MathIsimple

Two-Digit by Two-Digit Multiplication (With Regrouping)

Master the art of multiplying two-digit numbers with regrouping! Learn to handle multiple regrouping steps with confidence and accuracy.

Learning Scenario

Library Book Purchase Challenge! The school library is buying new books for the reading program. Each box contains 28 books, and they need to order 15 boxes. How many books will the library receive in total? Let's solve this using the powerful area model method!

What is the Area Model?

Definition

The area model represents multiplication as the area of a rectangle. We break down each number into its tens and ones, then create a grid to show all the partial products.

Think of it like this: It's like dividing a rectangle into smaller rectangles, where each smaller rectangle represents a part of the multiplication!

Why Use the Area Model?

  • Makes regrouping visual and easy to understand
  • Shows all partial products clearly
  • Helps organize multiple regrouping steps
  • Builds understanding of place value relationships

Area Model Example: 28 × 15

Step 1: Break Down the Numbers

28 × 15

Break down 28:

28 = 20 + 8

Break down 15:

15 = 10 + 5

Step 2: Create the Area Model Grid

20
× 10
200
20
× 5
100
8
× 10
80
8
× 5
40

28 (20 + 8) × 15 (10 + 5)

Step 3: Add All Partial Products

20 × 10 =200
20 × 5 =100
8 × 10 =80
8 × 5 =40
Total:420

Vertical Algorithm with Regrouping

Step-by-Step Process

28 × 15
28
× 15
140 ← 28 × 5 (ones)
280 ← 28 × 10 (tens) - shifted left
420 ← Final answer

Understanding the Regrouping

In this example, we don't have regrouping in the final addition, but let's look at a case where we do:

Example: 26 × 18
26
× 18
208 ← 26 × 8 (ones)
260 ← 26 × 10 (tens) - shifted left
468 ← Final answer

Notice: 26 × 8 = 208 (regrouping needed)

Example Problems

Example 1: 34 × 16

Using Area Model:

30
× 10
300
30
× 6
180
4
× 10
40
4
× 6
24
300 + 180 + 40 + 24 =544

Answer: 34 × 16 = 544

Example 2: 27 × 19

Using Vertical Algorithm:

27
× 19
243 ← 27 × 9 (ones)
270 ← 27 × 10 (tens) - shifted left
513 ← Final answer

Note: 27 × 9 = 243 (regrouping: 7×9=63, write 3, carry 6)

Answer: 27 × 19 = 513

Interactive Activities

Activity 1: Area Model Practice

Create an area model for 25 × 17:

Total:

Activity 2: Real-World Problem

A store has 23 shelves, and each shelf holds 18 items. How many items are there in total?

Common Mistakes to Avoid

❌ Forgetting to carry when regrouping

When a product is 10 or more, remember to carry the tens digit to the next column.

❌ Not shifting the second line properly

When multiplying by the tens digit, shift the result one place to the left.

❌ Adding partial products incorrectly

Double-check your addition when combining all the partial products.

Practice Problems

Problem 1

A factory produces 36 items per hour and operates for 14 hours. How many items are produced in total?

Problem 2

Use the vertical algorithm to solve: 29 × 17

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Key Takeaways

What You Learned

  • Area model makes regrouping visual and organized
  • Vertical algorithm requires careful regrouping steps
  • Both methods help understand place value relationships
  • Practice builds confidence with regrouping

Next Steps

  • Practice with more regrouping problems
  • Learn three-digit by one-digit division
  • Apply these skills to real-world problems