Category Archives: Activites

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.

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.

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.”

On Demand Webinar: Linear function algebra with TI-84 in 20mins.

Curriculum Inspirations Webinar Series

String Patterns (TI-84)

Presented: Thursday 5th September 2019
Duration: 20 minutes

In this Curriculum Inspiration, John explores creative ways to reinforce linear function algebra with Years 9–10 students and transformations of sets of such functions. The focus is on the students discovering what effect each parameter in the function (e.g. gradient & y-intercept) might have on the shape and location of the set of graphs. Speaker: John Bament

On Demand recordings will be available 48 hours after the live event.

On Demand Webinar: Driving further with your TI-Nspire in 20 mins.

Curriculum Inspirations Webinar Series

Driving Further (TI-Nspire)

Presented: Tuesday 3rd September 2019
Duration: 20 minutes

In this In this Curriculum Inspiration Bozenna will use golf driving distances from PGA and LPGA records to explore correlation between year and distance. The relationship shows a relatively strong positive correlation over a period of almost 20 years. Students use the relationship to predict driving distances into the future and compare the accuracy of these predictions against more recent data. Speaker: Bozenna Graham

On Demand recordings will be available 48 hours after the live event.

On Demand Webinar: Driving further with your TI-84 in 20 mins.

Curriculum Inspirations Webinar Series

Driving Further (TI-84)

Presented: Thursday 29th August 2019
Duration: 20 minutes

In this Curriculum Inspiration John will use golf driving distances from PGA and LPGA records to explore correlation between year and distance. The relationship shows a relatively strong positive correlation over a period of almost 20 years. Students use the relationship to predict driving distances into the future and compare the accuracy of these predictions against more recent data. Speaker: John Bament

On Demand recordings will be available 48 hours after the live event.

On Demand Webinar: TI-Nspire Technology Webinar

Introducing the binomial and normal probability distributions

Presented: Wednesday 28th August 2019
Duration: 60 minutes

In this webinar, effective ways of introducing the binomial and the normal distributions will be showcased. In particular, we aim to show how to construct easy-to-make TI-Nspire demonstration files that will allow the teacher to illustrate key features and important properties of both distributions. This webinar concentrates on developing conceptual understanding of both distributions.

On Demand recordings will be available 48 hours after the live event.

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