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Mathematics › 🎯 Grade 4 › Multiplication: The Elastic Area Model
🎯 Grade 4 · Lesson 2 of 9

Multiplication: The Elastic Area Model

Stretch a rectangle into tens and ones, and multiplication turns into simple addition.

Grade 4Elementary
Multiplication: The Elastic Area Model — illustration
💡
The big idea: Multiplying two numbers is the same as finding the area of a rectangle whose sides are those numbers. When the numbers are big, split each one into tens and ones, build a rectangle for every combination, and add up the areas. This area model turns one hard multiplication into several easy ones.
🎯 By the end, you'll be able to
  • Explain multiplication as building an array or the area of a rectangle
  • Split a two-digit number into tens and ones to multiply it by a one-digit number
  • Apply the distributive property to break a multiplication into partial products
  • Multiply two two-digit numbers by adding four partial products
📎 You should already know
  • Multiplication facts up to 10 x 10
  • Place value of tens and ones

Multiplication builds a rectangle

You already know multiplication as repeated addition: 4 × 3 means 4 groups of 3, or 3 + 3 + 3 + 3. There is another way to picture the exact same thing: draw a rectangle that is 4 squares tall and 3 squares wide. Count the squares inside, and you get the product.

This rectangle picture is called the area model, and it turns out to be the secret behind multiplying much bigger numbers too.

🔑 Area = length times width
For whole numbers, the area of a rectangle (the number of unit squares that fit inside it) is always its length multiplied by its width. Multiplying two numbers and building a rectangle are the same idea.
\[ \text{Area} = \text{length} \times \text{width} \]
A rectangle 4 units tall and 3 units wide holds 4 × 3 = 12 unit squares.
🎮 The Area Model of Multiplication LIVE
Drag the tens and ones of each number; the rectangle splits into partial products you add up — exactly the area-model steps.

Splitting one number by place value

To multiply a bigger number like 23 by a single digit such as 6, split 23 by place value first: 23 = 20 + 3. Now build two smaller rectangles instead of one big one: one that is 6 tall and 20 wide, and one that is 6 tall and 3 wide. Then add their two areas together.

📝 Worked example: Find 6 × 23 using the area model.
  1. Split 23 into 20 + 3.
  2. First rectangle: 6 × 20 = 120.
  3. Second rectangle: 6 × 3 = 18.
  4. Add the two areas: 120 + 18.
✓ 6 &times; 23 = <strong>138</strong>.
✨ This is exactly the standard algorithm
When you multiply 6 × 23 on paper the usual way, you multiply 6 by the 3 first, then 6 by the 20, and add. The area model is not a different trick: it is a picture of the very same steps.
\[ 6\times 23 = 6\times 20 + 6\times 3 = 120 + 18 = 138 \]
Splitting the 23 into 20 + 3 turns one hard multiplication into two easy ones you add together.

Two two-digit numbers: four rectangles

When both numbers have tens and ones, split them both. Multiplying 23 × 14 means splitting 23 into 20 + 3 and 14 into 10 + 4, then building four smaller rectangles: tens × tens, tens × ones, ones × tens, and ones × ones. Add all four areas for the final answer.

📝 Worked example: Find 23 × 14 using the area model.
  1. Split 23 into 20 + 3, and 14 into 10 + 4.
  2. Tens × tens: 20 × 10 = 200.
  3. Tens × ones: 20 × 4 = 80.
  4. Ones × tens: 3 × 10 = 30.
  5. Ones × ones: 3 × 4 = 12.
  6. Add all four: 200 + 80 + 30 + 12.
✓ 23 &times; 14 = <strong>322</strong>.
⚠️ Do not forget a rectangle
The most common mistake with two two-digit numbers is only building two rectangles (tens × tens and ones × ones) and skipping the two “cross” rectangles. All four partial products must be added, or the total comes out too small.

Check your understanding

1. In the area model, 23 is split into which two parts?
23 is split by place value into 2 tens and 3 ones, which are the numbers 20 and 3.
2. Using the area model, 6 × 23 is found by adding which two products?
23 splits into 20 + 3, so 6 × 23 = 6 × 20 + 6 × 3 = 120 + 18.
3. What is 6 × 23?
6 × 20 = 120 and 6 × 3 = 18, and 120 + 18 = 138.
4. When multiplying two two-digit numbers such as 23 × 14 with the area model, how many rectangles (partial products) do you need?
Each number splits into tens and ones, giving four combinations: tens×tens, tens×ones, ones×tens, and ones×ones.
5. What is 23 × 14?
200 + 80 + 30 + 12 = 322.
✅ Key takeaways
  • Multiplication is the area of a rectangle: length times width.
  • Split a factor by place value (e.g. 23 = 20 + 3) to turn a hard multiplication into easier pieces.
  • Splitting a factor and adding the pieces is the distributive property in action: 6 × 23 = 6 × 20 + 6 × 3.
  • Multiplying two two-digit numbers needs four partial products: tens×tens, tens×ones, ones×tens, ones×ones.
  • Add every partial product; forgetting one is the most common mistake.