Category Archives: STEM

Roller Coasters & Wine Glasses

 

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
 – Velocity
 – Time

Students can also explore how concavity affects the ride experience. For an easierwine 3 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
Wine glasses come in a range of shapes and sizes depending on their purpose. The tallwine 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.

Mathematics:
 – Standard drink = 175ml (wine) – This should occur at the widest section of thewine 2
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.

Modelling Rocket Launch with Calculus

This Activity was developed by T-Cubed Trainers Bozenna Graham and Stephen Broderick as part of their session What’s your Vector Victor? – Flying High with PSMT & TI-Nspire™ that they presented at the Brisbane Learn, Energise, Connect PD Day last November.

Context

Telemetry video data for SpaceX launches are readily available on the web. Analysis of this data verifies a number of different elements for the various stages of a launch such as acceleration, altitude, distance travelled and average speed. SpaceX has successfully launched a number of payloads into orbit including satellites and supplies to the International Space Station.

Task: Investigating a SpaceX rocket launch

Collect speed, altitude and time data for the first stage of the SpaceX launch contained in the stimulus link below. The first stage involves the time that the three rocket engines are firing. The end of the first stage occurs just before the rocket boosters are detached. You can add time data by pausing the video and noting the timestamp in the telemetry data.  Use regression analysis to determine equations for Stage 1 and Stage 2 burns of the launch. Stage two occurs sometime after the 27-minute mark. Use calculus techniques to investigate the mathematical models produced from the data. Compare the acceleration and average speed during the Stage 1 and Stage 2 burns and also use calculus to determine the altitude of the SpaceX rocket shortly after launch.

To complete this task

• use the problem-solving and mathematical modelling approach to develop your response

• respond with a range of understanding and skills, such as using mathematical language, appropriate calculations, tables of data, graphs and diagrams

• provide a response that highlights the real-life application of mathematics

• respond using a written report format that can be read and interpreted independently of the instrument task sheet

• develop a unique response

• use both analytic procedures and technology.

Stimulus

Below is a Youtube link for a SpaceX launch. The telemetry data for speed, altitude and time is located on the bottom of the screen.

www.youtube.com/watch?v=TXMGu2d8c8g

Possible Solution for task:

Introduction (formulate)

This task involves collecting SpaceX launch data during Stage 1 and Stage 2 of a launch.

The data collected includes time, altitude and speed of the SpaceX rocket. The starting times and duration of each stage will need to be determined from the video. Mathematical models for Stage 1 and Stage 2 burns will be determined and analysed with calculus techniques to determine the acceleration, distance travelled and average speed in each stage. The altitude in the early stages of the launch will also be approximated.

Results (Solve)

Some of the initial assumptions include:

  • Stage 1 involves the time when the three rocket boosters are firing and extends from t = 0 to t = 156 seconds
  • Stage 2 commences around 271/2 minutes into the launch and lasts approximately 85 seconds
  • Regression analysis will be used to develop mathematical models for both stages
  • Altitude and distance travelled are only the same in the initial stages of a launch when the SpaceX rocket is travelling vertically
  • Distance travelled by the SpaceX rocket can be approximated with calculus techniques
  • Since this is actual data, atmospheric friction (or wind resistance) is included in the mathematical models

The telemetry data was collected from the video. 

Table 1 includes the telemetry data for the first 156 seconds of the Stage 1 burn and includes time, speed (kilometres/hour) and altitude (kilometres). The speed in metres/second was added to the table by multiplying by the conversion factor (1000/3600)

Stage 1 Burn data

Time (seconds)Speed (kilometres/hour)Altitude (kilometres)Speed (metres/second)
81100.130.556
101550.243.056
121990.355.278
132390.466.389
163030.684.167
193600.8100
213881107.778
224101.1113.889
254761.6132.222
285221.9145
315752.3159.722
336252.8173.611
356693.1185.833
387333.7203.611
407694213.611
438364.7232.222
478985.6249.444
509346.4259.444
539777.3271.389
5710218.2283.611
6010689.1296.667
70132612.2368.333
90201919.9560.833
110294330817.5
130410541.91140.278
1505498551527.222
156586159.41628.056
Table 1: Stage 1 data for the SpaceX launch

The graph of time (seconds) versus speed (metres/second) for the Stage 1 burn in figure 1 is best represented by the cubic function below:

The shape of the graph suggests that the acceleration of the SpaceX rocket is increasing over the 156 second interval. The derivative of the speed function yields the acceleration function.

The reason for this difference is that the SpaceX rocket is not flying on a vertical trajectory at this stage; it has changed its pitch and is flying on an acute angle with the Earth’s surface. In actual fact, 81.258 km is the total distance that the SpaceX rocket has flown after 150 seconds.

The total distance travelled after the Stage 1 burn is illustrated in figure 4.

Stage 2 Burn data

The data in Table 2 was collected after 27:37 minutes of flight time. The second stage burn lasts for approximately 85 seconds.

Time (seconds)Speed (km/h)Altitude (km)Speed (m/sec)
0265721987381.111
10273771997604.722
20284592007905.278
30294192018171.944
40304282038452.222
50314372058732.5
60327602089100
70341572129488.056
80356472169901.944
853669422110192.778
Table 2: Stage 2 data for the SpaceX launch

The graph for the 85-second burn during Stage 2 starts after 27 minutes and 37 seconds into the launch and is shown below in Figure 5.

The data can be represented as a linear model. The correlation coefficient for the association is 0.99417, which means that the algebraic model is a close match to the empirical data.

The distance travelled (Figure 6) in the Stage 2 burn of 85 seconds is approximately 734 kilometres.

Conclusion

The trajectory of a SpaceX rocket varies considerably during a launch. In this task different mathematical models were used to represent Stage 1 and Stage 2 of the launch. A cubic equation with a correlation coefficient (r ) of 0.999537 was used to model Stage 1, while a linear equation with a correlation coefficient (r) of 0.99417 was used to model Stage 2.

The acceleration rates for Stage 1 and Stage 2 are quite different due to the effects of gravity. For Stage 1, the acceleration ranges between 4.79 m/sec2 and 23.99 m/sec2, whereas for Stage 2, acceleration is approximately 32.51 m/sec2 throughout the 85 second burn. This difference is due to the effects of gravity. As the SpaceX rocket gets further away from Earth, the effects of gravity decrease by a factor of 1/distance2 in accordance with the inverse-square law. This results in greater acceleration of the SpaceX rocket as it escapes the Earth’s gravity. The integration of the mathematical model for stage 1 can be used to determine the altitude when the rocket is launching vertically. There is a good agreement with the telemetry data for up to 40 seconds of the Stage 1 burn; however, after this period of time, the rocket’s trajectory follows a curved path making the determination of the altitude difficult. The integration of the models can be used to determine the total distance travelled during the Stage 1 and Stage 2 burn. The distances travelled by the SpaceX rocket in Stage 1 and Stage 2 are 90.744 and 733.383 km respectively. During Stage 2, the rocket travels 8 times further than it does in Stage 1 and in half the time. The average speeds for Stage 1 and Stage 2 are 0.5817 and 8.628 km/sec respectively. The SpaceX rocket is travelling nearly 15 times faster in Stage 2 than it is in Stage 1. Although average speeds are useful for looking at and comparing various stages of a launch, precise speeds are needed for docking with the International Space Station which travels at 7.9 km/sec.

Australian Trainers to present sessions at T-Cubed Instructor Training day in Dallas

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 12th 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 WorldMany 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 UnderCome 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.    

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.

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 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