Division as Repeated Subtraction - Third Grade Mathematics

Just as multiplication is repeated addition moving forward, division is repeated subtraction moving backward until nothing is left. When you need to divide a quantity, you can repeatedly take away groups of the divisor size until you hit zero, counting how many times you were able to subtract. In this chapter, you will master repeated subtraction with counters, equations, and number lines.

How Repeated Subtraction Works

When you divide 12 ÷ 3, you are asking: "How many times can I subtract 3 from 12 until I reach 0?"

Watch the step-by-step subtraction chain:

Starting Amount:  12
Subtract 3:       12 - 3 = 9   (1st time)
Subtract 3:        9 - 3 = 6   (2nd time)
Subtract 3:        6 - 3 = 3   (3rd time)
Subtract 3:        3 - 3 = 0   (4th time)

We subtracted 3 exactly 4 times!
Therefore: 12 ÷ 3 = 4

Every single subtraction step represents removing one complete group. The number of subtractions you perform is your quotient.

Modeling Division on a Number Line

On a number line, repeated subtraction means starting at the dividend and making backward hops of the divisor's size until you reach 0.

Example: Modeling 15 ÷ 5 = 3 on a Number Line

  1. Find 15 on the number line.
  2. Hop backward by 5 units each time.
  3. Count the total number of backward hops.
    Hop 3             Hop 2             Hop 1
  (-5 units)        (-5 units)        (-5 units)
<------------     <------------     <------------
0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
  • Hop 1 moves from 15 back to 10.
  • Hop 2 moves from 10 back to 5.
  • Hop 3 moves from 5 back to 0. You took 3 backward hops of size 5 to arrive at 0. Therefore, 15 ÷ 5 = 3!

Connecting Repeated Addition and Repeated Subtraction

Notice how multiplication and division are mirror images of each other on the number line:
- Multiplication (3 x 5 = 15): Start at 0, take 3 forward hops of 5, land on 15.

- Division (15 ÷ 5 = 3): Start at 15, take backward hops of 5 until you reach 0, count 3 hops.


Forward hops (Addition):   0 ---> 5 ---> 10 ---> 15  (3 jumps of 5)
Backward hops (Division): 15 ---> 10 ---> 5 ---> 0   (3 jumps of 5)

What Happens When There Are Leftovers? (A Preview of Remainders)

What if you try to divide 14 ÷ 4 using repeated subtraction?
- 14 - 4 = 10 (1st subtraction)

- 10 - 4 = 6 (2nd subtraction)

- 6 - 4 = 2 (3rd subtraction)
Now you have 2 left. Can you subtract another group of 4? No, because 2 is smaller than 4! You subtracted 4 three whole times, with 2 left over. In third grade, we focus primarily on numbers that divide evenly with zero leftovers, but recognizing leftovers prepares you for higher mathematics!

Chapter Practice Exercises

  1. Use repeated subtraction to solve each division problem. Write out each subtraction step: a. 16 ÷ 4 b. 20 ÷ 5 c. 18 ÷ 6 d. 21 ÷ 7
  2. Write the division equation represented by this subtraction chain: 24 - 8 = 16 16 - 8 = 8 8 - 8 = 0
  3. Draw or describe the backward jumps on a number line for 18 ÷ 3. Where does each backward jump land? How many jumps are there?
  4. A candle maker has 30 inches of wick. Each candle needs 6 inches of wick. Use repeated subtraction to find how many candles can be made.
  5. True or False: 25 ÷ 5 means subtracting 5 from 25 five times until reaching 0. Explain your reasoning.

Solutions and Step-by-Step Answers

  1. Subtraction steps: a. 16 ÷ 4: 16 - 4 = 12; 12 - 4 = 8; 8 - 4 = 4; 4 - 4 = 0 (4 times). Quotient = 4. b. 20 ÷ 5: 20 - 5 = 15; 15 - 5 = 10; 10 - 5 = 5; 5 - 5 = 0 (4 times). Quotient = 4. c. 18 ÷ 6: 18 - 6 = 12; 12 - 6 = 6; 6 - 6 = 0 (3 times). Quotient = 3. d. 21 ÷ 7: 21 - 7 = 14; 14 - 7 = 7; 7 - 7 = 0 (3 times). Quotient = 3.
  2. The number 8 was subtracted 3 times from 24 to reach 0. Equation: 24 ÷ 8 = 3.
  3. For 18 ÷ 3: Start at 18.
  4. Jump 1 lands on 15.
  5. Jump 2 lands on 12.
  6. Jump 3 lands on 9.
  7. Jump 4 lands on 6.
  8. Jump 5 lands on 3.
  9. Jump 6 lands on 0. There are 6 backward jumps of size 3, so 18 ÷ 3 = 6.
  10. Candle wick: 30 - 6 = 24 24 - 6 = 18 18 - 6 = 12 12 - 6 = 6 6 - 6 = 0 The candle maker can make 5 candles (30 ÷ 6 = 5).
  11. True. 25 - 5 = 20; 20 - 5 = 15; 15 - 5 = 10; 10 - 5 = 5; 5 - 5 = 0. Subtracting 5 exactly five times brings the total to 0, which confirms that 25 ÷ 5 = 5.