1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
Before now, students calculated area by counting the number of unit squares inside of a figure, and they learned that the area of a rectangle is equal to its length times its width. They also learned the properties of special types of triangles, quadrilaterals, and other polygons. Now it's time to take these ideas and weld them together—blowtorch not required.
The focus here is less on area formulas and more on developing a visual sense of calculating area—understanding that we can break up a complicated shape into less complicated ones and, as long as we put all the pieces back together, we'll have a shape with the same area. That's really the whole point.
For instance, we can take a parallelogram, snip a triangle off of one of the sides, and glue it back to the opposite sides to make a rectangle with the exact same base and height lengths.
This way, students can turn complicated shapes that they aren't familiar with (like parallelograms) into shapes that they're way more comfortable working with (like triangles, rectangles, and squares). Once the that's done, calculating the area ain't no thang.Again, the goal is not
to have the students memorize and apply lots of formulas for area; the goal is to have them understand
the concept of area and be able to find the area of lots of figures by cutting them up and rearranging them into simpler figures, like rectangles and triangles. That'll give them the visual understanding they'll need to truly understand all the crazy formulas they'll come across in the future. (See how we can derive the parallelogram area formula A
from this?)Also, notice we're talking about polygons only
, here. Stay away from circles. Stay far, far away. Finally, we can't let students off the hook completely until they've proven that they can apply this understanding to real-world situations. What is the area of home plate on a baseball field? For that matter, what is the area of the entire baseball field? What is the area of the Pentagon in Washington D.C.? Wherever students encounter polygons, we want them to be able to find the areas of those polygons by decomposing them into more convenient shapes.Though we'd discourage them from trying to decompose the Pentagon. That's kind of a national security threat.