I support an 8th grade math teacher in Inglewood, CA. Fabian is a first-year teacher with a lot of professional responsibilities on his plate (8th grade science classes, an environmental science class, advisory, and 8th grade math classes).

He is also a former student of mine. I taught him AP Calculus about 10 years ago. It’s been fun closing the circle (pulling back the curtain? bridging the gap?) between student and instructor.

About a month ago we were discussing the 8th grade geometry standards (transformations) and his teaching for the next unit. We looked at how MathLinks 8 sequences the geometry concepts for students and guides instruction for teachers. (Disclosure: I’m an author of the program.) While the MathLinks curriculum is great for conceptual development, it still felt too formal as a starting point for his students. We wanted to invite students into a conversation that would require them to use math. We didn’t want to start the unit with a conversation about math explicitly because we knew some students would be turned off.

So we turned to Robert Kaplinsky inviting, “low floor” lesson about the movements of Ms. Pac-Man. I used to work for a non-profit that used a free software program called GameMaker to teach students mathematical concepts through video game design. (I’m not sure where the project is at now, but it was called MathMaker and you can find more about it here.) Kaplinsky’s lesson is an example of the main philosophical thrusts of MathMaker: students can learn math not just by playing video games, but also by talking about video game design and making their own games using mathematical reasoning and simple drag-and-drop computer code. (I miss working on the project and have more to say about it’s value; geek out with me about it sometime in person!)

The goal of our intro-to-the-unit lesson was to get students engaged quickly and talking to each other about describing motion. We followed Robert’s write-up pretty closely. We played the first video a few times and chose the prompt: “Describe Ms. Pac-Man’s movements.” We let them talk in groups first and then started making a class list. To keep engagement and honor participation in thinking, we repeatedly used the statement: “Raise your hand if your group said a similar thing. High five each other!” To attend to precision, we followed up slightly imprecise statements with “Did anyone say the same thing in a different way?” We did not introduce “translate,” “reflect,” and “rotate” language. But we did unpack what they meant by move, flip, and spin (and the like). The definitions of their informal language ended up being recorded and we attached the formal vocabulary to their informal definitions in the next lesson. We did our best (but may have missed that mark wth some students) at only formalizing the math after conceptual understanding had been established.

Robert’s sequence of videos accurately predicted our student misconceptions. (The initial reflection was the hardest for students to see.) The videos allow students to see their mistake without feeling they are wrong or “bad at math.” The follow up questions are great. “How far did she move? Which way did she turn? What do you mean ‘flipped’?” were some questions we created too. If you use this lesson, I encourage teachers to go slow in the discussion and to take the time to unpack student thinking.

We concluded the exploring part of the lesson after we had flushed out all of their language about Ms. Pac-Man’s motion. Then we asked them what math they saw in their words. After a brief discussion, they wrote their responses on exit slips. We wanted to save Robert’s larger task for later when they had a stronger grasp of the language and math.

The rest of the lessons in the unit started with some sort of connection to video games whenever possible before launching into more formal conceptual and structural “patty-paper” type lessons for each rigid transformation.

Here’s a quick snap shot of some of the Warmups:

Note the addition of numbers. We were slowly adding in more structure.

Then we put in more structure using coordinates. (Not sure why Ms. Pac-Man is stretched in the image above. Her distortion was not intentional.)

And we used more formal Warmups as we progressed:

Eventually, Fabian created an assessment that was very similar to Robert’s task. In addition, he had 4 more formal problems asking students to perform or analyze rigid transformations on object.

Here are some of his statements in our reflection conversation today:

- Students of all levels got on board with the opening lesson. It’s such an easy thing to get kids talking about.
- Ms. Pac-Man makes the math so conceptually obvious. She needs to look where she’s going. Orientation matters!
- There was much less student push back when the test came. They all wanted to do it and every student completed the test. That was the first time this year that every student made a legitimate attempt.
- Only stronger students could use abstract notation and coordinates to describe motion in the coordinate plane.
- All students have room to improve their precision.

Some things we’re pondering moving forward:

- Most students still struggle to describe reflections mathematically. All students did well describing rotations and translations. We wondered how to leverage the Ms. Pac-Man task to do more reflections. Could we get a clip of here moving turning around vertically when she moved horizontally and horizontally when she moved vertically? We wondered about making one for next year.
- Also, how do we get her to reflect across an axis?
- How do we get students to attend to precision by checking to see if their answers are right?
- How do we get them to keep using the tools (namely, patty paper) when their use of mathematical structure isn’t accurate?
- How do we get them to embrace the notation more?
- A sequel idea was to have students pick a ghost in the image and describe how that ghost could get Ms. Pac-Man if she wasn’t moving. This would allow them to practice describing transformations in a more “open middle” way because there are several answers that yield similar results. Letting students pick their ghost also invites them to choose their own level of complexity. Some ghosts are a lot further than others.

Thanks Robert for making such a great resource! Kudos to Fabian for finding ways to make math more inviting and to keep all of his students engaged in math as a first-year teacher!

Teaching is an amazing journey.

This is a wonderful and thorough reflection (transformation pun is unintended!). Thank you for sharing how you used the lesson, what you tweaked, and even the teacher’s reflections. Very cool to read and I will definitely be adding this to the lesson for other teachers to read.