Category Archives: Article

Pole Dancing with the Stars

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:

  1. I started by importing the photo to a Graphs page of my TI-Nspire CAS.
  2. I then moved the axes to place the origin at where I estimated the SCP to be.
  3. 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.
  4. 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).
  5. 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).
  6. The CAS indicates the equations of the two lines and I extracted the gradient of each into a Notes page.
  7. 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)
  8. 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)
Photograph credit: Doug McVeigh. Australian Antarctic Division, Casey Station 2015

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.

Engineered for Education

I have witnessed and participated in in many changes in the educational landscape over my almost four decades in the business. Expressions such as flipped classroom, teacher-centred, student-centred, instructor, facilitator, coach, mentor, project-based learning, explicit teaching and more have contributed to the vocabulary used to describe this evolutionary process.

Through it all however, I believe an important constant to recognise is that teachers and students have different roles to play in education.

Subsequently, I have found the perfect technology solution for my maths classroom to be the marriage of TI-Nspire CAS software for me and handheld units for my students. OK, I also have the luxury of a Navigator system, but that’s really the icing on the cake. And sure, I’m also a big fan of the vast offerings from full computer technologies; laptops, tablets and phones with access to endless data via internet connectivity. I have reflected on this again while reading the VCE Mathematics Review and associated background papers. However, I find myself returning to the proven model of TI software/handheld partnership and conclude that the reason for this is that these are specific built-for-purpose tools rather than high-powered, but generic computing machines. It comes back to that difference in roles of teacher and student and, corresponding with that, a difference in tools. I have heard the argument of ‘why should students use a technology in their schooling that they are unlikely to use in their later working life’. The answer is because it is specifically designed for their schooling.

At every stage in the development of TI educational technologies, the engineers, designers and managers consult established educational research and seek feedback from practicing teachers. The happy tech model that I conveniently employ in my classroom is not just a lucky accident, but the result of careful design. 

For me, the TI-Nspire Premium Teacher Software is simply the best. I use it for demonstrating, generating discussion and collecting student assessment data (with the help of TI-Navigator). I also use it to distribute files to students and collect files and screen-shots back from them. Although the software is my primary tool-of-trade, I still use my handheld unit for straight out portability about the classroom. My standard lesson generally consists of some whole-class instruction (Teacher Software) followed by individual and small-group assistance about the classroom as students work on their assigned exercises (Handheld  unit). I also know that, although students primarily use their handheld units, they also utilise the student software, mostly in their home study, collecting screen-shots for their notes. I have seen this feature to be especially useful for students studying by distance education as they can screen-shot evidence of their working and paste into assignments.

I absolutely believe that the principal role of educational technology is for the enhancement of student learning. However, I also know from experience that what has become the defining factor for most schools and school systems is whatever technology is or isn’t permitted for use in senior exams. Here again, the handheld unit is the best solution (though as I have stated its educational value extends way beyond that of simply being an exam tool). Handheld calculators do NOT have student to student connectivity other than via a physical link cable. Even their capacity for Wi-Fi connectivity (TI-Navigator system) requires fitting a bright yellow Wi-Fi adaptor and communication is only between individual students and the teacher. Further yet, in jurisdictions that specify the lockdown of specified calculator functions for exams, the TI engineers have installed a ‘Press-to-Test’ mode especially for such situations. The point is that the calculators are deliberately designed for this. Computers are not. I have seen situations where schools have tried to use computers for exams – with disastrous consequences. School computer technicians have been challenged to lockdown laptops to prevent students from communicating with each other or the broader internet. The worst I’ve seen is where students tried to access online exams, with the school server collapsing under the stress and panic-stricken invigilators needing to quickly organise paper copies of the exams.

Yes, computers are more powerful machines than calculators, but they are not more powerful in this situation, because they are not primarily designed for the task. The conclusion is very simple – if you seek to incorporate a technology for education, select one that is specifically engineered for education. Want to know more?      https://education.ti.com/en-au        

Modelling Phases of the Moon and hiding planets

Queensland based T3 Instructor and hobbyist astronomer, Stephen Broderick, captured this amazing image of the occulation of Saturn last month. (Photo taken 12th August at 6:10 pm AEST)

Saturn, with its distinctive rings can be clearly seen as the Earth’s Moon is a about to hide it from view. Occulation of Saturn 12th August 2019. Photo: Stephen Broderick

For many years now, Stephen has shared his astronomical expertise with fellow maths and science teachers at various conference presentation both in Australia and abroad. His astrophotography skills are, without doubt, reaching to professional levels, yet Stephen has also demonstrated how school students and other budding amateur astronomers can easily and inexpensively start into this field. One simple example of this is pin-hole camera Solargraph photography, which Stephen has promoted in his presentations.

