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.
During the late 1990’s Fox studios approached a Las Vegas magician, Val Valentino, with a proposal for a new television series. The planned shows would expose the secrets behind popular magician’s tricks. Magicians follow a ‘code’, pledging never to reveal the secrets behind their magic, however Valentino agreed and soon became the ‘masked magician’.
At the conclusion to the series Valentino explained his motivation, “…revealing the secrets will encourage kids to try magic rather than discouraging them”. While the resurgence of interest in magic over recent years is more likely associated with the Harry Potter phenomenon than Valentino, the series was extremely popular and is often repeated.
Despite their exposure, magicians continue to amaze audiences with physical and optical illusions. A parallel group of magicians now entertain audiences creating illusions of computational wonderment, they’re called Mathemagicians. Just like their theatrical cousins, Mathemagicians have a number of techniques, or should that be techniques of number? Some of the Mathemagicians’ routines incorporate a mentalist’s persona, an entertainer whose performance appears to be based on psychic abilities. It is this type of illusion that engages students most immediately and intensely. Phenomenally fast mental computation is the other art of the Mathemagician. Art Benjamin is a wonderful Mathemagician exchanging sleight of hand for cerebral gymnastics in the name of entertainment equating him to a human calculator.
Treating students to examples of mathematical – magic can increase motivation and engagement. Only a handful of individuals have the ability to perform at the level of a Mathemagician; however most students enjoy learning some of these illusions so they may perform these tricks on their friends and family. Most of the routines involve basic number theory or a combination of numerical patterns that can be explained using basic algebra.
The secret of nines
As an educational consultant I spend many hours working in different schools. On one occasion I was working with a school helping introduce Computer Algebra Systems. (CAS) In the morning I worked with a class doing some mathematical magic, creating the illusion that I could read the student’s mind, the mentalist persona. After several examples the sceptics began to question the illusion and soon discovered a pattern. We discussed the pattern, formulated a conjecture and then proved the result algebraically. This magic trick is a lesson on algebraic representation, expanding and simplifying.
One of the teachers observing the lesson asked if I could do the same with her class later in the day. That afternoon I repeated the lesson as requested. Students enjoyed the first ‘illusion’ but we didn’t need to repeat it. One of the students in the class ‘exposed’ the illusion. The classroom teacher was disappointed; I proceeded to question the student. The student admitted that they hadn’t figured it out for themselves; they had been discussing the illusion at lunchtime with their friends. Initially their classroom teacher was annoyed, until I asked “how often do your students talk about mathematics at lunchtime?” Students are more likely to remember or develop a deeper understanding when they discuss their learning with others. “Thought is not merely expressed in words, but comes into existence through them.” [Vygotsky 1996]
Mathematics is one of the few subjects that require students to engage in pure thought. Student objection is often verbalised through the familiar challenge “where are we going to use this?” Teachers often struggle to find an immediate purpose or context. According to Piaget, students learn new concepts through assimilation and accommodation. Many junior level secondary students are still at the concrete learning stage, these abstract concepts prove challenging and therefore demand a higher degree of motivation. Motivation is provided through the mathematical illusions. Students ask “How do you do it?” Many of them want to learn so that they can perform the same illusions on friends and family. Students are more likely to learn when they have a purpose.
One of the most valuable questions asked by students during such lessons is: “Does it always work?” Number tricks generally involve too many possibilities, testing them all would be time consuming. The necessity and power of algebraic proof comes to light.
The illusion used with the students appears on many web sites. The same illusion used in the lesson (and demonstrated in the Mathemagicians workshop) can be created using a series of PowerPoint slides. For a copy of these slides email: firstname.lastname@example.org
Step 1: Select a two digit number. Example: 65
Step 2: Add the two digits together. 6 + 5 = 11
Step 3: Subtract this quantity from the original number. 65 -11 = 54
Step 4: Locate the result on a page containing a mixture of numbers and apparently random adjacent images.
Step 5: The Mathemagician correctly identifies the image the student has selected. Note that it is not the number the student selected, only the adjacent image.
The images adjacent to the numbers are not completely random; at least every image adjacent to a multiple of 9 is not random. Once students have identified the pattern, they develop a conjecture:
“I believe all the answers are multiples of 9.”
The objective is to prove their conjecture. The proof requires some relatively simple algebra and illustrates how algebra can be used for proof and generalisation. Too much of the algebra taught in schools relates to algorithmic manipulation, simplification and procedural recall through rote. Rote learning, by definition is learning without understanding.
