- Math Notes
- General Notes
- Fractions
- Ratios and Rates
- Proportions
- Variables and Real Numbers
- Solving Equations and Inequalities
- Exponents and Polynomials
- Factoring
- Measurements
- Geometry
- Graphing
- Systems of Equations and Inequalities
- Rational Expressions
- Radical Expressions and Quadratic Equations
- Functions
- 2: Check if the last number is divisible by 2.
- 3: Add each individual number and check if it's divisible by 3.
- This can be repeated
- 4: Check if the last two digits are divisible by 4.
- 5: Check if the last number is divisible by 5.
- 7: Take off and multiply the last number by 2. Subtract it from the
remaining digits. If it's 0 or a number divisible by 7, it is. Repeat if the
number is too large, until there's an easy to tell number.
- 203
- Take off 3 and multiply by 2 = 6
- Subtract 6 from 20 = 14
- 20 is not divisible by 7, so it's not divisible by 7.
- 8: If the last three digits are divisible by 8, then it is.
- If the first digit of the 3-digit number is even, then the entire number is divisible by 8 if the last two digits are divisible by 8.
- If the first digit of the number is odd, then subtract the number 4 from the last two digits and check to see if this new number is divisible by 8. If it is, then the entire number is too.
- If a number is divisible by the factors numbers that make it up,
then it is also divisible by that number
- 6: If it's divisible by both 2 and 3, it is.
- 9: If it's divisible by 3, it is. You can also add the numbers and see if it's divisible by 3.
Reciprocal: A number that, when multiplied by another number, equals 1.
- The reciprocal of 3/4 is 4/3
- Keep - Keep the first fraction the same
- Change - Change the division symbol to multiplication
- Flip - Flip the second fraction to get the reciprocal
- Ratio: compares two quantities by division, or describes the relationship between two quantities
- Rate: A ratio that compares quantities that are measured in different units
- Unit Rate: compares a quantity to one unit of measure
- The units must be the same. Convert if needed.
To check if a proportion is true, use cross multiplication.
- If they're equal, it's true.
- Constant: A number or symbol that represents a quantity that cannot change
- Variable: A letter or symbol used to represent a quantity that can change
- Expression: A mathematical phrase that contains operations, numbers, and/or variables.
- Counting/Natural Numbers: Positive numbers only
- Whole Numbers: All the natural numbers + zero
- Integers: All negative and positive numbers
- Absolute Value: A nubmer's distance from zero, regardless of its sign
- Displayed by using pipes one eacn side:
- |-3| and |3| are both the same: 3
- Displayed by using pipes one eacn side:
- Real Numbers: Contain all integers (negative and positive). Consists of
two sets:
- Rational Numbers
- Numbers that can be written as the ratio of two integers, where the
denominator is not zero.
- Integers
- Whole numbers
- Natural numbers
- A terminating or repeating decimal is a rational number.
- Numbers that can be written as the ratio of two integers, where the
denominator is not zero.
- Irrational Numbers
- Numbers that cannot be written as the ratio of two integers
- Irrational numbers do not repeat and are non-terminating decimals.
- Numbers that do not have a perfect square are also irrational.
- Rational Numbers
When adding real numbers, first change all the signs to be the same.
- Add their absolute values
- Assign same sign to sum
- Find the difference of the absolute values
- The sum gets the same sign as the number with the larger absolute value
The identity property of 0: Adding 0 to other numbers doesn't change their value.
Additive Inverses: Two numbers whose sum is 0.
When subtracting a negative number, use its additive inverse instead.
- Additive Identity: 0
- Multiplicative Identity: 1
- You can multiply any number by this without changing it.
- Multiplicative Inverse (Reciprocal): Two numbers that when multiplied equal the multiplicative identity.
Keep, Change, Flip
- Multiply the top numbers
- Multiply the bottom numbers
- Simplify
First get all the constants (The things that we know that are not going to change), then create the variable (The thing that you're trying to find out.)
The variable disappears on both sides, and you don't get a solution. Instead, you get either a True or False statement.
- If it's True, then there's infinite solutions.
- If it's False, then there are no solutions.
x + 5 = 5 + x5 = 5- This is true since 5 does equal 5.
The perimeter of Tina’s rectangular garden is 60 feet. If the length of the garden is twice the width, what are the dimensions of the garden?
2X = Y
4X + 2Y = 60X + X + X + X + 2Y = 60X + X + X + X + X + X = 606X = 60X = 10Y = 20
Inequality
a mathematical statement that shows the relationship between two expressions where one expression can be greater than or less than the other expression
- Inequalities can be solved similar to equations in that you can do the same thing to both sides.