Stephen’s own students are most fortunate to have this expert guidance available to them and he has also developed a range of activities and record-sheets to help students discover the maths and science behind real ‘out of this-world’ study.    

See below, one such example where Stephen uses circular functions for modelling the phases of the moon:

 

Modelling the Phases of the Moon    Author: Stephen Broderick

The following data was obtained from the U.S. Naval Observatory.

https://aa.usno.navy.mil/data/docs/MoonFraction.php

Capture3

The fraction of the Moon illuminated for January 2019 (in the orange box) was plotted to obtain a sinusoidal model for the phases of the Moon. The model can be used to determine the period of the lunar phases, predict when full Moons occur throughout the year and determine when the next blue Moon occurs. (A blue Moon is when two full Moons appear in the same month)

The sinusoidal Moon data model for the month of January 2019 is graphed below in Figure 1.

               Capture2

                                                 Figure 1

The model is represented by the function: f(x) = 0.495267 sin(0.213405 x – 3.0094) + 0.469134

From the model, the period for the phases of the Moon = = 29.44 days. The accepted value is 29.55 days, so the percentage error is 0.37%. ((29.55-29.44)/29.55) x 100 =0.37%)

How good is the model? A full Moon occurs on the 19th May which is day 139. Subbing 139 into the model:

f(139) = 0.495267 sin(0.213405 x 139 – 3.0094) + 0.469134

= 0.96, which means the Moon is 96% illuminated and so it still appears as a full Moon.

In 2019, there will be 12 Full Moons and 13 New Moons. The two New Moons occur in August 2019. When will the next blue Moon occur. (A blue Moon happens when two Full Moons occur in the same month) Thirteen Full Moons occur in 2020, so 2 Full Moons occur in the same month. The month in question is September 2020.

Sponsorship for Victorian Regional Teachers to attend LEC PD Day & MAV Annual Conference 2019!

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

Download the application form here

TI industry partner for MAV Maths Camp

Twenty-four enterprising Year 10 students made the selection for this year’s STEM-focused MAV Maths Camp held in the first week of July this year. Texas Instruments joined the program as a new industry partner this year and hosted all students for a site visit on the Wednesday. Students were provided an insight into the tasks of TI team members and particularly where STEM skills & knowledge are called upon. This was especially emphasised in a video conference meeting with Harshal S Chhaya from TI’s product development team in Dallas, Texas. Harshal discussed his STEM roles within TI and students had an opportunity to ask questions about task specifics and career path options.

Harshal remarked, “I was glad to interact with these students and talk to them about my work as an engineer. They were very curious to learn about TI. I was also impressed by the questions on the importance of ethics in technology”

Following this, all students had an opportunity to work with the TI-Innovator and completed an RGB activity. Most then worked on TI-Rover activities under the guidance of Peter Fox, while T-Cubed Trainers, Shelley Cross and Karleigh Nicholls mentored a focus group.

Other industry partners for the program were Ford, Reserve Bank of Australia, RMIT University and Victorian Space Science Education Centre. Over the week, all students visited each of these industry sites for a general overview. Additionally, student teams of 4 or 5 were partnered with each of the industries to be mentored through deeper research into a particular problem or project.

The TI project team, mentored by Shelley and Karleigh, investigated the problem of pets being left in hot cars (Pet Alarm Project) and presented their product and findings to the MAV (Mathematics Association of Victoria), mentors, teachers and fellow students at the end of the week. Along the way, this group completed 10 Minutes of Code activities, researched how the colour of the car can affect the internal temperature of a car and the biological effects of heat to different dog sizes. They completed the Pet Alarm project successfully by building their individuals cars, coding the TI-Innovator to flash LED lights, sound an alarm and using the servo motor open the car windows.

T-Cubed Manager, Daisy Patsias, commented:

“It was truly wonderful to see how excited they all were with their final product. Mentors Karleigh Nicholls & Shelley Cross did a wonderful job of working with these students throughout the week. They even provided the students with extension questions to investigate before their group presentations that were held at the end of the week. Peter Fox also did a wonderful job with the balance students working with TI-Rover. The students really loved this experience. The TI mentor group presented a wonderful report to the whole group about their project. They also showed one of the cars demonstrating how when the temperature reached 26 degrees the alarmed was activated.”

“In fact it was during the group’s presentation to their peers that I really saw how much they got out of the Pet Alarm project. I was very impressed with their presentation and their explanation of all STEM associations with in the project.

Overall there was a lot of work involved both prior to the event and during the event week but seeing what the students got out the program was very rewarding. It is easy to see how such a program can be life changing to Students.”