Step 1: Select a two digit number. 10a + b (Where a and b are single digit values)
Step 2: Add the two digits together. a + b
Step 3: Subtract this quantity from the original number. 10a + b – (a + b) = 9a
The result appears straight forward; however it is quite likely many students will not fully understand the significance without testing the result numerically. 9 x 1 = 9, 9 x 2 = 18 …
The inspiration for this problem comes from the American Mathematics Teachers Journal. The original problem is done by hand; the electronic and wireless medium of TI-Navigator makes the problem significantly more impressive. Students are asked to generate 8 random numbers. It is suggested to keep the numbers relatively simple as some mental computations will be required later, “perhaps restrict your numbers to two digit numbers”.
Students enter the random numbers on the calculator and then type the word “Magic”. The result is a matrix (which has been predefined) that appears to contain a selection of random numbers with little or no apparent relationship with the original random numbers. A sample of the magic square created is shown below:
Ask students to add the numbers in the magic square; “don’t use the calculator or let the calculator know which numbers you selected”. The teacher then tells the students: “I calculated everyone’s answers and sent it to your calculator. Write the word answer on your calculator.” Students are amazed that whilst they have different answers to each other, the calculator knows the answer they computed. It is left to the reader to figure out the secret behind this trick! The trick is suitable for students in junior secondary.
Developing Human Calculators – Perfect Squares and Difference of Perfect Squares
There are many ways to calculate the product of two numbers and even more for squaring numbers. Students are generally taught using traditional methods; however there are many other ways this can be done both visually and mentally.
The diagram shown opposite illustrates how drawing lines to represent the tens and units values can result in a very simple approach to multiplication. Vedic mathematics techniques can also be employed (Workshop demonstration). These techniques can also be validated using simple algebra.
Sample Vedic Mathematics result: (Reference number = 100)
98 x 96 100 – 98 = 2 & 100 – 96 = 4
2 x 4 = 8
94 x 100 = 9400 96 – 2 = 98 – 4 = 94
98 x 96 = 9408. 9400 + 8 = 9408
An algebraic proof of the above approach is left to the reader. The use of CAS means that students can focus on producing a ‘general solution’ rather than algebraic manipulation.
Difference of perfect squares can be applied to squaring numbers such as 15, 25, 35 … A wonderfully simple paper folding activity can help students remember how ‘difference of perfect squares’ works, then supported with the algebraic approach and applications to number.
(a + b) (a – b) = a2 – b2 Let a = 35 and b = 5
(35 + 5)(35 – 5) = 352 – 52 Substitute
40 x 30 = 352 – 25 Simple calculation 40 x 30 = 1200
1200 + 25 = 352.
Perfect Squares can also help, particularly for numbers such as 31, 41, 51 …
(a + b)(a + b) = a2 + 2ab + b2 Let a = 30 and b = 1
(30+1)(30+1) = 302 + 2x30x1 Substitute
312 = 900 + 60 + 1 = 961
With a little practice it is possible to square numbers such as 126.
(125 + 1)(125 + 1) = 1252 + 2 x 125 x 1 + 12
[Note: 1252 = (125 – 5)(125 + 5) + 25 = 120 x 130 + 25]
1262 = 15625 + 250 + 1 = 15,876
It may seem complicated to apply different approaches to multiplication problems when we have a ‘one algorithm suits all’ solution already. However, students already have valid techniques to calculate problems such as: 100 – 98 and for slightly more advanced students: 112 – 98. Think about the way that you calculate the answer. Asking students to describe their thinking, metacognition, when solving problems such as this is a very powerful learning tool in itself.
There are many more magic tricks that involve mathematics. There are card tricks which revolve around numbers with different bases and others that rely on divisibility tricks. Try a couple with your students and challenge them to learn the tricks to try on their friends and family. Students remember the mathematics as it is associated with the thing they want to remember, the magic trick. Of course it is all just a trick to get students to engage in mathematics!
Introducing Computer Algebra Systems (CAS) into mathematics classrooms is a lot more involved that simply identifying what buttons to press and what platform to use. Enormous pedagogical and paradigm shifts may need to take place. It can also be a challenging time for students, depending on the stage of their mathematical journey. Parents may also contribute to the equation. Then there’s assessment! This document refers to aspects of an 18 year journey through the continued introduction of CAS at the student, institutional, state and national levels. Specific interactions with students and educators will be used as waypoints along this journey.
