Review Guide Math

User Manual: Pdf

Open the PDF directly: View PDF PDF.
Page Count: 22

1
ACT Math Concepts to Know
Just Numbers
Terms
?
Concept
Explanation
Example/Visual
Undefined
An expression is undefined when the
denominator equals zero.
If f(x) =
, for what value of x would this
function be undefined?
Answer: c, because c - c = 0
Imaginary number
To take the square root of a negative
number, first take the square root of
the number as if it were positive, then
add “i”
i2 = -1
Solve:   
1) Square both sides 3x = -6
2) Divide both sides by 3 x = -2
Integers
Whole numbers, including negative
numbers and zero
-2, 0, 3, 7
Rational/ Irrational number
Rational – Can be expressed as a decimal or
fraction
Irrational – Cannot be expressed as a
decimal or fraction
Rational - .45, ¾
Irrational -   
Adding and subtracting negative
numbers
Adding a positive and a negative –
1) Ignore the signs
2) Subtract the smaller number from the
larger number
3) Put on the sign from the larger number
Subtracting negative numbers –
1) Change the subtraction into addition
-38 + 25
1) 38 - 25 = 13
2) -38 is larger than 25, so the answer is -13
13 - (-23) =
13 + (+23) = 36
Multiplying and dividing with
negative numbers
1) Ignore the signs and do the problem
without signs
2) If there is an odd number of negative
signs, add a negative sign to the final
answer
-3 x -5 x -2
1) 3 x 5 x 2 = 30
2) Odd number of negative signs -30
2
PEMDAS/ Order of operations
1) Parentheses
2) Exponents
3) Multiplication and Division
4) Addition and Subtraction
Absolute value
A number inside an absolute value sign
becomes positive
1) Do what's inside the absolute value sign
first
2) Make the result positive
|2 - 5| =
|-3| = 3
The number of integers from one
number to another
Subtract the two numbers and then add 1
We need to add 1 to include the first
number
How many integers are there from 12 to 25?
25 - 12 = 13
13 + 1 = 14
Divisibility
?
Concept
Explanation
Factors and Multiples
Factor – A number that divides into an
integer with no remainder
Multiple – A number that the integer
divides into with no remainder
Factors of 20:
Prime number
The only factors are 1 and the number.
Note: 1 is NOT a prime number
Prime Factorization
Keep breaking a number into factors until
all the factors are prime
Prime factorization of 48:
3
Least Common Multiple
The lowest number that is a multiple of
both numbers
1) Find multiples of the larger number until
you get to one that is also a multiple of the
smaller number.
The LCM of 12 and 15:
15, 30, 45, 60
60 is also a multiple of 12
The LCM is 60
Greatest Common Factor
The highest number that is a factor of both
numbers
1) Figure it out mentally
OR
1) Break each number into its prime factors
2) Multiple the prime factors they have in
common
The GCM of 16 and 24
Prime factorization of 16:
2 x 2 x 2 x 2
Prime factorization of 24:
2 x 2 x 2 x 3
16 and 24 have 2 x 2 x 2 in common
The GCM is 2 x 2 x 2 = 8
Even or Odd?
To find out if an answer will be even or odd,
just plug in simple numbers like 1 and 2
Will 2x3 + 1 be odd or even?
x = 1 2 + 1 = 3
x = 2 16 + 1 = 17
2x3 + 1 will always be odd
Is a number divisible by 2, 3, 4, 5,
9, and 10?
Divisible by…
2 – If the last digit is even
3 – If the sum of the digits is divisible by 3
4 – If the last two digits are divisible by 4
5 – If the last digit is 5 or 0
9 – If the sum of the digits is divisible by 9
10 – If the last digit is 0
36 the last digit, 6, is even
357 3 + 5 + 7 = 15 (15 is divisible by 3)
524 24 is divisible by 4
55 the last digit is 5
396 3 + 9 + 6 = 18 (18 is divisible by 9)
730 the last digit is 0
Remainder
The whole number left over after division
14/4 = 3 Remainder 2
Fractions and Decimals
?
Concept
Explanation
Reducing fractions
Cancel out all the factors that the
numerator (top number) and the
denominator (bottom number) have in
common
Adding and subtracting fractions
1) Find a common denominator
2) Add or subtract the numerators
4
Multiplying fractions
1) Multiply the numerators
2) Multiply the denominators
Dividing fractions
1) Flip the second fraction
2) Multiple the two fractions
Changing a mixed number to an
improper fraction
1) Multiply the whole number by the
denominator, then add the numerator
2) Put the number from #1 over the same
denominator
Changing an improper fraction to
a mixed number
1) Divide the denominator into the
numerator to get a whole number and a
remainder
2) The whole number remains a whole
number, and the remainder is the
numerator. The denominator stays the
same

