Multiplying by 1-Digit Numbers: Fifth Grade Mathematics

Multiplying multi-digit numbers by a 1-digit number forms the essential building block for all higher-level multi-digit multiplication algorithms.

Connecting Models to the Standard Algorithm

Before using the standard vertical algorithm, we can visualize multiplication by a 1-digit number using an area model. The multi-digit factor is expanded by place value, and the 1-digit factor multiplies each rectangle section.

Area Model for 4 * 2,365:

       2,000              300             60            5
    +----------------+--------------+-------------+-----------+
  4 |   4 * 2,000    |   4 * 300    |   4 * 60    |   4 * 5   |
    |   = 8,000      |   = 1,200    |   = 240     |   = 20    |
    +----------------+--------------+-------------+-----------+

Summing Partial Products: 8,000 + 1,200 + 240 + 20 = 9,460

The Standard Vertical Algorithm

The standard algorithm compresses the partial products into a single vertical process by recording digits and carrying regrouped values to the next column.

   Standard Vertical Algorithm for 4 * 2,365:

         1   2   2
         2 , 3   6   5
      *              4
      ----------------
         9 , 4   6   0

   Step 1: 4 * 5 ones = 20 ones. Write 0, regroup 2 tens.
   Step 2: 4 * 6 tens = 24 tens + 2 regrouped = 26 tens. Write 6, regroup 2 hundreds.
   Step 3: 4 * 3 hundreds = 12 hundreds + 2 regrouped = 14 hundreds. Write 4, regroup 1 thousand.
   Step 4: 4 * 2 thousands = 8 thousands + 1 regrouped = 9 thousands. Write 9.
   Final Product: 9,460.

Common Error: Adding Before Multiplying

A frequent mistake is adding the regrouped number before multiplying. For example, in Step 2 above, a student might incorrectly add 6 + 2 to get 8, and then multiply 4 * 8 = 32. Always remember the correct order: multiply the digits first, then add the regrouped amount.

Multiplying Numbers with Internal Zeros

When the multi-digit factor contains internal zeros, you must still multiply the zero and then add any regrouped value.

   Multiplying with Zeros:  7 * 6,008

            5
         6, 0 0 8
      *         7
      -----------
        4 2,0 5 6

   - 7 * 8 ones = 56 (write 6, regroup 5 tens).
   - 7 * 0 tens = 0 tens + 5 regrouped = 5 tens (write 5).
   - 7 * 0 hundreds = 0 hundreds (write 0).
   - 7 * 6 thousands = 42 thousands (write 42).

Exercises and Solutions

Exercise 1: Standard Algorithm Calculation. Compute the product 6 * 4,837 using the standard vertical algorithm. Show the carried regroupings and describe each step.

Solution 1: Write 4,837 with 6 below it. Multiply ones: 6 * 7 = 42; write 2 in the ones place and carry 4 above the tens place. Multiply tens: 6 * 3 = 18; add the carried 4 to get 22; write 2 in the tens place and carry 2 above the hundreds place. Multiply hundreds: 6 * 8 = 48; add the carried 2 to get 50; write 0 in the hundreds place and carry 5 above the thousands place. Multiply thousands: 6 * 4 = 24; add the carried 5 to get 29; write 29 in the thousands and ten-thousands places. The final product is 29,022.

Exercise 2: Multi-Step Real-World Word Problem. A school library purchases 8 new computer stations for a technology lab. If each computer station costs $1,209, what is the total cost of the purchase?

Solution 2: Multiply the unit cost ($1,209) by the quantity (8). Write 1,209 multiplied by 8 vertically. In the ones place, 8 * 9 = 72 (write 2, carry 7). In the tens place, 8 * 0 = 0, and adding the carried 7 gives 7 (write 7). In the hundreds place, 8 * 2 = 16 (write 6, carry 1). In the thousands place, 8 * 1 = 8, and adding the carried 1 gives 9 (write 9). The total cost of the 8 computer stations is $9,672.