- To check if an inequality is correct, substitute the solution value into the
inequality to check if it holds up.
- Check the endpoint of a Greater/Less than (>/<) by substituting the end point, and changing it to an equals (=) sign. Then check the point after and keep the sign the same.
- When multiplying by a negative number, remember to flip both sides.
-
When you multiply or divide both sides by a negative number, flip the inequality sign
-
To check answers, check the endpoint in the related equation, and check the inequality by checking one or more solutions of the inequality.
- Simplify
- Distributive property (Eliminate parentheses)
- Combine like-terms
- Move variable terms to one side (usually left side)
- Isolate variable term on one side by moving the numbers to the other side
- Isolate variable by multiplying or dividing
- Remember to flip the sign if multiplying or dividing by negatives.
two inequality statements linked together either by "and" or "or".
You can split a compound inequality into two and solve both, or solve as is:
7 < 3X + 4 < 10
- Split into two:
7 < 3X + 4and3X +4 < 10
- Solution: values that make both individual inequalities true
- Many and statements can be written as a 3-part statement:
X > 0 and x < 5->0 < x < 5 - Solve three-part statements by applying the properties of inequality to all three parts
- Solution: values that make either of the individual inequalities true
- Or statements cannot be written as a single statement
Exponential notation has a Base and an Exponent.
- The base is the number or variable, and the exponent is the exponent.
Any number (other than zero) raised to the power of 0 is one.
Any number with a negative exponent is the reciprocal of the number with a positive exponent.
- With positive exponents, you multiply by the base.
- With negative exponents, you divide by the base (as long as it isn't zero).
- Take the reciprocal and then attach the positive exponent to it.
- When multiplying variables in exponential notation, as long as they have the same base, you can add the exponents.
- When dividing variables in exponential notation, as long as they have the same base, you can subtract the exponents.
- To simplify a power of a power, you multiply the exponents, keeping the same base
(ab)x = ax * bx
- (3x)3 = 33 * x3 = 27x3
- When multiplying, the quotient applies to all values in the parenthesis.
(x/y)3 = (x/y) (x/y)(x/y) = x3 /y3
- When dividing, you apply it both the numerator and the denominator
Scientific Notation is a way to write very big or very small numbers so that they're easy to read.
Scientific notation consists of two parts:
- a * 10n
amust be greater than or equal to 1, but less than 101 <= a < 10
- No fractions or decimals
The exponent counts how many times the decimal moves left or right, not the amount of zero's.
- For Positive Exponents: If you have a number such as 5.5510,
then there will be
exponent - 2zero's after it: 55,500,000,000 - For Negative Exponents: If you have a number such as
5.55-10, then there will be
exponent - 1zero's before it: .000000000555
- 5.28 * 102 = 528
- 3 * 10-3 = .003
- (4.5 x 105) * (5.28 * 10-7)
- (4.5 * 5.28)(105 * 10-7)
- 23.76
If the number is higher than 10, move the decimal over and increase/decrease the exponent to account for it.
15x3
- 15 is the co-efficient
- x is the variable
- 3 is the exponent
- The entire thing is the term
Different Types:
- 15x: Monomial - One term
- 15x + 12h: Binomial - Two terms
- 15x + 12h + 10x: Polynomial - A monomial or sum of monomials
Polynomial terms cannot have:
- Negative exponents
- Fraction exponents
- Variables in the denominator
- Combine like-terms (terms that have the same variables with the same
exponents)
- Add the co-efficients and keep the variables the same
A rectangular picture frame has a length of 3a + 5 inches and a width of a + 2 inches. Find the perimeter of the picture frame in inches.
- Add the lengths and widths:
(3a + 5) + (a + 2) + (3a + 5) + (a + 2) - Regroup using the commutative and associative properties:
(3a + 3a + a + a) + (5 + 5 + 2 + 2) - Combine like terms to simplify:
8a + 14
It helps to FOIL when multiplying polynomials.
- F irst
- O uter
- I Inner
- L Last
(a + b)(c + d) = ac + ad + bc + bd
There are three special cases:
- Square of a Sum
- Square of a Difference
- Product of a Sum and a Difference
- (a + b)2 = (a + b)(a + b) = a2 + 2ab + b2
- (a - b)2 = (a - b)(a - b) = a2 - 2ab + b2
- (a+b)(a - b) = a2 - ab + ab - b2 =
a2 - b2
- The final formula is using subtraction, not multiplication.
These special cases are useful, because the final answers above will always be a formula you can quickly apply to get the answers.
- The sign is not taken into account as part of either a or b.
