Cone Connections

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Objective

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

Big Idea

Having grasped the concept of volume, students use this understanding to solve real world tasks, this time involving cones.

Launch / New Info

20 minutes

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.  

 

Activity 1: A straight forward problem

10 minutes

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, r2, 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

 


Activity 2: Applying the formula

20 minutes

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.