Technique explanation
Teaching Explanation
The crosswise method uses a symmetric star pattern to calculate digits of the product from right to left. For 4-digit numbers, we follow seven specific steps:
1. Right Vertical: Multiply the units digits.
2. Right 2-Digit Cross: Multiply and sum the units and tens digits diagonally.
3. Right 3-Digit Cross: Multiply and sum the units, tens, and hundreds digits.
4. 4-Digit Cross: Multiply and sum all four digits (outer pairs and inner pairs).
5. Left 3-Digit Cross: Multiply and sum the tens, hundreds, and thousands digits.
6. Left 2-Digit Cross: Multiply and sum the hundreds and thousands digits.
7. Left Vertical: Multiply the thousands digits.
Each step produces a sum; we record the units digit and carry any tens to the next step on the left.
Method Conditions and Fallback
This method fits any multi-digit multiplication and is particularly useful for mental or single-line calculations when the digits are small.
Standard-method fallback: Standard long multiplication (column method) using multiple rows of partial products.
Misconception and Correction
Misconception: Forgetting the inner pairs in the 4-digit cross step.
Correction: Visualize the step as two nested crosses. The outer digits $(1st \times 4th)$ and $(4th \times 1st)$ form the large cross, and the inner digits $(2nd \times 3rd)$ and $(3rd \times 2nd)$ form the small cross. Both must be summed.
Visual Overlay Requirement
A 7-frame diagram sequence showing two 4-digit rows ($A B C D$ and $E F G H$). Each frame highlights the digits being multiplied:
- Frame 1: $D \cdot H$
- Frame 2: $C \cdot H + D \cdot G$
- Frame 3: $B \cdot H + D \cdot F + C \cdot G$
- Frame 4: $A \cdot H + D \cdot E + B \cdot G + C \cdot F$
- Frame 5: $A \cdot G + C \cdot E + B \cdot F$
- Frame 6: $A \cdot F + B \cdot E$
- Frame 7: $A \cdot E$
Worked examples
Example 1: $1212 \times 1011$
- Steps:
1. $2 \times 1 = 2$
2. $(1 \times 1) + (2 \times 1) = 3$
3. $(2 \times 1) + (1 \times 1) + (2 \times 0) = 3$
4. $(1 \times 1) + (2 \times 1) + (1 \times 0) + (2 \times 1) = 5$
5. $(1 \times 1) + (2 \times 0) + (1 \times 1) = 2$
6. $(1 \times 0) + (2 \times 1) = 2$
7. $1 \times 1 = 1$
- Result: $1,225,332$.
- Independent Check: $1212 \times 1000 = 1,212,000$; $1212 \times 11 = 13,332$. Sum: $1,212,000 + 13,332 = 1,225,332$.
Example 2: $1111 \times 1111$
- Steps:
1. $1 \times 1 = 1$
2. $1+1 = 2$
3. $1+1+1 = 3$
4. $1+1+1+1 = 4$
5. $1+1+1 = 3$
6. $1+1 = 2$
7. $1 \times 1 = 1$
- Result: $1,234,321$.
- Independent Check: $1111^2 = (1000 + 111)^2 = 1,000,000 + 222,000 + 12,321 = 1,234,321$.
- Steps:
1. $2 \times 1 = 2$
2. $(1 \times 1) + (2 \times 1) = 3$
3. $(2 \times 1) + (1 \times 1) + (2 \times 0) = 3$
4. $(1 \times 1) + (2 \times 1) + (1 \times 0) + (2 \times 1) = 5$
5. $(1 \times 1) + (2 \times 0) + (1 \times 1) = 2$
6. $(1 \times 0) + (2 \times 1) = 2$
7. $1 \times 1 = 1$
- Result: $1,225,332$.
- Independent Check: $1212 \times 1000 = 1,212,000$; $1212 \times 11 = 13,332$. Sum: $1,212,000 + 13,332 = 1,225,332$.
Example 2: $1111 \times 1111$
- Steps:
1. $1 \times 1 = 1$
2. $1+1 = 2$
3. $1+1+1 = 3$
4. $1+1+1+1 = 4$
5. $1+1+1 = 3$
6. $1+1 = 2$
7. $1 \times 1 = 1$
- Result: $1,234,321$.
- Independent Check: $1111^2 = (1000 + 111)^2 = 1,000,000 + 222,000 + 12,321 = 1,234,321$.
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