##
* *Reflection: Connection to Prior Knowledge
Making Sense of Area Formulas for Triangles, Parallelograms, Trapezoids, and Kites - Section 2: Homework Review

I was pleased with my decision to discuss what can seem to be a very straightforward and simple triangle area problem. Despite the fact that several students got the right answer, I found that everyone had something to gain in discussing why the formula of a triangle is given by ½bh and how this relates to the area of a parallelogram and rectangle made by the same base and height.

I set the stage for the whole-class discussion by identifying three triangles that were commonly chosen by students as having the greatest area. I asked students, “what are some reasons for why these triangles would have been selected?” which surfaced common misconceptions (the triangles with slantier sides appear to be wider, which means they will have more area; the height of a triangle is measured along its slant, not the perpendicular from the vertex to the base) and commentary on how students could avoid those pitfalls.

I thought this discussion was much more interesting and thought provoking than simply launching into the triangle area formula, which many students have memorized but cannot explain. What I also appreciated was being able to project particular students’ work to value their sense making (like visualizing the square units within the triangle or copying the triangle to create a parallelogram with the same base and height) and, as a result, assign them status. Taking time to discuss the square unit approach helped students visualize area; connecting triangles to parallelograms allowed students to reason and connect prior knowledge about the diagonal of a parallelogram dividing it into two congruent triangles.

*The WHY Behind the Formulas is What Matters Most*

*Connection to Prior Knowledge: The WHY Behind the Formulas is What Matters Most*

# 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.

## Big Idea: By critiquing others' written explanations of triangle and trapezoid area formulas, students will develop strategies for determining areas of other 2-D shapes.

*85 minutes*

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.

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#### Homework Review

*10 min*

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.

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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**).

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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.

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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.

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#### Homework: Area Review

*5 min*

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.

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- UNIT 1: Creating Classroom Culture to Develop the Math Practices
- UNIT 2: Introducing Geometry
- UNIT 3: Transformations
- UNIT 4: Discovering and Proving Angle Relationships
- UNIT 5: Constructions
- UNIT 6: Midterm Exam Review
- UNIT 7: Discovering and Proving Triangle Properties
- UNIT 8: Discovering and Proving Polygon Properties
- UNIT 9: Discovering and Proving Circles Properties
- UNIT 10: Geometric Measurement and Dimension
- UNIT 11: The Pythagorean Theorem
- UNIT 12: Triangle Similarity and Trigonometric Ratios
- UNIT 13: Final Exam Review

- LESSON 1: Sectors of Circles
- LESSON 2: Making Sense of Area Formulas for Triangles, Parallelograms, Trapezoids, and Kites
- LESSON 3: Making Sense of Area Formulas for Regular Polygons and Circles
- LESSON 4: Strategies for Decomposing 2-D Figures
- LESSON 5: Sector Area Application: The Grazing Goat
- LESSON 6: Surface Area and Area Differentiation
- LESSON 7: Extreme Couponing: Pizza Edition
- LESSON 8: Area "Quest"
- LESSON 9: Introduction to Volume: Origami Boxes
- LESSON 10: Origami Boxes Gallery Walk
- LESSON 11: Volume Formulas, Cavalieri's Principle, and 2-D Cross-Sections
- LESSON 12: Real World Volume Context Problems
- LESSON 13: Ratios of Similarity and 3D Solids Generated by Revolving 2D Figures
- LESSON 14: Volume "Quest"