burchellmath

paper: tactile functions

2011-12-23T12:54:00.004+05:30

Abstract:

This article discusses the construction of tactile functionsin the dynamic geometry environment. Technology now allows us to visually/graphically define functions responding to informalcosmetic directives while inheriting the analytical properties of a functionobject in a software environment. The specific topic explored is the Fundamental Theorem of Calculus.

tactile functions - piecewise cubic

2011-12-18T20:35:00.000+05:30

Download: TactileFunction_PiecewiseCubic.gsp

This sketch follows an earlier experiment with piecewise linear tactile functions. These functions are defined using a small number of user-positioned independent points which use a simple convention to define a function. The linear one offered what I thought to be an interesting look at the antiderivative. This sketch shows a piecewise cubic function, meaning that consecutive independent points are joined with a segment of a cubic function.

Let me know what you think.

There is something called a Bezier curve that involves cubics and addresses some similar applications, but I am really not sure if my construction is different.

I decided that I wanted a manipulable function that was smooth on its domain. I could not do that with lines, and quadratics offer only the most unwieldy options if we insist on differentiability. I did make one with sinusoidal curves, but the first construction I tried demanded in horizontal tangents at each of the independent points, which was a bit limiting. In the end, I realized it needed to be cubic and I was pretty sure that a unique curve existed where endpoints and slopes (at the endpoints) were defined.

I'm not sure if it is good for anything.

In this sketch, the derivative (red) and antiderivative (blue) can be shown and hidden easily. The curve in the construction is really seven distinct objects, but it would be possible to create it as a single function object.

waiting for cab 1729

2011-12-01T10:31:00.000+05:30

Korean license plates have a few small Korean characters followed by four large digits. I have not devoted myself to the study of Korean, so the first few characters do little to distract me. Mostly I enjoy the numbers. A few weeks ago, it occurred to me that there may be a taxicab in this city with the plate "1729", which would be supercool. So now I can't stop looking for it.

The significance of 1729 as a taxi number is a legend of mathematics history. G.H. Hardy visited Ramanujan at a hospital and noted that his taxi was number 1729, an (unfortunately) uninteresting number. Ramanujan replied that 1729 is in fact interesting because it is the smallest number that can be written as the sum of two cubes in two different ways. 1729=1³ + 12³ = 9³ + 10³.

It would be most satisfactory to see it on a taxi, but I find myself checking other license plates just in case. I saw 1728 once, which is very close and is 12³, which is in fact part of why 1729 is special. It seems that this would be a disappointing game with occasional near misses. If I ever found cab 1729 it would surely be anticlimactic at this point.

Unexpectedly, it is the near misses that have proved to be the most stimulating part of the game. I was surprised to find myself coming up with arguments in favor of each number being somehow "close" to 1729. For example 1279, 1698, 3729, and 3458 would all be close to 1729 for different reasons. What if a license plate has two or three of the right digits? (2175, 9243). It's better if they are in the correct places (1754, 1889), or if the wrong digits somehow resemble the correct missing digit (1724, 1759, 1429). Given the variety, I started to wonder, what is the probability that a given four digit integer--at least a little bit--resembles 1729? After all, there is about an 87% chance that a license plate contains at least one of the digits {1, 7, 2, 9}.

I can hardly walk around Seoul anymore without getting sucked into this game, which is far from a voluntary obsession. It is really more like humming the Pororo song that plays on a loop at E-mart. Kind of catchy.

When I am in the US, I always try to find acronyms for the letters on license plates. I'm just throwing that out there in case you need any traffic obsessions of your own. Mercifully, I do not understand Korean enough for that one to take hold of me.

favorite number?

2011-11-30T06:58:00.001+05:30

I encourage my students to start thinking about a favorite number. Sometimes I ask if anyone has a favorite number, and I get a few students tossing up a 'random' number to make their point that any number is as good as the next. I get a few students staring blankly, like numerical favoritism is for nerds. And it is. I get some students who see that a number could be pleasing in some sense, and they are able to choose a favorite, or a few that stand out as candidates.

A number can be fun to write. Its numerical representation can possess some visual aesthetic quality. It can be just the right size. It can occur in interesting contexts. It can play a role in elegant statements describing a complexity that has chosen to highlight a particular number.

My own favorite number is 17. It is small, but not too small. It is prime, a Fermat prime in fact. There are 17 distinct wallpaper patterns. Gauss showed that you can construct a regular 17-sided polygon with compass and straightedge. The smallest number of clues necessary for a uniquely solvable Sudoku puzzle is believed to be 17. But I digress. I was just giving an example of a favorite number (existential instantiation), I am not trying to make you feel bad about your own favorite number.

slope field swarm

2011-11-16T13:04:00.001+05:30

Download (for GSP v4 or v5): Calculus_SlopeFieldSwarm_v4.gsp
This amused me for awhile today. I wanted to see a slope field constructed from a single perspective, not delivered as a completed product. I made a point with a segment whose slope was determined by a differential equation, and I enjoyed painting the slope field into being. This was time consuming, however, and I got the idea of animating and duplicating the point.

calculus - simpson's rule

2011-11-09T20:18:00.000+05:30

Download (GSP 5 file): calculus_SimpsonsRule.gsp

This sketch shows the area measured by Simpson's Rule using parabolic arcs. The area beneath a parabolic arc was known to the Greeks ages before calculus was used, and Simpson's Rule is a slick application of this fact. The interval is divided into an even number of sub-intervals and a parabolic arc is chosen for each sub-interval. The application of the rule does not of course involve actually finding the parabolas, but I found this graphical exploration illuminating.

word spelling

2011-10-11T19:38:00.001+05:30

It has occasionally bothered me that the standard Microsoft Word spelling dictionary knows about "dodecahedron" but not "icosahedron". It does acknowledge "semi-duality", which I suppose explains the deeper issue.

TIMPS

2011-09-15T07:15:00.003+05:30

Today I learned that the TI-89 will factor the Mersenne number 2⁷¹-1, which is equal to 228479*48544121*212885833. Mine probably took a minute or more, not blink-of-an-eye Wolfram|Alpha fast, but nonetheless dazzling in light of the enormity of the number. I would not have thought to try this, but a student made short work of a bonus question this way. I am sure that this functionality is not news to many who will read this, but I felt that if this blog is to be any sort of log of my nerdy amusements, I should probably make a note. Take it with a grain of salt, since I am easily amused, but this is a highlight of my day.

mourning collatz

2011-06-02T10:26:00.002+05:30

I just saw a post on The Endeavor about the solution of the Collatz conjecture. Here is the paper which claims to prove the conjecture. I can't understand it.

