Unit 1: Variables, Expressions & Integers
Quiz 1:
1.1 Expressions & Variables
1.2 Powers & Exponents
1.3 Order of Operations
1.4 Compare & Order Integers
1.5 Adding Integers
1.6 Subtracting Integers
1.7 Multiplying & Dividing Integers
Quiz 2:
2.2 Simplify Expressions: Distributive Property
2.3 Simplify Expressions: Combine Like-Terms
Unit 1 Test (all of the above)
Study Guide & Notes for each section are on the following pages.
They provide important vocabulary and key concepts.
Suggestions for studying before a quiz or test:
-Review notes and study guide
-Try practice problems in your textbook or online textbook
-Try practice problems IXL.com (see Ms. Botto’s website for a list of links to specific skills
required for this unit.
-See your math teacher before or after school.
1.1 Expressions & Variables
Expression: numbers, operations, and sometimes variables.
10
8(2)
x–4
5
3x + b
*Multiplication
*Fractions are also division
Equation: numbers, operations, sometimes variables, AND always an equal sign.
8(2) = 16
10
5
x – 4 = 10
=2
3x + b = 15
Variable: a letter to represent one or more numbers.
1.2 Powers & Exponents
53 = 125
Is the same as 5  5  5 = 125
Exponents are used to shorten repeated multiplication of the same number.
base: the number that is multiplied
exponent: how many times the base is multiplied
power: the base AND the exponent (“5 to the 3rd power”)
1.3 Order of Operations
1st Grouping Symbols
Parentheses: ( )
Radical: 
Brackets: [ ]
Absolute value:
Fraction Line (vinculum):
*If there are operations in numerator and/or denominator, solve as a group first, then divide.
Example:
8(2)+4
4+1
=
20
5
= 4
2nd Exponents
-52 = -25
*If there are no parentheses, do NOT drop the negative!
(-52) = 25
*If there are parentheses, multiply the base and its given sign according to the exponent:
(-5)(-5) = 25
3rd Multiply OR Divide **whichever comes first!
10 ÷ 5 (4)
Division comes before
multiplication, so divide FIRST,
then multiply:
2(4) = 8
5(7–3)÷2
First do the math in parentheses.
You would get:
5 (4) ÷ 2
Multiplication comes before
division, so multiply first, then
divide:
20 ÷ 2 = 10
𝟐𝟒
𝟑
●2
Division comes before
multiplication, so divide
FIRST, then multiply:
8 ● 2 = 16
LAST: Subtract OR Add **whichever comes first!
Save the easiest operations for last 
8–3+2
12 + 4 – 5
Subtraction comes before addition,
so subtract FIRST, then add:
5+2=7
Addition comes before subtraction,
so add FIRST, then subtract:
16 – 5 = 11
1.4 Comparing & Ordering Integers
Integers: all of the counting numbers, their opposites & zero
Absolute Value: a numbers distance from zero
-5 = 5
5 =5
Opposites: the same distance from zero, but on the opposite side of the number line.
The opposite of -5 is 5.
The opposite of 5 is -5.
 LESS- Left 
 MORE - Right 
1.5 – 1.7 Integers: Adding, Subtracting, Multiplying & Dividing
Adding Integers Rules
“Same signs keep and add…”
+
Positive
+
+
Positive
Add, answer is positive
–
Negative
+
–
Negative
Add, answer is negative
“…different signs subtract, take the sign of the higher number, then you’ll be exact.”
+
Positive
+
–
Negative
–
Negative
+
+
Positive
Subtract, answer takes the sign of the greater
absolute value
Subtract, answer takes the sign of the
greater absolute value
Examples:
10 + -5 = 5
Subtract: 10 – 5 = 5
Absolute values are 10 and 5
10 is greater, its sign is positive,
so the answer is positive
–8 + 3 = -5
Subtract: 8 – 3 = 5
Absolute values are 8 and 3
8 is greater, its sign is negative, so
the answer is negative
Subtracting Integers Rules
1) Insert an addition operation immediately after the first integer:
5 – 10 becomes: 5 + (– 10)
– 8 – 5 becomes: – 8 + (– 5)
Subtracting Negatives:
Subtracting a negative is the same as adding a positive.
Think of it as rearranging the 2 dashes to make a positive sign. (– – becomes + )
6 – -3 becomes: 6 + 3
-7 – (-4) becomes: -7 + 4
2) Follow the adding integers rule to solve.
*Sometimes parentheses are used to make negative signs easier to see.
Finding Change in Temperature or Elevation:
The word “change” implies subtraction.
Subtract the integers.
IF it was a drop in temperature or elevation, then the final answer is NEGATIVE.
If it was a rise in temperature or elevation, then the final answer is POSITIVE.
Multiplying & Dividing Integers Rules
-------------------------------------------------------------------------------------------------------------------------If the signs are the same, the product/quotient will be positive
+
Positive
and
+
Positive
Product/Quotient is positive
–
Negative
and
–
Negative
Product/Quotient is positive
---------------------------------------------------------------------------------------------------------------------------If the signs are different, the product/quotient will be negative
+
Positive
–
Negative
and
and
–
Negative
Product/Quotient is negative
+
Positive
Product/Quotient is negative
2.3 Combining Like-Terms
To simplify expressions, we can combine like-terms.
4x2 + 2 + 3x2 – 1 + x
Terms: parts of an expression; includes constants, variables & coefficients.
*In above expression, the terms are: 4x2, 2, 3x2, -1, and x
Constants: a number that has no variable
*In above expression, the constants are: 2 and -1
Variables: letter that represents an unknown amount.
*In above expression, the variables are: x2 and x
Coefficients: the number that multiplies a variable.
*In above expression, the coefficients are: 4, 3 and 1
(1 because there is “one” x, even if 1 is not written there.)
Like-Terms: the terms in an expression that “match”.
*In the expression above, the like-terms are:
4x2 and 3x2; 2 and -1 (x does not have another like-term.)
To simplify the above expression; combine the like-terms:
4x2 + 3x2 = 7x2
2 + -1 = 1
7x2 + 1 + x
2.2 Distributive Property
2(5x -3)
*Multiply the number outside parentheses by ALL TERMS inside the parentheses:
2(5x) and 2(-3) = 10x – 6
(4 + 7y) 3
*The multiplier can be before OR after the parentheses.
3(4) and 3(7y) = 12 + 21y
**Write expressions in ORDER, alphabetical, exponent value, followed by constants:
Final answer: 21y + 12
**It is NOT a multiplier if there is an operation between the number and parentheses, such
as (4 + 7y) + 3
More examples of Writing Expressions in CORRECT ORDER:
5x2 – 2x + 7
3x2 + 8x – 7 y + 2
-(3x + 2)
*If there is just a negative outside the parentheses, you can read it as “take the opposite
of all terms inside the parentheses”:
-3x – 2
*This is the same as multiplying -1 to all terms in the parentheses:
-1(3x) and (-1)(2) = -3x -2
-2x(6x + 3)
*Multiplying the same variables results in using exponents:
-2x(6x) = -12x2
-2x(3) = -6x
Answer: -12x2 – 6x
**When adding the same variables, use coefficients to show final count:
Examples: x + 5x = 6x
3y – 5y = -2y