Here’s how many of us were taught division:

“How many 25s are there in 3, non, carry and move on. ”

“How many 25s in 36? 1. Move on and take the remainder with you. ”

and so on.

How you dealt with the remainder depended on your teacher and whether the book wanted a decimal answer or not.

Is it straight forward? No not really, but the method is easy to figure out.

For example, from a child’s point of view, there’s one 25 in 36, and four 25’s in 115, therefore there’s fourteen 25’s in 365… eh?

It’s very hard to conceptualise for a child.

One 250 in 360, with 110 left over, and four 25’s in 115 equate to fourteen 25’s in 365 isn’t much better either.

Either way, that method isn’t very common anymore.

Where it is used, it looks more like this these days:

This is supposedly clearer.

The other more common method is known as ‘chunking’, or the ‘partial quotient method’ (most likely known as chunking!).

This is essentially a method whereby we break off pieces of the number and divided them by 25 separately. In the example above, we ‘know’ (we assume we know) that there are ten 25’s in 250, which is a big chunk of 365. So our answer has to be bigger than 10. We put 10 to the side (on the right in the picture above), and subtract 250 from our original number, 365. Now we have 115 left over. We ‘know’ that four 25’s are 100, so that chunk is dealt with too, and we subtract it from 115, and store the ‘4’ on the right hand side. So we have 10 + 4 groups of 25, and we’re left at this point with 15 left over from the original number 365. Well, 15 is less than our divisor 25, so it must be our remainder.

And you wonder why kids find division difficult to do, let alone comprehend!

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God knows why anybody needs to learn how to use division algorithms for BIG numbers these days.

“Hey, if we start down this road then adding fractions will be next.”

“And about time ,too!”

I was taught your second method in the 1950’s.

It is not an easy concept for students to get their head round, and I find most do not understand the method they know (even if they can apply it), when they get to us. However, the second method is useful when we later get to dividing polynomials, and those who use chunking struggle later on with this.

When I first encountered long division of polynomials I didn’t believe it. Looking at it as ” find the other factor in a multiplication” as a first introduction would help a lot. I mean for example :

(x-2)(x^3 + bx^2 + cx + d) = x^4 + 2x^3 + x -5

Multiply out and match coefficients.

Gives more application to systems of linear equations as well.

I also use that approach to answer questions, and introduce the topic with that method. I believe there is a lot more maths understanding shown in that method. However I still find some who struggle with it, as their understanding of division is so incomplete (by that I mean that it is the opposite of multiplication).