AUTUMN/WINTER ‘09

In this issue:

Crop circles Learning from other teachers’ experiencesFortifying FranceSquaring the circle

Welcome...to the Autumn Edition of *sight magazine! With *sight, we encourage

you to take a look at interesting articles, experiences, viewpoints and

suggestions which may inspire and enhance the time you spend together

with your pupils and colleagues.

In this Issue:

3 Crop circles

4 Learning from other teachers’ experiences - the TI-Nspire™ pilot evaluation project

7 Fortifying France

10 TI-Nspire goes wireless

13 Squaring the circle

A lot of press coverage has been given to

the ‘relevance’ of maths and science in daily

life, and how this can be captured in subject

teaching. Topics should be relevant, but

mathematical and scientifi c thinking for their

own sake can contribute positively in all kinds

of areas. Using technology can enhance our

ability to link pure mathematics with practical

applications and in this edition we explore

geometry with crop-circles!

Recent policy initiatives have focused on

promoting ‘interactive’ teaching in schools. What

exactly is meant by ‘interactive’ in the context of

a technology-enabled classroom? Relationships

between technical and pedagogical interactivity are

explored by Alison Clark-Wilson in an article about

researching use of the TI-Nspire platform with 14

teachers. One of the interesting fi ndings was that

using the technology helped teachers concentrate

on their students’ learning experiences.

New developments allowing purposeful interactivity

are featured in Cindy Hunt’s account of her

TI-Nspire Navigator pilot classroom. Read about

her experiences and how she is planning tasks

to enhance the learning and instant assessment

opportunities offered by the new technology.

Finding an innovative way to tackle a classic

problem is a challenge all teachers face. In

Adrian Oldknow’s ‘Squaring the Circle’, TI-Nspire

offers an interesting way for learners to tackle a

problem which cannot be exactly solved: how

to construct a square of exactly the same area

as a given circle. Using technology, a good

approximation of the ‘squared circle’ can be

found and a mathematical journey dating back

to the ancient Greeks discovered along the way!

Maths education benefi ts from an integration

of the history of mathematics, providing a

context for meaning and application. From the

ancient Greeks, we look ahead to more recent

applications in ‘Fortifying France’. Peter Ransom

has developed classroom activities themed

around building and breaching fortifi cations.

Using quadratics to fi re a cannonball into

defences could provide stimulating ideas and

an interesting cross-curricular project with your

history department.

I hope you enjoy reading this term’s *sight

and can take away at least one idea to use in

your classroom! We always want to hear from

our readers, so if you have an idea for an

article or comments you wish to share, please

email us at t3@ti.com.

Have an ‘*spiring’ term!

Andrea Forbes, Education Technology Group

Texas Instruments

*sight is edited by

Barrie Galpin

barrie.galpin@zen.co.uk

2009 has been a bumper year for crop circles which have been

appearing, to the consternation of farmers, all over the country. Whatever

the cause – extra terrestrial or just extraordinary vandalism – their

geometry offers plenty of scope for mathematical investigation. These two

TI-Nspire screen shots are based on crop circles which fi rst appeared in

2006 and are called “9-point Mandala” and “Wormholes” respectively.

The Internet provides a rich source of information

and photos of very ambitious creations. Go to

www.tinyurl.com/nsightmag for a sample of great

resources. Last year in Wiltshire, a crop circle

was based on the digits of pi and many other

mathematical topics can be involved.

For younger learners this could include line

and rotational symmetries, ruler and compass

constructions and even area, whilst older learners

could practise their skills by constructing crop

circle patterns using algebraic equations of circles,

parametric or polar equations.

CirclesCrop

By Jenny Orton, Maths Education Consultant

A challengeOur challenge to you is to create (or get your

students to create) some fantastic crop circles

using TI-Nspire. If you don’t have TI-Nspire

software, you can download a trial version

from our website at education.ti.com/uk. The

winning entry will receive a TI-Nspire ‘Teacher

Bundle’, which contains a handheld and

software. Email your entries to t3@ti.com by

30th November. The winner (as judged by the

editor) will receive the prize before Christmas.

