7th GRADE UNIT1: Algebraic Expressions and Integers
LESSON 1: Variables and Expressions
LESSON 2: The Order of Operations
LESSON 3: Writing and Evaluating Expressions
LESSON 4: Integers and Absolute Value
LESSON 5: Adding Integers
LESSON 6: Subtracting Integers
LESSON 7: Looking for a Pattern
LESSON 8: Multiplying and Dividing Integers
LESSON 9: The Coordinate Plane
LESSON 10: Review Day
LESSON 11: Test
LESSON 1: Variables and Expressions
OBJECTIVE: To write a variable expression for word phrases.
A VARIABLE
A VARIABLE EXPRESION
A NUMERICAL EXPRESSION
Examples: Identify each expression as a numerical or variable expression. For a variable expression,
name the variable.
a.) 5 – 5
b.) c + 8
c.) 7 x 3
d.) 4t
e.) 8 ÷ y
f.) d + 43 – 9
g.) 6(7) – 3
h.) 10 ÷ 2
KEY WORDS IN ALGEBRA
Operation
Addition
Subtraction
Verbal Phrase
Expression
a) The sum of five and a number
a)
b) Four more than a number
b)
c) A number plus eight
c)
d) A number increased by nine
d)
a) The difference of five and a number
a)
b) Four less than a number
b)
c) A number minus eight
c)
d) A number decreased by nine
d)
Multiplication
Division
Grouping ( ) or [ ]
a) The product of five and a number
a)
b) Four times a number
b)
c) A number multiplied by two
c)
a) The quotient of a number and three
a)
b) Six divided by a number
b)
a) Three more than the quantity five time a number
a)
b) A number x decreased by the sum of ten and the
square of a number y
b)
c) Four times the quantity three plus a number.
d) The quantity of three less than a number divided
two
c)
d)
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.
f.) 12 times b boxes.
g.) 6 less than d dollars.
h.) Dinner bill d dollars divided by 5 friends.
i.) The cost of a package of markers is d dollars.
j.) Nine students will hang t posters each.
Examples: Write an algebraic expression for each word phrase.
a.)11 time 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
e.) 4 plus a number f increased by 22.
d.) 12 plus 35 times a number n.
f.) 3 more than the difference of 5 and a number n.
Example: How many miles can you drive on ten gallons of gas? Write a variable expression for miles
per gallon.
Vehicle Type
Subcompact
Compact
Mid-size Sedan
Sport Utility
Pickup Truck
Miles
330
300
245
175
160
Gallons
10
10
10
10
10
Miles per Gallon
LESSON 2: The Order of Operations
OBJECTIVE: To use the order of operations.
BELL RINGER: Write a variable expression for each word phrase.
a.) the total of h and 56.
b.) three less than d.
c.) p decreased by three.
d.) a divided by 7.
ORDER OF OPERATIONS:
1.)
2.)
3.)
Examples: Simplify each expression.
a.) 4 + 15 ÷ 3
b.) 2 + 5 x 3
c.) 12 ÷ 3 – 1
d.) 3 ∙ 5 – 8 ÷ 4 + 6
f.) 5 + 6 ∙ 4 ÷ 3 – 1
e.) 4 – 1 ∙ 2 + 6 ÷ 3
g.) 24 ÷ [6 – (2 ∙2)]
h.) 20 – 3[(5 + 2) – 1]
i.)
84
 11
6
j.) 1 
Example: Use the order of operations to express the total area of the figure below.
3m
4m
2m
6m
10  2
4
LESSON 3: Writing and Evaluating Expressions
OBJECTIVE: To evaluate variable expressions.
BELL RINGER: Simplify each expression 3[9 ∙ 2 ÷ (10 – 4)].
To EVALUATE a variable expression,
Examples: Evaluate each expression.
a.) 4y – 15, for y = 9
b.) 4(t + 3) + 1, for t = 8
c.) 63 – 5x, for x = 7
d.) [3m + 1] ∙ 2, for m = 5
e.) 2xy – z, for x = 4, y = 3 and z = 1
g.) 3ab 
c
, for a = 2, b = 5, and c = 10
2
f.)
rs
, for r = 13 and s = 11
2
h.) 2ab 
c
, for a = 3, b = 4, and c = 9
3
Example: The store pays $29 for each case of drinks. Write a variable expression for the cost of c
cases. Find the cost of 5 cases.
