Teaching steps for long division with large numbers

Division as repeated subtraction - introduction to long division


Multiplication is repeated addition.  Division is the opposite of multiplication.  You can think of division as repeated subtraction.

Example.  Bag 771 apples so there are 3 apples in one bag.  How many bags are needed? You can start by putting 3 apples to one bag, which leaves you 768 apples. Then for each bag you subtract 3 apples and keep counting the bags you use, until you hit zero apples.   771 − 3 − 3 − 3 − 3 − 3 − 3 ... keep subtracting! 1 bag 1 bag 1 bag 1 bag 1 bag 1 bag ... keep counting bags! It just takes quite a long time, doesn't it?  Instead you can take a 'shortcut' and initially subtract 300 apples (taking 100 bags) or some other big multiple of 3.   771  −  300  −  300  −  30 ... 100 bags 100 bags 10 bags ...  Let's figure it out and keep count of the bags as we subtract (put in bags) the apples.   Method 1 - slower Apples Bags 771 - 300 471 100 bags - 300 171 100 bags - 30 141 10 bags - 30 111 10 bags - 30 81 10 bags - 30 51 10 bags - 30 21 10 bags - 21 0 7 bags Method 2 - quicker  Apples Bags 771 -  600 200 bags 171 -  150 50 bags 21 7 bags -  21 0 It's over 700 apples. In 100 bags would have 300 apples. In 200 bags would have 600 apples. In 300 bags would have 900 apples; too many bags. Subtracting the 600 apples from 771 leaves 171 still  to be bagged. It's over 170 apples still. In 10 bags I'll have 30 apples. In 20 bags I'd have 60 apples. In 50 bags I'd have 150 apples. In 60 bags I'd have 180 apples; that's too many. Subtract 150 apples from 171; 21 apples are left.  Need 7 bags for the last 21 apples.   So total needed 200 + 50 + 7 = 257 bags to bag all the apples.  And it all went even - no apples left over!  In other words, 771 ÷ 3 = 257.

Example 4.  You have 646 apples.  If you put 8 apples in one bag, how many bags will you need?   Apples Bags 686 -    46  -   6  Thinking first in whole hundreds:  In 100 bags I would have 800 apples but there's not that many apples, so won't need even 100 bags. In terms of whole tens then. In 10 bags I'll have ____ apples. In 50 bags I'd have ____ apples. In 80 bags I'd have ____ apples. In 90 bags I'd have ____ apples. So will need ____ bags.  Subtracting the ____ apples from 686 leaves 46 apples. Lastly for the ___ apples left  I need __ bags, and there are 6 apples left over.  So total needed _____ + ___ =           bags to bag all the apples and have 6 apples left over. In other words, 686 ÷ 8 = ______, R 6.

Example 5.  It won't matter even if you do the subtracting in smaller steps.  Compare the two ways to do the division 795 ÷ 3  by subtracting repeatedly.  You can think in terms of bagging apples if it helps. Dividend Quotient 795 -  300 100 495 - 300 100 195  - 30 10 165 - 30 10 135 - 30 10 105 Dividend Quotient 795 -  600 200 195 - 180 60 15 -  15 5 0 So total the quotient is _____ + ___  + ___ = _____, and division is even. In other words, 795 ÷ 3 = ______.

Example problems

1.  Fill in the bags/fruits at each step of bagging.

a.  Bag 610 apples; 5 apples in each bag. Apples Bags 610 - 500 110 - 100 10 -  10  0

b.  Bag 852 kiwis; 3 kiwis in each bag. Kiwis Bags 852 - 200   -  240 12 -  12  0

e.  Bag 162 pears; 6 pears in each bag. f.  Bag 495 cherries; 9 cherries in each bag. g.  Bag 429 mangos; 3 mangos in each bag.

h.  Bag 164 pineapples; 2 pineapples in each bag. i.  Bag 4613 guavas; 7 guavas in each bag. j.  Bag 1098 bananas; 9 in each bag.

