Cone Connections

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.  

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

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. 

• Task 1 Hour Glass
  
• Task 2 Lemonade

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.