Algebra foundations
Overview and history of algebra
Intro to the coordinate plane
Why all the letters in algebra?
Introduction to variables
Why aren’t we using the multiplication sign?
Evaluating an expression with one variable
Substitution and evaluating expressions
Evaluating expressions with two variables
Evaluating expressions with two variables: fractions & decimals
Intro to combining like terms
Combining like terms challenge problem
Simplifying expressions with rational numbers
Introduction to equivalent expressions
Why dividing by zero is undefined
The problem with dividing zero by zero
Undefined & indeterminate expressions
Solving equations & inequalities
Linear equations with variables on both sides
Why we do the same thing to both sides: Variable on both sides
Intro to equations with variables on both sides
Equations with variables on both sides: 20-7x=6x-6
Equation with variables on both sides: fractions
Equation with the variable in the denominator
Linear equations with parentheses
Equations with parentheses
Analyzing the number of solutions to linear equations
Number of solutions to equations
Worked example: number of solutions to equations
Creating an equation with no solutions
Creating an equation with infinitely many solutions
Linear equations with unknown coefficients
Linear equations with unknown coefficients
Inequalities with variables on both sides
Inequalities with variables on both sides (with parentheses)
Compound inequalities: OR
Compound inequalities: AND
A compound inequality with no solution
Compound inequalities examples
Intro to dimensional analysis
Same rate with different units
Working with units
Interpreting units in formulas
Interpreting units in formulas: novel units
Word problems with multiple units
Worked example: Rate problem
Multiple units word problem: road trip
Measurement word problem: running laps
Multiple units word problem: drug dosage (advanced)
Two-variable linear equations intro
Two-variable linear equations intro
Solutions to 2-variable equations
Worked example: solutions to 2-variable equations
Completing solutions to 2-variable equations
Positive & negative slope
Worked example: slope from graph
Graphing a line given point and slope
Calculating slope from tables
Worked example: slope from two points
Linear equations & graphs
Horizontal & vertical lines
Slope of a horizontal line
Horizontal & vertical lines
x-intercepts and y-intercepts
Intercepts from an equation
Applying intercepts and slope
Slope, x-intercept, y-intercept meaning in context
Slope and intercept meaning in context
Slope and intercept meaning from a table
Finding slope and intercepts from tables
Linear functions word problem: fuel
Intro to slope-intercept form
Intro to slope-intercept form
Slope and y-intercept from equation
Worked examples: slope-intercept intro
Linear equation word problems
Graphing slope-intercept equations
Graph from slope-intercept equation
Writing slope-intercept equations
Slope-intercept equation from graph
Slope-intercept equation from slope & point
Slope-intercept equation from two points
Constructing linear equations from context
Forms of linear equations
Intro to point-slope form
Point-slope & slope-intercept equations
Intro to linear equation standard form
Graphing a linear equation: 5x+2y=20
Clarifying standard form rules
Converting from slope-intercept to standard form
Summary: Forms of two-variable linear equations
Writing linear equations in all forms
Introduction to systems of equations
Systems of equations: trolls, tolls (1 of 2)
Systems of equations: trolls, tolls (2 of 2)
Testing a solution to a system of equations
Systems of equations with graphing: y=7/5x-5 & y=3/5x-1
Systems of equations with graphing: exact & approximate solutions
Setting up a system of equations from context example (pet weights)
Setting up a system of linear equations example (weight and price)
Interpreting points in context of graphs of systems
Solving systems of equations with substitution
Systems of equations with substitution: potato chips
Systems of equations with substitution: -3x-4y=-2 & y=2x-5
Systems of equations with elimination: potato chips
Systems of equations with elimination (and manipulation)
Worked example: equivalent systems of equations
Worked example: non-equivalent systems of equations
Equivalent systems of equations and the elimination method
Systems of equations with elimination: King’s cupcakes
Why can we subtract one equation from the other in a system of equations?
