Math 1 Review Sheet (Topics for Final Exam)

Order of Operations	Number Properties
PEMDAS (Please Excuse My Dear	Commutative: a + b = b + a , ab = ba
Aunt Sally)	Associative: a + (b + c) = (a + b) + c
P – Parenthesis (and other grouping symbols)	a(bc) = (ab)c
E – Exponents	Distributive: a(b + c) = ab + ac
MD – Multiplication and Division (Left to Right)	When there is a subtraction sign before a
AS – Addition and Subtraction (Left to Right)	parenthesis, you MUST distribute the negative.
	Additive Identity: a + 0 = a
Solving Equations	Multiplicative Identity (property of 1): a · 1 = a
	Additive Inverse: a + (-a) = 0
1) Simplify (Use distributive property, eliminate	Multiplicative Inverse: a · (1/a) = 1
denominators, combine like terms on either side.)	Zero Property (Multiplication): a · 0 = 0
2) Move Variable Terms to One Side (use
Addition/subtraction to “move” terms)	Word Problems
3) Move Constant Terms to Other Side
4) Isolate Variable (Use Multiplication/Division	Always define the variable: Let x =
to “undo” coefficient)
	Is means =
Square Roots	Decreased by means –
	Increased by means +
Principal square root is the positive square root.	Two times means 2x
	Half of means 0.5x or ½ x
There is also a negative square root. If asked for	Squared means x^2
± square root, give both positive and negative	Less than means –
roots.	Result is means =
Sets of Numbers	Consecutive Integers increase by 1
	Let x = 1^st CI
Natural Numbers (Counting): {1, 2, 3, …}	Let x + 1 = 2^nd CI (etc.)
Whole Numbers: {0, 1, 2, …}
Integers: {…, -2, -1, 0, 1, 2, …}	Consecutive Odd/Even Integers increase by 2
*When graphing the above, only graph the points	Let x = 1^st Cons. Even (or Odd) Integer
(no fraction or decimal parts!!!)	Let x + 2 = 2^nd Cons. Even (or Odd) Integer
Rational Numbers: Any number which can be	Probability
written as a fraction in the form a/b where b ≠ 0
(includes repeating decimals)	P(E) = # desired outcomes / # possible outcomes
Irrational Numbers: non-repeating, non-	P(impossible event) = 0 , P(certainty) = 1
terminating decimals. Ex: π = 3.14159……	0 ≤ P(E) ≤ 1
and square roots of non-perfect squares.
Real Numbers: all rational and irrational #s.	Tree Diagram – Shows all possible outcomes
*When graphing the above, darken in the line. Use	Counting Principal – Tells how many outcomes.
an open circle for < and >, use a closed circle for
≤ and ≥	*Watch for questions with/without replacement.
Statistics	NEW: P (A and B) = P(A) • P(B)
Line Plots: Draw number line, use x-s
Stem-Leaf Plots: Leaves must be 1 digit. *KEY	Frequency Tables: Can measure both regular and
	Cumulative (total to that point) frequency.
Measures of Central Tendency
Mean: Add all values and divide by # of values.	Histogram: Can measure regular or cumulative
Median: Middle value when numbers are in order,	frequency. Use horizontal axis for intervals,
[(# of values + 1) • 0.5] tells place where median	vertical axis for frequencies.
occurs
Mode: Most frequently occurring.	*NO space between bars, show break in axis if
	necessary. Use straightedge. Label graph!!!
Measures of Variation
Range: Distance from least to greatest	Use formula to find percentiles/quartiles:
Quartiles: Divides data into fourths.	(# of values + 1) • (decimal of percentile)
Lower (Q1) = 25^th % ile, Median (Q2) = 50^th % ile,	Ex: (40 + 1) • .4 = place where the 40^th % ile is.
Upper (Q3) = 75^th % ile.
	Box and Whisker Plot: Draw number line.
Inter-quartile Range (IQR): Distance between	Find Q1, Q2, Q3, Hi Value, Lo value, any outliers.
Q1 and Q3. *Magic Number = 1.5 • IQR	Boxes from Q1 to Q3, Whiskers out to Hi and Lo.
	(outliers do not count as Hi or Lo)
Outliers will be “The Magic Number” above Q3,
or “The Magic Number” below Q1.	*Can construct multiple plots on one number line*
Algebraic Expressions	Inequalities
Monomial: One term, a constant, variable, or product	Graphing on a number line
of a constant and variables. Ex: -3ab^2
	Greater Than, Less Than: Open Circle.
Degree of a monomial: Sum of the exponents on the
variables. Ex: 4a^2b^3c (2 + 3 + 1 = 6) Degree is 6.	Greater Than or Equal To, Less Than or Equal
	To: Closed Circle.
Adding or Subtracting monomials: Combine
ONLY Like Terms (same variables to same power)	AND inequalities: Graphed between 2 points
Remember, combining apples and apples still leave you with apples.	(except ex: greater than 5, less than 2…no such #)
Multiplying Monomials: a^n • a^m = a^n + m	OR inequalities: Usually graphed in opposite directions.