Example: Energy drinks come in cases of 24 bottles. Write a variable expression for the number of
cases a store should order to get b bottles of energy drink. Evaluate the expression for 120 bottles.
Example: The One Pizza restaurant makes only one kind of pizza, which costs $16. The delivery
charge is $2. Write a variable expression for the cost of having pizzas delivered. Evaluate the
expression to find the cost of having 2 pizzas delivered.
Example: An online music store charges $14 for each CD. Shipping costs $6 per order. Write a
variable expression for the cost of ordering CDs. Find the cost of ordering 8 CDs.
LESSON 4: Integers and Absolute Value
OBJECTIVE: To find opposites and absolute value.
To represent, graph, and order integers.
BELL RINGER: Evaluate 3c + 4 ÷ d, for c = 8 and d = 2.
Example: Seawater freezes at 28⁰F, or about 2 degrees Celsius below zero. Write a number to
represent the Celsius temperature.
THE NUMBER LINE
A number line can be used to compare numbers and arrange them in order.
The numbers ____________ in value from left to right.
To compare numbers use inequality symbols.
less than
less than or equal to
greater than
greater than or equal to
Examples: Compare the following numbers using a number line. Then order the numbers form least
to greatest.
a.) -1, 4, 0.5, -5
b.) 0, 2, -6, -2/3
c.) -2, -3, -4.5
VOCABULARY:
Examples:
INTEGERS-
OPPOSITES-
Find the opposite of:
a.) 7
b.) -16
The ABSOLUTE VALUE of a number is its _________ from __
on the number line.
Determine the absolute value:
Absolute Value is _________ __________.
a.) │9│
Example: Find the opposite of -6 using the number line.
Examples: Find the opposite of each number.
a.) -8
b.) 13
c.) -22
b.) │-5│
Examples: Find the absolute value using a number line.
a.) │-3│
b.) │2│
c.) │-5│
Examples: Order each set of numbers from least to greatest
a.) 3, -1, -4, and 2
b.) -5, 4, │3│, │-6│, and 2
LESSON 5: Adding Integers
OBJECTIVE: To add integers.
BELL RINGER:
a.) Simplify │-12│ and │8│.
b.) Write an integer to represent a debt of $50.
When you add opposites, the sum is _______. Opposites are called ADDITIVE INVERSES.
ARITHMETIC:
1 + (-1) =
-1 + 1 =
ALGEBRA:
x + (-x) =
-x + x =
Example: From the surface, a diver goes down 8 feet and then comes back up 2 feet. Use a number
line to find where the diver is.
Example: On two plays, a football team first gains 3 yards and then loses 8 yards. What is the result
of the two plays.
Examples: Use a number line to find each sum.
a.) 5 + (-4)
b.) -5 + (-2)
c.) -6 + 6
d.) -8 + 1
e.) -1 + (-7)
f.) -3 + (-8)
g.) 3 + 8
h.) 3 + - 8
i.) -3 + 8
RULES FOR ADDING INTEGERS:
SAME SIGN: The sum of two positive integers is __________. The sum of two negative numbers is
___________.
Examples:
3+5=
-3 + (-5) =
DIFFERENT SIGNS:
1.)
2.)
3.)
Examples:
-3 + 5 =
3 + (-5) =
SA DS
Examples: Determine the sum of the following using the Rules for Addition.
a.) 4 + (-7)
b.) -12 + (-6)
c.) -2 + 9
d.) 18 + 6
e.) -11 + 10
f.) -3 + 8
g.) -6 + -7
h.) -3 + 3
i.) 2 + (-1)
LESSON 6: Subtracting Integers
OBJECTIVE: To subtract integers.
BELL RINGER: Determine the sum of each.
a.) -3 + 4
b.) -12 + (-7)
c.) 8 + (-20)
To SUBTRACT an integer, ______ its ____________.
