7th Grade Review
The Number System
Rational Number: Any number that can be written as a fraction
Integer : positive, negative whole numbers and zero
Whole Number : { 0, 1,2,3,….. }
Counting Number : { 1, 2, 3, 4, …..}
Perfect Squares: { 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144…..}
Square Roots: √25 = 5 √36 = 6 − √49 = −7
√60 is between 7 and 8
Absolute Value: The distance on the number line from zero
|−5 | = 5 | 6 | = 6
− | 4 | = −4 ± | 8 | = ±8
Adding Integers: Same signs add and keep the sign, different signs
subtract and take the sign of the larger number. -3 + -6 = -9
-2 + 7 = 5
Subtracting Integers: Keep Change Change (only when needed)
And follow rules for addition
-5 – 9 = -14 don’t need
-6 – (-4) = - 6 + 4 = -2
Multiplying and Dividing Integers: Same signs answer is positive,
different signs the answer is negative
Combining fractions(add or subtract): Need common denominators
Multiplying Fractions: Fractions must be improper
Dividing Fractions: Fractions must be improper and then
Keep Change Flip
1
1
−7
9
−7 4
−14
−3 ÷ 2
÷
 =
2
** remember
4
−3
4
2
=
3
−4
4
= −
2
3
9
9
***
4
Complex fraction: fractions within the fraction(just means division)
2
9
1
6
2
=
9
÷
Additive Identity : a + 0 = a
Additive Inverse : a + (-a) = 0
Multiplicative Identity: a  1 = a
1
Multiplicative Inverse: a  = 1
𝑎
1
2
6
9

6
1
=
4
3
Expressions and Equations
Exponents: 43 = 4 4 4
64
( -3 )2 = (-3) (-3 )
9
60 = 1
- 32 = -1( 3)(3)
-9
Variable : a letter that represents a number
Constant : a number
Coefficient: usually a number that is being multiplied by a variable(s)
Expression: a math sentence with no equal sign
3x2 + 4
3 is the coefficient
x is the variable
2 is the exponent
4 is the constant
(you evaluate an expression) let x = -5
3x2 + 4
3(-5)2 + 4
3(25) + 4
75 + 4
79
Follow PEMDAS(Dorito!!)
Equation: a math sentence with an equal sign (you solve an equation)
3x + 7 = 12
Polynomials: a sum of monomials
Types of Polynomials:
monomials- a one term expression(not allowed to have a variable in
the denominator)
5x3y5
binomial: a two term expression
5x3y5 – 6xy7
trinomial: a three term expression 5x3y5 – 6xy7 + 12z
Like terms: math terms that have exactly the same variables and
exponents (you simplify by combining like terms)
3x – 5 + 2y – 6x + 10
-3x + 5 + 2y
Distributive Property:
Factor (using GCF):
Inequalities:
3( x – 6 ) = 3x – 18
4x + 24 = 4( x + 6)
less than <
greater than >
at least ≥
at most ≤
** remember when solving an inequality be careful.. if you multiply
or divide by a negative number you must FLIP the symbol !!! **
Graphing solutions of inequalities:remember open circle vs closed circle
x ≥ 4
x<4
Ratio and proportional relationships
Ratio: Comparison of numbers
Proportion: when two ratios are equivalent.
To solve a proportion equation, you cross multiply
𝑥+3
6
=
𝑥
5
Don’t forget to distribute
so 5(x + 3) = 6x
Unit rate: is the ratio of two measurements in which the
second term is 1
5
1
6
6
I drive of a mile in of an hr. What is my unit rate?
You can write a proportion or just divide
So the unit rate is 5 miles/hr
5
6
5
÷
6
𝑥
6
1
1
6
=5
Better Buy: Find the unit rate of each item and the better buy is the
cheaper price
A graph of a line will show a proportional relationship(that it varies
directly) if the line goes through the origin (0,0). The equation for the
line would be y = mx where m is the constant of proportionality (or the
unit rate, or the constant of variation.)