*sight 3

Between September 2007 and April 2008 fourteen teachers were

involved in a project entitled ‘Evaluating TI-Nspire in Secondary

Mathematics Classrooms’. Getting started with a completely new

technology can be a lonely process, but knowing that some of your

frustrations are experienced (and overcome) by other teachers and

fi nding out about their successes can provide inspiration and ideas.

In this article the project’s director, Alison Clark-Wilson, describes how

using the technology helped teachers concentrate on their students’

learning experiences, illustrating the point by describing a particularly

innovative lesson idea developed by one of the project teachers.

The New National Curriculum for secondary

mathematics is challenging those of us in England

and Wales to consider thinking less about our own

teaching and more about the experiences of our

students as they actually learn mathematics. The

teachers involved in the TI-Nspire Evaluation Pilot

were able to consider these recommendations

alongside the introduction of the TI-Nspire handheld

and software in their classrooms.

When planning lessons, having decided on the

curriculum content, a common starting point is

to think fi rst about what I as teacher am going

to say and do to enable to students to learn the

mathematical concept or skills. During the project,

we strove to turn this around slightly, focussing our

thinking on what the students were going to do

and say to enable them to engage deeply in the

concept or skills in a way that allowed them

develop their own mathematical understandings.

The following activity is a good example of how

this worked in practice.

Rebecca Davey, who at the time of the project

taught at St Andrews CE Boys School Worthing,

had already identifi ed the main objective of her

lesson with her able Year 10 boys. She wanted

them to be able to recognise and generate the

equations of lines perpendicular or parallel to

straight lines such as y = 2x – 3.

The students, who were working on TI-Nspire

handhelds, began with a new TI-Nspire document

and inserted a blank Graphs & Geometry page.

Rebecca showed the students how to enter a linear

function and asked the students to all generate

the line y = x by entering the function f1(x) = x.

She then asked them to come up with a second

line, perpendicular to this one. From prior knowledge,

they suggested y = -x and entered this as a second

function, f2(x) = -x.

Alison Clark-Wilson

teaches at The

Mathematics Centre,

University of Chichester

Learningfrom other teachers’ experiences

the TI-Nspire pilot evaluation project

4 *sight

Rebecca then showed the students how to measure

the angle between the two linear functions using

the Angle tool from the Measurement menu by

clicking on points on the lines and the point of

intersection. The result of course supported their

initial conjecture–– 90˚! (Note: the handhelds’

settings had previously been set to display degrees

rather than radians.)

The students were then asked to edit the function

f1(x) to produce a different line, such as y =2x.

Immediately, the angle between the lines changed

from 90° and, in order to restore this value, Rebecca

showed the students how to use a unique feature

of TI-Nspire: when they grabbed the graph of f2(x)

near one of its ends they were able to drag it and

rotate it about the origin.

*sight 5

As the angle between the lines had already been

defi ned and measured, there was instant feedback

about the angle between the two lines and the

students were quickly able to identify the function

so that the two lines were again perpendicular.

The students were then given time freely to investigate

the situation and were also encouraged to consider the

equations of lines that did not intersect at the origin.

After about 30 minutes’ activity, Rebecca was able to

discuss the fi ndings of the class, drawing them together

in a table of results which helped the students to

generalise about the relationship between the gradients.

Some students had recorded their fi ndings on a Notes

page that they had inserted into their document.

During the pilot project, all of the teachers used the

key processes detailed in the New Curriculum to

help them to evaluate their students’ mathematical

learning and, in this lesson, some of the outcomes that

Rebecca recorded were very detailed. For example, she

commented how some students predicted that, since

y = x and y = -x were parallel, then the equation of a line

perpendicular to y = 2x would be y= -2x. She observed

some students analysing their results and, having found

one counter-example (e.g. y = 2x and y = -2x are not

perpendicular), they moved on to look for an alternative

relationship. The feedback provided by the technology

enabled the students to test their conjectures quickly

and, as a result, they made rapid progress.