My first reaction was disappointment. I think because I always felt capable of enjoying the problem, yet it's pretty clear at a glance that I am not capable of understanding the proof. It was a very accessible unsolved problem, but as a proven statement, I suppose its simplicity will make it less interesting, reduced to a bit of trivia and not the sort of treasure hunt that fills up pages of my paper.

My second reaction is that Gerhard Opfer must have really enjoyed proving this, and I envy him that thrill. I've only ever proven things that other people already knew.

calculus: solids by cross section

2011-01-20T19:58:00.004+05:30

This is something I've been picking away at for a few years now, and I finally realized that I could probably use Nick Jackiw's 3D plotting toolkit from the Sketchpad site, which has many great resources that hint at some of the powerful uses of this software. I've had one that never was complete enough or correct enough to post, so it's nice to get one that works nicely. This is a fun sketch. Don't mess with the function, but the cross sections can be changed for some variety and the real utility here is that it gives a look at this family of solids which come standard in AP Calculus, but are not very easy to draw. This is a GSP 5 file. Let me know if there are issues downloading it, I have posted it in a hurry and I am somewhat out of practice.

Download (GSP 5 file): calculus_cross_section_solid2.gsp
Download (GSP 4 file): calculus_cross_section_v4.gsp

Update: I changed the file by taking out a page of scratch work (an accidental relic) and I have included a link button to the Advanced Sketch Gallery, where you can find the 3D plotting toolkit.

Update 2: I have added a version 4 file.

fractals

2010-11-13T18:45:00.006+05:30

After coming across a TED talk by Benoit Mandelbrot on Pat's blog, I had fractals on the brain and I eventually returned to an old problem of trying to construct the dragon fractal in Sketchpad. The construction of the dragon fractal was first described to me as follows: start with a line segment. Rotate it 90 degrees about one endpoint. Rotate the figure 90 degrees about its endpoint, always using the newest endpoint as the pivot. Repeat. This I tried several times in Sketchpad. I could do it one iteration at a time using the rotation, but I could not utilize the "iteration" construction because I couldn't figure out what iterated to what. If you would like to wrestle with this part of the problem, stop reading and give it a try.

It may in fact be possible to describe the iteration in terms of the rotation definition of the curve, but eventually I gave up and consulted the internet, where I quickly found this nice page by Shannon Umberger. I learned that the dragon fractal has another definition that is far simpler to employ in the statement of an iteration rule. After that, I started to poke around with some other fractals. In GSP, once you define an iteration rule, you can change the number of iterations using the + and - keys.

benford's law

2010-11-10T09:40:00.004+05:30

In Statistics recently we discussed Benford's Law, which says that data which are not dimensionless (so... monomensionless?) have first (most significant) digits which are not uniformly distributed. In other words, there are nine possible first digits {1, 2, 3, 4, 5, 6, 7, 8, 9} and we might expect that a nice distribution of values might have 1/9 or 11.1% of the numbers starting with 1, but in fact around 30% start with 1. Other digits are decreasingly frequent with small digits being more common. An interesting implication of this is that falsification of data by a careless attempt at making up numbers could result in an observable departure from a known pattern that is not very intuitive. Of course the cat's out of the bag, so now fraud is just taking on a more authentic appearance.

Data that is not dimensionless refers to numbers whose values are affected by the units used in measurement. That part confuses me.

Anyway, I wanted to take this data from the US Census Bureau on world populations by country and see if it follows Benford's Law. You should try it.

I did observe a distribution that Benford would have more or less predicted. I wondered if I had gotten lucky. The two biggest populations were in the 1 billions, and I thought that the frequency of 1's would benefit from spilling over into the billions but not reaching the other-than-1 billions, so I wondered what would happen to the distribution of first digits if the population of each country doubled. Or tripled. Or quadrupled. Guess what happened?

The part of the task that I most enjoyed was finding an excel formula for the first digit of x as a function of x. Can you find such a formula?

dynamic geometry challenge?

2010-09-26T19:19:00.002+05:30

I have recently been working on a few of the Project Euler problems (I just solved problem 171 {but I'm not telling the answer}) and I started wondering if there could be any dynamic geometry contests of a similar nature. Or any dynamic geometry contests at all. Each Project Euler problem deals with a fairly elementary concept, generally having to do with number theory. The problem may describe some property of a set of integers and you are asked to find the sum of the numbers below a billion having that property, for example. To solve the simple problems, a brute force approach may work, but the more difficult problems require some creativity and skill. I enjoy it immensely.

What could be done to create the same sort of stimulating set of really sound questions that would let people hone skills in the dynamic geometry environment? Questions that are easy to understand but whose answer could not be obtained without a fairly solid construction, the type of questions that would stimulate expertise with tools like Sketchpad.

If dynamic geometry is ultimately no more than an illustration or diagram, perhaps it is never really the best way to solve a problem. Are there (interesting) mathematical problems that can be resolved using Sketchpad which cannot be practically approached in other environments?

geogebra embedding: minor victory

2010-09-26T18:43:00.002+05:30

Well, I seem to have fixed my embedded GeoGebra applet for the Taylor Series (below a few posts) and I added the LSRL applet. The difficulty may have been in my blog layout, which was trying to scale the applet instead of expand to accommodate it. I am perplexed by XML, which is far less transparent than the HTML that I understand pretty well, but which is generally obsolete.

The embedded applet also seems to not work when you scale the page using a zoom tool in your browser. This means that other people may or may not have experienced the difficulties that I was seeing. It also means that no matter how I configure it, somebody else may have browser settings that mess it up. What is here at the time of this post works in both firefox and chrome... but it is a bit delicate, so don't click too hard or you might break it.

geogebra: LSRL

2010-09-26T18:38:00.002+05:30

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Nate Burchell, Created with GeoGebra

geogebra - lsrl

2010-09-02T14:07:00.009+05:30

While I would like to have a snazzy little Least Squares Regression Line applet on my blog (who wouldn't?), I have temporarily and figuratively thrown up my hands in frustration over the embedding geogebra wild goose chase. Instead, I have uploaded my GeoGebra file to be viewed on its own html page. I hope you enjoy it. The GSP file that appears elsewhere on my blog has more features, but of course requires the program.

This was a neat sketch to construct because I was able to use the spreadsheet view in GeoGebra, which is pretty slick.

