Q When tutoring a Year 11 pupil and revising 3D shapes, I created this diagram, asking her to ring the prisms.
She didn't select the cylinder so I deducted a mark. She then told me that her teacher had said that a cylinder wasn't a prism as it didn't have polygons at its ends. I have always taught that a cylinder is a prism, which makes sense when you consider that the volume of a prism is found by multiplying the area of the end section by the length, which is how you also find the volume of a cylinder. I then checked my maths dictionary by Brian Bolt and David Hobbs from the school of education in Exeter, published by Cambridge. It says: "a prism is a solid shape whose cross-sections parallel to the end are all identical." I hope you don't think my question too trivial - I felt very embarrassed that I didn't know how to reply correctly. Help please!
A One mark can be the difference between a pass or a fail, so your question is not trivial.
There was a discussion a while ago on The TES maths forum: type "polygon" in the search box. I was always taught, and have taught myself, that a cylinder is a prism with circular ends. I decided to research further for definitions.
The Qualifications and Curriculum Authority's glossary for teachers defines a prism as: "A solid bounded by two congruent polygons that are parallel (the bases) and parallelograms (lateral faces) formed by joining the corresponding vertices of the polygons." If the ends are polygonal, then the questions that follow are "are circles a kind of polygon?" and "what is a polygon?" Before answering these questions I searched further.
The BBC website says: "Any shape that has the same shape and area of cross-section all the way through is a prism. Therefore, a cylinder is a type of prism, as is a cuboid. A cylinder is a prism with a circular cross-section. It is a very simple object and can be cut (for mathematical purposes) into three polygons: two circles and a rectangle wrapped around them." www.bbc.co.ukdnah2g2A533189 I emailed the examination boards. OCR replied: "The maths team tell me the answer is yes - a cylinder is defined as a prism at OCR, one of the 'big three' awarding bodies."
Bob Childs, mathematical officer for the Welsh Joint Education Committee, replied: "The formal definition of a prism requires the shape of the cross-section to be polygonal. However, we consider a cylinder, for example, to be a special case as the number of sides approaches infinity.
The inter-board formula list for GCSE Maths has, in fact, a diagram of a solid with a uniform cross-section in the shape of a general curve to illustrate a prism when the formula for its volume is given."
The principal examiner for the higher papers at Edexcel replied: "a prism is a shape which has to have a constant cross-section. So on that idea a cylinder is a prism. The crucial discussion point is whether the cross-section has to be a polygon. In general, I believe that the accepted definition is that you have to have a polygon - in which case a cylinder is not a prism. The cylinder can be thought of as a limiting case of a prism with a cross section which is a regular polygon with n sides as n tends to infinity."
The word polygon is derived from Greek, "poly" meaning many and "gonia" meaning angles. In fact "gonia" it seems itself came from the Greek word "gonu", which means knee. So a polygon is really a many-angled shape. An angle is the measure of the amount of turn between two line segments where they are fixed at one end on a common point. So by this definition a circle is not a polygon in the true sense of the word. The definition of a circle is that the locus of all points is equidistant from a central point. To really understand the implications of limiting values a higher level of maths has to be studied.
In your example, I would show students the various definitions, just to let them know that your definition has been qualified by the awarding bodies.
Deeper understanding of definitions and their implications comes at A-level maths and at university.
l I thought this looked interesting, making unusual barrelsvases and calculating their volumes as an extension for bright pupils: www.math.nmsu.edubreakingawayLessonsbarrels_casks_and_flasksBarrels_Cask s.html Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.
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