Category Archives: VCE
Roller Coasters provide a wonderful context for mathematical modelling. The actual construction of a roller coaster is a marvellous example of engineering, mathematics and physics. From a technology aspect students can get visual feedback by placing an image in the background of the graphing screen. Students can build an understanding of how the degree of a polynomial determines features such as the quantity of turning points, the nature of odd and even degree polynomials and curvature.
In the example shown here a quartic function has been used to represent a section of the roller coaster. The x-axis was aligned approximately with the horizon and the image scaled using known information. The turning points on the function were used to model the peaks and troughs of the track which provided no freedom to adjust the curvature to align the remaining sections, even those within the domain of the model. Students are invited to consider other functions, including piecewise functions that may produce a more appropriate model with greater flexibility.
If an appropriate scale is set, students can also include calculations of:
– Ride length
– Gravitational potential energy
– Kinetic energy
Students can also explore how concavity affects the ride experience. For an easier starting context, students can consider water slides that typically have downhill sections only. There are numerous water slides around the world worth studying. “Summit Plummet” at Disney’s Blizzard theme park in Florida is a relatively simple design. Consisting of a starting height of approximately 40 metres and an almost vertical drop, riders can achieve speeds of approximately 100km/h. The radar located at the base of the drop displays rider speeds and boasts the highest speed of the day. The following curves are all modelled on a drop of 40 metres over a horizontal distance of 40 meters. The blue line on each graph represents the shape of the water slide; the red dotted line is the speed at each point on the slide, assuming zero resistance.
It is clear to see that curvature has an enormous impact on the average speed. As we have assumed that there is no resistance to the rider’s motion, the maximum speed would be the same for each ride. The velocity is calculated based on the assumption that all the gravitational potential energy is transferred to kinetic energy, when friction is taken into account this is not true. It is easy to incorporate resistance into these types of models, particularly where a program is being used. A very simplistic approach would be to use:
In reality there are many complications that include the volume of water flowing, the weight and dimensions of the rider, the type and area of the bathing costume (nylon has a lower drag) and indeed, the amount of sunscreen the rider is wearing! It is interesting to note that increasing the volume of water slows down heavier riders but accelerates lighter riders, so the water acts as a moderating effect. Rider speeds on Summit Plummet typically vary between 80km/h and 110km/h. When piecewise functions are used to model any of these types of rides, students need to ensure that curves are joined smoothly or at least understand that this presents limitations to their model. Even if students have not started calculus, they should already understand the concept of gradient, technology can be used to determine the gradient where two curves meet.
Wine glasses come in a range of shapes and sizes depending on their purpose. The tall stem is so that you don’t have to place your hands on the body of the glass, this is to avoid heating the wine with your hands. Red wine glasses tend to have a wider body to help the wine breathe and a relatively narrow rim to allow the concentrated aromatic flavours to rise and tantalise the pallet. The glass must also be designed to hold a standard drink (175ml for wine) and preferably at the widest point of the body. The rim to body area ratio should be around 1 : 1.5. Finally, stability plays a role in the design of the glass. If the stem is too long or the base too small the glass will be unstable and tip over too easily. There is a lot more to designing a good wine glass than first meets the eye. Considering some bottles of wine sell for $100.00’s and more, it makes sense to design the perfect glass from which to serve the wine. Students can design the wine glass to meet all the specifications, starting with a standard wine glass and using functions to model the curvature. To ensure the curves used to model the glass profile are functions, tip the glass on its side.
– Standard drink = 175ml (wine) – This should occur at the widest section of the
glass (turning point)
– Ratio between y ordinates at the turning points and top of the glass can be
used to help match the rim to body ratio.
– What is the ideal stem length? (Now it’s a STEM activity!) A piecewise function
could be used to ensure the stem and vessel curves join smoothly.
– To increase the complexity of the task the thickness of the glass could also be
included in the modelling. Is it okay to simply translate the function modelling
the outside of the glass or does this pose a problem with regards to glass thickness?
A range of calculus concepts can be dealt with in this task. If you are really adventurous, try using a 3D printer to produce a real
model of your perfect wine glass.
FREE support videos to help prepare for VCE Exams.
Short focused revision in each video set that best reflects each course outline and specified outcomes.
Quick and efficient use of your TI Nspire™ CAS Calculator can contribute to exam success.
Download sample TI-Nspire files that accompany many of the videos.
To encourage more regional mathematics educators to attend the MAV19 Annual conference, MAV has sought sponsorship from Texas Instruments to bring regional secondary teachers this fantastic opportunity for 3 days of PD!
We understand that it is a challenge for many regional teachers to attend events in Melbourne based on the additional cost for travel and accommodation.
Applications close Tuesday 8th October, with successful applicants advised by Monday 14th October. Download the application form here
TI is sponsoring the following for each of 5 successful teachers:
- Registration to LEC PD Day at Mantra Bell City on Wed 4th Dec
- Invitation to TI VIP Dinner on Wednesday night.
- Registration for MAV19 Conference on Thursday 5th and Friday 6th December 2019
- Travel allowance for use as required ($150)
- 3 nights accommodation (Tuesday, Wednesday and Thursday) at Break Free Bell City, including breakfast.
- $25 dinner allowance for two nights ($50)
See application form for full criteria & conditions
Students are provided with a series of functions consisting of polynomials, trigonometric and exponential functions and combinations thereof. The equations represent the derivative, students use their TI-Nspire to graph the original or primitive function and relate key features of the derivative to the primitive function.
The activity bundle also includes a teacher demonstration file to help students understand the concepts involved.
Author: Brian Lannen
I have been teaching senior mathematics since 1985 and also working on a range of curriculum consultancy projects since 1995.
My teaching has been mostly in Victoria and New South Wales in both government and independent school systems and my consultancy work in Australia, USA and south east Asia. I like stepping across the boundaries and am generally keen to embrace diversity and innovation.
Across my time in this profession I have seen numerous educational initiatives and trends come and go. Some have been good, some not so good, but always interesting.
Recently my inbox pinged with the arrival of a memorandum from the Victorian Curriculum and Assessment Authority (VCAA) inviting maths teachers and interested parties to comment on a suite of proposed structural changes for the new VCE senior mathematics Study Design. This took me a bit by surprise. Didn’t we just have changes only a few years ago? I phoned a few of my colleagues. Most had missed the memo. Generally they were comfortable in having just managed to implement the more recent changes and their focus right now is more on enjoying the Easter holidays than to contemplate further curriculum upheaval. But the Stage 1 survey response is due by midday 10th May. What rolls on from there may well inform the direction of VCE mathematics for the next decade. This matters! This is why I am blogging. I can’t afford full page ads in the paper. Who reads the paper these days anyway? But we need to be informed and we need to make comment. Here is my summation of the proposed structures on offer:
Structure A (both models) is reasonably familiar, as it is along similar lines to what we have already been teaching. Structure B (as far as it has been defined) looks like something that could be partly built from current curriculum plus options (all really yet to be defined) and Structure C is a significantly restructured commercially driven Wolfram-directed computer-based model.
I’d be interested in hearing your comments, more importantly so would the VCAA. Here is the link to their survey: Click here to take the survey.