Rebecca also commented on how the students

had really engaged with each other’s mathematical

reasoning and, although they each had access to

a TI-Nspire handheld, at times during the lesson

they stopped working individually and worked an

idea through together on one handheld. She said

that she was impressed by their resulting

mathematical discussions, which she felt were often

enhanced by the use of the correct mathematical

terminology that the menus on the handheld had

prompted. Rebecca concluded that using the

TI-Nspire had allowed the students to explore

the problem for themselves quickly and effi ciently.

At the end of the lesson, Rebecca asked the

students for their thoughts about using the

TI-Nspires and their comments included:

“ It was good to use IT in class – I liked playing around with graphs”

“ I liked being able to move the lines on the graphs and making notes on the handheld”

“We were able to explore more independently”

When Rebecca fi rst described this lesson to the pilot

project team, it generated a buzz of activity as we all

thought about how it could be adapted and extended.

Some of the suggestions that came up were:

• A Lists & Spreadsheet page could be inserted so

that the students could keep a record of the pairs

of gradients. This would allow for conjecture about

the relationship between the gradients to be tested

by inputting a formula into the spreadsheet.

• The lines could have been drawn as geometric

lines and then their slopes measured – this was

suggested as an alternative approach for students

who were less confi dent with function notation.

You might like to let me know about your own adaptations of Rebecca’s lesson – please email them to me at a.clarkwilson@chi.ac.uk.

And look out for more classroom examples from the TI-Nspire pilot project in future editions of *sight!

6 *sight

For many years Peter Ranson has been interested in the history

and application of mathematics and is well known for his work on

sundials (see the sundials case study on the Bowland maths DVD,

or download it from www.bowlandmaths.org.uk). He has developed

various ‘sun clocks’, both for the TI-84 family of graphics calculators

and for the TI-Nspire. This interest sparked off a chain of thought

and classroom activities in another area – military fortifi cation.

Fortifying France

Peter Ransom teaches

mathematics at The

Mountbatten School,

Romsey

peter.ransom@

mountbatten.hants.sch.uk

Last year I bought an old French mathematics

book Recueil de traités de mathématique,

a l’usage de messieurs les gardes de la marine

by P. de Chatelard (A collection of mathematical

treaties for marines, 1749) which contained a

treatise on sundials and another on fortifi cation.

The associated plates intrigued me (part of one is

shown below) and I thought this had great potential

in the classroom. What follows is a summary of just

part of the work that developed.

Scale model of the fortifi ed town of Gravelines

Vauban, my hero in this work, lived from 1633 to

1707 and was responsible for designing defences

and breaking through them. He was a military

engineer and fortifi ed a lot of French towns and ports,

especially near the borders with other countries.

His work on campaign forts, built by the army

overnight while on the move from place to place,

keeps them simple and symmetric so that defenders

in the fort can fi re upon attackers at the walls.

Although this adds to the perimeter of the fort,

the added strength of the defensive position

outweighs the extra work needed.

Starting with the diagram labelled fi gure 1,

students discuss the symmetries of the fi gures

and overleaf, fi gures 2 to 6 show how questions about

the forts have been put into a TI-Nspire document.

Using a Graphs & Geometry page, students are

given the opportunity to draw some of the campaign

forts to scale. It is very interesting to see how students

(and educators!) tackle drawing the simple starred

fort of f.23. Vauban recommended that the fort’s

format should be such that the indent of a side

should be 1–6 of the side’s length. Thus the distance

ef in f.23 is ac/6.

fi gure 1

*sight 7

To begin it is best to construct a square.

One approach is to draw a circle, construct

a diameter and its perpendicular and to

use the intersection points with the circle as

the vertices of the square. An alternative

approach with the TI-Nspire is to use the

Regular Polygon tool in the Shapes menu.

