"Notice & wonder" in K-8 world helps Ss develop the life skill of developing & refining natural next qus. In HS world let's not quelch NNQs! A curiosity like attached leads to: Change nmbr of pennies? Asymmetrical leapfrogging? 3D? Right-angled leaps? etc. And teaches curriculum. pic.twitter.com/1mpghA5KZc

After seeing the tweet, I couldn’t wait to do the project today. It turned out to be much more difficult than I was expecting, though. One of the difficulties that really caught me off guard was the time it took to make accurate measurements of the positions. The difficulty there turned what I mistakenly thought was going to be a really quick part of the project into maybe 90% of the project’s time.

Maybe a nice surprise with this project is that it could be a good introductory project for introducing kids to measurement.

So, although I’ll publish all 6 videos from today’s project, the main ideas are in the first and last videos.

Here’s the introduction to the project:

The boys weren’t totally sure what happened the first time around, so we tried again:

After two tries, they still weren’t sure what was going on, so we tried one more time:

After this third try, they had some interesting observations:

One idea they had in the last video was to see what would happen if the 3 points were on a line.

To wrap up, the boys wondered about a few other set ups. I was happy to hear that they were starting to think about new ideas.

I think Tanton’s problem is an absolutely great exercise for kids. I’m sorry that I misjudged the difficulty coming from the measurements, though.

One thought on “James Tanton’s penny and dime exercise”

I saw some of your posts on twitter. I only skimmed the first and last video here.

As two ideas: maybe consider talking about what happens to other points (not just a single red cube, but several at once) under the same operation, under various numbers of reflections. I think if you get a big sheet of butcher paper or whatever and then pick a different color to use for the points after each operation, you can maybe see what happens to e.g. 3 points in a little equilateral triangle after each reflection, how they transform relative to each-other.

Also maybe consider trying again using dots on a big grid paper? Or is that too much of a hint about one way to go about solving it? (You already seem to have gotten the idea of trying on just one axis.)

Have you folks done other transformation geometry problems, either with or without a coordinate system? You might get some useful tools for thinking about this kind of problem out of talking in general about translations, rotations, reflections through a point, reflections through a line, glide-reflections, etc.

I think transformation geometry is one of the really key conceptual ideas that people should work on (e.g. in middle/high school) before trying to learn newtonian mechanics, linear algebra, non-euclidean geometry, group theory, etc. In particular, figuring out how to relate algebraic expressions with geometric transformations is so powerful.

Anyway… I think you can profitably keep picking away at this and related problems for another few weeks.

Cheers.

[I think some glitch in the forum software ate my comment the first time]

I saw some of your posts on twitter. I only skimmed the first and last video here.

As two ideas: maybe consider talking about what happens to other points (not just a single red cube, but several at once) under the same operation, under various numbers of reflections. I think if you get a big sheet of butcher paper or whatever and then pick a different color to use for the points after each operation, you can maybe see what happens to e.g. 3 points in a little equilateral triangle after each reflection, how they transform relative to each-other.

Also maybe consider trying again using dots on a big grid paper? Or is that too much of a hint about one way to go about solving it? (You already seem to have gotten the idea of trying on just one axis.)

Have you folks done other transformation geometry problems, either with or without a coordinate system? You might get some useful tools for thinking about this kind of problem out of talking in general about translations, rotations, reflections through a point, reflections through a line, glide-reflections, etc.

I think transformation geometry is one of the really key conceptual ideas that people should work on (e.g. in middle/high school) before trying to learn newtonian mechanics, linear algebra, non-euclidean geometry, group theory, etc. In particular, figuring out how to relate algebraic expressions with geometric transformations is so powerful.

Anyway… I think you can profitably keep picking away at this and related problems for another few weeks.

Cheers.

[I think some glitch in the forum software ate my comment the first time]