Carefully Constructed Cones

Containing Cool Colorful

Candies Created Cooperatively

by Calculating "C"ids

Ben Huntington Vicki Miller Barry Nielsen Khris Willis

Susie Shafer Paula Lincoln Debbie Wilch

Class: Pre-algebra, General Math or Middle School

Materials: Graph paper, note paper, pencil/pen, protractor, compass, calculator, included student sheet, math book with formulas, measuring devices, scissors, construction paper, tape/glue, large number of gumballs (spheres) of varying sizes.

Goals: Students will make a judgement as to what shape of cone will hold the maximum number of gumballs.

Time: Two consecutive days and the evenings to work on calculations and report. Due the third consecutive day.

Background: Students will need to know how to work cooperatively, how to find volume, how to estimate, how to evaluate expressions, how to measure, how to use a compass, how to use a protractor.

Setting: Groups of students are going to try to find a cone which will hold the most candy. Candy will be spherically shaped (i.e. gumballs). Students will find the cone that holds the most candy--level or heaped. Students will make their cones by cutting a sector from a circle and taping the edges together.

Problems: The student sheet states three specific problems.

Evaluation: Scoring rubric:

5 point paper:

* Entire paper is neat and legible.

* Entire paper is relatively free of spelling, grammatical, and usage problems.

* Entire paper is well organized.

* Title page with all names of group is included.

* Clear statement of the problem as you understood it is made.

* Report lists all pertinent assumptions.

* Report clearly explains the plan and procedure followed in gathering data.

* Data and/or graphs are well organized; data should be appropriate and thoroughly analyzed.

* Correct calculations.

4 point paper:

* Same as 5 point paper, but lacking neatness or legibility.

3 point paper:

* Same as 4 point paper, but lacking organization and/or minor calculation errors.

2 point paper:

* Same as 3 point paper, but lacking clear explanation and/or several calculation errors.

1 point paper:

* Little effort shown; major calculation errors; no evidence of understanding the problem.

Extensions: The use of non-spherically shaped objects will enhance this cool lesson and give you goosebumps all over. Increase the size of the circle (11x17 paper or butcher paper) and see how close their predictions are at a larger scale. Try baseballs or tennis balls and see how the predictions hold up.

Teacher Notes: A good opener to this project might be to give the students a sheet of typing paper and have them compare the volume of cylinders formed rolling the paper lengthwise versus widthwise. Students may have the misconception that the same size paper will construct the same size volume.

Don't offer any suggestions or tools unless the students specifically ask for them. You may always substitute inedible objects (marbles, etc.) for the candies if that works better in your class. This project will lend itself to an oral group presentation rather than a written report, if so desired.

Carefully Constructed Cones

Containing Cool Colorful

Candies Created Cooperatively

by Calculating "C"ids

Each group must hand in a report solving these problems. The report must contain the following: cover page; list of any assumptions you make; calculations and formulas; data collection pages; explanations and justifications of your solution methods; your conclusions; sources of possible error; and your diagrams and models

Problem A:

You can form a cone by cutting a sector out of a circle and taping the edges together. Each member of your group should make a cone which he/she believes will hold the largest number of gumballs: filling to level and heaping the top. Keep track of the angle of the sector you cut out. Next, find the volume of one gumball and the volume of your cone. Combine the information on the group data collection sheet. All cones need to be made from the circles provided.

Problem B:

Each member of the group should repeat the process from Problem A with a different size gumball, but all members need to use the same size. Enter information on the group data collection sheet. All cones need to be made from the circles provided.

Problem C:

Your group will be given gumballs of a new size. Predict an angle θ, and construct the cones that you hope will hold the most gumballs.

C11 - Student Page

Data Collection Sheet

Name___________________

___________________

___________________

___________________

 angle θ of cut out sector Volume of gumball Volume of cone # filled level w/ top # filled when heaped Student 1 Student 2 Student 3 Student 4 Student 5

Group Discussion:

1. What angle θ would hold the largest number of candies - level

and heaped?

2. Find a formula so that if you know the angle θ, you can find

the number of candies that can be held.

3. Would you have to make any adjustments for other larger or

smaller shapes?

4. Repeat exercise with another size gumball.

 angle θ of cut out sector Volume of gumball Volume of cone # filled level w/ top # filled when heaped Student 1 Student 2 Student 3 Student 4 Student 5

Funded in part by the National Science Foundation and Indiana University 1995

Funded in part by the National Science Foundation and Indiana University 1995