3Blue1Brown is the best channel on youtube. Here is a selection of his amazing videos, and you can find the rest of his videos at the youtube page here. If you find a video worthy of recommendation, please let Mr Fletcher know.
Essence of Linear Algebra playlist (linked to matrices in further maths)
Visualising the Riemann Hypothesis
Why is the volume of a sphere four times its shadow?
The sum of reciprocals squared ... an interesting result
Collisions of blocks equals the digits of pi
Why do primes numbers make spirals?
All possible Pythagorean triples
Derivative formulae through geometry
Numberphile is an amazing youtube channel with loads of amazing videos, it's well worth having a look here. Below are some videos that Mr Fletcher recommends, and please inform him of any other videos that you find particularly interesting and worthy of being on this list for future students.
What's so special about the Mandelbrot Set?
Godel's Incompleteness Theorem
Fundamental Theorem of Algebra
BlackPenRedPen is the Youtube name of a great teacher in America, so you might find the terminology a bit different, but it well worth a look if you would like some extension questions. There's loads on his Youtube page, some of which have been recommended by Mr Fletcher below:
Introduction to Hyperbolic Trigonometry
Integration by Parts - The DI Method
A Taste of Real Analysis (adult calculus)
Find here the youtube channel for Oxford University Mathematical Institute - they have loads of excellent lectures on all sorts of different areas of mathematics. Some are examples of university lectures, but the majority are for public viewing. Below are some of Mr Fletcher's favourites:
Productive Generalization - Timothy Gowers
In February, the upper 6th take part in the annual Adams Integration Bee. This is a non-house event where serious integrators compete to become the grand integrator. In preparation, the MIT Integration Bee website here. Scroll to the bottom of their page for resources.
MIT 2006 Integration Bee Video
MIT 2015 Integration Bee Video
MIT Integration Bee Qualification Test 2013 - Video Solutions
MIT Integration Bee Qualification Test 2019 - Video Solutions
MIT Integration Bee Qualification Test 2020 - Video Solutions