Children CRAVE big numbers - and using big numbers gives a great deal of repetition without it being a worksheet drill.
We write the problem in a linear fashion at this level. The focus is on the sensorial experience of short division. Only one board is used (for the 1-digit divisor).
|When they have completed this work, they will have done|
the equivalent of 6 division problems on a worksheet.
Except that sensorially, they understand the
concept on a much deeper level.
They are actually DOING the division.
In elementary, whether they have done that work or not, they move into the racks and tubes fairly early (assuming a mostly full primary experience), using a 4-digit or 5-digit dividend and a 2-digit divisor. They then make up their own problems with as many digits in both dividend and divisor as the material allows (7 is the max for the dividend; 4 for the divisor).
- They work with combinations of zeroes in various places.
- They record their answers in the proper places using long division notation.
- The slowly build up through a series of exercises to writing more steps on paper.
- Finally they do the work on paper and only check with the beads (usually somewhere between age 8 and 10).
Some of my son's work when he first began this work at elementary:
|8,492 / 34|
He then extended into his own problems, including 3-digit divisors and just being plain silly.
9,999,999 / 99 - his first completely on his own (he was surprised at this answer!) - he only needed reminder where to place the answer; otherwise he covered all the steps himself.
7,657,776 / 214
8,222,743 / 1,234
This was ONE afternoon, not even the entire afternoon. How many 1st grade worksheet problems is this equivalent to? For most curricula - NONE. They don't do division in 1st grade. But for the sake of counting it up anyway, I'm still not sure - how do you quantify the number of individual division problems in 8,222,743 / 1,234 - technically, there are 4 distribution steps, but over 4 boards each time. Essentially we are looking at 12 division problems, but there is also the sensorial understanding of how it all works that just doesn't come from facing a worksheet of repetitive math problems.
My son made these numbers up himself. They mean something to HIM. And in these 4 division problems he has covered the concepts of 3 years of division work by most curricula.
And that's only exercise 1. Follow-up exercises introduce the writing out of the problem in stages.
HOW TO CHOOSE A DIVIDEND WITH NO REMAINDER:
You can see a lot of my own writings - helping him to find a problem that had no remainder just for the sake of focusing on a different skill at that particular moment - choose your divisor and quotient and multiply them; this provides the dividend; just don't share the quotient with the child until they've discovered it themselves - for some children this is like MAGIC! It's a fun game to play - "I have written a number on this piece of paper, I wonder if you will get the same number if you divide out this really large number, but I'm not going to show you until you've done the work - we'll see if we get the same number!" (if the child makes an error in the beginning, ah well, the numbers don't work and the magic wasn't active that day) :)
LONG DIVISION IS JOYFUL:
This work brought a lot of joy on an otherwise very rotten day. Who'd've thought that long division brings JOY!? :) This was big work, it was meaningful (he turned it into food distribution for a king whose populace were facing a drought and had to come to the king for a fair amount of food), it was long, it was challenging, but he was ready and capable. He was playing a game with numbers and finding some patterns.