This is an update to an earlier post.

Let’s see if we can get some information from the graph. Below is a screenshot of the GeoGebra applet.

Here’s a few questions:

• Which approximation gives the best estimate of total resource collected? Why?
• If the graph represents the instantaneous resource collection rate (in the context of this game), which sum will provide the best approximation?
• If the graph represents the average resource collection rate, which sum will provide the best approximation?
• The red line shows an increase followed by a decrease, what does that mean in context?
• What happens to the approximation as the number of columns is decreased? What happens when it is increased?
• Is there information that can help us determine if the graph displays instantaneous collection rate or average collection rate?
• Whats your best estimate about the amount of resources collected?

Below are 2 screenshots that show the amount of unspent resources by each team on the top right and the amount spent resources in the menu on the top left.

Let’s check our answers:

• Which sum provided the best approximation?
• What are some sources of error?

Now let’s connect the graph to what’s happening:

• Why does the graph of resource collection rate rise and fall?
• Why does the graph have humps and plateaus? What do they represent? (hint)
• How did the red player have a steeper rise on the second hump (after first plateau) than the blue player? (hint)
• What caused the drop-off in the red graph near the end? Does the drop off mean the red player is losing resources or gaining less resources?
• In the beginning why did the blue graph have dips in its graph? (hint)

I want thank Jamie and Mimi for their requests along with comments and suggestions.

I also want to thank the folks on twitter that helped promote this experiment or just gave me encouragement in general. @jybuell @wmcneary @jrykse @thescamdog @carnivalofmath @mathheadinc @scsocha.

I expected to have more applets completed, but I guess I didn’t take into account Hofstadter’s Law. I guess that’s part of learning. Much of the time was spent learning how things worked in GeoGebra. I spent a lot of time in GeoGebra Documentation and GeoGebra Forums. I also spent a lot of time just experimenting with commands and functions to see how they worked.

Below are links to all the posts along with links to all the applets for your convenience. For these applets press F9 to re-randomize values (e.g. Dotplot vs Boxplot and Plane from 3 Points in Space) and to start or stop animation on a single slider Right-Click on the slider and click on Animation On menu item to toggle the animation on and off. The play button will play and pause the animation on sliders that are turned on.
GeoGebra Applet Request started it all.

GeoGebra Applet Request – Statistics (Dotplot vs Boxplot) was Jamie’s request. There are multiple solutions to this. Try turning off the Easy Mode (integers only).
Dotplot vs Boxplot (applet link).

GeoGebra Applet non-Request – 3D Coordinate System was my first response to Mimi’s request. I was, um let’s say, a bit off the mark. It may still be useful for other things.
3D Coordinate System (applet link).

GeoGebra Applet Request – Plane from 3 Points in Space was my second response to Mimi’s request. I think this was closer to what she wanted.
Plane from 3 Points in Space version 1 (applet link).

Plane from 3 Poitns in Space version 3e (applet link).

Mimi :

“Fix” is a term (I guess used more commonly in science??) that means that a variable has taken on a particular value. Last year I tried to explain to kids, like, if you have a rigid plane (ie. cardboard paper) and you “fix” (ie. stop moving / assign a location to) two of its corners, the piece of paper can still flap around that hinge, but once you hold onto the third corner, the paper is now fixed in place and can’t rotate anymore. Does that make more sense?

Like, a door doesn’t move because it’s got three “fixed” points: doorstop, top of the door hinge, and bottom of the door hinge. If you take away the door stop, the plane is now free to move / become other planes.

I think this version is much closer to the original request than my last previous attempt where I was moving an airplane around. I have no idea why I thought that. I’m going to say it is because I don’t teach Geometry. For some reason, planes don’t conjure up “abstract surface which has infinite width and length, zero thickness, and zero curvature” or, if you prefer, “surface which lies evenly with the straight lines on itself.”

Some things I learned working on this applet.

