Below is an example of how a parent allowed his child to guide the learning. Good job Mr. Hansen!!
My 3rd grade son is deep into multiplication right now and our discussion about groups of things and multiplication eventually led to factors. And while he was pondering how many numbers can be multiplied together to get 12 or 16, he asked a very good question. He asked me "How come there isn't a division table?" and he went on to say "We could make a division table and sell it!"
I explained to him that division tables do not exist because once you put the numbers down the sides, the majority of the inside is empty because most numbers are not (evenly) divisible by the other numbers. I told him that when we do division mentally we actually use the multiplication facts in reverse, as we did with addition. But I could see in his eyes that my explanation didn't phase him much and he was still dreaming of selling millions of division tables to other 3rd graders. So I said "Let's make a division table!" Big smile.
So I opened a spreadsheet and we started making our table. I said to him "let's just go up to 20 for now and we can expand it later", otherwise we would be there for hours if we went all the way to 144 on both edges. So I numbered the rows and columns from 1 to 20 and then like a game of battleship I started calling off the rows and columns and waiting on him for the quotient, or as is the case in the majority of cells, his response "it doesn't divide".
It took less time than I imagined before he saw the light as to why it is difficult to make a good division table. Maybe it was all the dead ends with him responding "it doesn't divide", but he realized that most of the cells are empty and when you are talking about "whole" division you are actually talking about knowing which combinations of numbers are divisible in the first place which is essentially the multiplication table. But something neat happened. He noticed the diagonal of 1's (you can't really miss that) but then he spotted the 2's, so we studied that for a bit and I started coloring the 1's, then the 2's and then the 3's and so forth so that he could see how they each repeat and form a line. And then I pointed out that there were some numbers with no quotients at all, except for 1 and themselves. I colored those red and pointed out that every now and then those patterns of 2's and 3's and 4's line up in such a fashion that they skip a number entirely, we call those primes. They aren't divisible by any number, except 1 and themselves.
In the end (actually there is no end to this) I told him that what we are doing is turning the multiplication table inside out and if we go in and count all of the filled in cells there will be 144 of them (assuming we did the whole table), one for every entry in the multiplication table. You will find all of the 2's and 3's and 4's and so forth but spread out in a 144x144 table, instead of a 12x12 table.
Our division table is here...
It was a very productive evening to say the least