Assignment: Using the lattice method, solve the following: 1. 135x2, 2. 182x24, 3. 631x2, 4. 832x3, 5. 912x3, 6. 452x3, 7. 213x4, 8. 923x4, 9. 324x5, 10. 416x5, 11. 805x5, 12. 424x6

Here are the solutions to the multiplication problems using the lattice method.

The lattice method involves:

1. Drawing a grid (lattice) based on the number of digits in the multiplicands.
2. Multiplying individual digits and placing the tens digit in the upper triangle and the units digit in the lower triangle of each cell.
3. Summing the numbers along the diagonals, starting from the bottom right, carrying over any tens digits to the next diagonal.
4. Reading the result from the top left to the bottom right.

Since I cannot draw the lattice grids directly, I will describe the products within the cells and then show the diagonal sums. For a number like $AB \times C$, the lattice would have cells for $A \times C$ and $B \times C$. For $ABC \times DE$, it would be a $3 \times 2$ grid.

• For single-digit multipliers, the lattice has one row. The cells will contain the products of each digit of the first number with the single digit of the second number. Example: $135 \times 2$ Cells: $1 \times 2 = 02$, $3 \times 2 = 06$, $5 \times 2 = 10$. Diagonals (from right to left):

  1. Units: 0 (from 10)
  2. Tens: 1 (from 10) + 6 (from 06) = 7
  3. Hundreds: 0 (from 06) + 2 (from 02) = 2
  4. Thousands: 0 (from 02) Result: 270

• For multi-digit multipliers, the lattice has multiple rows. Example: $182 \times 24$ Cells (row 1 for multiplier 2, row 2 for multiplier 4): $1 \times 2 = 02$, $8 \times 2 = 16$, $2 \times 2 = 04$ $1 \times 4 = 04$, $8 \times 4 = 32$, $2 \times 4 = 08$

  Lattice (conceptual):

        1   8   2
      +---+---+---+
    2 |0/2|1/6|0/4|
      +---+---+---+
    4 |0/4|3/2|0/8|
      +---+---+---+
  

  Copy

  Diagonals (from bottom right to top left):

  1. Units: 8 (from 0/8)
  2. Tens: 4 (from 0/4) + 2 (from 3/2) + 4 (from 0/4) = 10. Write 0, carry 1.
  3. Hundreds: 0 (from 0/4) + 6 (from 1/6) + 3 (from 3/2) + 0 (from 0/4) + (carry 1) = 10. Write 0, carry 1.
  4. Thousands: 2 (from 0/2) + 1 (from 1/6) + (carry 1) = 4.
  5. Ten Thousands: 0 (from 0/2) Result: 4368

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1. $135 \times 2$

   • Cells: $1 \times 2 = 02$, $3 \times 2 = 06$, $5 \times 2 = 10$
   • Diagonal sums:
     • Units: 0
     • Tens: $1 + 6 = 7$
     • Hundreds: $0 + 2 = 2$
     • Thousands: 0
   • Result: $\boxed{\text{270}}$

2. $182 \times 24$

   • Cells:
     • Row 1 (for 2): $1 \times 2 = 02$, $8 \times 2 = 16$, $2 \times 2 = 04$
     • Row 2 (for 4): $1 \times 4 = 04$, $8 \times 4 = 32$, $2 \times 4 = 08$
   • Diagonal sums:
     • Units: 8
     • Tens: $4 + 2 + 4 = 10$. Write 0, carry 1.
     • Hundreds: $0 + 6 + 3 + 0 + 1 (\text{carry}) = 10$. Write 0, carry 1.
     • Thousands: $2 + 1 + 1 (\text{carry}) = 4$
     • Ten Thousands: 0
   • Result: $\boxed{\text{4368}}$

3. $631 \times 2$

   • Cells: $6 \times 2 = 12$, $3 \times 2 = 06$, $1 \times 2 = 02$
   • Diagonal sums:
     • Units: 2
     • Tens: $0 + 6 = 6$
     • Hundreds: $0 + 2 = 2$
     • Thousands: 1
   • Result: $\boxed{\text{1262}}$

4. $832 \times 3$

   • Cells: $8 \times 3 = 24$, $3 \times 3 = 09$, $2 \times 3 = 06$
   • Diagonal sums:
     • Units: 6
     • Tens: $0 + 9 = 9$
     • Hundreds: $0 + 4 = 4$
     • Thousands: 2
   • Result: $\boxed{\text{2496}}$

5. $912 \times 3$

   • Cells: $9 \times 3 = 27$, $1 \times 3 = 03$, $2 \times 3 = 06$
   • Diagonal sums:
     • Units: 6
     • Tens: $0 + 3 = 3$
     • Hundreds: $0 + 7 = 7$
     • Thousands: 2
   • Result: $\boxed{\text{2736}}$

6. $452 \times 3$

   • Cells: $4 \times 3 = 12$, $5 \times 3 = 15$, $2 \times 3 = 06$
   • Diagonal sums:
     • Units: 6
     • Tens: $0 + 5 = 5$
     • Hundreds: $1 + 2 = 3$
     • Thousands: 1
   • Result: $\boxed{\text{1356}}$

7. $213 \times 4$

   • Cells: $2 \times 4 = 08$, $1 \times 4 = 04$, $3 \times 4 = 12$
   • Diagonal sums:
     • Units: 2
     • Tens: $1 + 4 = 5$
     • Hundreds: $0 + 8 = 8$
     • Thousands: 0
   • Result: $\boxed{\text{852}}$

8. $923 \times 4$

   • Cells: $9 \times 4 = 36$, $2 \times 4 = 08$, $3 \times 4 = 12$
   • Diagonal sums:
     • Units: 2
     • Tens: $1 + 8 = 9$
     • Hundreds: $0 + 6 = 6$
     • Thousands: 3
   • Result: $\boxed{\text{3692}}$

9. $324 \times 5$

   • Cells: $3 \times 5 = 15$, $2 \times 5 = 10$, $4 \times 5 = 20$
   • Diagonal sums:
     • Units: 0
     • Tens: $2 + 0 = 2$
     • Hundreds: $1 + 5 = 6$
     • Thousands: 1
   • Result: $\boxed{\text{1620}}$

10. $416 \times 5$

    • Cells: $4 \times 5 = 20$, $1 \times 5 = 05$, $6 \times 5 = 30$
    • Diagonal sums:
      • Units: 0
      • Tens: $3 + 5 = 8$
      • Hundreds: $0 + 0 = 0$
      • Thousands: 2
    • Result: $\boxed{\text{2080}}$

11. $805 \times 5$

    • Cells: $8 \times 5 = 40$, $0 \times 5 = 00$, $5 \times 5 = 25$
    • Diagonal sums:
      • Units: 5
      • Tens: $2 + 0 = 2$
      • Hundreds: $0 + 0 = 0$
      • Thousands: 4
    • Result: $\boxed{\text{4025}}$

12. $424 \times 6$

    • Cells: $4 \times 6 = 24$, $2 \times 6 = 12$, $4 \times 6 = 24$
    • Diagonal sums:
      • Units: 4
      • Tens: $2 + 2 = 4$
      • Hundreds: $1 + 4 = 5$
      • Thousands: 2
    • Result: $\boxed{\text{2544}}$