Long Multiplication, Step by Step
Long multiplication is four small multiplications and one addition. The written method hides that, which is why the area model is worth meeting first.
Split, multiply, add
To multiply by 47, multiply by 40 and by 7, then add the two answers.
That is the entire method. Everything that looks complicated about long multiplication is bookkeeping laid on top of that one sentence, and a child who has been shown the bookkeeping without the sentence is following a procedure they cannot check. When they get a wrong answer, and everybody does, they have no way to tell whether it is wrong by a little or wrong by a factor of ten.
The area model
Draw the multiplication as a rectangle. 23 by 47 is a rectangle 23 tall and 47 wide, and if you cut it at the place values you get four smaller rectangles whose areas are four easy multiplications.
The picture is not a beginner's version to be discarded later. It is the reason the written method works, and children who keep it in mind spot their own errors, because a partial product of 8 where 800 belongs simply looks wrong on the rectangle.
It scales down too
6 x 14 is two rectangles rather than four, which makes it the right place to start if the four-way version is too much at once.
The written method
Two rows, then add
1. Multiply the top number by the ones digit of the bottom number. Write that row.
2. Put a zero in the ones column of the next row, then multiply by the tens digit.
3. Add the two rows.
Work out 23 x 47 the written way.
- 23 x 7: three sevens are 21, write 1 carry 2. Two sevens are 14, plus the carried 2 is 16. First row is 161.
- Write a 0 in the ones column of the second row, because we are now multiplying by 40, not 4.
- 23 x 4: three fours are 12, write 2 carry 1. Two fours are 8, plus 1 is 9. Second row is 920.
- Add: 161 + 920.
1,081
Line the two rows up against the four rectangles and they match: 161 is 140 plus 21, and 920 is 800 plus 120. The written method groups the four areas into two rows, which is why it is shorter and also why it is harder to see.
Why the second row gets a zero
This is the step children are most often told to do without being told why, and it is the step they most often forget.
The 4 in 47 is not four. It is forty. So the second row is not 23 x 4, it is 23 x 40, and that is ten times bigger. Every digit has to move one column to the left, which leaves the ones column empty, and a zero goes there to hold it. It is the same move as multiplying by ten, because that is precisely what it is.
What forgetting it costs
Leave the zero out of 23 x 47 and the second row is 92 instead of 920, so the answer comes to 253 instead of 1,081. It is not slightly wrong. A quick estimate of 20 x 50 = 1,000 catches it immediately, which is the best argument there is for estimating first.
Common mistakes
1. Missing the zero in the second row
By far the most common, and the most expensive, because it makes the answer wrong by hundreds rather than by ones. Writing the zero first, before doing any multiplying at all, removes the problem entirely.
2. Carrying from the first row into the second
A carry belongs to the row it was made in. When the second row starts, the old carries are finished with, and leaving them written above the numbers is what causes them to get used twice. Cross them out before starting the next row.
3. Adding the carry before multiplying
In 23 x 7, doing two plus the carried two first and then multiplying, giving 28 instead of 16. Multiply first, then add the carry. Saying it out loud in that order, every time, is what makes it stick.
Practice questions
Seven questions. One attempt each, and the explanation appears once the answer is in.
Check yourself
No sign up. Nothing is sent anywhere, the score stays on this device.
What to learn next
Division is the same relationship read backwards, and short division is the natural next step now that the times tables have somewhere to be used.
For parents
Ask for an estimate before the working, every single time. "About how big will the answer be?" takes five seconds and it is the habit that catches the missing zero, the misplaced carry and the times table slip. A child who estimates first will find their own mistakes, which is worth far more than being told about them.
Still stuck on this?
Send us the question and a teacher will answer it. No charge, and you do not have to be a student here.
What to learn next
Everything in the Learning Hub
Every topic is free and every one ends with practice questions. Start anywhere.
Fractions
Math · 9 topics
What Is a Fraction?Equivalent FractionsSimplifying FractionsComparing FractionsAdding and Subtracting Fractions With the Same DenominatorAdding Fractions With Different DenominatorsFinding a Fraction of a NumberMixed Numbers and Improper FractionsFractions to DecimalsDecimals and percentages
Math · 7 topics
What Are Decimals?Comparing and Ordering DecimalsAdding and Subtracting DecimalsRounding DecimalsWhat Are Percentages?Finding a Percentage of a NumberFractions, Decimals and PercentagesScience: matter and the water cycle
Science · 5 topics
What Is Matter?Solids, Liquids and GasesChanging StateThe Water CycleMixtures and SolutionsWhole numbers
Math · 10 topics
Place Value to 1,000,000Rounding Whole NumbersAdding and Subtracting With RegroupingTimes Tables That Actually StickMultiplying by 10, 100 and 1000Long Multiplication, Step by StepShort DivisionLong Division, Step by StepFactors, Multiples and PrimesOrder of OperationsForces and energy
Science · 6 topics
Forces and MotionFrictionGravity, Mass and WeightLevers and Simple MachinesEnergy and Its FormsSimple CircuitsStuck on this topic? Get one class, free.
A tutor works through it live with your child, one on one, and sends you a short written report afterwards. No card needed.