Chris Collins

Tools · math

Maths practice gym

Practise the skills from arithmetic to intro calculus — a skill tree of 33 skills, generated problems checked by equivalence, step-by-step checking that names your mistakes, timed drills, an adaptive placement test, and spaced-repetition review, with progress kept in your browser.

Added

How to use it

Exercise the skills on the road to precalculus and calculus. Every problem is generated — with nice numbers, always solvable, never degenerate — so there is always another one, and the same seed always gives the same problem, so a link can share one exactly.

How to use it

  • Skill map — 33 skills from arithmetic and number sense through fractions, powers, algebra, polynomials, quadratics, functions, exponentials and logarithms, trigonometry, sequences and vectors, to limits and derivatives. Lines join each skill to the ones that build on it. A skill opens when its prerequisites are mastered (a dashed box is still locked — you can practise it anyway, or turn on free browsing on the Progress tab). Each box fills with your mastery; an orange dot means it is due for review. Click a skill for its description, prerequisites, a link to its chapter of Mathematics, Reconnected and links into the explorer tools (the unit circle, graphing, algebra step by step, the CAS, calculus, vectors…). ⛶ Full screen (or F) shows the map on the whole screen; Esc leaves.

  • Practice — pick a skill and a level (★ marks the level your mastery suggests). The first time you open a skill you get a worked example before your first problem. Answers are checked by value, not by spelling: x^2 + 5x + 6, 6 + 5x + x² and y = x*x + 5x + 6 are all the same answer, and 3/4, 0.75 and 6/8 the same number. Where the task asks for a form the form is checked too: factor needs a complete factorisation (2(x − 3)(x + 3), not 2x² − 18 or 2(x² − 9) or (2x − 6)(x + 3)), expand needs no brackets and like terms collected, simplify needs a single fraction with no common factor, and some answers ask for lowest terms, an exact value (sqrt(3)/2, 2pi/3), scientific notation (3.2*10^5 or 3.2e5), a simplified surd (6sqrt(2)) or vertex form. An answer that is equal but in the wrong form gets "Equivalent, but …" and doesn’t count as a miss. Wrong answers are checked against the common mistakes for that problem — adding numerators and denominators, (a + b)² = a² + b², forgetting to flip an inequality, a root with the wrong sign, losing the second solution of sin x = ½, keeping a log answer outside the domain — and the mistake is named with a hint. Hint gives up to three graduated hints (each costs a little mastery); Show answer shows the answer and the worked solution. Some problems are multiple choice (the wrong choices are the common mistakes), some are graph sketches — drag two points to draw a line, or a vertex and a point to draw a parabola or |x| — and some are unit-circle drills: click the point at 5π/6.

  • Step mode — solve, expand or factor one line at a time. Each line is checked: for an equation, that it has exactly the same solutions as the line before; for an expression, that it is equal to it — and whether it is closer to the goal. The checker names sign errors (moving a term across the = without changing its sign), an exponent spread over a sum, multiplying only part of a bracket, cancelling terms instead of factors, dividing by something that can be zero (x² = 6x → x = 6 loses x = 0), losing the ± of a square root, and steps that add solutions. x − 1 = ±3 and x = 4 or x = −2 are fine lines. Show next step takes the next line of the worked solution from where you are (using the same engine as Algebra step by step).

  • Timed drill — one, two or five minutes against the clock, no hints; arithmetic level 1–3 makes a mental-maths round, the unit circle a quick-recall round of exact values. Your best score per skill, level and time is kept.

  • Placement test — an adaptive test that finds where to start: a binary search along the skill tree, about ten questions (two right in a row to pass a skill; one wrong moves down). Apply the result to mark what you know as mastered.

  • Review — spaced repetition (SM-2): a skill you get right comes back after 1 day, then 6, then the interval times an ease factor that grows when it is easy and shrinks when it isn’t (never below 1.3); a miss brings it back in 10 minutes and starts again. A second success on the same day doesn’t stretch the interval. Recently missed problems can be retried exactly.

  • Progress — skills mastered, problems answered, a 28-day activity chart, mastery and next review date per skill, an optional day streak, and Export / Import JSON.

Mastery is a running estimate between 0 and 100 %: each answer moves it 30 % of the way towards the score for that attempt (level 1 counts up to 70 %, level 2 92 %, level 3 100 %, less 20 % per hint; a miss or a revealed answer scores 0). A skill is mastered at 80 %, so it takes level 2 or 3.

Type powers with ^, roots with sqrt (or √), π as pi, and ≤ ≥ as ⇐ >=; the line under the box shows how your input was read. The chips under the box type the awkward symbols on a phone.

Your progress is stored only in this browser (IndexedDB). It is never put into share links and never sent anywhere; export it to keep a copy or move it to another browser.

Limits

Equivalence is decided by evaluating both expressions at a dozen sample points (with exact polynomial arithmetic for the form checks), as a CAS would numerically; two expressions that agree everywhere they are both defined count as equal, so ln(x²) and 2 ln x are accepted for each other. Factoring is over the rationals: x² − 2 counts as fully factored. Step mode handles equations in one letter and polynomial or rational expressions; it doesn’t follow trigonometric or logarithmic manipulations line by line. Mistake detection recognises the patterns listed above; other wrong answers get "not equivalent" with a value where they differ. The generators cover the standard exercises of each topic, not every kind of question a textbook asks. The SM-2 schedule is the classic SuperMemo 2 algorithm applied per skill (the problem is new each time).

Related: algebra step by step, symbolic algebra, graphing, unit circle and waves, calculus explorer, vectors, number theory step by step.

