Slope (an introduction) - Algebra

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Algebra

Geometry

Trigonometry

Algebra II

Calculus

Statistics

Learn about functions, graphs, lines, and polynomials. "Algebra" is the math for describing how different things are related.

SUBJECT
COMPLETION

Addition and subtraction

Number line

Locating integers and non-integers on the number line

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Addition

Using the number line to add numbers

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Subtraction

Moving left on the number line

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Adding negatives

What happens when you add negative numbers?

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Subtracting negatives

Turns out it's addition!

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Absolute value

The "size" of a number, or how far it is from zero

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Number line distance

Ways to find the distance between any two numbers

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Multiplication and division

Multiplication

Multiplying positive numbers, in any order

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Multiplying by 1 and 0

Discover what happens when you multiply by 1 and 0

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Division

Dividing positive numbers, and ways to think about it

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Dividing by 0

An operation you just can't perform...

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Order of operations

Is there more than one way to evaluate an expression?

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Distributive law

Getting rid of parentheses when you add and multiply

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Multiplying negatives

What happens when you multiply negatives together?

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Dividing negatives

What happens when you divide negative numbers?

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Distributing negatives

Distributing a negative flips the signs

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[Brackets] and {braces}

What to do when you have too many parentheses!

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Finding the average

Discover how to find the "average" of numbers

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Factoring

Factors

Dividing by a factor always gives you a whole number

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Prime numbers

Their only factors are 1 and themselves

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Factoring out

Pulling out common factors using the distributive law

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Factoring out negatives

Flip the signs when you factor out negative numbers

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Greatest common factor

The biggest factor that numbers have in common

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Relatively prime

When two numbers have no common factors

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Euclid's algorithm

A 2000-year-old way to calculate the GCF

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Multiples

What you get when you multiply numbers by integers

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Least common multiple

The smallest multiple numbers have in common

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Fractions

Intro to fractions

Draw and compare fractions, and convert to decimals

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Multiplying fractions

What's 1/2 times 3/5? Learn the trick in this lesson!

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Comparing fractions

How to tell when fractions are equal or which is bigger

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Cancellation

Simplifying fractions by removing common factors

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Adding fractions

Add and subtract fractions using common denominators

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Dividing fractions

Dividing by a fraction is really multiplying by its reciprocal!

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Reciprocals

A number's multiplicative inverse is 1 over the number

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Mixed fractions

Switching between mixed and improper fractions

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Distributing division

The distributive law works for multiplying, but also dividing

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When 'of' means multiply

Evaluating fractions of numbers (like 5/7 of 210)

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Powers and roots

Squaring a number

Multiplying a number by itself

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Raising to powers

Squaring, cubing, raising to the fourth, and so on...

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Square roots

Work backwards to find the number that was squared

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Roots of non-squares

How to find the roots of numbers that are NOT square

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Square roots of negatives

Do negative numbers even have square roots?

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Cube roots and beyond

Cube roots, fourth roots, fifth roots, and so on...

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PEMDAS

Order of operations, now with exponents!

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Rules for powers

Multiplying powers

A rule for multiplying one power by another

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Dividing powers

A rule for dividing one power by another

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Raising powers to powers

A rule for raising one power to another

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Distributing powers

It works for multiplication and division, but that's it!

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Raising to the zero

What happens when you raise a number to the zero?

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Negative powers

What does it mean to raise to a NEGATIVE exponent?

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Roots as powers

Taking a root is really the same as raising to a power

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Fractional powers

What does it mean to raise to the 2/3 power?

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Distributing roots

Roots are also exponents, so the same rules apply!

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Simplifying roots

Getting smaller numbers under the root

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Solving equations

Solving for unknowns

Solve for unknown values using the number line

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Algebraic expressions

Learn how to evaluate expressions with variables

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Solving by adding

Adding or subtracting on both sides of equations

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Solving by multiplying

Multiplying or dividing on both sides of equations

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Cross multiplication

A quick way to solve equations with fractions

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Coefficients

Numbers in front of variables

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Combining like terms

Adding/subtracting terms with matching variables

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Multi-step equations

Sometimes it takes more than one step to solve!

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Multivariable equations

Solving equations when they have more than one variable

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Unsolvable equations

Some equations have no solutions, others have plenty!

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Inequalities and simultaneous equations

Inequalities

Plotting inequalities on the number line

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Solving inequalities

Solving and plotting multi-step inequalities

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Negative inequalities

What happens when you multiply by a negative

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2 equations, 2 unknowns

Solve for both variables when you have two equations

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2 equations, no solution

Sometimes two equations have no solutions, or many!

