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# Proving the formula for the area of a trapezoid

Lesson 3 of 8

## Objective: SWBAT prove the trapezoid area formula using a decomposition argument.

#### Activating Prior Knowledge

*15 min*

In order to understand the trapezoid area formula proof students will need to know the definition of a trapezoid and they will also need to understand that all triangles inscribed between two parallel lines have equal areas. APK_Prove Trapezoid Area Formula is designed to have students recall the definition of trapezoid and to get them thinking about triangles inscribed between parallel lines.

I give the students 7 or so minutes to complete the handout independently. Then I give the students 5 minutes to share their responses with their partners and invite critique. While this is happening, I walk around listening to the arguments and looking for exemplar students. When the time elapses, I call up a few exemplars to come up and present at the document camera. Then it's time to move on.

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#### Proving the Formula

*25 min*

I begin by giving all students a copy of Trapezoid Area Formula. We read through the first page together, stopping when students are supposed to write or do something.

One of my main goals is to get students to experience those first stages of planning a proof when we really don't know what to do yet. I give them some help by asking them to analyze the structure of a form of the trapezoid area formula that suggests the strategy for the proof. So when we get to the students are asked to describe what they see, I give them enough time to think about it even if they don't get it at first. I walk around and if i see that a student has noticed that the right side of the formula is the sum of two triangle areas, I push them on to the second page where they will soon be working independently to write the proof.

After 10 minutes or so, if there are still students who don't get it, I'll stop the class and reveal the answer and push everyone forward to the second page.

Once everyone is writing the proofs, I'll give students a deadline by which they need to have their proofs completed. When time has run out, I'll have student exemplars come to the document camera to share their proofs with the class.

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- UNIT 1: Community Building, Norms, and Expectations
- UNIT 2: Geometry Foundations
- UNIT 3: Developing Logic and Proof
- UNIT 4: Defining Transformations
- UNIT 5: Quadrilaterals
- UNIT 6: Similarity
- UNIT 7: Right Triangles and Trigonometry
- UNIT 8: Circles
- UNIT 9: Analytic Geometry
- UNIT 10: Areas of Plane Figures
- UNIT 11: Measurement and Dimension
- UNIT 12: Unit Circle Trigonmetry
- UNIT 13: Extras

- LESSON 1: A Deeper Look at Area
- LESSON 2: Proving Area Formulas for Parallelograms, Triangles and Rhombuses
- LESSON 3: Proving the formula for the area of a trapezoid
- LESSON 4: Construct Regular Polygons Inscribed in Circles
- LESSON 5: Regular Polygons and their Areas
- LESSON 6: Areas of Regular Polygons Inscribed in Circles
- LESSON 7: Area Construction Challenge
- LESSON 8: Application: Geometric Integrals