Once the square is constructed there are many

ways of constructing the fort’s indentations using

refl ections and/or rotations.

By working in pairs all learners acquire new

skills, though it is important to invest time in

acquiring those skills – do not expect instant

results! However, students help each other and

it is recommended that they be allowed so to do.

Constructing the campaign forts is only one part

of the fun. Figure 7 shows another plate with the

profi le (profi l) and scale (echelle) of the walls of a

fortifi ed town. This can be constructed on another

Graphs & Geometry page.

Finally the students can do some work on

quadratics in order to fi re a cannonball into part

of the defences! Another TI-Nspire document

contains pages using sliders linked to quadratics.

These help students see how a change in a

parameter affects the graph. Starting with the graph

y = x2, students work with y = x2 + c, y = (x + b)2,

and y = ax2, to obtain an understanding of the

transformations. Figures 8 to 11 show some of

the screenshots.

Once students are aware of the transformations

caused by changing the parameters a, b and c

individually they are ready to use the graph of

y=a(x + b)2 + c and to attack the fort. Figures 12 and 13

are screen shots showing their instructions and one

unsuccessful cannon ball!

This work is now part of my Year 9 mathematics

masterclass that I do at various venues on Saturday

mornings. I also include it in the geometrical and

algebraic work with my students at school and it has

led onto some work on the mathematics of ancient

siege machines such as the ballista, onager and

trebuchet. In one activity students study similarity and

scale by making a pair of proportional dividers (an old

mathematical instrument used to enlarge or reduce

diagrams) and I enjoy seeing them work with both

this old handheld technology as well as the new!

I hope that I have conveyed the fun that this

work provides but, if we need to justify its use,

the new Key Stage 3 curriculum for mathematics

stresses the need to include the history of

mathematics and creativity. This work on

fortifi cation certainly involves both.

fi gure 2

fi gure 5

fi gure 4

fi gure 6

fi gure 3

fi gure 7

8 *sight

To begin it

One approa

a diameter

use the inters

the vertices of

approach with

Regular Polygo

Once the squa

ways of constru

refl ections and/

By working in p

skills, though it

acquiring those s

results! However,

it is recommende

Constructing the

of the fun. Figure 7

profi le (profi l) and

fortifi ed town. This

Graphs & Geometr

fi gure 11

fi gure 8

fi gure 12

fi gure 9

fi gure 13

fi gure 10

If you would like copies of all the TI-Nspire documents, worksheets, videos and notes on a CD-ROM please send a stamped addressed

envelope to me at The Mountbatten School, Romsey, SO51 5SY. Unfortunately there is too much material to send it all by email.

Is it Math or Maths? Greatest Common Divisor or Highest Common Factor? Dilation or Enlargement? Yes, the language of mathematics

is slightly different on opposite sides

of the Atlantic. Now you can make

your handheld speak UK English.

When you upgrade your handheld’s

operating system make sure that

you choose English (U.K.) The

latest update to 1.7 was released

in June 2009.

N.B. Newly created .tns fi les

in v1.7 cannot be opened

with an older TI-Nspire release.

*sight 9

OK, so we’ve had TI-Nspire for

a while now. We’ve got used to

using a class set of handhelds

as well as software running

on a PC and projecting onto

a whiteboard.

Now just imagine being able to project any one

of the handheld’s screens for the whole class to see!

Or imagine being able to see from the teacher’s

PC exactly what was happening on any student’s

handheld. And wouldn’t it be great to be able

to share tns fi les with the whole class without

the hassle of cables? Can you imagine the

new types of classroom interactions such a

system would provide?

TI-Nspire goes

Cindy Hunt is Subject

Leader for Mathematics

at Davison CE High

School for Girls

(Technology College),

Worthing, West Sussex.

wireless

10 *sight

Well, the imagined has become a reality!

