How do you teach long multiplication step by step?

Long multiplication is taught in three stages. Check the times tables first, build the problem as an area model, then compress that model into the standard algorithm. For 36 × 24, a child multiplies 36 × 4 = 144, writes a placeholder zero, multiplies 36 × 20 = 720, and adds the two partial products to get 864.

How do you multiply 2 digit numbers by hand?

Two-digit numbers are multiplied by splitting the bottom number into tens and ones. For 36 × 24, multiply 36 × 4 = 144 on the first line, write a placeholder zero at the start of the second line, multiply 36 × 20 = 720, then add the two partial products: 144 + 720 = 864.

What is the area model of multiplication?

The area model of multiplication draws each factor split by place value along the sides of a rectangle. For 36 × 24, the rectangle has four parts: 30 × 20 = 600, 30 × 4 = 120, 6 × 20 = 120, and 6 × 4 = 24. Those four parts add to 864, the same answer the standard algorithm gives.

Why does my child write a zero in the second row?

The zero in the second row holds the ones place, because that row multiplies by tens rather than by ones. In 36 × 24, the second row is 36 × 20 = 720, and the zero is what turns 72 into 720. A missing zero gives 144 + 72 = 216 instead of the correct 864.

Should my child learn the lattice method?

The lattice method gives correct answers and suits a child who keeps losing column alignment, although it sits outside the Common Core expectations and it hides place value. Treat it as a temporary support while handwriting settles, then move back to partial products or the standard algorithm.

How to Teach Long Multiplication, Step by Step

Q: What grade learns long multiplication?

Long multiplication starts in fourth grade under the Common Core standards. Standard 4.NBT.B.5 asks students to multiply two two-digit numbers using strategies based on place value and the properties of operations, and to illustrate the calculation with equations, arrays or area models. Fifth grade adds fluency with the standard algorithm in 5.NBT.B.5.

Long multiplication is where fourth grade stops forgiving a shaky times table. A child who can chant the sixes on a good day will lose 36 × 24 somewhere in the second row, and the number that comes back looks nothing like 864. The repair is rarely another worksheet of the same kind.

What long multiplication asks a child to do

Long multiplication asks a child to run three skills at once and in a fixed order, which is why it exposes gaps that single-digit work hides. One problem such as 36 × 24 needs four fact recalls, two carries, one deliberate shift of place value and a final three-digit addition. A weakness in any one of them looks, on the page, like a long multiplication problem.

The three skills it stacks

Three skills carry every long multiplication problem, and each one fails in a recognizable way.

Multiplication facts. The child answers 6 × 4 and 3 × 4 without pausing. Slow recall does not produce a wrong digit by itself; it produces a forgotten carry three seconds later.
Place value. The child knows that the 2 in 24 means 2 tens, so the second row of the calculation is worth 720 and not 72.
Carrying. The child multiplies first and adds the carry second. Reversing that order turns 36 × 4 into 204, and nothing downstream recovers.

Common Core standard 4.NBT.B.5 puts this work in grade 4: students "multiply two two-digit numbers, using strategies based on place value and the properties of operations". Grade 5 raises the bar to "fluently multiply multi-digit whole numbers using the standard algorithm" (5.NBT.B.5). A fourth grader who still draws a model is on schedule.

The area model of 36 times 24, with the four parts 600, 120, 120 and 24 adding to 864.

Why long multiplication fails before it starts

Most wrong answers begin below the algorithm, in the facts. A child who needs three seconds to reach 7 × 8 spends the whole problem on recall, and the carry is what gets dropped. Teaching a new procedure on top of that costs a week and fixes nothing.

The times-table check

The times-table check takes two minutes and decides whether to teach the algorithm this month or next month. Read out twelve mixed facts from the sixes, sevens, eights and nines, and time the gap before each answer. Two seconds is the working threshold.

Two or more slow answers means fluency work comes first. The What Works Clearinghouse practice guide Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades recommends that teachers "regularly include timed activities as one way to build fluency in mathematics". Its earlier companion, the response to intervention guide, puts roughly ten minutes a session on fluent retrieval of basic facts.

Ten minutes is small enough to survive a school week. The times tables game and the doubles facts game cover the sixes, eights and nines that carry most of the load, and the multiplication tables game collects the rest. If facts are the real blocker, the guide on how to learn the times tables fast is the better starting point.

How to teach long multiplication, step by step

The order that works runs from a picture the child can explain to a procedure the child can trust, and one problem carries the whole journey. Every step below uses 36 × 24.

Estimate before anything else is written. Round 36 to 40 and 24 to 25. Since 40 × 25 = 1,000, the answer will land near a thousand, and a written estimate is what makes a wrong answer visible ten minutes later.
Draw 36 × 24 as a rectangle. Split the sides by place value: 30 and 6 down one side, 20 and 4 along the other. The four parts are 30 × 20 = 600, 30 × 4 = 120, 6 × 20 = 120 and 6 × 4 = 24.
Add the four parts. 600 + 120 + 120 + 24 = 864. Set that against the estimate of 1,000 and the answer survives its first test.
Rewrite the four parts as a column. Stack 600, 120, 120 and 24 under one another and add them. Nothing new has happened; the same four products now live in a vertical list instead of a drawing.
Multiply by the ones digit. 36 × 4 = 144, said out loud as the child writes: 6 × 4 = 24, write the 4 and carry the 2; 3 × 4 = 12, plus the carried 2 makes 14.
Write the placeholder zero, then multiply by the tens digit. Put a 0 in the ones column of the second row, because that row multiplies by 20 rather than by 2. Then 36 × 2 = 72 fills the rest of the row, giving 720.
Add the two partial products and return to the estimate. 144 + 720 = 864, which sits comfortably under the 1,000 the child predicted in step 1.

