Snehalathak at 11:44, 5 October 2018

2018-10-05T11:44:04Z

Madhurig at 07:04, 28 September 2018

2018-09-28T07:04:51Z

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Vidhya at 13:42, 12 September 2018

2018-09-12T13:42:18Z

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Vidhya: Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on ''' Curve Fitting'''. |- | | '''Slide Num..."

2018-09-04T08:23:41Z

Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on ''' Curve Fitting'''. |- | | '''Slide Num..."

New page

{|border=1
| | '''Visual Cue'''
| | '''Narration'''

|-
| | '''Slide Number 1'''

'''Title Slide'''
| | Welcome to this tutorial on ''' Curve Fitting'''.
|-
| | '''Slide Number 2'''

'''Learning Objectives'''

Demonstrate an interactive '''PhET simulation'''
| | In this tutorial, we will demonstrate '''Curve Fitting PhET simulation'''.
|-
| | '''Slide Number 3'''

'''System Requirements'''

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' 60.0.2
| | Here I am using,

'''Ubuntu Linux OS''' version 16.04

'''Java''' version 1.8.0

'''Firefox Web Browser''' 60.0.2
|-
| | '''Slide Number 4'''

'''Pre-requisites'''
| | The learner should be familiar with topics in high school mathematics.
|-
| | '''Slide Number 5'''

'''Learning Goals'''

Lines '''y=ax + b'''

'''Quadratic polynomials y = ax2+bx+c'''

'''Cubic polynomials y= ax3 + bx2 + cx + d'''

'''Quartic polynomials y =''' '''ax4 + bx3 + cx2 + dx + e'''

'''Reduced chi squared statistic χr2''' and '''correlation coefficient r2'''
| | Using this '''simulation''' we will look at,

Lines of the form '''y=ax + b'''

'''Quadratic polynomials y equals ax squared plus bx plus c'''

'''Cubic polynomials y equals ax cubed plus bx squared plus cx plus d'''

'''Quartic polynomials y equals ax raised to 4 plus bx cubed plus cx squared plus dx plus e'''

'''Reduced chi squared statistic''' and '''correlation coefficient r squared'''
|-
| | '''Slide Number 6'''

'''Binomial Theorem'''

'''Binomial theorem''' states that if ''a, b'' ℝ, index ''n'' is a positive '''integer''', ''0 ≤ r ≤n, then,''

''(a + b)n <nowiki>= </nowiki>nC0 an + nC1 an-1 b1 + nC2 an-2 b2 + … + nCr an-r br + … + nCn bn''

'''Reminder:''''' nC1 = n!/[1! (n-1)!]''
| | '''Binomial Theorem'''

'''''a''''' and '''''b''''' are '''real numbers''', '''index''' '''''n''''' is a '''positive integer'''.

'''''r''''' lies between 0 and '''''n'''''.

'''Binomial theorem''' states that '''a''' plus '''b''' raised to '''n''' can be expanded as shown.
|-
| |
| | Let us begin.
|-
| | '''Slide Number 7'''

'''Link for PhET simulation'''

[http://phet.colorado.edu/ http://phet.colorado.edu]
| | Use the given link to download the '''simulation'''.

[http://phet.colorado.edu/ http://phet.colorado.edu]
|-
| | Show the '''Downloads''' folder.
| | I have already downloaded '''Curve Fitting simulation''' to my '''Downloads''' folder.
|-
| | Press Ctrl+Alt+T to the terminal.

Type '''cd Downloads''' >> press '''Enter'''.

Type '''java space hyphen jar space equation-grapher_en.jar'''.

Point to the opened '''file''' format.
| | To open the '''jar file''', open the '''terminal'''.

At the '''terminal prompt''', type '''cd Downloads''' and press '''Enter'''.

Type '''java space hyphen jar space curve hyphen fitting underscore en period jar'''.

'''File''' opens in the '''browser''' in html '''format'''.
|-
| | '''Cursor''' on the '''interface'''.
| | This is the '''interface''' for the '''Curve Fitting simulation'''.
|-
| | Point to the '''Help button''', the '''Functions''' box and the '''Data Points''' bucket in the first quadrant.

Point to '''Linear''' and '''Best Fit''' default selections.
| | Observe the '''Help button''', the '''Functions''' box and the '''Data Points''' bucket in the first quadrant.

In '''Functions''' box, '''Linear''' and '''Best Fit radio buttons''' are default selections.
|-
| | Click the '''Help button'''.
| | Let us click the '''Help button'''.
|-
| | Point to the legend for '''draggable error bars''' in the first quadrant.

Point to the data point bucket.
| | A legend for '''draggable error bars''' appears in the first quadrant.

The data points can be pulled out or put in the bucket.
|-
| | Point to the '''Best Fit equation''' in the 4th quadrant.