The TI-Nspire™ Navigator™ System, a wireless

classroom network, will be available in the UK in

the Summer term of 2010, following pilot trials in

various schools. Cindy Hunt was one of the willing

guinea pigs who piloted it in her classroom.

Here she describes the experience.

Well, we knew it was on the way! The ‘pushy-up

thing’ at the bottom of the TI-Nspire handheld

had to have some purpose!

The TI-Nspire Navigator System and

two guys from TI arrived in my classroom in

December 2008. With my Year 11’s I had been

waiting with bated breath! I did have the luxury of

an hour’s preparation and an empty room to get it

all up and running! There were cradles that attached

to the handheld by means of the pushy-up thing.

The cradles, which can be recharged in special

unit, become the power source for the handheld as

well as providing the wireless connection. I breathed

a sigh of relief––no more battery power issues!

Just an hour earlier my class had sat a GCSE module

exam so I didn’t know how they would react to

my insistence on doing something a bit different.

However, they were soon to start their Statistics

coursework so a peek at some related issues

was going to be benefi cial.

They began by registering each handheld, logging

in with their username. As they did so the Navigator

software allowed me to see all their handhelds on

my PC screen.

I had prepared a tns fi le containing a small

spreadsheet of data involving hand lengths and

reaction times of 2 groups. Using the Navigator

software I was able to send this fi le out to the whole

class and within seconds everyone had received it.

After a short time of looking at the data I interrupted

their thoughts with a “quick poll” question, sent

from my PC to every one of their handhelds, “What

hypothesis could we formulate with this set of data?”

The question was open-ended so the text responses

took a little time to put together. As the students

completed their responses I was able to read them

on my PC. We then projected the responses on the

whiteboard and scanned through them––this led

to some good discussions on the wording and

usefulness of each of the hypotheses.

Copies of 28 handheld screens were up on the

screen and the students insisted their usernames

were also displayed. There is an option for the

screens to appear anonymously but my students

wanted to be acknowledged!

I then demonstrated scatter graphs and box plots

and they were able to apply these techniques to

the data on their individual handhelds.

*sight 11

As the students used the technology to create

their own graphs, problems could be discussed

using the presenter mode––the named cradle

became the on-screen live presenter and this

proved a very effective way to share instructions

to help deal with technology issues. (How did

you make it do that?) At this point it did not occur

to me to become the live presenter––that was

a later development!

It’s now four months on and how have things moved

on since that fi rst tentative lesson?

TI-Nspire™ Navigator™ is a completely wireless classroom learning system that is designed to help teachers to:

Increase learner engagement• Select and present individual handheld screens anytime, from anywhere

in the room

• Project all handheld screens to let learners compare work in

an interactive environment

• View performance and guide learner work at any point during the lesson

Instantly assess each learner’s understanding• View learners’ handheld screens to check progress and time ‘on task’

• Take Quick Polls to know where learners are and what they are having

trouble with instantaneously

Maximize effi ciency of classroom learning time• Transfer fi les to the whole class at once – wirelessly!

• Save, share and record learners’ work in real time

• Take advantage of the TI-Nspire document structure in a

collaborative environment

I now prepare lots of tns fi les to broadcast to the

class, albeit some very simple pages that haven’t

taken long to prepare! If I send a fi le out through

the Navigator students can work on it during the

lesson and I can automatically retrieve the student’s

versions at the end of the lesson.

This means I am planning tasks very differently

to ensure that the technology will enhance the

teaching and learning process within my classroom.

I have used the “quick poll” facility very frequently

and it has created many opportunities for

discussion, allowing students to appreciate

other’s results. A recent example was with a top

set Y11: “Find 2 numbers which when squared

give a sum of 25”. All the results were displayed

and some interesting thoughts of negatives and

zeros were discussed. This was followed by my

sending out a small tns fi le consisting of a simple

pre-labelled spreadsheet and a graphs page.

After they had put in as many results as they

could think of it was time to project their screens.

“What have we got on page 1.2?” “Circles?”