Reciprocal
Flip the numerator and the denominator
Which fraction is greater?
1) Convert both fractions so they have a
common denominator
OR
2) Convert both fractions to decimals
or
 
 or

is greater
Converting fractions to decimals
Divide the top number by the bottom
number
Finding a particular digit in a
repeating decimal
1) Which digit are you trying to find?
2) How many digits are repeating?
3) Find the multiple of #2 that is closest to
the digit you are trying to find, and then
count up or down to find the digit you are
looking for
1) To find the 101th digit of

Percent formula
1) Change the percent into a decimal
(divide it by 100)
2) Change “of” into a multiplication sign,
and Change “is/are” into an equal sign
3) Complete the problem
70% of 50 is what?
5
Percent increase and decrease
Percent increase –
1) Convert the percent to a decimal, add 1
2) Multiply
Percent decrease –
1) Convert the percent to a decimal
2) Multiply
3) Check if you need to subtract your
answer from the original
$120 increased by 25%
1) 25% = .25
.25 + 1 = 1.25
1.25 x $120 = $150
$120 decreased by 25%
1) 25% = .25
2) .25 x $120 = $30
3) $120 - $30 = $90
Multiple increases and decreases
1) Start with 100 and then apply the
increases and decreases
A price is increased by 20% and then the new
price is decreased by 30%. What is the net
change?
20% increase 100 x 1.20 = 120
30% decrease 120 x .30 = 36
120 - 36 = 84
Net change 100 - 84 = 16
16% decrease
Ratios, Proportions, and Rates
?
Concept
Explanation
Setting up a ratio
1) Put the number after “of” on top
2) Put the number after “to” on bottom
3) Reduce
What is the ratio of 3 cats to 5 dogs?
Solving a proportion
1) Cross multiply

Solving rate problems
Rate x Time = Distance
Average Rate = Total Distance / Total Time
A car travels 294 miles in 6 hours. What is the
rate at which it is traveling?
6
Averages
?
Concept
Explanation
Average formula
Average = Sum of terms / Number of terms
Average of evenly spaced
numbers
Just average the smallest and largest
numbers
Using the average to find the sum
Sum = Average x Number of terms
Then, subtract the numbers you already
have to find the answer
Jim’s average score after four tests is 88. What
score on the fifth test would bring Martin’s
average up to exactly 90?
Counting the possibilities
Multiply the number of choices for the first
thing by the number of choices for the
second thing
John has 5 different shirts and 7 different pairs
of pants. How many different combinations of
shirts and pants can he have?
Probability
Number of items / Total number of items =
Probability/Percentage
In a group of 30 students, 12 are male. What
percentage of the group is male?
Roots
?
Concept
Explanation
Example/Visual
Simplifying square roots
1) Factor out the perfect squares
2) Put the square root of the perfect
square(s) in front of the radical

  
Adding and subtracting roots
If the number under the radical is the
same, you can add or subtract them
   
7
Multiplying and dividing roots
You can multiply two different roots by
first multiplying the numbers under the
roots
You can divide two different roots by first
dividing the numbers under the roots
  
 
Matrices – Adding and
subtracting
Simply add or subtract the spaces that
correspond to each other
8
Algebra
Algebraic Expressions
?
Concept
Explanation
Multiplying and Dividing Powers
To multiply powers with the same base:
add the exponents
To divide powers with the same base,
subtract the exponents
  
