Developing Formulas (2)

"I deeply worry about a curriculum that pushes students to results and not let mathematics be the organic conversation it deserves to be..." James Tanton

When I began teaching, I (like many others) simply gave students the formulas, and some worked examples, and expected this knowledge to stick. 
It did not work as well as I wanted to. 
When I started showing why the formulas worked, students were far more likely to recall and use the correct formula. When studying area, we would learn about rectangles, then triangles, then parallelograms, then trapezoids (or, if you will, trapeziums (or, if you will, trapezia!)) before getting stuck into circles. Recently, I have been wondering if a more logical order would be rectangles, then parallelograms, then triangles, then trapezoids. Fundamental to all of this is learning why the area of a rectangle is length times width and the best way to get students to develop this idea is to consider arrays (as touched upon in this earlier post ) How I then get the students to develop the formulas for parallelograms, triangles, and trapezoids can be seen below. I must point out, that I do not do all of this in one lesson!

Usually at this point, I am pretty confident that most students will now understand why the formulas work. Part of a balanced Math program must involve putting this knowledge to practice. This practice should involve a good balance of closed questions (the standard text book ones where a diagram is given with different measurements given) as well as open questions e.g. a trapezoid has an area between 60 cm² and 70cm², what could its dimensions be? The question below came from Anne Yeager and I have used the question with many grade 7 and grade 8 classes. It has always generated a lot of different solutions as well as great thinking and discussion amongst the students as they decide which formulas to use and when.

James Tanton (whose quote appears at the top) provides some fantastic resources for Math teachers. In particular, I love his curriculum videos and his Mathematical Essays. Do check out his site here.