On Monday, Drawing Area won the WGBH Interactive Math Challenge and will eventually become part of the Math at the Core: Middle School collection at PBS LearningMedia.
Derek, a co-organizer of an instructional design meetup I attend, asked me if I'd be willing to share my thinking behind Drawing Area with his instructional design students. I replied by joking that there wasn't much thinking behind it. But, of course, that isn't the case.
In Drawing Area, students find the area of a polygon by drawing the polygon on a grid with rectangles and right triangles. This happens in a browser window. The idea for Drawing Area came from a series of lessons I developed when I was a classroom teacher. For those lessons, students drew their polygons on sheets of graph paper.
To find the area of a polygon, it helps if you can decompose the polygon into rectangles and triangles. However, I found some of my students doing things like this:
This student has decomposed the polygon into three triangles, but the area of the middle triangle isn't going to be easy to find. This student either hasn't understood why he or she is decomposing polygons into rectangles and triangles, or is unable tell at a glance when a rectangle's or triangle's area is going to be easy to find.
You could ask the student to find the base and height for each triangle. And then, when the student can't find the base and height of the middle triangle, ask the student to try again by decomposing the polygon another way. However, that is going to get old pretty quickly.
You could give the student a rule to follow: only make horizontal or vertical slices starting from a vertex. Over time, the student may make sense of the rule and internalize it.
I elected to take a different approach. I decided to draw the polygons on a grid and ask the students to find the number of squares the polygon covered.
When you are counting squares, it is natural to divide a polygon along grid lines. You aren't decomposing the polygon into rectangles and triangles so that you can apply the area formulas for rectangles and triangles; it is simply easier and more efficient to count squares when they are grouped into rectangles (an array of squares) and right triangles (half of a rectangle). This instantly makes sense to students and is something that they come up with on their own.
Once a polygon has been decomposed into rectangles and triangles, some students have a hard time finding dimensions that are not given. This is not an issue when a polygon is drawn on a grid (the student can simply count the number of squares along an edge), but it is an issue when a grid is not provided. What is the length of the base of the triangle?
It is not uncommon for students to pick two numbers in the drawing and add or subtract them. They may see the 12 and the 7 and think: 12 − 7 = 5. It doesn't matter that this makes no sense whatsoever. They know they are suppose to either add or subtract two numbers to find the unknown length.
You could guide the students to focus on the relevant information and eliminate distractors. Since the student is trying to find the length of a horizontal line, have the student highlight all horizontal lines and eliminate the dimensions from all vertical lines:
Is there a way to use the information provided to figure out the length of the middle red line? Some students will make sense of and internalize this over time, but you are still subtly reinforcing the notion that you should be looking for things to add and subtract.
Since the students in my classroom are already comfortable finding the area of polygons drawn on a grid, I decided to leverage that. Can you take this polygon and draw it on a grid? Most students think that's easy. And once the polygon is drawn on a grid, they can find the area.
A student may start by drawing the rectangle to the left, since those dimensions are given.
The student knows that the height of the second rectangle is 6 cm, but the length of the base and its exact position relative to the first rectangle are unknown.
But once a rectangle with a height of 6 cm has been drawn, it can be moved so that its base is aligned with the base of the first rectangle.
Then it can be resized so that the width of the entire polygon is 15 cm.
The right triangle fits right in between the two rectangles, and now any dimensions that the student needs can be determined from the drawing.
After a while, I would encourage the student to predict the dimensions of the individual rectangles and right triangles before drawing them, and then confirm those predictions with the drawing. At no point is the student following rules or doing something that doesn't make sense. The emphasis is on figuring out the number of squares a polygon covers. Using this process, all students can reason through increasingly complex problems: