Prism Volumes TN
The cross-section measurements of the right-angled, triangular faced prisms are 3 by 4 cm, 6 by 4cm and 6 by 8cm. The largest is therefore four times the volume of the smallest.
The triangular 3 by 4 cm right-angled triangular prism has the same area and hence volume (all prisms have a depth/height of 6cm) as the trapezium, isosceles triangular prism and the obtuse scalene triangular prism. The area of the parallelogram, cuboid and cylinder are twice the volume of the others.
This allows the teacher to reinforce why the area of an obtuse scalene triangular prism is half that of the surrounding parallelogram (match them together and students can see for themselves that one is half the other!).
Why the trapezium is half the area of the volume (match them together and show students that if a second one was rotated onto the first two trapeziums would have the same area as the parallelogram!) and that the parallelogram has the same area as the cuboid (match them together to show that if they cut off a right-triangle and rotated onto the other end a parallelogram transforms into a rectangle!).
Short cuts to preparing your class set of Prisms
It is essential that you get your class doing the preparation work for you! Print out the templates provided onto card ready for your students to cut out and fold together in a prior lesson. As well as saving you hours of preparation time it provides a great resource for revision/learning of nets and surface area! Set the challenge (judgement by class vote and rewards for the winning pair) as to who can produce the most accurate final product. Make sure you have one or two students laminating the nets as students finish folding them. Once made, the prisms should last at least a couple of years or more (using sand will extend the life of your prisms to 3 or more years!). Read the “practical hints and tips” section below before presenting this task to your classes.
I'd also recommend investing in a professional, plastic, transparent set of relational 3D shapes. Safe-T products has a good range of options.
I haven't yet found a set of commercially made relational prisms that reinforce the relationship between cross-sectional area and volume (and revision of the link between the area of a triangle, parallelogram, trapezium and rectangle) so that students can discover this relationship, hence formula, for themselves (please email me if you know of a supplier).
Moreover, a demonstration set is great, but what I really want is a class set so that students can get hands-on, memorable experiences of the concepts and relations. Buying a enough sets of commercially made 3D shapes can be a barrier to access for some/many schools.
Practical Hints and Tips
I have purposefully not made these models too large so that the cost implications for use in school are not prohibitive. Of course, there is nothing to stop you blowing up the templates to A3 size or larger – maybe to make a giant, interactive display in your classroom or at the school’s reception for “Maths Week” . . .
. . is easier to use than water – provided you have enough trays (book well in advance with the science department) so that all sand stays in the tray at all times! A hoover will fix it if not! SAND also preserves the laminated prisms for longer as water will eventually find the gap in between the laminated faces and begin wetting the card.
Smaller scissors are better than larger ones.
Precision is very important to the success of these solids. The templates are mm accurate so that the relationships between them are demonstrable through practical experiment. It really helps student’s accuracy and easy of folding if they place their rulers along the fold lines and score the card with a pair of scissors before trying to fold each face, and the holding flaps, into place. They should laminate the card after having made these folds.
These should be placed on the outside of the main faces so as to not influence the volume of each prism.
Sticky tape is very useful for covering any small gaps that may remain. In my experience sand rarely flows through any tiny gaps that may remain, whereas when using water, students need to work fast not too lose any.
These are the hardest to put together. Students should tape the top circle to the curved surface little by little to get the right curvature and fit the faces precisely.
Displays take time, but they’re worth it! They provide a stimulating environment within which students are constantly reviewing previous knowledge and making new links and connections. Create a giant set with a huge sand pit at the front of reception for parents and students to experiment and investigate with, hang the solids on strings from your ceiling to catch students attention as they come into class. Create a pulley system (using screw hooks) to lower them to the floor as and when required!
A Mathematically very rich activity
This activity is great for going off in different directions: Pythagoras, scale factors of enlargement – lengths, area, volume, how do multiples and factors provide a useful strategy for finding numerous solutions etc. It would be great to hear about any inspiring/exciting moments, conversations, questions you have whilst working with your students: firstname.lastname@example.org.