2. Autograph by muhammad qamaruzzaman

Program, 48:

32:

16:

.agg File, 48:

32:

16:

Autograph 3.20 Button Icons MAIN

STATISTICS

2D GRAPHING

[Advanced Level only]

3D GRAPHING

MODES

New Statistics page

Relabel f-x

Reset x-y axes

Restore x-y-z orientation

Select object Marquee select points

New 2D Graph page

Relabel F-x

Reset x-t axes

Restore y-x orientation

Add point

New 3D Graph page

Relabel %-x

Reset x-axis only

Default scales

Scribble

Results Box

Relabel f(x)-x

Default scales

Black background

Rub-out

Key below

Relabel p-x

Cartesian grid

Medium background

Drag (3D: camera)

Key at right

Auto-scale

Polar grid

Light background

Zoom-in

x 1 snap

Discrete plotting

No axes

Show grid

Zoom-out

x 0.1 snap

Continuous plotting

Equal Aspect

Show bounding box

Zoom in ‘x’

Degrees

Enter Grouped data -

Enter XY data

Enter equation

Zoom out ‘x’

Radians

Enter Raw data -

Enter Equation

Enter vector line

Zoom in ‘y’

Edit Axes

Enter Box plot

Enter vector line

Enter vector plane

Zoom out ‘y’

Text Box

Enter Prob. Distribution

Enter Shape

Enter shape

Zoom in ‘z’

Constant Controller

Histogram

Enter 2D Coordinates

Enter 3D coordinates

Zoom out ‘z’

Animation Controller

Cumulative F.D.

Re-plot, and start-up

Re-plot

Rectangle zoom in

Whiteboard Mode

Box Plot

Pause /continue

Rectangle zoom out

Line thickness

Dot Plot

Fast forward

D. E. /Integral solution

Line colour

Sample Means

Slow Plot mode

MAIN

STATISTICS

2D GRAPHING

EXTRAS

3D GRAPHING

Fill colour

Mean and 3 SD

Gradient function

Area of a Circle

Zoom modes on/off

Line Plot

Integral function

Trigonometry

Equation History

Moving Average

Reflection in y=x

Monte-Carlo π

View Keyboard

Histogram Measure

Manage equation list

Dice Simulation

Undo

C.F.D. Measure

Define f(x) and g(x)

Confidence Intervals Poisson Grid

Prob. Distr. measure



File Save

Table of Stats

File Open

Statistics box

Right-Click option

Stem and Leaf diagram

Area under curve

Volume of Revolution

AUTOGRAPH 3.20 May 2007

Name:

Date:

Autograph Activity You are going to investigate the ‘c’ in the equation y=mx+c Open a new graph page in Standard Level Enter equation y= 2x + c and then . . .

choose equal aspect

Use the constant controller to investigate the effect of changing ‘c’ You can change ‘c’ using the up/down arrows and the ‘Step’ using L/R Next click on ‘options’ and then on ‘family plot’. Change the ‘Start’, ‘Finish’ and ‘Step’ values as shown below. Use

etc. to zoom as required . . .

Repeat for equations of the form y = 3x + c and then y = c – 2x Conjecture: Write down, in your own words, the significance of the value of ‘c’ in the equation y = mx + c:

Ask your teacher to check this now!

alan@catley.org 25/10/11

30ciny=mx+c3.20 (1).doc

Autograph Activity Quadratic Graphs - the ‘a’ in y=ax² Open a new graph page in Standard Level Enter equation y= ax² and then select

. . . the Constant Controller

Change the value of ‘a’ and investigate the effect on the graph. Use up/down arrows to change ‘a’ and the L/R arrows to change Step

Choose Options and then ‘Family Plot’ (see below) Change the ‘Finish’ to 2 . . . and ‘Step’ to 0.5 then OK

Choose Options again and this time, change ‘Start’ to -2 as shown above and also experiment with the Animation option! Write down, in your own words, the effect of changing the value of ‘a’ in the equation y = ax²:

ACTION: On the reverse side of this sheet draw two separate ‘sketches’ of the graph of y= x². On one of them add a sketch of the graph y = 0.5 x² and on the other sketch the graph of y = -3x². Label clearly each curve with its equation and give an indication of the scale on all the axes.

020y=ax2 3.20 (1).doc

Autograph Activity Quadratic Graphs - the ‘b’ in y=(x−b)² Open a graph page in Standard Level . . .

