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# Cone Connections

Lesson 3 of 7

## Objective: SWBAT understand and apply the cone volume formular to real world problems.

#### Launch / New Info

*20 min*

I enjoy teaching this lesson, particularly this Launch. It sets a positive tone for the entire lesson. My students always seem comfortable, yet challenged by the problems. We accomplish a lot in a few minutes. My students recall prior knowledge and I introduce new content.

I start the lesson by placing the Entrance Slip face down on each student's desk. I ask that they not turn it over until they've watched Deriving The Formula - Volume of Cone. The clip demonstrates how the volume of a cube is about one-third the volume of a cylinder, given equal height and base area measurements.

After we watch the video, I ask my students to read the Entrance Slip to themselves quietly, then answer the questions. There should be no pencil-paper calculations nor calculator use. After my students answer the questions I will employ the Think Pair Share strategy:

- Students pair with an elbow partner once they complete the slip
- Partners discuss the answers to the questions together
- Pairs get ready to be called on to share their thoughts with the class

Here are some possible answers to the Questions based on my experience teaching the lesson.

**Question 1**: My students usually do not memorize or have the cone volume formula at hand even if they've been exposed to it. After seeing that it took 3 cones of water to fill the cylinder, students should be able to state that the cone volume is 1/3 the cylinder volume. (V=1/3 B x h) where B = pi x radius squared)

**Question 2:** Students should recognize that the volume of Cylinder A is greater than that of Cylinder B, based on what they learned in the previous lesson on cylinders. I hope they reason that the cone inside Cylinder A holds more water because both cones are 1/3 the volume of the respective cylinder.

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Our next task, Activity 1, should not take more than 10 minutes. I ask that students remain with their current partners. Each pair of students will receive two cone problems to tackle, while I walk around monitoring and guiding students where needed. Each student will complete one of the problems -- they decide who completes which problem.

Once finished, the partners should switch papers and correct each other's work based on 3 criteria:

- Correct formula and diagram
- Work clearly shown
- Correctness of the answer

I give students a couple of minutes to discuss and correct any errors, before randomly calling on two students to write their work on the board.

As I am circulating during this task, I am looking to make sure that my students can solve a cone volume problem before going on to Activity 2. As a result the problems in this section are relatively straightforward:

**Problem 1**: Find the volume of a cone with a base diameter of 12cm and a height of 15cm.

**Problem 2**: Find the height of a cone whose volume is 225 cubic inches and has a base radius of 5 inches.

I am also looking for lingering issues with solving equations and simplifying expressions:

- Students who are struggling with solving equations with fractions may choose to round 1/3 to 0.3. I indicate that they can divide the product of pi, r
^{2}, and h, by 3. This gives a more exact answer. I am on the lookout for this problem. - In problem 2, students may overlook that multiplying both sides of the equation by 3 is a valid way to simplify the equation quickly. I may suggest this route to some students.
- Here's an interesting error that I see on a regular basis in this lesson:

225 = 1/3(3.14)(25)h

225 = 1/3(78.5)h

(3)225 = [1/3(78.5)h](3)

675 = (235.5)h **{Student incorrectly employed the Distributive Property.}**

See Reflection: Common issues in my class

#### Resources

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As a final activity for this lesson, I want my students to apply the cone formula to real world situations. Again, I'll probably pair students up, but groups of 3 will work as well. I like relatively homogeneous groupings for application tasks, so some group re-arranging may be necessary before we start. I made enough copies of each task for each pair so they could do both tasks, but I generally ask pairs to choose one task and to present their answer in detail.

I want to receive a paper from each student. Since we are working in groups, as I visit partners I make sure that both students are actively engaged. I announce several times that I am going to choose two students, at random, to come to the board and present one of the problems.

It helps my students when I am clear about the objectives for their work. So, I write on the board:

1. Draw diagram and show all math work on paper.

2. Discuss the problem among group members and be prepared to be called upon to share your work and/or answer questions.

3. Each student must have their own work written in their notebooks.

As usual, I allow students to tackle the problem and struggle through it on their own. I try to intervene only enough to guide, without revealing key points in the problem that they should discuss and figure out themselves.

I monitor the time so that I can ask two students to go to the board and discuss their work on one of the problems.

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- UNIT 1: Number Sense
- UNIT 2: Solving Linear Equations
- UNIT 3: Relationships between Quantities/Reasoning with Equations
- UNIT 4: Powers and Exponents
- UNIT 5: Congruence and Similarity
- UNIT 6: Systems of Linear Equations
- UNIT 7: Functions
- UNIT 8: Advanced Equations and Functions
- UNIT 9: The Pythagorean Theorem
- UNIT 10: Volumes of Cylinders, Cones, and Spheres
- UNIT 11: Bivariate Data