Technique explanation
Teaching Explanation
We can use a special algebraic pattern to check our multiplication. If two numbers are equally spaced around a "friendly" number (like 20 or 50), their product is always the square of that middle number minus the square of the distance to it. For example, to check 18 × 22, we see the middle is 20 and the distance is 2. The formula says the answer is 20² - 2², which is 400 - 4 = 396. If our original calculation also gave 396, we have a high degree of confidence that our answer is correct.
Method Condition
This method is applicable when: Two numbers with an even difference (ensuring the midpoint is an integer).
Standard Fallback
If this method is not suitable, use: Direct standard multiplication.
Misconception & Correction
Misconception: Subtracting the distance `y` instead of the square of the distance `y²`. Correction: The identity is `x² - y²`. You must square the distance from the midpoint before subtracting it from the squared midpoint.
Worked examples
Worked Examples
| Step | Example 1: 27 × 33 | Example 2: 14 × 16 |
|---|---|---|
| 1. Find Midpoint (x) | (27 + 33) / 2 = 30 | (14 + 16) / 2 = 15 |
| 2. Find Distance (y) | 33 - 30 = 3 | 16 - 15 = 1 |
| 3. Square Midpoint (x²) | 30² = 900 | 15² = 225 |
| 4. Square Distance (y²) | 3² = 9 | 1² = 1 |
| 5. Subtract (x² - y²) | 900 - 9 = 891 | 225 - 1 = 224 |
| 6. Independent Check | 27 × 33 = 891 | 14 × 16 = 224 |
Always perform an independent check to verify your result.
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