Tag Archives: apps

Mistakes are Good

Confession #1:  My answers on my last post were WRONG.

I briefly thought about taking that post down, but discarded that idea when I thought about the reality that almost all published mathematics is polished, cleaned, and optimized.  Many students struggle with mathematics under the misconception that their first attempts at any topic should be as polished as what they read in published sources.

While not precisely from the same perspective, Dan Teague recently wrote an excellent, short piece of advice to new teachers on NCTM’s ‘blog entitled Demonstrating Competence by Making Mistakes.  I argue Dan’s advice actually applies to all teachers, so in the spirit of showing how to stick with a problem and not just walking away saying “I was wrong”, I’m going to keep my original post up, add an advisory note at the start about the error, and show below how I corrected my error.

Confession #2:  My approach was a much longer and far less elegant solution than the identical approaches offered by a comment by “P” on my last post and the solution offered on FiveThirtyEight.  Rather than just accepting the alternative solution, as too many students are wont to do, I acknowledged the more efficient approach of others before proceeding to find a way to get the answer through my initial idea.

I’ll also admit that I didn’t immediately see the simple approach to the answer and rushed my post in the time I had available to get it up before the answer went live on FiveThirtyEight.

GENERAL STRATEGY and GOALS:

1-Use a PDF:  The original FiveThirtyEight post asked for the expected time before the siblings simultaneously finished their tasks.  I interpreted this as expected value, and I knew how to compute the expected value of a pdf of a random variable.  All I needed was the potential wait times, t, and their corresponding probabilities.  My approach was solid, but a few of my computations were off.

2-Use Self-Similarity:  I don’t see many people employing the self-similarity tactic I used in my initial solution.  Resolving my initial solution would allow me to continue using what I consider a pretty elegant strategy for handling cumbersome infinite sums.

A CORRECTED SOLUTION:

Stage 1:  My table for the distribution of initial choices was correct, as were my conclusions about the probability and expected time if they chose the same initial app.

App1

My first mistake was in my calculation of the expected time if they did not choose the same initial app.  The 20 numbers in blue above represent that sample space.  Notice that there are 8 times where one sibling chose a 5-minute app, leaving 6 other times where one sibling chose a 4-minute app while the other chose something shorter.  Similarly, there are 4 choices of an at most 3-minute app, and 2 choices of an at most 2-minute app.  So the expected length of time spent by the longer app if the same was not chosen for both is

E(Round1) = \frac{1}{20}*(8*5+6*4+4*3+2*2)=4 minutes,

a notably longer time than I initially reported.

For the initial app choice, there is a \frac{1}{5} chance they choose the same app for an average time of 3 minutes, and a \frac{4}{5} chance they choose different apps for an average time of 4 minutes.

Stage 2:  My biggest error was a rushed assumption that all of the entries I gave in the Round 2 table were equally likely.  That is clearly false as you can see from Table 1 above.  There are only two instances of a time difference of 4, while there are eight instances of a time difference of 1.  A correct solution using my approach needs to account for these varied probabilities.  Here is a revised version of Table 2 with these probabilities included.

App4

Conveniently–as I had noted without full realization in my last post–the revised Table 2 still shows the distribution for the 2nd and all future potential rounds until the siblings finally align, including the probabilities.  This proved to be a critical feature of the problem.

Another oversight was not fully recognizing which events would contribute to increasing the time before parity.  The yellow highlighted cells in Table 2 are those for which the next app choice was longer than the current time difference, and any of these would increase the length of a trial.

I was initially correct in concluding there was a \frac{1}{5} probability of the second app choice achieving a simultaneous finish and that this would not result in any additional total time.  I missed the fact that the six non-highlighted values also did not result in additional time and that there was a \frac{1}{5} chance of this happening.

That leaves a \frac{3}{5} chance of the trial time extending by selecting one of the highlighted events.  If that happens, the expected time the trial would continue is

\displaystyle \frac{4*4+(4+3)*3+(4+3+2)*2+(4+3+2+1)*1}{4+(4+3)+(4+3+2)+(4+3+2+1)}=\frac{13}{6} minutes.