Raising powers to powers
Multiply the exponents
  
Evaluating an algebraic expression
Plug in the values for the unknown
If f(x) = x3 - x2 + x, what is the value of f(-2)?
Adding and subtracting algebraic
expressions
Add and subtract like terms
2x + 3x = 5x
Multiplying monomials (one term
by one term)
Multiply the coefficients and the variables
separately
Multiplying binomials (two terms
by two terms) using FOIL
In this order, multiply the:
1) First terms
2) Outside terms
3) Inside terms
4) Last terms
(2x + 2)(x – 2) =
Factoring Algebraic Expressions
?
Concept
Explanation
Factoring out a common divisor
If all the terms have a common factor, it
can be factored out
2x2 - 8x =
Factoring the difference of
squares
The ACT likes to test this.
x2 - (number)2 = (x - number)(x + number)
x2 - 9 =
9
Factoring the square of a binomial
(a + b)2 or (a - b)2
The ACT likes to test this
If the last number is a perfect square, check
if the algebraic expression is a square of a
binomial
x2 + 8x + 16 =
(x + 4)2
Factoring other algebraic
expressions
Think about what binomials you could use
FOIL on to result with the algebraic
expression
1) What first terms could get you the
squared term?
2) What last terms could get you the
number?
3) What combinations of first and last
terms could get you the middle term?
6x2 - 16x + 8 =
1) 3 x 2 or 6 x 1
2) 4 x 2 or 8 x 1
3) (3 x 4) + (2 x 2) = 16
(3x2 - 2)(2x2 - 4)
Simplifying an algebraic fraction
1) Factor the numerator and denominator
2) Cancel out factors that are in both the
numerator and denominator
 
   
  
   
  
Solving Equations
?
Concept
Explanation
Example/Visual
Solving a linear equation
1) Add and subtract terms to get the x
terms on one side
2) Divide (or multiply) to solve for x
2x + 6 = 5x
* Subtract 2x from both sides
6 = 3x
2 = x
Solving “in terms of”
To solve for one variable in terms of
another, do the same thing above for
the variable you are solving for. The
other side will have the variable you are
solving in terms of.
Solve for x in terms of y: 2x - y - 4 = y
* add y to both sides
2x – 4 = 2y
* divide both sides by 2
x – 2 = y
10
Translating from English into
algebra
1) Break the word problem into parts
2) Write out the algebraic expression for
the different parts and then put them
together according to what the problem
is asking for
The toll for driving a segment of a certain freeway is
$1.20 plus 20 cents for each mile traveled. John paid
a $25.00 toll for driving a segment of the freeway.
How many miles did he travel?
1.20 + .20(m) = 25
.20(m) = 23.80
m = 119 miles
Solving a quadratic equation
1) Factor the algebraic expression
2) Find out what values will make the
expression equal zero
OR
1) Use the quadratic formula
   

x2 + 5x + 6 = 0
(x + 2)(x + 3) = 0
x = -2 or -3
   
 


