'ICT Bringing Algebra and Geometry to Life'



'ICT Bringing Algebra and Geometry to Life'

Adrian Oldknow a_oldknow@

The Royal Society and JMC have published reports on the teaching of both algebra and geometry in schools, which are clearly important aspects of the National Curriculum, and of the Key Stage 3 mathematics strategy. The plenary talk at IMECT3 focused on how ICT tools such as graphical calculators, dynamic geometry software and integrated graphing, algebraic and data-handling packages can support the development of algebraic and geometric reasoning, knowledge and modelling skills. It emphasised whole-class teaching approaches building on experiences gained with the DfES Year 7 MathsAlive! project.

The MathsAlive! project began in October 2000 with the identification of 20 pilot sites with Y7 classes. Building on the pilot, Research Machines (RM) have now developed a complete KS3 service called Maths Alive Framework Edition. The picture below show a typical project classroom. Whole class work is supported by the interactive whiteboard (Smartboard), ceiling mounted projector and PC. There is also a conventional OHP with an LCD pad (Viewscreen) used to display a teacher’s graphical calculator (TI-83 Plus SE) and data-logger (CBR). The graphical calculator can also be displayed via the video input of the projector using an interface (TI Presenter).

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A further two PCs in the corner of the room are available for group work, and there is a class set of pupil graphical calculators (TI-83 Plus) for individual or paired work.

As well as a range of software developed by RM to support the project, a couple of significant mathematics ICT tools were identified as the heart of the mathematics tool-kit, along with the graphical calculators. These are the dynamic geometry software package called the Geometer’s Sketchpad (Key Curriculum Press) and the integrated mathematics package called TI Interactive! (Texas Instruments).

Some illustrations from the KS3, 4 and 16-19 curriculum are used to give a feel for how these tools can be used to help develop geometric and algebraic reasoning. Those who know me will recognise my passion for geometry – so that’s where I’ll start. So here’s a question for you to consider:

What’s the locus of a point P which is the same distance from two fixed points A and B?

There are many ways of exploring this locus – for example Ann can hold one end of a piece of string and Brian the other, and Petra can hold the middle. As Ann and Brian both pull in the string together what will be Petra’s path? Using this basic result – that it is the perpendicular bisector of AB - we can get a long way in elementary geometry!

In the triangle ABC imagine the perpendicular bisectors of the sides AB and AC. Let them meet in D. What can you say about D?

In a static article it is hard to give a dynamic feel for using ICT to explore such a problem!

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Try first to imagine moving A nearer to BC – what will happen to D? Can it leave the interior of ABC? If so, for what sort of triangle ABC? Can D lie on BC? For what sort of triangle? At which point on BC? Can you move A so that D always stays at the same point on BC – what is the path of A – at which point on BC does D stay? As you move A around the plane what is the locus of D? This suggests a theorem – that D always lies on the perpendicular bisector of BC. Can you prove it? D lies on the perpendicular bisector of BC, so what sort of triangle is BDA? D lies on the perpendicular bisector of AC so what sort of triangle is ADC? So what can you deduce about triangle BDC? How does that prove the theorem?

The use of dynamic images, together with large clear diagrams using different colours, styles, weights and labels provides a powerful medium for making and exploring geometric conjectures in this way. But the geometry need not be abstract and divorced from the pupils’ world of experience.

Do you think the lower arch of Sydney Harbour Bridge is an arc of a circle? If so where would the centre lie?

Here we can import a digital image into Geometer’s Sketchpad and make constructions over it. Points A, B, C can be dragged into positions on the lower arch. Segments AB and BC will be chords of the circular model, so its centre will be the point of intersection D of their perpendicular bisectors.

We can also use our knowledge of perpendicular bisectors, and the properties of quadrilaterals to derive the standard `straight-edge and compass’ constructions which are in the curriculum. For example.

Can you imagine the quadrilateral whose diagonals are perpendicular and bisect each other? What sort is it?

In the picture below we want to construct the line through C which is perpendicular to the segment AB. P is any point on AB.

Can you think how to construct a rhombus with C and P as two of its vertices?

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The key idea here is the use of a compass to create points like Q on AB which are at the same distance from C as P, and a point like D which is both the same distance from P as C, and from B as C.

These ideas are not restricted to the geometry of the triangle, rhombus and circle!

In the picture below X’X is a segment on the x-axis and P is any point on it. F is a point on the y-axis.

Can you construct the point Q on the perpendicular to the x-axis at P such that FQ = QP? What is the locus of Q as a function of P?

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Well, Q has to be the vertex of an isosceles triangle QFP, so Q also lies on the perpendicular bisector of FP. The locus of Q with P is the desired curve. As point F slides on the y-axis, so the curve deforms continuously.

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Note: Here Q is the dependent variable, P is the independent variable, XX’ is its domain and the parabolic locus of Q is the graph of some quadratic function! If F is at (0,a) and Q at (x,y) you can use Pythagoras to find an equation connecting x and y and then plot it on the same axes to see if you’re correct.

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We can extend the idea of the parabola construction to that of an ellipse or hyperbola.

We can also extend the ideas, by using measurements, to make models and simulations with dynamic geometry.

In the picture below XY represents a piece of rope 10m long which is bent round to enclose a rectangle ABCD – where D is the midpoint of XY. As A slides on XD so the side length x =AD and area y = ABCD change. These are used to plot the point P (x,y) . The locus of P with A is the graph of the area against side for the rectangle.

For what position of A is the area a maximum? When is the area zero? Can you find an expression for the length of the side AB using x, and hence find the area y as a function of x? Plot the corresponding graph.

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Dynamic geometry software is a powerful exploratory medium at all levels - here are a couple of rather more advanced ideas. The figure below illustrates the iteration of the function f(x) = ax(1-x) . The parameter a can be adjusted using the `slider’. The fixed points of the iterations are the intersections of y = f(x) and y = x.

The point P represents the initial value of the iteration, and P can be slid on a segment of the x-axis. The picture below shows a `staircase’ which converges monotonically to the fixed point attractor. Changes in the values of P and a result in different forms of dynamic behaviour.

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The figure below shows the basis of the Mandelbrot iteration in the complex plane.

Z is any point.

Z2 is interpreted as a rotation of Z to Z’ about O through angle IOZ and a dilation to Z’’ with scale factor OZ/OI.

To complete the mapping to Z’’’=Z2 + c requires a translation by the vector OC.

Now we can apply the mapping repeatedly taking Z0 = C as the starting point.

In the figure below we can move the initial point P around the Argand model of the complex plane and track a large number of successive iterations to look for structure in the dynamic behaviour within the Mandelbrot set.

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We now turn to some hand-held technology: a data-logger which measures distances – the CBR (calculator-based ranger) – used with the TI-83 Plus graphical calculator. This is one of the new breed of GCs which has a large amount of `flash-ROM’ which acts like a hard-disk for storing a range of programs, known as `Apps’. Here we use the CBL/CBR App. to capture distances walked in a straight line.

We can use the left and right cursor keys to trace and read off values from the distance-time graph. We can make "qualitative" observations, such as "walking away", "standing still", walking towards", "constant speed", "faster", "slower", "longer", "shorter" etc. We can also take measurements from the graph and make "quantitative" remarks such as "in the first 4.4 seconds you moved from 0.46m to 2.09m away" or "your average speed was 0.37 m/s". A popular activity is to draw a target distance-time graph on the whiteboard, or OHT overlay, and to ask students to try to match it. When we choose to "QUIT" the time data (in seconds) are stored in list L1 and displacement data (in metres) are in list L2. Pressing [GRAPH] redisplays the last graph we saw, and by pressing [TRACE] we can take readings from it. So we can now try to "fit" a model to any part of the motion. For example you can enter a function in Y1 which you think might match the first part of the distance-time graph.

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The screens above show a first attempt – how could you improve on it?

Now think hard about the various terms in the definition of Y1 above. Y1(X) is a measurement in metres of the distance from the CBR at any time X in seconds. So when you use a function like Y1(X) = 0.37X+0.46 it is clear that 0.46 must also be measured in metres, and that Y1(0)=0.46 so that it is the distance away from the CBR when the time is 0, i.e. the starting, or "initial", displacement. Similarly 0.37X must be in metres, but X is in seconds, so 0.37 must be in metres/sec (or ms-1) and so is a speed (or, since it can be negative, strictly a "velocity"). Hence the gradient of the line represents the average, or constant, velocity.

The BECTa document "Data-capture and modelling in mathematics and science" gives several examples of the use of the CBR including pupils writing about how it helps them understand about straight lines, and the power of "kineasthesis". We will return to the CBR shortly.

For now, though we will look at some algebraic versions of parallel and perpendicular lines.

Enter the equation for a straight-line graph in Y1, plot it and trace it ("ZDecimal" will help to get the aspect ratio and cursor steps OK.) Enter a similar equation for Y2, but change the intercept - what do you think the pair of graphs will look like? Use the TI-83 to check your idea.

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Can you change the equation in Y2 so that the second line is perpendicular to (i.e. makes a right-angle with) the first line? Can you work out a rule so that whatever linear function is defined in Y1 you can make an equation for Y2 so that the two linear graphs are always perpendicular?

Below are some graphs drawn with a graphical calculator - can you find functions to match them?

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The remainder of these examples use the TI Interactive! software on the PC, which is compatible with the graphical calculators and data-loggers used so far. This package integrated features of graph-plotting software (like Omnigraph), data-handling software (like Fathom), symbolic algebra (like Derive) together with MS Office style tools such as a word-processor, spreadsheet, web-browser and e-mailer.

