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* *Reflection: Complex Tasks
Trapezoids - Section 2: Trapezoid Coordinate Geometry

I teach an honors geometry course. I have a lot of students who enjoy a good challenge. The challenge problem, the solution of which you can see in the video in this section, gave my students a chance to use their algebra, spatial reasoning, and problem-solving skills to try and earn some extra credit points.

Students spent a lot of time on the challenge and several of them got a partially correct solution. However, no student got both of the possible solutions. After they had turned their work in, this gave me an opportunity to show students how to get the second solution. Students were fascinated with the second solution and typically had a reaction like "Oh yeah! Why didn't I think of that?"

This just reminded me how every now and then it's nice to have a really tough problem that not everyone is expected to solve but everyone has an opportunity to solve.

*Challenging Students*

*Complex Tasks: Challenging Students*

# Trapezoids

Lesson 7 of 8

## Objective: SWBAT apply properties of trapezoids

#### Activating Prior Knowledge

*10 min*

The purpose of this section of the lesson is to give students the opportunity to process the definition of trapezoid, get familiar with the anatomy of trapezoids, and to begin making connections to prior learning (for example the polygon interior angle sum theorem and same side interior angles theorem.)

The resource for this section is Activating Prior Knowledge: Trapezoids. I have students collaborate with their A-B partners for 7-8 minutes. I advise students to make sure that (1) their answers are documented precisely on their papers, and (2) they understand what is on their paper because I will be randomly selecting students to present their answers to the class. I've found that telling students they will have to show their paper under the document camera if they are selected leads to higher quality work.

Next, I show answers for 1-8 under the document camera, explaining important concepts as I do. Then I ask A-B partners to discuss their answers to #9. Finally, I call three students up (one at a time) to present what they wrote for #9. After the three presenters have presented and all the mathematical discussions have been completed, we are ready to move on to the next section.

#### Resources

*expand content*

In this section, students will be using coordinate geometry to verify attributes of trapezoids. This parallels what they were asked to do in the preceding lesson on Using Coordinate Geometry to Prove that a Quadrilateral is a Parallelogram.

I designed Trapezoid Coordinate Geometry to develop students' ability to make sense of problems and persevere in solving them (MP1). For this reason, I have students get right to work on it without much build-up on my part. All of the information they need is on the paper. Their job is to decipher what's in the text and apply what they have learned previously to do what is being asked of them.

As students work, I walk around the room coaching. Most of my coaching is in the form of directing students to re-read, or having them to interpret, what the text says. For example, a few students generally ask me which coordinates they should "plug in" to the midpoint formula to create the midsegment. Usually these students either (A) haven't processed the text that says " the midsegment of a trapezoid is the segment that joins the midpoints of the legs of the trapezoid," or (B) they still don't know which sides are the legs of the trapezoid. So for these students I might have them use their index fingers to trace on the graph approximately where the midsegment should go based on the definition (which I have them re-read). This usually gets things on the right track.

In addition to helping students to move past stuck points, my coaching is also aimed at pushing them to produce precise and well-organized work. See the provided Trapezoid Coordinate Geometry KEY for an idea of the quality of work I aim to have students produce.

Some students tend to finish the main part of this activity more quickly than others. For them, the challenge problem is a good chance to do some substantial problem-solving that involves algebra skills that need to be kept sharp (it involves linear functions and quadratic equations students can solve with quadratic formula or by factoring). Even if students can not solve the problem algebraically, it is a valuable geometry experience for them to try to solve the problem through construction, measurement, etc.

See the Trapezoid Coordinate Geometry KEY for the answers to the challenge problem and check out these videos which demonstrate how dynamic geometry software can be used to solve the challenge problem (MP5).

** Case 1 Video:**

**Case 2 Video:**

*expand content*

In the Student-Created Trapezoid Problems activity, students will create their own problems involving what they know about trapezoids. They will also create solutions for these problems. I am careful to clarify that students are not to write coordinate geometry problems. I also clarify that their solutions should be on a page separate from their problems. Next, I go over the rubric with particular emphasis on what it takes to earn a 4.

The students take these home to complete and bring them to the next class meeting. Depending on time constraints I will either collect the assignment right away or give students time to work in pairs to solve each other's problems.

#### Resources

*expand content*

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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: Parallelogram Definition and Properties
- LESSON 2: Proving Properties of Parallelograms
- LESSON 3: Special Parallelograms
- LESSON 4: Proving Properties of Special Parallelograms
- LESSON 5: Proving that a Quadrilateral is a Parallelogram
- LESSON 6: Using Coordinates to Prove a Quadrilateral is a Parallelogram
- LESSON 7: Trapezoids
- LESSON 8: Quadrilaterals Transfer Task