Systems of equations with elimination: x-4y=-18 & -x+3y=11
Number of solutions to systems of equations
Systems of equations number of solutions: fruit prices (1 of 2)
Systems of equations number of solutions: fruit prices (2 of 2)
Solutions to systems of equations: consistent vs. inconsistent
Solutions to systems of equations: dependent vs. independent
Number of solutions to a system of equations
Number of solutions to a system of equations graphically
Number of solutions to a system of equations algebraically
How many solutions does a system of linear equations have if there are at least two?
Systems of equations
Systems of equations word problems
Age word problem: Ben & William
Age word problem: Arman & Diya
System of equations word problem: walk & ride
System of equations word problem: no solution
System of equations word problem: infinite solutions
Systems of equations with elimination: TV & DVD
Systems of equations with elimination: apples and oranges
Systems of equations with substitution: coins
Systems of equations with elimination: coffee and croissants
Checking solutions of two-variable inequalities
Testing solutions to inequalities
Testing solutions to systems of inequalities
Graphing two-variable inequalities
Intro to graphing two-variable inequalities
Graphing two-variable inequalities
Two-variable inequalities from their graphs
Intro to graphing systems of inequalities
Graphing systems of inequalities
Modeling with linear inequalities
Writing two-variable inequalities word problem
Solving two-variable inequalities word problem
Graphs of two-variable inequalities word problem
Interpreting two-variable inequalities word problem
Modeling with systems of inequalities
Writing systems of inequalities word problem
Solving systems of inequalities word problem
Graphs of systems of inequalities word problem
Worked example: Evaluating functions from equation
Worked example: Evaluating functions from graph
Evaluating discrete functions
Worked example: evaluating expressions with function notation
Inputs and outputs of a function
Worked example: matching an input to a function’s output (equation)
Worked example: matching an input to a function’s output (graph)
Worked example: two inputs with the same output (graph)
Inequalities (systems & graphs)
Obtaining a function from an equation
Interpreting function notation
Function notation word problem: bank
Function notation word problem: beach
Introduction to the domain and range of a function
Intervals and interval notation
What is the domain of a function?
What is the range of a function?
Worked example: domain and range from graph
Determining the domain of a function
Determining whether values are in domain of function
Examples finding the domain of functions
Worked example: determining domain word problem (real numbers)
Worked example: determining domain word problem (positive integers)
Worked example: determining domain word problem (all integers)
Functions
Recognizing functions from graph
Does a vertical line represent a function?
Recognizing functions from table
Recognizing functions from verbal description
Recognizing functions from verbal description word problem
Maximum and minimum points
Introduction to minimum and maximum points
Worked example: absolute and relative extrema
Intervals where a function is positive, negative, increasing, or decreasing
Increasing, decreasing, positive or negative intervals
Worked example: positive & negative intervals
Interpreting features of graphs
Graph interpretation word problem: temperature
Graph interpretation word problem: basketball
Introduction to average rate of change
Worked example: average rate of change from graph
Worked example: average rate of change from table
Average rate of change word problem: table
Average rate of change word problem: graph
Intro to inverse functions
Intro to inverse functions
Inputs & outputs of inverse functions
Graphing the inverse of a linear function
Finding inverse functions: linear
Introduction to arithmetic sequences
Intro to arithmetic sequences
Extending arithmetic sequences
Using arithmetic sequences formulas
Worked example: using recursive formula for arithmetic sequence
Constructing arithmetic sequences
Recursive formulas for arithmetic sequences
Explicit formulas for arithmetic sequences
Arithmetic sequence problem
Converting recursive & explicit forms of arithmetic sequences
Introduction to geometric sequences
Intro to geometric sequences
Extending geometric sequences
Using explicit formulas of geometric sequences
Using recursive formulas of geometric sequences
Explicit & recursive formulas for geometric sequences
Constructing geometric sequences
Converting recursive & explicit forms of geometric sequences
Evaluating sequences in recursive form
Graphs of absolute value functions
Shifting absolute value graphs
Scaling & reflecting absolute value functions: equation
Scaling & reflecting absolute value functions: graph
