Can Division equals Subtraction for Natural Numbers
Division as Repeated Subtraction
This is a complete lesson with teaching and exercises, showing how division can be seen as repeated subtraction. It is meant for third grade.
Students solve divisions by "subtracting" or crossing out equalsize groups from the total in the visual model, until there is nothing left. Examples show how divisions can be solved by repeatedly subtracting the same number (the divisor). Often, it is actually easier to add intead of subtract, and figure out how many times you will add the number (divisor) until you reach the dividend.
The lesson also shows how numberline jumps tie in with this concept: we jump backwards from the dividend, making jumps of same size (the size being the divisor), until we reach zero. The lesson also has several word problems to solve.
You drew _____ groups of four. 5 × 4 = 4 + 4 + 4 + 4 + 4 = 20. 
20 − 4 − 4 − 4 − 4 − 4 = 0 This is repeated subtraction. You subtract 4 repeatedly till you reach zero. 
1. Make groups, but in your mind 'move them away' or subtract. Write a subtraction sentence.
DIVISION can be solved by repeated subtraction:  
20 ÷ 4 = ??
20 − 4 − 4 − 4 − 4 − 4 = 0.  75 ÷ 25 = ??
I subtracted 25 three times,  84 ÷ 21 = ??
I subtracted 21 four times, 
Often, it is handier to actually add instead of subtract:
Since 13 + 13 = 26,  Since 25 + 25 + 25 = 75,  Since 21 + 21 + 21 + 21 = 84, 
2. Write a multiplication sentence AND a division sentence that fits the addition/subtraction facts.
a. 15 − 5 − 5 − 5 = 0 ___ × ___ = ______ _____ ÷ ___ = ____  b. ____ − 20 − 20 − 20 − 20 − 20 = 0 ___ × ___ = ______ ___ ÷ ___ = ___ 
c. _____ − 23 − 23 − 23 = 0 ___ × ___ = ______ ___ ÷ ___ = ___  d. ____ − 14 − 14 − 14 − 14 − 14 = 0 ___ × ___ = ______ ___ ÷ ___ = ___ 
3. Write a subtraction sentence for each division sentence.
a. 45 ÷ 15 = _______ 45 −  b. 32 ÷ 8 = _______ 32 − 
c. 100 ÷ 20 = _______ 100 −  d. 50 ÷ 10 = _______ 50 − 
e. 50 ÷ 25= _______ 50 −  f. 78 ÷ 26 = _______ 78 − 
Multiplication is like jumps on the number line.
5 × 4 = 20. Five jumps of 4 gets you to 20. 
Division is like making jumps of four backwards from 20 till you get to 0:
20 ÷ 4 = 5. 20 − 4 − 4 − 4 − 4 − 4 = 0 Five jumps of 4 gets you from 20 till 0. 
What division is illustrated here? 
4. Draw jumps backwards to illustrate the division sentences.
a. 
b. 
c. 
d.

e.

f.

g.

h.

i.

5. Solve using repeated subtraction OR adding up to the number being divided.
a. 90 ÷ 30 = ______ 30 ÷ 15 = ______  b. 34 ÷ 17 = ______ 69 ÷ 23 = ______  c. 32 ÷ 16 = ______ 72 ÷ 18 = ______  d. 90 ÷ 15 = ______ 90 ÷ 18 = ______ 
6. If 12 × 2 = 24, then 13 × 2 is _____ . How about division? Use the previous
problem to help you solve the next one.
a. 26 ÷ 2 = ______ 28 ÷ 2 = ______ 30 ÷ 2 = ______  b. 36 ÷ 2 = ______ 38 ÷ 2 = ______ 42 ÷ 2 = ______  c. 50 ÷ 2 = ______ 52 ÷ 2 = ______ 58 ÷ 2 = ______  d. 66 ÷ 2 = ______ 70 ÷ 2 = ______ 78 ÷ 2 = ______ 
7. Try the same kind of thing when dividing by 3.
a. 36 ÷ 3 = ______ 39 ÷ 3 = ______  b. 45 ÷ 3 = ______ 51 ÷ 3 = ______  c. 69 ÷ 3 = ______ 72 ÷ 3 = ______  d. 90 ÷ 3 = ______ 99 ÷ 3 = ______ 
8. Solve the problems.
a. Complete the tables for Alice's reading schedules, if
If her book has 235 pages and she wants to read it in two weeks, which reading schedule should she choose?
 
b. Jerry reads 25 pages a day. How many pages does he read in  
c. Jerry's book has 325 pages. How many days does it take him to read it? Use the previous exercise to help.
 
d. In a bookstore there are many copies of the same book on the shelf. One book is 2 cm thick. Fill in the table:
How many books can you fit on a shelf 66 cm long?

This lesson is taken from Maria Miller's book Math Mammoth Division 1, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.
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