02 / The beauty of a pattern
Mathematics
From number sense to calculus and statistics, we make each step mean something. Build the foundations, connect the ideas, and solve the next problem with confidence.
Try the graph lab Find your starting point
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The lowest point, the vertex, sits at (2, −1). The curve opens upward because the squared term is positive.
The gold line just touches the curve. Its slope is x − 2: negative on the left, zero at the vertex, positive on the right.
8 stages, from grade‑school number sense to Calculus II
2 extra tracks: Statistics and SAT/ACT
55 minutes per session
Try it
A taste of how we teach: see the idea move, then prove you own it on paper. The slider builds intuition; the pencil builds understanding.
Drag each slider and watch what that one letter does. Then take a challenge: match the dashed curve.
Turn on JavaScript to drag the sliders and draw the graph.
Pencil check: before you touch a slider, predict on paper what happens to the vertex when c goes up by 2. Then test it. Predicting first is what turns a cool animation into actual learning.
The solution unlocks one step at a time, and only after you've tried it yourself. That's how our sessions work too.
A rectangle's length is 3 cm more than its width. Its area is 40 cm². How wide is it?
w, so the length is w + 3.w(w + 3) = 40.w² + 3w − 40 = 0.(w + 8)(w − 5) = 0, so w = −8 or w = 5.Where AI fits: once you've solved it, ask an AI to write three more problems "like this one, answers hidden" — and solve those on paper too.
Personal plan
No two students get stuck in the same place. The first session is a diagnostic that finds the exact idea that didn't land; from it, the student gets a written plan — and the plan changes as they do.
We trace today's trouble back to its root — often a fraction or equation idea from years ago — and write down what we'll fix, in what order, and how you'll see it working.
Practice is chosen for this student and moves at their speed. Their interests become the examples: batting averages, game scores, music tempos, the price of the shoes they want.
Online or in person, evenings and weekends, weekly sessions or an exam-season sprint. Rescheduling is free up to 12 hours before, and you get a short note after every session.
One-on-one by default (small groups or siblings only if you ask for one), and never a rotating substitute. The person who found the gap is the person who closes it — and who notices the day it's closed.
Illustrative examples — the student's plan is written after their first session.
Real-world math
Lessons are led by trained professionals with in-industry experience, so the examples come from real work: data that has to be right, money that has to add up, and software that has to pass its tests.
A model that picks fight winners 67% of the time sounds impressive — until you compare it with the 50% you'd get by guessing. Students learn to ask that of every number: compared with what?
Budgets, discounts and interest on real numbers: why 20% off and then 20% on isn't the starting price, and what a credit card's interest rate does over a year.
A map's arrival time is distance over speed; a phone plan is a piecewise function; a game's level-up curve is exponential. Once students see the function, the graph stops being abstract.
If–then reasoning, proof by cases and counterexamples are the same moves engineers use to test code: check every case, then hunt for the one input that breaks it. Geometry proofs and discrete math become a skill that lasts well beyond the exam.
We teach the math we use at work — and it always starts on paper.
Taught by Mohammed Arafat — senior software engineer, 10+ years in industry. Meet your tutor
The track
Math is cumulative — a shaky idea in one stage becomes a wall in the next. Here's everything we teach, in order. Most students work in one stage at a time, with a light thread back to repair whatever is wobbly.
Where confidence with numbers is built — or quietly lost. We make place value and the four operations automatic, treat fractions as real numbers on a number line, and turn word problems into pictures before they become equations. Mostly hands-on: blocks, number lines, paper and talk.
The stage that decides how the rest of math feels. We rebuild number sense so fractions stop being guesswork, make negative numbers automatic, and get comfortable turning a sentence of English into an equation.
Where most students first hit real difficulty. We treat equations as a handful of moves instead of a page of rules, learn to read a graph the way we read a sentence, and build factoring reflexes that every later course quietly assumes.
The subject that teaches mathematical argument — which students have usually never seen before. We turn proofs from a mystery into a checklist, drawn by hand with a ruler and compass, and keep the algebra sharp for the next course.
The bridge to everything after it: the function zoo — exponentials, logs, rationals — plus the trigonometry that calculus leans on hard. Students who leave this stage fluent in functions find calculus much friendlier.
Consolidation and preview. We sharpen function analysis (domains, ranges, transformations, inverses), get fluent with trig on the unit circle, and take a first honest look at limits so calculus feels like a continuation rather than a cliff.