Graphing absolute value functions
Sequences
Introduction to piecewise functions
Worked example: evaluating piecewise functions
Worked example: graphing piecewise functions
Worked example: domain & range of step function
Worked example: domain & range of piecewise linear functions
Exponent properties review
Multiplying & dividing powers (integer exponents)
Powers of products & quotients (integer exponents)
Understanding square roots
Simplifying square roots (variables)
Simplifying square-root expressions
Exponential vs. linear growth
Intro to exponential functions
Exponential vs. linear growth
Exponential vs. linear models: verbal
Exponential vs. linear models: table
Exponential expressions word problems (numerical)
Initial value & common ratio of exponential functions
Exponential expressions word problems (algebraic)
Interpreting exponential expression word problem
Graphs of exponential growth
Exponential function graph
Graphs of exponential growth
Absolute value & piecewise functions
Exponential vs. linear growth over time
Exponential vs. linear growth over time
Exponential growth & decay
Graphing exponential growth & decay
Writing exponential functions
Writing exponential functions from tables
Writing exponential functions from graphs
Analyzing tables of exponential functions
Analyzing graphs of exponential functions
Analyzing graphs of exponential functions: negative initial value
Modeling with basic exponential functions word problem
Exponential functions from tables & graphs
Exponents & radicals
Exponential vs. linear models
Linear vs. exponential growth: from data
Linear vs. exponential growth: from data (example 2)
Multiplying binomials: area model
Multiplying binomials intro
Special products of binomials
Special products of the form (x+a)(x-a)
Squaring binomials of the form (x+a)²
Special products of the form (ax+b)(ax-b)
Squaring binomials of the form (ax+b)²
Exponential growth & decay
Introduction to factoring
Intro to factors & divisibility
Factoring with the distributive property
Factoring quadratics as (x+a)(x+b)
Factoring quadratics as (x+a)(x+b) (example 2)
More examples of factoring quadratics as (x+a)(x+b)
Factoring quadratics intro
Factoring quadratics by grouping
Factoring quadratics by grouping
Factoring quadratics: common factor + grouping
Factoring quadratics: negative common factor + grouping
Factoring difference of squares: shared factors
Factoring quadratics with difference of squares
Difference of squares intro
Factoring difference of squares: leading coefficient ≠1
Factoring difference of squares: analyzing factorization
Perfect square factorization intro
Factoring perfect squares
Identifying perfect square form
Factoring perfect squares: negative common factor
Factoring perfect squares: missing values
Factoring perfect squares: shared factors
Factoring quadratics with perfect squares
Strategy in factoring quadratics
Strategy in factoring quadratics (part 1 of 2)
Strategy in factoring quadratics (part 2 of 2)
Graphing quadratics in factored form
Quadratic word problems (factored form)
Solving and graphing with factored form
Quadratics: Multiplying & factoring
Solving by taking the square root
Solving quadratics by taking square roots
Solving quadratics by taking square roots examples
Solving quadratics by taking square roots: strategy
Quadratics by taking square roots: with steps
Graphing quadratics: vertex form
Quadratic word problems (vertex form)
Solving quadratics by factoring
Solving quadratics by factoring
Solving quadratics by factoring: leading coefficient ≠1
Solving quadratics using structure
Quadratic equations word problem: triangle dimensions
Quadratic equations word problem: box dimensions
Worked example: quadratic formula (example 2)
Worked example: quadratic formula (negative coefficients)
Using the quadratic formula: number of solutions
Proof of the quadratic formula
Worked example: Completing the square (intro)
Worked example: Rewriting expressions by completing the square
Worked example: Solving equations by completing the square
Worked example: completing the square (leading coefficient ≠1)
Solving quadratics by completing the square: no solution
Finding the vertex of a parabola in standard form
Graphing quadratics: standard form
Quadratic word problem: ball
Intro to rational & irrational numbers
Classifying numbers: rational & irrational
Worked example: rational vs. irrational expressions
Worked example: rational vs. irrational expressions (unknowns)
Sums and products of rational and irrational numbers
Proof: sum & product of two rationals is rational
Proof: product of rational & irrational is irrational
Proof: sum of rational & irrational is irrational
Sums and products of irrational numbers
Proof: √2 is irrational
Proofs concerning irrational numbers
Proof: square roots of prime numbers are irrational
Proof: there’s an irrational number between any two rational numbers
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