bythreedu · Math Tracks · Curriculum v1.0 · 2026

Variables, equations,
and the grammar of
mathematics.

A complete algebra course from the number system through quadratics, rational expressions, and exponential functions. Every rule is derived before it is used. Students who finish this track don't just know algebra — they understand it well enough to reconstruct it when they need to.

Duration~24weeks
Total Lessons24at your pace
Units7progressive phases
Assignments7+problem sets
DestinationGeometry &
Precalc Ready
00Approach
Course Philosophy

Rules Have Reasons

Every algebraic rule exists for a reason. Why does distributing multiplication over addition work? Why does the quadratic formula look the way it does? This course answers those questions before asking you to use the results. Students who know the reason can recover the rule. Students who only memorized it are stuck when they forget it.

Setup Before Solving

Most algebra errors happen before a single calculation — in how the problem is set up. This course puts heavy emphasis on translating situations into algebra correctly before touching any computation. Every word problem requires a written setup step. Every equation requires identifying what is being solved for.

01Curriculum
Learning Path
01
Unit 1 · Weeks 1–2
The Real Number System
3 lessons · Number sets · Properties · Order of operations
number setsproperties of realsorder of operations
+
L1
Number Classification
Integers, rationals, irrationals — what makes each set distinct. Why irrational numbers exist and what they mean. The real number line as a complete picture.
L2
Properties of Real Numbers
Commutative, associative, identity, inverse, and distributive properties as tools, not facts to memorize. Using properties to justify each step in a simplification.
L3
Order of Operations
Why the convention exists and what breaks without it. Multi-step expressions evaluated carefully. The most common mistakes — identified and corrected at the root.
Assignment 1
Evaluate 12 expressions using only the order of operations. For 5 of them, identify which property justifies each step. No calculators.
You understand the rules governing every manipulation you will do for the rest of the course.
02
Unit 2 · Weeks 3–5
Linear Equations and Inequalities
4 lessons · Solving equations · Inequalities · Systems · Applications
inverse operationssystemsinterval notation
+
L4
Solving One-Variable Equations
Inverse operations, multi-step equations, special cases (no solution, infinite solutions). The balance model applied rigorously.
L5
Inequalities
Solving inequalities — why the direction flips when multiplying or dividing by a negative. Interval notation. Compound inequalities and how to interpret "and" vs "or."
L6
Systems of Linear Equations
Substitution and elimination — when each is faster. Identifying special systems (no solution, infinitely many) before solving. Applications with two unknowns.
L7
Word Problems and Modeling
Translating word problems into equations. Writing the setup paragraph before any algebra. Checking whether the answer makes sense in context.
Assignment 2
Write and solve 5 word problems — each must include a written translation step and a context check before the answer is accepted.
You build and solve any linear model from a real-world description.
03
Unit 3 · Weeks 6–8
Linear Functions and Graphing
3 lessons · Slope · Three forms of a line · Graphing systems
slope as ratethree forms of a linesystem solutions
+
L8
Slope and Slope-Intercept Form
Slope as a rate of change — rise over run as a ratio, not a trick. Slope-intercept form as the most readable form. Interpreting slope and intercept in context.
L9
Point-Slope and Standard Form
When each form is the right tool. Converting between all three. Writing the equation of a line from two points or a point and a slope.
L10
Graphing Systems
Intersection as a solution, parallel as no solution, same line as infinite solutions — all three cases read geometrically before solving algebraically.
Assignment 3
Graph one system three ways: by hand, by table, and by algebra. Write one sentence on what each method reveals that the others don't.
You see a linear function as a relationship, not just a formula.
04
Unit 4 · Weeks 9–12
Polynomials and Factoring
4 lessons · Polynomial operations · Factoring methods · Factoring strategy
degree & leading coefficientfactoring methodsdifference of squares
+
L11
Polynomial Operations
Adding, subtracting, multiplying polynomials. FOIL as a special case of distribution, not a separate method. Degree and leading coefficient as descriptors.
L12
GCF, Grouping, and Factoring Trinomials (a=1)
Factoring as reverse distribution. Finding the GCF first. Grouping for four-term polynomials. Trinomial factoring when the leading coefficient is 1.
L13
Factoring When a ≠ 1
The AC method explained and applied. Trial and error as a backup strategy. Difference of squares and perfect square trinomials as patterns worth recognizing.
L14
Factoring Strategy
Reading a polynomial and choosing the right method. The four-step strategy: GCF first, then count terms, then pattern match. Knowing when a polynomial is prime.
Assignment 4
Factor 20 polynomials using the strategy checklist. Document which method you used for each and why. Any polynomial labeled "prime" must have a written justification.
You can factor anything and you know why factoring matters.
05
Unit 5 · Weeks 13–16
Quadratic Equations and Functions
4 lessons · Three solving methods · Discriminant · Graphing parabolas
discriminantvertex formcompleting the square
+
L15
Solving by Factoring and Square Roots
Zero product property derived from the multiplicative property of zero. Solving by square roots when the equation has no linear term. Knowing when each is fastest.
L16
Completing the Square
The derivation that makes the quadratic formula make sense. Why completing the square works geometrically. Converting from standard to vertex form.
L17
The Quadratic Formula and Discriminant
The formula derived from completing the square. The discriminant as a preview — what it tells you before you solve. Two real, one real, no real solutions read from b² - 4ac.
L18
Graphing Parabolas
Vertex, axis of symmetry, roots, direction, and y-intercept. Connecting the algebraic solution to the graphical picture. Vertex form as the most graphing-friendly form.
Assignment 5
Solve the same 5 quadratics all three ways — factoring, completing the square, and the quadratic formula. Note which method was clearly fastest for each and why.
You have three tools for quadratics and know when to reach for each.
06
Unit 6 · Weeks 17–20
Rational and Radical Expressions
4 lessons · Rational expressions · Radical expressions · Rational exponents
LCD for polynomialsrationalizingrational exponents
+
L19
Simplifying Rational Expressions
Factoring first, then canceling common factors — never before. Identifying excluded values. Why canceling terms (not factors) is the most common rational expression error.
L20
Adding, Subtracting, Multiplying, Dividing Rational Expressions
LCD of polynomials found by factoring each denominator. Multiplying and dividing — invert and multiply derived, not just stated. Complex fractions simplified by multiplying by the LCD.
L21
Radical Expressions
Simplifying radicals by factoring out perfect squares. Rationalizing the denominator — why it matters and how to do it. Adding and subtracting radical expressions by combining like radicals.
L22
Rational Exponents
The connection between radical notation and fractional exponents. a^(m/n) defined and applied. Properties of exponents extended to rational exponents.
Assignment 6
Simplify 10 expressions — 5 rational, 5 radical. Show every step and identify where the most common errors typically hide in each type.
You work with fractions and roots involving variables as fluently as with integers.
07
Unit 7 · Weeks 21–24
Exponential Functions and Capstone
3 lessons + capstone · Exponential growth · Function inverses · Final synthesis
exponential vs linearfunction compositioninverse functions
+
L23
Exponential Functions
Growth and decay models. Base e introduced. The key difference between linear and exponential growth — constant difference vs. constant ratio.
L24
Function Notation, Composition, and Inverses
f(g(x)) computed and interpreted. Finding the inverse algebraically. Domain restrictions that make an inverse a function. Verifying inverses using composition.
L25 (cap)
Capstone Review and Synthesis
Student-directed review of weakest units. A student-written 10-problem set covering every unit — solved, then reviewed and critiqued together in the final session.
Capstone
Build and solve a 10-problem set that touches every unit. Problems written by you, solved by you, explained in writing. The final session is a walkthrough and critique.
You can set up, solve, and explain any algebraic problem a gateway math course throws at you.
02Reference
Core Concepts
ax+b
Linear Equations
A linear equation in one variable has exactly one solution, no solution, or infinitely many solutions. The goal of solving is always to isolate the variable using inverse operations.
Inverse operations · Balance model · Special cases
m
Slope
Slope is the rate of change of y with respect to x — rise over run. A positive slope goes up left to right, negative goes down. Slope is constant for any two points on a line.
Slope-intercept · Point-slope · Standard form
△
Factoring Strategy
Factor out the GCF first. Then count terms: 4 terms suggests grouping, 3 terms suggests trinomial methods, 2 terms suggests special patterns. Always check for difference of squares.
GCF · AC method · Special patterns
b²-4ac
Discriminant
The discriminant b²-4ac tells you how many real solutions a quadratic has before you solve: positive → two real, zero → one real, negative → no real solutions.
Quadratic formula · Vertex form · Parabola
f(x)
Function Notation
f(x) means the output of function f when the input is x. It is not f times x. f(3) means substitute 3 for x everywhere in the formula.
Domain · Range · Composition
a^x
Exponential Growth
Exponential functions grow by a constant factor each period. Linear functions grow by a constant amount. The difference: constant ratio vs. constant difference.
Base e · Growth vs. decay · Inverses
03Capstone
Student-Built Final Review

