Algebra I

121 core topics + 41 prerequisite topics taught as needed · approximately 40 hours of instruction including spaced review

How the course runs

An adaptive diagnostic (up to 40 questions) places the student on the course's knowledge graph — topics already known are credited, and instruction begins exactly at the learning frontier. Every topic is taught with a worked-example lesson and auto-graded practice; a topic is mastered at 75%+ and then maintained through spaced reviews on an expanding schedule. A cumulative quiz follows every 6 lessons. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.

Core curriculum

Linear Functions · 10 topics

The Coordinate Plane [M]	Locating points with (x, y) pairs.
Slope of a Line [M]	Rise over run between two points.
Slope-Intercept Form [M]	y = mx + b describes a whole line.
Finding a Line from Points [H]	Reconstructing y = mx + b from data.
Systems of Equations (Substitution) [H]	Two equations, two unknowns.
Elimination: A First Look [H]	Add or subtract equations so one variable cancels.
Systems: Word Problems [H]	Translating two facts into two equations.
Absolute Value Equations [H]	Distance equations have two answers.
Arithmetic Sequences [H]	Add the same amount each step.
Geometric Sequences [H]	Multiply by the same ratio each step.

Quadratics & Polynomials · 13 topics

Adding & Subtracting Polynomials [E]	Combining polynomials by collecting like terms.
Multiplying Binomials (FOIL) [M]	Expanding products of binomials.
Factoring Out the GCF [M]	Undoing the distributive property.
Factoring Trinomials [M]	Reversing FOIL: finding two numbers that multiply to c and add to b.
Special Factoring Patterns [M]	Difference of squares and perfect-square trinomials.
Solving Quadratics by Factoring [M]	Zero-product property: if a·b = 0 then a = 0 or b = 0.
Solving x² = k [M]	Taking square roots of both sides — remembering ±.
Completing the Square [M]	Turning any quadratic into a perfect square plus a constant.
The Quadratic Formula [H]	x = (−b ± √(b² − 4ac)) / 2a solves any quadratic.
The Discriminant [M]	b² − 4ac tells you how many real solutions exist.
Vertex of a Parabola [M]	The turning point at x = −b/2a.
Graphs of Quadratics [M]	Intercepts and symmetry of a parabola.
Quadratic Models [H]	Projectile motion and other parabolic models.

Algebra I: Inequalities & Compound Statements · 13 topics

Two-Step Inequalities with the Flip [E]	Undo the constant, then divide — and flip the sign if the divisor is negative.
Multi-Step Inequalities [M]	Distribute and collect variables just like an equation — then mind the sign.
Which Way Does the Sign Point? [E]	Spot when the inequality flips — and when it doesn't.
Compound AND Inequalities [M]	A sandwiched inequality: do the same steps to all three parts.
Compound OR Inequalities [M]	OR keeps everything either piece allows — only the gap between them fails.
Absolute Value: Less Than [M]	|x − m| < k traps x within k of m — an AND sandwich.
Absolute Value: Greater Than [M]	|x − m| > k pushes x farther than k from m — an OR split.
Multi-Step Absolute Value Inequalities [H]	Isolate the absolute value first; only then split into a sandwich.
At Least / At Most Word Problems [M]	Translate a budget or goal into an inequality, then round the smart way.
Hitting an Average Target [M]	An average of at least T means the total must reach n times T.
Translating Words into Inequalities [E]	At most means ≤, at least means ≥, more than means > — order matters too.
Checking Points Against a System [M]	Substitute the point into each inequality and judge them separately.
All Real Numbers or No Solution? [M]	When the x-terms cancel, only a true-or-false number fact remains.

