BEd Mathematics, Cheltenham UK

After graduating from high school at the American School of Brasilia, I was able to attend a traditional Teacher Training College in England. This four year course prepared we well for teaching the Middle Years by insisting that I do five in-school practical teaching experiences of various lengths. The Mathematics department supplied me with an in depth understanding of the material I would later find myself using to challenge Middle School and Upper Elementary students in Mathematics. Here are some of the lessons, most in the form of investigations that I have used:

Fudge It : An investigation into the number of square pieces into which a 12 in by 12 in pan of fudge cat be cut. Students find the obvious solutions: 4, 9, 16, 25, 36...144 pieces; then they are told that the pieces do not necessarily have to be all the same size, but they must be squares. This opens up a whole new range of solutions and the discussion of possible outcomes. More advanced students will start to find patterns and rule out impossibilities. There is always the added bonus of making real fudge at the end!
Used with students grades 4 through 7.

My degree familiarized me with classic problems that are perfect for math enrichment. These include the Fibonacci Numbers, Graphs and Networks (including the Bridges of Koenigsburg Problem and the Five Room Problem),  The Rice and the Chessboard Story, and the Tower of Brahma Puzzle.
There are many sources with supplementary material if you know such topics exist and their value in teaching problem solving, investigating and critical thinking in mathematics.
Here is a site I use regularly with upper elementary enrichment students:
http://math.schaubroeck.net/school.html

I have used the platonic solids countless times with students as young as 2nd grade, as a beginning to thinking about and creating in the 3D.  Students learn quickly how to make their own nets to build a solid, then go on to create their own general rules and create their own, more advanced, 3D solids.

Tessellation is another great activity that can capture the attention of students in geometry, and focus on its application to pattern, space and design.


Finally, I did take some basic computer courses as part of my degree.  Although they are now very out-of-date, I did learn some programming. I see its value today, and have found some simple applications in geometry.
Here is a site that is easy for younger students to use:
http://www.mathplayground.com/mathprogramming.html