“Why do some appear more fi lled in?”

“What can you tell me about the circle?”

The system has transformed the way in which

I use TI-Nspire. The ease of transferring data/fi les

means I am much more willing to put together a fi le

where the students are engrossed in the mathematical

content straight away. Issues of technical know-how

have now taken a back seat and the beauty of the

maths is much more apparent.

Wires? What wires?

If you would like a demonstration of the system

or would like to arrange professional development

from T3, please send an email to t3@ti.com.

12 *sight

In this article Adrian Oldknow describes how the

classic geometry problem of “squaring the circle”

can be approached using TI-Nspire. In the fi rst part of

the article he shows how to fi nd a good approximation

to the squared circle, leading directly to an approximation

for π. Another approximation technique for calculating the

area of a circle is then explored.

A geometry problem studied by the ancient Greeks

was how to construct a square of exactly the

same area as a given circle. Nowadays we know

that this is an impossible task. However, it is a

myth that the Greek geometers only used abstract

methods. Archimedes, in particular, was a very

practical person, and he worked out many very good

approximate methods. In order to test whether the

area of one shape was greater or less than another

he would make accurate drawings, cut them out

from the same material and use a pair of balances

to compare their weights, and hence their areas.

Nowadays we can use computer software to calculate

areas for us, at least approximately. We can fi nd a good

approximation to the squared circle and the coloured

picture above gives the basic idea. In that diagram,

the green circle lies snugly within the largest square

shown, and the smallest square fi ts snugly inside the

circle. So there must be a square between these two

squares which has area equal to that of the circle. The

regions coloured red are the bits of the square that

are not inside the circle, and those coloured yellow are

the bits of the circle that are not inside the square. So

our aim is to adjust the middle sized square to make

the red and yellow areas as equal as possible. We can

then use the square to approximate π.

Below is an outline method for setting this up on

the TI-Nspire handheld. The named points are for

reference in this article but there is no need to enter

them on the screen unless you wish to do so.

Draw a circleStart with a blank Graphs & Geometry screen.

From the View menu choose Hide Entry Line.

From the Shapes menu use the Circle tool to

construct a circle centred at the origin.

Press /` and copy and paste the page, which

you will use later. Press · to return to page 1.1.

From the Measurement menu choose Length

and measure the radius of the circle.

Place the cursor over the measurement and store

it as variable r.

Notice that you can drag any point on the circle

and change the value of r.

Adrian Oldknow is

Emeritus Professor

of Mathematics and

Computing Education

at the University of

Chichester and visiting

fellow at the London

University Institute

of Education.

Squaring the

circle

fi gure 1

*sight 13

Draw the squareUse the Angle Bisector tool on the Constructions

menu to construct a line through the origin at 45˚

to the axes.

From the Points & Lines menu use the Points On

tool to construct the point V.

Construct the other three corners of the square

by refl ecting V in the x- and y-axes.

Use the Polygon tool to construct the square

through V.

Measure the length of the side of the square and

store it as variable x.

Notice that you can drag point V along the line

and change the value of x.

Equalizing areasAt this stage we can tidy up the fi gure by hiding unwanted

construction lines and points. Then it is time to see if

we can change the size of the circle and/or the square

until the two polygons have (as accurately as possible)

equal areas. It takes a little trial and improvement but one

possible solution is shown here.

Calculate the red and yellow areasWe will restrict ourselves to getting the handheld

to measure areas that we could calculate for

ourselves, such as squares, triangles and trapezia.

As a fi rst approximation we can defi ne some

polygons whose areas approximate the red

and yellow regions of our diagram.

First defi ne the quadrilateral (kite) AWBV as in fi gure 3 .

Use the Polygon tool to defi ne the polygon and

change its Fill Attribute to grey.

We could use the triangle BEU to approximate the

yellow area, but we could get more accuracy if we

construct another couple of points C, D on the circle

and use the polygon EDUCB instead.