Iterating:  So now I recognized there were 3 potential outcomes at Stage 2–a \frac{1}{5} chance of matching and ending, a \frac{1}{5} chance of not matching but not adding time, and a \frac{3}{5} chance of not matching and adding an average \frac{13}{6} minutes.  Conveniently, the last two possibilities still combined to recreate perfectly the outcomes and probabilities of the original Stage 2, creating a self-similar, pseudo-fractal situation.  Here’s the revised flowchart for time.

App5

Invoking the similarity, if there were T minutes remaining after arriving at Stage 2, then there was a \frac{1}{5} chance of adding 0 minutes, a \frac{1}{5} chance of remaining at T minutes, and a \frac{3}{5} chance of adding \frac{13}{6} minutes–that is being at T+\frac{13}{6} minutes.  Equating all of this allows me to solve for T.

T=\frac{1}{5}*0+\frac{1}{5}*T+\frac{3}{5}*\left( T+\frac{13}{6} \right) \longrightarrow T=6.5 minutes

Time Solution:  As noted above, at the start, there was a \frac{1}{5} chance of immediately matching with an average 3 minutes, and there was a \frac{4}{5} chance of not matching while using an average 4 minutes.  I just showed that from this latter stage, one would expect to need to use an additional mean 6.5 minutes for the siblings to end simultaneously, for a mean total of 10.5 minutes.  That means the overall expected time spent is

Total Expected Time =\frac{1}{5}*3 + \frac{4}{5}*10.5 = 9 minutes.

Number of Rounds Solution:  My initial computation of the number of rounds was actually correct–despite the comment from “P” in my last post–but I think the explanation could have been clearer.  I’ll try again.

App6

One round is obviously required for the first choice, and in the \frac{4}{5} chance the siblings don’t match, let N be the average number of rounds remaining.  In Stage 2, there’s a \frac{1}{5} chance the trial will end with the next choice, and a \frac{4}{5} chance there will still be N rounds remaining.  This second situation is correct because both the no time added and time added possibilities combine to reset Table 2 with a combined probability of \frac{4}{5}.  As before, I invoke self-similarity to find N.

N = \frac{1}{5}*1 + \frac{4}{5}*N \longrightarrow N=5

Therefore, the expected number of rounds is \frac{1}{5}*1 + \frac{4}{5}*5 = 4.2 rounds.

It would be cool if someone could confirm this prediction by simulation.

CONCLUSION:

I corrected my work and found the exact solution proposed by others and simulated by Steve!   Even better, I have shown my approach works and, while notably less elegant, one could solve this expected value problem by invoking the definition of expected value.

Best of all, I learned from a mistake and didn’t give up on a problem.  Now that’s the real lesson I hope all of my students get.

Happy New Year, everyone!

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Great Probability Problems

UPDATE:  Unfortunately, there are a couple errors in my computations below that I found after this post went live.  In my next post, Mistakes are Good, I fix those errors and reflect on the process of learning from them.

ORIGINAL POST:

A post last week to the AP Statistics Teacher Community by David Bock alerted me to the new weekly Puzzler by Nate Silver’s new Web site, http://fivethirtyeight.com/.  As David noted, with their focus on probability, this new feature offers some great possibilities for AP Statistics probability and simulation.

I describe below FiveThirtyEight’s first three Puzzlers along with a potential solution to the last one.  If you’re searching for some great problems for your classes or challenges for some, try these out!

THE FIRST THREE PUZZLERS:

The first Puzzler asked a variation on a great engineering question:

You work for a tech firm developing the newest smartphone that supposedly can survive falls from great heights. Your firm wants to advertise the maximum height from which the phone can be dropped without breaking.

You are given two of the smartphones and access to a 100-story tower from which you can drop either phone from whatever story you want. If it doesn’t break when it falls, you can retrieve it and use it for future drops. But if it breaks, you don’t get a replacement phone.

