Multi-Digit Multiplication

✖️ Multi-Digit Multiplication

When numbers get bigger, you cannot always solve them in your head. Instead, you use a step-by-step method called the standard algorithm. It breaks one big problem into a few small, easy multiplication facts.

The Big Idea

Multiplication is just a fast way of adding equal groups.

• $4 \times 23$ means "4 groups of 23."
• Instead of adding $23 + 23 + 23 + 23$, you multiply.

The secret to large numbers is to multiply one digit at a time, always working from right to left, and to carry when a product is 10 or more.

Place Value Is Your Friend

Every digit lives in a column: ones, tens, hundreds. Keeping digits lined up in their columns is what makes the algorithm work.

The $2$ is really 2 tens (20) and the $3$ is 3 ones. We multiply each part and put it back together.

Multiplying by One Digit

Let's multiply $23 \times 4$.

Step 1 — Multiply the ones. $3 \times 4 = 12$. That is too big for one column, so write the 2 and carry the 1 to the tens.

Step 2 — Multiply the tens. $2 \times 4 = 8$, then add the carried 1: $8 + 1 = 9$. Write the 9.

Step 3 — Read the answer. $23 \times 4 = 92$

Why carrying works

$3 \times 4 = 12$ is 1 ten and 2 ones. The 2 ones stay in the ones column, and the 1 ten moves over to wait with the other tens. That carried 1 is what we add in Step 2.

Multiplying by Two Digits

When the bottom number has two digits, you multiply twice and then add.

Example: $34 \times 26$

1. Multiply by the ones digit (6): $34 \times 6 = 204$
2. Multiply by the tens digit (2): $34 \times 2 = 68$. Because the 2 means 20, you shift one place left, so it becomes $680$.
3. Add the two partial products: $204 + 680 = 884$

So $34 \times 26 = 884$. The little 0 you write when shifting is a placeholder that keeps your columns honest.

Concept Check 🎯

Make sure the core idea makes sense before we practice.

📝 Worked Examples

Let's slow down and watch every step.

Example 1: $46 \times 3$ (one digit)

Step 1 — Ones: $6 \times 3 = 18$. Write 8, carry 1.

Step 2 — Tens: $4 \times 3 = 12$, plus the carried $1 = 13$. Since there are no more digits, write 13 in front.

$46 \times 3 = 138$

Estimate to check: $46$ is about $50$, and $50 \times 3 = 150$. Our answer $138$ is close, so it makes sense. ✅

Example 2: $52 \times 14$ (two digits)

Row 1 — Multiply by the ones digit (4): $52 \times 4 = 208$

Row 2 — Multiply by the tens digit (1): $52 \times 1 = 52$. The 1 means 10, so shift one place left and write a 0 placeholder: $520$.

Add the partial products: $208 + 520 = 728$

So $52 \times 14 = 728$.

Your Turn 🧮

Solve $63 \times 5$ one step at a time. Type a number in each box.

1. Ones step: $3 \times 5 = ?$

2. Tens step: $6 \times 5 = 30$, plus the carried $1$. What is the new total for the tens step?

3. What is the final answer to $63 \times 5$?

🧭 Guided Practice

Work through each problem carefully. Use the standard algorithm and check with estimation.

Build the Algorithm 🔍

Choose the correct words to complete the rules of multi-digit multiplication.

🌍 Real-World Multiplication

Multiplication shows up everywhere in daily life. The trick is spotting the equal groups.

Shopping

If one toy costs $24 and you buy $15$ toys, the total is $24 \times 15 = 360$, i.e. $360.

Area

A garden is $18$ feet long and $12$ feet wide. Its area is $18 \times 12 = 216$ square feet.

Equal Groups

If each class has $28$ students and there are $4$ classes, that is $28 \times 4 = 112$ students.

Tip: Underline the two numbers being multiplied, then set them up in the standard algorithm.

Solve the Word Problem 🧮

A school orders 6 boxes of markers. Each box holds 35 markers.

1. Ones step: $5 \times 6 = ?$

2. Tens step: $3 \times 6 = 18$, plus the carried $3$. What is the tens total?

3. How many markers are there in all?

Apply It 🎯

🏆 Review & Challenge

You now have all the tools for multi-digit multiplication. Here is the whole method in one place.

The Standard Algorithm, Step by Step

Remember

• Line up your columns carefully — ones under ones, tens under tens.
• Shift one place left when you multiply by the tens digit.
• Estimate first so a wrong answer stands out.

You've got this! 🌟

Mixed Challenge 📋

Pull together everything you've learned.