Following on from the success of TI Australia’s Online Student Revision Tutorials last year, as well as the success of this blog site, TI Australia has launched another blog specifically for Australian students using TI technology. Named the TI Australia Student Hang Out, it can be found at: www.tistudents.wordpress.com . It contains links for students to register for upcoming tutorial events (as have also been notified in this blog) as well as direct links to recordings of past exam preparation tutorials and instructional videos for students.

Tutorials are presented by T-Cubed Instructors who are teaching the associated courses. It’s a wonderful thing that these highly skilled teachers can share their expertise with students beyond just their own classrooms. Just in the past week, the “Get ahead of the Game” online tutorial series has reached hundreds of students, with those attending the live events actively utilising the question/answer and chat facilities.

Students have expressed gratitude in the post-event feedback surveys and suggestions from them and their teachers have been helpful in setting direction for what next to include in this program. If you have ideas for further development, please enter them in the comments section here. Of course, students are also encouraged to utilise their Hang Out blog to share comments, recommendations and ideas.

Two members of the Australian T-Cubed Instructor team have been invited to conduct sessions at his year’s Instructor Training Day in Dallas on 12^{th} March this year. The training day is only open to accredited T-Cubed trainers and immediately precedes the three-day T-Cubed International conference, which is predominantly for teachers.

Jody Crothers of Ridge View Secondary College, Perth, and John Bament of O’Loughlin Catholic College, Darwin, have both presented at the teachers’ conference before, but this will be their first for the Instructor Day. In fact, it is very rare for the Texas-based organisers to invite international presenters for this day. The (couple of hundred) T-Cubed instructors in attendance are mostly from the USA. These are the elite of technology-skilled math & science teachers. To be invited to facilitate further skills development for these people is indeed an honour and a recognition of the world-leading standard of our Trainer team here in Australia.

Australian teachers may have caught Jody or John in action at state conferences in recent months. Jody presented a couple of sessions at the MAWA Annual conference. His workshop on TI-Rover was especially well-received. He is quite rightly acknowledged as a world authority on Rover and leads a lot of the discussion on the TI-Innovator and Rover Google Group. Teachers might also like to check Jody’s Twitter feed https://twitter.com/jodstar2000 for updates from his classroom.

If you saw Jody at the MAV conference, again he ran a couple of sessions with Rover. You may also have found him at the Texas Instruments display stand where he invested a lot of his conference time at the Rover “play space”.

John Bament was also at the MAV conference. You may have seen him at TI’s STEM activity table. He set up and hosted this activity stand at the SASTA/MASA STEM conference (29 Nov 2019) as well as the LEC Preston conference (4 Dec 2019). John is a regular host of TI Australia’s Webinar Program as well as the brains and voice behind many of the videos to be found on TI Australia’s YouTube channel. You particularly might like to check out John’s video on String Art.

John’s session at the T-Cubed Instructor Training Day will be Tell the World – Many T3 instructors are doing something cool or amazing in their classroom or workshop on a regular basis, but who knows that you are? Why not tell the world. How do they share this with the world? In this practical workshop you will see how easy it is to create a video; using your phone, webcam and document camera and how to share it with your students, peers, parents and the world!

Jody’s session is Working with Rover Down Under – Come see some of the innovative things we are trying with Rover in Australia. We will cover our motion match activity and others!

A delegation of thirteen Australian T-Cubed Trainers will be making the trip to Dallas for this year’s International Conference.

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: p-fox@ti.com

Instructions:

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. (Refer presentation)

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 …

Magic Squares

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.

Example: 35^{2}

(a + b) (a – b) = a^{2} – b^{2} Let a = 35 and b = 5

(35 + 5)(35 – 5) = 35^{2} – 5^{2} Substitute

40 x 30 = 35^{2} – 25 Simple calculation 40 x 30 = 1200

1200 + 25 = 35^{2}.

Perfect Squares can also help, particularly for numbers such as 31, 41, 51 …

Example: 31^{2}

(a + b)(a + b) = a^{2} + 2ab + b^{2} Let a = 30 and b = 1

(30+1)(30+1) = 30^{2} + 2x30x1 Substitute

31^{2} = 900 + 60 + 1 = 961

With a little practice it is possible to square numbers such as 126.

Example: 126^{2}

(125 + 1)(125 + 1) = 125^{2} + 2 x 125 x 1 + 1^{2}

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.

