Here’s another Look at Trapezoids

Most calculus students do not realize that the area of a trapezoid is the average of the area of two rectangles.  Areas of trapezoids come up in calculus when approximating the value of a definite integral, where the focus of the lesson is on improving the approximation. As a side note, I sketch a diagram of the trapezoid with the rectangles along with the formula that students recall for the area of a trapezoid (b1+b2)h/2, and add (b1•h+b2•h)/2. “I never thought of it that way!” is a common student reaction.

It has been some time since I taught geometry, and I looked up an NCTM Illuminations resource for finding the area of a trapezoid. The resource suggests different proofs for the area formula, and incorporating the notion of averages can provide another perspective.

The simple diagrams can help geometry students build convincing arguments about the relationship between the areas of the trapezoid and the rectangles:

Screen Shot 2016-01-18 at 6.50.35 PM

In calculus, the trapezoids students use in approximating integrals have two right angles as shown below. It might be easier for geometry students to make observations using the following series of diagrams instead.

Screen Shot 2016-01-18 at 7.00.37 PM

Whether students model with paper by cutting and rearranging the pieces or use measurement as their tool, I believe it would help them get away from memorization and get closer to seeing formulas from a meaning-making perspective.

Reading Mathematics in High School Algebra

Very early during the school year, I write a statement like 2 < x < 7 on the board and ask my students in algebra 2 to read it. The most common response does not surprise me: “2 less than x less than 7″. It happened again this year, and I said: “that is correct, but now I would like you to think about what this statement says about some real number x”.  There was a long pause, and someone said with a questioning tone “2 less than a number less than 7?”. “What else?” I prompted them, “what if you were to say something about this number to a seven-year-old?”, to which someone responded “x is a number between 2 and 7″.  This was a good beginning to a classroom culture where we would try to find meaning in mathematics.

A discussion about meaningful reading of symbols is necessary to promote experiences in the classroom that might lead to understanding of abstract concepts. It is a difficult transition for most students. Prompting students to read the meaning in algebraic statements beyond the symbols is a challenging endeavor.

And, it can be rewarding. Following a lesson on sequences in the algebra 2 class, I asked the students to read a-sub-n = 3n + 2.  A student replied: “this is the rule for the nth term of an arithmetic sequence with common difference 3 and first term 5.”