teaching mathematics using ict notes on chapter 1

www.continuumbooks.com Teaching Mathematics Using ICT Continuum © Adrian Oldknow, Ron Taylor, Linda Tetlow 2010

Teaching Mathematics Using ICT

Notes on Chapter 1

Chapter 1 begins with a series of questions about a picture of a hanging chain. The questions fall into three categories

• Mathematical questions such as what mathematics can be found in the picture. • ICT questions such as how technology could help with exploring the mathematics. • Pedagogical questions such as how the activity could be used in a classroom with

students of different ages and aptitudes.

Some of the questions in chapter 1 are answered briefly later in the chapter. This document provides some additional information to help answer the questions. There are also useful examples later in the book. For example:

Chapter 3 looks at the variety of different types of software available. Chapter 4 looks at examples of ICT in use in the classroomChapter 5 explores in more depth how to use a variety of ICT tools to explore different

aspects of the mathematics curriculum.

The notes in this document are designed to get you started using different tools. There are more detailed examples including more sophisticated use of the software in chapter 5. Sample software files for chapter 1 are in the software section of the website.

1.2(a) Using Dynamic geometry software to fit a circle to the image of the chain

There is more detailed information and ideas on using images and obtaining data from them in Chapter 5 of the book in section 5.5 (c) Modelling using captured images.


Setting up the axes and inserting the image into a dynamic geometry file

Geogebra Cabri GSPSet up the axes

Use the right hand menu icon (11) to move the drawing padRight click on the drawing pad to select x:y axis ratio ( default is 1:1)

Use the right hand icon (11) and select ‘show axes’.

Right click on the drawing pad and select ‘square grid’.

Insert the image

Click on the 2nd from the right menu icon (10) and select ‘insert image’Click on the drawing pad then browse until you find the image you want to insert. Right click on the image and drag to your chosen position.Right click on the image and select ‘fix object’ to fix the position.

Right click on the drawing pad and select ‘background image’ and ‘from file’ Drag the origin of the grid to the corner of the image.

Open a document such as Microsoft Word and insert the image into it. Then ‘copy’ the image onto the clipboard. From the ‘Edit’ menu ‘paste the picture’ as the background. Right click the picture and select ‘properties’ – uncheck the box marked “arrow selectable” so that it can’t be dragged about by accident. From the ‘Graph’ menu select ‘Define coordinate system’ and drag the origin to a suitable point. The point (1,0) can also be dragged to adjust the scale.

Mark some points

Select the point icon 2nd from the left (2) and click on the image to mark some points on the chain. Right click on each point and select ‘properties’. This will enable you to change the colour and style (size) of each point.

Select the point icon 2nd from the left (2) and click on the image to mark some points on the chain. You can change the size and colour of the points by selecting ‘thickness’ or ‘colour’ on the Right hand (11) icon and then clicking on each point in turn.

Select the point tool on the left of the screen and click on the image to mark some points on the chain.


Using geometric constructions to find the centre of the circle

The construction to find the centre of the circle depends on the circle property that

A straight line from the centre of a circle to the mid-point of a chord is perpendicular to the chord.

Conversely this means that

The perpendicular bisector of a chord of a circle passes through the centre of the circle.

Therefore joining pairs of points to form chords and then constructing the perpendicular bisectors of these chords will produce lines that go through the centre of the circle.

Two such lines will intersect at the centre.


Geogebra Cabri GSPConstruct two chords

Use the ‘segment’ tool on icon 3 to join two points to form a chord. Repeat for a second pair of points

Use the ‘segment’ tool on icon 3 to join two points to form a chord. Repeat for a second pair of points

Use the ‘straight edge tool’ on the left of the screen to join two points to form a chord. Repeat for a second pair of points.

Construct their perpendicular bisectors

Use the ‘perpendicular bisector’ tool on icon 4 to construct the perpendicular bisectors of your 2 chords, by moving to the chord until it is highlighted and then selecting it.

Use the ‘perpendicular bisector’ tool on icon 4 to construct the perpendicular bisectors of your 2 chords, by moving to the chord until it says ‘this segment’ and then selecting it.

Use the ‘selection arrow tool’ on the left to highlight the two chords. Then from the ‘construct menu’ at the top select midpoint. Highlight both midpoints and chords and then from the ‘construct menu’ at the top select ‘perpendicular lines’

Mark the centre of the circle

Find the intersection points of your 2 perpendicular bisectors and mark this using ‘intersection point’ on icon 2

Find the intersection points of your 2 perpendicular bisectors and mark this using ‘intersection point’ on icon 2

Highlight the two perpendiculars and then from the ‘construct menu’ select ‘intersection’

Draw the circle.

