# Why does the long division algorithm work?

I am having trouble answering the following question. Any help would be greatly appreciated.

Use long division to solve the following problem. How would you explain this algorithm conceptually to a 5th grader? In other words, why does this method work?

For example, 462 ÷ 3

Relevance

It is just the distributive law. 462 = 300 + 150 + 12, so dividing by 3, or multiplying by 1/3, just take 1/3 of each part:

462/3 = 300/3 + 150/3 + 12/3 = 100 + 50 + 4 = 154.

The algorithm first finds the 100 and subtracts off the 300, leaving 162. That is the part that still needs to be split into thirds. The algorithm then does 162/3 getting 50 and subtracts off the 150 leaving 12. That last part is then also divided by 3.

• It's easy to show that 300/3 = 100. What you need to show after that is that (300 + 162) ÷ 3 is equal to (300 ÷ 3) + (162 ÷ 3). It's the distributive property, of course, but the usual 5th grader hasn't seen that.

I'd use a look-see proof for that, with a set of boxes. Rectangles of equal height and different width. (Maybe just two boxes here, one about twice as wide as the other, that you can label "300" and "162" later.)

Put those side-by-side to make a wide rectangle with the common height, and then draw horizontal lines to divide the combined box into 3 equal pieces. One of those wide stripes is clearly 1/3 of the whole, but it's also made up of two pieces...each being also obviously 1/3 of one of the two original rectangles. This illustrates (A + B)/3 = (A/3) + (B/3) in a picture.

Now point out that dividing 4/3 to get 1 means that 400/3 is at least 100, that 3*100 = 300 and the original number is equal to 300 + 162. Then explain that (300 + 162)/3 is 100 + 162/3. You have 1 digit of the quotient (the hundreds digit) and you can use the same method to find the first digit of 162/3. And so on...

Finally, explain that the long division layout does all that, but keeps the amount of brainwork down to a minimum by just working with enough leading digits to get the next digit of the quotient.

That probably won't all sink in, but it's almost at-level, it's mathematically correct, and the picture of the distributive property may pay dividend later. Your last resort is, of course, "...because it works, that's why!"

Best wishes...our kids are out of school now. :^)

• Anonymous
4 years ago

Long Division Algorithm