Using the two phones, what is the minimum number of drops you need to ensure that you can determine exactly the highest story from which a dropped phone does not break? (Assume you know that it breaks when dropped from the very top.) What if, instead, the tower were 1,000 stories high?

The second Puzzler investigated random geyser eruptions:

You arrive at the beautiful Three Geysers National Park. You read a placard explaining that the three eponymous geysers — creatively named A, B and C — erupt at intervals of precisely two hours, four hours and six hours, respectively. However, you just got there, so you have no idea how the three eruptions are staggered. Assuming they each started erupting at some independently random point in history, what are the probabilities that A, B and C, respectively, will be the first to erupt after your arrival?

Both very cool problems with solutions on the FiveThirtyEight site.  The current Puzzler talked about siblings playing with new phone apps.

You’ve just finished unwrapping your holiday presents. You and your sister got brand-new smartphones, opening them at the same moment. You immediately both start doing important tasks on the Internet, and each task you do takes one to five minutes. (All tasks take exactly one, two, three, four or five minutes, with an equal probability of each). After each task, you have a brief moment of clarity. During these, you remember that you and your sister are supposed to join the rest of the family for dinner and that you promised each other you’d arrive together. You ask if your sister is ready to eat, but if she is still in the middle of a task, she asks for time to finish it. In that case, you now have time to kill, so you start a new task (again, it will take one, two, three, four or five minutes, exactly, with an equal probability of each). If she asks you if it’s time for dinner while you’re still busy, you ask for time to finish up and she starts a new task and so on. From the moment you first open your gifts, how long on average does it take for both of you to be between tasks at the same time so you can finally eat? (You can assume the “moments of clarity” are so brief as to take no measurable time at all.)

SOLVING THE CURRENT PUZZLER:

Before I started, I saw Nick Brown‘s interesting Tweet of his simulation.

cw4fvbgwsaai00c

If Nick’s correct, it looks like a mode of 5 minutes and an understandable right skew.  I approached the solution by first considering the distribution of initial random app choices.

App1

There is a \displaystyle \frac{5}{25} chance the siblings choose the same app and head to dinner after the first round.  The expected length of that round is \frac{1}{5} \cdot \left( 1+2=3=4+5 \right) = 3 minutes.

That means there is a \displaystyle \frac{4}{5} chance different length apps are chosen with time differences between 1 and 4 minutes.  In the case of unequal apps, the average time spent before the shorter app finishes is \frac{1}{25} \cdot \left( 8*1+6*2+4*3+2*4 \right) = 1.6 minutes.

It doesn’t matter which sibling chose the shorter app.  That sibling chooses next with distribution as follows.

App2

While the distributions are different, conveniently, there is still a time difference between 1 and 4 minutes when the total times aren’t equal.  That means the second table shows the distribution for the 2nd and all future potential rounds until the siblings finally align.  While this problem has the potential to extend for quite some time, this adds a nice pseudo-fractal self-similarity to the scenario.

As noted, there is a \displaystyle \frac{4}{20}=\frac{1}{5} chance they complete their apps on any round after the first, and this would not add any additional time to the total as the sibling making the choice at this time would have initially chosen the shorter total app time(s).  Each round after the first will take an expected time of \frac{1}{20} \cdot \left( 7*1+5*2+3*3+1*4 \right) = 1.5 minutes.

The only remaining question is the expected number of rounds of app choices the siblings will take if they don’t align on their first choice.  This is where I invoked self-similarity.

In the initial choice there was a \frac{4}{5} chance one sibling would take an average 1.6 minutes using a shorter app than the other.  From there, some unknown average N choices remain.  There is a \frac{1}{5} chance the choosing sibling ends the experiment with no additional time, and a \frac{4}{5} chance s/he takes an average 1.5 minutes to end up back at the Table 2 distribution, still needing an average N choices to finish the experiment (the pseudo-fractal self-similarity connection).  All of this is simulated in the flowchart below.

App3

Recognizing the self-similarity allows me to solve for N.

