Abandoning Ship on Wolfram Alpha?

I am really getting fed up tired of having to explain Wolfram Alpha graphs to students.  For some reason, the default in Wolfram Alpha is to graph everything with imaginary numbers.  This results in bizarre-looking graphs and makes it near-impossible to use Wolfram Alpha as a teaching tool for undergraduate mathematics, a real shame.  Now that Google has entered the online graphing fray, I have a wary hope that the programmers at Wolfram Alpha might finally (after two years of waiting) fix the problem.

Here are a few examples.  I’ll show you the graph in Wolfram Alpha, on a TI-84 Plus emulator (TI-SmartView), from Google Search, and from Desmos Graphing Calculator.  These are all the “default” looks.  Wolfram Alpha consistently shows this confusing imaginary view as the default whenever working with graphs involving variables in radicals.

Example 1:

Example 2:

Example 3:

I was hoping to really teach my College Algebra students to use Wolfram Alpha next semester.  But, between the Logarithm Issues and this graphing issue, I’m afraid I’m going to have to abandon ship on using Wolfram Alpha as a teaching tool for students. Students simply don’t have enough mathematical sophistication to look at the graphs and realize that they aren’t seeing what they are supposed to be seeing and I’m seeing far too much confusion on assessments that are caused by the oddities in graphs and logarithms on Wolfram Alpha.  What a shame that we can’t work this out, huh?

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Collection of Math Games

To view the collections of Math Games, hover over the Games Menu, and go to one of the dropdown categories.

 

 

 

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Playing to Learn Math (new version)

I am at the Kansas City Math Technology Expo this weekend doing two talks.

Today’s talk was Playing to Learn Math? I gave this at TexMATYC in the spring, but just updated it to add some non-digital types of play that you can use in the classroom.  There are five great math games mentioned in this presentation. Direct links to these games are below:

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NYT Opinionator Series about Math

For a few months now, the NYT Opinionator Blog has been hosting a series of pieces that do a phenomenally good job of explaining mathematics in layman’s terms.

The latest article is about Calculus (with a promise of more to come): Change We Can Believe In is written by Steven Strogatz, an Applied Mathematician at Cornell University.

There are several other articles in this series, and if you haven’t been reading them, you really should go check them out.  Assign them.  Discuss them in your classes.

Given the discussions we’ve been having about teaching Series and Series approximations lately on Facebook, Twitter, and LinkedIn, I wonder if he’d consider writing an article explaining “Why Series?” to students.

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Logarithm Graphs in Wolfram Alpha

At the Wolfram Alpha Workshop at ICTCM, there was universal disappointment about the fact that you cannot get a graph of a logarithm that is only over the real numbers.  We tried everything we could think of to remove the complex part of the graph.

Personally, I have tried and tried and tried and tried to explain the problem with this in the feedback window for Wolfram Alpha, but been universally unsuccessful.   Every time I suggest a change, I am told that the “After review, our internal development group believes the plots for input “log(x)” are correct.” … yes, I know that … that doesn’t mean it’s the answer that most people will be looking for.

I find it ironic that “inverse of e^x” produces the graph we’d like to see, and even gives log(x) as an equivalent.

inverse-of-ex

But then ask for a graph of  log(x) or ln(x) and the graph will always include the solution over the complex numbers.

problem-with-logs3

What’s worse is that W|A inconsistently decides when to use reals only and when to use both complex and real numbers.  For example, the output for y=ln(x), y=x includes the complex numbered plot, while the output for y=ln(x), y=2x-3 includes only the Reals.  What!?!  Actually, I have some idea why this is … it seems that in some cases, if the extra graph intersects the real part of the log graph, then you get reals only.  If the graphs do not intersect, then you get real+complex.  For example compare the output for y=ln(x), y=2x-3 to the output for y=ln(x), y=2x+5.  On the other hand, when I tried to show a graph transformation, like y=ln(x) with y=ln(x)+4 (including the extra graph y=4x-3), I was back to getting the graphs with complex numbers again. Maddening.

log-maddening

We spend a LOT of time in the algebra and precalculus levels working with  transformations of graphs, understanding inverse functions, and specifics like the domain of a graph.  We can’t use Wolfram Alpha for any of these topics with regards to logarithms because of the way the graphs look.  I can live with the fact that W|A uses log(x) instead of ln(x) … it’s not great, and is confusing to students, but I can explain it and live with it.  But as long as the Wolfram Alpha graph includes the complex number system with no way to see the graphs on only the reals, we’ll have to pull out that old-fashioned graphing calculator to teach this section, and that’s a shame.

I’ve also heard the argument that we should just include the domain we want to see in the W|A input.  For example, y=ln(x), x>0.  But how is a student, learning logs for the first time, supposed to recognize that this is required?  After all, the graph they see when they first try W|A with y=ln(x) leads them to believe that y=ln(x) has a domain that includes all real numbers but zero.  This argument also means that to show graph transformations, we need to use much more complicated graphing commands, restricting each domain separately (to tell you the truth, I have not yet figured out a way to do it … although I suspect it’s possible).

It seems to me that there are two obvious solutions to this math teaching nightmare, and I can’t imagine why either one wouldn’t serve all parties using Wolfram Alpha (both high-level mathematicians, and the rest of us):

Solution #1: Use a toggle-able option to see the graph with only reals or both complex and reals  (I would prefer a default to the Real numbers graph, since my guess would be that the majority of the world’s population would be looking for that one).

log-toggle-option

Solution #2: Display TWO graphs.  Show a graph of the logarithm that is only on the real number system.  Then, below it, show a graph that includes both the complex and real number systems.

log-two-graph-solution

That’s all – end of rant.  This is the one thing I absolutely hate about Wolfram Alpha.  And I’m guessing that I’m not alone here.  Please can’t we just find a solution without hearing “After review, our internal development group believes the plots for input “log(x)” are correct.” again?

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