On the graph of a line that shows a proportional relationship the unit
rate can be found at the point (1, unit rate). The unit rate is found in
𝑦
the graph shown as ( 1, 20 ) so the unit rate is 20/1 (remember ). The
𝑥
equation for that graph is y = 20x
Percents
Percent of a number:
Finding a percent:
6.35% of 24 = .0635 x 24
𝑖𝑠
𝑜𝑓
=
%
𝑝𝑎𝑟𝑡
100
𝑤ℎ𝑜𝑙𝑒
=
%
100
Discount : an amount you save
Sale Price: The price after it is discounted
There are two ways to find discount
30% off of $200
This is the discount
200 x .30 = 60 then 200 - 60 = 140
OR
This is the sale price
you need to get 70% of the price 200 x .70 = 140
Sales Tax: an amount that you add on to a price
7.5% tax on $45
There are two ways to find the price plus tax
This is the tax
45 x .075 = 3.375 then 45 + 3.375 = 48.375 which is $48.38
OR
45 x 1.075 = 48.375 which is $48.38
This is the price plus tax
Commission: amount of money you make on sales of a product
Sell $2,500 worth of shoes and make a 3% commission
2500 x .03 = $75
This is how much you earn
Gratuities(Tip): amount you live a person of service
A restaurant bill is $50 and you leave a 20% tip
50 x .2 = 10 so you would leave a $10 tip
Simple Interest: The amount of money you pay or earn depending on
the principle(amount of money) time(in years) and the
rate(percentage)
time must be in years
You borrow $600 at a 3.25% interest rate for 3 months (3/12 of a yr)
I = PRT
You owe $4.88 in interest so you must
=(600)(.0325)(.25)
pay back a total of $604.88
= 4.875
Percent Increase or Decrease: Find the percent of change
The price went for $24 to $30. What is the percent of markup?
𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒
𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙
=
𝑝𝑒𝑟𝑐𝑒𝑛𝑡
100
so
6
24
=
𝑥
100
cross multiply
x = 25 so there is a 25% increase
Percent of error: Find the percent of error in a measurement
𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒
𝑝𝑒𝑟𝑐𝑒𝑛𝑡
=
𝑎𝑐𝑡𝑢𝑎𝑙
100
I took the temperature outside to be 50 degrees but it was
actually 45 degrees. What was my percent of error?
5
45
=
𝑥
100
cross multiply x = 11
1
9
1
so the percent of error was 11 %
9
Geometry
Perimeter of any shape: add all the sides
Area of any parallelogram = (base)(height)
1
Area of any triangle= (base)(height)
2
Area of a trapezoid=
1
2
(base1 + base2)(height)
Area of a circle = πr2
r = radius
Circumference of a circle = 2πr
or
πd
d= diameter
If the Circumference is 16π, what is the Area?
So C = πd
16π=πd so d = 16 and so the radius = 8
A = πr2
= π(8)2
= 64π
Statistics
1) Statistics: deal with collecting, organizing and interpreting data
2) Survey: a method of collecting information
3) Data: The information that is gathered
4) Population: The entire Group
5) Census: A survey of the entire population
6) Bias: an unfair survey/sample
7) Unbiased sample: a survey/sample that is selected so that it accurately represents the entire
population
8) Simple Random Sample: Each item or person in the population is equally likely to be
chosen
(pick a name out of a hat)
9) Systematic Random Sample: The items or people are selected according to a specific time
(every 5th person on a list)
10) Convenience Sample: consists of members of a population that are easily accessed
11) Voluntary Response Sample: involves only those who want to participate
12) Measures of Central Tendency: Statistics that involves those around the center
(mean,median,mode)
13) Mean: Average of the data
Median: the number that is in the middle of the data
Mode: the number that occurs the most
14) Five Number Summary: Minimum, Lower Quartile, Median, Upper Quartile, Maximum
Minimum: smallest number in the data set Maximum: largest number in the data set
Lower Quartile: The median of the lower half of data
Upper Quartile: The median of the upper half of the data
15) Variation: The spread of the data
16) Measures of Variation: Statistics that involves the variation (Range, Interquartile Range
and Mean Absolute Deviation)
17) Range: The difference between the largest and the smallest
Interquartile Range: The difference between the upper and lower quartiles
Mean Absolute Deviation: find the mean, find the difference (absolute value) between the
mean and every piece of data and then find the mean of those numbers
15) Outlier: a value that is really large or really small compared to the rest of the data
Probability
Complement of an Event: All outcomes that are NOT the event
P( rain) = .4 P(not rain) = .6
P(snow) =
2
5
P(not snow) =
3
5
Theoretical Probability: Is based on theory. shows what should happen
in an experiment.
Experimental probability shows what actually happened.
Independent Events : Two events are said to be independent if the
result of the second event is not affected by the result of the first
event.
If A and B are independent events,
P(A and B) = P(A) • P(B).
Ex:
Two dice are tossed. What is the probability of getting an even then a 2
3
1
3
P( even, 2) = 𝑥 =
6
6
36
Dependent Events : Two events are said to be dependent if the second
event is affected by the results of the first event.
Ex: Two numbers are picked from 0-9 without replacement. What is
the probability the first is even and the second is even.
{ 0,2,4,6,8}
P( even, even) =
5
10
𝑥
4
9
=
20
90
Counting Principle - If a first event has n outcomes and a second event
has m outcomes, then the first event followed by the second event has n
times m outcomes.
m X n = mn
If Laquisha can enter school by any one of three doors
and the school has two staircases to the second floor,
in how many different ways can Laquisha reach a room on the second floor?
3 x 2 = 6 ways