Point to the '''display''' boxes for '''a''' and '''b'''.

Point to the equation '''y = a + bx'''.

Point to '''r2'''.
| | In the fourth quadrant, '''Best Fit equation''' is seen with the '''display''' boxes for '''a''' and '''b'''.

The equation is '''y equals a plus bx'''.

Below the '''display''' boxes is the '''correlation coefficient r squared'''.
|-
| | Point to the '''χr2''' scale.

Point to the formula in the '''Help''' box.
| | In the 2nd and 3rd quadrants is a scale for the '''reduced chi squared statistic'''.

The formula for the '''chi squared statistic''' is given in the '''Help''' box.
|-
| | Point to the '''conditions for fit''' in the '''Help''' box.
| | Below the formula, we see the '''conditions for fit'''.

Good or very good fit of data with the equation is seen with a '''chi squared statistic''' of or below 1.
|-
| | Click on '''Hide Help'''.
| | Let us click on '''Hide Help''' to hide these boxes.
|-
| | Drag three data points out of the bucket.

Place them at '''(-10, -4)''', '''(-4, 4)''' and '''(5, 10)'''.

Place the mouse on the '''co-ordinates''' to show them.
| | Drag three data points out of the bucket.

Place them at '''-10 comma -4''', -'''4 comma 4''', and '''5 comma 10'''.

Placing the mouse on them will show their '''co-ordinates'''.
|-
| | Point to the equation '''y = 6.07 + 0.912 x'''.

Point to '''r2 '''<nowiki>= 0.9616. </nowiki>

| | Note that the equation for the best fit line drawn is '''y equals 6.07 plus 0.912 x'''.

The '''correlation coefficient r squared''' for the '''best fit line''' is 0.9616.

The closer the '''r squared''' value is to 1, the better is the prediction of '''variance''' in '''y''' from '''x'''.
|-
| | Point to '''χr2''' = 6.74 and the red bar.

Click on '''Help''' and to conditions for a poor fit.
| | Note that the '''reduced chi statistic''' is 6.74 but the bar is red.

Click on '''Help''' and note that this means that the fit is poor.
|-
| | Drag another data point and place it at '''(0, 11)''' on the '''y axis'''.

Point to the '''best fit line''', '''y = 7.51 + 1.004 x'''.
| | Let us drag another data point and place it at '''0 comma 11''' on the '''y axis'''.

Note that the '''best fit line''' becomes '''y equals 7.51 plus 1.004 x'''.
|-
| | Point to the slope of the '''best fit line''', 1.004.

Point to the '''y intercept''' of 7.51.

Point to the data point '''(0, 11)'''.

Point to '''r2'''<nowiki>= 0.8529. </nowiki>

| | The slope of the '''best fit line''' has increased slightly from 0.912 to 1.004.

The '''y intercept''' has also increased from 6.07 to 7.51.

The data point '''0 comma 11''' is further away from the '''best fit line''' than the other points.

Note how the '''r squared''' value decreases from 0.9616 to 0.8529.

The prediction of '''variance''' in '''y''' from '''x''' with this equation has become less reliable.
|-
| | Point to the '''χr2''' of 18.66.
| | Note also how the '''reduced chi squared statistic''' has increased from 6.74 to 18.66.
|-
| | Drag the data point from '''(0, 11)''' to '''(0, 6''').

Point to the equation '''y = 6.05 + 0.911 x'''.

Point to '''r2''' = 0.9635 and''' χr2''' of 3.37.
| | Drag the data point from '''0 comma 11''' to '''0 comma 6'''.

Note how the equation becomes '''y equals 6.05 plus 0.911 x'''.

The '''r squared''' value increases to 0.9635 and the '''reduced chi squared statistic''' falls to 3.37.
|-
| | Drag the data point from '''(-4, 4)''' to '''(-4, 3.5)'''.

The '''r2''' value increases to 0.9772 and '''χr2 '''falls to 2.12.

Point to the green bar.

Click on '''Help''' and to the green zone indicating good fit.

Click on '''Hide Help'''.
| | Drag the data point from '''-4 comma 4''' to '''-4 comma 3.5'''.

The '''r squared''' value increases to 0.9772.

The '''reduced chi squared statistic''' falls to 2.12.

The bar now becomes green.

Click on '''Help'''<nowiki>; the green zone shows good fit. </nowiki>

Click on '''Hide Help'''.

A true '''best fit line''' explains all the data and gives a good prediction of '''y''' values from '''x''' values.
|-
| | Click '''Adjustable Fit'''.

Drag '''sliders a''' and '''b''' to values close to 0.

Show space where erased line was seen.

Point to the line that is now parallel to the '''x axis'''.

