Technique explanation
Teaching Explanation
This lesson extends the base method to numbers above 100 (using surpluses) and mixed cases. For $103 \times 104$, the surpluses are +3 and +4. We cross-add ($103 + 4 = 107$) and multiply ($3 \times 4 = 12$) to get 10712. In "across" cases like $102 \times 97$, we have a surplus (+2) and a deficiency (-3). The cross-operation still works ($102 - 3 = 99$), but the product is negative ($-6$). We treat the left side as hundreds ($9900$) and subtract the 6 to get $9894$.
Method Conditions
Numbers must be near 100. For "across" cases, the product of differences should be easily subtractable.
Standard Fallback
Standard long multiplication.
Misconception & Correction
Misconception: In mixed cases, forgetting to treat the left side as "hundreds" before subtracting (e.g., $99$ and $-6$ becoming $984$ instead of $9894$).
Correction: The left side represents the number of hundreds. A negative right side must be subtracted from the total value ($9900 - 6$).
Module
Worked examples
Worked Examples
- Calculate $103 \times 104$. Step 1: Surpluses are +3 and +4. Step 2: Left side: $103 + 4 = 107$. Step 3: Right side: $3 \times 4 = 12$. Result: 10712. Verification: $103 \times 104 = 10712$.
- Calculate $102 \times 97$. Step 1: Differences are +2 and -3. Step 2: Left side: $102 - 3 = 99$. Step 3: Right side: $2 \times -3 = -6$. Step 4: Final adjustment: $9900 - 6 = 9894$. Verification: $102 \times 97 = 9894$.
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