At this point in the unit, I want to make sure students have a variety of strategies for thinking about and tackling area problems, especially problems that require them to persevere (MP1). Like all warm-ups in my class, I ask students to work quietly on their own. I circulate the room to see the strategies students are trying and to encourage good reasoning when I see it. When it appears that everyone has at least tried one strategy, I ask groups to begin discussing their methods, seeing what they agree and disagree with.
Many students tend to want to add a diagonal to the quadrilateral to find the area of each triangle—they are often stumped, however, as the base and height of each of these triangles is not clearly evident. While students often perceive this to be a dead end, I want to publicly validate the good reasoning here (the idea of finding a base and height) before we share out more successful methods. Ultimately, student volunteers share out, either decomposing the quadrilateral into four right triangles and a rectangle, or, building a rectangle around the quadrilateral and subtracting the extra areas gained by the four right triangles in the corners of the rectangle.
In this group task, students will investigate regular polygons of different side lengths to ultimately generalize a formula for determining the area of any regular polygon.
I launch the task by having students construct a regular hexagon inscribed in a circle. Then, we review the mathematical terms regular, congruent, isosceles, and apothem as they see these terms in their construction. I then pass out the Regular Polygon Area task cards so students can carry out the investigation. Ultimately, I hope that each group will successfully arrive at the conjecture that the area formula for any regular n-gon is A=(1/2 as)n or A=1/2 aP where P=sn.
After the group investigation, I facilitate a whole-class discussion in which I show students a circle that has been divided into 16 congruent sectors. I rearrange the sectors so that they form a parallelogram-like figure. I then give students about 2-3 minutes in their groups to see if they can make sense of why the area formula for a circle, which they will then share out with the whole class.
Because I want students to have a variety of ways to make sense of all of our area formulas, I show students the video below, which they watch silently, twice, before talking to a partner about their ideas. I then get the attention of the whole class and call on several students to explain how they made sense of the video and why the area formula for a circle is r2.
Source: https://www.youtube.com/watch?v=whYqhpc6S6g (Accessed July 30, 2014)
For me, there is almost nothing better than seeing my students build on their prior knowledge to make meaning out of a previously nonsensical idea. Today was the day we rearranged our circle into a parallelogram-like figure, which helped many students to visually see why the area of a circle is given by pr2. Students seemed to be able to grasp the notion that the “base” of our “parallelogram” was equal to half the circle’s circumference and that the “height” of our “parallelogram” was equal to the radius of the circle. I was happy to have called on several students who were able to summarize the strategy and explain their understanding to the class.
While facilitating this discussion, I decided to ask the class, “what questions might a skeptic ask us?” This question provoked all kinds of responses, like “how do we know this is a parallelogram?” or “why can we think of this as a parallelogram when the base is curvy and not straight?” or even, “if you really drew the height of the ‘parallelogram,’ which must be perpendicular to the base, wouldn’t it be less than the radius?” I responded to this last question with another question, “what could we change to make our circle even more like a parallelogram?” which set the stage for students to consider slicing the circle into more and more sectors until the number of sectors approached a million or even infinity; this was a great moment in that students began to develop in their heads the idea of a limit.
Like the last lesson, I want students to have time to explain the area formulas for regular polygons and circles in their own words. After students have written their explanations, they show them to me and get the Area Application, which they can begin in class.
Although area measurement is not a major feature of the High School Geometry Common Core standards, it is important that students get opportunities to solve real-world area problems. In the Area Application Homework, students also get to explore how changing the dimensions of similar figures impacts their areas, which is often a challenging concept for students to grasp.