ARITHMETIC:
2–5=
2 – (-5) =
ALGEBRA:
a–b=
a – (-b) =
Examples: Subtract.
a.) -4 – 2
b.) -4 – (-2)
c.) 13 – (-5)
d.) 2 – 48
e.) 6 – (-8)
f.) -15 – (-15)
g.) -40 – 66
h.) 32 – (-3)
i.) -12 – (-1)
Example: The temperature in Caribou, Maine, was 8⁰F at noon. By 10:00 p.m. the temperature had
dropped to -4⁰F. Find the change in the temperatures.
Example: You jump from a cliff 22 feet above sea level and go 8 feet below sea level. What is your
change in altitude?
LESSON 7: Look for a Pattern
OBJECTIVE: To find number patterns.
BELL RINGER: Determine the difference of each.
a.) -3 - 4
b.) -12 - (-7)
c.) 8 - (-20)
Examples: Write a rule for each pattern. Find the next three numbers.
a.) 8, 11, 14, 17, …
b.) 1, 5, 4, 8, 7, …
c.) 3, 5, 10, 12, 24, …
d.) 1, 4, 7, 10, …
Example: News spreads quickly at Selinsgrove Middle School. Each student who hears a story
repeats it 15 minutes later to two students who have not yet heard it and then tells no one else.
Suppose one student hears some news at 8:00 am. How many students will know the news at 9:00
am?
Example: Suppose each student who hears the story repeats it in 10 minutes. How many students
will know the news at 9:00 am?
LESSON 8: Multiplying and Dividing Integers
OBJECTIVE: To multiply and divide integers.
BELL RINGER: Each student on a committee of five students shakes hands with every other
committee member. How many handshakes will there be in all?
Example: A balloon descends at a rate of 4 ft/min for 3 min. To multiply integers, think of
multiplication as repeated addition
3(-4) = (-4) + (-4) + (-4) = - 12
the balloon descends 12 ft.
You can use number lines to multiply integers.
The product or quotient of two integers with the same sign is ______________.
The product or quotient of two integers with different signs is ______________.
The product of 0 and any integer is ____.
Division by zero is ____________.
Examples: 3(4) =
3(-4) =
-3(4) =
-3(-4) =
3(0) =
10/5 =
-10/5 =
10/-5 =
-10/-5 =
0/5 =
5/0 =
Examples: Multiply of Divide the following expressions.
a.) -5 x 3
b.) -3(-4)
c.) 7 ∙ (-2)
d.) 2(-6)(-4)
e.) 0/-6
f.) -24 ÷ 24
g.) 120/-3
h.) -49/0
Examples: Multiply of Divide the following expressions.
a.) 6(-8)
b.) -48/-6
c.) -14(-2)
d.) -32 ÷ 8
e.) -8(0)
f.) 72/-6
g.) 6(-2)(-3)
h.) -3 ∙ 5(-4)
Example: Find the average of 4, -3, -5, 2, and -8.
LESSON 9: The Coordinate Plane
OBJECTIVE: To name and graph points on a coordinate plane.
BELL RINGER: Simplify.
a.) -36 ÷ -9
b.) -6 ∙ (-2) x (-1)
c.) 21(-2)
Coordinate Plane
XY-Plane
A _________________ PLANE is a grid formed by a _____________ number line called the
_________ and a ___________ number line called the ____________.
The axes divide the plane into four ____________.
The ______________ is where the axes intersect.
An ___________ ______ (x, y) gives the location of a point.
The first number of an ordered pair is the ___________________. It tells the number of
_____________ units a point is from the origin.
The second number is the __________________. It tells the number of __________ units a point is
from the origin.
Examples: Name the coordinates of each point.
A
B
C
D
E
F
Examples: Plot the given points on the graph below. In which quadrant does each point lie?
a.) A (7, 2)
b.) B (4, 5)
c.) C (-2, 3)
d.) D (0, 8)
e.) E (-5, -3)
f.) F (3, -2)
g.) G (-2, 0)
h.) H (0, 0)
Example: Graph the points and describe the figure that results: K(3, 1), L(-2½, 1) and M(-2, -4).
Example: Name the coordinates necessary to complete a square with sides 4 units, one vertex at
(6, 6).