Point to the '''display''' boxes for '''a''' and '''b''' and to '''sliders a''' and '''b'''.

Point to the data points.

Point to the red bar and '''χr2'''.

Point to the '''r2''' value of 0.
| | Click '''Adjustable Fit radio button'''.

Drag '''sliders a''' and '''b''' to values close to 0.

Observe how this erases the line drawn earlier.

A line parallel to the '''x axis''' is seen.

'''Slider a''' and '''b''' values will be displayed in the boxes.

The data points are still where we placed them.

But the '''reduced chi square statistic''' is very high and in the red zone.

And the '''r squared''' value is 0, meaning poor correlation.
|-
| | Click '''Best Fit '''again.

Note down the values for '''a''' and '''b''' (5.94 and 0.918).

Again, click '''Adjustable Fit'''.

Now drag '''sliders a''' and '''b''' and point to the line.

Point to the line.

Drag '''slider a''' to 6 and '''b''' to 0.97.

Point to the line, '''r2''' (0.9709) and '''χr2''' (2.23).
| | Click '''Best Fit radio button''' again.

Note down the values for '''a''' and '''b''' (5.94 and 0.918).

Again, click '''Adjustable Fit radio button'''.

Now drag '''sliders a''' and '''b''' from end to end.

Observe the effects of these changes on the line.

Drag '''slider a''' to 6 and '''b''' to 0.97.

The line looks like the '''best fit line''' we saw earlier.

Note '''r squared''' and the '''reduced chi squared statistic'''.
|-
| | Check '''Show deviations''' and click '''Best Fit'''.

Point to the vertical lines from the data points to the '''best fit line'''.
| | Check '''Show deviations''' and click '''Best Fit'''.

The vertical lines from the data points to the '''best fit line''' show the deviations from the line.
|-
| | Drag the data points at '''(-4, 3.5)''' and '''(0, 6)''' into the bucket.

Point to the line and the two points.

Point to '''r2 '''and '''χr2'''.

| | Drag the data points at '''-4 comma 3.5''' and '''0 comma 6''' into the bucket.

Note how the line now passes through the two points.

'''R squared''' approaches 1 and the '''reduced chi squared statistic''' becomes 0.

The fit has become too good because a line is defined by two points.

Without a third point, there is no question of the line being anything but the '''best fit line'''.
|-
| |
| | Now, we will look at some information for you to graph a '''quadratic polynomial'''.
|-
| | '''Slide Number 9'''

Quadratic polynomials

FIGURE

y = a + bx + cx2

Degree = 2; quadratic

Maximum 2 roots

(-9, 10), (-7, 2), (2.5, -2.5), (5, 10)

a = -7.89, b = 1.495, c = 0.396

r2 = ?,''' χr2''' = ?

Adjustable Fit
| | '''Quadratic polynomials'''

'''Quadratic polynomials''' are of the form '''y equals a plus bx plus c x squared'''.

The degree of the '''polynomial''' is 2, hence, it is called '''quadratic'''.

The '''function''' can have a maximum of 2 roots.

Drag and place data points at the following '''co-ordinates'''.

'''-9 comma 10''', '''-7 comma 2''', '''2.5 comma -2.5''' and '''5 comma 10'''

Note the '''r squared''' and '''reduced chi squared statistic''' values. (0.9794, 4.23)

Also, click '''Adjustable Fit''' and see effects of '''a''', '''b''' and '''c''' on the fit.
|-
| | Show the '''best fit'''graph for the '''quadratic polynomial'''.
| | This is what the '''best fit''' graph for this '''quadratic polynomial''' will look like.
|-
| |
| | Now, we will look at some information for you to graph a '''cubic polynomial'''.
|-
| | '''Slide Number 10'''

Cubic polynomials

FIGURE

y = a + bx + cx2 + dx3

Degree = 3; cubic

Maximum 3 roots

(-9, 10), (-7, 2), (-6, -4), (5, 10), (13, 2)

r2 = ?,''' χr2''' = ?

Adjustable Fit
| | '''Cubic polynomials'''

'''Cubic polynomials''' are of the form '''y equals a plus bx plus c x squared plus d x cubed'''.

The degree of the '''polynomial''' is 3, hence, it is called '''cubic'''.

The '''function''' can have a maximum of 3 roots.

Drag and place data points at the following '''co-ordinates'''.

'''9 comma 10, -7 comma 2, -6 comma -4, 5 comma 10''' and '''13 comma 2'''

Note the '''r squared''' and '''reduced chi squared statistic''' values.