   
Solving a system of equations
Combine the equations in a way that one
of the variables cancels out
Elimination Substitution
Solving an equation that has
absolute value signs
There will be two different answers: one
that results in a positive number and one
that results in a negative number
11
Solving an inequality
1) Solve for the variable as if it is a linear
equation
2) If you multiply or divide by a negative
number, flip the inequality sign
Graphic inequalities on a
number line
Use a solid circle if the point is included
and an open circle of the point is not
included
Graphing inequalities on a grid
1) Choose a coordinate to plug into the
equation
2) If the coordinate makes the inequality
true, shade in that side of the line/curve.
If not, shade in the other side.
12
Coordinate geometry
?
Concept
Explanation
Example/Visual
Finding the distance between two
points
1) Use the Pythagorean Theorem
(x1 - x2)2 + (y1 - y2)2 = c2
OR
2) Use special right triangles
Finding the midpoint
Using two points to find the slope
Slope = Change in y / Change in x
13
A line perpendicular to the slope
Take the reciprocal of the slope and change
the sign
Using an equation to find the
slope
Use the slope-intercept form
y = mx + b
m is the slope, b is the y-intercept
Using an equation to find the
intercept
To find the y-intercept: Plug in 0 for x and
solve for y
To find the x-intercept: Plug in 0 for y and
solve for x
Equation for a circle
(x - h)2 + (y - k)2 = r2
r = radius
(h,k) is the center of the circle
14
Equation for a parabola
y = ax2 + bx + c
Factored: (x + 1)(x - 3)
Equation for an ellipse
x2/a2 + y2/b2 = 1
a = x-intercepts
b = y-intercepts
15
Plane Geometry
Lines and Angles
?
Concept
Explanation
Example/Visual
Intersecting lines
When two lines intersect:
Adjacent angles, or angles next to each
other, are supplementary and add up
to 180 degrees
Vertical angles, or angles across from
each other, are equal
∠a + ∠b = 180°
∠b = ∠d
Parallel lines and transversal (a
line that crosses through parallel
lines)
Forms four equal acute angles and four
equal obtuse angles
Triangles – General
?
Concept
Explanation
Example/Visual
Interior angles of a
triangle
The three angles add up to 180
degrees
16
Exterior angle of a
triangle
An exterior angle equals the
sum of the two angles it is not
next to
The three exterior angles of a
triangle add up to 360 degrees
∠a + ∠b + ∠c = 360° ∠a = ∠C + ∠D
Similar triangles
Have the same shape, but different
size
Corresponding angles are
equal
Corresponding sides are
proportional
Area of a triangle
Area = ½(Base)(Height)
For non-right triangles, the
height is inside the triangle
Right Triangles
?
Concept
Explanation
Example/Visual
Pythagorean Theorem
(leg1)2 + (leg2)2 = (hypotenuse)2
17
Special right triangles
3-4-5 (and multiples)
5-12-13 (and multiples)
30-60-90 – 1::2
45-45-90 – 1:1:
Other Polygons
?
Concept
Explanation
Example/Visual
Rectangle
Has 4 right angles
Opposite sides are equal
Diagonals are equal
Area = Length x Width
Parallelogram
Has 2 pairs of parallel sides
Opposite sides are equal
Opposite angles are equal
Angles next to each other add up to
180 degrees
Area = Base x Height
Square
Rectangle with 4 equal dies
Area = (Side)2
Trapezoid
One pair of parallel sides
One pair of non-parallel sides
Area = 
 
18
Interior angles of a polygon

 
o n is the number of sides
for a polygon with 8 sides
Circles
?
Concept
Explanation
Example/Visual
Circumference (outside) of a circle
Circumference = 2r
Area of a circle
Area = r2
Length of an arc (piece of the
circumference)
Arc = n/360 * (2r)
n = the arc’s central angle
19
Area of a sector (piece of the area
of a circle)
Sector = n/360 x (r2)
n = the sector’s central angle
Solids
?
Concept
Explanation
Example/Visual
Surface area of a rectangular
solid
Volume of a rectangular solid
Surface area = 2(lw + wh + lh)
Volume = lwh
Volume of a cylinder
Volume of a cylinder = r2H
Volume of a cone
Volume of a cone = 1/3r2H
20
Volume of a sphere
Surface area of a sphere
Volume of a sphere = 4/3r3
Surface area of a sphere = 4r2
21
Trigonometry
?
Concept
Explanation
Example/Visual
Sine, Cosine, and Tangent
of acute angles
SOH-CAH-TOA
Sine = Opposite/Hypotenuse
Cosine = Adjacent/Hypotenuse
Tangent = Opposite/Adjacent
Cotangent, Secant, and
Cosecant of acute angles
These are the reciprocals of Sine, Cosine,
and Tangent
Cotangent = 1/Tangent =
Adjacent/Opposite
Secant = 1/Cosine = Hypotenuse/Adjacent
Cosecant = 1/Sine = Hypotenuse/Opposite
Trigonometric functions of
other angles
For angles greater than 90 degrees
1) Draw a circle with radius 1 centered on
the coordinate grid
2) Rotate the appropriate number of
degrees counterclockwise
3) Draw an acute triangle depending on
where the angle lands
4) Find the answer
Simplifying trigonometric
expressions
sin2x + cos2x = 1
Also, use the trigonometric properties to
simplify
22
Graphic trigonometric
functions
x-axis angle
y-axis value of the trigonometric
function
Use special angles like 0, 30, 45, 60, 90,
etc. to plot key points
Radians
π = 180°
2π = 360°
π = 180°
π/2 = 90°

Navigation menu