• determinant[] was useful for this application. Just give it a matrix and GeoGebra will do the calculations.
• A matrix in GeoGebra is a list of lists. It was much easier to use than expected.
• Sliders can be switched from oscillating to increasing or decreasing. Use a trigonometric function like sine or cosine and increase the number of periods by some random integer factor and offset the trigonometric function and we get a randomized but periodic motion starting at randomized locations. F9 key which recalculates the GeoGebra applet will also recalculate the random function to give a new value.
• Polygon[] takes points and creates a polygon with those points as vertices. A nice feature to have for a future version will be for the Polygon[] command to be able to take a list of points instead. It would’ve been helpful in this case when I wanted to draw a polygon for the portion of the plane under the XY plane. I used a clunky if[] statement to test for all 3 possible scenarios.

Remember to right-click on the slider and click on Animation On. Double-click or right-click and edit object properties of the slider if you want the slider to go faster or slower. It’s the textbox under Animation, Speed.

I highly recommend that you save the applet and run it from your computer. Click below for the applet.

You can determine for yourself if the applet is an effective visual.

UPDATE: Here’s the updated applet with those additional controls along with options to keep the applet window decluttered. Version 2 is no longer available. For some reason it wasn’t loading properly. Try version 3 below instead.

UPDATE2: Click below for version 3. Just click play button on the bottom left. Once you’re ready, try the other drawing options and experiment with the slider controls.

Mimi :

Not sure if you’d be interested in making this (since it’s only 9th-grade math), but I’d really like an applet where you have a plane that moves around uncontrollably, until you fix one of its points, then another, then another. Each time you fix one of the points, the plane should decrease its degree of freedom, but still rotate around the fixed point(s). This would help me illustrate why 3 points in space are required to define / fix a plane!

Thanks,

Mimi

I completely misunderstood Mimi’s request. So the first thing that came to my mind when plane was mentioned was an airplane! I imagined an airplane moving in the air where the movement along one dimension is restricted at a time. I thought it was unusual to have an airplane moving that way. An UFO, hummingbird, or dragonfly would have been better. It wasn’t until I was close to finishing the applet that I finally (I think) understood what Mimi’s request was. I imagine that’s how students get distracted in class. You talk about a term and their brain is thinking of something else.

Press play button on the lower left to start the animation. Right-click on the slider bar in the bottom of applet and click on the checked Animation On menu item to stop the motion along that dimension.

Click below for the applet.

I might update this later to display octants as the point moves around in space.

Jamie :

How about one for statistics?

Have a dot plot (line plot) on top and a box plot below. As a user adds points to the dot plot the box plot changes to match the data. Help students see connection between the two. Also can help them see why a whisker might be longer on one side compared to other side. See connection between shape of dot plot and shape of box plot.

It would be nice if when a student adds a point to the dot plot they can click a spot on the dot plot to add a point instead of adding a value to a table. To make it easier to do some investigations, only allow integer values or integer values and halves?

Getting really fancy, put a second box plot on the bottom that is generated randomly. Student adds points to their dot plot to make their box plot change to try and match the second box plot.

This was a much tougher to make than I had anticipated. However, I learned a lot while building this applet.