Derivatives are a way of asking "how fast is this changing?", and integrals reverse the question. Once that idea clicks, the algebra becomes practice rather than panic. Strong focus on AP Calculus AB topics and first-semester college work.
The techniques course: integration becomes a toolbox, and infinite series become a way of writing functions no formula can capture. This is where organised handwritten work and pattern recognition matter most — and where a shaky grade is most often turned around.
Also on the syllabus
High-school stats, AP Statistics, and intro college statistics: distributions, sampling, confidence intervals, hypothesis tests, and regression. Taught with real data wherever possible — statistics is much easier when it's about something that matters to the student.
At college level, the same care extends to discrete math, linear algebra (intro) and college statistics courses.
Targeted prep: timed section work, the question patterns these tests repeat, and the specific content gaps that cost the most points. Usually four to six sessions alongside regular class work, not instead of it.
Test prep
From a Grade 3 state test to a college final, we prepare the same calm way: find what is costing points, fix it, then practise under the real conditions.
A real past paper or official practice test, taken cold, so we know exactly where the points are going.
We rank the gaps by the points they cost and rebuild those ideas first, by hand — not another hundred random questions.
Full timed sections under the exam's own rules: on paper where the exam is on paper, with exactly the calculator it allows.
Calm practice on the skills the state test samples, plus the unit tests and quizzes that make up the school grade.
The question patterns the digital SAT repeats, the built-in calculator used on purpose, and the content gaps that cost the most points.
Pacing practised as a skill, across the test's whole range from pre-algebra to trigonometry.
Function modelling, and the written justifications free-response graders look for.
Free-response answers written out in full — the work earns the points, not just the final number.
Interpreting results in context, the four-step inference write-up, and the calculator rules of the real exam.
Placement tests, so the first college course is the right one, and finals in calculus, statistics or discrete math.
Pencil & AI in math
Math is the subject where AI shortcuts hurt most: an app can produce the answer, but the test asks for the steps. So we build the steps by hand — and then show students how to make AI a sparring partner.
My math lessons begin with hands-on thinking: pencil, paper, ruler and compass, a whiteboard, and conversation.
Screens come out for a reason — to check a graph, to see an idea move — and go away again.
Inside a session
Every session follows the same shape, whatever the level. The student does the mathematics, pencil in hand; we ask the questions that make the thinking visible. When they can explain a problem out loud — including where it gets hard — we know the idea is actually theirs.
Notes and a short summary come to you after every session: what we covered, what clicked, and the one thing to watch this week.
We look at a recent assignment or test together and find the precise skill the trouble traces back to.
That earlier idea gets re-explained from scratch — with the pictures and examples the textbook skipped.
Short, targeted problems, ordered from easy to exam-style, with "why?" asked at every step.
The student solves a fresh problem alone and explains it back. That's the bar for "we've got it".
Questions
That sentence almost always means one or two specific gaps — often fractions or equation-solving from years ago — never a lack of ability. We find the gap, fill it, and the confidence usually follows within a few weeks.
They can get answers that way, but tests and later courses ask for the reasoning — and that only comes from doing it. We teach students to use those tools after an honest attempt: to check work, find a mistake, or generate more practice. Used like that, they're genuinely helpful.
It's tight but workable. We'd focus on the highest-yield topics for that test and skip everything else. Even two or three focused sessions usually move a grade meaningfully — tell us the date in the form.
Homework help is fine and often where we start, but the goal is the student not needing us for the next set. So alongside the homework we're usually reteaching the idea underneath it. That's what makes grades stick.
AB is roughly one college semester: limits, derivatives, and integration. BC covers a year, adding advanced integration techniques and series. We teach both.
Not unless the course demands it. The calculator isn't the problem — understanding what to type is. We keep arithmetic sharp, but we don't manufacture struggles the exam won't ask for.
Very common, and very workable. Sessions move at the student's pace, mistakes are treated as information rather than failure, and we check in on how it feels, not just how the answers look.
Keep exploring
Tag the parts of a real paragraph and watch a sentence get revised in three drafts.
Step through sorting algorithms and predict what code prints before you run it.
Why we teach on paper — and how we teach AI as a tool. Try the "tool or crutch?" quiz.
Next step
A sentence is enough to start. Describe the topic, the test date if there is one, and what's been frustrating — we'll reply within one business day with what we'd suggest.