# Monthly Archives: March 2013

## Still imbalanced

‘Blogging and Twitter are all about sharing ideas, and hopefully growing as a result.  In this, my 100th post, I share a creation I made in response to Paul Salomon‘s Imbalance Challenge.  As he notes on his ‘blog, these challenges emerged when he wondered how he could shift his young students to some deeper thinking from their conversations about equivalence.

From the image below, can you rank circle, square, and triangle from lightest to heaviest?

Thanks to those who share with me.  I hope some of my offerings help others think and grow, too.

## Algebra Explorations Before Algebra

Here’s a short post to share two great tools for students to learn algebra–long before any formal algebra course and without seeming like you’re even learning algebra.

ACTIVITY ONE:  The first is a phenomenal recent set of posts on Imbalance Problems (here and here) from Paul Salomon.  If you’re on Twitter, interested in math or math education, and haven’t already, you should definitely follow him  (@lostinrecursion).

Paul is (or was) hosting an Imbalance Problem competition (mentioned at the end of his first post on  imbalance problems), but the following image from his post gives the general idea. Can you figure these out?  Better yet, can you write some of your own?  Can you encourage your students (or kids) to create some?  The process of thinking about the values of the unknown measures of the circles, squares, and triangles necessary to create these puzzles lies at the very heart of the concept of unknown variables that is so critical to algebraic reasoning.  Best of all, this feels like a game, and puzzle solvers don’t even realize they’re learning algebra.

ACTIVITY TWO:  Several months ago, a Westminster alumnus (Thanks, Phillip!) suggested an iPad app that my 3rd grade daughter instantly fell in love with–Dragon Box.  The app is available for $5.99 on both iOS and Android platforms. Jonathan H. Liu (@jonathanhliu) wrote such a great overview of the play on DragonBox for Wired. Rather than trying to imitate his post, I recommend you read his review: DragonBox: Algebra Beats Angry Birds. The play of the game is great for teaching algebraic skills, again without ever seeming like it’s teaching as much as it is. As a parent, I was THRILLED to see an educational game pull my daughter in so effectively and completely. A high point came when I was driving carpool last week and my daughter recommended to a friend who was wondering what app would be fun to play for the ride, “Try DragonBox. It’s fun!” As a math teacher, I certainly can appreciate the game’s end-of-level challenges to get the box alone (solve for a variable) in the right number of moves (efficiency) using the right number of cards (also efficiency). Still, there were a few times when I noted that a particular scenario could have been solved using a different sequence of equally-efficient moves that were not appropriately acknowledged as such by the software. My teacher side wasn’t particularly pleased with the only-one-way-earns-top-recognition approach of the app, especially when other alternatives are equally valid. Too many times, I fear students are faced with similar scenarios in their math classes. Efficiency and elegance are certainly valuable skills in mathematics, but I think we too often try to impose rigor on young learners long before they have achieved basic understanding. My grousing aside, my daughter and her friends weren’t bothered at all by the rare times where I had identified alternatives–I don’t think they even noticed. Back to my point about too much emphasis on “the ‘right’ way” too early, I decided not to mention it to them. As Mark Twain noted, I decided not “to let schooling get in the way of [their] education.” CONCLUSION: I hope you get a kick out of Paul’s imbalance problems (no matter what your age) and DragonBox if you have some younger kids around. As always, make learning fun and not obvious–your charges will learn in spite of themselves! ## Nspire Apps Update Thanks to a great comment from Mr. Schirles on my post on the new Nspire app from earlier today, I’ve learned something new. When you are using the math keyboard in the app, notice that several of the keys have overlines along their tops. When you tap-and-hold those keys, you get access to several related keys. For example, • holding the x-key gives access to y, z, and t. • Holding the r-key gives $\theta$. • Holding > gives the other three inequalities. • Nicely, holding the sine button button reveals the related arcsine, cosecant, and arc-cosecant options. Same for the other two circular trig buttons and their functions. (See note below for a wish.) • Holding the angle button reveals options for radian measures and DMS modes. • Importantly, the comma key shows options for _ (underscore–for units), | (the constraint or substitution character, critical for many solving and CAS features), and quotation marks. • Holding the parentheses keys reveals options for brackets and braces. In short, there are MANY nice hidden options. I’m very impressed and completely withdraw my earlier concern. But … NEW WISH: I did notice that hyperbolic trig functions aren’t included anywhere, and it isn’t easy (as far as I can tell) without navigating keyboards to add an “h” to the end of one of the trig functions to create the hyperbolics. I know I’m being picky and most high school math teachers don’t touch hyperbolic trigonometry, but it sure would be nice if either the trig buttons, or perhaps the $e^x$ button revealed some hyperbolic function options. My concern about having to navigate between keyboards for necessary characters was unwarranted. TI had already incorporated the fix. Nice. Thanks, Mr. Schirles! ## 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.