Update: the link now takes you to lsrl9.html, which features, as requested by Pat, a way to view the point (x, y), which is always on the LSRL. Thanks, Pat for the suggestion.

embedded geogebra applet

2010-06-02T22:08:00.006+05:30

DISCLAIMER: Since posting this, I have tried to do more with GeoGebra in blogger with very dismal results. I tried to do my bit and contribute to the collective wisdom of the internet, but I'm afraid that embedding GeoGebra in blogger can be frustrating. For one thing, the size of the applet seems tricky. I adjusted mine and now the cursor location on the screen doesn't match the cursor location in the applet. The scaling is off or something and I wasn't able to fix it. Also, once I tried to "Edit" the post just to snag the code (because I wondered if it was different from that in "View Source") and the post changed itself in some vital manner without my consent. So proceed at your own risk as I continue to experiment.

I have seen a few interactive dynamic geometry applets in blogs from Kate Nowak (using GeoGebra) and squareCircleZ (using JSXGraph), the latter of which I just learned about from Ross Iseneggar's comment in my post from earlier today.

Kate's page gives simple instructions for embedding a GeoGebra applet in a blogger post, but I ran into a problem when I tried it. My sketch (I'm not sure if I'm allowed to call GeoGebra files 'sketches' but this is really disorienting for me as I am really more of a Sketchpad guy) was not complete, all but the Taylor polynomial showed up.

The export process refers to the .ggb file by name, but in the embedded applet you will need to (as Kate noted) provide the full url to the .ggb file. Additionally, the code from the GeoGebra-generated html file needed one or both of these two adaptations before it completely worked.

I changed:
codebase=http://www.geogebra.org/webstart/3.2/unsigned/
to read:
codebase="./"
and I changed the line in mine that said:
archive="geogebra.jar"
to match what Kate's says:
archive=http://www.geogebra.org/webstart/geogebra.jar

I'm not sure why the export code doesn't cooperate. I'm also not sure if my eventual solution works in other browsers, but for now I shall call it a minor victory.

geogebra applet - taylor polynomials

2010-06-02T21:51:00.017+05:30

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Nate Burchell, Created with GeoGebra

geogebra

2010-06-02T16:59:00.002+05:30

I've been monkeying around with GeoGebra for a couple of days. It is like Sketchpad, but free and open-source. So far I like GSP better. On the other hand, GeoGebra is free and my students could have it at home... And of course if I can get the applet to work in my previous post, I might have a way to post GeoGebra files on webpages as applets, something that GSP doesn't really offer. We'll see how this works out.

activity - polar curves

2010-05-06T10:59:00.002+05:30

This activity introduces students to the graphing of polar curves. A few of my students made use of (played with) the polar-rectangular sketch while working on this activity. The first part of the activity, creating a dynamic value with a slider, is a repetition of the parametric activity that my students did, so the technicalities were streamlined. There is no sketch that goes with this activity, since the students construct all of it.

Download Worksheet (.doc file): GSPActivity_PolarCurves.doc

activity - parametric curves

2010-05-06T10:42:00.003+05:30

This activity introduces students to the graphing of parametric curves in GSP. Detailed instructions walk students through the construction of a parameter whose value is determined by a point on a segment. This parameter is central to the investigation, but the construction could be streamlined by using a slider tool from the Tool Folder that comes with Sketchpad. The rest of the investigation asks students to consider the meaning of function as it relates to curves which might intersect themselves or fail the vertical line test. Skills include formulation of parameterizations for circles and lines/segments. There is no sketch with this as students construct the activity on their own.

Download Worksheet (.doc file): GSPActivity_ParametricCurves.doc

function - polar graphs

2010-03-05T21:43:00.003+05:30

Download GSP5 file: Function_Polar_Rectangular.zip
Download GSP4 file: Function_Polar_Rectangular_2.zip

This sketch has been on my mind for awhile. I wanted to show the rectangular graph of a function fwoosh into a polar graph of the same function, illuminating the patterns and behaviors of the function which are visible in either type of graph. This sketch allows you to move in slow motion between rectangular and polar graphs, zipping or inching or pausing or otherwise enjoying the fluid movement of some fantastic math. Change the function and watch the familiar features of your favorite rectangular graph move smooothly into their polar places. Try functions like g(x)=1, g(x)=1/sin(x), g(x)=.3sinx+cos3x.

The GSP4 version is fully functional and it looks nearly as good.

gsp trick: ordered values

2010-02-26T11:20:00.007+05:30

The last sketch I posted on this site deals with the topic of the cumulative area function. I wanted to make a dynamic and tactile function whose cumulative area function (anti-derivative) could be observed alongside the original function. I have already made sketches that graph a polynomial through a given number of independent points, but these polynomials are slippery critters and the process of manipulating them to a desired shape is neither as simple nor as intuitive as one might expect. First, the movement of one point affects the entire curve and not just a local region, which is perhaps beneficial for observing the shape and behavior of polynomial curves, but it does not allow one to easily mold the function toward a desired shape. Secondly, as the degree gets higher the function is difficult to scale, since the curve between roots can take on very large values. For these reasons, I decided to make my function a piecewise continuous function consisting of seven line segments defined by eight points. This function has a cumulative area function which is a differentiable curve of quadratic and linear sections.

The implementation included a particular challenge as I wished to construct the area using the points in order of their x-values. That is, the polygon area is always representative of an area between a function and the x-axis and does not allow the user to manipulate it otherwise. Seven sides of the polygon connect the eight independent points. Two sides connect the points with the highest and lowest abscissae to the base, which is an interval of the x-axis. The polygon sides connect the point with the smallest abscissa to the point with the second smallest abscissa to the point with the third smallest abscissa and so on. This is different than a polygon whose sides connect point A to point B and point B to point C. Most notably, it does not allow the top seven edges of the polygon to violate the vertical line test.

Finding the Maximum or Minimum of two values in GSP:

I have occasionally made use of a calculation to determine the maximum or minimum of two numbers. Here are examples of such calculations that are supported in GSP:

max(a,b) = ((sgn(a-b)+1)*a+(sgn(b-a)+1)*b)/2
min(a,b) = ((sgn(a-b)+1)*b+(sgn(b-a)+1)*a)/2

Figure out why it works, then verify that it works when a=b. It is a handy calculation that I have used recently to construct the interval (a,b) for a Riemann integral sketch; while a and b can switch orders, the smaller value of the two defines the initial point of the construction. I did use this some time ago in my LSRL sketch to scale residual plots so that the dimensions of the graph were each determined by the maximum and minimum of seven values. My approach then was to construct a tree-like set of max/min equations and define a tool to find the maximum of seven numbers by clicking on the seven values. This was a bit tedious, but effective if I only needed the largest or smallest. This does not easily address ordering the points or identifying the second largest of a list.

