Program, 48: 32: 16: .agg File, 48: 32: 16:

Autograph 3.20 Button Icons [Advanced Level only]

MAIN STATISTICS 2D GRAPHING 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

Pause /continue

Rectangle zoom out

Line thickness

Dot Plot

Fast forward

Fast forward

D. E. /Integral solution

Line colour

Sample Means

Slow Plot mode

Slow Plot mode

MAIN STATISTICS 2D GRAPHING 3D GRAPHING EXTRAS

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

Prob. Distr. measure

Poisson Grid

File Save

Table of Stats

File Open

Statistics box Right-Click option Right-Click option

Stem and Leaf diagram

Area under curve

Volume of Revolution

AUTOGRAPH 3.20 May 2007

Name: Date:

alan@catley.org 25/10/11 30ciny=mx+c3.20 (1).doc

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 020y=ax2 3.20 (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.

alan@catley.org 25/10/11 022y=(x+b)2 3.20.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.org25/10/11 024y=x2+c 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

25/10/11 400threeDtrig3.20.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

25/10/11 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

© alan@catley.org 25/10/11 20gradstline3.20.doc

Autograph Activity

The Gradient Function of a Straight Line

Follow these instructions to help

you understand the notation:

dyGradient

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

y mx c

dym

dx

© alan@catley.org 25/10/11 10NumGrad3.20.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 y ( 2x )

Diff in y’s

yy 4

Diff in x’s

xx 2 Gradient

x

y

3 9

2.9 8.41 8.41 - 4 = 4.41 2.9 – 2 = 0.9 9.041.4 = 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 25/10/11 30diffquad3.20.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

dyGradient

dx

Open a new 2D page (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: 2y x x

dy

dx

Use the Constant Controller to

change values of a, b and c

For EACH new graph write down both

equations: y and dy

dx

Complete the box opposite by

giving the general result for

‘differentiating’ any quadratic

function.

Ask your teacher to check this!

2y ax bx c

dy

dx

© alan@catley.org 25/10/11 40diffcubic3.20.doc

Autograph Activity

The Gradient Function of a Cubic Graph

You should, from your work with quadratic functions, know that if:

2y ax bx c then 2dy

ax bdx

Open a new 2D page (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: 3 2y x x x

dy

dx

Use the Constant Controller to

change values of a, b, c and d

For EACH new graph write down both

equations: y and dy

dx

Complete the box opposite by

giving the general result for

‘differentiating’ any cubic.

Ask your teacher to check this!

3 2y ax bx cx d

dy

dx

25/10/11 45gradgraphsinx.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:

cosdy

a xdx

where ‘a’ is approximately 0.02

Select ‘Radians’

Choose ‘Default Scales’

Complete this statement for

angles measured in Radians:

sindy

y xdx

Complete the table below by inserting the correct angles in degrees:-

Radians 0

6

4

3

2

2

3

3

4

3

2

2

Degrees 0 360

Name: Date:

25/10/11 47gradgraphstrig.doc

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: cos

dyy x

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: sin 2

dyy x

dx

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:

sindy

y kxdx

cosdy

y kxdx

Ensure you get your teacher to check these results!

© alan@catley.org 25/10/11 100AreaLinear3.20 (1).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 =

Rectangle(+)

Area =

Exact AREA

under graph =

0 1 2.6

0 2

0 3

1 3

1 2

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?

© alan@catley.org 25/10/11 110QuadArea3.20.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!

0 1 3.333

0 2

0 3

1 3

1 2 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.

© alan@catley.org 25/10/11 130IntQuad3.20.doc

Autograph Activity

Integration of Quadratics

Before you start this you need to

understand that the notation:

2

1

2 dxx 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 Area 3 x Area Upper Limit3 = Upper Limit3 3

1

0

2dxx

2

0

2dxx

3

0

2 dxx

4

0

2 dxx 21.33

(using 5000

rectangles!)

64

(63.99)

43 = 64

3121364

10

0

2 dxx

Observe from your results that 3

3

0

2 bdxx

b

- You now know that:

THE INTEGRAL FUNCTION of y = x² is the function 3

3x i.e.

3

31 x

Using the rule of differentiation you will see that:

differentiating this “integral function” 22

31

3

31

3 xxdx

dy

xy

confirms that “integration” is the REVERSE of “differentiation”

We write cx

dxx 3

32

. . . but why „c‟? – Ask now if you don‟t know!

alan@catley.org25/10/11 100volrevWS3.20.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:

22

0( 1)

x

xx dx

Compare your answer to the volume shown in the above diagram

© Alan Catley alan@catley.org 25/10/11 102VolsRevLP3.20.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 2y dx between limits. Discuss how to

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!

25/10/11 600photos.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’ . . . !

25/10/11 030IntegerData3.20 (1).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’)

alan@catley.org 25/10/11 040GroupedData3.20.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.org25/10/11 050GpdDataTable3.20 (2).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. x f 2x f

160 ≤ x < 170 165 10 0 0 170 ≤ x < 180 175 10 1 1 180 ≤ x < 190 185 10 4 5 190 ≤ x < 200 195 10 6 11 200 ≤ x < 210 205 10 6 17 210 ≤ x < 220 215 10 9 26 220 ≤ x < 230 225 10 6 32 230 ≤ x < 240 235 10 9 41 240 ≤ x < 250 245 10 1 42 250 ≤ x < 260 255 10 2 44 260 ≤ x < 270 265 10 0 44 270 ≤ x < 280 275 10 0 44 280 ≤ x < 290 285 10 3 47 290 ≤ x < 300 295 10 0 47 300 ≤ x < 310 305 10 3 50 310 ≤ x < 320 315 10 0 50

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.org 25/10/2011 070MovingAves3.20.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

M.J.Nixon 2006

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)

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.

