a/k/a Annuli Follow-up
Is there any advantage to these images over the analogous problem in a textbook?
I vote "definitely, yes." The first four of these questions offer two enormous bonuses on top of the fifth while assessing the same skills.
The first is that you can guess them intuitively before you answer them mathematically.
What do you think? 500 tickets? 5,000 tickets? 50,000 tickets?! Give me a wrong answer. Give me an answer you know is too high. Give me an answer you know is too low.
I spent six years looking for high-yield techniques to draw students who hate and fear math into conversations and then calculations about math. Given another six, I'm sure I'd find something more effective but that right there is the best I have. It costs you nothing and it gets them talking. It gets them interested in an outcome. It gets them interested in the tools to determine that outcome.
The other advantage to this curriculum is that the student doesn't need the teacher to verify the answer.
I usually envy all the fun ELA instructors get to have with their students. Not here, though. ELA instructors have to grade essays using subjective measures of form and content. "Was my thesis coherent?" the student wonders. "Was my essay persuasive?" The student waits for the instructor to render judgment. This is necessary, I suppose, but it's also adversarial and it forces the teacher to double down elsewhere to restore a spirit of collaboration to the relationship between teacher and student.
Meanwhile, my math student wonders, "Was my original guess correct? Is my math right?" to which I can respond, "Beats me, man. Let's find out." And we count up the tickets. Or I show them the playlist from which I burned that CD. Or we measure the toilet paper. Or we look at the front of the dental floss container.
Every answer but the last one disposes the student to see that math makes sense on its own terms, that math coheres to the world, that math exists apart from her teacher's say-so. Her teacher doesn't determine the correctness of her answers.
This did wonderful things for my relationship to my students. At our very best, we became peers, collaborators, and co-conspirators in the creative exercise of mathematics.
2012 Mar 12: "It’s Killing Me. I Gotta Know."
2013 Feb 14. Mr. Ward has another illustration.
2013 Feb 27. The conclusion of the Barbie Bungee activity has students testing out their predictions, making sure their bungee cord is long enough for Barbie's head to come close to the ground but short enough that it doesn't touch the ground. Kids flip for this, apparently. Here are examples from different teachers' classrooms: