I read an article on three ideas for math teachers to do this summer. I think it should have had four ideas. Yes technology, relevancy, and inspiration are all important but we forget connectivity. We need to connect the topics in math so students no longer see each section as independent of each other.
1. The distributive property.
This diagram can be used to show students something like 16 x 14. The large square represents the 10 x 10 or 100 units. The small rectangles are 10 x 10 or 100 units and the smallest squares represent ones or 24. So the total is 100 + 100 + 24 or 224. If you calculate 16 x 14, it does equal 224. This is a good visual way of showing the distributive property.
It can also be used for multiplication of binomials such as (x + 6)(x + 4). the largest square represents X times X = x^2. Each rectangle represents 1X so you have 10X and each small square represent 1 x 1 or 24 total units. If you look at everything, you end up with x^2 + 10X + 24. You can use the same drawing but use blue for positive and red for negative and you can show (x - 6)(x + 4) in a visual manner so students see what is happening. This is a nice way to show a connection between the two.
2. Right angle triangles.
A. Show the triangle on a coordinate grid to show students the idea that the legs are actually x and y values. This sets things up to jump over to the unit circle, to basic trig values.
B. The triangle on the grid sets students up to see the Pythagorean theorem.
1. Use the grid with obtuse and acute triangles and the variations of the Pythagorean theorem formula to show why a triangle is right, acute or obtuse.
3. Unit circle and trig graphs.
A. Show the correlation between the values on the unit circle and the sin or cos waves.
B. Show the correlation between the sin waves and the oscilloscope.
These are just a few ideas for showing connections between topics and ideas in math. Too many of my students see each idea or topic in math as isolated and unrelated to anything else. I need a way to show them the relationships. I think this might help their ability to transfer knowledge among situations. I plan to try it this coming year and we'll see how it works.