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* *Reflection: Vertical Alignment
Solids of Revolution - Section 3: Solids of Revolution Problem Solving

I love how the Common Core has caused us to be much more conscious of progressions in the math curriculum than we were before. Having taught Calculus and seeing how students in Calculus struggled with Geometry concepts has made me more intentional about priming students as early as possible for the type of thinking they'll do in more advanced classes. Of course we can always scale things down to make them appropriate to where students are developmentally, but just being aware that I am preparing students for something beyond my class has made me much more intentional about which topics I decide to teach and how I choose to teach them. As I tell myself, Pre-Calculus is all math before calculus.

*Preparing for Calculus*

*Vertical Alignment: Preparing for Calculus*

# Solids of Revolution

Lesson 5 of 6

## Objective: SWBAT visualize 3-D solids of revolution and find their volumes

The idea of rotating a two-dimensional region to create a three-dimensional solid is a new concept for most students. Even when the concept is explained well, it is still difficult for many students to visualize it. Thankfully, we have technology that makes our jobs as teachers a lot easier.

I use one such technology aid to start this lesson. It is a resource from Wolfram Alpha and its full citation is shown here:

http://demonstrations.wolfram.com/SolidOfRevolution/

The following video explains how I use this demonstration to introduce this lesson.

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To begin this section, I give each student a copy of Guided Practice_Solids of Revolution. The problems on this handout deal with rotating rectangles about their sides and right triangles about their legs. The more complex problems students will do later in the lesson involve rotations that are merely compositions of these fundamental rotations.

The two problems on the handout each have an (a) portion and a similar (b) portion. I model the (a) portion for students and then I give them time to complete the (b) portion on their own. After they have completed the (b) portion, I have them compare answers and process with their seat partners before I show the correct process and answer.

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In the previous two sections, I've been careful to equip my students with the fundamental knowledge and skills they will need in order to tackle the problems in this section.

My intent in this section is for students to gain experience with the process of problem-solving, which someone once described as what you do when you don't know what to do. For this reason, I will not be giving answers to the problems during this classroom session. I want students to take risks, collaborate, make sense of things, persevere, etc. If they know I'll soon be giving answers, some students hesitate to take the full onus of the problem-solving upon themselves.

So I have my students move the desks so that they are seated side by side with their seat partners. Then I give each student a copy of Problem Solving _Solids of Revolution I give the students 40 minutes to work collaboratively on the problems. I walk around the room looking for students who are seriously stuck. I'll try to ask these students open ended questions to try and move them forward. For example, I will usually ask, "What lengths can we infer on the diagram besides what's been given?" or "How might you compose or decompose this figure into simpler figures?" and sometimes it just "Explain to me how this figure is going to rotate."

In any case, as I said, I will allow students to struggle through the problems and use each other as resources. If they don't finish in class, they will need to take the problems home to complete. When students have had enough time to finish (inside and outside of class), I will post Solids of Revolution Worked Solutions online for students to learn from and to check their work.

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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: Constructing an Argument for the Circumference Formula
- LESSON 2: Constructing an Argument for the Circle Area Formula
- LESSON 3: Argument for the Volume Formula of a Cylinder
- LESSON 4: Arguments for Volume Formulas for Pyramids and Cones
- LESSON 5: Solids of Revolution
- LESSON 6: Solving Problems involving Volumes of Solids