Twelve rungs of arithmetic, in the order they build
One path from counting to ratio. Each rung has a tool you can push around, three worked examples, the mistakes worth catching early, and a set of questions it feeds into.
The 12 rungs, and why they sit there
Every rung is a faster way of doing something the rung above it already does slowly. Start at the top and work down, or jump to the one you are teaching.
- Counting Saying how many, one thing at a time
- Addition Putting two amounts together into one
- Subtraction Taking away, and measuring the gap
- Multiplication Equal groups, arrays and the times tables
- Division Sharing fairly, and what is left over
- Place value What each digit in a number is worth
- Rounding Choosing the nearest landmark number
- Fractions Equal parts of one whole
- Decimals Tenths and hundredths, written with a point
- Percentages Parts out of every hundred
- Negative numbers Numbers below zero
- Ratio Comparing two shares in order
Practice
Fresh questions for any rung, marked as you answer, with the strategy named whenever an answer is wrong. No account, no timer, nothing kept but your own best streak in this browser.
Questions
What order should arithmetic be taught in?
Counting comes first, because every other operation is done by counting: addition counts on, subtraction counts back or counts the gap, multiplication counts in equal groups, division counts how many groups fit. Addition comes before subtraction so the sums to twenty are already known, multiplication is repeated addition, division is multiplication backwards, and place value is what lets all of them work beyond twenty. The rungs after that — rounding, fractions, decimals, percentages, negatives and ratio — each depend on the ones before them, which is why they are taught in this order rather than by school year.
What age is this for?
It fits the years in which arithmetic is first built: most learners meet counting around five or six and reach ratio by ten or eleven. But each page is written for whoever is learning or teaching, so an older student filling a gap can start at the rung that is missing rather than the one that matches their age.
How long should a rung take?
A few weeks of short sessions, not a weekend. Ten minutes most days beats an hour once a week, because the aim is recall that arrives without effort. The practice set is the honest check: if a rung's questions are quick and certain, move up; if not, the rung below is usually where the gap actually is.
Does this cost anything, and is anything tracked?
Nothing costs anything, there is no account and no sign-in. Scores are kept in your own browser, sound is off until you turn it on, and no answer you type is sent anywhere.