Enter equation y=(x−b)²

Edit Axes to x: -4 to 4 and y: 0 to 4 . . . Choose

Equal Aspect

Select the Constant Controller Change the value of ‘b’ and investigate the effect on the graph. Up/Down arrows change value of ‘b’ and L/R arrows change Step

Next click on ‘Options’ and then on ‘family plot’ (see below) Investigate the ‘Animation’

Write down, in your own words, the effect of changing the value of ‘b’ in the equation: y = (x−b)². Illustrate with labelled sketch graphs over this page.

alan@catley.org 25/10/11

022y=(x+b)2 3.20.doc

Autograph Activity Quadratic Graphs - the ‘c’ in y = x²+ c Open a new graph page in Standard Level Edit Axes to x: -5 to 5 and y: 0 to 15 Enter equation y= x²+c . . . and then select the Constant Controller Investigate the effect on the graph of ‘manually’ changing the value of ‘c’ Use Options then Family Plot to display the graphs shown opposite Select any one of the ‘family’ of curves Click on Text Box and tick ‘Show Detailed Object Text’ – it is also a good idea to use one of the ‘Preset Styles’ (Ice Blue here) Zoom in as shown oppposite Add a single point to one of the curves – right click and add ‘Vector’ as shown Change the ‘Snap Settings’ to 0.1 Select the point and move it along the curve using L/R arrows. Also move the point Up/Down! Write down, in your own words, the effect of changing the value of ‘c’ in the equation y = x²+ c. Illustrate with labelled sketch graphs over this page

alan@catley.org25/10/11

024y=x2+c 3.20.doc

Autograph Activity Getting going with 3D Trigonometry Using the diagram opposite, which of these statements are correct? The Length of LN is 15cm The length of XP is 11.9cm The angle between the lines LN and NY is 90° The angle between the lines MP and XP is 19.7° The angle between planes PNYZ and PNXW is 61.8°

To reproduce a dynamic image of the above diagram: Open a new 3D-page on Autograph Use ‘Enter 3D Coordinates’ to add the points L, M, N and P making up the base. Zoom out in order to see all 4 points Select only the points L and P then right click and choose ‘Line Segment’ Repeat this for the other 3 edges of the box as shown opposite

Notes: 1. To enlarge the image hold the Ctrl key and move the mouse 2. To adjust the position of the image hold the ‘Shift’ key and move mouse Repeat the above instructions for the points W, X, Y and Z (the top of the box) Check out the options under ‘Edit Axes’ In the diagram shown opposite the ‘bounding box’ has been removed

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400threeDtrig3.20.doc

Here the axes have been removed to show clearly the dynamic version of the diagram at the top: Click here to open this Autograph file

Here the dynamic image has been viewed from underneath to show the line LN. This ‘Line Segment’ is added by selecting the appropriate points and the a right click

Here the ‘Line Segments’ MP ands XP have been added as they are required to calculate XP. Also the edges of the box that are not required for the calculation of XP have all been made thinner! Again this image can be viewed from any chosen elevation

The two images below show two views of the plane PNYZ and the line PW which helps identify clearly the trigonometry required to calculate the angle between planes PNYZ and PNXW. In order to show the plane shaded (as seen on the left) select three points then right click and choose ‘Group to Shape’ . . . twice!

Click here for the Autograph file of the final image

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400threeDtrig3.20.doc

Autograph Activity The Gradient Function of a Straight Line Follow these instructions to help you understand the notation:

dy  Gradient dx Open a new 2D page (Advanced Level) Add Equation y = mx + c Set Equal Aspect Select the graph and then . . . Text Box Tick the box to ‘Show Detailed Object Text’ . . . also a good idea is ‘Select Preset Style’ (e.g. Ice Blue)

Use the Constant Controller and set: c = - 1, m = 2 as shown below In ‘Slow Plot’ select the . . . Gradient Function Write down why the ‘Gradient Function’ is this graph …………….. Change ‘c’ and then ‘m’ using the Constant Controller and observe the Gradient Graph. What happens when ‘c’ is changed? Why? You should now understand the result shown opposite that for any straight line in the form:

y = mx + c . . .

the ‘Gradient Graph’ (dy/dx) is given by:

dy/dx = m

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y  mx  c dy m dx 20gradstline3.20.doc

Autograph Activity Numerical Approach to the Gradient of a Curve Open a new 2D page

(Advanced Level)

Add Equation y = x² Edit Axes to x: 0 to 4 y: 0 to 10 Attach points (1, 1) and (3, 9) to the curve – these must be attached to the curve! Select both points then from the Object menu select ‘Gradient’ Note the information given in the Status Bar (bottom left of page) Select only the point at (1, 1) and move it along the curve to x = 2 Now select only the point at (3, 9) and move this point to x = 2.9 Use the Zoom Box to enlarge  Confirm the second row of the table below Move the right hand point to help you complete the table below:

( x ) 2

2.9

8.41

Diff in y’s

Diff in x’s

y  4  y

x  2  x

8.41 - 4 = 4.41

2.9 – 2 = 0.9

Gradient 

4.41 0.9

y x

= 4.9

2.5 2.1 Try holding the shift key then the control key when moving the point! 2.05 2.01

4.0401

0.0401

0.01

4.01

2.001 © alan@catley.org

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10NumGrad3.20.doc

Autograph Activity The Gradient Function of a Quadratic Graph You should, from your work with linear functions, understand this notation: Let us now consider the Gradient Function of curves Open a new 2D page

dy  Gradient dx

(Advanced Level)

Add Equation y = x² In ‘Slow Plot’ mode click on . . . . . . Gradient Function. Use the ‘Pause’ button (or the Spacebar) to continue when it stops! Write down the equation of the gradient graph in the space provided The general quadratic function is: y = ax² + bx + c Select the graph of y = x² then use the Object Menu (right click) to Edit Equation as shown here . . . . . . Select Edit Constants - change c to 0 Work out and write down below the equation of the gradient graph: y  x2  x dy  dx

Use the Constant Controller to change values of a, b and c For EACH new graph write down both dy  equations: y  and dx Complete the box opposite by giving the general result for ‘differentiating’ any quadratic function. Ask your teacher to check this!

