Multiplying Multi-Digit Numbers

Grade 4 Number and Operations in Base Ten: 4.NBT.B.5

In this lesson:

• Break big multiplication problems into easy parts
• Show your work with an area model, not just a number

Learning Objectives for This Lesson

You will learn to:

1. Decompose a factor into partial products by place value
2. Use an area model to organize partial products
3. Multiply up to four digits by one digit
4. Multiply two two-digit numbers with an area model
5. Explain why the strategy works

6 Boxes of Pencils — About How Many?

• A school orders 6 boxes. Each box has 214 pencils.
• Estimate: 6 × 200 = 1,200 — so the exact answer should be a bit more.

Can you find the exact number without a calculator?

Two Quick Recalls Before We Start

• What is 6 × 200? Say it as "6 × 2 hundreds = 12 hundreds = 1,200."
• Is 6 × 7 the same as 6 × 5 plus 6 × 2?

Decomposing 214: 200 + 10 + 4

6 × 214 = 6 × 200 + 6 × 10 + 6 × 4 = 1,200 + 60 + 24 = 1,284

A bit more than the 1,200 estimate — reasonable.

The Area Model for 6 × 214

Three sections, three partial products — together they cover the whole rectangle.

A Bigger Number, Same Routine: 3 × 1,472

3 × 1,472 = 3 × 1,000 + 3 × 400 + 3 × 70 + 3 × 2 = 3,000 + 1,200 + 210 + 6 = 4,416

Four place-value parts instead of three — same decompose-multiply-add routine.

Your Turn With a Partner: 7 × 385

Work with a partner. Decompose 385, draw the area model, and write the partial products equation.

How many sections will your rectangle need?

Predict First: 120 or 1,200?

Is 4 × 300 equal to 120, or 1,200? Predict before advancing.

Say the Unit, Not Just the Digits

• 300 is 3 hundreds. 4 × 3 hundreds = 12 hundreds.
• 12 hundreds = 1,200, not 120.
• Losing track of the unit is the single most common multiplication error.

Partial Products, Two Digits Through Four

1. 3 × 24
2. 5 × 348
3. 6 × 3,207 (a zero digit — how many sections does this need?)

Show both the area model and the partial products equation on every item.

What If Both Factors Have More Digits?

You just decomposed one factor at a time.

What happens when both factors have more than one digit?

The 2-by-2 Grid: 14 × 12

100 + 40 + 20 + 8 = 168. You could check this by skip-counting — it matches.

Predict First: Tens Times Tens Enough?

For 23 × 15, is 20 × 10 plus 3 × 5 the whole answer? Predict before advancing.

Four Boxes, Four Products: 23 × 15

20×10 + 20×5 + 3×10 + 3×5 = 200 + 100 + 30 + 15 = 345.

A Harder Grid: 23 × 36

600 + 90 + 120 + 18 = 828.

Your Turn, Independently: 34 × 27

Draw the area model, fill in all four partial products, then add.

Write all four products before you start adding any of them.

Filling and Reading the Grid

1. 42 × 16 (fill in a blank grid)
2. A completed grid shows 20×20, 20×4, 3×20, 3×4 — write the matching equation.
3. Why are there exactly four partial products when both factors have two digits?

Computing Is Only Half the Standard

You can compute both types of problems now.

But the standard doesn't just ask you to compute — it asks you to explain.

Three Views of One Problem: 5 × 347

• Model: area model with three sections — 300, 40, 7.
• Equation: 5×300 + 5×40 + 5×7 = 1,500 + 200 + 35 = 1,735.
• Explanation: "5 groups of 347 is 5 groups of 300, plus 40, plus 7 — added together."

Your Turn: Model, Equation, Explanation

For 26 × 18, produce all three: draw the area model, write the equations, and write or say a short explanation.

Share your explanation with a partner and refine it together.

A Preview: Next Year's Shortcut

• Next year, you'll learn the standard algorithm — a faster, more compact method.
• It computes the exact same partial products — just organized differently.
• Understanding partial products now is why that shortcut will make sense later.

Find the Error in This Solution

A friend says 4 × 52 = 4×5 + 4×2 = 20 + 8 = 28. What did they do wrong?

Now You Choose the Model Yourself

Every model so far has been drawn for you, or you knew which type of grid to use.

From here on, nobody tells you which model fits — you decide.

Which Model? How Many Sections?

8 × 53 (?), 24 × 31 (?), 6 × 1,250 (? — watch the zero)

A Farmer's Rows of Seeds: 8 × 435

A farmer plants 8 rows with 435 seeds in each row. How many seeds in total?

Estimate first. Then model, compute, and check.

A School's Desks and a Truck's Packages

• A school has 24 classrooms with 32 desks each. How many desks?
• A delivery truck makes 6 trips, carrying 1,250 packages each trip. How many packages?

Your Turn: Choose the Model, Solve It

A recipe makes 6 batches of 218 cookies each. A second recipe makes 34 batches of 22 cookies each. Which recipe makes more, and by how much?

Nobody is telling you which model each part needs.

Find and Fix the Mistake

A worked solution for 5 × 218 shows: 5×2 + 5×1 + 5×8 = 10 + 5 + 40 = 55. Find the mistake, explain it, and give the correct answer.

Four Common Mistakes to Double-Check

Decompose into place values (200), not bare digits (2).

A 2-by-2 grid needs all four boxes — cross products are easy to skip.

Say the unit: "4 times 3 hundreds," not just "4 times 3."

If you can't explain a section, you're following steps, not understanding.

Key Takeaways From This Lesson

✓ Decompose a factor by place value before multiplying.
✓ An area model proves you covered every part with no gaps.
✓ Two decomposed factors need four partial products, not two.
✓ Illustrating and explaining are part of the math, not extras.

Coming Up Next: Multi-Digit Division

Multiplication breaks a factor apart to make it easier. Division breaks the number you're dividing apart the same way.