**Special relativity is one of those branches of physics that wows students. For some, it can even provide that spark of inspiration that draws them down a path towards pursuing physics in college, but can we do it justice at the HS level?**

Like many others, I was first drawn to physics after learning about special relativity. When I entered college I came in as an undeclared major. During my first quarter, I took an astronomy course with the name "Overview of the Universe." During the last class of the term, our professor introduced special relativity and taught us about time dilation, length contraction, and the relativity of simultaneity by going through some classic thought experiments. I was enticed. Who knew that learning about physics could cause me to question the intuitions & assumptions that I held most dear about the world!? By the next term, I had changed my major to physics and enrolled in the remedial math courses that I needed to get my math skills up to snuff for what was to come.

The most beautiful experience we can have is the mysterious. It is the fundamental emotion that stands at the cradle of true art and true science.- Albert Einstein

I wanted to make this branch of physics somewhat accessible to my students. Not just for them to follow the logic behind it, but to OWN it. I wanted them to use the framework of special relativity to reason through problems on their own.

But aren't the concepts too demanding for high school students? Einstein doesn't seem to think so. To quote him again (and I think I have the right to do so on a SR post) "if you can't explain it to a six year old, you don't understand it yourself." The challenge was in figuring out the right way to scaffold the concepts, the right emphasis of qualitative vs. quantitative understandings, and the right depth.

Since Special Relativity is not in APP1 or APP2 curriculum, I chose to dedicate four 80 minute blocks to it with my APP2 students during our modern physics unit. Below, I hope to briefly summaries day by day what I did in this unit.

**Day 1: Einstein's Postulates, Galilean vs. Lorentzian Relativity, and the Light Clock**

What better way is there to start a special relativity unit for HS students than learning from another HS student? My classroom is mostly flipped, so to motivate our unit we watched *Why You Can Never Reach the Speed of Light: A Visualization of Special Relativity* by Kadi Runnels. If you haven't come across this video, watch it! It is fantastic.

In the video, Runnels introduces the concept of a boost and uses grid geometry to qualitatively explain why boosting in hyperbolic space doesn't ever allow you to perform a boost that would allow you to observe an object to br moving faster than the speed of light (If you don't know what I mean--watch the video!). He also introduces Einstein's Postulates: (1) All inertial reference frames are equally valid and (2) The speed of light is the same in all inertial frames.

We followed up this video with in-class discussions where I went through some basic thought experiments. We did some velocity transformations (boosts) with Galilean relativity and talked about how if you do velocity transformations with special relativity you never observe another object going faster than the speed of light, because with hyperbolic geometry (like in the video) no boost will ever push an object's tile out of the edge of the circle. I put together some examples utilizing tiles similar to those in the video, but I actually showed students galilean and lorentzian boosts where I had calculated the numbers ahead of time. I did this to really emphasize lorentzian boosts never give you speeds faster than the speed of light (I didn't actually show them how I calculated these velocities, however. I just showed them the results to give them a *feel* for how equal boosts don't necessarily mean equal changes in observed velocity.)

**Deriving the Lorentz Transformations:**

I don't usually do derivations for APP1 or APP2, but the derivations for the time dilation and length contraction equations are so elegant and interesting that I thought my students would benefit from it. It's just so cool that you can derive these equations using nothing but Einstein's postulates, v = x/t, and the pythagorean theorem.

The derivation begins with introducing the concept of a light clock. The light clock when viewed from different reference frames forces you to conclude that the time between ticks on a light clock viewed from different frames will be different. With a bit of math you can quantify this difference.

Here are some quick videos I made of the derivations:

This is a lot for HS students to stomach, but the next two days were much more student-driven and didn't really introduce any new concepts. Instead they were geared towards applying what was discussed on the first day.

Students struggled a bit, but we moved slowly and they were engaged & involved, asking questions right up until the end of class. Overall, the day definitely seemed to be successful.

**Day 2: Qualitative Thought Experiments**

On day two we did only qualitative thought experiments. I handed out a list of statements they needed to PROVE using Einstein's postulates and then gave them the framework for the thought experiments they needed to walk through for each qualitative proof.

The statements began with proving time dilation. This was really just a re-hashing of the light-clock proof we did the previous day, but without any math. The thought experiments increased in difficulty eventually incorporating relativity of simultaneity through the classic train/lightning and ladder/barn paradoxes.

Students worked collaboratively on dry erase boards at their tables (as we usually do). I told them to raise their hands when their group was ready to explain a proof (knowing that I would pick a group member at random to take me through their explanation).

In retrospect, this worked fairly well, but I had too many student groups getting stuck. I ended up needing to pull multiple groups aside at the same time to go over thought experiments and work on reasoning through them together. In the future, I might do this differently, because I struggled to support student groups that were moving at such varied paces.

**Day 3: Quantitative Approaches to Special Relativity**

For day three we revisited the Lorentz transformations and started actually solving some problems quantitatively. We looked at some simple examples, but then went on to examine some of the same thought experiments from before, but rather than just qualitatively explaining what's happening, we used the Lorentz transformations to do quantitative examples of them. These tasks culminated in a quantitative example very similar to the train/lightning paradox that they discussed the previous day.

I worried that students would struggle with these, but the general attitude was much more comfortable than the previous day. Students like being able to fall back on equations and get a numeric answer that is right/wrong, so working on tasks that allowed them to do this gave them confidence and a sense of ownership over the equations.

**Day 4: Space-Time Diagrams and the Twin Paradox**

As a college student, when I studied special relativity in my modern physics class, I remember spacetime diagrams and boost grids being the key to developing a real intuition for concepts in special relativity. Even though we only had one more day, I really wanted them to get some exposure to these diagrams.

This day was much more lecture-based. We discussed light cones, then I walked them through how to create axes for a boosted frame on top of another frame. (I have yet to find a really great introduction to spacetime diagrams, but this minutephysics playlist uses a pretty interesting apparatus to introduce the idea).

I then projected boost grids onto my board at the front of the room and used it to explain why two observers moving relative to each other both see the other's clock as ticking slower, why moving objects contract, and last of all, we redid the barn/ladder paradox using a spacetime diagram.

To sum up everything I posed the twin paradox for them, gave them a minute to digest it, and then walked through this explanation using spacetime diagrams from Cal Poly professor John Mallinckrodt.

We had a bit more time at the end of the block so I gave them a chance to play around with Velocity Raptor, a game that forces you to exploit the effects of special relativity to solve puzzles.

**Looking Back:**

There's no way to really do special relativity justice in just a few days, but if I have time to do this again next year, I think the big thing I would do differently would be introducing spacetime diagrams earlier and giving students a chance to work through problems using boost paper on their own. As we all know real learning comes from doing, not just watching and listening!

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