Factor Flamboyantly! Paris and Sea Creatures!

 My seventh graders are learning about prime numbers, divisibility rules, factorization, Greatest Common Factor, and Least Common Multiple.  I love this content.  They get a real sense of the feel and weight of numbers and some tools to use for less familiar numbers.  And, we get to talk about monomials... but another day.  Back to the factoring.  We had worked on finding all the factors of numbers, and finding the prime factorization of numbers, and had applied our newly acquired divisibility rules throughout.  Then came....

 I was thinking of a number that the students could really dig into to with their factoring tools.  Why not 504?  It has promise.. So, we started making a factor rainbow to find all the factor pairs of 504.  We all started off leaving plenty of room, anticipating quite a few factors. We enjoyed the rainbow, watching it fill, enjoyed all the opportunities to divide.  Then, the rainbow was full!  We weren't finished!  What to do!?!

 

 

One of the students dropped a vertical line down through the rainbow and we continued, a "rainbow annex" of sorts.  There couldn't be that many more factors; we were closing in on the half way mark... so we made the "annex" optimistically a bit narrower.  (oops.)

 Two pair later, 4 simple factors and we had exceeded the "annex".  Someone said it looked like a factor umbrella at an outside table.  So, naturally, we called it the "Factor Cafe Table in Paris Where We Sit Outside and Eat Croissants While Drinking Cafe au Lait" (wouldn't you?)  Some of us spontaneously began speaking with French accents while we continued exploring.  Were we finished?

Of course not.  However, 12 and 42 were getting pretty close to one another in factoring terms, so we dropped another vertical, made "Annex le Deux", (that's Annex 2 to the language challenged among you).  More factoring, more dividing.  We were in uncharted territory here.. our divisibility rules had been left in the dust, all used up.  We were flying without a net, a drift in a sea of possible factors... we. were. terrified. and intrepid.

They found fourteen, and eighteen and their factor partners.  We filled Annex le Deux.  But then... using some exceptional logic, one student opined, "Hey, 3 and 7 are factors.  Doesn't that work like 6 with 2 and 3?  Is 21 a factor?"  They tried it, and there it was.. we needed Annex le Troisieme (You guessed it, Annex 3).
We finished.  It was long, it was exhausting, it was beautiful.  It was...  a factor Man o' War!!

Whew.  Exhausted, but beaming, we all agreed that 504 is our new, absolute favorite number... for now.