# 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 equal-size 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 number-line 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.

 MULTIPLICATION has to do with many groups of the same size.Here is one group of four. Draw another group of four to the picture.And another group. And another. And one more group.

You drew _____ groups of four.   5 × 4 = 4 + 4 + 4 + 4 + 4 = 20.

 Let's reverse the process.  Start out with 20 sticks.Make one group of four. In your mind, “move it away” from the picture. Form another group of four. Again, “move it away”, or subtract it from the picture.Keep forming groups of four till you have none left.  How many groups did you make?  ______ | | | | | | | || | | | | | | || | | |

20 − 4 − 4 − 4 − 4 − 4 = 0

This is repeated subtraction.  You subtract 4 repeatedly till you reach zero.
Each subtraction is a group of 4.

How many groups? _____  How many times did you subtract? _____
That is the answer to the division problem 20 ÷ 4.

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.
I subtracted 4 five times,
so 20 ÷ 4 = 5.

75 ÷ 25 = ??

 7 5 − 2 5 1 5 0 − 2 5 1 2 5 − 2 5 1 0 3

I subtracted 25 three times,
so 75 ÷ 25 = 3.

84 ÷ 21 = ??

 8 4 − 2 1 1 6 3 − 2 1 1 4 2 − 2 1 1 2 1 − 2 1 1 0 4

I subtracted 21 four times,
so 84 ÷ 21 = 4

 Since 13 + 13 = 26, 13 goes into 26 two times. So, 26 ÷ 13 = 2 Since 25 + 25 + 25 = 75,  25 goes into 75 three times. So, 75 ÷ 25 = 3 Since 21 + 21 + 21 + 21 = 84,  21 goes into 84 four times. So 84 ÷ 21 = 4

2. Write a multiplication sentence AND a division sentence that fits the addition/subtraction facts.

 a. 5 + 5 + 5 = 1515 − 5 − 5 − 5 = 0 ___ × ___ = ___________ ÷ ___ = ____ b. 20 + 20 + 20 + 20 + 20 = _______ − 20 − 20 − 20 − 20 − 20 = 0 ___ × ___ = _________ ÷ ___ = ___ c. 23 + 23 + 23 = ________ − 23 − 23 − 23 = 0 ___ × ___ = _________ ÷ ___ = ___ d. 14 + 14 + 14 + 14 + 14 = _______ − 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 = 0Five jumps of 4 gets you from 20 till 0. What division is illustrated here?

4. Draw jumps backwards to illustrate the division sentences.

 a. 30 ÷ 5 = _______ b. 16 ÷ 4 = _______ c. 27 ÷ 3 = _______ d. 28 ÷ 4 = _______ e. 42 ÷ 6 = _______ f. 48 ÷ 3 = _______ g. 26 ÷ 2 = _______ h. 39 ÷ 13 = _______ i. 48 ÷ 4 = _______

5. Solve using repeated subtraction OR adding up to the number being divided.

 a. 40 ÷ 20 = ______90 ÷ 30 = ______30 ÷ 15 = ______ b. 52 ÷ 13 = ______34 ÷ 17 = ______69 ÷ 23 = ______ c.88 ÷ 22 = ______32 ÷ 16 = ______72 ÷ 18 = ______ d. 45 ÷ 15 = ______90 ÷ 15 = ______90 ÷ 18 = ______

6. If 12 × 2 = 24, then 13 × 2 is _____ . How about division? Use the previous

 a. 24 ÷ 2 = ______26 ÷ 2 = ______28 ÷ 2 = ______30 ÷ 2 = ______ b. 32 ÷ 2 = ______36 ÷ 2 = ______38 ÷ 2 = ______42 ÷ 2 = ______ c. 48 ÷ 2 = ______50 ÷ 2 = ______52 ÷ 2 = ______58 ÷ 2 = ______ d. 60 ÷ 2 = ______66 ÷ 2 = ______70 ÷ 2 = ______78 ÷ 2 = ______

7. Try the same kind of thing when dividing by 3.

 a. 30 ÷ 3 = ______36 ÷ 3 = ______39 ÷ 3 = ______ b. 42 ÷ 3 = ______45 ÷ 3 = ______51 ÷ 3 = ______ c. 60 ÷ 3 = ______69 ÷ 3 = ______72 ÷ 3 = ______ d. 81 ÷ 3 = ______90 ÷ 3 = ______99 ÷ 3 = ______

8. Solve the problems.

a. Complete the tables for Alice's reading schedules, if
• she reads 12 pages a day
 Day 1 2 3 4 5 6 7 8 9 10 11 12 Pages read 12 24

• she reads 15 pages a day
 Day 1 2 3 4 5 6 7 8 9 10 11 12 Pages read 15 30

• she reads 20 pages a day
 Day 1 2 3 4 5 6 7 8 9 10 11 12 Pages read 20 40

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:

 Books 1 2 10 20 30 40 50 60 80 100 Shelf space 2 cm 4 cm

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.