Long Division: Lesson Ideas

The lesson ideas below can be given with or without Montessori materials. The preliminary exercises could be done with simple multiplication or division boards or with the bead frame. The later lessons could be done wth racks and tubes.

Preliminary Lessons

To start, you might present a problem like this:

Here we started with zero and added 4 six times to get twenty four. This is the same as saying:

You might want to do some work here to prepare children for work with remainders, modular arithmetic, and bases other than base ten. One approach might be to work with a calendar. "How many days in a week? How would you write 16 days in weeks and days (answer: 2 weeks and 2 days)? Let's look at the calendar for September 2008. Let's add the first number for Monday to the first number for Tuesday (the first Monday is September 1 and the first Tuesday is September 2). One plus two is three, and the third is a Wednesday. So we added a number from the Monday column to a number in the Tuesday column and got a number in the Wednesday column. Does that always happen? What if we take the 8 from the Monday column and add it to the 16 in the Tuesday column. That gives us 24. Yup, 24 is in the Wednesday column." Children can work through lots of examples. When October comes, we find that if we a Monday date to a Tuesday date, we get back a Monday date. Why the change?

We can do further investigations using bead bars of seven to represent a week. Continuing investigation can get us to modular arithmetic (e.g., 4 mod 7 + 6 mod 7 = 3 mod 7) and work in different bases (4 base 7 + 6 base 7 = 13 base 7).

After children can produce the multiplication and division problems for the "repeatedly adding to zero" problem, you can do subtractions to get down to zero:

As before, children relate this problem to both multiplication and division problems. The problem above corresponds to:

As they do more of them, make sure they notice that they can subtract multiples of a number to make their work go faster. For example, for the problem above, we could use the fact that three sevens are twenty-one and two sevens are fourteen in order to get to zero faster:

First we subtracted three sevens (21), then we subtracted two sevens (14), for a total of five sevens, so 35 divided by 7 is 5.

This next problem could be done with racks and tubes if you have them.

Here we subtracted 200 thirty-fours, then 60 thirty-fours, then 3 thirty-fours, for a total of 263 thirty-fours. This is the notation used by Hans Freudenthal in Didactical Phenomenology of Mathematical Structures on page 130.

The next step after this is learn the traditional notation for the long division:

I expect the experience with the Freudenthal notation will help kids better understand why we are doing the subtractions we do and why, e.g., the 2 goes in the hundreds place, the six goes in the tens place and the 3 goes in the units place.

I also have some thoughts on doing this stuff with Montessori materials.