Cassie’s Story – 2003 [Year 9: 14 to 15 years of age]
The first topic was Pythagoras’s theorem. Cassie didn’t complete the problem solving task, investigation and performed poorly on her test. The second topic was algebra, solving linear equations. It was very clear that Cassie was already in trouble. Speaking with Cassie directly, she mentioned that she had failed mathematics in Primary school (Elementary), Year 7 and Year 8. “I’m just not good at maths” was her self-diagnosis. Fortunately I had a TI-92Plus calculator loaded with the Symbolic Maths Guide. I generated some questions, showed her how to use the calculator and left it with her. Cassie worked solidly for the remainder of the lesson. She was happy that she could ‘solve’ the problems, but commented “But I still won’t be able to do the test”. I advised Cassie that she could use the calculator on her test and in fact she could use the same technology on her state level exams should she do mathematics in Year 12. Other students were intrigued and commented that using this calculator was ‘cheating’, evidence of how students perceive mathematics already at this level.
“What’s the first step you would do Cassie”, I asked. “You could multiply both sides by 4” she replied. Cassie worked through the problem step by step as I (the teacher) worked just like the calculator. Cassie solved the problem much to the dismay of her fellow students. Cassie went on to score more than 90% in her test, the first time she has passed a mathematics test in more than 5 years! She tackled mathematics with a new found confidence and optimism.
CAS can be used to help support student learning by shifting focus and providing supportive scaffolds that catch incorrect algebraic manipulation.
Chelsea’s Story – 2003 [Year 11: 16 to 17 years of age]
Four of the girls in this class would always sit together of which Chelsea was one. The group had been in the same class for the past four years and were all very studious. They were happy to do every question from the textbook and their exercise books were immaculate. These girls seemed to derive pleasure from getting questions correct and essentially racing one another through each exercise. The girls however were part of one of the first “CAS enabled” senior mathematics classes. Some of the students in the class did not like using the CAS, it was ‘cheating’, initially, Chelsea held this belief. As the course continued students started learning calculus. Most classes would start with a ‘problem to solve’, often an opportunity to discuss important concepts.
Students started working through the problem, books open, calculators on and heads down. Chelsea was just sitting there, staring at the board where the question was written. I asked Chelsea “Do you need some help?” “No, I’m done” she replied. I could see her book wasn’t open and her calculator wasn’t even switched on. “Okay, can you show everyone how you solved the problem?” “Sure” said Chelsea. She took her place at the board, drew a well labelled diagram, as you would expect from Chelsea.
Then she proceeded to announce: “I’ve drawn a line between the origin and a point P (x, y) on the curve. I would use the distance formula to work out an equation involving d, x and y; then substitute the equation for y. Now I have an equation involving just d and x which I’ll define as d(x). Now I can differentiate, as I need to find the minimum value for d, I’ll solve the derivative equal to zero; that would give me a value for x. Now I can substitute that value into d(x) to get the distance.” I was impressed! “So, what did you get for your answer?” I asked. “Do you want me to press the buttons?” she replied.
You can change student’s perceptions of mathematics.
Lizzie’s Story – 2004 [Year 12: 16 to 17 years of age]
A delightful mathematics problem involves paper folding. An A4 piece of paper oriented in ‘landscape’ mode is folded such that the corner of the page just touches the base of the page forming a triangular region.(Shown) Determine the location of the fold and the corresponding maximum area of the triangle.
The physical act of folding the paper and collecting the data is a great way to start this problem. Students generally let the ‘height’ of the triangle be represented by x. They can see that the hypotenuse of the triangle is therefore 21 – x. This gives them two sides of a right angled triangle so they are able to generate an expression for the base of the triangle and subsequently an expression for the area in terms of x. Using calculus students find that the maximum area is generated when x = 7.
Lizzie was very astute and asked “Will this always be the case? The optimum height is 1/3 of the height of the paper.” The beauty of CAS in this case is that students can work through the same problem but swap out the 21 (page height) for h and see the solution occurs when x = h/3 . For me, this is a beautiful relationship. Lizzie however followed up with the question: “Why?” “What do you mean?” I asked. Lizzie continued, I can see that now, but why? She could see the general solution. Even when the shape of the original paper changed, Lizzie could see that, within certain limitations, the optimal solution would be when the x = 1/3 the height of the paper. She wanted to know, why 1/3?
When you focus on conceptual understanding, students can ask some really tough questions!