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y  ax 2  bx  c dy  dx 30diffquad3.20.doc

Autograph Activity The Gradient Function of a Cubic Graph You should, from your work with quadratic functions, know that if:

y  ax2  bx  c Open a new 2D page

then

dy  2ax  b dx

(Advanced Level)

Add Equation y = x3 In ‘Slow Plot’ mode click on . . . . . . Gradient Function. Use the ‘Pause’ button (or the Spacebar) to continue when it stops! Write down the equation of this graph in the space provided The general cubic is: y = ax3 + bx2 + cx + d Select the graph of y = x3 then use the Object Menu to Edit Equation Use Edit Constants and change d to 0 Work out and write down below the equation of the gradient graph: y  x3  x 2  x dy  dx

Use the Constant Controller to change values of a, b, c and d For EACH new graph write down both dy  equations: y  and dx Complete the box opposite by giving the general result for ‘differentiating’ any cubic. Ask your teacher to check this!

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y  ax3  bx 2  cx  d dy  dx

40diffcubic3.20.doc

Autograph Activity The Gradient Graph of y = sinx and an introduction to Radians Open a New Graph Page . . . must be in Advanced Level Select ‘Degrees’ Enter Equation y = sinx Choose ‘Default Scales’ Select ‘Slow Plot’ Choose ‘Gradient Function’ Think about why this Gradient Graph is the shape it is – and what the function could be! Zoom in on the y-axis only You should see that the Gradient Graph is a function of the form:

dy  a cos x dx

where ‘a’ is approximately 0.02 Select ‘Radians’ Choose ‘Default Scales’ Complete this statement for angles measured in Radians:

y  sin x 

dy  dx

Complete the table below by inserting the correct angles in degrees: Radians 0 2 2 3 3     3 4 2 6 4 3 2 Degrees

360

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45gradgraphsinx.doc

Name:

Date:

Autograph Activity The Gradient Graph of other Trigonometric Functions Open a New Graph Page . . . must be in Advanced Level Enter Equation y = cosx Select ‘Radians’ Choose ‘Default Scales’ Select ‘Slow Plot’ Choose ‘Gradient Function’

Complete the statement opposite for angles measured in Radians:

y  cos x 

dy  dx

y  sin 2 x 

dy  dx

Open another New Graph Page Enter Equation y = sin2x Select ‘Radians’ Choose ‘Default Scales’ Choose ‘Gradient Function’

Complete the statement opposite for angles measured in Radians:

Experiment with the ‘Constant Controller’ and the functions y = sinkx and y = coskx until you can . . . Complete the statements below for angles measured in Radians:

y  sin kx 

dy  dx

y  cos kx 

dy  dx

Ensure you get your teacher to check these results!

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47gradgraphstrig.doc

Autograph Activity The Area under a Straight Line Graph On paper sketch the graph of y = −x + 3 and shade the area between: the x-axis, the graph and the vertical lines through x = 1 and x = 2. Calculate this shaded area

Area =

On a new (Advanced) Autograph page . . .

Add Equation y = −x + 3

Edit the axes so that both the x and y axes are from –0.5 to +3.5 Attach a point to the graph precisely at (0,3) and another at (1,2) Return to ‘select’ mode and select both points Right Click and choose ‘Find Area’ from the menu

Confirm (bottom left) the value 2.6 in the table below: FROM x =

TO x =

Rectangle(-) Area =

2.6

Rectangle(+) Area =

Exact AREA under graph =

Complete the other shaded boxes above as follows: ‘Double click’ at one of the rectangles in the area from x = 0 to x = 1 Change to ‘rectangle(+)’ and OK – note the effect and add 2.4 to the table Select only the point at (1,2) and use the keyboard arrow to move to (2,1) Complete the above table – including the ‘all important’ final column! Finally – use either ‘rectangle(-)’ or ‘rectangle(+)’ with 500 divisions (instead of 5) to confirm the last column in your table! You might like to experiment with other equations?

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100AreaLinear3.20 (1).doc

Autograph Activity Estimating the Area under a Curve On paper sketch the graph of y = x² + 3 and state clearly on your sketch the exact y-coordinates of the points on the graph at x = 0, 1, 2, and 3. Shade the area under your graph from x = 1 to x = 2 and write down an ‘educated guess’ of the size of this shaded area. Compare your ‘educated guess’ to the one given in the table below. Complete the ‘educated guess’ column below using your diagram to help you: FROM x =

TO x =

‘Educated guess’ of the area

Area obtained Autograph

To be completed later!

3.333

5.5

On a new Autograph page . . .

