bythreedu · Math Tracks · Curriculum v1.0 · 2026

Numbers, operations,
and the start of
mathematical thinking.

A ground-up pre-algebra course built for students who need to own the fundamentals before moving on. No rushing. No assuming. Every concept is explained from first principles so the student builds a foundation that actually holds when algebra starts.

Duration~20weeks
Total Lessons20at your pace
Units7progressive phases
Assignments7+problem sets
DestinationAlgebra I
Ready
00Approach
Course Philosophy

No Assumptions About What You Already Know

Pre-algebra students often have patchwork knowledge — solid in some areas, shaky in others. This course starts from the number system itself and builds forward. If something was taught badly the first time, we fix it here before it becomes a ceiling in algebra.

Understanding Before Procedure

Every rule in pre-algebra has a reason. Why do two negatives make a positive? Why does order of operations work the way it does? Students who know the reason can reconstruct the rule when they forget it. Students who only memorized the rule are stuck.

01Curriculum
Learning Path
01
Unit 1 · Weeks 1–2
The Number System
3 lessons · Integers · Fractions · Decimals and place value
number setsintegersplace value
+
L1
Integers and the Number Line
What integers are, where they live on the number line, and what negative numbers actually mean in context. Absolute value as distance, not just "remove the sign."
L2
Fractions: What They Actually Are
A fraction as a division problem and as a part-of-a-whole. Equivalent fractions derived from multiplying by 1. Simplifying by finding the GCF, not by guessing.
L3
Decimals and Place Value
Place value extended past the decimal point. Converting between fractions and decimals in both directions. Repeating decimals explained, not just noted.
Assignment 1
Place 15 numbers on a number line — mix of integers, fractions, and decimals. Order them from least to greatest. Write one sentence explaining how you decided where each went.
You know what every type of number is and where it lives relative to every other type.
02
Unit 2 · Weeks 3–5
Operations with Integers and Fractions
3 lessons · Integer operations · Fraction arithmetic · Order of operations
integer rulesfraction arithmeticorder of operations
+
L4
Integer Operations
Adding, subtracting, multiplying, and dividing integers — each rule derived from the number line or from patterns, not handed down as laws. Why a negative times a negative is positive, explained.
L5
Fraction Arithmetic
Adding and subtracting fractions with unlike denominators using the LCD. Multiplying and dividing fractions — why dividing by a fraction means multiplying by its reciprocal.
L6
Order of Operations
PEMDAS as a convention, not a rule handed down from on high. Why the convention exists. Multi-step expressions evaluated carefully. Common mistakes identified and corrected.
Assignment 2
Evaluate 15 expressions involving integers and fractions. For each, write the operation you performed at each step and why it came before the others. No calculators.
You compute any arithmetic expression correctly and know the reason behind every step.
03
Unit 3 · Weeks 6–8
Ratios, Proportions, and Percents
3 lessons · Ratios · Proportional reasoning · Percent problems
ratiosunit ratespercent
+
L7
Ratios and Unit Rates
What a ratio compares and how to express it three ways. Unit rates as ratios with denominator 1 — and why they're the most useful form for comparison.
L8
Proportions and Cross-Multiplication
Setting up a proportion correctly. Cross-multiplication derived from multiplying both sides by both denominators — not as a magic trick. Scaling problems and map problems.
L9
Percents
Percent as "per hundred" — converting between percent, decimal, and fraction. Finding the percent of a number, finding what percent one number is of another, and finding the whole when the percent is known.
Assignment 3
Solve 5 real-world proportion problems and 5 percent problems. Each must include a written setup sentence before any calculation. Show units throughout.
You solve any ratio, proportion, or percent problem by setting it up correctly before calculating.
04
Unit 4 · Weeks 9–11
Introduction to Variables and Expressions
3 lessons · Variables · Algebraic expressions · Combining like terms
variablesexpressionslike terms
+
L10
What a Variable Is
A variable as a placeholder for an unknown quantity — not a mystery letter. Writing expressions from word descriptions. The difference between an expression and an equation.
L11
Evaluating and Simplifying Expressions
Substituting values into expressions. The distributive property applied to algebraic terms. Identifying and combining like terms — why unlike terms can't be combined.
L12
Writing Expressions from Situations
Translating English into algebra. The vocabulary of operations: sum, difference, product, quotient, twice, increased by. Avoiding the most common translation mistakes.
Assignment 4
Write and simplify expressions for 8 word descriptions. Then evaluate 6 expressions for given variable values. Show all substitution steps explicitly.
You translate any arithmetic situation into algebra and work with expressions fluently.
05
Unit 5 · Weeks 12–14
Solving One-Step and Two-Step Equations
3 lessons · One-step equations · Two-step equations · Word problems
inverse operationsbalance methodequation setup
+
L13
One-Step Equations
The balance model: whatever you do to one side, you do to the other. Solving using inverse operations for all four operations. Checking solutions by substituting back.
L14
Two-Step Equations
Unwrapping the equation in the right order — why you undo addition/subtraction before multiplication/division. Special cases: equations with fractions, negative coefficients.
L15
Setting Up Equations from Word Problems
Identifying the unknown, defining the variable, writing the equation, solving, and answering the question asked — not just finding x. The five-step word problem process.
Assignment 5
Solve 10 equations — 5 one-step, 5 two-step. Check every answer. Then write and solve 3 word problems using the five-step process. Show each step explicitly.
You solve any one- or two-step equation and can set one up from a description.
06
Unit 6 · Weeks 15–17
Geometry Foundations
2 lessons · Area and perimeter · Angles and basic geometry
area & perimeteranglesgeometric reasoning
+
L16
Area and Perimeter
Area and perimeter of rectangles, triangles, parallelograms, and circles. Each formula derived geometrically, not memorized. Why pi appears in circle formulas.
L17
Angles and Geometric Relationships
Types of angles. Complementary and supplementary relationships. Vertical angles. Angle sums in triangles and quadrilaterals. Setting up and solving angle equations.
Assignment 6
Find area and perimeter for 6 figures from given dimensions. Then find missing angles in 5 diagrams, showing the equation you set up for each.
You apply geometric formulas with understanding and set up equations to find missing measurements.
07
Unit 7 · Weeks 18–20
Data, Graphs, and Capstone
2 lessons + capstone · Reading graphs · Basic statistics · Final review
mean median modegraphsdata interpretation
+
L18
Mean, Median, Mode, and Range
Each measure of center and spread defined and computed. When each is the most useful summary. What an outlier does to the mean vs. the median.
L19
Reading and Interpreting Graphs
Bar graphs, line graphs, pie charts, and scatter plots. What each is designed to show. Reading values correctly and identifying misleading graphs.
L20
Capstone Review and Gap Work
Student-directed review of the two weakest units. Full diagnostic problem set covering all 7 units, reviewed together with written corrections on every missed item.
Capstone
A comprehensive problem set covering all 7 units. Every missed problem gets a written correction — not just the right answer, but where the reasoning went wrong and how to fix it.
You walk into Algebra I with every pre-algebra concept solid and no gaps left to paper over.
02Reference
Core Concepts
−
Negative Numbers
A negative number represents a value less than zero — a distance below the starting point on the number line. Absolute value is the distance from zero, always positive.
Integers · Opposites · Number line
%
Percent
Percent means 'per hundred.' 45% is 45 out of 100, or 0.45 as a decimal, or 9/20 as a fraction. All three forms represent the same value.
Percent ↔ Decimal ↔ Fraction
x
Variables
A variable is a placeholder for an unknown quantity. It is not a mystery — it is a number we don't know yet. Writing x + 3 = 7 is asking: what number plus 3 equals 7?
Expressions · Equations · Translation
a/b
Fractions
A fraction represents division: a/b means a divided by b. Equivalent fractions represent the same value — multiplying numerator and denominator by the same number is multiplying by 1.
LCD · Simplifying · Mixed numbers
△
Area vs. Perimeter
Perimeter is the total distance around a figure — a length. Area is the amount of surface inside — measured in square units. They are different quantities with different units.
Units matter · Composite figures · Circles
●
Mean vs. Median
The mean is the arithmetic average — sensitive to outliers. The median is the middle value — resistant to outliers. For skewed data, the median is usually the better summary.
Mode · Range · Outliers
03Capstone
Diagnostic Review and Algebra Readiness

