My 8th graders are learning about rigid transformations. I want to add a bit more complexity to what our book is asking the kids to do. For example, the book is having them reflect a shape mainly across the *x*-axis or *y*-axis, or on a rare occasion, reflect it across “the horizontal line that goes through *y* = 3.” Well, right before this chapter, we’ve been working with writing and graphing linear equations, so I want kids to reflect a shape across *any* line, including one that may cut through the shape itself.

The book surprisingly has very few examples and exercises with rotations. And from what I can find, all these rotations happen about the origin or about a point coinciding with one of the vertices of the shape. Again, I want kids to be able to rotate a shape about *any* point, including one that’s inside the shape. (I used a playing card — number 7 works well because it’s asymmetrical — poked my pencil through it as the center of rotation, and turned the card. I think this helps them see what I keep referring to as the pivot or anchor point.)

Then I give each student this task:

- Draw a shape that has between 5 to 8 sides with no curved edges.
- Transform your shape through at least 3 rigid transformations of rotation, translation, and reflection — in any order.
- On grid paper, give your teacher your complete work on this, including the written directions for the transformations.

- On grid paper, give your teacher
**only**the original shape and the written instructions. Your teacher will give this paper to a random classmate to follow your written directions to arrive at the intended location of the final image.

For students who want more challenge, they may ask for a copy that has just the original shape and its final image without the written directions. The task will then be to figure the appropriate transformations that connect the two images.

I really believe that it’s good practice to always give kids more than what we believe they can handle. Let kids tell us when it’s too much for them — and we find out soon enough. An ounce of struggle on something hard is worth a pound of completion on something easy.

(And I’m hoping to update this post with pictures of kids’ work when they turn them in on Friday.)

## 8 Comments

Wow!! I am so stealing this later this year. Thanks so much for writing about it.

I love having kids make the problems. It is a different kind of insight, and leads to a different and better feel to the problem-solving.

I like the playing card demo too.

It’s really nice to get kids to make their own problems for their mates to solve. You can challenge them to make it as difficult as they can, with the condition that they themselves must be able to solve it. This of course applies to any kind of math problem, not just transformations.

I like what you are doing.

Now that they can do reflections in any line you could lead them to see that a reflection followed by another reflection gives a rotation.

Also they should be able to see that all of this can be done without coordinate axes.

I have several posts on this stuff, back in August or September.

Expanding on howadat58’s comment, can they create any rigid transformation with just reflections?

What about with just rotations? Or just translations? Why/why not?

Before launching into this investigation, I would be tempted to have a discussion about which transformation seems the most powerful and flexible. how does it feel to them?

reflection does for translation. just reflect in the halfway perpendicular line.

I am still always mystified that the textbooks …now animated with technology…don’t tie animations into all of this transformation stuff. From there its a quick slide to 3d transformations that kids are doing on their smart (or not so ) smart phones. I guess it is now time to look at that numberphile that talks about animation. Glad to see you are stretching and shearing the kids minds.

I love using Geogebra and asking students to predict what will happen when I reflect a shape over a line other than the axes, or rotate around a point other than the origin. Or predict if I reflect a shape over a line of reflection within the shape or a point of rotation within the shape. The predictions they provide gives me a great insight on their ability to actually understand what those transformations do. I then make it happen and have them compare and contrast to their predictions to what actually happened. The best is when we get to dilations and I ask them to predict about a dilation by a factor of -2. Because “since -2 is less than 0, of course it will shrink”

Thanks for sharing!! I am going to adapt this for my high school geometry class.