Course Eleven · The Complete Micro-Course

College algebra,
one idea at a time.

The whole college algebra canon, rebuilt as 263 numbered videos — each one a single idea, built in the order the ideas depend on each other. Not a problem set with the reasoning removed: the rules themselves, and why each one has to be true given the last.

263 lessons across nine units0 of 263 live

How the course works

Every idea has a number. Every number knows what comes before it.

Algebra is the subject where students most often learn the steps without the reason, and then cannot tell a rule from a habit. This course is built the other way. Each new kind of number is introduced so the old rules stay true; polynomials behave like integers because they were built to; every logarithm law is an exponent law seen from the other side. Nothing appears before the thing it depends on, and every dependency is a number you can follow back.

Three kinds of video: atoms (one definition, rule, or distinction, about twenty seconds), procedures (one method, worked start to finish — classifying a number, solving a system), and capstones (several ideas run together on one problem).

A whole family of entries exists to name the error and put it beside the rule it breaks — the invisible-parentheses error, the zero-power error, the negative-exponent sign error, the radical split. Naming a mistake precisely is usually what stops it.

The whole map

All 263 videos, in order.

New videos post daily until the course is complete. Live ones link straight to the short; the rest are on their way.

Unit 1

Real Numbers, Exponents, and Radicals

001 The Real Numbers Atom Soon
002 Natural Numbers and Whole Numbers Atom Soon
003 The Integers Atom Soon
004 Rational Numbers Atom Soon
005 Irrational Numbers Atom Soon
007 Classifying a Real Number Procedure Soon
009 Exponential Notation Atom Soon
010 The Order of Operations Atom Soon
011 Exponents Before Negation Atom Soon
012 The Invisible-Parentheses Error Atom Soon
014 The Commutative Properties Atom Soon
015 The Associative Properties Atom Soon
016 The Distributive Property Atom Soon
017 The Identity Properties Atom Soon
018 The Inverse Properties Atom Soon
022 The Product Rule of Exponents Atom Soon
023 The Quotient Rule of Exponents Atom Soon
025 The Power Rule of Exponents Atom Soon
026 The Power-Product Confusion Atom Soon
028 The Zero Exponent Rule of Exponents Atom Soon
029 The Zero-Power Error Atom Soon
031 The Negative Rule of Exponents Atom Soon
032 The Negative-Exponent Sign Error Atom Soon
034 The Power of a Product Rule of Exponents Atom Soon
035 The Power of a Quotient Rule of Exponents Atom Soon
037 Simplifying with the Exponent Rules Procedure Soon
039 Scientific Notation Atom Soon
040 Absolute Value Atom Soon
041 The Principal Square Root Atom Soon
042 The Square Root of a Square Atom Soon
043 The Dropped-Absolute-Value Error Atom Soon
045 The Product Rule for Simplifying Square Roots Atom Soon
046 The Split-Sum Error Atom Soon
048 The Quotient Rule for Simplifying Square Roots Atom Soon
050 Simplifying a Square Root Procedure Soon
052 Like Radicals Atom Soon
054 Rationalizing a Denominator Procedure Soon
056 The Principal nth Root Atom Soon
057 Rational Exponents: the 1/n Power Atom Soon
058 Rational Exponents: the m/n Power Atom Soon
060 From Counting to Rational Exponents Capstone Soon
Unit 2

Polynomials, Factoring, and Rational Expressions

061 Polynomials Atom Soon
062 Degree and Leading Coefficient Atom Soon
063 Monomials, Binomials, Trinomials Atom Soon
065 Adding and Subtracting Polynomials Atom Soon
067 Multiplying Polynomials Atom Soon
068 FOIL Atom Soon
070 Squaring a Binomial Atom Soon
071 The Missing-Middle Error Atom Soon
073 The Difference of Squares Atom Soon
081 Factoring Atom Soon
082 The Greatest Common Factor Atom Soon
083 Factoring Out the GCF Procedure Soon
085 Factoring x² + bx + c Atom Soon
086 Factoring ax² + bx + c by Grouping Procedure Soon
088 Factoring a Perfect Square Trinomial Atom Soon
090 Factoring a Difference of Squares Atom Soon
091 The Sum of Squares Atom Soon
093 The Sum and Difference of Cubes Atom Soon
095 Choosing a Factoring Strategy Procedure Soon
101 Rational Expressions Atom Soon
103 Only Factors Cancel Atom Soon
104 The Canceled-Term Error Atom Soon
106 Simplifying a Rational Expression Procedure Soon
108 Multiplying Rational Expressions Atom Soon
109 Dividing Rational Expressions Atom Soon
111 The Least Common Denominator Atom Soon
112 Adding and Subtracting Rational Expressions Procedure Soon
114 Splitting a Fraction over Its Numerator Atom Soon
115 The Denominator-Split Error Atom Soon
117 Simplifying a Complex Rational Expression Procedure Soon
120 The Arithmetic of Polynomials Capstone Soon
Unit 3