• GeoGebra has many upcoming features in GeoGebra 4. It’s currently in beta and is a work in progress. I would recommend it only to the more adventurous users who want to play with it to learn about upcoming features. There are still unfinished features and (probably) bugs and there isn’t enough documentation to help you if you’re stuck. Though I must say that from reading the beta forums, I found the folks at GeoGebra very helpful in their replies to questions from users.
• BoxPlot[] – GeoGebra actually has boxplot features built in the current version. You can give it a list, or give it a 5 number summary and some basic settings and it’ll do the rest.
• Object[] – If you give the name of an object as string and pass it to Object[] it allows you to access that object. While creating this, one thing I used while experimenting with the commands and features was Object["A"+j]. I wrapped it inside a sequence[] command to build a list sequence[Object["A"+j],j,1,100]. That command created a list with the contents inside cell A1, A2, A3,…, A100. Later I found quicker way to do this.
• CellRange[] – CellRange[A1,A100] essentially does the same thing as the above sequence-object command.
• CountIf[] and KeepIf[] – I haven’t found a good way to count the number of items in a column. So I created a list using CellRange[] and then counted the number of numbers that are greater than zero. Not a great solution, but works for now.
• I wasn’t initially how to re-randomize the settings. The Online GeoGebra Documentation was very helpful. To recalculate values in an applet, simply press F9. If a random function is used in the applet, it is recalculated to a new random value.
• RandomBetween[] – It takes 2 integers and find a random number between those 2, inclusive (meaning RandomBetween[2,11] could potentially give you 2 or 11).
• min[], q1[], median[], q3[], max[] – These commands do what you think they do statistically. Among other things, useful for when you need to create boxplots
• Textfield can also take if[] statements. if[mymin==targetmin && myq1==targetq1 && mymedian==targetmedian && myq3==targetq3 && mymax==targetmax, "Good Job!", "Keep Trying!"] will congratulate you when your boxplots match.

Other Observations

• It was helpful to use the a Check Box to show and hide objects and also sliders while working on this. It made it easy to move elements on the screen around.
• In this version of GeoGebra, I couldn’t find a good way to switching between 2 settings like in a radio box where options are mutually exclusive. Checkbox is not useful here. So I repurposed a slider and added some text to allow for switching between settings.
• GeoGebra has (as of version 3.2.44) limited interactivity capability. This might be because GeoGebra is a work in progress (where features will be added later) or because it was designed that way (GeoGebra may be designed to meet only certain needs so that it does what it does really well). However, I think that some things that GeoGebra doesn’t do well yet can be made up using JavaScript. One issue that came up for me was how to reset many values in the applet to some original/default state. I’ll have to learn how to set up a GeoGebra applet to work with client-side JavaScript. Another issue was how to capture a click on the applet so that I add a data point in to the list automatically depending on where it was (as suggested by Jamie). So I used sliders.

You can see for yourself if it’s dynamic enough or useful to demonstrate how change in data can affect the dotplot and boxplot. Don’t be put off by the visual noise in the user interface (UI). I’m not a UI designer and elements can be moved around the screen and simplified.

If all goes well, you get an applet for use in class and I get to learn a little more GeoGebra. I’ll do as many as I can in the next week. I’m guessing one applet a day if it’s hard, more applets if I can get them done quickly. If I can’t do it, I’ll reply with a comment after giving it an attempt and hopefully someone more capable can help you out.

Send me your request(s) by placing a comment below,  tweeting or direct messaging @mrhodotnet, or sending me an e-mail through my Contact page. Here is a sample of what I’ve done so far with GeoGebra.

UPDATE1: Jamie’s suggestion for an applet on dotplot vs boxplot can be found here.
UPDATE2: My first attempt at Mimi’s suggestion can be found here.
UPDATE3: I think I got it right this time. Mimi’s suggested applet is here.
UPDATE4: Thanks for the help/support/encouragement. Last post is here.

A great mix of Statistics concepts:

• @3:14 On problem with polls on the Internet
• @16:45 On probability of data corruption from cheating when 2 players play ESP Game (a game that labels images)
• @23:50 On statistically significant difference when measuring play time in A/B Testing
• @39:10 On Sampling Bias. Population of gamers vs. population of all (gamers and non-gamers)

@3:14 On problem with polls on the Internet

A few years ago Slashdot, which is a very popular website, put up this poll in their site asking which is the best computer science graduate school in the United States. This is a very dangerous question to ask over the web. As with most online polls, IP addresses of voters were recorded to make sure that each person could only vote at most once. However, as soon as the poll went up, students at CMU wrote a program that voted for CMU thousands and thousands of times. The next day, students at MIT wrote their own program, and a few days later the poll had to be taken down with CMU and MIT having like a gazillion votes and every other schools having less than a thousand. I guess the poll worked in this case. In general, this is a huge problem because you simply cannot trust the results of an online poll because anybody could just write a program to vote for their favorite option thousands and thousands of times.