Ordering Values in GSP:

To order a list of eight values {a, b, c, d, e, f, g, h}, I made the following calculations*, one for each value:

rankA = sgn(a-b)+sgn(a-c)+sgn(a-d)+sgn(a-e)+sgn(a-f)+sgn(a-g)+sgn(a-h)
rankB = sgn(b-a)+sgn(b-c)+sgn(b-d)+sgn(b-e)+sgn(b-f)+sgn(b-g)+sgn(b-h)
rankC = sgn(c-a)+sgn(c-b)+sgn(c-d)+sgn(c-e)+sgn(c-f)+sgn(c-g)+sgn(c-h)
rankD = sgn(d-a)+sgn(d-b)+sgn(d-c)+sgn(d-e)+sgn(d-f)+sgn(d-g)+sgn(d-h)
rankE = sgn(e-a)+sgn(e-b)+sgn(e-c)+sgn(e-d)+sgn(e-f)+sgn(e-g)+sgn(e-h)
rankF = sgn(f-a)+sgn(f-b)+sgn(f-c)+sgn(f-d)+sgn(f-e)+sgn(f-g)+sgn(f-h)
rankG = sgn(g-a)+sgn(g-b)+sgn(g-c)+sgn(g-d)+sgn(g-e)+sgn(g-f)+sgn(g-h)
rankH = sgn(h-a)+sgn(h-b)+sgn(h-c)+sgn(h-d)+sgn(h-e)+sgn(h-f)+sgn(h-g)

The calculation rankA would have a value of 7 if a is the maximum in the set and -7 if a is the minimum. If a is the second smallest, then whenever we subtract the minimum we will get a positive "sgn" and the other six terms in the calculation will be negative, making rankA=-5. Thus, the eight calculations will map the eight values to {-7, -5, -3, -1, 1, 3, 5, 7}, provided that no two of the values are equal**. I could then plot the points (rankA, a), (rankB, b), (rankC, c), etc.*** At that point I could find, say the second highest value by looking for the y-value of the plotted point whose x-value was 5. Now a human user can see the plotted points as an ordered ascending sequence, but the computer identifies each point with its name, so determining which point was the maximum involves an inverse approach. For this I defined a polynomial f(x) passing through the eight points. Now f(5) gives the second largest value, f(7) gives the maximum, f(-3) gives the sixth largest value, no matter which point possesses that distinction. These values I used for my construction, so that if the independent points change order the points defining the construction still maintain their ordering.

This solution, once constructed is of course tailored to eight points and a different number of points would need to be individually approached. The ability to define custom tools is really what makes this possible, since the construction of a seventh degree polynomial through eight points is not something I would be inclined to do more than once. I appreciated the task for its logical demands and further exposure to the ideas of generality in dynamic geometry.

*This type of repeated tedious calculation can streamlined by carefully defining a custom tool that does most of the work.

**For my purposes in the construction described, it is desirable that two equal values should lead to an undefined ordering. I am trying to use the ordering as a function and if two values share a rank, my construction is ambiguous.

***For the sake of discussion, this is the simplest way to describe the plotted points, but in fact if one looks at the construction in the file I am describing, I have done something slightly different. To keep them out of the way of my sketch, I actually multiplied each of the "rank" calculations by 0.1 and subtracted 2, so that they ranged from -2.7 to -1.3 before plotting.

calculus: cumulative area function

2010-02-18T17:03:00.011+05:30

Download Sketch (GSP5 file): Calculus_CumulativeArea_7.zip
Download Sketch (GSP4 file): Calculus_CumulativeArea_8.zip
Download Worksheet (.doc file): GSPActivity_CumulativeArea.doc

This sketch offers a look at a simple antiderivative. The sketch shows a shaded region below a continuous "function" defined by seven line segments. The region can be adjusted by moving any of the eight independent points, and the arrangement of segments will always be a function. That is, the seven segments which define the area connect the eight independent points according to the order of their abscissae. Each segment represents a quadratic or linear increase in the cumulative area function, which is represented as a continuous string of parabolic loci.

My calculus students seemed to enjoy this activity. I took them to the computer lab to work with this sketch and worksheet after a discussion of an AP Calculus free-response question (2005B, #4) which deals with similar concepts.

Update 1: I was not initally able to make this GSP4-compatible since the construction relied on a new feature of GSP5, namely the ability to find an intersection of a line and a function. I was able to find a way around it and the GSP4 file is now available too.

Update 2: I have now included a worksheet (.doc file) in the zip file for the GSP5 version. The GSP5 sketch now includes a set of buttons to view eleven different graph overlays. Try to move the points around so that the graph of the cumulative area function matches the picture.

Update 3: I have put the graph overlay feature in the GSP4 file as well.

Update 4: I have now put a Hide/Show button for the cumulative area function in both versions of the sketch. Additionally, I have changed two of the questions on the worksheet.

calculus - volumes by revolution

2010-01-13T23:40:00.013+05:30

Please Note: This file marks the first time I have posted a GSP5 file, which will not work in earlier versions of the program. GSP5 allows you to save a file to be GSP4-compatible (provided the sketch does not make use of new features) and I will try to do this when it is possible.

This sketch may be a handy tool for depicting the volume that is generated by revolving an area about a vertical or horizontal axis. The volume is shown as a wire-frame "family of curves", a locus of a locus, and one of several fantastic new capabilities in GSP 5. This sketch will not work in GSP 4.

The volume can be tilted at any angle, and the movement through a small range of angles will allow one to observe the shape of the volume. The vertical and horizontal axes can each be moved. The area can be 'swept' around the axis.

The default functions define an area that can be integrated with respect to x, but not (explicitly)with respect to y. Therefore, one axis forces us to use cylindrical shells while the other direction of revolution will allow only discs. Discs and cylindrical shells can be shown for each individual value of x, as determined by a point moving through the interval that defines the area.

I have now updated the file by adding another page. The new sketch shows an example which could be integrated by shells with respect to x or by discs with respect to y. I hope that the sketch will work nicely as an illustration. The functions in the new page of the sketch can be changed, but the example is not general but rather discusses a case in which both functions are one-to-one and can therefore be negotiated with either variable.

