Multi-Digit Multiplication

✖️ Multi-Digit Multiplication

Part 1 of 5 — Place Value: The Secret Behind Big Multiplication

Topics in This Part

🔑 Key Concept: Every big multiplication problem is really just a bunch of small multiplications added together. The trick is place value — knowing that the $4$ in $\,341\,$ is worth $40$, not $4$. Master that, and the rest is easy.

Breaking Numbers Apart by Place Value

A number like $\,247\,$ is built from three pieces:

$247 = \underbrace{200}_{\text{hundreds}} + \underbrace{40}_{\text{tens}} + \underbrace{7}_{\text{ones}}$

We call this expanded form. Pulling a number apart this way is the whole idea behind multiplication — you multiply each piece, then add the results back together.

Try Reading the Places

💡 Heads up: The digit and its value are different. In $914$, the digit is $1$, but its value is $10$ because it sits in the tens place.

Concept Check 🎯

Multiplying by 10, 100, and 1,000

This is the fastest shortcut in all of multiplication. To multiply by a power of ten, count the zeros and tack them on the end:

$6 \times 10 = 60 \qquad 6 \times 100 = 600 \qquad 6 \times 1{,}000 = 6{,}000$

It works for bigger numbers too:

$23 \times 10 = 230 \qquad 23 \times 100 = 2{,}300$

🔑 Why it works: Multiplying by $10$ shifts every digit one place to the left — ones become tens, tens become hundreds. The new $0$ just fills the empty ones place.

⚠️ Watch out: $40 \times 30$ is not $120$. Multiply the non-zero parts ($4 \times 3 = 12$), then add the two zeros back: $40 \times 30 = 1{,}200$.

Zero Power-Up 🧮

Use the count-the-zeros shortcut.

1) $8 \times 100 = \,?$ 2) $52 \times 10 = \,?$ 3) $40 \times 30 = \,?$ (multiply $4 \times 3$ first, then add both zeros)

One Last Place-Value Skill

Before we start multiplying big numbers, make sure you can name the value of any digit on sight. This is the single skill every method below depends on.

$\underset{\text{value }600}{\boxed{6}}\,\underset{\text{value }30}{\boxed{3}}\,\underset{\text{value }8}{\boxed{8}} \;=\; 638$

Read the place, multiply by what it's worth ($100$, $10$, or $1$), and you have its value.

Match the Place Value 🔽

Choose the correct value of each underlined digit.

Putting It Together

Place value is the engine. When you see $\,34 \times 6$, your brain can split it:

$34 \times 6 = (30 + 4) \times 6 = (30 \times 6) + (4 \times 6) = 180 + 24 = 204$

That splitting move is called the distributive property, and it's exactly what every method in this lesson is built on. In Part 2 we'll turn it into a quick, reliable procedure.

✖️ Multi-Digit Multiplication

Part 2 of 5 — Multiplying by a One-Digit Number

🔑 The Idea: To multiply a big number by a single digit, multiply one place at a time, starting from the ones. When a column overflows past $9$, you carry the extra into the next column.

The Standard Algorithm (One Digit)

To compute $\,47 \times 8$:

1. Ones first: $8 \times 7 = 56$. Write the $6$, carry the $5$.
2. Tens next: $8 \times 4 = 32$, then add the carried $5$: $32 + 5 = 37$. Write $37$.
3. Read the answer: $376$.

$47 \times 8 = 376$

We can check with place value:

$47 \times 8 = (40 \times 8) + (7 \times 8) = 320 + 56 = 376 \;✓$

⚠️ The #1 mistake: Forgetting to add the carried digit after multiplying the next column. Multiply first, then add the carry.

Concept Check 🎯

A Three-Digit Example

Compute $\,218 \times 6$. Work right to left:

1. Ones: $6 \times 8 = 48$ → write $8$, carry $4$.
2. Tens: $6 \times 1 = 6$, plus carry $4$ → $10$. Write $0$, carry $1$.
3. Hundreds: $6 \times 2 = 12$, plus carry $1$ → $13$. Write $13$.

$218 \times 6 = 1{,}308$

✅ Check: $(200 \times 6) + (10 \times 6) + (8 \times 6) = 1200 + 60 + 48 = 1{,}308$ ✓

Order the Steps 🔽

You're computing $\,57 \times 6$. Choose what happens at each stage. ($6 \times 7 = 42$ and $6 \times 5 = 30$.)

You've Got the Rhythm

The whole one-digit method is a steady beat: multiply, write, carry — repeat.

🔑 At each column: multiply the two digits, add any carry from the previous column, write the ones digit of that result, and carry the rest. When you reach the last column, write the whole number.

Now run the beat yourself on the problems below.

Carry Practice 🧮

Multiply. Remember to carry when a column goes past $9$.

1) $63 \times 7 = \,?$ 2) $84 \times 5 = \,?$ 3) $347 \times 9 = \,?$

✖️ Multi-Digit Multiplication

Part 3 of 5 — The Box (Area) Method for Two-Digit × Two-Digit

🔑 The Idea: When both numbers have two digits, split each number by place value into a grid. Multiply every box, then add. The grid makes it almost impossible to lose a piece.

