Long Multiplication

Introduction

Long multiplication is a written method for multiplying numbers with more than one digit. It is also called the standard multiplication algorithm or grade-school multiplication. The method works because each digit has a place value. You multiply one place-value part at a time, record the partial products, and add those partial products to find the final product.

You will use skills you already know: basic multiplication facts, place value, addition, and regrouping. Long multiplication is useful when mental math would be slow or difficult.

By the end of this aiMOOC, you should be able to explain why the method works, multiply whole numbers accurately, estimate before calculating, find and fix common errors, and use multiplication in real-life problems.

Key Vocabulary

A factor is a number that is being multiplied. The product is the answer to a multiplication problem. In a written calculation, one factor may be called the multiplicand and the other the multiplier, but both are factors.

A partial product is one product found during a step of long multiplication. Regrouping means exchanging groups of ten between place-value columns. For example, 56 ones can be regrouped as 5 tens and 6 ones.

Knowing your multiplication facts helps you focus on place value and regrouping instead of stopping at every basic fact.

Why Long Multiplication Works

Our decimal number system is based on place value. In 27, the 2 means 20 and the 7 means 7. That means:

34 × 27 = 34 × 20 + 34 × 7

The two products, 680 and 238, are partial products. Adding them gives 918. Long multiplication writes the same mathematical idea in a compact column format.

This connection is important: the second row is not moved left just because of a rule to memorize. It is moved because that row represents multiplication by tens, hundreds, or another larger place value.

From a Grid to the Standard Algorithm

A grid method can show the place-value parts clearly. For 34 × 27, you can think of 34 as 30 + 4 and 27 as 20 + 7. The four smaller products are 30 × 20, 4 × 20, 30 × 7, and 4 × 7. Their sum is the same product you get from long multiplication.

The standard algorithm groups these small products into rows, so it takes less space while keeping the same place-value reasoning.

Step-by-Step Method

For a whole-number multiplication problem, line up the factors by place value. Start with the ones digit of the bottom factor. Multiply it by every digit of the top factor, moving from right to left and regrouping when needed. Then move to the next digit of the bottom factor. Because that digit has a larger place value, start its partial product in the correct column. Continue until every digit of the bottom factor has been used, then add the partial products.

Keeping the columns straight is part of the mathematics, not just neat handwriting.

Worked Example: 468 × 37

First multiply 468 by 7. Seven times 8 is 56, so write 6 in the ones place and regroup 5 tens. Seven times 6 tens is 42 tens; add the 5 regrouped tens to get 47 tens. Write 7 in the tens place and regroup 4 hundreds. Seven times 4 hundreds is 28 hundreds; add the 4 regrouped hundreds to get 32 hundreds. The first partial product is 3,276.

Next use the 3 in 37. That digit means 30, not 3. Multiply 468 by 3 to get 1,404, then shift the value one place to the left because you are really multiplying by 30. The second partial product is 14,040.

Finally, add the partial products: 3,276 + 14,040 = 17,316.

	468
×	37
	3,276
+	14,040
Product	17,316

The image shows the same calculation with the partial products and final sum.

What Does the Zero Do?

When you multiply by a tens digit, many written versions of the algorithm place a zero in the ones position of the second partial product. This zero is a place-value placeholder. It shows that the row represents tens.

For example, in 52 × 24, the first partial product is 52 × 4 = 208. The next partial product is 52 × 20 = 1,040. Then 208 + 1,040 = 1,248.

You can also think of this as shifting the digits one place to the left when multiplying by tens. The important idea is place value, not the shape of the written rule.

Estimation and Checking

Estimate before you calculate. For 468 × 37, rounding to 500 × 40 gives an estimate of 20,000. The exact answer, 17,316, is reasonably close to that estimate. An answer such as 1,731 or 173,160 would be a warning that a place-value error may have happened.

You can also check the ones digit. Since 8 × 7 ends in 6, the final product must end in 6. A second check is to use the inverse operation: if your division skills are ready, 17,316 ÷ 37 should give 468.

Common Errors and How to Fix Them

Misaligned partial products: Make sure each digit is written in the column that matches its place value.

Forgetting regrouped amounts: Write small regrouping notes clearly and add them at the correct step.

Treating a tens digit as ones: Remember that the 3 in 37 means 30.

Skipping a partial product: Check that you used every digit of the multiplier.

Adding incorrectly at the end: Keep the addition columns aligned and check the sum separately.

Making your own worked-example poster can help you explain the method to someone else and notice where each place-value step belongs.

Guided Practice

Try 326 × 4 first. Multiply from right to left and regroup when a product is 10 or greater. Your answer should be 1,304.

Next try 326 × 24. The ones partial product is 326 × 4 = 1,304. The tens partial product is 326 × 20 = 6,520. Add them to get 7,824.

Before accepting an answer, ask yourself three questions: Did I use every digit? Are the partial products aligned by place value? Is the result close to my estimate?

Real-Life Uses

Long multiplication can help with shopping, planning, measurement, and repeated groups. If 28 students each need 36 craft sticks, the total is 28 × 36 = 1,008 craft sticks. If a theater has 24 rows with 135 seats in each row, it has 24 × 135 = 3,240 seats.

In real situations, estimation matters too. It helps you decide whether the exact product makes sense before you use it to buy materials, plan a budget, or report a result.

