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The speedometer in this video is broken.

Can you (or your students!?) fix it? Be careful: there are a couple of interesting ways to get this one wrong.

Also: what would the graph of speed v. time and position v. time look like here?

Let us know how you’re thinking about it in the comments.

2015 Oct 17. Updated to include the answer video and answer graph. You can also download these files at 101questions.

I received about one hundred loop-de-loops from teachers, parents, and students from several different countries. It took me an hour to take in all the awesome eye candy, which included dioramas, videos, 3D loop-de-loops made from snap cubes, and more. I pulled out my five favorites and sent them to three judges who I think embody the best of creativity in mathematics.

The Judges

  • Malke Rosenfeld, who uses dance and choreography to explore mathematical thinking.
  • George Hart, a research mathematician who also sculpts using geometry as his medium.
  • Michael Serra, author of Discovering Geometry, a geometry textbook infused from the front cover to the back with Michael’s love for math and art.

Five Finalists

Autumn, from Angela Ensminger’s class:


Theo, from Alice Hsiao’s class:


Trish Kreb’s seventh grade student:


John Grade & his daughter:


Maddie Bordelon and her math art team, “Right Up Left Down”:


[BTW. In an early draft of this post, I reversed the second and third prize winners. Mistakes were made. Apologies have been issued.]

Third Prize

Third prize, which is a medium-intensity high five delivered if we ever meet, and one copy of Weltman’s book, goes to Maddie Bordelon and her math art team, “Right Up Left Down.”


Second Prize

Second prize, which is sustained applause in a crowded, quiet room, and five copies of Weltman’s book, goes to Theo from Alice Hsiao’s class:


One judge wrote:

[E] completely holds my attention. The coloring choices pull me in and highlight the patterns and structure in a way that fascinates me. The long bands of white, blue and grey make a fantastic contrast to the brighter colors closer to the middle, which are also the shorter segments in the design. And, the bold outlines pull out the structure even more. I don’t know if it was intentional, but the overall effect of hand-coloring plus scanning the image made for a lovely final effect.

First Prize

First prize, which is 40 copies of Anna Weltman’s awesome book, goes to John Grade & his daughter.

[2015 Oct 12. John Grade is graciously passing his first prize down to the second prize winner.]


Our judges wrote about John Grade’s loop-de-loop:

It is very well constructed, brilliant use of color, and the number pattern chosen is pretty special.

A nice experiment to try Pi and see if a visible pattern emerges.

Congratulations, everybody.

Honorable Mention

I loved seeing students conjecturing mathematically about loop-de-loops, asking each other which ones converge and diverge, trying to predict the patterns they’d find in different strings of numbers. (See: Denise Gaskin’s comment for one example.)

Also, The Nerdery really sank its teeth into this assignment. This blog’s collection of programmer-types produced some great loop-de-loop visualizations:

Here is the original Malcolm Swan task, which I love:

Draw a shape on squared paper and plot a point to show its perimeter and area. Which points on the grid represent squares, rectangles, etc? Draw a shape that may be represented by the point (4, 12) or (12, 4). Find all the “impossible” points.

We could talk about adding a context here, but a change of that magnitude would prevent a precise conversation about pedagogy. It’d be like comparing tigers to penguins. We’d learn some high-level differences between quadripeds and bipeds, but comparing tigers to lions, jaguars, and cheetahs gets us down into the details. That’s where I’d like to be with this discussion.

So look at these four representations of the task. What features of the math do they reveal and conceal? What are their advantages and disadvantages?

Paper & Pencil

You’ve met.


Dan Anderson’s Processing Animation

Hit run on this sketch and watch random rectangles graph themselves.


Scott Farrar’s Geogebra Applet

Students click and drag the corner of a rectangle in this applet and the corresponding point traces on the screen.


Desmos’ Activity

277 people on Twitter responded to my prompt:

Draw three rectangles on paper or imagine them. Choose at least one that you think that no one else will think of. Drag one point onto the graph for each rectangle so that the x-coordinate represents its perimeter and the y-coordinate represents its area.

Resulting in this activity on the overlay:


Again: what features of the math do they reveal and conceal? What are their advantages and disadvantages?

September Remainders

Quick programming note: our Loop-de-Loop contest ends 10/6 at 11:59 PM Pacific Time.

New Blog Subscriptions

New Twitter Follows

Commenters I Wish Had A Blog / Twitter Account / Zine / Etc.

2015 Oct 14. Announcing the winners.


You should buy Anna Weltman’s new math book, This is not a Math Book.

You should buy several, probably, for all the little people in your life who are deciding right now what they think about math and what math thinks about them. If they’re taking their cues on that decision from someone who dislikes math or who dislikes little people, consider using This is not a Math Book for counterweight.

You’ll find dozens of pages of math art, math sketches, math reasoning, and math whimsy. I read it in one sitting outside a coffee shop one afternoon, big dumb smile on my face the whole time. Actually finishing the book, fully participating in Weltman’s assignments of creativity and invention, will take many more afternoons.

I’d like to send one of you a class set of Weltman’s book. Here is how you get it:

  • I love Weltman’s Loop-de-Loop assignment. It lends itself to some of my favorite mental mathematical acts around prediction, sequencing, transformation, and questions like “what if?” So you or your students or all of the above should make an awesome Loop-de-Loop. (Here is Weltman’s instruction page and her student work page, but any piece of graph paper will work.)
  • Scan and send it to
  • I’ll pick my five favorites and ask some of my favorite math artist friends to pick the winner from those five. Winner takes all, which is to say 40 copies of This is not a Math Book, from me to you.
  • Contest ends 10/6 at 11:59 PM Pacific Time.

Drawings, color, character work, mixed media, it’s all fair game. I can’t wait.

BTW. Over the next several days, Weltman is blogging interesting questions to ask your students about Loop-de-Loops.

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