	(either one can be true for the whole thing to be true)
Power to Power: (a^n)^m = a^n • m
	Solving
Dividing Monomials: a^n / a^m = a^n - m	Just like solving equations BUT if you multiply or
	divide by a negative number, you MUST change
*Any number to the zero power (a^0) = 1	the direction of the inequality.
*If denominator = 0, then the number is UNDEFINED
	Percents, Ratios, Proportions
Polynomials – The sum (or difference) of two or more
monomials.	Part / Whole = % / 100
Binomial: 2 Terms Trinomial: 3 Terms	Part / Whole = Decimal Part
Degree: The degree a polynomial is the same as the	Proportion: 2 equal ratios. Cross Multiply to solve.
term with the highest degree.
	Direct Variation: If x and y vary directly, they can
Standard Form: Descending order of degree.	be put into a proportion.
Adding Polynomials: Add ONLY like terms.	Pythagorean Theorem: a^2 + b^2 = c^2
Parenthesis can be ignored.	Used to solve for a missing side in a RIGHT
	triangle. c is ALWAYS the hypotenuse.
Subtracting Polynomials: Distribute subtraction
across ENTIRE polynomial.	Simplifying Radicals:
	√a • b = √a • √b
Multiplying a Polynomial by a Monomial:
Distribute (remember multiplication rules)	Factor base into at least 1 perfect square
	Ex: √ 72 = √36 • 2 = √36 •√2 = 6√2
Binomial times Binomial: F O I L
First, Outer, Inner, Last (remember to combine outer	*Radicals can only be combined (added and
and inner)	subtracted) if they are like (same bases)
Dividing Polynomials by a Monomial: Divide	TIP – Use your notes and old tests/quizzes
EACH TERM by the monomial.	To find practice problems for yourself.
Geometry
	Angle Properties
Angles: Acute – Less than 90^o, Obtuse – Greater than
90^o, Right – Equal to 90^o, Straight – Equal to 180^o	Supplementary Angles – Add up to 180o
	Complementary Angles – Add up to 90o
Perpendicular lines form right angles. Straight lines
Contain 180^o	Interior Angles of a Triangle – Add up to 180^o
	Interior Angles of a Quadrilateral – Add up to 360^o
* Angles formed by intersecting lines OR Parallel
lines cut by a transversal. (see diagram)
*Angles will either be Supplementary or Congruent.
(both acute, they will be congruent. Both obtuse, they
will be congruent. If one is acute and one is obtuse,
they MUST be supplementary.)	Quadrilaterals
Triangles: Area = ½ bh	Parallelograms: Area = bh
Equilateral – All sides equal	Opposite angles are congruent, opposite sides too.
Isosceles – 2 sides are equal, 2 angles are equal.
	Rhombus: Parallelogram with all sides equal.
Circle: Area = πr^2, Circumference = πd or 2πr
Diameter is double the radius.	Trapezoid: Area = ½ (b1 + b2)h
Area of shaded portion = Area (big) – Area (small)	Square and Rectangle: Area = lw (or s^2 for square)
	*Perimeter of all figures = sum of all sides.
Scientific Notation – used for very big or very	Graphing Review
small numbers.
	Coordinate Axes: Label x and y…use pencil
2.3 x 10^7 = 23,000,000
2.3 x 10^-7 = .00000023	Quadrants (see back)
Factoring Polynomials (reverse of multiplying)	Slope (m): Rise over Run
	(see back for formulas and examples)
Try to factor the GCF first (opposite of distributing)
2x^3 + 4x^2 – 6x = 2x(x^2 +2x – 3)	*A line represents all points (x and y values) which
	make an equation true.
Try to factor into binomials next (opposite of FOIL)
2x(x^2 +2x – 3) = 2x(x + 3)(x – 1)	Graphing Lines: Solve for y
	Either:
Always check your work!!!!!	a) Make a table using at least 3 convenient x values.
	OR
Solving Polynomial Equations:	b) Know that y = mx + b form gives you the
1) Set equation equal to zero.	y-intercept (b), and the slope (m)
2) Factor.
3) Set all factors equal to zero.	Once points are plotted, use a ruler to connect them,
4) Solve for each variable.	remember to label the line.
5) CHECK FOR EACH SOLUTION
6) Eliminate unreasonable answers.	x = (some #) ALWAYS A VERTICAL LINE!
	y = (some #) ALWAYS A HORIZONTAL LINE!
Solving Systems of Equations
	Graphing Inequalities on the Coordinate Plane
A. Graphically: Graph each equation. Solution is	1) Graph as though it were an equation.
where the lines intersect. Check that the x and y	*Greater than/Less than – Dotted Line
value make BOTH equations true.	*Greater than or equal to/Less than or equal to - Solid
	2) Shade above the line for greater thans, below for
B. Algebraically: Substitution or Elimination	less thans. (right for greater thans if line is vertical)
Check your notes!!!! If you don’t have notes, you
have no one to blame but yourself!	Systems of Inequalities: Solutions will lie where
	shadings overlap.