Add Equation y = x² + 3

Edit the axes to give: x-axis from –0.5 to +3.5 and y-axis from –1 to 14 Attach one point to the graph precisely at (0,3) and another at (1,4) Select both points, Right Click then Find Area… Choose Trapezium Rule and change Divisions to 50

Confirm (bottom left of screen) the value 3.333 given in the table above.

Complete the rest of the ‘Area obtained using Autograph’ column in the above table as follows: Select only the point at (1,4) - use the keyboard arrow to move to (2,7) Repeat until all except the last column is completed in the table above.

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110QuadArea3.20.doc

Autograph Activity Integration of Quadratics Before you start this you need to understand that the notation: 2



x 2 dx

is used to represent the:

area under the graph of y = x² from x = 1 (known as the lower limit) to x = 2 (known as the upper limit) Use Autograph to obtain the areas in column 2 then complete the table: Integral 1



 



0 3 0

21.33

(using 5000 rectangles!)

(63.99)

Upper Limit3 =

Upper Limit3 3

x 2 dx 

10 0

3 x Area

x 2 dx 

Area

43 = 64

64  3  21 13

x 2 dx  b3 Observe from your results that  0 x dx  - You now know that: 3 b

THE INTEGRAL FUNCTION of y = x² is the function

x3 3

1 i.e. 3

Using the rule of differentiation you will see that: y  13 x 3

differentiating this “integral function”

dy  3  13 x 2  x 2 dx

confirms that “integration” is the REVERSE of “differentiation” We write © alan@catley.org



x 2 dx 

x3  c . . . but why „c‟? – Ask now if you don‟t know! 3 25/10/11

130IntQuad3.20.doc

Autograph Activity Introducing ‘Volumes of Revolution’ Open a new 3D (Adv) Graph Page Restore x-y Orientation Enter Equation y = x + 1 ensuring that you select “Plot as 2D-equation”

Attach two points to the graph at x=0 and at x=2 Select both points and right click to . . . Find Area – use Rectangle (-) with only 2 Divisions

Calculate and write down the volume that would be generated by rotating each of these rectangles through 360° around the x-axis:

a) Using the smaller rectangle b) Using the larger rectangle

Hence write down the total volume when the shaded area is rotated 360° about the x-axis

Total Volume = Continued on next page . . .

alan@catley.org25/10/11

100volrevWS3.20.doc

Checking your answer . . . Using „Slow Plot‟ . . . Select only the Shaded Area Right Click to Find Volume . . . This is known as the:

Volume of Revolution

The „Status Box‟ will display the value of this volume

The shape is shown opposite

It is now possible to change the equation and/or the position of the 2 points on the graph. Similarly rectangles can be changed for trapezium and/or the number of „divisions‟ can be increased.

Finally calculate the value of:



x 2

x 0

 ( x  1)2 dx 

Compare your answer to the volume shown in the above diagram

alan@catley.org25/10/11

100volrevWS3.20.doc

Introducing Volumes of Revolution – A possible ‘Lesson Plan’ Before the lesson set up Autograph and project onto the front board as follows: Open a new 3D graph page and

change to ‘y-x orientation’ (click on the arrow)

Edit the axes as follows: 0 to π for x and −2 to 2 for y (‘Alt P’ gives π) Enter the equation y = sin2x then ‘attach points’ to the graph at x = 0 and x = π/2 This can be done by selecting the graph and using the right click option Enter Point on Graph. Now select the two points and right click to enable ‘Find Area’ – use Trapezium Rule with 5 divisions. The image below should now be displayed. Student action – Each student should now consider the 3D shape that will be produced when the shaded area shown is rotated fully about the x – axis. They should draw a sketch and give a rough estimate of the size of the volume as a decimal and also in terms of π. Teacher Action – introduce the concept of how to find the exact volume using

  y dx between limits. Discuss how to 2

solve the appropriate integration before returning to Autograph. Restore ‘x-y-z orientation’ Choose ‘Slow Plot’ mode Select only the area shown in pink then Right Click and choose ‘Volume’ The ‘Status Box’ (below) will display all relevant details which can be used to confirm values that have been estimated and also calculated using integration.

A whole host of ‘questions’ can now be investigated using the ‘Animation’ options. When animating ‘Volume’ ensure the Slow Plot is selected. Use the ‘zoom’ options to get a closer look!

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102VolsRevLP3.20.doc

’Getting Going with Autograph’ Activity Adding photos using ‘Insert Image’

Open a new Graph Page (Standard Level) Set ‘Equal Aspect’ Use the Object Menu to ‘Add Image’ You will need to have the image you wish to insert saved – for example here (opposite) is an image taken from Multimap which is ideal for work on Scales and Bearings. Add a point, north vector and a line segment. Select the three points to show dynamic angle measuring the bearing) Click here for Autograph file shown Edit Image . . . Double click on an inserted image and choose whether or not you wish to ‘Scale Image with Axes’ as is chosen opposite. Also – drop the Transparency down to about 40% allows viewing the grid behind the image. Other options available as shown here Modelling the Outside World . . . The picture below of the Tyne Bridge in Newcastle shows how mathematics can be used to model the outside world. There are many examples of such engineering structures, buildings, water fountains etc. that can be used in such a way.

