Illustration of the four level model for arguing, reasoning and proof around surface areas.

Surface Area - Trapezoid

In Austria at grade 7 students deal with the surface areas of rectangles and triangles. The math curriculum at this grade demands that students should argue formulas for surface areas. This is a very broad field for arguing, reasoning and proof with various possibilities to use a paper and pencil approach as well as technology.
Let me now focus on the trapezoid and the derivation of the formula for the surface area.
I assume that the students already know the properties of the trapeze and the formula of the parallelogram before the start to deal with the area of the trapeze.
With an applet like the following students can easily and autonomously find the formula for the surface area of the trapeze. They "only" have to write down what they see and will find the formula autonomously. So with this applet student’s work at level 4.
If the students are asked to communicate and explain the formula of the surface of a trapeze they can also use the applet. And so they work at level 3.

As you can imagine there are a lot of analogues applets for each rectangle and the derivation of area formulas. The main point is, that the applet should help to find the formula autonomously and it should support students by explaining the derivation.