Robbie’s Story – 2005 [Year 12: 17 to 18 years of age]
Dynamic environments provide for some amazing visuals that include the option for students to explore, collect data and generalise. One of the most common introductory calculus questions is identifying the maximum volume of a box cut from a rectangular piece of card.
Students were provided with pieces of card, scissors and rulers. As students created their open boxes they would measure the dimensions and volume and write their results on the board.
Robbie asked the question: “What height would you like me to measure?” I didn’t really understand the question; I thought it was obvious; the box has an open top. “Place your box on the table, with the open section facing up and measure the height. Make sure you ruler starts at zero or you will have to move your box to the edge of the table.” Robbie had already done this apparently, but didn’t know which ‘height’ to measure. I went over to look at Robbie’s box, he had cut rectangles from the corners of his original card; the ends of his box were higher than the sides! I sorted this out and made some mental notes. In subsequent years I would give students some dotted lines to cut along so they didn’t make the same mistake. I later decided it was okay if students made this mistake. What if they didn’t understand the seemingly obvious requirements for the original cut out?
It’s okay to make mistakes; it’s part of the learning process. Sometime dynamic diagrams can deprive students of the opportunity to make mistakes!
Teacher Development – 2006 [National Teacher Training]
I was providing teacher training in a country where they were piloting CAS. This particular country decided that they would introduce CAS at the start of student’s high school journey. The justification was based on a number of factors, mostly on the notion that it would not interfere with ‘high stakes’ exams, therefore teachers would be more likely to take ‘risks’ and be willing to explore. One of the problems was that many of the teachers involved in the pilot were not mathematics trained! The shortage of mathematics teachers in the education system resulted in a lot of out of field teachers taking junior level mathematics classes. These teachers could step students through algorithms, follow textbook examples and mark student work, but they didn’t really have a strong understanding of mathematics.
Activities were written to assist teachers, but teachers struggled themselves with some of the content. For example, a ‘traditional’ approach to helping students build algebraic expressions is to provide them with blocks, build specific constructions, record the quantity of blocks and identify a rule relating to the number of blocks and the build.
Problems similar to the ones above are included in textbooks and are relatively straight forward. Students write rules relating the number of blocks (squares) to the pattern number and then use the rule to predict how many blocks are required for pattern 100. What happens when you reverse this scenario?
Some teachers struggled with relatively simple variations. More conceptually demanding questions posed an even greater challenge for the teachers to solve.
Teacher’s knowledge of mathematics needs to extend beyond procedural.
Teacher Development – 2007 [Assessment Items]
As the pilot study continued, some national assessment items were being compiled. I worked with teachers and the relevant curriculum authority to help write examination questions. Teachers were struggling to write assessment items that could not be trivialised through the use of CAS. I suggested teachers look through past examinations and identify any questions that required ‘something more’ than an algorithm to solve, something that challenged conceptual understanding. Three years of papers and nothing surfaced, other than the realisation that assessment had focused very much on procedural knowledge rather than mathematical understanding.
Understanding this change is very challenging. Even textbook companies have struggled with the idea of CAS. One textbook company tried to encourage me (my school) to ‘update’ to their latest version of their textbook. I sat down with the company representative and we checked over the questions. Previous question items were all the same as their new book (CAS enabled). “What is the difference?” I asked. “We’ve put screen shots and calculator instructions in the new series” was the response. Some questions that were previously instructed to solve ‘by hand’ now stated ‘use CAS’; but the question remained the same.
A range of categories exist for classifying questions in a CAS enabled environment:
- CAS enabled
- CAS neutral
Trivial questions are those where a student should be able to solve the problem without CAS or that CAS provides no benefit due to the simplicity of the question. CAS enabled questions are such that appropriate use of CAS provides a significant benefit or makes them more accessible. CAS neutral means that the CAS provides no benefit as attempts to ‘solve’ reveal little or no information about the solution.
Assessment in CAS enabled environments forces us to reflect more on the actual purpose of a specific question. What is it trying to measure? CAS may be driving higher quality assessment items.
Martin’s Story – 2015 [Mathematical Accuracy]
The very nature of computer input is that it should avoid ambiguity. Similarly, the output is generally very precise. CAS can act as a mathematical litmus test.
These are just some of the stories that have contributed to my experience with CAS. No doubt there is much more to learn. One of the biggest challenges is to ensure that the learning continues and permeates into all classrooms so that it becomes a transformative tool rather than a misused one.