## Friday, October 9, 2015

### More Prime Factors

When we last discussed prime factors, the boys were using Table C to find all of the factors of a number and break them down into prime factors.  I did finally make peace with why oh why we do this when I realized the information can be used to find the lowest common multiple (upcoming lesson).   Last week the boys worked on another way to find all of the factors and break them down into prime factors, this time using the pegboard.

Why oh why do we use the pegboard for this lesson?  If we were only using the green pegs I would guess it would be so the child could visually determine which prime numbers were factors of the non-prime factors.  But, right out of the gate we are looking at a two-digit number and are using the blue pegs which doesn't give you any visual cue.  I also wonder why this lesson comes before divisibility.  It would make sense to me to do all of the prime factor work after divisibility.  Maybe it doesn't come before divisibility.  Maybe I just think it does.

At any rate, the boys have a hard time buying anything I'm selling poorly.  Since I was clearly uncommitted to this method, so were they.  "Why are we putting pegs on the board mom?"  "Can I just tell you the factors?" So, we moved right a long to doing it the way I did it when I was a kid.

They put any non-prime numbers in a black rectangle and branched the factors of each rectangle, starting with the lowest prime factor for each, below.  Prime factors are in the red circles.  The decided what the factors are using the facts in their heads.  They enjoyed this and each did about six number a day for a week.  I planned on showing them how to use this to find the lowest common multiple this week, but the stomach flu knocked out a few of our school days.

Me Too, in particular, really enjoyed choosing his own numbers and finding all of the factors.  It was his idea to find the prime factors for one million.  He did most of the calculating  in his head because the numbers are easy enough especially when the prime factors are 2 and 5 like this.  I had to help him find the factors for 31,250 because he couldn't keep track of the remainders in his head.  Again, for lower numbers doing this before divisibility is fine because the kids can imagine what the multiplication equation will be that arrives at the original number.  However, with a number like 31250 you are really determining divisibility.  So, perhaps we will revisit this again after divisibility.

1. I didn't remember this from school. I looked it up...but apparently you can find square and cube roots with prime factors.

As for the divisibility, I'd say that this would still be more of an exploration, and then we work to derive the "rule" for divisibility that the later lesson...but you know where my kids are in the sequence, and they are no where near where yours are. Mine need all the exploration they can get. Maybe for you all the sequence is backward.

1. Thanks. It makes me happy that we will use the prime factorization for more than LCM and GCF. Me Too LOVES to make prime factor trees. He made eight of them yesterday just for fun when we were working on GCF and LCM.

As for the divisibility, exercise 3 (finding prime factors with the pegboard) even mentions divisibility, "Start with the lowest prime number (2) to determine if it can be divided into that number." Since the kids haven't done divisibility yet, it shold say "Start with the lowest prime number and determine if it is a factor of that number." Now, someone who understands divisibility will determine if it's a factor by dividing. Someone who hasn't been introduced to divisibility will try to "think of " a factor that can be paired with two that make the necessary product. The kids can use their multiplication facts for anything under 81. No problem. For anything over they estimate and try some things out. Makes much more sense to do larger number AFTER divisibility.

But, I can see what Jessica means when she says there is a lot of multiplication fact practice built into the other work.