Symmetry in Nature . . . The Autograph file below shows a set of points added to the right wing. Select all points then ‘Convert to Data Set’. Double click on the Data Set then choose ‘Join Points’. Finally select the Data Set then use the Object Menu and ‘Reflect in y-axis’ . . . !

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600photos.doc

Autograph Activity Analysing Integer Data – Bar Graphs, Box Plots etc. 

Start by opening all 3 applications – Autograph/Excel/Word

   

Open a new Statistics Page on Autograph Collect your data in Excel and then highlight the column required Tip! - Use the top (one!) cell in the column to name the data set Copy (use Ctrl C) and switch to Autograph (use Alt Tab) Click ‘Enter Grouped Data’ and choose ‘Use Raw data’ – these windows open: Select ‘Integer Data’ Paste the data (Ctrl V)

Nothing appears to happen – but lots of options light up on the ‘Tool Bar’ Histogram

Box Plot etc.

Use Drag/Zoom as required! - these buttons give access to the data values etc. ,

Note - there is an option to enter Raw Data ‘ungrouped’ – if this option is chosen then the data can be grouped later (‘Right Click’ on the ‘Data Set’ in the ‘Key’)

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030IntegerData3.20 (1).doc

Autograph Activity Analysing ‘single variable’ data –

Histograms, Box Plots etc.

Start by opening all 3 applications – Autograph/Excel/Word Open a new Statistics Page on Autograph Collect your data in Excel and then highlight the column required Tip! - Use the top (one!) cell in the column to name the data set Copy (use Ctrl C) and switch to Autograph (use Alt Tab) Click ‘Enter Grouped Data’ and choose ‘Use Raw data’ and then ‘Edit’ Paste the data (Ctrl V) and, if required, tick the ‘Column Header’ boxes as shown. Think about appropriate ‘Class Intervals’ rather than accept those given!

Nothing appears to happen – but lots of options light up on the ‘Tool Bar’ Histogram

Box Plot

Cumulative Frequency etc.

. . . Use the Drag/Zoom options as appropriate to get a clear picture! ,

- these buttons access the data values, tabulated data etc.

Ctrl C / Ctrl V – Diagrams can be copied to Word using the usual ‘copy’ and ‘paste’ The Statistics Box etc. can be copied to Word using ‘Alt Prt Sc’ as shown above It is also possible to copy Tables of Data to Word (covered in another document!)

alan@catley.org 25/10/11

040GroupedData3.20.doc

Autograph Activity Producing a ‘grouped’ data table in Word First you will need to input your data as described in: Autograph Activity – Analysing ‘single variable’ data – Histograms, Box Plots etc Use the Table of Statistics option to open up the results box Highlight the data table (as shown)  ‘Copy’ (Ctrl C)  Switch to Word (‘Alt Tab’)  ‘Paste’ (Ctrl V) To put the data into a neat table: Highlight it in Word and then from the drop down menu use: Table – Convert - Text to Table This should produce a table: Class Int.

Mid. Int. (x)

Class Width

Freq.

Cum. Freq.

160 ≤ x < 170

165 175 185 195 205 215 225 235 245 255 265 275 285 295 305 315

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

0 1 4 6 6 9 6 9 1 2 0 0 3 0 3 0

0 1 5 11 17 26 32 41 42 44 44 44 47 47 50 50

170 ≤ x < 180 180 ≤ x < 190 190 ≤ x < 200 200 ≤ x < 210 210 ≤ x < 220 220 ≤ x < 230 230 ≤ x < 240 240 ≤ x < 250 250 ≤ x < 260 260 ≤ x < 270 270 ≤ x < 280 280 ≤ x < 290 290 ≤ x < 300 300 ≤ x < 310 310 ≤ x < 320

x f

x2  f

Teacher Note - The above table can be edited (e.g. adding columns as shown) and used as a ‘projected’ teaching resource to explain the techniques of, for example, estimating the mean, median, quartiles using approaches such as the coding method and interpolation. Using real data (collected by students) enhances understanding and, of course, correct answers are readily available in the

Statistics Box! alan@catley.org25/10/11

050GpdDataTable3.20 (2).doc

Autograph Activity Moving Averages / Time Series Collect data in Excel then highlight the two columns where e.g. column A is the month and column B is frequency. Open a new Statistics page Choose ‘Enter Grouped Data’ Select ‘Use (x,f) Table’ from the window that opens up then . . . Paste in the data

You will have to ensure that the ‘Data Type’ is marked as Discrete!

Before the Moving Average can be shown you will have to display the Line Graph so: First select Line Graph Then . . . Select Moving Average

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070MovingAves3.20.doc

Using Autograph: Histograms and Frequency Polygons Remember the following:  To copy an Autograph page, use ‘Ctrl+C’. To paste this page into Word use ‘Ctrl+V’  Ensure that your name is included in the footer of any Word files that you print. Do not print direct from Autograph.  Do not get out of your seat to get things from the printer – I will check regularly and hand them out.