\displaystyle N = \frac{1}{5} \cdot 1 + \frac{4}{5} \cdot N \longrightarrow N=5

FINAL ANSWER:

Number of Rounds – Starting from the beginning, there is a \frac{1}{5} chance of ending in 1 round and a \frac{4}{5} chance of ending in an average 5 rounds, so the expected number of rounds of app choices before the siblings simultaneously end is

\frac{1}{5} *1 + \frac{4}{5}*5=4.2 rounds

Time until Eating – In the first choice, there is a \frac{1}{5} chance of ending in 3 minutes.  If that doesn’t happen, there is a subsequent \frac{1}{5} chance of ending with the second choice with no additional time.  If neither of those events happen, there will be 1.6 minutes on the first choice plus an average 5 more rounds, each taking an average 1.5 minutes, for a total average 1.6+5*1.5=9.1 minutes.  So the total average time until both siblings finish simultaneously will be

\frac{1}{5}*3+\frac{4}{5}*9.1 = 7.88 minutes

CONCLUSION:

My 7.88 minute mean is reasonably to the right of Nick’s 5 minute mode shown above.  We’ll see tomorrow if I match the FiveThirtyEight solution.

Anyone else want to give it a go?  I’d love to hear other approaches.

New Nspire Apps PLUS Weekend Savings

TI finally converted its Nspire calculators to the iPad platform and through this weekend only in celebration of 25 years of Teachers Teaching with Technology, they’re offering both of their Nspire apps at $25 off their usual $29.99, or $4.99 each.  This is a GREAT deal, especially considering everything the Nspire can do!  Clicking on either of the images below will take you to a description page for that app.  

nspirenspire_cas

In my opinion, if you’re going to get one of these, I’d grab the CAS version.  It does EVERYTHING the non-CAS version does plus great CAS tools.  Why pay the same money for the non-CAS and get less?  You aren’t required to use the CAS tools, but I’d rather have a tool and not need it than the other way around.  If you read my ‘blog, though, you know I strongly advocate for CAS use for anyone exploring mathematics.

Now, on to my brief review of the new apps.

MY REVIEW:  From my experimentations the last few days, this app appears to do EVERYTHING the corresponding handheld calculators can do.  I wouldn’t be surprised if there are a few things the computer version can do that the app can’t, but I haven’t been able to find it yet.  In a few places, I actually like the iPad app better than either the handheld or computer versions.  Here are a few.

  • When you start the app, your home page shows all of the documents available that have been created on the app.  It’s easy enough to navigate there on the handheld or computer, but it’s a nice touch (to me) to see all of my files easily arranged when I start up.

ti3

  • A BRILLIANT addition is the ability to export your working files to share with others.  Using the standard export button common to all iPad apps with export features, you get the ability to share your current doc via email or iTunes.
  • The calculator history items can now be accessed using a simple tap instead of just arrow key or mouse navigation.

ti1

  • Personally, I find it much easier to access the menus and settings with conveniently located app buttons.  I prefer having my tools available on a tap rather than buried in menus.  A nice touch, from my perspective.

ti4

  • Moving objects is easy.  I was easily able to graph y=x and the generic y=a\cdot x^2+b\cdot x+c with sliders for each parameter.  It’s easy to drag the slider values, and after a brief tap-and-hold, a pop-up gives you an option to animate, change settings, move, or delete your slider.

ti2

  • Also notice on the left side of the three previous screens that you have thumbnails of your currently open windows.  With a quick tap, you can quickly change between windows.
  • One of the best features of the Nspire has always been its ability to integrate multiple representations of mathematical ideas.  That continues here.  As I said, the app appears to be a fully-functional variation of the pre-existing handheld and computer versions.
  • The 3D-graphing option from a graphing page seems much easier to use on the iPad app.  Being able to use my finger to rotate a graph the way I want just seems much more intuitive than using my mouse.  As with the computer software, you can define your 3D surfaces and curves in Cartesian function form or parametrically.

ti5

  • A lovely touch on the iPad version is the ability to use finger pinch and spread maneuvers to zoom in and out on 2D and 3D graphs.  Dragging your finger over a 2D graph easily repositions it.  Combined, these options make it incredibly easy to obtain good graphing windows.

