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U03 • Lesson 21 of 105

Long Division Algorithm

Develops the long division algorithm step by step, connecting division to repeated subtraction and the inverse of multiplication. Students learn the divide-multiply-subtract-bring down cycle, progressing from 2-digit / 1-digit to 4-digit / 1-digit, and interpret remainders in context.

Upper Elementary • 3-5

Prerequisites: E05, U02

Key Concepts

  • divide-multiply-subtract-bring down cycle
  • estimating quotients using compatible numbers
  • interpreting remainders in context
  • checking division with multiplication

Long Division Algorithm

Division answers the question: "How many equal groups can I make?" The long division algorithm is a step-by-step process that works for any division problem, no matter how big the numbers are.

The Four-Step Cycle

Long division repeats four steps over and over:

  1. Divide: How many times does the divisor go into the current number?
  2. Multiply: Multiply the quotient digit by the divisor.
  3. Subtract: Subtract to find what is left over.
  4. Bring down: Bring down the next digit from the dividend.

Memory Helper

Remember the steps: Divide, Multiply, Subtract, Bring down. Repeat!

Worked Example 1: 84 divided by 4

Set up: 4 ) 84

  1. Divide: 4 goes into 8 how many times? 2 times. Write 2 above the 8.
  2. Multiply: 2 x 4 = 8. Write 8 below the 8.
  3. Subtract: 8 - 8 = 0.
  4. Bring down: Bring down the 4. Now we have 04.
  5. Divide: 4 goes into 4 how many times? 1 time. Write 1 above the 4.
  6. Multiply: 1 x 4 = 4.
  7. Subtract: 4 - 4 = 0. No remainder!
84 / 4 = 21

Worked Example 2: Division with a Remainder

Find 47 / 3.

  1. Divide: 3 goes into 4 once. Write 1 above.
  2. Multiply: 1 x 3 = 3.
  3. Subtract: 4 - 3 = 1.
  4. Bring down: Bring down 7, making 17.
  5. Divide: 3 goes into 17 five times (3x5=15). Write 5 above.
  6. Multiply: 5 x 3 = 15.
  7. Subtract: 17 - 15 = 2. This is the remainder.
47 / 3 = 15 R 2

Check: 15 x 3 + 2 = 45 + 2 = 47. Correct!

Worked Example 3: Larger Dividend

Find 1,236 / 6.

  1. 6 goes into 1? No. Try 12: 6 goes into 12 twice. Write 2. Multiply: 2x6=12. Subtract: 12-12=0. Bring down 3.
  2. 6 goes into 3? Zero times. Write 0. Bring down 6, making 36.
  3. 6 goes into 36 exactly 6 times. Write 6. Multiply: 6x6=36. Subtract: 36-36=0.
1,236 / 6 = 206

Interpreting Remainders

What you do with a remainder depends on the real-world situation:

Drop the remainder:

"You have 25 stickers and put 4 on each page. How many full pages?" 25/4 = 6 R1. Answer: 6 full pages.

Round up:

"25 students need rides. Each car holds 4. How many cars?" 25/4 = 6 R1. You need 7 cars (cannot leave 1 student behind!).

Common Mistake

If your remainder is equal to or larger than the divisor, your quotient digit is too small! For example, if you divide by 5 and get a remainder of 6, increase your quotient digit by 1 and try again.

Check Your Work

Always verify: quotient x divisor + remainder = dividend. If the equation does not balance, recheck your steps.

Practice Problems

1. Divide: 96 / 8

Show Answer

96 / 8 = 12. 8 goes into 9 once (remainder 1), bring down 6 to get 16, 8 goes into 16 twice. Check: 12 x 8 = 96.

2. Divide: 73 / 5

Show Answer

73 / 5 = 14 R 3. 5 into 7 = 1 R2. Bring down 3 to get 23. 5 into 23 = 4 R3. Check: 14x5+3 = 73.

3. Divide: 2,415 / 5

Show Answer

2,415 / 5 = 483. 5 into 24 = 4 R4. Bring down 1 = 41. 5 into 41 = 8 R1. Bring down 5 = 15. 5 into 15 = 3. Check: 483 x 5 = 2,415.

4. 43 students are split into teams of 6. How many full teams? How many students are left over?

Show Answer

43 / 6 = 7 R 1. There are 7 full teams with 1 student left over.

5. You need to transport 50 boxes. Each trip carries 8 boxes. How many trips do you need?

Show Answer

50 / 8 = 6 R 2. You need 7 trips (round up, because the remaining 2 boxes still need a trip).

Lesson Summary

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