Algebra I: Exponential Functions · 12 topics

Evaluating Exponential Functions [E]	Plug integer inputs into f(x) = a·bˣ — including 0 and negatives.
Growth Rates & Growth Factors [E]	Growing r% per step means multiplying by 1 + r/100 each step.
Decay Rates & Decay Factors [E]	Losing r% per step means multiplying by 1 − r/100 each step.
Modeling Percent Growth & Decay [M]	Turn a percent rate into a factor, then multiply once per time step.
Doubling Contexts [M]	Doubling every period is y = a·2ᵗ with t counting periods, not hours.
Halving & Half-Life [M]	Halving every period is y = a·(1/2)ᵗ — count the halvings first.
Exponential Patterns in Tables [M]	Equal steps in x: equal differences mean linear, equal ratios mean exponential.
Graphs of Exponential Functions [M]	y-intercept a, growth when b > 1, decay when 0 < b < 1, floor at y = 0.
Linear vs. Exponential Growth [M]	A linear head start never survives repeated doubling.
Simple vs. Compound Interest [H]	Simple interest grows linearly; compounding earns interest on interest.
Same-Base Exponential Equations [M]	Match the bases, then set the exponents equal.
Exponential Equations with a Base Change [H]	Rewrite both sides over one common base before equating exponents.

Algebra I: Descriptive Statistics · 13 topics

Computing the Mean [E]	The mean is the total shared out equally — and totals work backwards too.
Median & Mode [E]	Sort first: the median is the middle, the mode is the most frequent.
Outliers and the Mean [M]	One extreme value drags the mean toward it; the median barely moves.
Choosing a Measure of Center [E]	Skewed or outlier-heavy data wants the median; symmetric data, the mean.
Mean Absolute Deviation [M]	MAD is the average distance of the data from its own mean.
Variance of a Data Set [M]	Square each deviation from the mean, then average the squares.
The Five-Number Summary [M]	Min, Q1, median, Q3, max — five landmarks that sketch a whole data set.
Range, IQR & the Outlier Fence [M]	The IQR measures the spread of the middle half — and builds the outlier fence.
Reading Box Plots [M]	Each piece of a box plot — whisker, half-box, half-box, whisker — holds about 25% of the data.
Two-Way Relative Frequency [M]	Percent of what? The denominator — a row, a column, or everyone — changes the answer.
Interpreting a Fitted Line [M]	Slope is the predicted change per unit of x; the intercept is the prediction at x = 0.
Residuals [M]	A residual is actual minus predicted — the line's miss at one point.
Correlation vs. Causation [E]	Association alone never proves cause — look for lurking variables or a randomized experiment.

Algebra I: Literal Equations & Modeling · 10 topics

Solving a Literal Equation [M]	Isolate a variable using letters instead of numbers.
Rearranging Formulas [M]	Solve a familiar formula for the piece you need.
Consecutive-Integer Problems [M]	Name them x, x+1, x+2 and solve.
Coin & Mixture Problems [M]	One equation for count, one for value.
Percent-Mixture Setups [M]	The amount of an ingredient is percent times total.
Rate–Time–Distance [M]	Distance equals rate times time.
Two Movers [M]	Combine the speeds when they move apart or together.
Geometry Setup Problems [M]	Translate a geometry description into one equation.
Unit Analysis [E]	Multiply by a conversion factor to change units.
Piecewise Cost Models [M]	Different rules apply in different ranges.

Algebra I: Polynomials & Factoring in Depth · 12 topics

Polynomial Sums: Hunting a Coefficient [E]	Add or subtract polynomials and report one requested coefficient.
Monomial Times a Polynomial [E]	Distribute a single term across a trinomial and read off a coefficient.
Binomial Products: Any Term You Like [M]	Expand (ax+b)(cx+d) and pick out the leading, middle, or constant term.
Binomial Times a Trinomial [M]	Expand a binomial against a trinomial and locate a single term.
Squaring a Binomial [M]	Apply (ax±b)² = a²x² ± 2abx + b² and read a coefficient.
The Sum-Times-Difference Product [M]	Recognize (ax+b)(ax−b) = a²x² − b² — the middle term vanishes.
GCF Factoring: the Leftover Trinomial [M]	Pull out the greatest common factor and inspect the quotient's coefficients.
Factoring x² + bx + c to a Factor [M]	Split a monic trinomial and hand back one exact binomial factor.
Factoring by Grouping [H]	Group a four-term polynomial and extract the shared binomial factor.
Factoring a Difference of Squares [M]	Reverse a²x² − b² into the sum and difference of its square roots.
Perfect-Square Trinomials [M]	Recognize a²x² ± 2abx + b² and write it as a single squared binomial.
Factoring ax² + bx + c with a > 1 [H]	Factor a trinomial with a leading coefficient above 1 and return a factor.

