Really, really understanding the area of a triangle (Part 2)

Continuing from Part 1, we look for an elegant and convincing way to demonstrate to students that our triangle ideas also apply to the infamous (and well named) obtuse triangle. 

How can we show that the area of this triangle is the same as half the area of the rectangle made using the base and height of that triangle? I struggled with this - and it turns out drawing a rectangle was not the best way - the secret is to turn this triangle into a parallelogram. 


Here is a neat trick:

By rotating our obtuse triangle, and putting it on top of our original obtuse triangle, we now have a parallelogram. The parallelogram area is base times height, so the area of our obtuse triangle is just half that! Woot!

So long as we are solid in the knowledge that the area of a parallelogram is the base times the height - we are there : clearly the area of even the obstuse triangle is half the base times the height!

Just one small problem ... have we taught parallelograms yet?

Should we change the order we teach area?
Realising how central the parallelogram is to gaining a deep understanding of quadrilaterals and now even triangles, has led me to wonder about the classic teaching sequence for this topic.

The classic sequence for area is:

We start with a very specific rectangle 'the square', then extend to the rectangle, switch to a completely different shape, then are back to another quadrilateral. No wonder we have these puzzled looks when we say a square really is a rectangle, especially when in some textbooks the formulas are even written using different letters, and students are puzzled why some area formulas have that half factor and others don't. And we haven't even looked at the trapezium yet - which uses the half in yet another way!

So - with the goal of students better seeing the relationship between all these shapes and their areas, I'm going to try this sequence next time:

Why?

• Rectangles make sense to most students, the math is very simple - so we can focus on the real meaning of area as a measure of 2 dimensional space. Understanding what area is, and how it is different from length and volume is non-trivial - it's quite a deep teaching - and well worth quality time.
• We look at squares as an aside - so it's clear the square is 'just' a special rectangle.  This is time for a discussion of 'all squares are rectangles, but not all rectangles are squares". My students liked this version: "All crows are birds, but not all birds are crows".
• We then move to parallelograms.
  
  • What makes it a parallelogram?
  • Now we focus on what happens to the side lengths as we change the angles while the height remains constant. What happens to the side length?  How does it compare to the height?
  • We draw a bounding rectangle and show that so long as the height and base stay unchanged, the area is also unchanged. We can show this with guide lines, and through demonstration or student exploration with a dynamic geometry tool such as GeoGebra.  Keep the focus on the height - this will pay off soon.
  • We then observe that when the angle is 90 degrees we have a rectangle - so a rectangle is just a special parallelogram - which explains why the area formula is the same!
  • ... and when the side lengths are just right, we have a square - wow- that's a parallelogram too! Same formula.
• And now we are ready for triangles.
• Compare triangles to quadilaterals - so it's clear how they are different.
  
• Explore the area of an acute triangle using the rectangle as the base to develop the 'half' part of the formula, and emphasise the triangle height.  Don't start with a right triangle - it allows students to think the height is the same as the side length!
  
• Extend the idea to show how triangle area can be explained in terms of a parallelogram made by reflecting the triangle on a side. Now we can show the area calculation is also true for obtuse triangles.
  
• Now hammer home the calculation methods - with clear focus on the height versus the length of the side. Make sure students do the right thing, even when given a triangle with all three side lengths and a height provided. And now is the time to look at algebra issues students may be having
• And finally, go back and look at our rectangle again:
  
• Time for a deeper understanding of what happens when you change the units of measurement,
  
• And time to get a good physical sense for how big a square metre actually is - and why you can fit 10,000 square centimetres into just 1 square metre.

So in this sequence, we use the parallelogram as our central organising concept.

The rectangle and the square are special parallelograms - all using the same area formula : base times height. We can see triangles in terms of being two halves of a rectangle (this is the simplest), with the fallback of knowing the parallelogram is the secret for understanding why this also works for the obtuse triangle. And with the parallelogram well understood, and our technique of seeing areas in terms of rectangles, we are ready for our next set of friends: the trapezium, the rhombus and the kite.