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Coordinates

The coordinate plane

Plotting points and finding their coordinates

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Quadrants

The four quadrants, and the coordinates they contain

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Finding a midpoint

Given two points, which point is exactly halfway between?

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Slope

Slope (an introduction)

A number that tells you the steepness of a line

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Slope formula

How to calculate the slope between any two points

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Negative slopes

Discover what it means when slope is negative

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Zero slope

What are the slopes of horizontal and vertical lines?

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Find the slope of any line

Two steps that will always give you the correct slope

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Parallel slopes

Explore why parallel lines have the same slope

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Perpendicular slopes

Discover how slopes of perpendicular lines relate

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Lines

Graphing an equation

Turn an equation into your very first graph!

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Slope-intercept form

Turning an equation into a line, and vice versa

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Point-slope form

A formula for when you know the slope and coordinates

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Reading graphs

Forget equations, let's just look at the graphs!

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Intercepts

Figuring out where lines cross the x- and y-axes

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Solving for intersections

Given two lines, can you determine where they cross?

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Graphing absolute value

Exploring a graph that's not a line

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Inequalities (2 variables)

Simplify inequalities with two variables and test solutions

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Graphing inequalities

Graph inequalities on the coordinate plane

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Polynomials

What are polynomials?

Learn about a common type of algebraic expression

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Adding polynomials

Computing sums and differences of polynomials

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Multiplying monomials

Multiply and divide polynomials with one term

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Multiplying binomials

Multiply binomials together using FOIL

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Multiplying polynomials

How to multiply polynomials with many terms

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Difference of squares

A neat trick for simplifying certain binomials

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Quadratics

Quadratic equations

Sometimes, equations can have two solutions

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Graphing a parabola

Plot points to draw your very first parabola

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Graphing quadratics

Every quadratic graph boils down to these 3 numbers

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Factoring quadratics

An extremely useful trick for solving some equations

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The quadratic formula

A method for solving ANY quadratic equation!

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Discriminants and roots

A quick way to tell how many solutions a quadratic has

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The mathematics of lines, shapes, and angles. Essential stuff for describing the world around you.

SUBJECT
COMPLETION

Lines and angles

Lines, rays, & segments

Learn about lines, rays, and line segments

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Angles and degrees

Learn what angles are, and how to measure them

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Right, acute, and obtuse

Learn the names for angles of all sizes

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Parallel lines

Lines that never, ever cross

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Perpendicular lines

Lines that cross, forming right angles

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Naming angles

With these rules, you know which angle you mean

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Perpendicular bisectors

The equidistant points also cut segments in two

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Related angles

Combining angles

Adding and subtracting adjacent angles

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Complementary angles

Learn about complementary and supplementary angles

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Coterminal angles

What about angles bigger than 360 degrees?

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Vertical angles

Angles on opposite sides of intersecting lines

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Corresponding angles

Parallel lines form congruent angles in matching places

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Alternate interior angles

Congruent angles INSIDE parallel lines

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Alternate exterior angles

Congruent angles OUTSIDE parallel lines

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Triangles

Polygons

A fancy name for shapes with straight sides

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Congruence

When line segments, angles, or shapes are the same

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Triangles

Introducing triangles and three different types of them

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180 degrees in a triangle

See if it's really true, and then prove it!

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Triangles by angle

Learn about right, acute, and obtuse triangles

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Altitudes

Perpendicular line segments in triangles

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Medians of triangles

Line segments connecting vertices and midpoints

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Isosceles triangles

They have two equal sides, but what about their angles?

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Triangle inequality

The rules a triangle's side lengths always follow

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Big angles, longer sides

In triangles, sides and their opposite angles are related!

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Triangle congruence and similarity

Similarity

When they have the same shape, but different sizes

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Similar ratios

Use similarity to find unknown side lengths!

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SSS Postulate

Using sides to see if triangles are congruent

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SAS Postulate

Congruence check using two sides and the angle between

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ASA Postulate

Congruence check using two angles and the side between

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AAS Postulate

The final congruence check for triangles

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AA Postulate (Similarity)

Triangles are similar when they have matching angles

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SSA (ambiguous case)

Depending on the sides, you can have 0, 1, or 2 triangles!

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Polygons and quadrilaterals

Quadrilaterals

Learn about rectangles, rhombi, and more!

