The purpose of this activity is for students to create open cylinders and calculate their volume. In doing so students consider the relative effects of increasing and decreasing height and radius on the volume.
The background knowledge presumed for this task is outlined in the diagram below:
This activity should be used in a ‘free exploration’ way with an expectation that students will justify the solutions that they find.
The procedural approach (show more)
- The student makes each cylinder and calculates the volumes using measurements.
Students who use a procedural approach solve the problem by practical measurement. Look to see if they recognise that their answer is expressed in units of volume.Click on the image to enlarge it. Click again to close.
Choice of unit for the height and radius presents a dilemma. Millimetres are more accurate but produce a large number for the volume in mmsmall>3. Centimetres are a more sensible choice as the unit of volume, cmsmall>3, is more commonly used.
Procedurally focused students are less likely to use the relationships between radius, diameter and circumference of a circle, or the ratios of side lengths for standard paper sizes.
The conceptual approach (show more)
- The student applies relationships among the radius, diameter, and circumference of a circle, between side lengths of standard paper sizes, and conversions between units of volume.
Students who adopt a conceptual approach are likely use the given measurements of A4 paper to calculate the volumes without needing to make the cylinders. In doing so they recognise that the side length used to make the circumference of the circular surface must be divided by 2π to obtain the radius of the cycle.Click on the image to enlarge it. Click again to close.
Cubic millimetres are awkward units for volume, so students may convert the measures to cubic centimetres.Click on the image to enlarge it. Click again to close.
Algebra can also be used to prove that for the volumes of cylinders to be equal the length and width of the rectangle must be equal, i.e. the rectangle is a square.Click on the image to enlarge it. Click again to close.