Optional · never graded
Wonder Trails
Optional side-paths for the curious. They never count against you, and they always lead back to where you were.
- K–2Boxes inside boxesWhat is inside a box that contains only an empty box? Is it empty? And can every description you can think of make a collection?Starts from Collections →SetsRussell's paradox and why sets need axiomsZFC: axiomatic set theoryFollow the trail
- Grades 6–8How many infinities?There are infinitely many counting numbers and infinitely many infinite 0–1 sequences. Are these two infinities the same size?Starts from Sets →Countability and Cantor's diagonal argumentFollow the trail
- Grades 6–8Clock arithmeticIt's 9 o'clock. What time will it be in 5 hours — and in 29 hours? Can a clock teach us a whole new kind of arithmetic?Starts from Integers →Clock arithmetic: remainders and modular arithmeticCongruencesQuotient groups and quotient ringsFollow the trail
- Grades 6–8Inverses and structureEvery integer can be added to another integer to get . So why can almost no integer be multiplied by another integer to get ?Starts from Integers →Algebraic structures: monoids, groups and ringsGroupsRings, ideals and fieldsFollow the trail
- Grades 6–8Remainder familiesSort every integer — positive, negative and zero — into families by its remainder after dividing by 3. If you add a member of one family to a member of another, does the family of the answer depend on which members you chose?Starts from Integers →Relations, equivalence relations and partitionsCongruencesQuotient groups and quotient ringsFollow the trail
- Grades 9–12The product, by its mapsCan you describe the set of pairs without ever looking inside it — only by how functions go into it?Starts from Functions →Universal propertiesLimits and colimitsFollow the trail
- Grades 9–12Arrows that composeWhat do functions between sets, maps between groups and continuous maps between shapes have in common? What if we studied only the arrows?Starts from Composition →CategoriesFunctorsFollow the trail