Making Sense of Area Formulas for Triangles, Parallelograms, Trapezoids, and Kites
Lesson 2 of 14
Objective: Students will be able to make sense of and explain basic area formulas.
We begin today's lesson by reviewing the work from yesterday's lesson on Sectors of Circles. Some groups will finish reviewing Sectors of Circles earlier than others; for early finishers, I direct their attention to a few reflection questions I have written on the whiteboard and ask the students to prepare what they will share with the class. The questions include:
- How did you determine the radius of each circle? Give an example.
- What patterns did you notice and what strategies did you use to solve each domino? Give examples.
- What surprised you? Give examples.
When debriefing this activity, I have the groups that finished early share out about the reflection questions. Then, if possible, I ask each of the remaining groups choose one member of the group to present how they solved one of the dominos. The groups who finished early then present the “order” in which the dominos should be connected so the entire class can check their work.
During homework review, I give students time to see if they can come to an agreement about which triangle as the biggest area. All of the triangles have the same base, so when I circulate the room, I listen to see which students think the triangles with a longer slant height have a bigger area. As a whole class, we discuss this common misconception, which gives me a chance to reinforce the notion that the height of a figure is measured along a perpendicular to a base.
Since I haven’t explicitly taught triangle area, or parallelogram area for that matter, I try to get students to share their ideas of how they make sense of these areas. Most of the time, one or more students will recall methods they have seen in the past, and will use tracing paper to show how triangle, parallelogram, and rectangle area relate to each other under the document camera--this is important because developing students' understanding of making an informal argument for the area of a circle depends on their understanding of triangle and parallelogram area.
In this section, my goal for this area lesson is to make sure students understand why the different area formulas look the way they do. In this task, my students look at two methods for determining the area formula for a trapezoid, which I had taken from student work in previous years.
I give my students about 5-6 minutes to read over the methods silently to themselves. I encourage my students to make sense of what they are reading by underlining and circling key words, asking questions, and paraphrasing what they are reading. I ask students questions like, “what do you think the student is thinking?” or “to what extent do you agree with the student?” and ask students to share their ideas with the whole class. This activity gives students the opportunity to critique someone else’s reasoning in writing and in speech (MP3).
I display a general picture of a kite for students and ask them to develop at least two methods for determining the area of a kite. I ask them to use words and pictures to explain their ideas. I circulate the room as students share their ideas, trying to find unique methods.
I then facilitate a whole-class discussion, trying to involve as many students as possible. I ask students to raise their hands if a method I share makes sense to them:
- Finding the area of each of the four triangles and summing them together
- Finding the area of the bigger triangle on top, the smaller triangle on the bottom, then summing together
- Finding the area of the triangle at right, the triangle at left, then summing them together
Then, I ask for silence and no hands as I project some of the unique methods that emerged from groups. After about 15-20 seconds, I allow students who think they understand the method to explain in their own words. I see if others in the class agree or see the method another way, then check in with the group who generated the method. We repeat this process until we have explored and made sense of the various methods in the class, which usually include building a rectangle around the kite and taking half the area of the rectangle, or rearranging areas of the kite to make a rectangle with the height of one of the diagonals and the base half the length of the diagonal that has been bisected.
At the end of today's lesson, we usually formalize the big ideas of the day in a whole-class discussion while taking notes in our notetakers. Since the goal of today's lesson is to make sure students can make sense of the area formulas and explain them in their own words, I give students the last 10-15 minutes of class to write down their own ideas of what each part of each formula means. I encourage students to use color to connect parts of the figure with the corresponding parts of the area formula and their own words.
When students finish explaining these area formulas, they bring them to me so I can quickly assess their understanding and make sure they do not have any other questions. After I check their explanations, I give them the Area Review worksheet, which they can work on until the end of class.
Homework: Area Review
In this homework assignment, students solve problems by using area formulas for parallelograms and trapezoids. Problem #1 gives students a variety of ways to visualize the shaded area while problem #2 allows students to extend their area understanding to coordinate geometry. I like the last problem because students work with both area and perimeter, as well as unit conversion, in order to make sense of and solve the problem.