Arithmetic Methods

Updated 7 October 2026

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.

Where does addition start?

Addition starts with amounts small enough to see and count, and the first rule is to count on from the bigger number. Adding Within Twenty begins there. For 3 + 9, start at 9 and count on three more: 10, 11, 12, so 3 + 9 = 12. Starting from the bigger number works because order does not change a total, and three counts are safer than nine. Taking Away is the same idea run backward: for 12 − 5, start at 12 and count back five steps to 7. When a sum crosses ten, as 8 + 5 does, split the smaller number so that its first part makes ten: 8 + 2 = 10, then 3 more makes 13. These sums stay inside twenty, so a learner can check each one with counters. A subtraction also checks an addition, since 13 − 5 = 8 takes you back to where you started.

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.

What are number bonds to ten, and what do they do?

A number bond to ten is a pair of whole numbers that add up to ten: 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5, and 0 and 10. There are only six of them, and a child who knows them by heart can cross ten in two small steps instead of counting on one at a time. For 7 + 6, the 7 needs 3 more to make ten, so split the 6 into 3 and 3. Then 7 + 3 = 10, and 10 + 3 = 13. Number Bonds to Ten teaches the pairs with a ten frame, a row of ten boxes. Put 7 counters in the frame and three boxes stay empty, so 7 + 3 = 10. The empty boxes are the partner, and they can be seen without counting the full ones.

Taking away through ten runs the same way in reverse. For 16 − 9, take 6 away first to land on 10, then take the other 3 away to reach 7, so 16 − 9 = 7. A missing number works the same way: in 3 + ? = 10 the answer is the partner of 3, which is 7. Each answer can be checked with the fact it came from, because 7 + 6 = 13 also gives 13 − 6 = 7 and 13 − 7 = 6. The three facts belong to one family.

"How many more" is a subtraction too, even though it does not look like one. With 12 marbles in one row and 8 in another, match the marbles in pairs, one from each row. Four marbles in the longer row have no partner, so there are 4 more in that row, and 12 − 8 = 4 says the same thing. More and Fewer starts with the matching, because it shows which row is longer without counting either one. For larger numbers, count up from the smaller number instead: from 8 to 10 is 2, and from 10 to 12 is 2 more, so the difference is 4.

How do you add in your head?

To add in your head, change the sum into an easier sum that has the same answer, then do the easy one. The most common change is to split each number into tens and ones. For 58 + 26, add the tens, 50 + 20 = 70, then the ones, 8 + 6 = 14, and put the two parts together: 70 + 14 = 84. This works because regrouping the parts of a sum never changes its total. When one number sits just below a multiple of ten, rounding it is faster than splitting. For 58 + 29, add 30 instead and take back the extra 1: 58 + 30 − 1 = 87. The five moves below are all versions of this idea. A quick estimate catches a slip: 58 + 29 is close to 60 + 30 = 90, so 87 is a sensible answer, and 77 or 97 is not.

Reorder. Adding in Any Order is the license 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 — $46+37\to 40+30=70$46 + 37 → 40 + 30 = 70, 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: $8+5\to 8+2=10$8 + 5 → 8 + 2 = 10, 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: $198+47\to 200+47=247$198 + 47 → 200 + 47 = 247, then take back the $2\to 245$2 → 245.

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?

Column addition works because each column can hold only one digit, and ten of anything in one column is one of the next column along. Lining Up the Columns is the first step: ones under ones, tens under tens and hundreds under hundreds, so only like is added to like. Then add each column, starting from the right. For 365 + 278, the ones give 5 + 8 = 13, so write 3 and carry 1 ten. The tens give 6 + 7 + 1 = 14, so write 4 and carry 1 hundred. The hundreds give 3 + 2 + 1 = 6, and the answer is 643. The carried 1 is not a trick: 13 ones really are 1 ten and 3 ones. With numbers of different lengths, line them up by place and not by their left edges, so 57 sits under the last two digits of 1,204. Subtracting undoes the sum and checks it: 643 − 278 = 365.

Carrying in a two-digit sum: 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?

To multiply or divide in your head, reshape the numbers first and calculate second. Multiplying by Ten and a Hundred is the anchor: multiplying by 10 moves every digit one place to the left, so 47 × 10 = 470. The rule is about place value; adding a zero fails once a number has a decimal point. For harder products, split a number into tens and ones. For 6 × 47, work out 6 × 40 = 240 and 6 × 7 = 42, then add: 240 + 42 = 282. Splitting works because 47 is 40 and 7 together, so six groups of 47 are six groups of 40 and six groups of 7. Division splits too, but only the number being divided: 84 ÷ 4 = 80 ÷ 4 + 4 ÷ 4 = 20 + 1 = 21. Splitting the 4 into 2 and 2 gives a wrong answer. Multiplying back checks a division: 21 × 4 = 84.

Multiplying by Splitting is the workhorse: $7\times 34\to 7\times 30=210$7 × 34 → 7 × 30 = 210, 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: $96÷4\to 80÷4=20$96 ÷ 4 → 80 ÷ 4 = 20, 16 ÷ 4 = 4, so 24. And Rounding to Estimate is the habit that catches the wrong answer before you have finished believing it — $38\times 52\approx 40\times 50=2$38 × 52 ≈ 40 × 50 = 2,000, 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?

Long multiplication multiplies by one digit at a time and writes each result, called a partial product, on its own row. Multiplying in Columns does this once, and Two Digits by Two Digits does it twice. For 23 × 45, the first row is 23 × 5 = 115. The second row is 23 × 40 = 920, so it starts one place left. Adding the rows gives 115 + 920 = 1,035. The shift is there because that row multiplies by tens. Long division runs the other way, one place at a time from the left: divide, multiply, subtract, then bring down the next digit. When the divisor is bigger than the first digit, use the first two digits instead. For 952 ÷ 7, 7 goes into 9 once with 2 left, into 25 three times with 4 left, and into 42 six times, so the answer is 136. Multiplying back checks it: 136 × 7 = 952.

$24\times 36\to 24\times 6=144$24 × 36 → 24 × 6 = 144, 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.

1. Divide: how many times does the divisor fit into the part you are looking at?
2. Multiply: that digit times the divisor, written underneath.
3. Subtract: what is left over from the part you have handled.
4. Bring down the next digit, and go back to step one.

$852÷4\to 4$852 ÷ 4 → 4 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?

Multiplication comes before addition because a multiplication describes a single amount, not a step waiting its turn. In 5 + 2 × 6, the 2 × 6 means two sixes, which is one amount, 12. So the calculation is 5 + 2 × 6 = 5 + 12 = 17. Picture a pencil case holding 5 loose pencils and 2 boxes of 6 pencils. You count what the boxes hold, 12, and then add the loose ones. The rule makes everyone who reads the expression get the same answer. Division is the same kind of step as multiplication, so it also comes before adding and subtracting: 20 − 12 ÷ 4 = 20 − 3 = 17. When you really do want the addition first, brackets say so: (5 + 2) × 6 = 42. To test your reading, say the expression as a story and make sure the story gives the same number.

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.