I always wanted to understand Galois Theory enough that I know why there’s no formula with for the roots of a general quintic in terms of radicals. While I was on strike for a week, I read some notes and livestreamed the process of learning.
Who: I usually run this with Year 8 or Year 9 students who are keen on mathematics (keen enough to give up a Saturday morning to come to an Ri masterclass!). I've also run versions of this activity with older children. It's very hands-on.
How long: Two hours for the full masterclass, with plenty of time to work on the main game. Removing the motorcross intro and skipping the discussion at the end can get the session down to one hour.
Aims: Introduce vectors, vector addition, and the rough idea of integration as the sum of small parts. Demonstrate how applied mathematicians try to explain things in the real world. Proof of the formula for triangle numbers.
The first few slides go through a multiple-exposure photo of a motocross jump (Ronnie Renner setting a world-record quarter-pipe jump in 2009).
I ask the group what will happen next, and usually get some good descriptions of how he'll go up and come back down again.
Introduce Newton and say that Newton had an idea for how to make this precise. Show the photos of Ronnie doing the jump.
Show the table and introduce the idea of adding velocity to height, and adding acceleration (negative number for "down") to velocity.
Ask them to fill in their own copy of the table, and plot a graph of height against time in any way they want to (they usually make some nice line graphs or bar charts).
Show my version and overlay it onto the jump; looks like a pretty accurate prediction, and it's just adding, it's not rocket science! (This joke is important for later).
Next we're going to play a racing game (you may have seen these other games, but unfortunately we're playing a game where I made all the graphics).
Introduce the idea of a velocity vector because we need to keep track of your velocity in 2D for this game. Some examples of moves (vectors).
Explain the rule about changing your velocity. Let them play. They usually ask about crashing into each other; I usually suggest "blocking rules" where the car behind is not allowed to enter an occupied square, and must make a different choice (which is almost always possible).
About three-quarters of them will crash on the first corner by failing to brake, so have lots of spare copies of the racetrack ready. I like to do a show of hands to ask how fast they were going when they crashed off the track (and it's an excuse to introduce "speed" as the length of their velocity vector, using Pythagoras).
Eventually some of them will make it around, and I start a leaderboard of how many turns it took them (like racing against ghost versions of each other in a time trial).
Keen pairs of students may try to optimise the route and search for the fastest lap; at this stage working together is a good idea for speed and to avoid blocking. Note that changing your mind about a previous move isn't simple because it might affect later choices, so it's not easy to tweak a route.
The rules of the game quite naturally give curves a bit like racing lines around the corners, and it's quite fun to walk over to a group and commentate on how it's going as if it's a race.
Later slides look at braking distance via triangle numbers; there's a visual proof of the formula for triangle numbers.
Then there are several slides on the fastest possible lap; I taught my computer to play (time trial style, ghost cars) and got it to check all possible routes (removing the ones that crashed or went the wrong way around the track).
Each red circle is a car or multiple cars, and the computer tries out all possible moves, creating lots of cars at each time step. A short route of length 20 is shown.
Finally, there are some examples of other places where forces are important, and I try to say something vague about calculus ("we can make this more realistic with smaller pixels and more frequent checks of the controller input").
Mention 3D forces on F1 cars (downforce!) and early work on planetary orbits. Oh, and rocket science. This was rocket science all along.
Shapes
Context: I made this as a Maths taster session for the University of Cambridge. It's not interactive at all. The last activity features the number 2015, which dates the session a bit.
How long: half an hour, at most.
Who: I ran this with Year 10 or Year 11 students who were visiting the University for a general visit, and who weren't necessarily interested in maths.
Aims: Introduce three purely visual mathematical proofs.