Can Do Maths

CAN CUTTING

Purpose

Students work out how to cut out as many can shapes as possible from a given area of aluminium sheet using a paper and pencil simulation.

Activity

Students measure an aluminium beverage can and make a paper template.

Using this template, try to trace as many can shapes as possible onto a piece of broadsheet newspaper.

Discussion

Discuss the value of cutting aluminium sheet efficiently.

For secondary students

Work out the area of paper wasted.

Note: It would be worth explaining to students that aluminium beverage cans (unlike other cans) are not produced in three sections like this but the base and sides are extruded from a single piece of aluminium and the top is added as a second piece from different quality aluminium. Scrap aluminium sheet is recycled so there is no waste.

THE IDEAL CAN

Levels: 7-10

Purpose

To develop open-ended problem-solving skills. Students should be encouraged to solve this problem by trial and error or spreadsheet simulation. Higher level students may solve it by calculus but this removes the problem-solving challenge.

Activity

What's the best shape for a cylindrical can (height and basal radius) in order to maximise volume per area of aluminium used?

Students consider problems such as concave bases (called domes) on cans and ridges around top. (For this exercise it's easiest to assume that these don't exist.)

Give students as much or as little detail in the explanation as you think the students require in order to start the activity. Reinforce that this is a problem solving exercise - they are not expected to come up with the correct answer but to demonstrate an understanding of the concepts involved and to develop, trial and evaluate one or more methods for solving the problem.

It may be useful for students to start by drawing the can in cut-out view: [Andrew, I lost the diagram when the file was corrupted. Please copy and paste from email version I sent you earlier.]

Students should be shown (or derive for themselves) that total surface area is dependent upon only two variables:

r = radius of the can base (top), and h = can height.

Calculate the area of aluminium used:

Area = 2pr² + hy, where y = 2pr (circumference of the base)

= 2pr² + 2prh

= 2pr (r + h)

Calculate the volume of the can:

Vol = pr²h

Higher level students should be able to derive this for themselves, lower level students may need the equations to be provided and explained.

Once the equations are derived or supplied, students can then model the problem by trying different heights and base radii to try to maximise can volume per area of aluminium used.

3. Shrinking cans

Levels: all

Purpose

To analyse data relating to the reduction in the amount of aluminium used to make a beverage can.

Activity

Graph aluminium beverage can mass against year from the following:

1992 = 16.55 g

1993 = 16.13 g

1994 = 16.10 g

1995 = 16.00 g

1996 = 15.60 g

1997 = 15.50 g

Add in your figure = (See Activity 8)

Calculate the number of cans per kg for each year.

For secondary students

Students find out the official price of aluminium today (London Metal Exchange web site or newspaper) or use the recycling price for aluminium beverage cans. They calculate how much is saved by making lighter cans, based on the 1992 and 1996 figures.

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