On Mathematics 24x7 I initiated a discussion on How do you introduce coordinate geometry and relate it with daily life examples?
I mention some alternative approaches so that you have a choice. The Cartesian Coordinate System could be introduced from a historical perspective, mentioning the works of René Descartes who invented the system. Discuss why it was novel: it connected geometry and algebra, and why it was important: it enabled the work of Isaac Newton, to begin with. You could compare the rectangular framework of the Cartesian system with the circular framework of the Polar Coordinate System, explaining how they differ. I'm not suggesting that you try these approaches; I'm glad that I learned the chemistry of oxygen without being confused by the Phlogiston Theory. In teaching Physics to young adults, little introduction was necessary. I used rectangular and polar coordinates by way of examples, the trajectory of a cricket ball or of the International Space Station. I explained some basic conventions for describing the laws of motion. The y-axis usually represents the up-direction, but not always. The equations for a ball rolling off a table are easier if the origin is at the edge of the table, and the y-axis points downwards. All of this was explained using diagrams on the whiteboard, and I encouraged students to come to the white board and draw the diagrams from their text book.
The answers may lie in biology and psychology rather than in mathematics. According to two recent articles in New Scientist, fiddler crabs are good at trigonometry and humans can't navigate their way out of a paper bag.
Maria D said...
First, I introduce grid reasoning through combinatorial grids - both art projects (shapes horizontally, colors vertically, cool combinations) and, with older kids, guessing games like "guess my rule hangman" on a grid (e.g. 2x+y can be the rule, and the person who made it up tells answers to grid cells until others can guess the rule).
Coordinates themselves get introduced through map-based games, such as Battleship or Maze
Kids are just told to give pairs of numbers and see the results, and because games are goal-driven, the feedback loop does its trick.
The problem with coordinate geometry is that kids don't have strong coordinate reasoning foundations, especially the idea of CO-VARIATION. These early activities help with that.
Equations of lines is a good topic for a start of coordinate geometry, and the Green Globs game is great for introducing it: http://web.mac.com/ihor12/Math2.0/
For a first "areas through coordinates" task I would probably ask kids to program anything (Excel, Scratch, GeoGebra) to calculate areas of rectangles by coordinates of end points. This comes in cartography applications. I can imagine it as a game mechanic in games about conquests of territories.
That crab article is my favourite suggestion so far. Display a picture of a Hermit Crab from the CC collection on flickr, share this little snippet of info in the article, walk the class through the problem, then (working in collaborative groups) tell them they're hermit crabs and they have a problem to solve: "Swiper the Hermit Crab is about to steal their home ..."
With a little modification this could be used to introduce the Pythagorean Theorem, the distance formula, trigonometry, analytic geometry, or, if we include something about the rates at which they (as crabs) and Swiper can move this problem can be explored in calculus classes as well.
My experience of teaching the coordinate geometry is to begin it when children already know the connection between the position of point and it`s coordinates. It is necessary that they will know about locus ( maybe intuitively ) - from euclidian geometry. I think that the coordinate geometry goes after euclidian geometry and basics of function. Many things are clearly illustrated on "checked" paper - parallel and perpendicular lines, slopes -
are taught as problems on construction. We may make use of similarity - in order to conclude about equal angles and the consequent parallelism of two straight lines. As a matter of fact - similarity already is somewhat algebraic. After this pupil became more convinced about the usefullness of algebraic method and can begin to speak about geometrical facts in the language of algebra.