The capstone isn't a test for a grade. It's a diagnostic that shows exactly where you are before moving into algebra. Every gap gets addressed before the track ends so you start Algebra I from a position of actual readiness, not assumed readiness.

○
Gap Identification
Student identifies the two units they feel least confident in. We do a targeted review before the final problem set.
○
Comprehensive Problem Set
20 problems covering all 7 units. Completed independently, then reviewed together.
○
Written Corrections
Every missed problem corrected in writing — the reasoning, not just the answer.
○
Algebra I Readiness Map
A final session mapping what you've built to what Algebra I will expect 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 to a plan.

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 delivered 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
$2,600
20 lessons · 1.5 hours/lesson
~5 months · save vs. monthly
Online + In-Person available
  • Full course arc — all 7 units
  • Session notes delivered after every lesson
  • Capstone review + written corrections
  • Algebra I readiness session at completion

The complete Pre-Algebra arc in one commitment.

Enroll now
In-person sessions are available on Monthly and Full Track plans at $800/mo and $3,600 respectively. In-person is available within NYC. Per-hour sessions are online only.
05Contact
Let's Talk

Not sure where to start? One session is enough to figure out exactly where the gaps are and what the path forward looks like.

Is pre-algebra only for younger students? +
Not at all. Adult returners who are going back to school for nursing, business, or tech often need a solid pre-algebra foundation before anything else. The content is the same; the examples and pacing adjust to who's in the room.
How do I know if I need pre-algebra or algebra? +
If fractions, negative numbers, or basic equation solving feel shaky, start here. If those feel solid but graphing lines and solving systems are the challenge, Algebra is the right track. We can run a quick diagnostic in the first session if you're unsure.
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.