Boxes and Cylinders
TEACHER NOTES
The Right Stuff: Appropriate Mathematics for All Students Promoting the use of materials that engage students in meaningful activities that promote the effective use of technology to support mathematics, further equip students with stronger problem solving and critical thinking skills, and enhance numeracy. Overview This is a hands‐on classroom activity. Students, working in groups, will construct a box, with top, using a given volume. Using different dimensions, they will find that the surface area is a function of the dimensions. That idea will be carried over to a right circular cylinder for which they will find a function to model the surface area as a function of the radius and dimensions of the cylinder to minimize the surface area. • Data – Students will collect data from boxes constructed by groups. • Modeling – Students will be able to interpret the data to find a minimum surface area for the box. • Algebra – Students will use the formulas for the volume and surface area of a right circular cylinder to find a function that models the surface area as a function of the radius. Supplies and Materials • Student Worksheet 20.1 • Excel file 20.1 • Scissors, ruler, tape, and pattern (on 8 ½ x 11‐in. card‐stock) (one for each group) Pre­requisite Knowledge Students must be able to work in teams and use formulas from geometry. Instructional Suggestions 1. Place students in groups, assigning groups as #1 through #8. Each group will construct a box with a lid from the template provided. However, each group must use, as one of the dimensions of the box, their group number and the template must fit on one 8 ½ by 11‐inch piece of paper. 2. As groups finish with the construction, they enter the data in the table (dimensions, volume, and surface area). 3. Discuss why the surface area changes with the dimensions of the box even though the volume is constant. 4. Allow each group time to talk about the challenges they faced as they did the construction. Assessment Ideas Change the volume of the box to 30 (or 32, or 36) cubic inches. Which groups can still construct a box using only one piece of paper? Extension Ask each group to write the surface area of the box as a function of the height keeping the length (the group number) and the height (their selection) fixed. 1
Module 20
This material is based upon work supported by the National Science Foundation under Grant No. DUE 0632883
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily
reflect the views of the National Science Foundation.
Boxes and Cylinders
TEACHER NOTES
Instructions 1. Use the pattern that is provided to construct a box with a top so that the result has the volume of 24 cubic inches. At least one of the dimensions of your box must be equal to the length assigned to you by your instructor. (a) Discuss the dimensions that will result in the given volume and then how those dimensions can be used with the pattern to fit on the 8 ½‐ by 11‐inch piece of paper provided. (b) Carefully draw the pattern on the card‐stock paper provided. Discuss how you can use the edges of the paper to make the angles closer to being exactly 90 degrees. (c) Use the scissors to cut out the pattern, make the folds, and construct the box using tape to secure the sides together. Take note of how much “scrap” you had relative to other groups. (d) As the box is being taped together, enter the dimensions, volume and surface area, in the table provided by the instructor. Discuss the goals of this activity: (1) Construct a box with a top and with a volume of 24 cubic inches using the pattern provided. (2) Collect data on the dimensions and surface area of each box, (3) Analyze the data. If possible, split the class into eight groups. Each group’s number must be used as at least one dimension of the box. For example, Group #1 must use 1­inch as one of the dimensions of the box. Obviously, groups 5 and 7 have the tougher tasks. Allow them time to problem solve. If a group is having problems with the dimensions and placing it on the card stock, suggest that they put the dimensions right on the pattern. This will help them find the right place to put the longest side in order for the pattern to fit on the card stock. Using the edges of the paper helps make the angles 90 degrees. As the groups are taping the box together, have one member enter the dimensions, volume, and surface area in a spreadsheet (or on the board). Discuss the dimensions that result in the least amount of surface area. The conclusion should be that a perfect cube, each side equal to 2 7/8 inches (approximate), will result in the least amount of surface area. (See Excel file for further information.) Discuss why this is a practical problem. Minimizing the amount of material in order to enclose a given volume is one that many businesses must solve. 2. What are the dimensions of a right circular cylinder with the least amount of surface area? Lead the class through a discussion about right circular cylinders. (Another option is to continue to allow the groups to work independently on this task.) (a) What are the three plane figures that make up a right circular cylinder? (b) Given the radius and height of the cylinder, what is the volume? The surface area? (c) If the volume is fixed: 64 cubic inches, express the height as a function of the radius. (d) Using the expression in (c), find an expression for the surface area as a function of the radius. (e) Graph SA(r) and find the dimensions of the cylinder that produce the least amount of surface area. (See 21.3 Excel.) Discuss why this is a practical problem and why soup cans, for example, aren’t constructed to minimize the surface area. (Dimensions fit the golden ratio so the displays are pleasing to the eye, and for packaging concerns.) 2
Module 20
This material is based upon work supported by the National Science Foundation under Grant No. DUE 0632883
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily
reflect the views of the National Science Foundation.
Boxes and Cylinders
TEACHER NOTES
Pattern 3
Module 20
This material is based upon work supported by the National Science Foundation under Grant No. DUE 0632883
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily
reflect the views of the National Science Foundation.
Boxes and Cylinders
TEACHER NOTES
Box with a top L W H Volume Surface Area 1 2 3 4 5 6 7 8 4
Module 20
This material is based upon work supported by the National Science Foundation under Grant No. DUE 0632883
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily
reflect the views of the National Science Foundation.