2.  Do the divisions using the repeated subtraction.  You can still think in terms of bagging fruit if it helps you.  Also, you are encouraged to even try doing it mentally.  Note: some of these divisions are even, some have remainder.

a.  555 ÷ 3 b.  750 ÷ 5 c.  257 ÷ 5 d.  464 ÷ 8

m. 472 ÷ 5 n.  340 ÷ 2 o.  537 ÷ 9 p.  994 ÷ 7

q. 670 ÷ 5 r.  750 ÷ 6 s.  238 ÷ 9 t.  294 ÷ 7

Long division and why it works

The standard long division algorithm We compare here the repeated subtraction of the previous lesson and the conventional long division 'corner'.  The steps are the same, just written out differently.  For clarity's sake, we will initially write out the subtracted numbers with all the zeros included.  Also, for clarity and for easy comparison, we will write the parts of the quotient above each other.  As an example, we study  789 ÷ 3.  You can think of it as 789 apples that you are bagging in bags of 3 apples, wanting to know how many bags you need. First write the dividend 'inside' the corner, and the divisor outside:   3    7  8  9  Then let's divide! Continued subtraction Dividend (the apples) Quotient (the bags) 789     -  600 200 189 - 180 60 9 3 -  9 0   263 2 0 0 3    7  8  9  -  6  0  0 1 8 9     6 0 2 0 0 3    7  8  9  -  6  0  0 1 8 9 -  1  8  0   9   3   6 0 2 0 0 3    7  8  9  -  6  0  0 1 8 9 -  1  8  0   9 -  9 0 First step is the hundreds, finding out which multiple of 300 will fit into 789.  It is 600.  In terms of apples and bags, one has now used 200 bags to bag 600 apples. Second step, the tens.  Which multiple of 30 will fit into 189?  That is 180, meaning one uses 60 bags to bag 180 apples.  So 60 is added to the quotient. Lastly look at the ones. There are still 9 apples left, which means one needs 3 bags more.  Add 3 to the quotient. The final answer is 263. The last step is to check the division by multiplication: whether 3 × 263 is 789.  263 ×  3 789

Why it works Comparing the division to the continued subtraction probably has already let you see why it works.  In the conventional way of writing the long division, it's not so easy to see the process.  The key is that in each step, one does NOT actually divide by the actual divisor but by a multiple of it.  Just like in the apples/bags examples, you don't start out by subtracting 3 apples each time, but first 'hit it hard' by subtracting multiples of 300 apples if possible, then multiples of 30, then 3.  In essence, you first divide by 300, then by 30, then by 3. Also, in the conventional long division, you only place one digit into the quotient in each step, not with all the zeros.  The digits shown in gray are not usually written out in the conventional long division algorithm. Hundreds "How many 3's in 7?" (How many 300's in 789?) Tens "How many 3's in 18?" (How many 30's in 189?) Ones "How many 3's in 9?" 2 0 0 3    7  8  9  -  6  0  0 1 8 9     2 6 0 3    7  8  9  -  6  0  0 1 8 9 -  1  8  0    0 9     2 6 3 3    7  8  9  -  6  0  0 1 8 9 -  1  8  0    0 9 -   9 0 To get the hundreds digit in the quotient, one asks the question:  "How many times does 300 go into 789", or the division 789 ÷ 300!  You are not dividing by 3 because you try to 'hit it hard' and subtract as many multiples of 300 as possible.  Since 300 is a whole hundred, the tens and ones digits in the 789 won't matter when you are finding how many times 300 goes into 789.  So the thing can be done easier by calculating  7 ÷ 3, or thinking "How many times does 3 go into 7". The remainder from first step (what is left after subtraction) is in reality 189.  But since the ones digit (9) won't be important in the next step (which deals with the tens digit), in the traditional way, you only subtract 7-6 and then you 'drop' down the tens digit 8 from the dividend. To get the tens digit, similarly one asks the question: "How many times does 30 go into 189", or does the division  189 ÷ 30.  Again, since you're dividing by a multiple of ten, the ones digit '9' in the 189 does not affect the division at all.  The important thing is to look at the whole tens in the number 189, which is 180. So to find the answer to the division 189 ÷ 30, you can think of the division 180 ÷ 30, which is the same as thinking 18 ÷ 3:  "How many times does 3 go into 18?" The last step is simple since it is dealing with ones digits, how many times does 3 go into 9.

 

Examples of long division

These examples show how long division is done, with all of the dropping down of digits and such.  It is important to keep the rows and columns lined up.

850 ÷ 2 = ?   4 2    8  5  0  -  8 0 4 2 2    8  5  0  -  8 0 5 -  4    4  2    2    8  5  0  -  8 0 5 -  4  1 0 Drop down the 0 of the 850 next to the 1. Then divide 2 into 10. In the hundreds digits, divide 2 into 8. Ask, 'How many 2's in 8?" That is EXACTLY 4 times. Multiply 4 × 2 = 8 and subtract that from 8 to find the remainder which is of course 0. Then drop down the tens digit 5 and divide 2 into 5.  2 goes into 5 two times but the division is not exact.  So multiply 2 × 2 = 4, place 4 underneath the 5 and subtract to find the remainder. Then multiply 5 × 2 = 10 and place the result under the 10 and subtract.  Since the result is zero and there are no more digits to drop from the dividend, the division is over. 4  2 5 2    8  5  0  -  8 0 5 -  4 1 0 -  1  0 0 1   425 ×  2 850 Check the division by multiplication.