@16:45 On probability of data corruption from cheating when 2 players play ESP Game (a game that labels images)

One thing that some of you may be wondering about is… uh… what about cheating? For instance, could you try to cheat to screw up the label? Something like “my office mate and I could try to log in to the game at exactly the same time maybe we’ll get paired with each other and if we pair with each other we can agree on any word for any image” or even worse someone could go on Slashdot and say “Hey everybody, let’s play ESP Game and let’s all agree on the word a for every image.” Could happen. Fortunately, I thought about this and the ESP game has several mechanisms that fully prevents cheating. Let me tell you a few of the things that we do to prevent cheating. Here’s one. At random, we actually give players test images. These are just images that are there to test whether the players are playing honestly or not. And what they are is they are images for which we know all the most common things that people enter for them, and we only store a player’s guesses and the words that they agreed on if they successfully label the test images. So, if you think about it, in a way this gives sort of a probabilistic guarantee that a given label is not corrupt. What’s the probability that a given label is corrupt given that the players successfully label all their test images. This probability can be boosted by next strategy which is repetition. So we only store a label after N pairs of players have agreed on it. Where N is a parameter [here he means parameter as in input variable to a function not the sample statistic vs population parameter] that can be tweaked. So every now and then we actually delete all the taboo lists for the images and we put the images back into the game afresh. We only store a label after N pairs of players have agreed on it. So if we let X be the probability of label being corrupt given that players successfully labeled test images, then after N repetitions the probability of corruption is X to the N. This is assuming that N repetitions are independent of each other. But if X is small, X to the N is really really small.

@23:50 On statistically significant difference when measuring play time in A/B Testing

We continually do measurement to try to figure out what are the things that make people play longer. So let me explain one of these measurements to you. At some point in the history of the game, I added this very small message in the corner of screen alerting you whether your partner had already entered a guess or not. So a very tiny message, this just a magnification of it, just tells you if your partner has already entered a guess or not. When this was added to the game, it wasn’t added to all the players, it was just added to a small random subset of the players, and then we measured whether the players who have this feature played longer than those who didn’t. It turns out that those who had this feature played a whopping 4% longer than those who didn’t. You might not think that 4% is very large, but actually is a statistically significant difference. And, if you think about it, it’s just a very tiny message in the corner of screen makes people play 4% longer.

@39:10 On Sampling Bias. Population of gamers vs. population of all (gamers and non-gamers)

Audience question: Does it concern you at all that the fact that you are using a game will automatically give you a very biased population of people that are giving us answers to problems that we want answers to. and this population of people are people who have way more time on their hands and are not motivated to maybe get a job or do something… [indistinct]

Luis von Ahn: Very good question. Very good question. It’s true that the population is biased. There’s no questions about that. But for alot of really simple things, I mean.. anyone can do it. So it is true, but it is true that the population is biased. That’s definitely true.

Audience question: Have you seen any results. Any…

Luis von Ahn: I don’t know of any.. I can tell you that the population is biased, but I have not see anything that really tell me “Oh.. Because we are using gamers it’s like this is happening instead of the general population. I have not seen that.”

This is a great video to start discussion. One problem for high school students is Luis von Ahn’s rate of speech, it’s a little too fast for many students. Also, there might not be enough context to understand the concepts from the video. It’s probably better if shown after learning the concepts.

There’s also many rate/work related data:
@5:05 Spam companies use of sweat shops to solve CAPTCHAs.
@7:00 Interesting data on human-hour (a more gender neutral term). Solitaire played in 2003 (9 billion human-hours). Empire Estate Building (7 million human-hours). Panama Canal (20 million human-hours).
@15:10 Estimate on labeling the entire web. Probably a good tie-in to estimation on URL Shortener Math.