Use the ‘circle with centre through point’ tool on icon 6 to draw a circle. First select the centre point and then a second point on the chain.

Use the ‘circle’ tool on icon 6 to draw a circle. First select the centre point and then a second point on the chain.

Use either the ‘compass tool’ on the left or highlight the centre point and one point on the chain then choose ‘circle by centre + point’ from the Construct menu.

Notes Geogebra gives the coordinates of the points and equations of the perpendicular bisectors and of the circle in the Algebra view section of the screen

To find the coordinates of points or equations of lines or the circle go to icon 9 and select ‘coordinates or equation’ then select the points, lines or circle and the equation will appear. You can change the format of the circle equation by going to ‘preferences’

Right clicking on a point, line or circle will give options to find coordinates, equations or even the circumference of a circle.


Estimating the length of the chain

Using an arc of a circle as an approximation to the shape of the chain, two pieces of information are needed in order to estimate the length of the chain.

• the angle subtended by the arc at the centre of the circle • the radius of the circle

Dynamic geometry software (DGS) can be used to find the angle at the centre and to measure the radius of the circle using the scale on the software diagram. Being able to estimate the actual radius and hence the length of the chain depends on knowing or being able to estimate one measurement in the image. It would be possible at the time of taking the photograph to measure the highest point of the chain at one of the posts. Alternatively we could estimate the height of the man in the picture as 180cm, but how should this measurement be adjusted to take account of the fact that he is further back in the picture than the chain?

Geogebra Cabri GSPMeasure the angle at the centre

Use the ‘segment’ tool (icon3) to join the two end points of the chain to the centre of the circle. Use icon 8 to measure the angle at the centre by selecting the two line segments you have just constructed.

Use the ‘segment’ tool (icon3) to join the two end points of the chain to the centre of the circle. Use the angle measuring tool on icon 9 to measure the angle at the centre by selecting one of the line segments you have just constructed, then select the centre point and then the second line segment.

Use the ‘straight edge tool’ tool on the left to join the two end points of the chain to the centre of the circle. Use the arrow selection tool to highlight in order first one end point of the chain, then the centre of the circle, finally the other end point of the chain. Then select ‘angle’ from the ‘measure’ menu.

Measure the radius

Use icon 8 to measure the length of the line segments. These will be approximately the same length as the radius of the circle. You might decide to take an average.

Use the distance measure on icon 9 to measure the length of the line segments. These will be approximately the same length as the radius of the circle. You might decide to take an average.

Use the arrow selection tool to highlight the two line segments you have just constructed. Then select ‘length’ from the ‘measure’ menu. These will be approximately the same length as the radius of the circle. You might decide to take an average.


Finally Estimate the actual radius: First mark two points at the ends of a known measurement; join these to form a line segment; measure the length of this. This might be - head to toe on the man or for the chain – where the chain joins the post and the foot of the post. Then use the DGS to measure this distance and work out the scale of the DGS image. Use this scale to work out the actual radius of the circular arc fitting the chain.

Calculate the circumference of the circle then estimate the length of the chain by using the angle at the centre of the circle to find the fraction of the circumference that approximates to the chain.

Important Note: The dynamic geometry software enables the investigation of features such as formulas for mid points or equations of circles in advance of where they would normally be studied in the school curriculum. You might like to reflect on this and how to take advantage of these features in the classroom with the help of later chapters in this book especially chapter 4.

However none of this would be possible without the knowledge of circle properties or of the construction needed to find the centre of the circle. Dynamic geometry software requires the user to think geometrically. This has significant implications for encouraging its use in the teaching of mathematics.


1.2(b) Fitting a quadratic function to the image of the chain.

The Geogebra, GSP and Cabri files created in 1.2A both contain coordinate grids so these could also be used for fitting quadratic functions. You might like to consider questions such as • Which form of a quadratic function to use such as or • The questions that you would ask learners in order to help them think about how to

organise their trials of different functions to fit the curve. • What you would like your learners to get out of the activity- for example increased

understanding about how different transformations affect both the graph and the equation of a quadratic function. How is this question related to the previous two questions?

Geogebra Cabri GSPTo draw the graph of a particular function

Type the equation of the function into the input line. (^ can be used for powers) and press ‘enter’.

