If fractions make you pale, then scale, scale, scale! OR When kids invent algorithms

My goofy title is in response to the annoying rhyme you may have learned "Don't ask why, just invert and multiply".

In the course of writing a Professional Development module for teachers, and I've spent hours downloading papers about algorithms that kids invent for division of fractions.

Here's a summary of a not-so-common, but very useful method for dividing fractions and the representation that facilitates its discovery. I'll call this the scaling algorithm because it is born out of considering areas of rectangles and how the ratio of area to length of a side is preserved under scaling.  The algorithm is:

For example:

What representation would lead to this algorithm's discovery?  Ask the question in the context of area:

"What is the width of a rectangle whose area is 2 sevenths and whose length is 3 fourths?"

To answer this, let's make seven copies of the rectangle -- Why? Because integers are easier to deal with and we all know that 7x(2/7)= 2.  The question has now been reduced to "What is the width of a rectangle whose area is 2 and whose length is 7x 3 fourths?" 

Since 7x3=21, we have 21 fourths as one side of the rectangle.   We have exchanged ? x (3/4) = 2/7 for ?x (21/4) =2.  We may also notice that scaling each factor by the same number preserves the answer!

Let's scale again!!!  Take four copies of the big rectangle, which gives you a total area of 8, and a total side length of 21.
We have now replaced all fractions with integers and traded the original question in for the simpler sounding:
"What is the width of a rectangle whose area is 8 and whose length is 21?" At this point, we see that the answer is 8 divided by 21 or 8/21!  Woohoo!!!

Do kids invent this scaling algorithm?  Indeed they do if they are given the area context and their brains are not stuffed with "this is how you do it" algorithms.  Want to read a paper that talks about this?  Jaehoon Yim from South Korea has a 2009 paper that's fascinating -- Children’s strategies for division by fractions in the context of the area of a rectangle -- I'm not sure if you'll be able to access it for free.  But, the general gist is that ten and eleven year old students were able to "make the width equal to 1, make the area equal to 1, and change both area and width to natural numbers".  The strategy outlined above is of the last variety.  She even researched whether children could formalize their pictorial drawings to create a numeric algorithm -- and surprise, surprise -- they could!  Granted, these were students who were picked because they had a "positive attitude towards mathematics".

Incidentally, if the numbers are nice like 6/20 divided by 3/4, then they can use the area model to see that the answer is 2/5 since 6 divided 3 is 2 and 20 divided by 4 is 5.  In other words,

Here's a picture that shows this.  Can you see how?  The strategy is a little different from the one above as it starts from a unit square.

Six pointed stars -- just in time for the holidays!

So I'm on vacation in Maine where it was snowing today, and we took my little one for a walk in the snow.  A perfectly formed little snowflake fell on his eyelash, and we all just stood around and stared at him for a few minutes while it melted.  It was so beautiful!  I'd never been able to see that symmetry so well with my naked eye.  We took a picture, but it just didn't do it justice.
   Anyway, it reminded me of a book I'd seen once that would make a great gift.  The book contains snowflake photos taken and explanations by Caltech physicist Kenneth Libbrecht.  Apparently photographing snow is an expensive hobby, costing over a thousand dollars, not to mention you need to keep your camera warm!
   Then I just happened to see this wonderful video made by George Hart, Vi Hart's father and also one of the designers of many of MoMath's new exhibits.  The key in understanding what is obtained when you slice the Menger sponge in half along its diagonal is that we can view a star of David as being the union of three Rhombi, not just two equilateral triangles.  This different way of constructing the Star of David was exploited by an artist I recently saw at Tucson's 4th Avenue Street Fair.  Unfortunately, I can't find his site!  But the general idea can be seen in this basic picture I made.
It's just amazing to me how the things I see in everyday life sometimes link together in these nice ways... Anyway, there's a lot of six fold symmetry going on just in time for the holidays.