- Divide the coefficients
- Divide the variables
- The monomials are rewritten from negative exponents to fractions (the reciprocal)
- Factor out the coefficient
- Factor out the variables
If you cannot factor out a polynomial, use grouping to separate the polynomial into groups. Then factor out the individual groups.
- An example is below.
If ab = 0 then a or b, or both = 0.
- If one of the sides does not have a zero, just change both sides so that it does.
randsare used as examples in finding the integers to use for this setup.- The integers will always have a product of
cand a sum ofb.
- The integers will always have a product of
In setups where you cannot factor out the coefficient, this is the difference:
- Some trinomials cannot be factored.
- Sum of Cubes: a3 + b3 = (a + b)(a2 - ab + b2
- Difference of Cubes: a3 - b3 = (a - b)(a2 + ab + b2)
In the US Customary Measurement System, the units are:
- Mile - 5280 feet
- Yard - 3 feet
- Foot - 12 inches
- Inch
In order to convert units, multiply by a conversion factor.
- A conversion factor is the number 1 expressed as a ratio of two equivalent units.
- When converting, put the new units over the old units.
- The old units cancel out.
The three steps to multiplying and converting are:
- Find equivalent (12 inches = 1 foot)
- Put into ratio, new unit/old unit
- Cancel out units and multiply
In the US Customary Measurement System, the units are:
- Tons - 2000 pounds
- Pounds - 16 ounces
- Ounces
In the US Customary Measurement System, the units are:
- Gallon - 4 Quarts
- Quart - 2 Pints
- Pint - 2 Cups
- Cup - 8 Fluid Ounces
- Fluid Ounce
Metric System: A system of measurement based on 10s used in most countries outside the US, and in the US as well.
- Plane: 2-Dimensional object that continues on forever
- A plane is defined by 3 points.
- Line: 1-Dimensional object that only has length and stretches in both
directions forever.
- A line is defined by 2 points
- Often written alphabetically
- The common end point is the vertex.
- The angles are:
- Acute < 90
- Right = 90
- Obtuse > 90
- Straight = 180
You can classify triangles by both their angle and their sides.
- Equilateral: All sides are even length
- Isosceles: Two sides are even length
- Scalene: No sides are even length
The sum of the measure of all 3 angles of a triangle is 180 degrees.
- Congruent Triangles: Two triangles that have the same length sides and the same angles.
- Similar Triangles: Two triangles that have the same angles, but different
length sides.
- They are proportional
- Polygons: Closed plane figures with three or more straight sides
- They are not allowed to cross over eachother
- Triangles: Polygons with three sides
- Quadrilaterals: Polygons with four sides
- The internal angles add up to 360 degrees.
- Any quadrilateral can be divided into two triangles
Has two pairs of parallel sides
- The opposite angles are congruent
- The equals sign with the tilde above it is the sign for congruent.
- Rectangles, squares and rhombuses are parallelograms
An example of a quadrilateral that's not a parallelogram is a Trapezoid.
Area is expressed in squared units: units2
With parallelograms, the area is B * W where B is base (left - right) and W is width (up - down).)
- Radius: Distance from the center to the circle.
- Diameter: A line that passes through the circle with endpoints on the circle.
- Circumference: Distance around a circle
- Pie: ~= 3.14 or 22/7
- The Origin is the point where the two axes meet.
- The quadrants start with the top right one as Quadrant 1 and count counter-clockwise.
- Ordered Pair: (x, y)
- Linear Equation: An equation with two variables, and whose ordered pairs graph as a straight line.
- Intercepts: The points where the line meets the axes.
- Rise: The vertical movement
- Run: The horizontal movement
- Slope is used to describe and compare steepness. It is the ratio between
the horizontal and vertical movements.
- Slope/m = rise/run = y2 - y1/x2 - x1
- When calculating Rise and Run, you get the distance between two integer points on the graph, where it is on an integer (not a decimal) for both the X and Y axes.
- When the Run is left-to-right, it's positive. When it's right-to-left, it's negative.
y = mx + b
If you don't know the intercept but have the slope and the coordinates for a single pair, then:
- Substitute the coordinate and slope into the formula
- Solve for b (The Y intercept)
If you don't know the slope or intercepts, but have two points:
- Find the slope using the two points
- Substitute the slope and one of the coordinates into the formula
- Solve for b (The Y intercept)
- Parallel: Two lines have the same slope, but different intercepts
- Never meet
- Parallel: Two lines have opposite reciprocal slopes
- y = 5/3x + 5 and y = -3/5 - 2
- Meet at 90 degrees
When a problem cannot be put into a simple equation, we put it two or more linear equations that share two or more unknowns.
