _______________. You can use
Addition to ______ Subtraction.
Examples: Solve the following equations.
a.) x – 104 = 64
b.) x – 34 = -46
c.) d – (-126) = 98
d.) 9 + c = -10
e.) y + 3.14 = 11.89
f.) t – 4.83 = 13.12
Example: A hardcover book costs $19 more than its paperback edition. The hardcover book costs
$26.95. How much does the paperback cost?
Example: Your friend’s mountain bike costs $245 more than his skateboard. His mountain bike costs
$290. How much did your friend’s skateboard cost?
Example: Describe a problem situation that matches the equation s – 286 = 74. Then solve the
equation.
Example: Describe a problem situation that matches the equation n + 41 = 157. Then solve the
equation.
Examples: Fill in the missing number to solve each equation.
a.) b + 12 = 39
b.) y – 8 = 35
b + 12 – ___ = 39 – ___
y – 8 + ___ = 35 + ___
Example: Annie and Michael tried to solve the equation x + 4 = -9. Who solve the equation correctly?
Explain.
Annie: x + 4 = - 9
Michael: x + 4 = -9
+4 +4
-4 -4
x=-5
x = -13
DAY 2: Solving Equations by Multiplying or Dividing
BELL RINGER: Solve each equation.
a.) 7 + a = 46
b.) b – (-10) = -4
c.) 104 + d = 75
MULTIPLICATION PROPERTY OF EQUALITY:
If you ___________ each side of an equation by the _______number, the two sides remain ______.
Example:
Algebra:
DIVISION PROPERTY OF EQUALITY:
If you ________ each side of an equation by the ________ nonzero number, the two sides remain
_________.
Example:
Algebra:
MULTIPLICATION and __________ are __________ _________________. You can use
Multiplication to ______ Division.
Examples: Solve the following equations.
a.) 4p = 22.68
b.) -3j = 44.7
c.) 3x = -21.6
d.) -12y = -108
f.) 2c = -92
e.) 104x = 312
Example: The Art Club must buy 84 pieces of poster board. There are 6 pieces in a package. How
many packages must the Art Club buy?
Example: Suppose you and four friends go to a baseball game. The total cost for five tickets is $110.
Write and solve an equation to find the cost of one ticket.
Example: Your cell phone bill shows that you were charged an extra $8.58 this month for going over
your allotted minutes. The company charges $0.39 for each extra minute. How many extra minutes
did you use?
Examples: Solve the following equations.
t
a.)
 5
 45
c.)
m
 27
3
b.)
w
 15
26
d.)
y
8
12
Example: Find the mistake. Then solve the equation and find the correct answer.
x
 3
9
x
9
  3  9
9
x  27
LESSON 6: Exploring Two-Step Problems
OBJECTIVE: To write and evaluate expressions with two operations and to solve two-step equations
using number sense.
BELL RINGER: Solve each equation.
n
a.) -35 = -7b
b.)  13
5
c.) d + 9 = -12
d.) f – 8 = 15
Example: Suppose you are ordering roses online. Roses cost $5 each, and shipping costs $10. Your
total cost depends on how many roses you buy. What two operations are involved?
Example: Define a variable and write an algebraic expression for the phrase $10 plus $5 times the
number of roses ordered? Evaluate the expression for 12 roses.
Example: Define a variable and write an algebraic expression for “a man is two years younger than
three times his son’s age.” Evaluate the expression to find the man’s age if his son is 13.
Example: To rent a bicycle, you pay a $12 basic fee plus $2 per hour. Write an expression for the
total cost in dollars of a bicycle rental. Then evaluate the expression for an 8-hour bicycle rental.
Examples: Solve the following equations using number sense.
a.) 5g + 8 = -48
d.)
n
 20  31
3
g
42
6
b.) 9k – 4 = 14
c.)
e.) 4y – 11 = 33
f.) 3m + 9 = 21
Example: Suppose you buy a jumbo lemonade for $1.50 and divide the cost of an order of chicken
wings between you and a friend. Your share of the total bill is $5.50. Write and solve an equation to
find the cost of the chicken wings?
LESSON 7: Solving Two-Step Equations
OBJECTIVE: To solve two-step equations using inverse operations.
BELL RINGER: The Trometter family wants to buy a DVD player that costs $200. They have $80
saved. How much will they have to save per month for six months in order to have the whole cost
saved?
For many equations, you can undo addition or subtraction ________. Then you can multiply or divide
to get the variable ________.
Examples: Solve the following equations.
a.) 5n – 18 = -33
d.)
a
 4  10
5
b.)
n
 ( 11)  16
3
e.) -8y – 28 = -36
c.) 6r + 19 = 43
f.)
x
 35  75
5
Example: At the library sale, there was a $2.50 admission charge. Old paperbacks were on sale for
$0.25 each. If you only bought paperbacks and you spent $4.50 in all, how many paperbacks did you
buy?
Example: On weekday afternoons, a local bowling alley offers a special. Each bowling game costs
$2.50, and shoe rental is $2.00. You spend $14.50 total. What is the number of games that you
bowl?
Example: Find the mistake. Then solve the equation to find the correct answer.
2x – 6 = 8
-6 -6
2x = 2
x=1
Examples: Solve each word problem.
LESSON 8: Solving Equations Involving the Distributive Property
OBJECTIVE: To solve equations of the form p(x + q) = r using the distributive property.
BELL RINGER: Solve each equation.
a.) 3t – 4 = 44
b.)
x
 6  1
5
DISTRIBUTIVE PROPERTY:
When you multiply a factor by a sum (or difference), the product is the same as multiplying the factors
by each term and then adding (or subtracting) the products.
Algebra:
Examples:
Example: Suppose you have 42 feet of fencing to enclose your garden. The width of the garden is
8.2 feet. What length must the garden be to use all of the fencing?
Example: What is the width of a rectangle with a length of 17 centimeters and a perimeter of 58
centimeters?
Example:
Examples: Solve each equation.
a.) 10(a – 6) = -25
b.) -4.5 = -3(b + 15)
c.) 8½(c – 16) =340
d.) -2(j – 12) = 18
e.) 4(5h + 9) = 86
f.) 3(15 – m) = 12