Module 1: Numbers and Whole Numbers

Example:

${"version":"1.1","math":"241 \times 187 =$                                   

 

1. Align the pace value digits into columns.

   ${"version":"1.1","math":"\begin{array}{r} &241\\ \times\!\!\!\!\!\!&187\\ \hline & \end{array}"}$                                     

    

2. Start at the units digit of the bottom number and multiply it by the digit in the same column.
   1. When the product is greater than 10, transfer the tens digit to the top next column as done in addition.
   2. Next, multiply the top digit from the next column with the units digit of the bottom number
      1. If there is a digit transferred to the top of the column, add that number to the just solved product.

      ${"version":"1.1","math":"\begin{array}{r} &241\\ \times\!\!\!\!\!\!&187\\ \hline &7 \ \end{array} \;\;\ \rightarrow \;\; \begin{array}{r} &^2241\\ \times\!\!\!\!\!\!&187\\ \hline &87 \ \end{array}\;\; \; \rightarrow\; \begin{array}{r} &241\\ \times\!\!\!\!\!\!&187\\ \hline &1687 \ \end{array} "}$

      Multiplication Part 1 Description

      • Box 1: Multiply ${"version":"1.1","math":"7\times1"}$ to get 7. This product is only in the units, so there is no digit to transfer to the top of the next column.
      • Box 2: Multiply ${"version":"1.1","math":"7\times4"}$ to get 28. This product has a tens digit, so the 2 must be transferred over top of the next column.
      • Box 3: Multiply ${"version":"1.1","math":"7\times2"}$ to get 14. Since there is that 2 residing above the column, add it to the product: ${"version":"1.1","math":"14+2=16"}$.
3. Once the units digit of the bottom number has been multiplied to all the digits from the top number, move over to the tens digit of the bottom number and conduct the same steps, keeping in mind the “over 10” rule.
   1. Since you are multiplying the tens digit to the top number, write those products below the units digit product and start with a 0.

    

   ${"version":"1.1","math":"\begin{array}{r} &241\\ \times\!\!\!\!\!\!&187\\ \hline &1687\\ &80 \end{array} \;\;\ \rightarrow \;\; \begin{array}{r} &^3241\\ \times\!\!\!\!\!\!&187\\ \hline &1687\\ &280 \end{array}\;\; \; \rightarrow\; \begin{array}{r} &241\\ \times\!\!\!\!\!\!&187\\ \hline &1687\\&19280 \end{array} "}$

   Multiplication Part 2 Description

   • Box 1: When writing out the product using 8 now instead of 7, start the line with a 0 in the units position. Then, multiply ${"version":"1.1","math":"8\times1"}$ to get 8. Write this product in the tens position line.
   • Box 2: Multiply ${"version":"1.1","math":"4\times8=32"}$. Add 2 to the product line and place 3 above the next column.
   • Box 3: Multiply ${"version":"1.1","math":"2\times8=16"}$ and add ${"version":"1.1","math":"3=19"}$. Place that number into the correct product line.

    
4. Repeat these steps with the hundreds digit of the bottom number.
   1. For this product line, start with two 0.
   2. Continue until all bottom number digits are used.

    

   ${"version":"1.1","math":"\begin{array}{r} &241\\ \times\!\!\!\!\!\!&187\\ \hline &1687\\ &19280\\ &100 \end{array} \;\;\ \rightarrow \;\; \begin{array}{r} &^3241\\ \times\!\!\!\!\!\!&187\\ \hline &1687\\ &19280\\&4100 \end{array}\;\; \; \rightarrow\; \begin{array}{r} &241\\ \times\!\!\!\!\!\!&187\\ \hline &1687\\&19280\\ &24100 \end{array} "}$

   Multiplication Part 3 Description

   • Box 1: Start with the two 0. Multiply ${"version":"1.1","math":"1\times1=1"}$. Add it to the correct product line.
   • Box 2: Multiply ${"version":"1.1","math":"4\times1=4"}$. Add it to the correct product line.
   • Box 3: ${"version":"1.1","math":"2\times1=2"}$. Add it to the correct product line.

    
5. Finally, add all the product lines up as per vertical addition, being mindful of alignment.

   ${"version":"1.1","math":"\begin{array}{r} &241\\ \times\!\!\!\!\!\!&187\\ \hline &^11^1687 \\&^119280 \\&24100 \\ \hline&45067\end{array}"}$                                     

Example:

${"version":"1.1","math":"285 \times 47"}$

 

1. Begin by making a grid based on how many digits are present in the multiplication statement.
   1. The first number is the horizontal axis of the grid while the second number is the vertical axis of the grid.
   2. Write out the digits of the number along the boxes of their respective axis.
   3. Add diagonal lines from the top right to bottom left to cross out each box of the grid.

   

2. Fill in the boxes by multiplying the outer numbers corresponding to the box edges. For products that are over 10, place the tens digit in the top half and the units digit in the bottom half.

   

   ${"version":"1.1","math":"5\times4=20"}$. As this product is over 10, place the 2 in the top half of the box and the 0 in the bottom half.

   In the box to the left, the corresponding numbers are 8 and 4. Thus, multiply 8 and 4 to get 32. 3 is added to the top half of the box and 2 is added to the bottom half of the box.

   Continue to complete the rest of the grid.

3. Add up the numbers in each diagonal space within the grid and write the resulting sum of each at the bottom of their respective diagonal (starting from the right).
   1. Follow the “over 10” rule for addition and pass the tens digit to the diagonal over to the left.

   

   In the first diagonal, the only number from the grid is 5. Thus, place 5 at the bottom of the diagonal.

   In the next diagonal over, the numbers are 0, 3, and 6 with a sum of 9.

   In the next diagonal, ${"version":"1.1","math":"2+2+5+4=13"}$. Since the sum is over 10, write 3 at the bottom and place the 1 at the top of the next diagonal. Add that 1 to the sum of the next diagonal.

   Continue to complete the rest.

4. Read the numbers out to get the final answer.

   ${"version":"1.1","math":"285 \times 47 = 13395 "}$

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