When you first open Autograph it gives you a ‘2D Graphing Page’. To use Autograph for most statistics you need to have a ‘1D Statistics Page’ open. Do this by clicking on the icon shown here: Task One: Constructing a frequency diagram (or histogram with equal class widths) Age (years) 0 ≤y< 20 20 ≤ y < 40 40 ≤ y < 60 60 ≤ y < 80 80 ≤ y < 100

Frequency 36 48 20 28 15

Look at the data on the left. 1. To enter the data into Autograph you need to click on the icon here ‘Enter Grouped Data’

2. Choose ‘Enter Manually’ in both places, and type in the relevant information. Click on OK.

It’s done already here – you just need to copy

You can name the data

And here … Remember – it is continuous data

3. You are now ready to construct graphs of the data. There is a box at the bottom left of the screen containing ‘Data Set 1’ (or whatever you chose to call the data in point 2). Right click on this to get the options shown on the right. 4. Choose ‘Histogram’ and select the options as shown in the box on the left. Click on OK to draw the frequency diagram.

M.J.Nixon 2006

5. The axes now need adjusting to make them more sensible. Go to ‘Axes’, ‘Edit Axes’ and you will get the options here. Change the ‘x’ maximum to 120, and the ‘y’ maximum to 50. You can also label the axes by selecting ‘Labels’ and typing in the relevant information (‘Age’ and ‘Frequency’ in this case). Click on OK and you will have a much more sensible graph. Copy into Word, and move on to the next task.

Task Two: Constructing a histogram (with unequal class widths) Age (years) 0 ≤y< 20 20 ≤ y < 30 30 ≤ y < 40 40 ≤ y < 50 50 ≤ y < 70 70 ≤ y < 100

Frequency 28 36 48 20 30 15

Using the data here, follow steps 1 to 3 as in task one. But at step 4, choose the options differently – select ‘Frequency Density’ instead. Click OK and adjust the axes to suit the graph. You should end up with a histogram that looks similar to this one.

Task Three: Constructing a frequency polygon For each of the sets of data in tasks one and two construct a frequency polygon. You will need to select the ‘Draw Frequency Polygon’ option. If you leave ‘Draw Histogram’ selected, it will draw both together.

Task Four: Investigating ‘using raw data’ Acle Ashi Aylebridge Aylsham Barney

91.6 80.8 74.8 91.4 82.4

Barton Bawdswell Beccles Besthorpe Blakeney

84.7 73.2 73.7 73.5 76.1

Braconash Bradenham Briston Brundall Burgh Castle

63.1 58.4 91.5 68.6 76.9

Burnham Mk Burnham Thp Buxton Carbrooke Clenchwarton

63.0 42.2 85.3 93.1 56.0

Coltishall Costessey North Creake Dereham Ditchingham

87.0 74.6 80.2 85.8 70.6

Using the rainfall data above, investigate how to enter this ‘raw data’ into Autograph. Use the program to group the data into intervals of width 10. Construct a frequency diagram with superimposed frequency polygon to represent the data.

M.J.Nixon 2006

Using Autograph: Cumulative Frequency Diagrams

and

Box Plots

Remember the following:  To copy an Autograph page, use ‘Ctrl+C’. To paste this page into Word use ‘Ctrl+V’  Ensure that your name is included in the footer of any Word files that you print. Do not print direct from Autograph.  Do not get out of your seat to get things from the printer – I will check regularly and hand them out.

When you first open Autograph it gives you a ‘2D Graphing Page’. To use Autograph for most statistics you need to have a ‘1D Statistics Page’ open. Do this by clicking on the icon shown here: Task One: Using Autograph to group data The table below shows some countries of the world and their birth rate in 2005 (the number of births per 1000 of the population). You are going to use Autograph to group this data and plot some diagrams to represent it. Afghanistan Albania Algeria American Samoa Andorra Angola Anguilla Antigua & Barbuda Argentina Armenia Aruba Australia Austria

47.0 15.1 17.1 23.1 9.0 44.6 14.3 17.3 16.9 11.8 11.3 12.3 8.8

Azerbaijan Bahamas Bahrain Bangladesh Barbados Belarus Belgium Belize Benin Bermuda Bhutan Bolivia Bosnia & Herzegovina

20.4 17.9 18.1 30.0 12.8 10.8 10.5 29.3 42.0 11.6 34.0 23.8 12.5

Botswana Brazil Brunei Bulgaria Burkina Faso Burma Burundi Cambodia Cameroon Canada Cape Verde Cayman Islands Central African Republic

23.3 16.8 19.0 9.7 44.2 18.1 39.7 27.1 34.7 10.8 25.3 12.9 35.2

Chad Chile China Colombia Comoros Congo Congo, (The …) Costa Rica Côte d’Ivoire Croatia Cuba Cyprus Czech Republic

46.0 15.4 13.1 20.8 37.5 27.9 44.4 18.6 35.5 9.6 12.0 12.6 9.1

1. To enter the data into Autograph you need to click on here ‘Enter Grouped Data’ 2. Choose ‘Use Raw Data’ and Click on ‘Edit’. You can name the data

3. Type the information into the table as shown below. You can use copy and paste from a table or a spreadsheet to speed things up a bit.