Conclusion

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!

For many of us, Term 4 is the busiest; with exams, reports, step-up and transition classes as well as the various other end-of-year activities and commencement planning for the following year. It is also the time that several state maths associations hold their annual conference and the Teachers Teaching with Technology team host a range of Learn, Energise, Connect PD events.

For me, this involved the preparation and presentation of eight sessions across conferences in Perth (MAWA), Brisbane (LEC) and Melbourne (LEC & MAV). Although the online PD of webinars has become a popular and effective medium for our learning & communication, there is still a unique value in the collegiality and direct connection of face-to-face, hands-on conference workshops. All four of the events that I attended, were well-organised and enthusiastically embraced by the teachers that joined us.

This blog has already published two papers that were delivered at the MAWA conference by T-Cubed trainer, Peter Fox. See Lessons from CASand Mathemagicians Exposed. I will also gradually add some articles from the sessions that I conducted: Crashing Robot Cars for Simultaneous Linear Equations, TI-84PlusCE™ Fundamentals, TI-84Plus CE™ – Know your Limits, Efficient and Effective use of TI-84PlusCE™ in the Mathematical Methods and Specialist Mathematics Exams, Widgets for GM & FM Exams, Mathematical Investigations for the new VCE structure and Tips for using TI-Nspire effectively in Further Mathematics.

I invite contributions from any other trainers who presented at these events. Please either send me an article for publishing or contribute to the discussion via the comments section.

Meanwhile I leave you with a collage of photos gathered from the gatherings and wish all readers a relaxing, healthy & enjoyable holiday.

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:

Trivial

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.

A few years ago I received this photo from my cousin, Doug,
who at the time was living at Casey Station, Antarctica. Doug is a
communications technician with the Australian Antarctic Division. Over the past
decade he has mostly lived and worked at one or another of the four A.A.D. research
bases. This year he is at Macquarie Island. He sends a weekly newsletter,
adorned with magnificent photos of pristine icy landscapes, extraordinary
wildlife and colourful and mind-blowing captures of the night sky that include
Aroura Australis and time-lapse star-trails like the photograph here.

I find all his stories and photography amazing, but the mathematician in me drew special interest in this South Pole time-lapse specimen. There is no set point in the picture that identifies exactly where the south celestial pole (SCP) lies, but it’s not too difficult to pose a guess when considering the circle centre from all these arcs. The arcs themselves are quite interesting, because they are the actual stars, with their movement recorded over a slice of time. From our perspective, it would seem that any given star would complete a full circle around the SCP in 24 hours. However, we know that what’s really rotating is our Earth against the fixed background of stars. Here then comes the obvious maths problem: For how long was the camera shutter open in order to take this photo?

Here’s how I did it:

I started by importing the photo to a Graphs
page of my TI-Nspire CAS.

I then moved the axes to place the origin at
where I estimated the SCP to be.

Next I selected one of the star-trails. You can
see that my selection is towards the lower-left of the photo, but really any of
them would do.

I graphed f1(x)=1 and then used the
line-rotation feature until the line ran across one end of the star-trail arc
(blue line).

I graphed f2(x)=1 and used the line-rotation
feature again until the line ran across the other end of the star-trail arc (red
line).

The CAS indicates the equations of the two lines
and I extracted the gradient of each into a Notes page.

Continuing on the Notes page I used the two
gradients and a known formula to compute the angle of the arc. (Note that my
document is set to degrees)

This angle as a fraction of 360 degrees equates
to the exposure time as a fraction of a full day. I have made my calculation of
the exposure time in seconds further down the Notes page. (1593 s in this
example – about 26.5 mins)

And now the best part of it all: I can check my result another way! Remember that my cousin Doug had sent me this digital photo, which as a JPG file, I can not only view it, import it into calculator software, but I also have the original metadata. Here is the details page for the file. Note that among bountiful information, the exposure time is indicated: 1652 s. My calculation was but 59 s out.

Want to see more classroom resources from the deep south? Classroom Antarctica is a comprehensive online teaching resource produced by the Australian Antarctic Division, with lesson plans aimed at grades 3 to 8. Ideas contained in Classroom Antarctica will stimulate your students’ interest in real-world applications for science, mathematics and studies of society and environment, inspiring and engaging your students in learning.