To construct point C draw the perpendicular bisector

of BF and mark its intersection with the circle at C.

Then D can be constructed as the refl ection of C in

the y-axis.

Defi ne the Polygon EDUCB and shade it.

Finally use the Measurement tool to display the area

of the two shaded polygons.

An approximation for πWe could go further with geometric explorations,

but now we can just use some simple algebra and

arithmetic. For a square and a circle of equal area,

is there a simple relationship between the side of the

square and the radius of the circle? The side of the

square has length x, so its area is x2. Let us assume

that the area of the circle is a constant p times r2.

To fi nd p we must solve the equation pr2 = x2 for p,

i.e. p = (x/r)2. We can use the Text tool to write the

expression on the page and the Calculate tool to

substitute values for variables as in fi gure 5. Hence we

fi nd that the solution to squaring the circle is roughly

when p = 3.10387.

The challenge now is to make the two shaded

polygons have even more equal areas and the value

of p closer to the value of π.

fi gure 2

fi gure 3

fi gure 4

14 *sight

So we have used computing power, and our brains,

to short-cut Archimedes’ painstaking weighings.

Of course we’re not sure how TI-Nspire actually

calculates its areas – and so in the end we are

using it as a “black box”. However this approach

does give us a feeling for how we can use trial and

improvement to solve problems approximately.

An n-sided regular polygonTo fi nish let us set up a simple visual demonstration

of another way to fi nd the approximate area of a

circle. Here the circle is divided into n equal triangles

whose area can be calculated. As n increases the

approximation gets better and better.

Now move to page 1.2 and use the circle and

square copied earlier. We can use a slider to give

us counting numbers as big as we like. From the

Actions menu choose Insert Slider and assign to it

the variable n. With the cursor on the slider press

/b and change the settings to run from 1 to

40 with a step size of 1 and an initial value of 5.

You need to check that the Document Settings

(in the Tools File menu) are set for angles in

Degrees. Then divide 360 by the value of n to

obtain the value of an angle.

From the Actions menu choose Text and enter 360/n.

Then from the Actions menu choose Calculate

and evaluate the expression.

Now we can use the slider to increase the value of n

and investigate how the area of the n-sided polygon

gets closer to the area of the circle.

The diagram below shows a further construction of

a triangle OTU, part of an n-sided polygon sitting

just outside the circle. The area of this polygon, b.n,

provides an upper bound for the area of the circle.

As n is increased a.n and b.n get closer to each

other and to the true area of the circle.

This example gives a feel for the kind of limiting

processes used regularly by Archimedes, and which

Newton and Leibniz formalised into the calculus

techniques we know today. Why not fi nd out some

more about the geometric techniques the Ancient

Greeks developed to solve problems and see if you

can use TI-Nspire to model them?

The completed fi le from this article is available to

download from www.tinyurl.com/nsightmag.

Choose Rotation from the Transformations menu

and select, in turn, the x-axis, the angle 72˚ and the

origin, O. Mark the points of intersection R and S and

draw the triangle ORS.

Shade the triangle, measure its area and assign it to

the variable a.

If we multiply this area by n we will fi nd the area of a

n-sided regular polygon that lies inside the circle. Use

Text and Calculate to add this to the sheet.

fi gure 5

fi gure 6

fi gure 7

fi gure 8

*sight 15

You can’t tell this book by its cover!

What TI technology is available?Our range of technology for schools

includes software and handheld

devices designed specifi cally for

education, classroom networking tools,

graphics calculators and a variety of

sensors and probes for data logging

activities (to meet the STEM agenda).

Our software is designed to integrate

with existing classroom projection

systems to enhance the learning and

teaching experience.

