Unit 7 — Multi-Digit Multiplication

Welcome back!

Multiplication becomes even more useful when working with larger numbers.

Scientists, engineers, builders, business owners, and many others multiply large numbers every day.

Today you will learn efficient methods for multiplying multi-digit numbers accurately.

Reviewing Multiplication

Multiplication is repeated addition.

Example:

6 × 4

means:

6 + 6 + 6 + 6

or

4 + 4 + 4 + 4 + 4 + 4

As numbers become larger, repeated addition is no longer practical.

We use multiplication algorithms instead.

Multiplying by One Digit

Example:

      3,428
×         6
------------

Multiply from right to left.

Regroup whenever necessary.

Keep digits aligned by place value.

Always estimate your answer before solving.

Multiplying by Two Digits

Example:

       482
×       36
------------

Multiply first by the ones digit.

Then multiply by the tens digit.

Remember that the second row begins one place to the left because it represents tens.

Finally, add the partial products.

Organizing your work carefully makes multiplication much easier.

Partial Products

Some students find it helpful to write each partial product separately.

Example:

248 × 34

First:

248 × 4

Then:

248 × 30

Finally:

Add the two results.

Understanding partial products helps explain why the standard algorithm works.

Estimating Products

Estimate before multiplying.

Example:

398 × 21

Estimate:

400 × 20 = 8,000

Your exact answer should be close to 8,000.

Estimation helps identify calculation mistakes.

Solving Word Problems

Example:

A farmer plants 248 trees in each of 36 rows.

How many trees are planted?

Step 1:

Identify the important information.

Step 2:

Choose multiplication.

Step 3:

Estimate.

Step 4:

Solve carefully.

Step 5:

Check whether the answer is reasonable.

Checking Your Work

You can often verify multiplication by using division.

If:

248 × 36 = 8,928

Then:

8,928 ÷ 36 = 248

Checking increases confidence in your answer.

Activity

Solve five multiplication problems involving two-digit multipliers.

For each problem:

• estimate first
• solve exactly
• check your answer
• explain each step

Compare your method with a classmate's.

Practice

Answer these questions.

Why is estimation useful before multiplying?

Why is the second partial product shifted one place to the left?

What are partial products?

How can division help check multiplication?

Why is careful organization important?

Review

Today you learned:

• multiplication extends repeated addition
• multi-digit multiplication uses organized algorithms
• partial products explain the multiplication process
• estimation helps identify mistakes
• division provides an effective way to check answers
• careful alignment improves accuracy

Mastering multiplication prepares you for many advanced mathematical ideas.

Looking Ahead

Next we will study long division, learning how large quantities can be divided accurately into equal groups and how division helps solve real-world problems.