Study also the following examples with your teacher.
 

Thousands digit Hundreds digit Tens digit Ones digit How many 7's in 1? How many 7's in 15? How many 7's in 11? How many 7's in 42? (0) 7    1 5  1  2     2 7    1  5  1  2  - 1   4     1     2  1 7    1 5  1  2   - 1  4   1  1  -  7  4   2  1  6 7    1 5 1   2   - 1  4   1 1  - 7 4  2 -  4 2 0

1.  Divide using long division.  Check by multiplication.

a.   5   8 6 0 ×  5

c.   2   3 7 8 ×  2

e.   4   6 3 2 ×  4

g.   6   7 5 0 ×  6

Long division with remainder - Remainder in long division

Do you remember? 14 bananas divided between 3 people gives 4 bananas to each and 2 bananas that cannot  be divided. 14 carrots divided between 5 people gives 2 carrots to each and 4 carrots that cannot be divided. 14 ÷ 3 = 4, remainder 2. 3 × 4 + 2 = 14.    14 ÷ 5 = 2, remainder 4. 5 × 2 + 4 = 14. The remainder is always LESS than the divisor.  Otherwise we could divide some more. For example, if in the banana situation the remainder was more than 3, we could continue dividing the remaining bananas between 3 people. How do we check the division if it is not exact?  Look at the pictures above.  If we end up with 3 people each having 4 bananas and 2 bananas left over, then the total amount of bananas is 3 × 4 + 2 = 14.  Or if 5 people have 2 carrots each and there are 4 carrots left over, then all together we have 5 × 2 + 4 = 14 carrots. So to check the division that was not exact, multiply your result by the divisor as normal, and then ADD THE REMAINDER.  You should get the original dividend.

When using long division, the division is not always exact either. At this point we don't have any more digits to drop down from the dividend.  In the last subtraction we end up with 6, which is the remainder.  So 125 ÷ 7 = 17, R 6. Note that the remainder 6 is LESS THAN 7, the divisor         1 7 7  1  2  5  -     7 5 5 -  4  9 6        4   17 ×  7 119 119 + 6 = 125  To check we multiply the answer (17) by the divisor (7), and then add the remainder (6). We get the original dividend (125).

Example problems

1.  Do these problems in your notebook.  Write down here the result and the remainder.  Check each division by multiplying and then adding the remainder.

321 ÷ 2 = 532 ÷ 9 =

221 ÷ 3 = 922 ÷ 6 =

490 ÷ 4 =  324 ÷ 6 =

2.  Do the word problems in your notebook.

a.  While playing with matches, Annie had 204 matches.  She divided them into piles of 8 matches. How many piles dud she make?  How many matches were left over?

b.  If she divides them into piles of 20, how many piles does she get now?  Don't use long division but just think (or use real matches to help).

3.  Remember?  Multiply by 10, and make a division sentence.

10 × 21 =  ___ ÷ 10 = 21

10 × 90 =  ___ ÷ 10 = ___

10 × 87 =  ___ ÷ 10 = ___

4.  Based on the previous exercise, divide the following numbers by 10.

780 ÷ 10 = ___ 150 ÷ 10 = ___

450 ÷ 10 = ___ 120 ÷ 10 = ___

460 ÷ 10 = ___ 440 ÷ 10 = ___

5.  Based on the previous exercise, divide the following numbers by 10 and indicate the remainder.

787 ÷ 10 =           151 ÷ 10 =

452 ÷ 10 =           126 ÷ 10 =

463 ÷ 10 =           982 ÷ 10 =

Long division is one of my least favorite things to teach, but as I tell my children, “It doesn’t have to be fun, it just has to be done.”

So, I charge ahead every couple of years showing one of my fresh faced offspring the maze of steps.
One of the difficulties for them is remembering what step to do next.
Here is a handy little rhyme to remind them of the order:
Daddy, Mother, Sister, Brother
I point out that the beginning letters will remind them to
1)Divide 2)Multiply 3)Subtract 4)Bring Down
For example:

Let’s start with this simple problem.

Since “Daddy” comes first, we divide first.

Next comes “Mother”, so we multiply.

“Sister” means we subtract.

Last comes “Brother”, so we bring down and start the process again.
If that helps just one other person get through the grueling process of teaching long division, then it was worth every head banging, hair pulling, tear inducing minute.