The mix of software and handheld technologies – perfect partnership!

When I started teaching in the mid 80’s, it was evident then that emerging computer technology was poised to play a significant role in the future educational landscape. I had interacted with my first computer at university in 1980. Although I never actually saw it, I was assured that it existed in a large room somewhere behind the wall where I studiously typed my code. I first saw a personal computer in 1983 and by the time I started in schools a couple of years later, they were reasonably commonplace. Initially school software packages were mostly fixed programs, generally little more than a novelty, but then the first word processors arrived and my typewriter became redundant. For a maths teacher however, the big leap forward was the spreadsheet; a virtually endless page of inter-related computations. This definitely had potential. I remember using this combined with BASIC and LOGO in attempt to build some sort of algebraic function grapher. My endeavour at that only produced limited success. However by the end of the decade, there became various software packages available that achieved the basics of what I wanted.

Although not absolutely ideal, I managed to roster my students access to this useful facility and it greatly enhanced their capacity to realise links between the geometry and the algebra. But wouldn’t it be wonderful if they could access such technology on a full-time 1:1 basis?! Enter the 1990s and you know what happened.

TI-81 introduced 1990.

I was at a regional maths conference when somebody placed a miniature (calculator-sized) pedagogically-dedicated device in my hand. Not only did it have an advanced but interactive function-plotter, but also onboard was a powerful and demonstrative statistics package. This handheld device would have been a TI-81 or TI-82.

Within a couple of years the partnership was completed when I got hold of an LCD panel that sat upon an overhead projector while attached to the teacher’s calculator. I could demonstrate, students could follow and explore, and we could all discuss. At about the same time, laptop computers joined the mix but they were incredibly expensive. While the standard practice of many other teachers was to book their class into “the computer room”, I was to be seen walking along the corridor with the LCD panel under my arm as I headed for my next class. I should point out that at that stage the handheld technology was not yet permitted for use in exams. I simply saw it as a pedagogical tool – one that was much more flexible and superior to the computer room model.

By mid 90s the handheld technology had advanced to the point of incorporating a Computer Algebra System (CAS) and it became clear that our examination system of memory tests and algorithmic processes needed to change. The bar was raised and, with the first CAS-active examinations accompanying the new millennium, students were required more than ever before to demonstrate their ability to apply mathematical understanding and real problem solving.

Within the next decade many schools were implementing 1:1 laptop programs, with the computer power and connectivity of these machines far exceeding that of the handheld devices. I thought that perhaps that might spell the end for my beloved handheld device (which by then was a TI-Nspire).

But remember that partnership thing. By utilising a projector screen to share demonstrations from laptop software (TI-Nspire CAS) and having my students simultaneously extend their own explorations with what is essentially the same software but on smaller, more portable handheld devices, we now have demonstration, shared discussion and student investigation all perfectly catered for. This now is the ideal partnership and I am not exaggerating when I say that this is the model that I have now employed in every lesson of every day for nearly all of the past decade. The underlying reason for the success of this ultimate model is simple – the technology I am using was designed and developed specifically for education. Computer and laptop technology, while certainly very useful, are not in themselves task specific and purpose-built education tools. Computer technology will continue to advance but will never match TI handheld devices for their value in portability, pedagogy, and simple convenience as an examination tool.     

The Happy Rover – Girls in STEM Day

As the focus on STEM continues to gain momentum, so also did TI-Rover when it was put to the test by over 200 girls who attended a Girls in STEM day that was held recently at Ivanhoe Girls Grammar (9 Aug 2019). Hosted by the Mathematical Association of Victoria and attracting sponsorship from Ford, Texas Instruments, Engineers Without Borders, Aurecon and others, the successful event is now in its third year.

True to the theme of ‘Inspired by Curiosity’ the girls were receptive to presentations from GHD Group, SORA Architecture, Quantum Market Research, and the Bureau of Meteorology. Following morning tea and a panel discussion, the girls were given their chance for real hands-on STEM-task engagement in a two-part activity challenge that was facilitated by Texas Instruments and Engineers Without Borders.

Accredited T3 Instructors Shelly Cross and Karleigh Nicholls, from St Hildas School on the Gold Coast, led the TI component which assigned students the task of coding TI-Rover to race against other Rovers from fellow student teams; forward to a finish line, turn around and then return to the start line. Delighted T3 Manager, Daisy Patsias, observed “The girls were very engaged and loved ‘playing’ with TI-Rover. Presenters Shelly and Karleigh as always did a fabulous job and the girls appeared to love the activity. There was a lot of competitive spirit displayed on the day.”

Racecam video can be viewed at: https://youtu.be/M4BbVomPrU4

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