Factor Trees

BatBoy and SpiderGirl have been playing with multiplication lately.  The Beast Academy workbook (3B) has many problems requiring the use of multiplication and BatBoy is learning a good portion of the lower tables just through use.  In the car one evening, he was talking to me about finding all the combinations that would give a product of 12 (I think) and so I offered to show him factor trees.  SpiderGirl immediately wanted to know all about it too.

When I was in school, factor trees looked like this:

So I shouldn't be surprised when I google images of factor trees and find things like this:

But in this image, the tree grows up and the factors branch off downwards.  The visual, doesn't quite work, does it?

Our factor tree is not as pretty, but the factoring is integral to the tree:

I didn't actually get around to showing the kids the factor tree that evening, as I had said I would.  We got home, made dinner, and the kids wanted to get their music homework done so that they would have time to play in the snow in the morning.  It wasn't until later in the afternoon, when SpiderGirl was rehearsing with her choir, that I remembered about the factor trees I had said I would show them.  

BatBoy got into it, and quickly wanted to try large numbers, like 300.  We played around with multiplying the factors in different ways and I showed him how he could use it to find new multiplication facts.  25 x 12 = 300.  That was something we didn't know before.  We talked about why we don't use 1 in a factor tree, and I showed him what might happen if we did allow 1:  The tree could keep growing forever and it gets difficult to get any useful information from it.  It doesn't show in the picture, but we also talked about how the multiplication equals the "root" of the tree at every level.  He thought that was pretty neat.  Then he wanted to factor 998.  He of course had no idea what two numbers would multiply to make 998 and SpiderGirl was finishing rehearsal, so we stopped there.

When we got home, SpiderGirl wanted to know what she had missed, so I showed her also, with 12 as an example.  BatBoy was interested in listening in as well, so I threw in some new information.  We talked about putting the prime factors in order in the final statement, to make it easy to see if we've missed any factors and to make it easy to manipulate them.  SpiderGirl was impressed that she could use this way to find ALL the factors of 12.  She wanted to see another example right away (which, come to think of it, is fairly unusual for SpiderGirl), and we took a look at 100.  Unlike BatBoy, who likes me just to scribe for him, SpiderGirl likes me to do the work while she watches when first encountering a new concept.  And then she finished my tree off with some bushy leaves.

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