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* *Reflection: Developing a Conceptual Understanding
A Deeper Look at Area - Section 2: Collaborative Problem Solving

It's funny how sometimes you teach something and realize that there ends up being a bigger learning outcome that the one(s) you had planned. That happened in this lesson. The big idea that came out of this lesson was that area problems can be approached by composition or by decomposition. These are general heuristics that are applicable to a wide range of problems besides the ones in this lesson. So now when I teach the lesson, I'm really intentional about calling attention to the general strategies we're using and how they might be useful with future problems instead of just focusing on how to do the problems in this lesson.

*Not just content, but heuristics*

*Developing a Conceptual Understanding: Not just content, but heuristics*

# A Deeper Look at Area

Lesson 1 of 8

## Objective: SWBAT explain the conceptual meaning of area and the area formulas for rectangles and triangles; SWBAT solve challenging problems involving areas of rectangles and triangles.

#### Concept Development

*30 min*

While I understand that students are currently learning many of the concepts from this lesson in the fourth grade, fourth graders are also learning cursive, and our students have no idea how to do that when they get to high school. I'd argue the concept of area is a lot more important than cursive, equally mysterious to students, and, therefore worth going over again. The way we treat area, though, I promise will have plenty of rigor by the time we're done.

So I start the lesson with the Conceptualizing Area PowerPoint show. The ideas are simple enough, but also important. The idea of partitioning, for example, and counting squares will be useful to students who take AP Calculus.

The PowerPoint establishes the basis for the rectangle area formula. Once we have established that, I use what we've established about rectangles to prove the triangle area formula. I use a somewhat unconventional visual proof of the triangle area formula. I figure students have seen the old cut a parallelogram in two trick before, so I try to come at it with more of a high school flavor. The proof I use incorporates topics from the high school geometry course like the triangle midsegment, parallel lines cut by a transversal, and transformations. See the video below.

After demonstrating the visual proof for the triangle area formula, I have students try to replicate the proof using paper and scissors. Their success depends on how precisely they draw the midsegment. No problem if they don't get it the first time. They can keep trying until they get it. That will just mean more practice and reinforcement.

When students have had adequate time to grapple with the visual proof of the triangle area formula, I move on to another demonstration. This is the demonstration of the useful fact that all triangles with a particular base and height have the same area. See the video for an inside look at what that demonstration is like.

As a quick check after this demonstration, I give students a ruler and ask them to to draw five distinct triangles that have base length, 10 cm and height, 6 cm. Then I have students come to the document camera and showcase what they came up with. They must identify the base and height of each triangle and explain how the area is calculated.

#### Resources

*expand content*

During the first part of the lesson, students have been able to sit back, for the most part, and be spectators as I took them on a guided tour of important concepts and properties of rectangle and triangle area. By contrast, students are in the driver's seat during this section. They must apply the concepts they have seen in the lesson to challenging, non-routine problems.

For the first 15 minutes, students work independently. I want to see what they would have been able to come up with on their own if there was no prospect of getting any outside help, so I frame it this way. I say something like, "You will be working by yourself on a problem for the next 15 minutes. The problem may be challenging. You may get the feeling that you are stuck and have no idea what to do. You may even feel like it's impossible for you to solve the problem. This is ok. It is a natural part of the problem-solving process. Students before you who have successfully solved the problem have often started out feeling the same way. The important thing is to think seriously about the problem and actually try something that makes sense to you. If that doesn't work, try something else. Try. Try. Try. And don't give up. (MP1)

So after that little pep talk, I hand out the Shaded Triangle student resource.

Helping students during this time is out of the question. The most I can do is encourage them to try and remind them that there is no penalty for getting the problem wrong. Other than, that, I just make sure that the classroom is quiet and everyone is focused on the task at hand.

After the 15 minutes have expired, I walk around to see what students have come up with. Then I have students get into groups (for 15-20 minutes) so that they can discuss their strategies and efforts. Putting the students into groups also allows me to have more close interactions with a greater number of students. As students are collaborating to solve the problem, I come around ask questions like:

1. How might we find the area of the shaded triangle indirectly instead of directly?

2. What difference does it make, if any, that the triangle is inscribed in a rectangle?

3. How many triangles do you see in the diagram?

4. How might you find the missing segment lengths on the sides of the rectangle? (Try imagining the given measurements as constants instead of variable expressions if the variables are giving you trouble.)

My hope is that as students begin to get it, they will work with other students, and the ideas will spread throughout the classroom until everyone gets it. When the student conversations start to die down (kind of like knowing when to take popcorn out of the microwave), I ask for volunteers to come to the document camera and show their solutions.

Next, while students are still in their groups, I hand out the Shaded Triangle_Decomposed student resource. I explain that this is the same figure as the one in the previous problem only it has the dashed lines inserted so that we can see the problem in a different way (assuming no one did it this way in the first place). So I tell students to try solving the problem again, but not the same way as we did it the first time. This time, make use of the dashed lines to come up with an alternate solution strategy. This will most likely run until the end of class, so I allow students to take the papers home and bring them back to the next class meeting completed.

*expand content*

- 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