Building the Box

To compute $\,23 \times 44$, split each number:

• $23 = 20 + 3$
• $44 = 40 + 4$

Draw a $2 \times 2$ grid and multiply each row-piece by each column-piece:

Now add all four boxes:

$800 + 120 + 80 + 12 = 1{,}012$

$23 \times 44 = 1{,}012$

💡 Why four boxes? Two pieces times two pieces makes $2 \times 2 = 4$ products. The box method just keeps every partial product organized so nothing gets skipped.

Fill the Box 🔽

You're computing $\,36 \times 25$ using the area method. Split: $36 = 30 + 6$ and $25 = 20 + 5$. Choose the value for each box.

One More: $\,64 \times 53$

Split: $64 = 60 + 4$ and $53 = 50 + 3$.

Add the four partial products:

$3{,}000 + 200 + 180 + 12 = 3{,}392$

$64 \times 53 = 3{,}392$

✅ Check by rounding: $64 \approx 60$ and $53 \approx 50$, so the answer should be near $60 \times 50 = 3{,}000$. Our $3{,}392$ is close — that's a good sign it's right.

Concept Check 🎯

The Box Method in Four Moves

1. Split each two-digit number into tens + ones.
2. Draw a $2 \times 2$ grid.
3. Multiply to fill all four boxes.
4. Add the four partial products.

💡 The hardest box is always tens × tens — but it's just a single-digit fact with zeros tacked on (like $30 \times 20 = 600$). Try the full method on the two problems below.

Box It Up 🧮

Use the area method. Enter the final product.

1) $24 \times 32 = \,?$ 2) $51 \times 46 = \,?$

✖️ Multi-Digit Multiplication

Part 4 of 5 — The Standard Algorithm (Two-Digit × Two-Digit)

🔑 The Idea: The stacked "standard algorithm" is the box method written compactly. You multiply by the ones digit, then by the tens digit, and add the two rows. The key is the placeholder zero on the second row.

Stacking It Up: $\,73 \times 46$

Step 1 — Multiply by the ones digit ($6$):

$73 \times 6 = 438$

Step 2 — Multiply by the tens digit ($4$, which is really $40$):

Write a placeholder $0$ in the ones spot first, because you're really multiplying by $40$, not $4$:

$73 \times 40 = 2{,}920$

Step 3 — Add the two partial products:

$438 + 2{,}920 = 3{,}358$

$73 \times 46 = 3{,}358$

⚠️ The most common mistake is forgetting the placeholder $0$ on the second row. Without it, you'd be multiplying by $4$ instead of $40$ — and your answer would be ten times too small.

Concept Check 🎯

It Works for Bigger Numbers Too

The exact same two-row recipe handles a three-digit number on top. You still make a ones row, then a tens row with a placeholder $0$, then add.

🔑 No matter how long the top number is, multiplying by a two-digit number always gives you exactly two rows to add — one for the ones digit, one for the tens digit. Let's walk through one step by step.

Order the Steps 🔽

You're computing $\,125 \times 32$ with the standard algorithm. Choose what happens at each stage.

A Three-Digit × Two-Digit Example

Compute $\,236 \times 18$.

Ones row: $236 \times 8 = 1{,}888$ Tens row (placeholder $0$): $236 \times 10 = 2{,}360$

Add them:

$1{,}888 + 2{,}360 = 4{,}248$

$236 \times 18 = 4{,}248$

✅ Estimate to check: $236 \approx 240$ and $18 \approx 20$, so expect roughly $240 \times 20 = 4{,}800$. Our $4{,}248$ is in the right neighborhood. ✓

Standard Algorithm Drill 🧮

Use the stacked method (don't forget the placeholder $0$!).

1) $46 \times 27 = \,?$ 2) $312 \times 24 = \,?$

✖️ Multi-Digit Multiplication

Part 5 of 5 — Word Problems, Estimation & Mastery Check

You can now (1) use place value, (2) multiply by one digit with carrying, (3) use the box method, and (4) run the standard algorithm. Let's apply it to real situations and finish with a mastery quiz.

Multiplication in Real Life

Word problems hide a multiplication inside a story. Look for the phrase "each" or a rate — that's your signal to multiply.

Example

A school orders $\,24\,$ boxes of pencils. Each box holds $\,36\,$ pencils. How many pencils in all?

The phrase "each box holds $36$" means equal groups, so multiply:

$24 \times 36 = ?$

Using the standard algorithm: $24 \times 6 = 144$ and $24 \times 30 = 720$.

$144 + 720 = 864$

So there are $864$ pencils.

💡 Estimate first: $24 \approx 25$ and $36 \approx 35$, so expect about $25 \times 35 \approx 875$. The exact answer $864$ is close — that confirms it.

Estimation Check 🎯

A Plan for Any Word Problem

1. Read and find the two numbers.
2. Spot the signal: words like each, per, every, or rows of mean equal groups → multiply.
3. Estimate so you'll know if your answer is reasonable.
4. Compute with the standard algorithm.
5. Label your answer (pencils, seats, boxes…).

💡 If the story says "each ___ has ___," you are almost always multiplying. Use the plan on the two problems below.

Word Problem Drill 🧮

Solve each. Enter just the number.

1) A warehouse has $\,45\,$ shelves. Each shelf holds $\,28\,$ boxes. How many boxes total? 2) A bus seats $\,52\,$ people. How many people can $\,16\,$ full buses carry?

Quick Reference

⚠️ Top mistakes to avoid: forgetting to add the carry, dropping the placeholder $0$ on the tens row, and skipping a box in the area method.

Exit Quiz ✅

Answer all three to finish the lesson.