I usually pick weird days when many students (usually at least half) are called out for testing or other various reasons to show these clips. Usually followed by us actually playing those games with a purpose to see how they work. This is not so much a video for a lesson than it is providing the motivation for learn the stuff that we do. Great example for #whenwilliusemath.

This is what I found online (video below found on NBC website).

Can you spot the difference between the two videos above (aside from the intro)? Can you spot other mistakes?

Beyond the math mistakes. Compare the following:

Option A – Showing a video that has the following story followed by discussion and creating questions as a group.

Tom Costello: Alex Bogusky’s eureka moment for his book the 9-inch diet came when he bought a 1940s house and couldn’t get his new plates into the old kitchen cabinets.

Alex Bogusky: and i just thought, “you know, who.. what kind of idiot makes cabinets that don’t fit a normal size plate?” and then that stuck in my head for a second and I thought “No idiot does that. The size of our plates must have changed!”

Tom Costello: In fact they’ve grown a full 3 inches, from 9 inches in the 1960s to 12 inches today.

Option B – Just the following facts on a worksheet followed by a word problem.

In the 1960s, plates measured 9 inches in diameter on average. Plates today measure 12 inches in diameter.

How would you use this in the classroom?

Anywho. The following are things I learned about GeoGebra for this applet:

• It’s not possible to style individual Sequence[] items: That or I just haven’t figured out how. The nice folks at GeoGebra.org Technological Questions forum pointed me in the right direction regarding using spreadsheets to style objects for this one. There is Dynamic Colors that can change an object’s color depending on some variable, but I couldn’t get individual segments (created using Sequence[]) to retain a different color from its neighbor while its dimension and location changed. Maybe someone else can. But all this means that…
• Spreadsheet View: I had to learn how to use spreadsheets in GeoGebra for this one. Took me a few days of experimentation. Cells in spreadsheets function like objects. Right-click on a cell created using Segment[I2,I3]…
  
  … and we can edit the segment’s property.
  
• Dynamic Colors: Even though I don’t have the colors change dynamically to some variable, I still used the Dynamic Colors feature. I set them to the colors (in RGB, Red-Blue-Green) in cells A2 through C2. I didn’t do this initially and found that right clicking to go back into the properties was time consuming. Now, if I want to change colors, I just change the values in the cells in columns A to C and in rows 2 to 9) and the colors change automatically.
  
  Note: The color values range from 0 to 1. I assumed the 0 to 255 values and wasn’t getting the colors I wanted. A quick search for colors in the GeoGebra Documentation yielded:

  Note: In each of these text boxes, you can enter a function with range [0, 1].

  What it means is that not only will it take actual values, it can also take functions with range [0 , 1]. Linear, Quadratic, Cubic, Polynomial, Trigonometric, Exponential, Logarithmic, Rational, etc. Just limit it to [0, 1].

• Fill Handle: The fill handle (square on the bottom-right corner of the active cell) functions like it does in Excel. I created points to be drawn on a circle for each additional radian subtended by an arc. After entered the definition in the cell and dragged the fill handle down the column…
  
  … the cells adjust accordingly. Parts of the formula that depend on D3 gets changed to D4 automagically.
  
  This feature is not limited to just definitions. The Condition to Show Object and Dynamic Colors also change accordingly, as in here…
  
  …and here.
  
• Boolean Commands and Operators: I learned about IsDefined[]. And is &&. Or is ||. Not equal is !=. Comparison equal is ==. GeoGebra will automatically swap easier to understand symbols for each operator. When I first started using these operators, I thought it was a bug in GeoGebra that kept changing the expression I entered into the property text boxes.
• Check Box to Show/Hide Objects: Nice feature to slowly expose the complexity in an applet. Too many sliders can be intimidating.

Before we start, let’s look at the standard graphic for radian measure created using this applet.

Here’s the applet for Radian Measure. Click on play to start animation or drag the slider on the bottom of the applet. Check out the ruler first. Try the other check boxes and try different settings on the sliders.

Too much information? Too little? Too distracting? Bad colors? Comments are appreciated.