The lattice method is a technique that is some- times used for multiplication. Figure 4.40 shows how to use this method to multiply a 2-digit number by a 2-digit number. a. Use the lattice method to calculate 38 X 54 and 72 X 83. b. Use the partial-products method to calculate 38 X 54 and 72 X 83. c. Explain how the lattice method is related to the partial-products method. d. Discuss advantages and disadvantages of using the lattice method. 23 23 123 14 12A1A 7 18/ Draw a lattice as Multiply each digit Starting at the bottom shown in the figure. along the top with right, and moving left, Place the numbers each digit along the then up, add numbers you want to multiply side. Write the an- in diagonal strips, on the top and down swers in the cells carrying as necessary. the side. as shown. The answer is 1081. Figure 4.40 Using the lattice method to show that 23 X 47 = 1081.
The Correct Answer and Explanation is :
a. Lattice Method for 38 × 54 and 72 × 83
Let’s break down the lattice method for both calculations.
1. 38 × 54:
- Draw a 2×2 lattice grid. Label the top with 3 and 8, and the side with 5 and 4.
- Multiply each pair of digits and write the results in the cells of the grid. For example, multiply 3 × 5 = 15, and write the tens digit (1) in the top left cell and the ones digit (5) in the bottom right.
- Add the numbers in the diagonal columns to find the total product.
| 3 | 8 | |
|---|---|---|
| 5 | 15 | 40 |
| 4 | 12 | 32 |
Now, add the diagonals:
- First diagonal: 0
- Second diagonal: 5 + 1 = 6
- Third diagonal: 1 + 4 + 2 = 7
- Fourth diagonal: 3 + 3 = 6
So, the final product is 2052.
2. 72 × 83:
- Draw a 2×2 lattice grid. Label the top with 7 and 2, and the side with 8 and 3.
- Multiply each pair of digits and write the results in the cells of the grid.
| 7 | 2 | |
|---|---|---|
| 8 | 56 | 16 |
| 3 | 21 | 6 |
Add the diagonals:
- First diagonal: 6
- Second diagonal: 5 + 1 = 6
- Third diagonal: 6 + 2 + 1 = 9
- Fourth diagonal: 7 + 3 = 10
So, the final product is 5996.
b. Partial-Products Method for 38 × 54 and 72 × 83
1. 38 × 54:
Using the distributive property, break it down:
[
(30 + 8) \times (50 + 4)
]
Now, multiply:
- 30 × 50 = 1500
- 30 × 4 = 120
- 8 × 50 = 400
- 8 × 4 = 32
Add the partial products:
[
1500 + 120 + 400 + 32 = 2052
]
2. 72 × 83:
Using the distributive property, break it down:
[
(70 + 2) \times (80 + 3)
]
Now, multiply:
- 70 × 80 = 5600
- 70 × 3 = 210
- 2 × 80 = 160
- 2 × 3 = 6
Add the partial products:
[
5600 + 210 + 160 + 6 = 5996
]
c. Relationship between the Lattice Method and Partial-Products Method
Both methods break the multiplication process into smaller, more manageable steps by decomposing the numbers into place values. The lattice method organizes the partial products in a grid, helping visually track the intermediate results. In contrast, the partial-products method does this algebraically, adding the products of each place value directly. The lattice method is essentially a visual representation of the partial-products method, where each diagonal represents a set of partial products that are then added together to get the final result.
d. Advantages and Disadvantages of the Lattice Method
Advantages:
- Visual Aid: It can help visual learners by organizing the multiplication in a clear and structured way.
- Organization: The grid helps keep the multiplication process organized, ensuring no step is missed.
- Reduced Cognitive Load: The method divides the multiplication into smaller chunks, making it easier to focus on one small part at a time.
Disadvantages:
- Complexity: It can be more time-consuming compared to simpler methods like partial products, especially for larger numbers.
- Space Usage: The lattice grid requires more space to write, which can be cumbersome.
- Learning Curve: It can be harder for beginners to grasp initially, especially if they are not familiar with grid-based methods.