The standard algorithm is the usual written column method for adding, subtracting, multiplying or dividing multi-digit numbers. You line up digits by place value, work one place at a time, and regroup (carry or borrow) when a place overflows or runs short. For 478 + 365 it gives 843 in three short steps.
In Spanish
algoritmo estándar
Also called
traditional method, column method, carrying and borrowing
The standard algorithm adds one place at a time from right to left. Each regrouped 1 moves ten of a place into the next place.
Formal definition and notation
A computation algorithm is a fixed set of steps that, carried out correctly, always gives the right result. The standard algorithms for whole numbers process digits by place value: addition and subtraction from the ones place leftward with regrouping, multiplication by partial products of one digit at a time, and division by long division from the highest place down. The Common Core asks for fluency with them by grade 4 (addition and subtraction), 5 (multiplication) and 6 (division).
Key facts
Line up the ones digits, not the first digits, so every column has one place value.
Addition: when a column totals 10 or more, write the ones digit and regroup 1 into the next column (8 + 5 = 13).
Subtraction: when the top digit is too small, regroup 1 from the next place: 52 becomes 4 tens and 12 ones.
Multiplication: multiply by each digit of the bottom number, shifting one place left each time, then add the partial products.
Division: the standard algorithm is long division, which works from the left, not the right.
Other methods, such as the area model and partial quotients, show the same place-value ideas; the standard algorithm is the most compact.
Try it: place value behind the columns
Interactive
Each column of the standard algorithm is one place in this chart. Look at 843: 8 hundreds, 4 tens and 3 ones, the answer to 478 + 365.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Standard Algorithm examples, step by step
Example 1: Subtract with regrouping
Warm-up
Use the standard algorithm to find 704 − 268.
Ones: 4 is less than 8. The tens place is 0, so regroup from the hundreds: 704 becomes 6 hundreds, 9 tens, 14 ones.
Ones: 14 − 8 = 6.
Tens: 9 − 6 = 3.
Hundreds: 6 − 2 = 4. Check: 436 + 268 = 704.
Answer:704 − 268 = 436
Example 2: Multiply two-digit numbers
Core
Use the standard algorithm to find 47 × 36.
Multiply by the ones digit 6: 47 × 6 = 282.
Multiply by the tens digit 3 (really 30): 47 × 30 = 1,410. Write it shifted one place left.
Add the partial products: 282 + 1,410 = 1,692.
Answer:47 × 36 = 1,692
Example 3: Find the error
Test-style
Ana computed 356 + 87 and got 1,226. What went wrong, and what is the correct sum?
Ana lined up 87 under the 35 instead of under the 56, so 87 was treated as 870.
Long division is the standard algorithm for division specifically. The term standard algorithm covers all four operations.
Quick check: 3 questions
Question 1 of 3
Using the standard algorithm, what is 5,009 − 1,375?
Show answer and explanation
Answer: B. Regroup across the zeros: 5,009 becomes 4 thousands, 9 hundreds, 10 tens, 9 ones, so 9 − 5 = 4, 10 − 7 = 3, 9 − 3 = 6, 4 − 1 = 3: 3,634. 4,374 takes the smaller digit from the larger in each column; the others slip one regrouping.
Question 2 of 3
In the standard algorithm for 63 × 48, what is the second partial product?
Show answer and explanation
Answer: C. The tens digit 4 means 40: 63 × 40 = 2,520. 504 is 63 × 8, the first partial product; 252 forgets the shift; 3,024 is the final product.
Question 3 of 3
When adding 5,678 + 2,456, how many times do you regroup?
Show answer and explanation
Answer: D. Ones: 8 + 6 = 14 (regroup). Tens: 1 + 7 + 5 = 13 (regroup). Hundreds: 1 + 6 + 4 = 11 (regroup). Thousands: 1 + 5 + 2 = 8. Three regroupings; the sum is 8,134.
Standard Algorithm: frequently asked questions
What does the standard algorithm mean in Common Core?
It means the familiar column methods: stack the numbers by place value and compute one place at a time with regrouping. Common Core expects students to add and subtract this way fluently in grade 4, multiply in grade 5 and divide in grade 6, after first learning strategies such as area models that explain why the steps work.
Is carrying the same as regrouping?
Yes. “Carrying” (in addition) and “borrowing” (in subtraction) are older names for regrouping. When the ones add to 13, you regroup 10 ones as 1 ten and carry it to the tens column. When you borrow, you break 1 ten into 10 ones. Teachers prefer “regroup” because it describes what really happens to the place values.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 4.NBT.B.4, CCSS 5.NBT.B.5, CCSS 6.NS.B.2
Cite this entry
Effortless Math Team. “Standard Algorithm.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/standard-algorithm/
Flashcard
TermStandard Algorithmline up, one place at a time, regroup
DefinitionThe column method for multi-digit arithmetic: line up place values, work one column at a time, regroup as needed. 478 + 365 = 843.
Place ValuePlace value is the value of a digit based on its position in a number. In 3,507 the 5 is worth 500, because it sits in the hundreds place. Each place is worth 10 times the place to its right: ones, tens, hundreds, thousands, and after the decimal point, tenths and hundredths.Number System · Grade 3, Grade 4, Grade 5Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3RegroupingRegrouping is trading 10 of one place value for 1 of the next larger place, or 1 for 10 of the next smaller place, so that column addition or subtraction works. In 47 + 38, the 15 ones become 1 ten and 5 ones; in 52 − 27, 1 ten becomes 10 ones. It is also called carrying and borrowing.Operations · Grade 1, Grade 2, Grade 3MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5Long DivisionLong division is a step-by-step written method for dividing large numbers. You work through the dividend one digit at a time from the left, repeating four steps: divide, multiply, subtract, bring down. For 872 ÷ 4, the steps give the quotient 218 with a remainder of 0.Operations · Grade 4, Grade 5, Grade 6NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Area ModelAn area model is a rectangle used to multiply or divide, with each side broken into place-value parts. To find 23 × 14, split the sides into 20 + 3 and 10 + 4, find the area of each smaller box (200, 30, 80, 12) and add these partial products: 322.Operations · Grade 3, Grade 4, Grade 5QuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5DecimalA decimal is a number written with a decimal point to show parts of a whole. Decimals use place value to represent tenths, hundredths, thousandths and more: in 3.14, the 1 means 1 tenth and the 4 means 4 hundredths. So 0.75 = 75100, and 0.5 is one half.Number System · Grade 4, Grade 5, Grade 6EstimateAn estimate is an answer that is close to the exact one, and to estimate is to find an answer that is close enough, usually by rounding or using friendly numbers. For example, 49 × 21 is about 50 × 20 = 1000. Estimation helps you check whether an exact answer is reasonable.Problem Solving · Grade 3, Grade 4, Grade 5Expanded FormExpanded form is a way of writing a number as the sum of the values of its digits. Each digit is multiplied by its place value, so 4072 = 4000 + 70 + 2, or 4 × 1000 + 7 × 10 + 2 × 1. It shows what every digit is worth. Zeros add nothing, so they are usually left out.Number System · Grade 2, Grade 3, Grade 4