Also, click '''Adjustable Fit''' and see effects of '''a, b, c''' and '''d''' on the fit.
|-
| | Show the '''best fit''' graph for the '''cubic polynomial'''.
| | This is what the '''best fit''' graph for this '''cubic polynomial''' will look like.
|-
| |
| | Now, we will look at some information for you to graph a '''quartic polynomial'''.
|-
| | '''Slide Number 11'''

Quartic polynomials

FIGURE

y = a + bx + cx2 + dx3 + ex4

Degree = 4; quartic

Maximum 4 roots

(-9, 10), (-7, 2), (-6, -4), (5, 10), (9, 3) (13, 2)

r2 = ?,''' χr2''' = ?

Adjustable Fit
| | '''Quartic polynomials'''

'''y equals a plus bx plus c x squared plus d x cubed plus e x raised to 4''' is a '''quartic polynomial'''.

The degree of the '''polynomial''' is 4, hence, it is called '''quartic'''.

The '''function''' can have a maximum of 4 roots.

Drag and place data points at the following '''co-ordinates'''.

'''-9 comma 10, -7 comma 2, -6 comma -4, -3 comma -8, 5 comma 10, 9 comma 3 and 13 comma 2'''

Note the '''r squared''' and '''reduced chi squared statistic''' values.

Also, click '''Adjustable Fit''' and see effects of '''a, b, c, d''' and '''e''' on the fit.
|-
| | Show the '''best fit''' graph for the '''quartic polynomial'''.
| | This is what the '''best fit''' graph for this '''quartic polynomial''' will look like.
|-
| | '''Slide Number 12'''

'''Assignment'''
| | As an '''assignment''',

Change the data points and their number.

Follow the steps shown earlier to get '''best fit''' graphs for all the '''polynomials'''.
|-
| | '''Slide Number 13'''

'''Summary'''
| | In this '''tutorial''', we have demonstrated the

'''Curve Fitting PhET simulation'''
|-
| | '''Slide Number 14'''

'''Summary'''
| | Using this '''simulation''', we have looked at:

Lines of the form '''y=ax + b'''

'''Quadratic polynomial functions''' of the form '''y= ax2+bx+c'''

'''Cubic polynomial functions''' of the form '''y = ax3 + bx2 + cx + d'''

'''Quartic polynomial functions''' of the form '''y = 'ax4 + bx3 + cx2 + dx + e'''

'''Reduced chi square statistic χr2''' and '''correlation coefficient r2'''
|-
| | '''Slide Number 15'''

'''About the Spoken Tutorial Project'''

Watch the video available at http://spoken-tutorial.org/ What_is_a_Spoken_Tutorial

It summarizes the Spoken Tutorial project

If you do not have good bandwidth, you can download and watch it

| | The video at the following link summarizes the '''Spoken Tutorial project'''.''' '''

Please download and watch it
|-
| | '''Slide Number 16'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project '''team conducts workshops using '''spoken tutorials''' and gives certificate courses to learn the use of open source software.

For more details, please write to us.
|-
| | '''Slide Number 17'''

'''Forum for specific questions:'''

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them
| | Please post your timed queries in this forum.
|-
| | '''Slide Number 18'''

'''Acknowledgement'''
| | This project is partially funded by '''Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching'''.
|-
| | '''Slide Number 19'''

'''Acknowledgement'''
| | '''Spoken Tutorial Project''' is funded by '''NMEICT''', MHRD, Government of India.

More information on this mission is available at this link.
|-
| |
| | This is '''Vidhya Iyer''' from '''IIT Bombay'''.

Thank you for joining.
|-
|}

@@ Line 83: / Line 83: @@
 Type '''java space hyphen jar space equation-grapher_en.jar'''.
 Point to the opened '''file''' format.
@@ Line 90: / Line 92: @@
 Type '''java space hyphen jar space curve hyphen fitting underscore en dot jar'''.
 '''File''' opens in the '''browser''' in '''html''' format.
@@ Line 165: / Line 169: @@
 Click on '''Help''' and to conditions for a poor fit.
-||Note that the '''reduced chi statistic''' is 6.74 but the bar is red.
+||Note that the '''reduced chi squared statistic''' is 6.74 but the bar is red.
 Click on '''Help''' and note that this means that the fit is poor.
 |-
 ||Drag another data point and place it at '''(0, 11)''' on the '''y axis'''.
@@ Line 498: / Line 503: @@
 |-
 ||
-||This is '''Vidhya Iyer''' from '''IIT Bombay'''.
+||This is '''Vidhya Iyer''' from '''IIT Bombay''' signing off.
 Thank you for joining.
 |-
 |}

PhET/C3/Curve-Fitting/English - Revision history

Snehalathak at 11:44, 5 October 2018

Madhurig at 07:04, 28 September 2018

Vidhya at 13:42, 12 September 2018

Vidhya: Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on ''' Curve Fitting'''. |- | | '''Slide Num..."