Throughout my career I have been very fortunate to be able to work on many interesting problems in mathematics. Of all the areas I have worked on, though, some of the most enjoyable and memorable pursuits have been when posing, and exploring, a variety of mathematical problems that lend themselves to relatively elementary mathematics and which are accessible to students in schools and colleges. I have put some of these ideas together in the document A mathematician’s miscellany, and an apology.
My first foray into outreach was giving a talk at a local school based on one of my articles: Mathematics for the bath: a model and its solution. Teaching Mathematics and its Applications, 9 (1). pp. 15-18, (1990). ISSN 1471-6976 doi: 10.1093/teamat/9.1.15.
I followed this up with a Royal Institution Mathematics Masterclass entitled 'Torches, Teacups and Satellite TV' - no prizes for guessing what that was about! I did this for 5 years and had lots of fun doing it. There were lots of 'props' and plenty of hand-on activities for the students to enjoy (and learn something!). I plan to make the 'Lectures' and Exercises & Activities available soon.
I have other articles that could form the basis of masterclasses, enrichment sessions etc.
Having become interested in conic sections I wrote a module for my first year undergraduates on this topic, and had much enjoyment learning Geogebra and investigating existing and new results, some of which can be found here. The materials for this can be downloaded via the 386MB compressed file. (There are hyperlinks links to Geogebra, postscript and Shockwave files which are contained in subfolders - by maintaining the file structure will ensure the hyperlinks will be followed correctly.) A subset of these materials were made available as part of our article: Promoting problem solving through parabolas. Mathematics in School, 45. pp. 17-24. ISSN 0305-7259 (2016), via the following links: Background material for Promoting Problem Solving Through Parabolas, Problems for Promoting Problem Solving Through Parabolas, Problems and selected solutions for Promoting Problem Solving Through Parabolas, Geogebra files for Promoting Problem Solving Through Parabolas.
Other enrichment materials and activities I have put together have been used in outreach activities and events, including: Mathematics Masterclasses; Widening Participation Summer Schools; Enrichment Days, and through being a STEM Ambassador for many years. These resources include topics on: prime, transcendental, and Fibonacci numbers; proof; as well as codes, code-breaking, and internet security.