“Insanity is doing the same thing over and over again and expecting a different result.” So what are you going to do different with your students?

The new TI-Nspire Premium Teacher software incorporates the TI-Navigator software. The TI-Navigator system is a powerful teaching and learning tool that increases student engagement, provides instant feedback and provides for fast and efficient assessment of student work.

In this webinar you will see how to:

Set up a class

Get students to login

Send Quick Polls including instant data collection

Have students present live to the class without leaving their seat

Save precious time and learn from the experts. This student focused webinar will provide time saving strategies, tips, tricks and shortcuts to setup your TI-nspire calculator for success. Speakers:Bozenna Graham

YouTube video will be available 48 hours after the live event.

Save precious time and learn from the experts. This student focused webinar will provide time saving strategies, tips, tricks and shortcuts to setup your TI-84CE Calculator for success. Speakers:Jim Lowe & Peter Fox

YouTube video will be available 48 hours after the live event.

Save precious time and learn from the experts. This student focused webinar will provide time saving strategies, tips, tricks and shortcuts to setup your TI-nspire calculator for success. Speakers:Rodney Anderson and Melissa Hourigan

YouTube video will be available 48 hours after the live event.

Save precious time and learn from the experts. This student focused webinar will provide time saving strategies, tips, tricks and shortcuts to setup your TI-nspire CAS calculator for success. Speakers:Danijela Draskovic and Craig Browne

YouTube video will be available 48 hours after the live event.

Save precious time and learn from the experts. This student focused webinar will provide time saving strategies, tips, tricks and shortcuts to setup your TI-nspire CAS calculator for success. Speaker:Stephen Crouch and James Mott

YouTube video will be available 48 hours after the live event.

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 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 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.

Mathematics: – 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.

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.

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 27^{1}/_{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)

8

110

0.1

30.556

10

155

0.2

43.056

12

199

0.3

55.278

13

239

0.4

66.389

16

303

0.6

84.167

19

360

0.8

100

21

388

1

107.778

22

410

1.1

113.889

25

476

1.6

132.222

28

522

1.9

145

31

575

2.3

159.722

33

625

2.8

173.611

35

669

3.1

185.833

38

733

3.7

203.611

40

769

4

213.611

43

836

4.7

232.222

47

898

5.6

249.444

50

934

6.4

259.444

53

977

7.3

271.389

57

1021

8.2

283.611

60

1068

9.1

296.667

70

1326

12.2

368.333

90

2019

19.9

560.833

110

2943

30

817.5

130

4105

41.9

1140.278

150

5498

55

1527.222

156

5861

59.4

1628.056

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)

0

26572

198

7381.111

10

27377

199

7604.722

20

28459

200

7905.278

30

29419

201

8171.944

40

30428

203

8452.222

50

31437

205

8732.5

60

32760

208

9100

70

34157

212

9488.056

80

35647

216

9901.944

85

36694

221

10192.778

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/sec^{2} and 23.99 m/sec^{2}, whereas for Stage 2, acceleration is approximately 32.51 m/sec^{2} 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/distance^{2} 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.

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!