Often times the concrete examples are explained using the application or effect it has in a situation. The Learning Scientists in their blog use the example of scarcity to demonstrate its meaning using airlines as their concrete example.

So in Mathematics, concrete examples often manifest themselves as real world examples such as the use of the parabolic flight path in Angry Birds. You have to choose the correct path so you wipe out the enemy. This is really easy to find concrete examples. When studying ellipses, you only have to look at planetary orbits but finding a concrete example for hyperbolas are a bit more difficult. However, circles are fairly easy by talking about certain types of buildings.

Ratio and proportions are all over the place from reading scale models and finding how far a place is based on a map. Percentages, mark-ups, discounts, are found all over the place in the newspapers, on the television, and so many other places. In addition, finding applications for area and volume are just as easy but what about some of the other topics like multiplying binomials.

One concrete idea I have is based on the idea of planning a square room and determining the change in the area by changing the measurements say adding three feet to one side and subtracting 2 feet from the other so you have a rectangular shaped room. Perhaps you are planning to build a house on a narrow lot and you need the house to be so many feet from the sides and back. I think you could use binomials to help figure out the area of the house. This is a much harder topic to find concrete examples for.

On the other hand, trig is seen in surveying, flight, and so many other real world applications that you really don't have to search. When I look for concrete examples, I try for ones that make sense to my students and seem real rather than made up. I always hated those train problems. You know the ones where one train leaves at a certain time and a second train leaves two hours later from the opposite direction. They never made any sense to me. If someone had explained I needed to know that so they could have the switch ready so the trains didn't crash, I would have been happy but the problem itself made no sense to me without the context of why.

Think about it. Most railway lines are only made for trains to go one at a time, so if you have two trains using the same track, you have to figure out where to have them pass safely. Linear equations are nice because business provides us with activities. Unfortunately some of our older examples such as rental cars are falling into the past. It used to be they would charge a daily rate plus an unlimited mileage rate but now its usually a flat rate with unlimited miles.

That is the only thing with concrete examples. My concrete examples often change due to companies changing the way they do things. It makes life interesting.

I am still deciding if manipulatives provide concrete examples or if they provide concrete visualization for students. I'm off for the day, have a great weekend.