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Regular polygons

Polygons with equal sides and angles

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Perimeter

Tricks for finding the distance around a shape

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Angles in polygons

Find a formula for the sum of the angles in any polygon

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Parallelogram angles

Opposite angles are congruent in parallelograms

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Parallelogram sides

Opposite sides are congruent in parallelograms

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Paralleogram diagonals

In parallelograms, diagonals always bisect each other

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Rectangle diagonals

Rectangles always have two congruent diagonals

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Rhombus diagonals

In rhombi, the diagonals are always perpendicular

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Trapezoid angles

Supplementary pairs of angles in trapezoids

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Area of polygons

Intro to area

Measuring how much space is inside a shape

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Rectangle area

Discover the formula for the area of a rectangle

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Parallelogram area

Discover the formula for the area of a parallelogram

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Triangle area

Discover a formula for the area of any triangle

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Trapezoid area

Discover the formula for the area of a trapezoid

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Rhombus area

Find a rhombus' area using its diagonals

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The Pythagorean theorem

Pythagorean theorem

How the sides of right triangles are related

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Pythagorean triples

When whole numbers are the sides of right triangles

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Generating triples

Ways to write out every last Pythagorean triple

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Distance formula

Find the distance between any two points

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Equilateral area

Finding the area of equilateral and isosceles triangles

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Heron's formula

Another way to find the area of a triangle

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Proving Heron's formula

You might know the formula, but where does it come from?

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Circles, ellipses, and their areas

Circles

An intro to circles, radii, diameters, and chords

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Circumference and Pi

Find the distance around a circle (and then eat some pi)

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Size of the Earth

Use shadows to measure the Earth's circumference!

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Ellipses

What you get when you "stretch" a circle

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Area of a circle

Using circumference to find a circle's area

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Cavalieri's principle

Comparing shapes to find areas of the unusual ones

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Ellipse area

Calculate how many people can fit on the Ellipse in DC

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Areas of complex shapes

Add and subtract the ares of simpler shapes!

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Angles in circles

Central angles and arcs

As you'll see, there are 360 degrees in every circle

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Inscribed angles

Discover how inscribed angles are related to central angles!

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Inscribing the diameter

Prove that inscribed angles drawn to diameters are right

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Arc length

Finding a formula for the length of any arc

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Sector area

How to find the area of any slice of a circle

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Lines in circles

Secant and tangent lines

Some lines cross circles twice, others touch them only once

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Congruent tangents

Prove that tangents from the same point are congruent

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Circumscribed angles

Discover how angles between tangents are related to arcs

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Parallel chords

Prove that the arcs between them are always congruent

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Intersecting chords

Discover a rule for how intersecting chords relate

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Cyclic quadrilaterals

See how their opposite angles are related!

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Intersecting secants

Like chords, intersecting secants are also related!

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Inscribed circles

Calculate the radius of any inscribed circle

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Volume

Intro to solids

Learn the names and features of 3D shapes

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Rectangular prisms

Discover how to find the volume of any box!

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Prism volume

Find the volume of prisms and cylinders

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Cavalieri's Principle (3D)

Instead of matching widths, you'll be matching areas!

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Pyramid volume

Starting with a cube, discover the pyramid volume formula

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Cone volume

Find the volume of any cone, and learn about slant height

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Sphere volume

Use Cavalieri's principle to find the volume of a sphere

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Surface area

Prism surface area

Learn why ice is commonly shaped into cubes

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Pyramid surface area

Find a pyramid's surface area using squares and triangles

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Cylinder surface area

Unroll any cylinder to find its surface area!

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Cone surface area

Discover and use the formula for a cone's surface area!

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Sphere surface area

Find the the total area of a sphere's curved surface

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Internal diagonals

What's the distance between opposite corners of a cube?

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Transformations

Translation

Moving points and shapes in the coordinate plane

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Rotation

Rotating points and shapes around the origin

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Reflection

Flipping across different axes (and the origin)

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Dilation

Stretching shapes in one or two directions

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Preserving congruence

See which transformations keep things congruent

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Symmetry

When reflection or rotation doesn't change anything

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Geometric optics

Why light bends

Using Fermat's principle to understand refraction

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Rainbows explained

Use refraction to understand rainbows and their features

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Trigonometry helps you understand any topic that involves distances, angles, or waves. The trig functions (sin, cos, and tan) show up all over science and engineering.