Algebra I: Quadratic Functions & Graphs · 13 topics

Evaluating a Quadratic Function [E]	Substitute a number for x and simplify to get the function's output.
The Vertex: x = −b/(2a) [M]	The turning point's x-coordinate comes straight from a and b.
The Vertex y-value [M]	Plug the vertex's x back into the function to get its y.
The Axis of Symmetry [E]	The vertical mirror line runs midway between the two x-intercepts.
The y-Intercept of a Quadratic [E]	A parabola meets the y-axis at its constant term c.
x-Intercepts by Factoring [M]	The graph crosses the x-axis where each factor is zero.
Maximum or Minimum Value [M]	The vertex's y-value is the largest or smallest output the function reaches.
Does It Open Up or Down? [E]	The sign of the leading coefficient decides maximum versus minimum.
Transformations of y = x² [M]	Shifting y = x² moves the vertex from the origin by the shift amounts.
Reading a Shift from Vertex Form [M]	Vertex form spells out exactly how the parent parabola was moved.
Projectile Motion: Time of Peak [M]	A launched object peaks at the vertex time t = v/32.
Projectile Motion: Height at a Time [M]	Evaluate the height model at a given instant.
Comparing Two Parabolas [M]	Read vertex form to compare position, width, and extreme value.

Algebra I: Functions, Domain & Radicals · 12 topics

Evaluating f(x) [E]	f(x) names a rule; f(c) means substitute c for every x and simplify.
Solving f(x) = k [E]	Set the rule equal to the output value and solve the linear equation for x.
Interpreting f(a) = b in Context [E]	f(a) = b says the input a yields the output b — mind which is which.
Reading Function Tables [E]	A table pairs inputs with outputs; a constant step reveals the missing value.
Rate of Change from a Table [M]	Average rate of change is the change in output over the change in input.
Domain & Range of Ordered Pairs [M]	The domain is the set of x-values; the range is the set of y-values.
Domain from a Restriction [M]	Division by zero and negative radicands carve values out of the domain.
Is It a Function? [E]	A relation is a function only if each input maps to exactly one output.
Simplifying Square Roots [M]	Pull the largest perfect-square factor out front: √(k²·m) = k√m.
Adding Like Radicals [M]	Like radicals combine like terms: a√m + b√m = (a + b)√m.
Multiplying Square Roots [M]	Combine under one root — √a · √b = √(ab) — then simplify what results.
Rational Exponents [M]	A fractional exponent is a root: x^(p/q) is the q-th root of x, to the p.

Algebra I: Sequences, Data & Probability · 13 topics

Arithmetic Sequences: Any Term [M]	Reach a distant term, find a term's position, or recover the common difference.
Modeling with Arithmetic Sequences [M]	Fixed steps up or down are arithmetic — translate the story into a₁ and d.
Geometric Sequences: Any Term [M]	Multiply, don't add: the nth term uses a whole-number ratio raised to n − 1.
Modeling with Geometric Sequences [M]	Repeated multiplying — doubling, tripling — is geometric growth.
Recursive vs. Explicit Rules [M]	A recursive rule leans on the previous term; an explicit rule jumps straight to term n.
Building Terms from a Recursive Rule [M]	March one term at a time — even when no explicit shortcut exists.
Joint Relative Frequency [M]	A joint relative frequency is one inner cell divided by the grand total.
Marginal Relative Frequency [M]	A marginal relative frequency uses a whole row or column total over the grand total.
Theoretical Probability of Simple Events [M]	Favorable outcomes over equally likely total outcomes, reduced to lowest terms.
Probability of Compound Events [M]	Multiply for independent 'and'; add for non-overlapping 'or'.
Experimental Probability from Data [M]	Count what actually happened over the number of trials — then predict.
Using a Data Display [E]	Read the graph, then combine the values the question actually asks about.
Correlation vs. Causation [E]	Association is not proof of cause — hunt for a lurking variable or a random assignment.