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.

And

here …

Remember – it is

continuous data

It’s done already here – you just

need to copy

You can name the

data

Age (years) Frequency

0 ≤ y < 20 36

20 ≤ y < 40 48

40 ≤ y < 60 20

60 ≤ y < 80 28

80 ≤ y < 100 15

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)

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 91.6 Barton 84.7 Braconash 63.1 Burnham Mk 63.0 Coltishall 87.0

Ashi 80.8 Bawdswell 73.2 Bradenham 58.4 Burnham Thp 42.2 Costessey 74.6

Aylebridge 74.8 Beccles 73.7 Briston 91.5 Buxton 85.3 North Creake 80.2

Aylsham 91.4 Besthorpe 73.5 Brundall 68.6 Carbrooke 93.1 Dereham 85.8

Barney 82.4 Blakeney 76.1 Burgh Castle 76.9 Clenchwarton 56.0 Ditchingham 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.

Age (years) Frequency

0 ≤ y < 20 28

20 ≤ y < 30 36

30 ≤ y < 40 48

40 ≤ y < 50 20

50 ≤ y < 70 30

70 ≤ y < 100 15

M.J.Nixon 2006

Look here …

Remember – it is

continuous data

Select this …

Click on OK when you are finished – and return to the

box on the left

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 47.0 Azerbaijan 20.4 Botswana 23.3 Chad 46.0

Albania 15.1 Bahamas 17.9 Brazil 16.8 Chile 15.4

Algeria 17.1 Bahrain 18.1 Brunei 19.0 China 13.1

American Samoa 23.1 Bangladesh 30.0 Bulgaria 9.7 Colombia 20.8

Andorra 9.0 Barbados 12.8 Burkina Faso 44.2 Comoros 37.5

Angola 44.6 Belarus 10.8 Burma 18.1 Congo 27.9

Anguilla 14.3 Belgium 10.5 Burundi 39.7 Congo, (The …) 44.4

Antigua & Barbuda 17.3 Belize 29.3 Cambodia 27.1 Costa Rica 18.6

Argentina 16.9 Benin 42.0 Cameroon 34.7 Côte d’Ivoire 35.5

Armenia 11.8 Bermuda 11.6 Canada 10.8 Croatia 9.6

Aruba 11.3 Bhutan 34.0 Cape Verde 25.3 Cuba 12.0

Australia 12.3 Bolivia 23.8 Cayman Islands 12.9 Cyprus 12.6

Austria 8.8 Bosnia & Herzegovina 12.5 Central African Republic 35.2 Czech Republic 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.

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.

Task Two: Using Autograph to construct a cumulative frequency diagram

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?

You are now ready to construct graphs and charts, and calculate statistics for

this data

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.

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’.

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 5.17 India 5.29 Lebanon 4.56 Syria 4.88

Armenia 3.03 Indonesia 3.85 Macao 3.42 Taiwan 3.23

Azerbaijan 2.48 Iran 5.48 Malaysia 4.69 Tajikistan 4.57

Bahrain 3.96 Iraq 5.19 Maldives 5.48 Thailand 3.63

Bangladesh 5.90 Israel 3.52 Mongolia 4.70 Turkey 4.74

Bhutan 4.56 Japan 2.63 Nepal 4.49 Turkmenistan 3.41

Brunei 5.05 Jordan 5.67 Oman 3.36 UAE 6.41

Burma 3.74 Kazakstan 3.54 Pakistan 7.22 Uzbekistan 5.77

Cambodia 6.16 North Korea 5.30 Philippines 4.94 Viet Nam 3.32

China 3.45 South Korea 2.76 Qatar 3.90 Yemen 4.80

Cyprus 2.99 Kuwait 6.39 Saudi Arabia 5.91

Georgia 3.75 Kyrgyzstan 5.69 Singapore 3.45

Hong Kong 3.23 Laos 5.33 Sri Lanka 5.41

Europe Albania 2.88 Finland 2.16 Liechtenstein 2.55 Romania 2.95

Austria 2.36 France 2.41 Lithuania 2.51 Russian Federation 2.69

Belarus 2.50 Germany 2.11 Luxembourg 2.87 Serbia & Montenegro 3.22

Belgium 2.36 Gibraltar 3.62 Malta 3.38 Slovakia 2.52

Bosnia & Herzegovina 3.62 Greece 2.81 Moldova 4.38 Slovenia 2.88

Bulgaria 2.67 Hungary 2.73 Monaco 2.50 Spain 2.71

Croatia 3.04 Iceland 2.31 Netherlands 2.31 Sweden 2.14

Czech Republic 2.69 Ireland 2.96 Norway 2.27 Switzerland 2.22

Denmark 2.19 Italy 2.53 Poland 2.86 Ukraine 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.

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: Entering raw data

1. You can enter data as a simple list by clicking on ‘Enter Raw Data’.

2. The box below will appear on screen.

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.

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.

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.

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

alan@catley.org25/10/11 010ScatterGraphs3.20.doc

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 020LstSqReg3.20.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’