The lattice method uses partial products to multiply by breaking down the numbers into their place values and multiplying each digit separately. These partial products are then arranged within a grid (the lattice), and finally, they're summed along the diagonals to arrive at the final product.
Here's a breakdown of the process:
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Create the Lattice: Draw a grid with rows and columns corresponding to the number of digits in each factor. Divide each cell in the grid diagonally.
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Calculate Partial Products: Multiply each digit of one factor by each digit of the other factor. Write the tens digit of the product above the diagonal and the units digit below the diagonal within the corresponding cell.
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Sum Along Diagonals: Starting from the bottom right, add the numbers along each diagonal. If the sum is a two-digit number, write the units digit and carry over the tens digit to the next diagonal.
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Read the Result: Read the digits along the left and bottom of the lattice, starting from the top left, to obtain the final product.
Example: Multiplying 28 by 57
| 5 | 7 | |
|---|---|---|
| 2 | 1/0 | 1/4 |
| 8 | 4/0 | 5/6 |
- 2 x 5 = 10: 1 is written above the diagonal and 0 below.
- 2 x 7 = 14: 1 is written above the diagonal and 4 below.
- 8 x 5 = 40: 4 is written above the diagonal and 0 below.
- 8 x 7 = 56: 5 is written above the diagonal and 6 below.
Now, sum along the diagonals:
- Bottom right: 6
- Next diagonal: 4 + 5 + 0 = 9
- Next diagonal: 1 + 0 + 4 = 5
- Top left: 1
Reading from left to right: 1596. Therefore, 28 x 57 = 1596.
Key aspects of how partial products are utilized:
- Decomposition: The method decomposes the multiplication problem into smaller, manageable multiplications (the partial products).
- Organization: The lattice provides a structured way to organize these partial products based on place value.
- Summation: The diagonal summation effectively combines the partial products, taking into account their place values, to produce the final answer.
- No Carrying During Calculation: Carry-overs only happen during the diagonal summation, simplifying the individual multiplications.
The lattice method is particularly helpful for visualizing the multiplication process and minimizing errors, especially with larger numbers.
