Math and natural sciences

How to learn math from scratch

An adult coming back to math rarely needs to start with the times tables — usually the foundation breaks in one particular place: fractions, negative numbers, equations. The plan below starts by finding that place and then leads through arithmetic and algebra to functions and geometry, without school pressure or grades.

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First, find where the foundation broke#

School math is a staircase: percentages do not work without fractions, equations do not work without negative numbers, and graphs do not work without equations. So the first session is a diagnosis. Take a pre-algebra workbook and solve two or three problems from every chapter. Where the doubts begin is your starting point. This saves weeks: adults often spend effort on what they have known for years and skip over the one gap that makes everything after it collapse.

Stage 1: the arithmetic you will use forever#

Order of operations, fractions, decimals, percentages, negative numbers, exponents. It sounds dull, but this is the part you need in everyday life — working out a discount, an interest rate, a ratio — and in all the math that follows. The readiness test is simple: you can do fractions with unlike denominators and turn a percentage into a decimal without a calculator and without pausing to remember the rule. If you hesitate, the pages on operations with decimals and percentage word problems are the place to go back to.

Stage 2: algebra — letters instead of numbers#

Expressions with variables, expanding brackets, special products, linear equations and inequalities, systems of two equations, quadratic equations. The key skill of this stage is seeing an equation as a balance: whatever you do to one side, you do to the other — the idea behind how to solve linear equations. Write each step on its own line; the temptation to do two transformations in your head is the main source of lost minus signs.

Stage 3: functions and graphs#

The linear function, the parabola, the hyperbola, what a domain is and how to read a function's behaviour from its graph. Plot several graphs by hand from a table of points, then check yourself in a free graphing tool such as Desmos. Functions are the bridge to physics, statistics and calculus, so do not rush this stage.

Stage 4: geometry and trigonometry#

Areas and volumes, similar figures, the Pythagorean theorem, sine and cosine in a right triangle, then the unit circle. Learn geometry with a pencil: every problem starts with a careful drawing, and half the solution is often visible on it already. Area and perimeter is a good first check that the basic formulas are in place.

How much to study and how to keep the pace#

Aim for 30–45 minutes a day, four or five times a week. At that pace arithmetic and algebra usually take two or three months, and functions and geometry about as long again. Short daily sessions work better than one long one at the weekend: math fades quickly when you leave it alone. Start every session with two problems from the previous topic — that is free review.

How to tell a topic is learned#

Reading a solution and agreeing with it is not the same as solving. A topic is done when, a few days later, you solve a new problem on it without peeking and can explain every step out loud. If you are stuck for more than twenty minutes, look at only the first step of the solution and carry on yourself.

Where adults go wrong most often#

Adults rarely go wrong where they expect to. The trouble is seldom "no talent for math" — usually it is a minus sign lost when expanding brackets, dividing by an expression that could be zero, or percentages taken of different bases: a price went up 20%, then down 20%, and ended up lower than at the start. Keep a list of these traps and reread it before every new topic. In a month it will be shorter, and that is the best sign that arithmetic and algebra have settled into place.

Step-by-step plan

  1. Week 1 — diagnosisSolve a few problems from every chapter of a pre-algebra workbook and find where the mistakes begin.
  2. Month 1 — arithmeticFractions, percentages, negative numbers, exponents and order of operations without a calculator.
  3. Months 2–3 — algebraExpressions, linear and quadratic equations, systems, inequalities — every step on its own line.
  4. Months 4–5 — functionsGraphs of a linear function, a parabola and a hyperbola by hand, then checked in a graphing tool.
  5. Months 5–6 — geometryAreas, similar figures, the Pythagorean theorem, sine and cosine; a drawing for every problem.

Start learning this in your own space

The plan goes into your repository: tick off stages, keep notes — the change history shows how far you have come.

Start the plan

Check yourself

1.Work out 3/4 + 1/6

2.Solve 2x + 5 = 17. What is x?

3.What is 15% of 240?

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