The algebra capstone flips the script: you write the problems. Building a problem set that covers every unit forces you to think about the material from the inside out. The final session is a walkthrough and honest critique of what you built and how you solved it.

○
Gap Review Sessions
You identify the two units you're least confident in. We revisit them before the capstone problem set.
○
Student-Written Problem Set
10 problems, one per unit, written by you. Forces active recall and reveals how well you understand each topic.
○
Walkthrough and Critique
The final session reviews every problem — your solutions and your problem choices. Honest feedback on both.
○
Geometry and Precalc Readiness
A final mapping of your algebra skills to what the next track will require from day one.
04Investment
Pricing Options
per Hour
$85
1-hr lesson
Flexible · No commitment
Online only
  • Flexible scheduling
  • 1-on-1 focused attention
  • No long-term commitment

Best for trying the curriculum before committing.

Book a session
Popular
per Month
$650
4 lessons · 1.5 hours/lesson
1 lesson/week · book more to move faster
Online + In-Person available
  • Session notes after every lesson
  • Async DM support between sessions
  • Priority scheduling

Booking 2x per week draws down lessons faster — no extra charge.

Get started
Full Track
$3,120
24 lessons · 1.5 hours/lesson
~6 months · save vs. monthly
Online + In-Person available
  • Full course arc — all 7 units
  • Session notes after every lesson
  • Capstone critique session
  • Geometry/Precalc readiness review

The complete Algebra arc in one commitment.

Enroll now
In-person sessions available on Monthly and Full Track plans at $800/mo and $4,320 respectively. NYC only. Per-hour sessions are online only.
05Contact
Let's Talk

Not sure if this is the right level? One session is enough to run a quick diagnostic and figure out exactly where to start.

Is this Algebra I or Algebra II? +
This track covers Algebra I completely and includes the core Algebra II topics that matter for precalculus: rational expressions, radical expressions, exponential functions, and function composition. If you're specifically preparing for an Algebra II class, let me know and we'll adjust the focus accordingly.
What do I need to know coming in? +
Comfortable arithmetic with integers, fractions, and decimals. If those feel shaky, the Pre-Algebra track is the right starting point. We can figure out which is right in the first session.
Are lessons online or in-person? +
Per-hour sessions are online only. Monthly and Full Track plans include the option for in-person sessions in NYC. In-person is priced slightly higher — see pricing above.
What if I need to reschedule? +
Give 24 hours notice and we'll find another time. Monthly and Full Track students get priority rebooking.