This is one of my favorite lessons because it focuses on conceptual understanding, reassures students that memorization is not necessary, and it makes connections between formulas that students didn’t know were related. A lot of people take formulas for granted. They didn’t just appear out of nowhere; it had to be proven.
I first start with a little non-graded “quiz.” The purpose of this is because as stated in a previous blog post, I strongly believe that when students make mistakes first, it makes the learning much stickier. I ask them for the area formulas of a rectangle, parallelogram, triangle, trapezoid, kite, regular hexagon, regular octagon, and a circle.
I then reiterate the definition of area. The main noun of area is the *space* inside of a 2-d figure, measured in square units. Colloquially when we ask for the area though, we are asking for an amount. New: through a tweet yesterday from Bobson Wong (@bobsonwong), I learned that I should say “calculate the area” never “find the area” so there won’t be any smart aleck answers of students pointing or circling the figure when we ask about the area.
I write on the board “Wait, ALL area formulas are related to the area of a rectangle?!” and then we derive each formula together.
To find the area of a rectangle, we would multiply the base by the height to find the amount of square units that would fit in the figure. Note: some books may use length and width, but I use base and height to emphasize that these two dimensions are perpendicular to each other.
I then move on to parallelogram to see that we can manipulate the figure into a rectangle. Boom. Parallelogram -> rectangle. (“->” will mean “relies on the area of”)
I have students draw a non-right triangle and ask them how we can use what we know to find the area of the figure. Students would say that we could cut the triangle into smaller triangles and “copy” those triangles to make a rectangle. So boom. Triangle -> rectangle
For trapezoids, I have students draw a non-isosceles trapezoid. Some students said that the trapezoid is composed of 2 triangles and others said that we could copy the trapezoid, rotate it, to create a parallelogram, which we know the area formula. Boom. Trapezoid -> parallelogram -> rectangle
I do the same discovery approach with kite. I draw a kite (with the diagonals) and ask them how we can use what we already know to find the area formula of a kite. Some say that we can manipulate the triangles to make a rectangle with base 1/2 d_1 and height d_2 and others see it as the image below. It surprises the students that we don’t even need to know the side lengths of the kite to find its area. Boom. Kite -> rectangle
For a regular hexagon, I break the figure into 6 triangles, “unroll” them, then place half of the triangles in the crevices of the other triangles to create a parallelogram. In order to find the area of this parallelogram, we need to know the height of the triangle, which we call the apothem. Boom. Regular hexagon -> parallelogram -> rectangle
(found this pic on http://pediaa.com/how-to-find-the-area-of-regular-polygons/)
I then have students find the area of a regular octagon with their groups. They see that with the same process, it is the same area formula as a regular hexagon (1/2 x apothem x perimeter)! Boom. Regular octagon -> parallelogram -> rectangle
For circles, I do two approaches, one being very similar to how we found the area of a regular hexagon and octagon, and the other “cutting” the circle so it becomes triangular. Boom. Circle -> parallelogram -> rectangle or Circle -> triangle -> rectangle. Check out these animations to see what I mean:
At the end, I have students reflect on their “quiz.” I reiterate that math isn’t about memorizing formulas; it’s about relational understanding.
Thank you for reading.