Prerequisite material — taught automatically when the diagnostic finds gaps

Arithmetic Foundations · 8 topics

Adding & Subtracting Whole Numbers	Multi-digit addition and subtraction.
Multiplication	Multiplying whole numbers.
Division	Dividing whole numbers.
Order of Operations	Parentheses first, then multiplication/division, then addition/subtraction.
Negative Numbers: Adding & Subtracting	Working with numbers below zero on the number line.
Negative Numbers: Multiplying & Dividing	Sign rules for products and quotients.
Exponents	Repeated multiplication in shorthand.
Square Roots	Undoing a square.

Fractions · 4 topics

Equivalent Fractions	Different fractions can name the same amount.
Simplifying Fractions	Reducing a fraction to lowest terms.
Multiplying Fractions	Multiply straight across.
Dividing Fractions	Multiply by the reciprocal.

Decimals, Percents & Ratios · 4 topics

Fractions ↔ Decimals	Converting between the two notations.
Percent of a Number	Percent means per hundred.
Percent Increase & Decrease	Applying a percent change to a quantity.
Ratios & Proportions	Two quantities that scale together.

Expressions & Equations · 7 topics

Evaluating Expressions	Substituting a value for a variable.
Combining Like Terms	Adding the coefficients of matching variable parts.
The Distributive Property	Multiplying across a sum.
One-Step Equations	Undoing a single operation.
Two-Step Equations	Undo addition/subtraction first, then multiplication.
Multi-Step Equations	Equations needing distribution or variables on both sides.
Linear Inequalities	Solving with <, >, ≤, ≥.

Radicals & Exponentials · 4 topics

Product Rule for Exponents	Multiplying powers of the same base adds the exponents.
Quotient & Power Rules	Dividing powers subtracts exponents; a power of a power multiplies them.
Zero & Negative Exponents	Anything (nonzero) to the 0 power is 1; a negative exponent flips to a reciprocal.
Exponential Growth & Decay	Quantities that multiply by the same factor each time step: y = a·bᵗ.

Functions & Algebra II · 2 topics

Function Notation & Evaluation	Reading f(x) notation and plugging in inputs.
Transformations of Functions	How f(x − h) + k slides a graph around the plane.

Ratios, Data & Geometry (Middle School) · 9 topics

Absolute Value & Distance	Absolute value is distance from zero.
Unit Rates	Per-one comparisons: dollars per item, miles per hour.
Ratio Tables & Equivalent Ratios	Scaling both parts of a ratio keeps it equivalent.
Solving Proportions	Cross-multiply to find the missing value.
Percent Applications: Tax, Tip & Discount	Real-world percents: discounts, tips, and tax.
Mean, Median & Range	Three ways to summarize a data set with one number.
Reading Data Displays	Pulling answers out of dot plots, tables, and bar graphs.
Probability Basics	Favorable outcomes over total outcomes.
Compound Probability	Independent events multiply.

Grade 7: Proportions, Geometry & Statistics · 2 topics

Two-Step Inequalities	Solve like an equation; flip when you multiply by a negative.
Simple Interest & Percent Change	I = P·r·t; balance = principal + interest.

Grade 8: Functions, Exponents & Geometry · 1 topics

Scatter Plots & Association

Positive, negative, or no association between two variables.

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