SUBJECT
COMPLETION

Radians

What's a "radian"?

Discover another way to measure angles.

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Radians and degrees

How to switch back and forth between units for angles

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Lengths of circular arcs

A formula for arc length in terms of radius and angle

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Sectors

Learn what a sector is, and a formula for finding its area

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Trig functions

Sides of right triangles

Your friendly hypotenuses, opposites, and adjacents

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Sine

Your very first trig function

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Cosine

Along with sine, cosine is a fundamental trig function

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Tangent

Another important trig function (like sine and cosine)

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Components (and cops)

Find legs of right triangles, and don't get pulled over!

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Sec, Csc, and Cot

Secant and friends (three more trig functions)

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Special angles

45 degrees

Calculate the sine, cosine, and tangent of 45 degrees

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30/60 degrees

Calculate the sine, cosine, and tangent of 30 and 60 degrees

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0/90 degrees

Calculate the sine, cosine, and tangent of 0 and 90 degrees

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The unit circle

Trig for larger angles

What's sin(250)? Use the unit circle to tackle bigger angles

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Trig for negative angles

Using the unit circle for trig with negative angles

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Trig for coterminals

How trig functions are related for coterminal angles

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Inverse trig functions

Inverse sine

A function that takes you from side ratios to angles

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Inverse cosine

Another inverse trig function

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Inverse tangent

And one last inverse trig function!

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Trig identities

The tools for simplifying trig expressions

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Addition identities

Simplify trig functions of sums

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Negative angle identities

Simplify trig functions of negative angles

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Subtraction identities

Simplify trig functions of differences

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Double angle identities

Trig functions of twice an angle

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Triple angle identities

Trig functions of three times an angle

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Pythagorean identities

Relating squares of trig functions

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Half angle identities

Trig functions of half an angle

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Triangle formulas

Law of sines

Relating angles and opposite sides for any triangle

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Triangulation

Real-world application of the law of sines to find distances

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Law of cosines

Relating three sides and an angle for any triangle

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Triangle area: SAS

Find a triangle's area using 2 sides and the angle between

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Take the next step in Algebra! Here we'll introduce the most common functions you're likely to see, and different ways to describe functions.

SUBJECT
COMPLETION

Real numbers

Integers

Whole numbers, positives, and negatives

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Rationals

Numbers that are quotients of integers

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Irrationals

Numbers that are NOT quotients of integers

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Real numbers

All the rationals and the irrationals

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Imaginary and complex numbers

Imaginary numbers

Their squares are negative!

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Working with imaginaries

Add, subtract, multiply, and divide imaginary numbers

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Complex numbers

Combining the real and the imaginary

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The complex plane

A way to graph complex numbers

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Powers of i

Raising i to positive and negative integer powers

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Functions

Relations

Sets of (x,y) coordinates

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Functions

Relations where each input gives exactly one output

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Functional notation

Explore what f(x) means, and evaluate functions

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Continuity and smoothness

Some functions end and others keep going

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Describing functions

Make your own functions, and live to tell the tale

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Concavity

Math-talk for "curvy"

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Odd and even functions

Turns out functions can be "odd" and "even" too

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Exponents

Logarithms

Playing with functions

Piecewise functions

Franken-functions. They're alive!

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Compositions

Functions of functions

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Conic sections

Rational functions

Polar functions

The mathematics of change. And it'll use just about every branch of math you've learned before it.

SUBJECT
COMPLETION

Limits

Limits to infinity

What happens to functions as x gets really, really big?

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Vertical limits

Sometimes y goes off to infinity

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Finite limits

Limits when both x and y stay finite

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One-sided limits

Limits that come from only the left or right side

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Continuity (for real)

Dive deeper into continuity, using one-sided limits

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Splitting limits

Tricks for simplifying limits (and when they don't work!)

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Derivatives

Tangent lines

Finding the "slope" of a curve

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Drawing derivatives

Learn what a derivative is, and draw your own

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Integrals

Series

The math of data collection and analysis. Very useful for finding trends and making predictions!

SUBJECT
COMPLETION

Sensitivity analysis

Sensitivity and specificity

Quantifying whether tests for diseases are any good

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Predictive value

If you test positive/negative, what are your chances?

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Cutoffs and ROCs

Sensitivity vs. specifity, cutoff selection, and ROC curves

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Prevalence

How rare a disease is, and how it affects PPV and NPV

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Distributions

Regression analysis

Hypothesis testing

Algebra › Slope (an introduction)