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Here we have defined two linear functions f, g and their product h.

Can you explain why the graph of h is negative for x between -2 and 4? Or positive for x greater then 4 or less than -2. Why is it zero at x = 4 and x = -2?

Can you find an equivalent expression in x whose graph is identical with that of h ?

This is an example where the content (factorisation) is approached from the opposite way than would be conventional.

Now we moved onto quadratic function we could return to logging experimental data to provide examples. The next experiment with the CBR is described in the BECTa book and it is to try and draw a parabola - this time not displaying the graph in "real-time", but over say a 4 second interval. Here the data is captured directly into the TI Interactive! software. The scattergram shows my own (pathetic) attempt.

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Here we are using a quadratic function written in the form:

q(x) = a (x - b)2 + c and varying the "parameters" a, b and c to transform the basic quadratic

y = x2 to fit the data as closely as possible. This is a notation which is better for the purpose than the conventional y = ax2 + bx + c form.

To conclude this brief tour of software support for geometry and algebra here are some examples of the use of TI Interactive! as a Computer Algebra System (CAS) – something which RS/JMC report “Teaching and Learning Algebra pre-19” was still unsure about. Here is an example of some steps in rearranging a linear equation.

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And an example of an approach to solving simultaneous equations by elimination and back-substitution.

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In these examples there are no right or wrong answers – just sets of equivalent expressions. Rather like the use of Logo, the software is just giving dispassionate feedback – learners can adjust their strategy (`debugging’) to `fix’ their errors.

I have tried to illustrate the elements of what I call a `basis tool-kit’ which between them cover all aspects of the content of the KS3 framework, KS4 national curriculum, GCSE and post-16 qualifications including Further Mathematics, and which can be used flexibly for whole class, group, paired and individual work as well as for work outside the classroom. The power of this approach is that each of the elements (dynamic geometry, graphical calculators, data-loggers and integrated mathematics software) spans the whole age and ability range, and can be used by both teacher and learner.

References:

BECTa Curriculum Software Initiative: Mathematics, BECTa, Coventry, 2000

.uk/technology/software/curriculum/reports/maths.html

BECTa Graphical Calculators helpsheet:

.uk/technology/infosheets/html/graphcalc.html

Oldknow, A., ICT and Joined-Up Thinking in the Key Stage 3 Strategy TCT Virtual Conference paper, 2000 .au/tct/papers/week1/oldknow.htm

Oldknow, A., Let’s Integrate: hand-held and conventional ICT, InTegrate, ACITT, Issue 31, Spring 2001

g2fl..uk/acitt/resources/pubs/Integrate%2031/calcs.htm

Oldknow, A. (ed), Teaching and Learning Geometry 11-19, RS/JMC, London 2001 ISBN 0 85403 5656

royalsoc.ac.uk/education/index.html

Oldknow, A. & Taylor, R., Data-capture and modelling in mathematics and science, BECTa, Coventry, 1998 ISBN 1 85379 429 5



Oldknow, A. & Taylor, R., Engaging Mathematics, TCTrust, London 1999 ISBN 873882459 .uk/resources/publicationsp.html

Oldknow, A. & Taylor, R., Teaching Mathematics with ICT, Continuum, London 2000 ISBN0 8264 4806 2 continuum-

Ransom, P., Cross Curricular work using hand-held technology, TTA,London 2000 .uk/publications/community/research/grant/98-99/Peter_Ransom.pdf

Sutherland, R. (ed), Teaching and Learning Algebra pre-19, RS/JMC, London 1997

royalsoc.ac.uk/education/index.html

Notes about the tools:

For information about the RM MathsAlive Framework Edition for KS3 mathematics contact RM plc: Tel: 08709 086700 e-mail: rmls@

RM Maths Alive! secondary/Products/Product.asp?cref=PD7370

The graphical calculator illustrated is the TI-83 Plus Silver Edition which has about 1.44Mb of so-called flash ROM for storing applications such as a spreadsheet, geometry package, science data-base etc. This, together with the pupils’ TI-83 Plus GCs. the CBR (range-finder) and CBL (versatile data-logger), the Viewscreen and Presenter display devices, the TI Interactive! software etc. can be obtained from: Oxford Educational Supplies: tel:01869 344500 fax: 01869 343654

Site licences for Sketchpad, Cabri and TI Interactive! cost around £250-£300. As well as Oxford Educational, suppliers include:

QED Books: tel: 01494 772973 fax: 01494 793951

Chartwell-Yorke: tel: 01204 811001 fax: 01204 811008

Support for schools and colleges wishing to use technology such as illustrated above may be obtained from the Teachers Teaching with Technology (T-cubed or T3) programme administered by the Mathematical Association. For further details see the MA web-site: m-.uk or contact the co-ordinator, Ros Hyde: hyde@tcp.co.uk

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