Under the Weather Math: Edges, Reflection and Scale

The girl was low energy, snuffy, and generally under the weather.  She wanted to play some games, but after the first one made a pointed request to "play some games where no one wins."  She went to the game shelf and pulled down this...not sure what it's called, but it's been on the shelf for years without much use -- until today. 

At first she wants, uncharacteristically, to do the pre-printed puzzles that came with it, but after a time we moved to more free form designs. 

"What's this?" I ask, holding up a white piece.  "A parallelogram," she promptly replies, surprising me. 

Yes, and also a rhombus.  There's a larger blue rhombus too.  And the red piece is called...oh, I always forget.  Oh yeah, a trapezoid.  It's a triangle with its top cut off.  And, not an octagon, a hexagon, with six sides.  It's yellow. 

We play around a little.  Did you notice that a blue rhombus and a green triangle can combine to make a trapezoid?  And two trapezoids can make hexagon?  The girl discovers that she can place six trapezoids around the yellow hexagon and make a larger hexagon,and that she can take out the yellow center and fill it in with different combinations of green triangles and blue rhombuses, but is more interested in making a 'quilt design'.  "Remember to match your pieces edge to edge if you want them to fit together," I say.




















We collaborate on this white and red 'quilt block' a little, but the basic structure and design is hers.

Somehow all our designs seem to come out balanced.  "Look!" I say, "You can draw a line down the middle of this design and you'll find the same shapes in the same places on both sides!"  Although that is not completely accurate.  The hexagons are both about three inches from that line I mentioned, but one is 3" to the right, the other to the left.  A small but important point that seems to come out most clearly when I'm showing fourth graders how to reflect their Jump Patterns.

Speaking of reflection symmetry, the girl wants to do the butterfly puzzle together.  "Let's take turns," she says, "you do one side and I'll do the other.'  We take turns, but I get bored.  I suggest quite enthusiastically, "Why don't we try to make one just like it, but bigger?  Why don't we make one that is twice as big?" 

I ask, "How many orange squares in the first design?  If we're doubling that number how many do we need?"

We are both quite engaged as we work on the design.  We have the 'rules' to work with -- each piece on the original butterfly has to be doubled, but it isn't always clear how to include them in the design.

"We made it twice as big," she declares, "and twice as beautiful!"