2021 Term 3
Term 3 --- Week 1
8 : Zoltan : "Horrible game" with a quadratic equation.
1991 AMC.
9 : Tamiru : Intervals - closed, open, 1/2 open, infinite, finite, ...
Linear inequalities in one and two variables.
10 : Celina : Fundamentals of propositional logic and its relationship with algebra.
11/12: Chris : Discussed orthogonal polynomials and discussed their zero sets.
Term 3 --- Week 2
8 : Zoltan : Horrible game with y=x(x+1)(x+2)
AMC 1991 Q30. How many interior diagonals in a truncated cube?
9 : Celina : Propositional logic and its relationship with algebra.
10 : Elizabeth : Four short prime and factorisation themed questions.
Polite numbers and Problems 2 4 6 from Nrich.
11+12: Ralph : Platonic solids, planar graphs, complete graphs,
n-gonal prisms and Euler's proof of the fact that V-E+F=2.
Term 3 --- Week 3
8 : Zoltan : Number Wall. Horrible game with 2^x % 100
AMC 2013 J30.
9 : Tamiru : Inequalities.
10 : Chris : Fundamental theorem of calculus.
Weak derivatives.
Wheat derivatives.
11 : Ralph : Cubic curves classification. Singular points of a curve.
Elliptic curves. Addition via chord-tangent process.
Rational points on a line, circle, elliptic curve.
12 : Peter : Geometric variational problems & calculus of variations.
Fermat's principle of least time for light --> Snell's Law
*and* --> Light travels in straight lines.
Generalise to minimal area (e.g. soap films).
Mechanics (Hamilton's principle),
Euler-Lagrange equation is F=ma.
Term 3 --- Week 4 (zoom)
8 : Zoltan : In and out of the club game.
Worked on AMC problem sheets.
9 : Tamiru : Solved some problems involving inequalities.
Discussed "power of a point" theorem for circles and solved
a related problem.
10 : Ralph : Prove that the sum of any two consecutive integers is odd.
Tommi asked for Groups, Rings and Fields ... so we considered ...
Definition of a group [Closed, associative, identity, inverses].
<N, +> is not a group (no identity, or inverses),
<N U {0}, +> is not a group (no inverses).
<Z, +> is a group.
<Q \ {0}, x> is a group.
Students tried to produce groups with 1 and 2 elements .
We showed that <Z/2Z, +> is isomorphic to <{-1,1}, x>.
11&12: Peter : Introduction to and summary of calculus.
Term 3 --- Week 5 (zoom)
8 : Zoltan : In and out of club problems, namely
(a) is 2 x prime
(b) has 3 as one of its digits.
This led to a discussion of the density of subsets of Z
and a comparison of the sizes of integer sequences.
For example, there are as many evens as there are integers.
Discussed comparison of infinite sets.
9 : Tamiru : Proved the cosine rule and the sine rule.
Discussed mathematical induction and used it to prove
(i) 6^n-1 is divisible by 5, and
(ii) 1+4+7+...+(3n-2) = n(3n-1)/2.
10 : Ralph : Studied cyclic groups as groups of rotations of regular n-gons
and as abstract groups. Computed orders of elements in a group.
Distinguishing when groups are the same or not.
Compared cyclic group of order 4 and Klein-4-group.
(Searched for #occurrences of the word group in a random maths paper.)
11&12: Peter : Applications of calculus in maths, science, engineering,
medical research, finance, music, etc (part 1).
Term 3 --- Week 6 (zoom)
8 : Ralph : Edge minimisation series.
Count the minimum number of unit edges required to construct
1,...,12 unit squares when sharing of edges is allowed.
Discussed an algorithm to always find this number for any n.
Proved that if a circle and square have the same area
then the circle has a smaller perimeter.
Discussed the relevance of the Taxicab metric.
9 : Tamiru : Proofs by induction.
1) Prove 3^n > 2^n for all natural numbers n.
2) Consider the famous Fibonacci sequence {x_{n}}, where n = 1,2,3,...
defined by
x_{1} = 1, x_{2} = 1, and x_{n} = x_{n-1} + x_{n-2} for n>2
(a) Compute x_{20}.
(b) Use the extended Principle of Induction to show that for n>0,
x_{n} = 1/sqrt{5} [((1+sqrt{5})/2)^n - ((1-sqrt{5})/2)^n].
(c) Use the result of part (b) to compute x_{20}.
10 : Asilata : Triangulations and pseudo-triangulations.
We looked at triangulations of convex n-gons, and discussed how to generalise
to n points in general position in the plane (not necessarily convex). At the
end we briefly mentioned n points not necessarily in general position in the plane.
11&12: Peter : Applications of calculus across science, engineering, finance, medicine, music, etc.
Term 3 --- Week 7 (zoom)
8 : Zoltan : ???
9 : Tamiru : Let ABC be a right triangle with hypotenuse AC.
Let G be the centroid of this triangle and suppose
that we have AG^2 + BG^2 + CG^2 = 156 ... find AC^2.
10 : Ralph : Proof of Euler's theorem V-E+F=2.
Discussion of graph theory, planar graphs,
connected graphs, genus and relation between
Euler characteristic. Embedded K_5 into torus
without any crossings.
11&12: Peter : Weak derivatives and distributions in relation
to dual spaces.
Term 3 --- Week 8 (zoom)
8 : Zoltan : ???
9 : Tamiru : Three circles with radii 23, 46, 69 are tangent to
each other. Shade the region between the 3 circles.
Find the radius of the largest circle that can fit
inside the shaded region.
Stewart's theorem (about the length of a Cevian of a
triangle in terms of the sides and base segments)
was used to solve the question.
Proof of sine rule and cosine rule.
10 : Ralph : Classification of closed, orientable, 2-D surfaces.
Considered representing a sphere, cylinder, torus,
Moebius band, Klein bottle and real projective plane
as a square with various edge identifications.
Proved that MB + MB = KB.
11&12: Asilata : Triangulations and pseudo-triangulations."
We looked at triangulations of convex n-gons, and
discussed how to generalise to n points in general
position in the plane (not necessarily convex).
At the end we briefly mentioned n points not
necessarily in general position in the plane.