Mathematics Models

Paleologos Cone: Overview

Pre-Assignment

Prior to the beginning of the unit, a bulletin board will be assembled.  The title of the bulletin board is “Sanford Middle School’s Ice Cream Parlor” and it features a lady that is saying, “To get the most for your money, try the Paleologos Cone.”

Each student is to provide a written response as to what he/she thinks this means.

Day 1

Introduction:
Students begin class in their preassigned groups of four students.  
Discussion: Have students share their responses to the pre-assignment.  Lead the students to the idea that the Paleologos cone must hold a lot of ice cream and therefore we are dealing with volume.

Present the problem:

WHAT IS THE VOLUME OF THE PALEOLOGOS CONE?

Briefly relate to the students that we don't have enough information to solve this problem right now.  We need to make a plan.  What are some things that we need to know?  Given this information what do we need to do?  Allow students to make suggestions.  Help them to develop the following plan. 
1.  Become familiar with the different parts of a cone.
2.  Be able to compute the volume of a cone using the volume formula.
3.  Determine the restrictions on the size of the cone.
4.  Given the restrictions, what size cone has the greatest volume?
Point out to the students that we will be working on numbers 1 & 2 today, and numbers 3 & 4 tomorrow.

Step 1: Parts of a Cone
(whole class instruction)  Each student should be given a copy of "The Parts of a Cone" worksheet.  Go through the parts of the cone using an actual cone and the diagram.  
1.  Base
2.  Radius of the base
3.  Vertex
4.  Height
5.  Slant height
Have students label the parts on their sheets.  

Talk about the difference in the slant height and the height.  Emphasize the difference in the two. 
 
Share the example dealing with measuring a person’s height. When measuring a person’s height, you measure from the top most point to the bottom (perpendicular) and not the distance along the side of the person’s body.

Demonstrate how the pipe cleaners can help us in measuring the height of our cones.
 

Medial Summary
 

Step 2: Volume of a Cone
(Group work)  Each group will receive one set of instructions.

They are to gather the necessary materials and follow the directions.  This activity involves pouring sand from a cone to a cylinder in order to compare their volumes.  The students are able to generate the volume formula for a cone and then compute the volume of several actual cones.

Summary:
Summarize today’s work:
- Restate the problem.
- Go through the parts of the cone.
- Restate the volume formula for a cone.

Assignment:
Each student should receive the assignment handout.

If-Time:
Students will be allowed to choose another cone to measure and compute the volume.  The volumes of the objects may be compared.  Discuss why some groups may have computed different volumes for the same objects (error in measurement).

Day 2

Introduction:
(Whole class instruction) Students are in their groups.  Each group should have a paper cone with which to work and the corresponding activity sheets.  

Question for students: What kind of restrictions might there be on the size of the cone?  Why do we need restrictions?

Step 3: Restrictions
According to the Make-Believe Cone Company, the slant height of an ice cream cone must be 8 cm.

 
Each group has been given a paper cone.  Together, let's measure the slant height.  It should be 8 cm.

Direct the students through the following:

Slowly move the cut edge from letter A to B to C, and so on.

1.  Does the radius increase, decrease or remain the same?

2.  What happens to the height?

3.  What does the slant height do?

Medial Summary

Step 4: Given the restrictions, what size cone has the greatest volume?
(Group Work) Students work through their activity packets.  This activity involves forming different sized cones from the paper cone and computing the corresponding volumes.  These volumes are recorded on a graph so that they may be compared.

Summary:
Tie the solution from step 4 back to the original problem.

Assignment:
Each student should receive a copy of the assignment handout.

If-time:
Make a frequency table of the numbers of students choosing the various lettered positions (indicating the maximum volume).  Have students share why they chose certain letters.

Day 3

Assessment:
Students should complete the assessment individually.

Ice Cream for everyone!