Download GSP5 file: Calculus_Revolution_2.zip

Keywords: calculus, volumes, integration, revolution, GSP 5, family of functions

dynamic geometry paper

2009-12-05T23:21:00.004+05:30

Last year at this time I was putting the finishing touches on a paper: "Some Topics Which are Fundamentally Enhanced by the Use of Dynamic Geometry". Yesterday I had the opportunity to give a presentation on this paper at the Technology and Innovation in Mathematics Education conference at IIT Bombay, and it should appear in the proceedings for the conference. Enjoy.

gsp trick - booleans

2009-10-30T13:29:00.006+05:30

A boolean variable (see Java class Boolean) in computer programming can be thought of as true/false or 1/0 or on/off, to be used as a switch. The boolean variable is an element of Boolean Algebra, named for George Boole, an English mathematician who died in 1864 after making significant contributions to the foundation of computer science.

The boolean variable is an indispensible concept in programming and it can be created and used within Sketchpad.

Most recently, I employed this trick in my Riemann Sum sketch to define a single function plot which shows one of three different functions, as selected by clicking one of a set of three action buttons. Here are directions for such a construction.

Construct a line segment BC with a point A on the segment, as shown in the screenshot. Now select (in this order) C, B, A and from the Measure menu, select Ratio. Move point A back and forth to make sure the ratio measurement goes between 0 and 1. If the points are selected in a different order, a different ratio will be measured, which could range from 0 to infinity. We want 0 to 1, and this ratio measurement will be the boolean variable.

Now create two action buttons to "Move A->B" and "Move A->C". On the "Move" tab for the button properties, set the speed to "instant".

Next, create two presentation buttons. Select the "Move A->B" and "Move A->C" buttons (order matters) and create a presention. Make sure it is sequential rather than simultaneous. Construct the second presentation button after selecting "Move A->C" and then "Move A->B". Test the presentation buttons to make sure you know which sets the variable to one and which sets it to zero. Label the presentation buttons accordingly. The variable is complete and its values will be 0 or 1, provided it is changed only using the buttons.

Here is a neat application of the idea. Select the objects of the construction and define a tool to do the work. Using that tool, Sketchpad will define three new boolean variables with just six clicks, since the only independent elements of the construction are the two segment endpoints.

Next, I have renamed the variables f_boolean, g_boolean, and h_boolean. For each variable, I renamed the two presentation buttons as, for example, "f=0" and "f=1" after carefully making sure that I knew which was which.

Define three functions, for example: f(x)=x, g(x)=x², and h(x)=x³. Now plot the function k(x)=f(x)*f_boolean + g(x)*g_boolean + h(x)*h_boolean. The plot of k(x) can be changed by making three presentation buttons whose labels can be "show f", "show g", and "show h". The presentations should be constructed by selecting these buttons:

"show f" is the presentation of f=1, g=0, h=0
"show g" is the presentation of f=0, g=1, h=0
"show h" is the presentation of f=0, g=0, h=1

Here order is not important and a "simultaneous" presentation is acceptable. Also, it makes a smoother transition to change the six movement buttons {f=1, f=0, g=1, g=0, h=1, h=0} to have medium speed. This is mathematically different from a boolean and in some applications it may be confusing or incorrect to indicate a continuum of values from 0 to 1, but here it looks really cool. Also note that the function k(x) will have the greatest common domain of f, g, and h.

This is a nifty trick that lets you switch back and forth between a few functions for some variety without all of the typing.

the illusion of randomness

2009-10-23T10:15:00.006+05:30

This post and its ideas are, like so many of my endeavors, probably pretty obscure and generally of little use. That said, I have occasionally wished that Sketchpad contained a random variable mechanism. A truly random variable is not systematically obtainable for the simple reason that if it was the product of a system, it would not be random. However, if random is not available we will settle for a simulation of random, meaning a quantity that sporadically changes with no discernible pattern. This "discernible" is a very subjective criterion and much scrutiny would make the purportedly random variable even more difficult to declare "random". For my purposes in Sketchpad, I needed the variable to lack visually discernible order.

The application was the sketch on the previous post which iteratively defines a non-uniform partition of an interval into a variable number of sub-intervals. The intent was to show that the Riemann sum is a simpler concept by definition than the LRAM, MRAM, and RRAM special cases where uniformity is imposed without necessarily providing an optimal estimate. I have felt that students treat the Riemann definition, with its Δx_k's and c_k's as being more complicated for its generality. That's a shame.

The fascinating cognitive riddle behind my musings is that generality is simpler in essence but impossible to portray. Any representation must necessarily assign values to the parameters that define the construction. Making a series of decisions to do this will consume time and inevitably result in a construction that is not after all random.

The dynamic geometry environment entitles a construction to a sort of generality in that the construction can be observed through an apparent continuum of its parameters. Sometimes we want those parameters to vary at the whim of the observer, and other times it may be more effective to allow the user to witness the randomness that may potentially arise within a family of constructions. With the Riemann sums, a student given the opportunity to partition and calculate an estimate will quickly opt for uniformity and automation, but the simpler idea to not restrict Δx_k's and c_k's is more elusive.

Anyway, the function I set upon that was my first moderate success was this:
I am not entirely certain that a similar construction would even provide a uniform random variable, but it is sporadic and it has the range [0,1) or [0,1], I am not sure which. The inclusion of 1 relies upon the architecture and data type. This works for my purposes and it can obviously be improved but I think with a greater cost in calculating energy. For example, a greater exponent would make the mantissa more frantic for small movements of x, but my own application would not be appreciably improved. Here I will end my musings with the observation that "The Frantic Mantissas" would be an excellent name for a band.

calculus - riemann sum

2009-10-22T15:46:00.007+05:30

Download: Calculus_Riemann_Sum_f.zip

This sketch is, I hope, an improvement on my previous integration sketch. The concept I tried to capture here is the generality of the Riemann sum construction. The use of Rectangular Approximation Methods LRAM (left), MRAM (midpoint), and RRAM (right) involves the uniform partition of [a,b]. The Riemann sum defines an estimate using an arbitrary partition, which is more eloquent as it does not assume equal sub-intervals. The equal sub-intervals of RAM are not logically optimal, just more easily computed.

For me, this sketch was more of an exercise in the use of dynamic geometry to show generality and randomization. While sketchpad does not contain a random variable calculation, this sketch does utilize a function that affects sufficiently sporadic behavior for the purposes.
There are buttons that move the settings to LRAM, RRAM, MRAM, all with variable numbers of sub-intervals. The function can be changed as well as the interval.

