Let’s T.I.N.K.E.R. DIVISION : Blog Series #2 -Hacking Long Division Algorithm

7 min readMar 29, 2018

When children encounter Division around grade 3–4, they are assumed to be well versed with Place Value System, Addition, Subtraction, Multiplication, Number Bonds, ….. technically. If these foundations are laid strong, Division seems to be handy, as grouping, equal distribution, but LONG DIVISION ALGORITHM…is difficult to grasp and master it. There is no visualization, mere set of calculations, movements of numbers, and pre-learned Times Table.

It’s no doubt the most effective, optimized, generalized algorithm to process division, designed by adult expert mathematicians to teach children. But put yourself in children’ shoes and look from their vision, What’s actually happening ?

Look at the above long division of a number 4851 by 3 and 7 respectively. There rises few questions in child’s mind as we start proceeding with LDA:

Why is it, that when we divide by 3 we underline, just 1 digit, but when we divide by 7 we underline 2 digits. That’s the way my teacher and my mom told, if first digit of dividend is bigger than or equal to divisor then use it , else use first 2 digits of dividend. WHY ?
If I divide by divisor, do I have to always remember times table or inverse-time-table or brute force multiplication to carry on division ?
What if the remainder is bigger than divisor ?
If there is not big enough number in the intermediate steps, WHY do I need to put zero in quotient and pull down next number ?
What if divisor or dividend has zero as the last one more digits ?
I have nicely learnt TIMES TABLE till 10 (or may be 20) but if divisor is more than that, may be like division by 28 !!! What should I do now ?

Let’s begin hacking this division process, and connect dots with what kids already know to what they need to know as division process. Let’s introduce these following prerequisites:

Children should be clear with DIGITS: 0–9, and all the place value holders (Ones, Tens, Hundreds, Thousands…) can only hold them, nothing more, nothing less.
Let’s add flash cards and see how many do we need to move from one place value to the next (that’s the recipe EVERYONE SHOULD MASTER)

When we start from ONES place, (forming numbers 3,6,9,12,15,18,21,24,27) we can put maximum 9 cards there and as soon we have 10 cards of 3 , we do something interesting :

Use ONE card of 30, in-place of 10 cards of 3

Instead of 10 cards of 3, we pull up 1 card of 30, and move ahead. Now as we move further, we use cards of 30 and 3. We’ll keep using cards of 30 and again maximum we could use 9 cards (forming numbers 30,60,90,120,150,180,210,240,270) with different combinations of ( 3,6,9,12,15,18,21,24,27)

For Instance, lets see what are following numbers made of :

27 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 (9 : 3's)

30 = 30 (1 : 30's)

42 = 30 + 3 + 3 + 3 + 3 (1 : 30’s, 4 : 3's)

87 = 30 + 30 +3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 (2 : 30’s, 9 : 3's)

105 = 30 + 30 + 30 +3 + 3 + 3 + 3 + 3 (3 : 30’s, 5: 3's)

168 = 30 + 30 + 30 + 30 + 30 + 3 + 3 + 3 + 3 + 3 + 3 (5 : 30’s, 6: 3's)

270 = 30 + 30 + 30 + 30 + 30 + 30 + 30 + 30 + 30 (9 : 30’s)

297 = 30 + 30 + 30 + 30 + 30 + 30 + 30 + 30 + 30 +3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 (9 : 30’s, 9: 3's)

Now we’re exhausted with all 30’s and 3’s so as soon we add one more 3 :

10 : 3’s become 30 and move to next place, making 10 : 30’s eventually making 300 !

Use ONE card of 300, in-place of 10 cards of 30 or in-place of 100 cards of 3

300 = 100 cards of 3 = 10 cards of 30 = 1 card of 300

and similarly we can create maximum 2700 by using 9 cards of 300 and as we move ahead we need to use 10 card of 300, instead we use 1 card of 3000.

In all we have to keep handy, 9 cards of each :

Flash Cards can be downloaded here !

Now let’s start comparing Long Division Algorithm to method by using Flash Cards, which makes use of Place Value System.

Let’s divide 4,851 by 3, that’s equivalent of saying find the quantity that EQUALLY goes in 3 groups…

4851 = 4000 + 800 + 50 +1

> We need 1 card of 3000 (which means 3 groups of 1000)

(4000–3000) + 800 + 50 +1

= 1000 + 800 + 50 + 1

= 1800 + 50 + 1

> We need 6 cards of 300 (which means 3 groups of 600)

= (1800-1800)+ 50 + 1

= 50 + 1

> We need 1 card of 30 (which means 3 groups of 10)

= (50–30) + 1

= 21

> We need 7 cards of 3 (which means 3 groups of 7)

Now, if we browse up, we could find following groups of 3 at different steps :

1000 + 600 + 10 + 7 = 1617

Now if we compare the Long Division Algorithm done traditionally, to the one used by appreciating Place Value System we could have better understanding looking below :

Let’s divide 4,851 by 7, that’s equivalent of saying find the quantity that EQUALLY goes in 7 groups…

4851 = 4000 + 800 + 50 +1

Now, we CANNOT use the cards of 7000, as there are not enough Thousands to make group of 7. So we combine 4000+800 to find groups of 700. Hence we use 4800 at first place.

Remember, why we use 2 digits to start with, when first digit is smaller than divisor.

Let’s start with 4851 = 4800 + 50 +1

> We need 6 cards of 700 (which means 7 groups of 600)

(4800–4200) + 50 +1

= 600 + 50 + 1

> Now, we CANNOT use the cards of 700, as there are not enough hundreds to make group of 7. So we combine 600+50 to find groups of 70.

= 600+ 50 + 1

= 650 + 1

> We need 9 card of 70 (which means 7 groups of 90)

= (650–630) + 1

= 20 + 1

> We need 3 cards of 7 (which means 7 groups of 3)