Linear Equations, Models, and Inequalities

121 The Cartesian Coordinate System Atom Soon
122 Graphing by Plotting Points Atom Soon
124 Intercepts Atom Soon
126 The Distance Formula Atom Soon
127 The Midpoint Formula Atom Soon
131 Linear Equations in One Variable Atom Soon
132 The Solution of an Equation Atom Soon
133 Solving a Linear Equation Procedure Soon
135 Identity, Conditional, and Inconsistent Equations Atom Soon
137 Rational Equations Atom Soon
138 Checking in the Original Equation Atom Soon
140 Solving a Rational Equation Procedure Soon
142 The Slope of a Line Atom Soon
143 The Slope Formula Atom Soon
145 Slope-Intercept Form Atom Soon
146 The Point-Slope Formula Atom Soon
148 Writing a Line from Two Points Procedure Soon
150 Parallel Lines Atom Soon
151 Perpendicular Lines Atom Soon
153 Horizontal and Vertical Lines Atom Soon
156 Interval Notation Atom Soon
158 Solutions of Linear Inequalities Atom Soon
159 The Addition Property of Inequality Atom Soon
160 The Multiplication Property of Inequality Atom Soon
161 The Kept-Sign Error Atom Soon
163 Solving a Linear Inequality Procedure Soon
165 Compound Inequalities Atom Soon
167 Absolute Value as Distance Atom Soon
168 Absolute Value Inequalities: Less Than Atom Soon
169 Absolute Value Inequalities: Greater Than Atom Soon
170 The Swapped-Cases Error Atom Soon
172 Solving an Absolute Value Inequality Procedure Soon
180 The Line and the Balance Capstone Soon
Unit 4

Complex Numbers and Quadratic Equations

181 The Imaginary Unit Atom Soon
182 Complex Numbers in Standard Form Atom Soon
184 Adding and Subtracting Complex Numbers Atom Soon
185 Multiplying Complex Numbers Atom Soon
187 The Complex Conjugate Atom Soon
188 Dividing Complex Numbers Procedure Soon
190 Square Roots of Negative Numbers Atom Soon
191 The Powers of i Atom Soon
196 Quadratic Equations Atom Soon
197 The Zero-Product Property Atom Soon
198 Solving a Quadratic by Factoring Procedure Soon
200 Keeping Every Factor Atom Soon
201 The Divided-Away Solution Atom Soon
203 The Square Root Property Atom Soon
204 The Lost-Root Error Atom Soon
206 Completing the Square Procedure Soon
208 The Quadratic Formula Atom Soon
209 The Discriminant Atom Soon
211 Solving with the Quadratic Formula Procedure Soon
213 Complex Solutions of Quadratics Atom Soon
216 Radical Equations Atom Soon
217 Checking After Raising to a Power Atom Soon
218 The Unchecked-Squaring Error Atom Soon
220 Solving a Radical Equation Procedure Soon
222 Equations Quadratic in Form Atom Soon
224 Equations with Rational Exponents Atom Soon
226 Absolute Value Equations Atom Soon
230 Every Quadratic Surrenders Capstone Soon
Unit 5

Functions: Definition, Domain, Graphs, Transformations

231 Relations Atom Soon
232 Functions Atom Soon
234 Function Notation Atom Soon
235 Evaluating a Function Atom Soon
236 Evaluating at a Sum Atom Soon
237 The Assumed-Linearity Error Atom Soon
239 Evaluating versus Solving Atom Soon
241 One-to-One Functions Atom Soon
242 The Vertical Line Test Atom Soon
243 The Horizontal Line Test Atom Soon
245 Determining Whether a Relation Is a Function Procedure Soon
249 The Domain of a Function Atom Soon
250 The Range Atom Soon
252 Finding a Domain from an Equation Procedure Soon
254 Piecewise-Defined Functions Atom Soon
256 The Absolute Value Function Atom Soon
263 The Average Rate of Change Atom Soon
265 Increasing and Decreasing Functions Atom Soon
266 Local Maxima and Minima Atom Soon
273 The Algebra of Functions Atom Soon
275 Function Composition Atom Soon
276 Evaluating a Composite Function Procedure Soon
278 Composition Order Matters Atom Soon
281 Vertical Shifts Atom Soon
282 Horizontal Shifts Atom Soon
284 Reflections Atom Soon
285 Even and Odd Functions Atom Soon
287 Vertical Stretches and Compressions Atom Soon
288 Horizontal Stretches and Compressions Atom Soon
290 Graphing by Transformations Procedure Soon
293 Inverse Functions Atom Soon
294 The Reciprocal-Inverse Error Atom Soon
296 The Composition Test for Inverses Atom Soon
297 Finding an Inverse Algebraically Procedure Soon
299 Graphs of Inverse Functions Atom Soon
300 The Function Machine Capstone Soon
Unit 6