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.

• 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.

• 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.

• 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.

• 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.

• 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).

## Polar Derivatives on TI-Nspire CAS

The following question about how to compute derivatives of polar functions was posted on the College Board’s AP Calculus Community bulletin board today.

From what I can tell, there are no direct ways to get derivative values for polar functions.  There are two ways I imagined to get the polar derivative value, one graphically and the other CAS-powered.  The CAS approach is much  more accurate, especially in locations where the value of the derivative changes quickly, but I don’t think it’s necessarily more intuitive unless you’re comfortable using CAS commands.  For an example, I’ll use $r=2+3sin(\theta )$ and assume you want the derivative at $\theta = \frac{\pi }{6}$.

METHOD 1:  Graphical

Remember that a derivative at a point is the slope of the tangent line to the curve at that point.  So, finding an equation of a tangent line to the polar curve at the point of interest should find the desired result.

Create a graphing window and enter your polar equation (menu –> 3:Graph Entry –> 4:Polar).  Then drop a tangent line on the polar curve (menu –> 8:Geometry –> 1:Points&Lines –> 7:Tangent).  You would then click on the polar curve once to select the curve and a second time to place the tangent line.  Then press ESC to exit the Tangent Line command.

To get the current coordinates of the point and the equation of the tangent line, use the Coordinates & Equation tool (menu –> 1:Actions –> 8:Coordinates and Equations).  Click on the point and the line to get the current location’s information.  After each click, you’ll need to click again to tell the nSpire where you want the information displayed.

To get the tangent line at $\theta =\frac{\pi }{6}$, you could drag the point, but the graph settings seem to produce only Cartesian coordinates.  Converting $\theta =\frac{\pi }{6}$ on $r=2+3sin(\theta )$ to Cartesian gives

$\left( x,y \right) = \left( r \cdot cos(\theta ), r \cdot sin(\theta ) \right)=\left( \frac{7\sqrt{3}}{4},\frac{7}{4} \right)$ .

So the x-coordinate is $\frac{7\sqrt{3}}{4} \approx 3.031$.  Drag the point to find the approximate slope, $\frac{dy}{dx} \approx 8.37$.  Because the slope of the tangent line changes rapidly at this location on this polar curve, this value of 8.37 will be shown in the next method to be a bit off.

Unfortunately, I tried to double-click the x-coordinate to set it to exactly $\frac{7\sqrt{3}}{4}$, but that property is also disabled in polar mode.

METHOD 2:  CAS

Using the Chain Rule, $\displaystyle \frac{dy}{dx} = \frac{dy}{d\theta }\cdot \frac{d\theta }{dx} = \frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}$.  I can use this and the nSpire’s ability to define user-created functions to create a $\displaystyle \frac{dy}{dx}$ polar differentiator for any polar function $r=a(\theta )$.  On a Calculator page, I use the Define function (menu –> 1:Actions –> 1:Define) to make the polar differentiator.  All you need to do is enter the expression for a as shown in line 2 below.

This can be evaluated exactly or approximately at $\theta=\frac{\pi }{6}$ to show $\displaystyle \frac{dy}{dx} = 5\sqrt{3}=\approx 8.660$.

Conclusion:

As with all technologies, getting the answers you want often boils down to learning what questions to ask and how to phrase them.