On icon 9 select ‘expression’. Click in the drawing pad and key in an expression in x. Then on icon 8 select ‘ evaluate expression’ move to the expression and select ‘this expression’ then move to the x-axis and select ‘this axis’.

From the ‘graph’ menu select ‘new function’ and enter the function using the keypad shown. Then from the ‘graph’ menu select ‘plot function’


Using TiNspireTM

• Open a ‘lists and spreadsheet’ page and use the first two columns for the x and y coordinates of points found from DGS software. Label the columns x and y.

• Insert a ‘graphs and geometry’ page, select menu 4 and appropriate window settings such as ‘zoom square’ for equal scales on the axes. Then from menu 3 select scatter plot and select ‘x’ for the x coordinate and ‘y’ for the y coordinate.

• From menu 3 select ‘function’ and enter your choice of function to fit the scatter plot. • You could systematically change the function you enter to get a better fit but in

TiNspireTM it is possible to select certain types of graph and drag them to fit. If this is done slowly you can see the changes that take place to the equation of the function as it moves. Move towards and grab close to the turning point to translate the graph. Move towards and grab the arms of the graph to increase or decrease the slope of the graph. Is this cheating? What can you learn from doing this?


Using Autograph

• From the axes menu select ‘edit axes’ and then ‘equal aspect’. You can also move the origin and adjust the scales.

• Right click on the drawing pad and select ‘insert image’. Use the pointer tool to select the image and click and drag to move it to the desired position on the grid.

• From the ‘equation’ menu select ‘enter equation’ and enter this using the box provided.


1.2(c) Using data handling features to fit a quadratic function.

Using TiNspireTM

• Open a ‘lists and spreadsheet’ page and use the first two columns for the x and y coordinates of points found from DGS software. Label the columns x and y.

• Select and highlight the x and y columns and from menu 3 select ‘quick graph’ • Adjust the ‘window setting’ menu 5 or drag the axes to a suitable size and scale. • From menu 4 select ‘regression’ and ‘show quadratic’,

Other examples using different software can be found in Chapter 5 sections 5.5 and 5.6


1.2(d) Further research on hanging chains

The theoretical shape that a hanging chain or cable will assume when supported at its ends and acted on only by its own weight is called a catenary. What more can you find out about catenaries from an internet search or from the hyperbolic cosine function associated with them? Can you find any interesting real-life applications?

Graphing calculators or software will allow anyone to input functions such as hyperbolic cosines (cosh x) whether or not they have any further understanding about them. What could be learned from this? How do these functions respond to transformations? What general principles are involved?

This time TiNspireTM does not allow the function to be grabbed and dragged so a trial and improvement method is called for. Here is one attempt. Can you do better? What about using exponential functions?

The other functions tried have been hidden by right clicking/ selecting them and choosing ‘hide/show’. This feature can be used in reverse to reveal the functions tried on the website file. Can you explain how the changes in the function relate to the changes in the graph?


Finding the length of the chain.

If a function has been used to approximate the shape of the hanging chain then the length of the chain could be found using Calculus. If the length of a small section of arc is ds, then

Thus in the limit and the arc length is given by s where a and b are the values of x at either end of the chain.

The integrals are not always straightforward but hyperbolic functions may be easier than you expect.

Using TiNspire: TiNspireTM has arc length included as an application Insert a Calculator page and go to ‘menu 4 Calculus’ and select Arc length. Four inputs

are required in the bracket, separated by commas. They are: (function,x,lower limit,upper limit)

This screen shot shows a comparison for the arc length found using two different functions and also fitting an arc of a circle.


Further references

The ‘Pupil’s Entitlement to ICT in Mathematics’ is available in a document from either of the following links. This gives a variety of examples of different types of mathematical activities using ICT with different software and gives a rationale for using them.

The BECTA website has a link at http://schools.becta.org.uk/index.php?section=cu&catcode=ss_cu_ent_02

In addition: The entitlement document is being hosted on behalf of BECTA on the Teacher Net website: http://www.teachernet.gov.uk/teachingandlearning/subjects/ict/bectadocs/

The National curriculum for England is available from http://curriculum.qcda.gov.uk/key-stages-3-and-4/subjects/index.aspx

Chapter 4 The last few pages of chapter 4 contain details and an evaluation by both pupils and their teacher of an activity about parabolas and the path of a projectile using a catapult.

Chapter 5 Section 5.5 in particular has more information on using data handling tools and also on modelling using captured images. Section 5.6 has examples using applications from other subject areas.

teaching mathematics using ict notes on chapter 1

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