Arithmetic Methods
There are two ways to do a calculation, and fluency is mostly about choosing between them quickly. A mental method reshapes the numbers into ones that are easier — it is fast, but it depends on the numbers cooperating. A written method ignores what the numbers look like and grinds through them at a steady cost.
Both rest on the same idea: place value. Carrying, borrowing, splitting and doubling are all the same bundling and unbundling of tens, done in different notation. This page walks the whole ladder, with a worked example at each rung and the reason it works.
Where does addition start?
It starts with amounts small enough to see. Adding Within Twenty and Taking Away are the first pair, and they are deliberately kept inside the range where a child can check the answer by looking rather than by trusting a rule.
Multiplication and division start the same way, as arrangements rather than facts. Equal Groups establishes that multiplying counts equal-sized groups, Rows and Columns lays those groups out as a rectangle so that 3 × 4 and 4 × 3 are visibly the same rectangle turned sideways, and Sharing Equally introduces division as the question the rectangle answers backwards. The times tables guide covers making that recall automatic, which everything below assumes.
How do you add in your head?
By changing the sum into an easier sum that has the same answer. There are five moves worth owning, and between them they cover almost everything.
Reorder. Adding in Any Order is the licence for all of it: 7 + 24 + 3 becomes 7 + 3 + 24 = 34, because addition does not care what order it is given.
Split. Adding by Splitting breaks a number along its place values — , then 6 + 7 = 13, so 83. This is column addition done aloud, left to right.
Bridge. Bridging Through Ten uses ten as a stepping stone: , then 3 more = 13. It is the single most useful move in early arithmetic, because ten is where the notation changes gear.
Double. Doubles and Near Doubles turns 7 + 8 into 7 + 7 + 1 = 15. Doubles are recalled faster than any other addition fact, so anything within one of a double is nearly free.
Round and adjust. Rounding and Adjusting is the one adults use most: , then take back the .
Subtracting without taking away
Subtraction has its own two moves, and neither is "take away". Counting Up to Subtract reads 82 − 67 as the gap from 67 to 82: three to reach 70, then twelve more, so 15. That is how shopkeepers gave change before tills did it.
The Same Difference is the more powerful one. Slide both numbers by the same amount and the gap between them is untouched, so 82 − 67 = 85 − 70 = 15. Choose the slide that makes the second number round and the subtraction stops being work.
How does column addition actually work?
It works because a column can hold only one digit, and ten of anything is one of the next thing along. Lining Up the Columns is the step people skip and then pay for: ones under ones, tens under tens, so that only like is ever added to like.
Carrying is what happens when a column overflows. 47 + 38: 7 + 8 = 15 ones. Ten of those bundle into one ten, carried across; 5 stays. Then 4 + 3 + 1 = 8 tens. Answer 85.
Carrying in More Than One Column shows the cascade — a carry can trigger the next carry, and nothing about the rule changes when it does.
Borrowing, and borrowing past a zero
Borrowing is carrying run backwards: unbundle one ten from the next column into ten ones, so the subtraction in this column becomes possible. 52 − 27: 2 − 7 will not go, so take a ten: 12 − 7 = 5, then 4 − 2 = 2. Answer 25.
Borrowing Past a Zero handles the case that stalls people. 300 − 176: the tens column has nothing to lend, so the hundred comes across and is unbundled twice — 2 hundreds, 9 tens, 10 ones — and then the subtraction runs normally to give 124. The row of nines that appears is not a trick; it is what one hundred looks like when it is broken into tens and ones.
How do you multiply and divide in your head?
The same way: reshape first, calculate second. Multiplying by Ten and a Hundred is the anchor — every digit shifts one place left, and the reason is place value rather than "add a zero", which is a rule that fails the moment decimals arrive.
Multiplying by Splitting is the workhorse: , 7 × 4 = 28, total 238. Doubling to Multiply reaches ×4 and ×8 by doubling twice or three times, and pairs with Halving to Divide, which turns ÷4 into halving twice.
Dividing by Splitting breaks the dividend into friendly chunks instead: , 16 ÷ 4 = 4, so 24. And Rounding to Estimate is the habit that catches the wrong answer before you have finished believing it — ,000, so an answer near 200 or 20,000 is a misplaced digit. The division guide goes further into the mental side of dividing.
How do you do long multiplication and long division?
Multiplying in Columns takes the splitting method and writes it down, so nothing has to be held in the head. Two Digits by Two Digits then does it twice and adds the two partial products, which is why the second row is shifted one place left — that row is being multiplied by tens, so it must be written where tens are written.
, and 24 × 30 = 720, so 864.
Long Division is the one with a reputation, and the cure for it is knowing what each cycle is doing: it handles exactly one place value, from the left, and the four steps repeat unchanged.
- Divide: how many times does the divisor fit into the part you are looking at?
- Multiply: that digit times the divisor, written underneath.
- Subtract: what is left over from the part you have handled.
- Bring down the next digit, and go back to step one.
into 8 goes 2; bring down 5, 4 into 5 goes 1 remainder 1; bring down 2 to make 12, 4 into 12 goes 3. Answer 213.
When the last subtraction does not reach zero, what is left is a remainder, and Remainders is about deciding what to do with it — round up, round down, or keep it as a fraction — which is a question about the situation, not about the arithmetic. Thirteen children needing four to a car is three cars with a remainder of one, and the answer is four cars.
Why does multiplication come before addition?
Because a multiplication is a single packaged quantity, not an instruction waiting its turn. In 2 + 3 × 4, the 3 × 4 means three fours — one amount, twelve — so the sum is 14.
Multiply and Divide Before Adding and Subtracting makes that argument properly, and Brackets First introduces the notation for asking the other question on purpose: (2 + 3) × 4 = 20. Brackets with All Four Operations then puts every rung together, including the left-to-right trap where 8 ÷ 2 × 4 = 16 rather than 1. There is a dedicated order of operations guide if that is the question you came for.
The four mistakes worth naming
- Lining up the ends instead of the places. In a column sum, the columns are the whole point; ragged right-hand ends are fine, misaligned place values are not.
- "Add a zero" for multiplying by ten. It works for whole numbers and breaks immediately for 3.5 × 10. The digits shift; the point stays put.
- Choosing a written method for friendly numbers. 198 + 47 in columns is slower and more error-prone than rounding and adjusting.
- Discarding a remainder without asking. What a remainder means depends entirely on what is being divided up.
Where this leads next
These methods all rest on the columns described in whole numbers and place value, and they extend right past the decimal point without changing — the decimals and percentages ladder picks them up there. Asking which divisions come out exactly leads into factors, multiples and primes.
Practise it in the game
Math Challenge is a mental-math game, and every step above has an illustrated lesson behind it — a worked derivation with pictures, then try-it problems that re-teach the exact question you missed. These sit inside a catalog of 800+ lessons spanning the full ladder.
For drill rather than reading, the Multiplication topic adapts to whichever facts keep costing you time, and the Daily Challenge needs no account.
Your turn
Three to try — tap what you get.
47 + 38
82 − 47
34 × 5