8 : Elizabeth: Graph Theory,
Eulerian paths, Eulerian circuits,
Hailtonian paths, Hamiltonian circuits.
9 : Tamiru : Set Theory
10: Ralph : Infinity, size of sets, 1-1 correspondence,
Cantor diagonal proof,
|evens|=|N|=|Z|=|Q| < |R|.
11: Zoltan : 2017 AMC Senior Q26-Q30.
12: Peter : Fourier transforms.
8 : Tamiru : Sets, subsets, number of subsets, proper subsets
9 : Elizabeth: Discussion of Polite Numbers which led to lots
of interesting side talks.
See Polite Numbers
Question 6 from NRICH 9340 also led to lots of side talks.
10&11: Ralph : p-adic numbers, example of 5-adics,
&12 integers, addition, multiplication, valuation,
3,33,333,3333,... --> -1/3 in 5-adics.
8 : Peter : Pascal's Triangle
9 : Zoltan : Horrible game y=2(x-1)^2+1.
Discussion of Number Wall and why Delta^2(f) = constant 2
leads to f = x^2 + ...
Steps towards realising that the solutions to a recurrence
relation form a vector space.
10 :Ralph : Compute first 25 primes.
Write them as a sum of two squares.
Factorise them as Gaussian integers.
Give criterion for when a prime is expressible
as a sum of two squares.
11&12:Tamiru : Worksheet.
8 : Zoltan : Horrible game y=(x-4)^2+64=(x-10)(x+2)+100
y=3^x div 17
Struggled with Tamiru's hard question ... basically
2018^(2019^2020) mod 2020.
Danger : Why does 2^(n+2020) != 2^n mod 2020?
9 : Peter : Place value.
Pascal's triangle, patterns of values 0 mod k.
For p prime, pth row = 1 0 ... 0 1 mod p.
Complex numbers (Fundamental theorem of algebra)
10+11+12:Ralph: Finite fields, in particular GF(7).
Cubic curve types.
Elliptic curves over infinite field picture.
Elliptic curves over finite fields points.
8 : Ralph : Compute the first 25 primes. (Discuss why 1 is not a prime.)
Write them as a sum of exactly two squares of integers.
Find the pattern for when a prime is expressible
as a sum of two squares.
Factorise them as Gaussian integers.
9 : Elizabeth: Graph theory introduction.
Started with two logic problems, then defined graphs.
Euler's theorem on a plane and a torus ... plus some
other fun facts.
10: Tamiru : Worksheet.
11+12: Angus : Eggs are usually bought in cartons of 12.
I found a picture of a carton of 7 (hexagon + centre).
What amounts of eggs can you make?
--- student driven research. ---
8 : Zoltan : The Horrible Game, y=(40x+7) div 99.
Proved that multiplication "commutes" with residues mod 9.
Proved divisibility rules for 9, 11, 7.
9 : Ralph : Compute the first 25 primes. (Discuss why 1 is not a prime.)
Write them as a sum of exactly two squares of integers.
Find the pattern for when a prime is expressible
as a sum of two squares.
Discussion of complex numbers.
Factorise primes = 1 mod 4 as Gaussian integers.
10: Elizabeth: Nrich factors, multiples and primes.
Worksheet 1, Q2 and Q4. Worksheet 2, Q3.
(Same as Term 1, Year 8, Week4 ... but more depth)
11+12: Angus : Eggs are usually bought in cartons of 12.
I found a picture of a carton of 7 (hexagon + centre).
What amounts of eggs can you make?
--- student driven research continued. ---
8 : Angus : The egg puzzle. Given cartons of 7 and 12 eggs can
you make 50? 51?
From there ask your own questions!
Some saw some patterns, had some methods.
They were very excited to figure out how to make 69420.
9 : Peter : Pascal's Triangle part 2/2.
10+11+12:Tamiru: Worksheet, Aug 19, 2022
8 : Tamiru : Worksheet, Aug 19, 2022
9 : Peter + : The Horrible Game. y=10(-1)^x +2^(x-5)
Zoltan : Recurrence relations and some deductions.
10 : Angus : The egg puzzle.
11 :Elizabeth: Graph Theory. planar graphs, complete graphs, etc.
+ Theorems about planar graphs.
12 Graphs on a torus. Euler's characteristic.