Linear and Quadratic Functions as Models

301 Linear Functions Atom Soon
302 Direction from the Slope Atom Soon
304 Slope as a Rate Atom Soon
306 Building a Linear Model Procedure Soon
308 A Linear Model from Two Data Points Procedure Soon
310 Where Two Lines Meet Atom Soon
312 The Zero of a Linear Function Atom Soon
321 Quadratic Functions and Parabolas Atom Soon
322 The Vertex and Axis of Symmetry Atom Soon
324 General Form and Standard Form Atom Soon
325 Reading the Vertex from Standard Form Atom Soon
326 The Vertex-Sign Error Atom Soon
327 The Vertex Formula Atom Soon
329 Converting General to Standard Form Procedure Soon
331 Maximum and Minimum Values Atom Soon
333 Zeros, x-Intercepts, Solutions Atom Soon
335 The Full Quadratic Workup Procedure Soon
337 Optimizing with a Quadratic Procedure Soon
350 The Parabola Dossier Capstone Soon
Unit 7

Polynomial and Rational Functions

351 Power Functions Atom Soon
352 End Behavior Atom Soon
354 End Behavior from the Leading Term Atom Soon
356 Zeros and Multiplicity Atom Soon
358 Turning Points Atom Soon
360 Sketching a Polynomial Procedure Soon
364 Polynomial Long Division Procedure Soon
366 Synthetic Division Procedure Soon
368 The Remainder Theorem Atom Soon
369 The Factor Theorem Atom Soon
371 Finding Zeros with the Factor Theorem Procedure Soon
376 Rational Functions Atom Soon
377 Vertical Asymptotes Atom Soon
378 Removable Discontinuities Atom Soon
379 The Erased-Hole Error Atom Soon
381 Horizontal Asymptotes Atom Soon
383 Analyzing a Rational Function Procedure Soon
386 Radical Functions and Their Domains Atom Soon
388 Restricting a Domain for an Inverse Atom Soon
391 Direct Variation Atom Soon
392 Inverse Variation Atom Soon
393 Joint Variation Atom Soon
395 Solving a Variation Problem Procedure Soon
410 The Anatomy of Higher Functions Capstone Soon
Unit 8

Exponential and Logarithmic Functions

411 Exponential Functions Atom Soon
412 Growth and Decay Atom Soon
414 Linear versus Exponential Change Atom Soon
416 Evaluating Exponential Functions Atom Soon
418 Compound Interest Procedure Soon
420 The Number e Atom Soon
421 Continuous Growth and Decay Atom Soon
423 Graphs of Exponential Functions Atom Soon
426 The Logarithm Atom Soon
427 Logarithms and Exponentials Are Inverses Atom Soon
429 Common and Natural Logarithms Atom Soon
431 Converting Between Forms Procedure Soon
433 Evaluating Logarithms Mentally Atom Soon
435 Graphs of Logarithmic Functions Atom Soon
446 The Product Rule for Logarithms Atom Soon
447 The Split-Sum Log Error Atom Soon
448 The Multiplied-Logs Error Atom Soon
450 The Quotient Rule for Logarithms Atom Soon
451 The Divided-Logs Error Atom Soon
453 The Power Rule for Logarithms Atom Soon
455 Expanding and Condensing Logarithms Procedure Soon
457 The Change-of-Base Formula Atom Soon
463 The One-to-One Property of Exponential Functions Atom Soon
465 Solving Exponential Equations with Logarithms Procedure Soon
467 The One-to-One Property of Logarithms Atom Soon
468 Solving Logarithmic Equations Procedure Soon
473 Doubling Time Procedure Soon
480 The Inverse Kingdom Capstone Soon
Unit 9

Systems, Counting, and the Binomial Theorem

481 Systems of Linear Equations Atom Soon
482 The Substitution Method Atom Soon
484 Solving by Substitution Procedure Soon
486 The Addition Method Atom Soon
488 Solving by Addition Procedure Soon
490 Independent, Dependent, Inconsistent Atom Soon
492 A System from a Story Procedure Soon
496 Systems in Three Variables Atom Soon
498 Solving a Three-Variable System Procedure Soon
501 Nonlinear Systems Atom Soon
503 Solving a Nonlinear System Procedure Soon
511 The Multiplication Principle Atom Soon
513 Factorials Atom Soon
515 Permutations Atom Soon
516 Combinations Atom Soon
518 Binomial Coefficients Atom Soon
520 The Binomial Theorem Atom Soon
522 Expanding with the Binomial Theorem Procedure Soon
524 A Single Term of an Expansion Atom Soon
530 Many Unknowns, Counted Ways Capstone Soon
536 The Error Families Capstone Soon
538 The Course, One Thread Capstone Soon
540 Where the Numbers End Capstone Soon

The receipts

Check my work.

Behind the course sits a reference document in which every rule and worked example is traced to its source. The anchor is an openly licensed university text — OpenStax’s College Algebra — and mathematics has an advantage the other courses do not: most claims here can be checked by derivation rather than citation. Every worked example is recomputed and logged before its video is made.

Spot an error? Tell me the video number and the step — corrections get made in the open. In a subject where the answer is checkable, a wrong step is a wrong step, and it gets fixed.

Follow along

New videos daily.

The course releases in order on YouTube — one idea at a time until all 263 are live. Subscribe to catch each one, or leave your email for an occasional note when a course finishes.