Select this … Click on OK when you are finished – and return to the box on the left Look here …

Remember – it is continuous data M.J.Nixon 2006

4. Now you need to tell Autograph how you want the data grouped. The data ranges from 8.8 (Austria) to 47 (Afghanistan). It makes sense to group the data into intervals of width 5, with a minimum of 5 and a maximum of 50. Do this as the diagram here shows, then click on OK. You are now ready to construct graphs and charts, and calculate statistics for this data

Task Two: Using Autograph to construct a cumulative frequency diagram 1. There is a box at the bottom left of the screen containing ‘Data Set 1’ (or whatever you chose to call the data in point 2). Right click on this to get the options shown on the left. Choose ‘Cumulative Frequency Diagram’. 2. Select ‘Cumulative Frequency’ and ‘Linear Fit’ (this will give you a c.f. diagram rather than a c.f. curve). Click on OK to finish 3. The axes now need adjusting to make them more sensible. Go to ‘Axes’, ‘Edit Axes’ and you will get the options here. Change the ‘x’ maximum to 100, and the ‘y’ maximum to 100. You can also label the axes by selecting ‘Labels’ and typing in the relevant information (‘Birth Rate’ and ‘Cumulative Frequency’ in this case) Click on OK and you will have a much more sensible graph. Move on to the next task.

Task Three: Using Autograph to construct a box and whisker diagram (or box-plot) Repeat step 1 of task two. Then select ‘Box and Whisker Diagram’. Leave ‘Raw Data’ selected (to get a more accurate diagram) and click on OK to finish. You can click and drag the diagram (vertically) if you need to. Task Four: Interpreting the data (using dot-plots to help) 1. The box and whisker diagram shows a positive skew. Thinking back to the original data, can you suggest a reason why this might be? 2. You can mark on all the individual pieces of data to help: right click on ‘Data Set 1’ and choose ‘Dot Plot’. Click on OK to get the set of diagrams as shown here. 3. You can now see clearly the pieces of data causing the skewness. Refer back to the original table and identify the countries with high birth rates. Why do you think the data is positively skewed?

M.J.Nixon 2006

Task Five: Comparing data The tables show the number of children per family in 2004 for all countries in Europe and Asia. Enter the data into Autograph and use box and whisker diagrams to compare the information. A thorough comparison should include comments on (a) how the averages compare, (b) how the inter-quartile ranges compare, and (c) any skewness that occurs. Try to give reasons for any observations. You will need to think about the best way in which to group the data. Asia Afghanistan Armenia Azerbaijan

5.17 3.03 2.48

India Indonesia Iran

5.29 3.85 5.48

Lebanon Macao Malaysia

4.56 3.42 4.69

Syria Taiwan Tajikistan

4.88 3.23 4.57

Bahrain

3.96

Iraq

5.19

Maldives

5.48

Thailand

3.63

Bangladesh Bhutan

5.90 4.56

Israel Japan

3.52 2.63

Mongolia Nepal

4.70 4.49

Turkey Turkmenistan

4.74 3.41

Brunei Burma Cambodia

5.05 3.74 6.16

Jordan Kazakstan North Korea

5.67 3.54 5.30

Oman Pakistan Philippines

3.36 7.22 4.94

UAE Uzbekistan Viet Nam

6.41 5.77 3.32

China Cyprus

3.45 2.99

South Korea Kuwait

2.76 6.39

Qatar Saudi Arabia

3.90 5.91

Yemen

4.80

Georgia Hong Kong

3.75 3.23

Kyrgyzstan Laos

5.69 5.33

Singapore Sri Lanka

3.45 5.41

Europe Albania Austria Belarus Belgium Bosnia & Herzegovina Bulgaria Croatia

2.88 2.36 2.50 2.36 3.62 2.67 3.04

Finland France Germany Gibraltar Greece Hungary Iceland

2.16 2.41 2.11 3.62 2.81 2.73 2.31

Liechtenstein Lithuania Luxembourg Malta Moldova Monaco Netherlands

2.55 2.51 2.87 3.38 4.38 2.50 2.31

Romania Russian Federation Serbia & Montenegro Slovakia Slovenia Spain Sweden

2.95 2.69 3.22 2.52 2.88 2.71 2.14

Czech Republic Denmark

2.69 2.19

Ireland Italy

2.96 2.53

Norway Poland

2.27 2.86

Switzerland Ukraine

2.22 2.40

Estonia

2.35

Latvia

2.94

Portugal

2.74

United Kingdom

2.36

NOTE: You could investigate the ‘stem and leaf’, ‘statistics box’ and ‘results box’ capabilities to help you

M.J.Nixon 2006

Using Autograph: Statistical Analysis In this session you need to use the ‘World Statistics’ spreadsheet Remember the following:  To copy an Autograph page, use ‘Ctrl+C’. To paste this page into Word use ‘Ctrl+V’  Ensure that your name is included in the footer of any Word files that you print. Do not print direct from Autograph.  Do not get out of your seat to get things from the printer – I will check regularly and hand them out.

a) Copy and paste the data for highest points in Europe from the ‘World Statistics’ spreadsheet. b) Type in a name for the data c)

Click on OK

You could now construct box and whisker diagrams and/or dot plots – in the same way as shown on the ‘cumulative frequency’ worksheet. You can also use Autograph to calculate statistics for you, as shown in the next tasks.

