Wednesday, June 10, 2009

Right Angle Trigonometry Meta-language

My prac student is to deliver the right angle trigonometry unit and I was thinking about all the little things that I like to emphasize to students.

The first thing I would like him to consider is to design the course to make connections to previous courses of work (prior learning) especially with regard to language. Mathematics has such a rich history of precise language that it is important to identify ideas correctly to students in a manner that ties together topics to promote abstraction.

For instance before actually doing anything, we need to remind students what the LHS and RHS of an equation is, and how to identify the subject of the equation that we want to work with. Then we might think about how to eliminate coefficients and pronumerals/variables from one side on an equation, transpose terms/pronumerals/variables and multiplying through/simplifying to remove a denominator. What can we substitute into an equation? How do we solve the equation? Do we need simultaneous equations? We need to use every opportunity to reinforce concepts learned in previous algebra topics.

We have geometry prior learning to consider, three internal angles = 180°, a right angled triangle has one internal angle 90°. Line properties give us complementary, corresponding, co-interior, supplementary, adjacent, vertically opposite, exterior, alternate angles. Also the types of triangles, isosceles, equilateral and scalene help us find other angles. Circle geometry gives us tangents, subtended angles, cyclic quadrilaterals. Properties of 3D shapes!

Only after we consider possible connections to prior learning can we think about actually teaching the relatively small amount of new material. Without these connections we are just teaching students a new trick that will be forgotten immediately after assessment (a key issue exacerbated by the increased assessment required by OBE reporting requirements).

We have to introduce a range of new ideas such as opposite, adjacent and hypotenuse for right angled triangles. We have equations such as Pythagoras' and the three trigonometric ratios. .. and the dreaded bugbear bearings (until vectors makes bearings look easy!!)

We have conventions such as labelling the hypotenuse 'c' and the remaining sides a, b for Pythagoras' theorem problems and opp, adj and hyp in trigonometric problems.

We have good practice such as always writing the symbolic form of a trigonometric equation before substituting values, labelling diagrams, identifying right angles.

We have acronyms to help us remember trigonometric ratios SOH, CAH, TOA.

What is the correct sequence for introducing the material? What resources can we use or have available?

So now prac student, your job is to help students see how their prior learning is necessary to solve these problems!