For now, I see two drawbacks, but I can easily deal with both given the other advantages.

  1. This concern has been resolved.  See my response here. At the bottom of the 3rd screenshot above, you can see that variable x is available in the math entry keyboard, but variables y and t are not.  You can easily grab a y through the alpha keyboard.  It won’t matter for most, I guess, but entering parametric equations on a graph page and solving systems of equations on a calculator page both require flipping between multiple screens to get the variable names and math symbols.  I get issues with space management, but making parametric equation entry and CAS use more difficult is a minor frustration.
  2. I may not have looked hard enough, but I couldn’t find an easy way to adjust the computation scales for 3D graphs.  I can change the graph scales, but I was not able to get my graph of z=sin \left( x^2 + y^2 \right) to look any smoother.

As I said, these are pretty minor flaws.

CONCLUSION:  It looks like strong, legitimate math middle and high school math-specific apps are finally entering the iPad market, and I know of others in development.  TI’s Nspire apps are spectacular (and are even better if you can score one for the current deeply discounted price).

Air Sketch iPad app

I’ve rarely been so jazzed by a piece of software that I felt compelled to write a review of it.  There’s plenty of folks doing that, so I figured there was no need for me to wander into that competitive field.  Then I encountered the iPad Air Sketch app (versions: free and $9.99 paid) last Monday and have been actively using in all of my classes since.

Here’s my synopsis of the benefits of Air Sketch after using it for one week:

–Rather than simply projecting my computer onto a single screen in the room, I had every student in my room tap into the local web page created by Air Sketch.  Projection was no longer just my machine showing on the wall; it was on every student machine in the room.  Working with some colleagues, we got the screen projections on iPhones, iPads, and computers.  I haven’t projected onto Windows machines, but can’t think of a reason in the world why that wouldn’t happen.

–In my last class Friday, I also figured out that I could project some math software using my computer while maintaining Air Sketch notes on my kids’ computers.  No more screen flipping or shrunken windows when I need to flip between my note-taking projection software and other software!

–When a student had a cool idea, I handed my iPad to her, and her work projected live onto every machine in the room.  About half of my students in some classes have now had an opportunity to drive class live.

–This is really cool:  One of my students was out of country this past week on an athletic trip, so he Skyped into class.  Air Sketch’s Web page is local, so he couldn’t see the notes directly, but his buddy got around that by sharing his computer screen within Skype.  The result:  my student half way around the globe got real-time audio and visual of my class.

–This works only in the paid version:  We reviewed a quiz much the way you would in Smart Notebook—opened a pdf in Air Sketch and marked it live—but with the advantage of me being able to zoom in as needed without altering the student views.

–Finally, because the kids can take screen shots whenever they want, they grabbed portions of the Air Sketch notes only when they needed them.  My students are using laptops with easily defined screen shot capture areas, but iPad users could easily use Skitch to edit down images.

–Admittedly, other apps give smoother writing, but none of them (that I know) project.   Air Sketch is absolutely good enough if you don’t rush.

By the way, the paid version is so much better than the free, allowing multiple colors, ability to erase and undo, saving work, and ability to ink pdfs.

Big down side:  When  you import a multi-page pdf, you can scroll multiple pages, but when creating notes, I’m restricted to a single page.  I give my students a 10-15 second warning when I’m about to clear a screen so that any who want cant take a screen shot.  It would be annoying to have to save multiple pages during a class and find a way to fuse all those pdfs into one document before posting.  The ad on the Air Sketch site was (TO ME) a bit misleading when it showed multiple pages being scrolled.  As far as I can tell, that happened on a pdf.  Perhaps it’s my bad, but I assumed that could happen when I was inking regular notes.  Give me this, and I’ll drop Smart Notebook forever.  Admittedly, SN has some features that Air Sketch doesn’t but I’m willing to work around those.

Overall, this is a GREAT app, and my students were raving about it last week.  I’ll certainly be using it all of my future presentations.