• TI-Nspire™ – the award-winning

handheld and software ICT

platform for maths and science

with additional options:

• Teacher Edition Software - includes

an emulator of the TI-Nspire handheld

and enhanced functionality

• TI-Nspire Navigator™ System - the

wireless classroom network for TI-Nspire

• TI Connect-to-Class™ - document and

fi le sharing for TI-Nspire handhelds

• The TI-84 Plus™ and TI-83 Plus™

family of graphics calculators

• TI-SmartView™ - the software emulator

of the TI-84 Plus graphics calculator

• CBL 2™, CBR 2™, EasyTemp™ and

EasyData™ with support for more than

30 probes and sensors.

• Cabri Junior and a host of other APPS

available on the TI-84 handheld device

• TI-Nspire CAS, TI-89 Titanium and

Voyage™ 200, our CAS (Computer

Algebra Software) solutions.

TI Technology Loan – to support evaluation of our technology

and your CPD activities.

Using our free loan service, you can fi nd

out more about how TI technology can

enhance your pupils’ learning. It’s an ideal

way for you to get TI products for teacher

workshops and in-service training or to

borrow individual handhelds for class

evaluation. Loans are available for up

to three weeks.

What services do we offer?

T3 (Teachers Teaching with Technology™)T3 has delivered professional development

for mathematics and science teachers since

1992. T3 trainers are practising teachers

with experience and depth of knowledge

in subject teaching and learning who

aim to promote the appropriate use of

technology in the classroom by providing

CPD that helps teachers to develop effective

practices through pedagogy and technical

confi dence. The courses they run place

the emphasis on sharing good use of ICT

in the classroom. T3 offers opportunities

for teachers to get together with others

to explore teaching and learning with

TI Technology.

In addition to offering a range of CPD

opportunities, T3 members also support

research and pilot projects, author books

and create and share a wide range of

activities on topics for secondary level and

above. T3 is an international organisation and

sponsorship from Texas Instruments enables

T3 to deliver free courses and materials.

For further details, including dates and

venues, please visit www.tcubed.org.uk or

email t3@ti.com.

Volume Purchase Programme– free TI technology for volume purchases

through our educational suppliers.

With every purchase of a TI-Nspire

handheld device or graphics calculator,

you could obtain free TI technology –

from as little as purchases of 20 devices.

For more information, visit education.ti.com/uk

All handheld devices available in Europe are manufactured under ISO 9000 certifi cation. Cabri Log II is a trademark of Université Joseph Fourier. All trademarks are the property of their respective owners. Texas Instruments reserves the right to make changes to products, specifi cations, services and programs without notice. Whilst Texas Instruments and its agents try to ensure the validity of comments and statements in this publication, no liability will be accepted under any circumstances for inaccuracy of content, or articles or claims made by contributors. The opinion published herein are not necessarily those of Texas Instruments.

©2009 Texas Instruments

Available from Chartwell-Yorke

(www.cymaths.co.uk), £29.

You know that new technology has

really arrived when book publishers start

producing support materials – remember

all the books devoted to learning

mathematics with the TI-83 and TI-84

calculators? The US publisher Key

Curriculum Press seems to be fi rst off the

mark for the TI-Nspire with a collection

of 30 activities consisting of handouts,

teacher notes and tns documents. The title,

“Exploring Algebra 1”, may turn off some

UK teachers but this would be a shame

because the content of this book is certainly

not exclusive to the US curriculum.

There are some really interesting ideas

here that could be very useful in UK

classrooms. Many of the activities start

with data collected from the real world

and help students analyse it using the

TI-Nspire’s powerful graphing and statistics

facilities. Some of these data sets can

be collected by students themselves

(e.g. pencil rolling, pulse rates, paper

folding). Others are provided in the tns fi les

(e.g. high-jump records, life expectancies

and geographical data).

However, actually using the student

handouts as they stand may present

problems, because it was written for the

2007 version of TI-Nspire. Since then

commands and menus have changed,

so that some of the instructions are no

longer valid. There are also some contexts

(e.g. proportion of military veterans in

various States) and use of language

(e.g. “you still had to eyeball the slope”)

that do not transfer readily to this side of

the Atlantic. Nevertheless there are some

great ideas that you could use to create

your own handouts.