Interactive Tasks

Quiz: Test Your Knowledge

What does long multiplication help you find? (The product of numbers with many digits) (!The remainder of a division) (!The perimeter of a shape) (!The value of a fraction only)

What is the partial product of 34 multiplied by 7? (238) (!68) (!204) (!340)

What value does the 2 represent in 27? (20) (!2) (!200) (!7)

Why is a tens partial product shifted one place to the left? (It represents multiplication by tens) (!It makes the answer look larger) (!It changes addition into subtraction) (!It removes the ones digit)

What is 46 multiplied by 3? (138) (!128) (!136) (!148)

What is the ones partial product in 52 multiplied by 24? (208) (!104) (!1040) (!1248)

What is 52 multiplied by 20? (1040) (!104) (!520) (!1240)

What is 52 multiplied by 24? (1248) (!1148) (!1048) (!1448)

Which estimate is best for 398 multiplied by 21? (8000) (!800) (!80000) (!4000)

What should the ones digit be in the product of 468 multiplied by 37? (6) (!5) (!7) (!8)

Memory Game

Factor	A number being multiplied
Product	The answer to a multiplication problem
Partial product	A result found during one multiplication step
Regrouping	Exchanging groups of ten between place-value columns
Place value	The value a digit has because of its position
Estimate	A nearby value used to check reasonableness

Drag and Drop

Match the correct terms.	Topic
ones place	starting column for the first partial product
tens place	starting column for a partial product made with tens
regrouping	exchanging groups of ten between columns
partial product	result from one place-value multiplication step
estimate	quick check for whether an answer is reasonable

Match each mathematical term to the phrase that explains its job in long multiplication.

Crossword Puzzle

Product	What do you call the answer to a multiplication problem?
Factor	What do you call a number that is being multiplied?
Regrouping	What process exchanges groups of ten between place-value columns?
Estimate	What nearby value can help you check whether an answer is reasonable?
Multiplier	What can you call the factor whose digits are used to form partial products?
Algorithm	What word means a step-by-step method for solving a problem?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Place value model: Draw a place-value chart for 34 × 27 and show how 27 can be split into 20 and 7 before you calculate.
Multiplication story: Write a short real-life word problem that can be solved with a two-digit by two-digit multiplication, then solve it using long multiplication.
Error detective: Create one long-multiplication example with a deliberate place-value error, exchange it with a partner, and explain how to correct the mistake.
Fact family cards: Make a small set of study cards for multiplication facts that you find difficult and add one tip for remembering or checking each fact.

Standard

Maths in a shop: With a teacher or adult, visit or examine a local shop or online catalog, choose an item sold in equal packs, and calculate the total number or cost for several packs using long multiplication.
Grid and algorithm comparison: Solve the same two-digit multiplication with a grid method and long multiplication, then write how the partial products match.
Peer interview: Interview a classmate about the step they find hardest in long multiplication, record the explanation, and design one practice example that targets that difficulty.
Tutorial video: Produce a short teaching video that explains a two-digit by two-digit example, including regrouping, place-value alignment, and an estimate.

Advanced

Algorithm investigation: Solve three multi-digit problems with both partial-products notation and the standard algorithm, then compare which method makes place value easier to see.
Efficiency experiment: Time yourself solving a small set of similar problems with two written methods, check every answer, and explain whether the faster method was also the more accurate one.
Budget project: Plan a simple class event or project with repeated quantities, use long multiplication to calculate at least three totals, and explain how estimation helps you check the budget.
Teaching resource: Design an infographic or mini-poster that teaches long multiplication to a younger learner, including a worked example, a common mistake, a checking strategy, and two practice questions.

Learning Assessment

Explain the shift: Use place value to explain why the second partial product in a two-digit multiplier begins one column to the left of the first partial product.
Diagnose an error: Analyze a worked multiplication in which the tens partial product was not shifted, identify the first incorrect step, and repair the whole calculation.
Choose a method: Compare a grid method and long multiplication for the same problem and argue which method you would choose for accuracy and why.
Real-world transfer: Create and solve a situation that requires multiplying a three-digit number by a two-digit number, then interpret the product in context.
Estimate and justify: Estimate a product before calculating exactly, compare the two results, and explain why the exact answer is reasonable or unreasonable.
Teach the reasoning: Explain a regrouping step and a place-value shift to a learner who knows multiplication facts but has never used long multiplication.

Evidence of Learning

Strong evidence of learning includes knowledge of place value, multiplication facts, partial products, regrouping, and estimation. You should be able to explain why each row of the standard algorithm has its position.

Your skills should include accurate multi-digit multiplication, clear alignment of place-value columns, correct regrouping, addition of partial products, estimation, and error checking.

Useful products can include a correct worked example, a place-value model, an error-analysis explanation, a word problem, a poster, or a short teaching video.

Transfer means using the method beyond a practice page. You show transfer when you choose long multiplication for a real problem, explain the meaning of the answer, estimate to check it, and adapt your strategy when the numbers change.

OERs on the Topic

The English Wikipedia article below gives background on multiplication algorithms, including long multiplication and other methods. Some parts are advanced, so focus first on the sections about multiplying by hand.

Linked Learning Areas

Long multiplication connects multiplication facts with place value, addition, estimation, problem solving, and later work with decimals and division.

Long Multiplication

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