Companion to Mathematics, Reconnected — every chapter from 1 (numbers) to 17 (the derivative), plus 26 (vectors): practise the skills before a chapter’s mastery check.

Linking to an example

Share link (under the title) copies a URL that reopens the tool on the same tab with the same skill, level and problem. Your progress is never included.

You can also write a link by hand. Add the tab after #: map, practice, steps, drill, placement, review or progress.

Parameter Meaning Example

skill

Practice and step mode: the skill (ids below)

skill=quad-factor

level

1, 2 or 3

level=2

seed

Which problem: the same skill, level and seed always give the same problem

seed=42

dskill, dlevel

Timed drill: skill and level

dskill=arith&dlevel=3

secs

Timed drill length in seconds: 60, 120 or 300

secs=120

example

A preset (below); opens its tab

example=ch08-divide-by-x

For example ?skill=expand&level=2&seed=7#steps opens one expansion in step mode, and ?dskill=unit-circle&dlevel=2&secs=60#drill a one-minute unit-circle drill.

Skill ids, in the order of the tree:

skill= Skill Needs

arith

Arithmetic & mental maths

number-sense

Primes, factors, GCD and LCM

Arithmetic

integers

Integers & order of operations

Arithmetic

fractions

Fractions

Arithmetic, Primes & factors

exponents

Exponent laws

Integers & order

percent

Decimals & percentages

Fractions

ratio

Ratio & proportion

Fractions

roots

Roots & surds

Exponents, Primes & factors

expressions

Expressions: substitute & collect

Integers & order, Fractions

sci

Scientific notation

Exponents, Decimals & %

linear-eq

Linear equations

Expressions

expand

Expanding polynomials

Expressions, Exponents

inequalities

Linear inequalities

Linear equations

lines

Linear functions & graphs

Linear equations

factor-gcf

Factoring: common factors & squares

Expanding, Primes & factors

systems

Systems of equations

Lines & graphs

factor-tri

Factoring trinomials & grouping

Factor: GCF, squares

domain

Functions: evaluate, domain & range

Lines & graphs, Inequalities, Roots & surds

unit-circle

Unit circle & radians

Roots & surds, Lines & graphs

sequences

Sequences & series

Lines & graphs, Exponents

rational

Rational expressions

Factor: trinomials, Fractions

quad-factor

Quadratics by factoring

Factor: trinomials, Linear equations

compose

Composition & inverses

Domain & range, Expanding

vectors

Vector basics

Unit circle, Roots & surds

quad-square

Completing the square & vertex form

Quadratics: factor

log-laws

Exponentials & logarithm laws

Exponents, Compose & invert

trig-eq

Trig identities & equations

Unit circle, Quadratics: factor

limits

Limits

Rational expressions, Domain & range

quad-formula

Quadratic formula & discriminant

Vertex form, Roots & surds

transform

Transformations of graphs

Domain & range, Vertex form

log-eq

Exponential & log equations

Log laws, Quadratics: factor

derivatives

Derivative basics

Limits, Compose & invert

trig-graphs

Trig graphs

Unit circle, Transformations

Presets for Mathematics, Reconnected (?example=<name>):

example= Opens

ch01-mental

a 60-second mental arithmetic round (drill)

ch01-tricks

mental shortcuts: 99 × n, 25 × n, squaring numbers ending in 5 (practice)

ch01-primes

write a number as a product of primes (practice)

ch01-order

order of operations, including −a² against (−a)² (practice)

ch02-fractions

fraction arithmetic, answers in lowest terms (practice)

ch02-percent

percentage change (against the original) (practice)

ch02-ratio

sharing in a ratio (practice)

ch03-exponents

exponent laws, e.g. (3x²)³ (practice)

ch03-surds

simplify √n to k√m (practice)

ch03-scientific

multiply in scientific notation (practice)

ch04-expressions

expand and simplify, one line at a time (steps)

ch05-linear

solve a linear equation in step mode (sign errors named) (steps)

ch05-inequalities

an inequality where the sign flips (practice)

ch05-sketch-line

drag two points to draw y = mx + b (practice)

ch06-systems

a pair of linear equations (practice)

ch07-expand

expand a square — (a + b)² is not a² + b² (steps)

ch07-factor

factor completely (difference of squares) (practice)

ch07-trinomial

factor ax² + bx + c in step mode (steps)

ch07-rational

simplify a rational expression by cancelling factors (practice)

ch08-factoring

solve a quadratic by factoring, in step mode (steps)

ch08-divide-by-x

x² = kx: dividing by x loses a root (steps)

ch08-vertex-sketch

sketch a parabola from its vertex form (practice)

ch08-formula

the quadratic formula with irrational roots (practice)

ch11-domain

domain of a square root or a fraction (practice)

ch11-compose

composition and inverses (practice)

ch11-transform

sketch a transformed parabola or

x

(practice)

ch12-logs

evaluate logarithms (practice)

ch12-log-equations

a log equation with an answer outside the domain (practice)

ch13-unit-circle

click an angle on the unit circle (practice)

ch13-exact-values

a two-minute drill of exact trig values and radians (drill)

ch13-equations

solve sin x = k on [0, 2π) — two answers (practice)

ch13-graphs

period and midline of a sinusoid (practice)

ch14-sequences

nth terms and sums (practice)

ch16-limits

a 0/0 limit by factoring (practice)

ch17-derivatives

the power rule (practice)

ch26-vectors

length, dot product and angle (practice)

placement

the placement test: where should I start? (placement)

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