Update: I just updated the file to include three different buttons to quickly change the function. Any of the three functions can be altered individually and the buttons allow quick changes from one function to the next.

Update: (10/23/09) The new version includes a smoother transition between functions when using the action buttons to switch.

polynomials - bounded roots

2009-09-25T14:13:00.004+05:30

Download: Polynomial_Zero_Bounds.zip

This sketch examines the test for an upper or lower bound of real zeros for a polynomial function. The test uses Descartes' Signs Rule that gives a maximum number of zeros according to the number of times the polynomial terms change signs. The user can alter the quartic polynomial, which is defined by five independent points. (There is also a tool in the sketch for constructing a quartic polynomial from five points).

This sketch shows the synthetic division of f(x) by (x-k). Particularly interesting is the chance to observe that some bounds will not be recognized as such by this test. Under what conditions will the test fail? How can we employ this idea for a polynomial with a negative leading coefficient? What must be true in order for the the least upper bound to equal the greatest root? How do complex roots affect the result? Why is there a restriction that k ? 0 in order to demonstrate that k is an upper bound? Could an adaptation be made to allow us to test a negative k as an upper bound? Does a horizontal translation of the function affect the difference between the least upper bound and the greatest root?

polynomials - multiplicity

2009-09-21T13:57:00.006+05:30

Download: Polynomial_Multiplicity.gsp

This sketch is perhaps a handy tool for presenting the idea of multiplicity of polynomial zeros. The sketch shows a sixth degree polynomial defined by its roots, which are independent points on the x-axis. The dynamic environment allows us to see a continuum of functions as two roots are dragged closer to one another. In this way, one can develop an understanding of what a polynomial does near roots according to the multiplicity of that root. A root of multiplicity three, for example, can be recognized as the convergence of three roots. One can witness multiplicity as a special case in a coherent continuum.

The zeros on the sketch can be dragged with the arrow tool or directed to predetermined locations using buttons.

polynomials - six point quintic

2009-09-18T10:58:00.009+05:30

Download: Polynomial_Quintic.zip

This sketch is a follow-up to my sketch about polynomials through points, which showed dynamic quadratic and cubic functions calculated from the coordinates of independent points in the plane. My approach in that sketch used matrices to solve systems of equations, but I felt I had reached the end of my tether with the inversion of a 4x4 matrix, as you will see if you "show all hidden" in the previous sketch. This sketch uses a different method that streamlines the process considerably, so I was able to plot the quintic without too much pain. Actually, the quintic took me about three minutes of tinkering with the quartic sketch, and extending to a sixth degree polynomial would be a task of similar magnitude.

I suggest that producing such a sketch would be a challenging and insightful exercise for a curious pre-calculus student. When does this polynomial fail to exist? How could we extend this construction to a higher degree? Can we employ a similar concept to find any of the roots? All of the roots?

The calculation process used in this sketch is pretty neat. I do not know what the process is called, but I do not doubt that the principle is familiar to someone. To me, this approach seems to keep the process tidier and more intuitive than would be the case with matrices, although it accomplishes a nearly equivalent task. Matrices would be handy in a single static case with numerical values, but solving a 5x5 matrix in this environment is far less practical than the method which I have employed.

We are trying to find f(x), a polynomial of degree n. Now f(x) passes through (is defined by) n+1 known points, the first of which we will call (a,f(a)). Next, we define the function fT(x)=f(x)-f(a), so fT(x) is the vertical translation of f(x) with a zero at x=a. Now (x-a) is a factor of fT(x) and our task of finding f(x) is finished if we find fT(x), one of whose linear factors we know. In this way, we recursively lower the degree of the polynomial that we seek. When finally we find ourselves seeking a linear function, the task is of little difficulty and we have only to retrace our steps and piece together our dismantled polynomial.

functions - transformations and sinusoids

2009-09-01T07:00:00.001+05:30

Download: Transformation_3.zip

Note: This is an old post, but I just discovered that I had linked to the wrong file.

This sketch is essentially a remake of a previous transformations sketch. After using the first one as a presentation tool in class on several occasions, I have finally gotten around to making it more presentation-friendly. The design is simpler, featuring more tabs that deal with vertical/horizontal dilations and translations individually. The basic function can be easily changed and the scales for each of the parameters can be edited. The sinusoidal axis is always visible on the transformed function. There is a box drawn around one period, and this can be hidden with a button. The unaltered sine curve is visible the entire time, and the transformed function can be returned to sin(x) with the click of a button. I have tried to design this with a tablet or interactive whiteboard in mind.

Keywords: sinusoid, transformation, dilation, shift, stretch, horizontal, vertical, input, output, translation, period.

gsp trick - shading areas using iteration

2009-08-10T15:04:00.005+05:30

Using Iteration to Shade Areas in Sketchpad

These directions should assist in the construction of a trapezoidal estimate of the area under a function f(x) on an arbitrary x-interval (a,b).

1: Begin by plotting the new function f(x) = (x^2)/8. A function that is linear or very steep will not be helpful in the trouble-shooting of this type of construction. The function can be changed after the construction. Create a parameter (t[1] in the diagram) and give it a value of 5. This parameter will be used to define the number of sub-intervals of area. The value can be changed later to produce a better-fitting area estimate, but if it is too big, the widths will be inconveniently small during the rest of the construction process. Create a second parameter equal to t[1]-1. This will be the number of times that we want to iterate the first trapezoid, since the original trapezoid will not be part of the iteration. Create a third parameter equal to 1/t[1]. This will be the width of each interval. Next, create two independent points (A and B in the diagram below).

2: Select both points A and B and the x-axis and construct parallel (horizontal) lines through the points. Next, select both points and the x-axis and construct perpendicular lines through the points, as shown. Create the intersection C (as shown) and dilate that point by a factor of 1/t[1] with A as the center. Mark the vector from A to the dilated point. Now create a point D on the vertical line passing through A and translate D by the marked vector. D is the point we will iterate, and any structure dependent on D will be iterated to the specified depth.

3: Select D and the translation of D (Sketchpad will call this D') and go to "Measure" - "Abscissa (x)". It will create two new measurements, the x-values of each point. Now "Calculate" f(x[D]) and f(x[D']). Plot the two points ( x[D], f(x[D]) ) and ( x[D'], f(x[D']) ). Create a new parameter with value 0 and plot the two points ( x[D], 0 ) and ( x[D'], 0 ). These four plotted points are the vertices of our first trapezoid.