Play free in your browserTimes TablesGrade 3+

Where families skip a stage

Families skip steps 2 to 4 because the compact method is quicker to demonstrate, and the child is then left with no way to tell 864 from 216. The four-part drawing costs two lessons and pays for itself the first time a partial product goes missing. Teach the drawing until the child can produce it from memory, then retire it.

Four ways to multiply two two-digit numbers

Four written methods turn up in US classrooms, and all four give 864 for 36 × 24. The difference is how much place value each one leaves visible on the page.

Method	How it works	Best for	Limits	Where it fits
Area model	Splits both factors by place value and draws the four products as parts of a rectangle	First exposure; seeing why the answer is 864	Slow to draw; a 3-digit × 3-digit problem needs nine parts	Grade 4 strategy (4.NBT.B.5)
Partial products	Writes the same four products in a vertical column, then adds them	The bridge from the drawing to the algorithm	More lines than the compact method	Grade 4 into grade 5
Standard algorithm	Multiplies by the ones digit, writes a placeholder zero, multiplies by the tens digit, adds	Speed, larger numbers, later algebra	Place value is hidden, so a slip leaves no trace	Grade 5 fluency (5.NBT.B.5)
Lattice	Multiplies every digit pair inside a boxed grid, then adds along the diagonals	A child who keeps losing column alignment	Needs a drawn grid; hides place value; outside the standards	Temporary support

Which one to teach at home

The class method wins at home, whichever of the four it happens to be. A second method taught in parallel splits a child's attention at the exact moment the first one is fragile, so ask the teacher which layout the class uses and copy it. Show an alternative only once the class method comes out right three times in a row.

The drawing is worth protecting for its own sake. The practice guide Improving Mathematical Problem Solving in Grades 4 Through 8 rates "teach students how to use visual representations" as a recommendation with strong evidence behind it.

Free printable worksheetMultiplication — 4th GradersWorksheet · PDF

Moving from the area model to the standard algorithm

The standard algorithm is the area model with the drawing removed and two pairs of products already added together. That sentence is the whole transition, and a child who hears it stops treating the two methods as rivals.

Look at where the four parts go. The two products that involve the 20 are 30 × 20 = 600 and 6 × 20 = 120, and 600 + 120 = 720, which is the second row. The two products that involve the 4 are 30 × 4 = 120 and 6 × 4 = 24, and 120 + 24 = 144, which is the first row.

Once a child can point to those pairs, the compact layout is a shorthand rather than a new rule. The NRICH article Developing Number Fluency: What, Why and How argues that "fluency demands more of students than memorising a single procedure", and the pairing above is what turns the procedure into something a child can rebuild after forgetting it.

The placeholder zero in the second row

The placeholder zero holds the ones column so the second row lines up with the tens it represents. Without it, 36 × 20 gets written as 72 and the total collapses to 216. Some teachers write a small "× 20" beside the row for a few weeks so the zero has a visible reason.

The column layout of 36 times 24, with the partial products 144 and 720 and the placeholder zero circled.

How to check your child has it

Four observable signals tell you whether long multiplication has landed, and none of them is a score on a worksheet.

Writes an estimate unprompted. The child rounds 36 × 24 to 40 × 25 = 1,000 before starting, without being asked.
Explains the zero. Asked why the second row begins with a 0, the child says that the row multiplies by 20, not by 2.
Finds the broken step alone. Given a wrong answer of 216, the child locates the missing zero instead of restarting the problem.
Transfers to a new shape. The child handles 36 × 240 and a missing-factor question such as 36 × ? = 864 with the same reasoning. The missing factor game turns that into two minutes of practice.

A two-minute self-check

The self-check runs on one problem and three questions. Give the child 36 × 24 on blank paper, then ask: what is your estimate, where is the zero, and how do you know the answer is right? A child who answers all three owns the method.

Common mistakes and edge cases

Four mistakes account for most wrong long multiplication answers, and a written estimate catches three of them on the spot.

The missing placeholder zero. The child writes 144 and then 72, adds them to 216, and moves on. The estimate of 1,000 makes the error obvious, since 216 is nowhere near it. The correct second row is 720.
Carrying into the wrong column. A carried digit written above the wrong column joins the wrong multiplication. Squared paper, one digit per box, removes the problem faster than any explanation.
Forgetting to add the carry after the last digit. In 36 × 4, the child writes 4, carries 2, works out 3 × 4 = 12, and records 124 instead of 144. Saying "twelve plus the two I carried" out loud fixes it within a week.
A zero inside the number being multiplied. In 306 × 24, the first row is 306 × 4 = 1,224, where 0 × 4 = 0 still has to collect the carried 2. The second row is 306 × 20 = 6,120, and the total is 7,344. A child who drops the 0 entirely works out 36 × 24 and lands on 864 instead of 7,344.

Your long multiplication checklist

A short checklist keeps the sequence honest over the weeks that the method takes to settle.

Facts before algorithm. Twelve mixed facts under two seconds each, or fluency practice comes first.
One problem, drawn and written. The same 36 × 24 appears as a rectangle, as four partial products and as the compact layout.
The estimate is written first. Every problem starts with a rounded prediction on the page, never in the child's head.
The zero is explained, never only copied. The child says why the second row starts with 0 before writing it.
Errors are located, not corrected for the child. Point at the row, name nothing, and let the child find the step.
Division follows. Once 36 × 24 = 864 is secure, the guide on how to teach long division uses the same numbers in reverse, and the multiplication worksheet for 4th graders supplies the practice set with its answer key.

MathIt's free iOS and Android app has adaptive difficulty, so multiplication practice gets harder only after the easier facts come out right, and 22 of its roughly 90 game types play free in a browser with no sign-up.