Category Archives: PD

new student tutorial series on string graphs – ideal for year 10

This unit on string graphs is inspired by the amazing works by Architect, Engineer and Artist, Santiago Calatrava.

The four-part tutorial series will be conducted on Wednesday mornings from 11:00 to 11:30 am and presented by T-Cubed trainer, Roger Wander. Roger was a secondary mathematics teacher in Victoria, South Australia and the USA for over 30 years. His M.Ed studies at the University of Melbourne led to his move in 2008 to the Mathematics Education group at the Melbourne Graduate School of Education as a Senior Research Officer in the Texas Instruments-sponsored New Technologies for Teaching Mathematics project, which investigated the use of TI-Nspire CAS (Computer Algebra Systems) in middle secondary classrooms. He is currently a Clinical Specialist for the Master of Teaching graduate course at the MGSE, working with teacher candidates in secondary schools. He also works in the undergraduate subject School Experience as Breadth as the STEM stream lecturer.

Register now to ensure your place in this fascinating and curriculum relevant tutorial series. CLICK HERE

Wednesday, July 22, 11:00 am Part 1: Gradients, Intercepts and Parameters

This first tutorial explores a family of linear graphs to produce a pattern similar to the one found on the Chords Bridge. Equations will be formed using information about the gradient and y-intercept. Each equation will be linked to form a family using something called a parameter.

Wednesday, July 29, 11:00 am Part 2: Points, lines and Parameters

This tutorial literally turns the first session on its side. Continuing the theme, more patterns will be formed linear functions. In this session we stitch lines to y = x and y = -x. Just like a time-lapse of the Milwaukee Art Museum, you can explore what happens when structures move. Equations will be formed using two points. Each equation will be linked to form a family.

Wednesday, Aug 5, 11:00 am Part 3: Working Simultaneously

In this tutorial we combine one and two to get three. The results from the first two tutorials will be used to further explore the patterns formed when these linear functions intersect. You will learn how to solve equations simultaneously, appreciate the use of exact values over decimal approximations and value the functionality of different mathematical representations.

Wednesday, Aug 12, 11:00 am Part 4: A table with a difference

This final session brings everything together. The patterns created and observed can be described mathematically. A simple tool called a ‘difference table’ will help identify the patterns. The curves can be described using simple expressions and graphed nicely using your TI-nspire CX CAS. You may even want to dive into further opportunities to extend your journey.

Register now to ensure your place in this fascinating and curriculum relevant tutorial series. CLICK HERE

Register now to ensure your place in this fascinating and curriculum relevant tutorial series. CLICK HERE

Live Webinar

Navigating the new Premium Teacher Software

Date: Tuesday 7th April
Time: 7:30pm (Melbourne)
Duration: 1 Hour

“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
  • Download or Create assessment resources
  • Automatically mark student work
  • Review and analyse student results

Speaker: Peter Fox

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

Live Student Webinar: 20th February 2020

Get ahead of the game. IB: Analysis and Approaches SL and HL with TI-nspire

Start the year off feeling empowered!

Date: Thursday 20th February
Time: 7pm (AEDST)
Duration: 40 mins

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.

Live Student Webinar: 20th February 2020

Get ahead of the game. Methods and Specialist with TI-84PlusCE

Start the year off feeling empowered!

Date: Thursday 20th February
Time: 7pm (AEST)
Duration: 40 mins

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.

Live Student Webinar: 19th February 2020

Get ahead of the game. Methods and Specialist with TI-nspire (non-CAS)

Start the year off feeling empowered!

Date: Wednesday 19th February
Time: 7pm (AEST)
Duration: 40 mins

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.

Live Student Webinar: 19th February 2020

Get ahead of the game. General and Further Mathematics with TI-nspire CAS

Start the year off feeling empowered!

Date: Wednesday 19th February
Time: 7pm (AEDST)
Duration: 40 mins

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.

Live Student Webinar: 18th February 2020

Get ahead of the game. Methods and Specialist with TI-nspire CAS

Start the year off feeling empowered!

Date: Tuesday 18th February
Time: 7pm (AEDST)
Duration: 40 mins

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.

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.    

Mathemagicians Exposed

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:  352

                   (a + b) (a – b) = a2 – b2             Let a = 35 and b = 5

                   (35 + 5)(35 – 5) = 352 – 52        Substitute

                   40 x 30 = 352 – 25                     Simple calculation 40 x 30 = 1200

                   1200 + 25 = 352.                       

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

Example: 312

                   (a + b)(a + b) = a2 + 2ab + b2    Let a = 30 and b = 1

                   (30+1)(30+1) = 302 + 2x30x1    Substitute

                   312 = 900 + 60 + 1 = 961

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

Example: 1262

                   (125 + 1)(125 + 1) = 1252 + 2 x 125 x 1 + 12                   

[Note: 1252 = (125 – 5)(125 + 5) + 25 = 120 x 130 + 25]

                   1262 = 15625 + 250 + 1 = 15,876

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!

Conference Round-up rounding off the year

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 CAS and 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.

Mathematical Association of Western Australia Conference, Perth 18-19 Nov
T-Cubed trainers Brian Lannen, Jody Crothers, Peter Flynn, Peter Fox and Megan West
Learn, Energise, Connect, Brisbane 29 Nov
T-Cubed trainers: Peter Fox, Melissa Hourigan Jim Lowe, Tammy Mize (Dallas office), Rodney Anderson and Curtis Browne (Dallas)
Learn, Energise, Connect, Melbourne 4 Dec
T-Cubed trainers: Jim Lowe, John Bament, Neale Woods, David Tynan, Peter Fox
Mathematics Association of Victoria conference 5-6 Dec
Photo at top shows WA’s Jody Crothers with Texas Instruments Education & Technology President, Peter Balyta, Tammy Mize (Dallas) and Curtis Browne (Dallas)

          

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