Des-blog

Years ago, the Friday Fave attended a conference session by Zalman Usiskin who claimed there that all parabolas are similar to each other.

The Fave is convinced of the truth of this claim in a formal way, but still thinks it sounds preposterous and obviously false. If you squish a square in one direction it is no longer a square. If you squish a rectangle in one direction, you still have a rectangle but it is no longer similar to the original. So clearly a parabola described by f(0.5x) cannot be similar to the one that f(x) describes.

It is in the spirit of such claims about the relationships among graphs of functions under transformations that the Friday Fave offers up a new What’s My Transformation? in which students transform parangulas using function notation.

What is a parangula? you ask!

Well, my friends, a parangula—like a line or a parabola—is a geometric object described algebraically, which you may transform by translating, stretching, squishing, and reflecting in order to learn some general algebraic tools for working with these in the future.

What is more, this parangulas activity is featured in one of TWO new activity bundles over at teacher.desmos.com. The Fave is pleased to share with all of you a new Conics Bundle, and a new Function Transformations bundle.

So head on over and get your students started with parangulas. After all, there is just one parangula in the world, and just one line, and just one parabola. Usisikin was right about that.

We asked the Desmos Fellows to give this classic transformations activity an upgrade by adding one screen. The Fellows blended student background knowledge, goals for learning, and design principles to create these screens to enhance the original activity.

Reverse the Direction

• The original activity gives students a graph and asks them to generate a function. After practicing this skill, Bob Lochel and Anna Scholl like to see if students can go in the reverse direction by asking them to generate a sketch given a function. Asking students to describe their process allows teachers to see if students can decompose a function transformation into stages.
  

Delay Feedback for Reflection

• Paul Jorgens noticed that a student could complete most of the screens by guessing and checking answers. He added a cart sort so that students would have a chance to do some reasoning after the initial exploration.
• Sarah Blick Vandivort suggests adding a screen where students can predict what a function will look like using words and a sketch, before having the chance to check their thinking. Similarly, Scott Miller asks students to explain their reasoning for the composition of transformations of f(x). Analysis of responses allows the teacher to see what level of understanding, vocabulary and connections students are making.
  

Opportunities to be Right and Wrong in Different, Interesting Ways

• Linda Saeta added a screen that asked students for a second way to match a transformed graph. This flexible thinking about transformations and the ability to recognize them in a graph helps students with some of the problems they’ll encounter in Calculus, so having students look for them in earlier classes is good practice.
  
• Nick Corley asked students to create their own function similar to the ones encountered in previous screens, and to predict what the graph would look like before having a chance to check their thinking. This type of screen also offers a way to provide closure on the activity. Teachers can select several student generated functions to compare in words, with the teacher revealing each graph after students have had the opportunity to discuss similarities and differences.
  
• Shelley Carranza asked students to transform f(x) so that it would go through a given point. Using the dashboard to share solutions and ask questions such as “Which of these graphs used the MOST transformation types to go through the blue point and how do you know?” can make for rich conversation and formative assessment.
  

Extensions

• Meg Craig added the screen below to provide an additional challenge for Precalculus or Algebra II students. This challenges students to make connections to other function types (absolute value) and also brings up function composition, offering students the chance to think about what happens if we reverse the composition order of the solution.
• Patty Stephens expanded on this idea and asked students to explore f(abs(x)).
  

What are some other ways we can upgrade this or other practice activities? Let us know on Twitter @desmos.

One of our most used activities is The (Awesome) Coordinate Plane Activity. We think its popularity speaks both to the ingenuity of its author, Nathan Kraft, and also to the math student’s great need for graphing practice. Math teachers know what students need and Nathan knew how to help.

Since Nathan created that activity last year, we’ve upgraded our internal toolbelt and also our internal style guide for making great activities.

One of those principles is that practice should also involve elements of strategy. Practice should propel students towards procedural fluency, but that fluency should propel students towards more than just more practice.

So in our current version of Nathan Kraft’s classic, students receive feedback on exercises like this.

But later they’re asked to work without the axes.

And then later they’re asked to throw the darts.

So use Nathan’s activity as you’re helping students review or practice coordinate graphing. But ask yourself also how you can inject strategy into your existing practice sets – whether they’re on computers or on paper.