4: Create a quadrilateral interior using the four plotted points. Now select the point D and the parameter t[1]-1. Click "Transform", hold down the shift key and select the "Iterate To Depth" option that will appear. The pre-image D should be sent to D' in order to construct the iteration. I generally go into the structure menu and tell it to iterate "Non-Point Images Only" and I un-check "Tabulate Iterated Values".

5: Now the construction is in place and the function can be changed. The number of intervals can be changed (just change t[1] and the other parameters will respond accordingly) to give a smoother picture.

6: To shade the area between two functions, use the points ( x[D], g(x[D]) ) and ( x[D'], g(x[D']) ) instead of ( x[D], 0 ) and ( x[D'], 0 ).

7: One of the most common mistakes that I make is to iterate A, the point which determines the width of the interval. As the width of each trapezoid is a fraction of the horizontal distance between A and B, this will decrease with each iteration. First it will use AB/t[1], then A'B/t[1], then A''B/t[1]... and the iterated image will never get all the way across the interval. Instead iterate D, which will not change the distance between A and B, so the intervals will have constant width.

Note: This method creates an area that stretches and conforms predictably to any adjustment of the axes. I have made tools for this (available in some of the sketches on this page) and it should be noted that there are some complications regarding the square/rectangular axes when you try to generalize this process as a tool. This is because one sketch may contain multiple coordinate systems.

keywords: gsp, geometer's sketchpad, iteration, shaded area, integral area, integration, iterative, iterate, trapezoidal area approximation, step-by-step, tutorial, iterate to depth

gsp trick - "erase traces" button

2009-08-05T11:16:00.003+05:30

Sketchpad has a keyboard shortcut (ctrl+B) for erasing traces created by an animation. This trick offers a slight improvement, allowing you to include an "erase traces" button in the sketch so that the traces can be removed by clicking with a mouse or stylus rather than by turning to the keyboard. It seems to disrupt the interaction less, especially when Sketchpad is used on an interactive whiteboard or tablet.

To construct the button, I used the fact that any presentation button includes the option of erasing any traces before beginning the presentation. Essentially, the idea shown here is to create an innocuous (and invisible) presentation and erase the traces before beginning that presentation.

Create three points, here they are labeled A, B, and C. Select C, then A and click on the Edit menu --> Action Buttons --> Movement. The Movement Action Button dialogue should appear. The speed can be instant, since the movement is unseen and will not affect anything else in the sketch. This creates a button with the default label "Move C -> A". Create a similar button by clicking on C and then B, and it will be called "Move C -> B".

Now select the two buttons (the order will be of no consequence) and click on the Edit menu --> Action Buttons --> Presentation. Select "Sequentially" and check the "Erase Any Traces" box. Now click on the "Label" tab in the same dialogue to change the name; I have changed this button to read: "Erase Traces".

Finally, you will have this simple collection of objects.

Most of these can be hidden, giving a very tidy and efficient "Erase Traces" button. Once you have constructed one button in one sketch, it can be copied and pasted into other sketches without repeating the above steps.

calculus - integral

2009-07-20T13:51:00.003+05:30

Download: Calculus_Integral.zip

This sketch explores the idea of an integral as an area beneath a curve. Riemann Sums are brought to life as the number of sub-intervals on a user-defined (a,b) interval can be varied between 1 and 1000 using a sliding parameter indicator. Comparisons can be made between right sums, left sums, midpoint, and trapezoid techniques. The user can see the errors diminish as the sum of simply calculated areas approaches the exact area beneath a curve. The sketch includes a tool for shading the region between two curves, whose arguments are 2 points followed by two functions.

Keywords: calculus, Riemann sum, right sums, RRAM, left sums, LRAM, midpoint, MRAM, trapezoid, trapezium, converge, area, integral, integration, subintervals, definite integral

calculus - derivative

2009-07-20T13:50:00.001+05:30

Download: Calculus_Derivative.zip

This sketch shows the tangent line to a function at a moveable point a. The slope of the tangent line is depicted as a length, which reveals the visual connection to the derivative f'(x). As a is moved around, the slope/derivative can be observed with or without tracing. The sketch includes linear, quadratic, cubic, quartic, and sine functions, but the function can be redefined easily on any of the pages.

Keywords: calculus, derivative, differentiation, slope, gradient, rate

trigonometry - six functions, unit circle

2009-07-20T13:48:00.002+05:30

Download: Trigonometry_UnitCircle_SixFunctions.zip

This sketch shows the six trigonometric functions (Sine, Cosine, Tangent, Cosecant, Secant and Cotangent) as distances on a unit circle diagram. The angle is defined by the positive x-axis and a point on the circle that can be moved manually or animated. The unit circle is depicted adjacent to a Cartesian coordinate system of the same scale, for an easy visual translation between two important representations of trigonometric functions. Individual functions can be hidden or shown separately.

Keywords: trigonometry, trig functions, sine, sin, cosine, cos, tangent, tan, cosecant, csc, secant, sec, cotangent, cot, geometric, radians, degrees, unit circle diagram, period, animation

calculus - slope fields

2009-07-20T13:46:00.001+05:30

Download: Calculus_Slope_Field.zip

There are three levels of visualization to be gleaned from this illustration. The segments themselves give a low-resolution picture of the family of curves which satisfy the differential equation. Secondly, the solution curves can be viewed with the click of a button. Finally, a set of tracing points can be sent along their respective curves, illustrating the concept of initial values. The sketch is simple, but it covers the basics in a way that any visual learner will appreciate. The sketch works nicely when dy/dx is a function of x, but if dy/dx is implicitly defined, the construction is more complex and the differential equation is not easily changed.

Keywords: slope fields, direction fields, anti-derivative, anti-differentiation, integral curves, solution curves, family of solutions, differential equations, dy/dx

statistics - standard normal curve

2009-07-20T13:43:00.001+05:30

Download: Statistics_Normal_Distribution.zip

This sketch can be used to discuss various aspects of the normal curve. Normal curves have a specific nature to them that is brilliantly captured by Sketchpad. Every combination of Mean (mu) and Standard Deviation (sigma) yields a graph with the same area, and this sketch provides a look at the changes that take place in order to satisfy such a constraint. The calculations of area were made using trapezoidal estimates and summed iteratively as described in Sketchpad's sample calculus sketches. There are four pages: Normal Distribution with sliders, Standard Normal Distribution with z-scores, Confidence Intervals, and Standard Normal Distribution with a calculation of the shaded region corresponding to P( a < Z < b ).

