MathIsimple

Two-Digit by Two-Digit Multiplication (No Regrouping)

Master the art of multiplying two-digit numbers without regrouping! Learn powerful strategies that make large number multiplication easy and fun.

Learning Scenario

School Uniform Ordering Challenge! The school needs to order new sports uniforms for the students. Each uniform costs $23, and there are 32 students who need uniforms. How much money will the school need to spend? Let's help them calculate this using the powerful decomposition method!

What is the Decomposition Method?

Definition

The decomposition method breaks down a multiplication problem into smaller, easier parts. Instead of multiplying 23 × 32 all at once, we break it into simpler multiplications and then add the results together.

Think of it like this: It's like breaking a big job into smaller tasks that are easier to handle, then combining all the results at the end!

Why Use Decomposition?

  • Makes large multiplication problems easier to understand
  • Helps you see the place value relationships
  • Reduces the chance of making errors
  • Builds a strong foundation for more complex problems

Step-by-Step Process

The Decomposition Method

1

Break down the second number

Split 32 into 30 + 2 (tens and ones)

2

Multiply by each part separately

23 × 30 = 690 and 23 × 2 = 46

3

Add the results together

690 + 46 = 736

Visual Representation

23 × 32 = ?
Break down 32:
32 = 30 + 2
Multiply each part:
23 × 30 = 690
23 × 2 = 46
Add the results:
690 + 46 = 736
Answer: 736

Vertical Algorithm (Standard Method)

Setting Up the Problem

When we write the multiplication vertically, we need to be careful about place value alignment. The second line (when multiplying by the tens digit) should be shifted one place to the left.

23
× 32
46 ← 23 × 2 (ones)
690 ← 23 × 30 (tens) - shifted left
736 ← Final answer

Why Shift the Second Line?

When we multiply by the tens digit (3 in 32), we're actually multiplying by 30, not 3. That's why we shift the result one place to the left - it represents the tens place value.

23 × 2 = 46 (ones place)

23 × 30 = 690 (tens place - shifted left)

Total: 46 + 690 = 736

Example Problems

Example 1: 14 × 21

Using Decomposition Method:

Break down 21:21 = 20 + 1
14 × 20:14 × 20 = 280
14 × 1:14 × 1 = 14
Add results:280 + 14 = 294

Answer: 14 × 21 = 294

Example 2: 12 × 33

Using Vertical Algorithm:

12
× 33
36 ← 12 × 3 (ones)
360 ← 12 × 30 (tens) - shifted left
396 ← Final answer

Answer: 12 × 33 = 396

Interactive Activities

Activity 1: Decomposition Practice

Solve these problems using the decomposition method:

15 × 22 =
13 × 31 =
24 × 12 =

Activity 2: Real-World Problem

A bookstore has 18 shelves, and each shelf holds 25 books. How many books are there in total?

Common Mistakes to Avoid

❌ Forgetting to shift the second line

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

❌ Adding instead of multiplying

Make sure you're multiplying each part, not adding the digits together.

❌ Not checking your work

Always verify your answer by using estimation or the decomposition method.

Practice Problems

Problem 1

A school is ordering 16 boxes of pencils. Each box contains 24 pencils. How many pencils will they receive in total?

Problem 2

Use the vertical algorithm to solve: 22 × 13

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Key Takeaways

What You Learned

  • Decomposition method breaks large multiplication into smaller parts
  • Vertical algorithm requires careful place value alignment
  • Both methods give the same correct answer
  • Practice helps build confidence and accuracy

Next Steps

  • Practice with more two-digit multiplication problems
  • Learn multiplication with regrouping
  • Apply these skills to real-world problems