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# Solving Problems involving Volumes of Solids

Lesson 6 of 6

## Objective: SWBAT solve complex problems involving volumes of prisms, cylinders, pyramids, cones, and spheres.

#### Heavy Modeling

*180 min*

As the section title implies, this section is all about modeling. Solving problems of the type and level we'll be solving is entirely new to my students at this juncture. There are various expert skills that are required to solve these problems. My goal is to reveal these skills and impress them upon students through demonstration, thorough explanation and repetition.

I basically go through almost all of the challenging problems in the chapter and demonstrate their solutions for my students. I'm including Modeled Problems here for your reference, mainly to show the sheer volume (no pun intended) of problems I model for students.

I model these problems very methodically, being careful not to go too fast, and taking care to explain my reasoning as well as my procedure.

I tell my students not to write if writing interferes with their ability to pay attention. I will post all of my work online, so they will be able to take notes on it later.

This modeling spans three class periods. This is one of the only times during the year that I spend entire class periods talking and demonstrating. The students' job is to observe, make sense, and ask questions whenever they have them.

#### Resources

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#### Transfer Problems

*120 min*

Now that students have had a crash course in solving complex problems, now it's time for them to practice solving problems on their own.

Students will be working on two problem sets on two separate days. The first problem set is Book Problem Remixes and the second is geometry_unit_12_modified_book_problems. I purposely made these problem sets to correlate with Book Problems that I have already modeled in class.

On both days, students will have the entire period to work on the problems. I allow them to collaborate with their classmates as well. The only thing I don't do is give answers or hints. These are problem-solving days and being stuck/bewildered/confused/frustrated are all part and parcel of real problem-solving.

What students don't finish in class, they will continue at home. I will post unit_12_remix_problem_solutions and unit_12_modified_problem_answers_and_solutions on line so that students can learn and check their work.

All of this is training for the upcoming unit test, which will consist of problems similar to the ones students had to solve in this section.

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#### Assessment

*60 min*

At this point students have had problems modeled form them ad nauseum. They have completed two rounds of practice problems on their own. After these practice rounds, they have been able to study worked solutions and they have had time to have any remaining questions/problems addressed in class.

Now it is time for them to demonstrate their problem-solving skills when it really counts.

Students will take the Unit 12 Assessment Free Response. It is composed of problems similar in nature and level to the problems we've been solving in the lesson. Because this test is so close to the final exam, I will post unit_12_version_a_test_solutions online as soon as everyone has taken the test. That way, students can use the solutions to study for the final exam.

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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