Technique explanation
Teaching Explanation
Every square number can be broken down into smaller, easier-to-calculate parts. Imagine a large square with a side length of 13. We can split it into a 10+3 side. This creates four areas inside the big square: 1. A large 10×10 square (100). 2. Two identical 10×3 rectangles (30 + 30 = 60). 3. A small 3×3 square (9). Adding these together (100 + 60 + 9) gives us 169. This structure, `a² | 2ab | b²`, works for any two-digit number. We just need to remember that `a` represents the tens and `b` represents the units.
Method Condition
This method is applicable when: Squaring any two-digit number.
Standard Fallback
If this method is not suitable, use: Standard multiplication (x × x).
Misconception & Correction
Misconception: Forgetting to double the product of the digits (using `ab` instead of `2ab`). Correction: A square expanded as `(a+b)²` always results in two identical rectangles of area `ab`. You must include both to get the correct total area.
Worked examples
Worked Examples
| Step | Example 1: 24² | Example 2: 31² |
|---|---|---|
| 1. Split Digits | a = 2, b = 4 | a = 3, b = 1 |
| 2. Square Tens (a²) | 2² = 4 (represents 400) | 3² = 9 (represents 900) |
| 3. Double Product (2ab) | 2 × (2 × 4) = 16 (represents 160) | 2 × (3 × 1) = 6 (represents 60) |
| 4. Square Units (b²) | 4² = 16 | 1² = 1 |
| 5. Combine | 400 + 160 + 16 = 576 | 900 + 60 + 1 = 961 |
| 6. Independent Check | 24 × 24 = 576 | 31 × 31 = 961 |
Always perform an independent check to verify your result.
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