Saturday, August 25, 2012

Moves in translation with Miss Anna Parabola

Some concepts are so powerful in mathematics, they just keep popping in your course again and again - like old friends. Such an idea is the translation of a curve in the number plane. In my class, this old friend has a name: Miss Anna Parabola. Anna has been making an appearance throughout my course, starting off with an introduction to the quadratic function.

 Miss Anna Parabola demonstrates $y = x^2, y = -x^2, y = x^2 + k$. Ballerina: Alicia Alonso in 1944, photographed by Gjon Mili for Life magazine

I will admit to raising and lowering my arms in the different ballet positions in class, standing up on chairs (against OH&S regulations ....) - but I vehemently deny donning a tutu.

I knew I was onto a good thing when I started teaching the topic "Locus and the Parabola". One of our textbooks spends an arduous 35 pages (no kidding) going through all the iterations of the different orientations and translations of the parabola - but I realised with our class understanding of Miss Anna's dance moves, we could collapse the entire thing into two lessons: one lesson to cover the different orientations, one to cover the translation.

 The Four Standard Orientations of the Parabola - as interpreted by Miss Anna Parabola (aka Alicia Alonso) (Click image for a larger view)
 Translating the vertex.

And it worked : my students can now do this effectively and efficiently. They connected our previous work on shifting curves like $x^2 + y^2 = 25$ to $(x-2)^2 + (y+4)^2 = 25$ to this work on shifting the parabolas. We cracked what would otherwise be a very arduous (and boring) part of the topic by focusing on the key idea of 'moves in translation'.   I'm a big fan of creating characters and story to build a narrative in the course, so I was thrilled to see the work from previous topics developing Anna Parabola pay off like this.

Oh - and in case you haven't heard of him, Anna has a new friend: Billy the goat. Billy helps develop the idea of locus : if you tied him to a fence, he would happily devour everything around him, following the locus constraint imposed on him.  And yes - I do admit to tying myself to a desk and 'acting the goat'.  How am I going to live this down....?

PS: I'm not sure this trick would work at a boys' school.... Might have to invent a rugby player in motion...