Task Two: Calculating statistics (Using the ‘statistics box’) 1. Click on the icon shown here: ‘View Statistics Box’. 2. The statistics box will open on screen. In this case – as we are using raw data – the left hand box contains several useful statistical measures. 3. To copy this information, first click on ‘Transfer to Results Box’. Then go to ‘View’, ‘Results Box’. You can copy information from here and paste into Word.

NOTE: The semi-interquartile range is half of the interquartile range.

M.J.Nixon 2006

Task Three: Calculating statistics – grouped data (and demonstrating why unequal class widths and careful consideration of the data are sometimes needed) 1. Enter the largest lake data as grouped data in the same way as shown on the ‘cumulative frequency’ worksheet. (‘Enter Grouped Data’, ‘Use Raw Data’, ‘Edit’, copy and paste, ‘OK’). 2. Construct a histogram, and play about with the axes - you will get something like the graph shown on the left. It isn’t very sensible (and a box plot is totally useless – try it and see!) 3. A glance through the original data suggests that it would be sensible to alter the width of the groups. Double click on ‘Data Set 1’ or whatever you chose to call it. Under class intervals choose ‘Enter Manually’. Type ‘0, 250, 500, 1000, 5000, 60000’ into the box, and redraw the histogram. Adjust the scales to get a more sensible graph. 4. However, a dot plot shows that there are some pieces of data having a drastic effect on the graph (Lake Michigan: 57866km2, Lake Huron: 36001km2, Lake Baykal: 31500km2, Lake Victoria: 30960km2, Lake Malawi: 24400km2). Delete these obvious outliers from the data set and redraw the histogram again – changing the intervals to ‘0, 250, 500, 1000, 5000, 18000’. You should get a graph similar to the one here: It is still an unusual graph, but it does show clearly that there are very few lakes with an area greater than 5000km2, and the distribution of lakes with areas in the other groups is more clear. Zooming in on the histogram shows this (below left). Compare it with a frequency diagram (below right) which is very misleading

5. The results box will contain relevant statistics again, and will include a grouped frequency table which can be copied into Word. Task Four Using the spreadsheet state some hypotheses and test them. Use a variety of graphs and charts to back up your arguments.

M.J.Nixon 2006

Below are a series of Box Plots illustrating how five different classes performed in the same exam. There are 23 students in each class. For each class: (i) Estimate the top and bottom score (ii) Estimate the median and upper and lower quartiles (iii) Write out, in ascending order, possible scores for all the students in each class. Comment on the marks achieved by each class from the numbers you have written down and relate these observations to the shape of the Box Plots

Autograph Activity Scatter Graphs, Line of Best Fit and Correlation Start by opening all three applications: Excel/Autograph/Word Collect the data in Excel – use one cell (top of each column) as a ‘name’ In Excel highlight the 2 columns Copy (Ctrl C) then switch to Autograph Open a new Graph Page Click ‘Add Data’ Paste the data as shown here (Ctrl V) Use ‘column headers’ as axes labels Show Statistics? – see below! The example opposite shows the scatter graph with its Line of Best Fit - added using the Object menu. To ‘Show Statistics’ tick the box (see above) Note (also above) the option to ‘Perform Autoscale’ (it’s your choice!) In Advanced Level the ‘Show Statistics’ option provides lots more detail - as shown opposite. Try holding the ‘Ctrl’ key and moving one of the data items Observe the effect on these values as the point is moved.

alan@catley.org25/10/11

010ScatterGraphs3.20.doc

Autograph Lesson Plan Least Squares Regression – a possible lesson plan Set up Autograph (which is to be projected to the front board): Open a new 2D Graph page – use Standard Level Edit axes – x from 0 to 12 and y from 0 to 8 Choose Equal Aspect – you want squares not rectangles! The image on the board is now ready for action Student Action – Invite 3 students to place a coloured counter at a point of their choice (suggest integer coordinates!). Note you could use 4 or 5 if you prefer but no more than 5 for a theory lesson – 3 works well. A brief discussion can then ensue about correlation and, depending on the level the students are working towards; they can be directed to make calculations of estimated line of best fit, correlation coefficient, regression line. Teacher Action – At the computer: Use ‘Point Mode’ to place a point on Autograph at each point chosen by the group Use the Object Menu to ‘Select All Points’ Use the Object Menu to ‘Convert to Data Set’ Select the Data Set then right click and choose the Centroid option Place a different coloured counter over the Centroid to make it stand out Add another point at some random point in a corner of the graph Select both this point and the Centroid then use the Object Menu to add Straight Line through both points. Ensure ‘snap settings’ are 0.1 Select the random point which can then be moved – discuss various positions of the line with the group. Select the line and the data set - right click. Choose the option ‘y-on-x Residuals’ and these can be displayed either as lines or (as shown) as squares. Move the line as before discussing the changes in ‘Residuals’

alan@catley.org25/10/11

020LstSqReg3.20.doc