A System of Equations:
- Two or more equations that share two or more unknowns
- Solution must satisfy all the equations in the system
One method of solving a system of linear equations is graphing. It's not always the best method, but it is a way to tell if there even is a value that satisfies both equations.
- Graph out each linear equation
- This is easiest when you solve for the intercepts for each equation, and then graph those.
- Find the point where the lines intercept. This is the answer.
- Substitute the values into the equation and check that it's true
Substitution is another way to solve a system of linear equations, without graphing.
- Isolate a variable in one equation
- Substitute that variable's expression into the other equation.
- Solve for that variable
- Substitute this value back in to find the remaining variable
When you get a normal result back indicating the value of the variable.
If you get a false statement such as 0 = 1, then that means that there are
no solutions.
- A parallel line
If you get a True statement for all values of a variable, it means that there are an infinite number of solutions, because they're the same line.
- Also called the addition method.
- Optional: Multiply one of the equations by -1
- Optional: Multiply or divide one side so that it has an equal amount of at least one variable as the other equation.
- Add the two equations once you have opposite terms that will eliminate a variable.
- Substitute that variable's value into the other equation to get the answer.
A fraction that has a polynomial for either a numerator, denominator, or both is called a Rational Expression.
- Rational here refers to the fact that it's a ratio.
The first step in dealing with rational expressions is to find the Domain.
Domain: All possible values for the variables.
- This means finding and excluding the values that make the denominator equal to zero, since you can't divide by zero.
- Find the domain of each denominator
- Simplify
- Multiply
- Find the domain of each denominator
- Multiply by the reciprocal of the divisor
- Find the domain of each denominator
- Simplify
- Multiply
To find the least common denominator for denominators with no common factors, it will be the product of the two denominators.
- Same denominator: add or subtract numerators and keep the denominator.
- Different denominator: rewrite to have a common denominator.
- Multiply one or more rational expressions by 1 in a form that gives a common denominator.
- Try to find the least common denominator.
- Always define the domain
- Rewrite complex rational expressions that are stacked to be horizontal rational expressions instead.
- Find the domains
- Perform the multiplication or division steps
Rational equations are often used to solve work problems, such as how long it will take to do something.
- Check for excluded values in the domain.
- Find the common denominator (LCD is best).
- Multiply by 1 in a form to get common denominators then rewrite the equation
with just the numerators.
- OR Multiply both sides by least common denominator.
- Simplify and solve the remaining equation.
- Check answers in the original equation.
- Check that answers aren't excluded values.
There are three formulas for calculating work problems:
- W = The amount of work done
- r = The rate of work
- t = Time spent working
- W = rt
- t = W/r
- r = W/t
Formulas that are expresses as rational equations are called Rational Formulas.
- They have input / output. The way that the output varies in relation to the input is called Variation.
y = kx
yis the output (what you want to figure out)xis the input, a changing valuekis the constant, a known input
y = k/x
- It's called inverse, because the output changes inversely to the input.
y = kxz
- Similar to direct variation, only that it requires two inputs
(xz)rather than one.
To find the square root of something, break it down into it's factors and then simplify:
- You can also use approximations, down to the tens decimal.
- The Radical symbol is used to indicate square root.
- Don't use a negative sign before a radical sign.
- Factor the term, looking for squares inside the term.
- Rewrite with multiple variables
- Always enclose the end result in the absolute value sign if the value is even.
Index is the little number above the radical, it indicates what kind of root you need to find.
- If there is no index, it's square root.
xmust be a positive integer.
Treat them like you would variables. The variables, indexes, and variables need to be the same.
It can be useful to treat radicals like variables. Rewrite the formula
substituting the radicals with x.
- Use the distributive property or the FOIL method
- Combine like terms
Getting rid of the radical in a denominator is called rationalizing the denominator, and is the first step in solving an equation that has one.
You can multiply a radical with more than just the radical in the denominator by its conjugate:
Always check your answers when dealing with radicals, as there are sometimes extraneous solutions that must be discarded.
- Extraneous solutions are solutions that do not work in the equation.
- Isolate the radical expression
- Square both sides of the equation to remove the radical
- Solve for the unknown
- Check all your answers
- When you get two answers, only one of them will work.
A function is a relation where for any input there is only one output.
y = 2x
- x = number of bicycles = independent variable = Input
- y = number of wheels = dependent variable = Output
- It depends on the number of bicycles that come in
- If there are multiple possible
yvalues, then it is not a function.
For functions, instead of putting y = x + 1, you'd write f(x) = x + 1
- Different letters instead of
fcan be used if there's multiple functions, to avoid confusion. - Given f(x) = x + 1, find (2) just means to find the value if x is 2.
- Substitute the input for x.