Keywords: normal distribution, normal curve, N(0,1), mu, sigma, mean, standard deviation, z-score, probability, density curve, standardize, standard normal, normal random variable, confidence interval, CI, z*, statistics

statistics - z-tests, type i & ii errors, power

2009-07-20T13:39:00.001+05:30

Download: Statistics_Ztest.zip

This sketch gives some accompanying visuals for a discussion about significance testing. This could be particularly valuable to students finding it difficult to grasp the quantities described by a Type I Error (rejecting the Null Hypothesis when it is correct) or a Type II Error (Accepting the Null Hypothesis when a particular alternative is correct). The sketch allows the student to change the value of alpha and the value of standard deviation sigma, and thereby discover the costs of controlling either type of error. This sketch includes the confidence interval sheet from the "Normal Distribution and Z-scores" sketch.

Keywords: normal distribution, normal curve, mu, sigma, mean, standard deviation, z-score, probability, density curve, standardize, standard normal, normal random variable, confidence interval, CI, z*, statistics, statistically significant, p-value, alpha, one-sided, two-sided, z-test, null hypothesis, H0, Ha, alternative hypothesis, type i error, type 1 error, type ii error, type 2 error, power

geometry - pizza theorem

2009-07-20T13:34:00.001+05:30

Download: Geometry_Pizza_Theorem.zip

The Pizza Theorem: If a circular pizza is cut by four straight cuts into eight slices whose tips have the same angle and come to the same point somewhere on the surface of the pizza, then the sum of the areas of alternating slices is equal to half the area of the pizza. This sketch gives a 'visual proof' of this neat idea. I have used it as part of a discussion of mathematical proof. While this is not a proof, it is a very convincing display and it can certainly offer some amount of insight into the problem. The evidence put forth by this sketch is more accessible and interesting to most people than any analytical approach. The sketch is dynamic in that the 'center' can be moved and the rays can be rotated to allow the exploration of different scenarios. The proof relies on the tilting or shearing of triangles, which is briefly explained on an additional page of the sketch.

Keywords: gsp presentation, animation, pizza theorem, visual proof, shearing

calculus - polar integration

2009-07-20T13:33:00.001+05:30

Download: Calculus_Polar_Integration.zip

This sketch allows a brief look at the mechanism of polar integration. The concept is essentially identical to integration with Riemann sums in Cartesian coordinates, but with wedges instead of rectangles. The polar setting brings up some novel facets of the concept of integration. For example, the integral area can overlap itself (see the page of this sketch dealing with the Archimedes Spiral). The construction was tricky, but I recommend it for the experience with the polar coordinates. The theta-interval and the number of sub-intervals are both easily changed to highlight the simplicity and precision of this process as could not be conveyed by a stationary picture.

Keywords: polar integration, polar integral, polar calculus, limacon, cardioid

calculus - euler's method

2009-07-20T13:31:00.001+05:30

Download: Calculus_Euler.zip

This sketch explores Euler's method for approximating a solution to a differential equation. It includes a brief explanation of the concept and a simple example of the calculations, as well as a depiction of the method with a variable interval and a variable number of iterations. The slope field cannot be adjusted very easily (to reflect alternate differential equations) but the axes can be differently scaled to explore a variety of scenarios. The software does not allow a function of two variables, so the construction is not easily generalized. The featured slope field goes with the differential equation dy/dx=x^2+y.

Keywords: euler, euler's method, euler's approximation, slope field, differential equation, local linearity, solution curve

calculus - newton's method

2009-07-20T12:01:00.001+05:30

Download: Calculus_Newton.zip

Newton's Method for root approximation can go unappreciated by people who get bogged down in a repeated calculation toward an inexact (and sometimes elusive) solution. By outsourcing the calculations to GSP, this sketch offers a new look at a truly impressive little trick. The sketch looks at the premise of the algorithm while drawing attention to its strengths and limitations.

Keywords: calculus, Newton, Newton's method, approximation, root, zero, function, iterative, converge, diverge, local linearity

calculus - taylor and maclaurin series

2009-07-20T11:59:00.001+05:30

Download: Calculus_Taylor_Maclaurin.zip

This sketch shows the polynomial which represents the partial sum of the infinite series for a function. The sketch allows quick and continuous movement between each of the first ten polynomials centered at a (Taylor) or 0 (Maclaurin). The function can be changed, but Sketchpad calculates ten derivatives for each new function, which may be time consuming for some functions. By adding one term at a time, the user can watch as the approximating polynomial clings to curve of the original function and increases in its ability to describe the shape.

Keywords: calculus, Taylor, Maclaurin, series, infinite series, partial sum, polynomial, approximation, degree, derivative, power series

functions - polynomials through points

2009-07-20T11:53:00.001+05:30

Download: Polynomials_Through_Points.zip

A group of n arbitrary points on the x-y plane uniquely defines a polynomial function of degree n-1 (provided that no two points share the same x-value). While other sketches explore the curve's response to a change in parameters, this tactile picture offers what is probably a more aesthetic understanding of the nature of the curve. The math behind such a picture is a learning experience in itself, as it (perhaps unexpectedly) involves the calculation of inverse matrices. There are tools included to construct a cubic with four mouse-clicks and a parabola with three, and an additional version of each tool that includes the function equation.

Keywords: polynomial, cubic curve, parabola, line, function, regression, linear algebra, matrices, system of equations, inverse matrix

statistics - scatterplot, lsrl, correlation

2009-07-20T11:49:00.001+05:30

Download: Statistics_LSRL.zip

Download: Statistics_LSRL.gsp
Related Download: Statistics_Ztest.gsp
Related Download: Statistics_Normal_Distribution.gsp

This sketch has been my greatest GSP challenge to date, and it has expanded my impression of what GSP might be used to show. Seven independent points on a scatterplot can be manually moved around or sent to predetermined arrangements. The user can enhance an understanding of correlation and the implication of changes that is elsewhere not granted in such a dynamic and continuous setting. The sketch focuses on a visual examination of concepts whose calculations are otherwise prohibitively time-consuming and distracting.

Update: I have added a link for the .gsp file and a couple of related files (from other posts) after hearing that the zip link is bad. The zip link seemed to work on my mac, hopefully this fixes it.
Keywords: scatterplot, LSRL, least squares regression line, correlation, residuals, residual plot, transformation of nonlinear data, modeling, statistics