More about intermediate sup-maths courses

Course 4-A

Prerequisites

This is the first intermediate course, thus does not require a great deal of knowledge. Attending the junior level courses would of course be a great help and provide all the requisite background. The main thing needed is awareness of basic concepts related to graphs. We expect this course to be attended by students with at least 7 years of formal education. Please be advised that the lessons for immediate will last about 2 hours and 15 minutes, a bit longer than the junior ones, so "diligence" is also prerequisite :)

About the content

We will start by revisiting the topic of graphs and study** trees**, which are graphs which contain no cycles. This provides a good starting point to understanding more complicated concepts in graphs, which are widely used throughout the STEM fields.

Next, we will introduce the idea of mathematical induction and sequences, dealing with recursive definitions and how to use information about the current state to deduce something about the next state. In maths and computer science, variables and states are often defined recursively, so it is important to be able to handle these, even if it is not possible to write down explicitly a formula or completely describe the current state.

Finally, we will push students' logical reasoning abilities harder, with more challenging iterations of previously seen topics from the junior courses, as well as an introduction to true tables. In the final lesson, some interesting logical paradoxes will be given to pick their brains and test that they have really understood the material.

This course should be of interest to anyone who might have an interest in pursuing a natural science, as the ideas are widely applicable and are especially common in computer science.

Next, we will introduce the idea of mathematical induction and sequences, dealing with recursive definitions and how to use information about the current state to deduce something about the next state. In maths and computer science, variables and states are often defined recursively, so it is important to be able to handle these, even if it is not possible to write down explicitly a formula or completely describe the current state.

Finally, we will push students' logical reasoning abilities harder, with more challenging iterations of previously seen topics from the junior courses, as well as an introduction to true tables. In the final lesson, some interesting logical paradoxes will be given to pick their brains and test that they have really understood the material.

This course should be of interest to anyone who might have an interest in pursuing a natural science, as the ideas are widely applicable and are especially common in computer science.

Topics

Course 4-B

Prerequisites

This course has almost no technical prerequisites, as the material will be built up from scratch. However, it will be difficult precisely because of that! Strong reasoning skills are a must, since most lessons will involve deducing lots of results from a relatively limited set of assumptions.

About the content

This is an 8 week course in Euclidean geometry. Historically, the Greeks proved everything they knew from just five starting assumptions. Quite a feat!

Since we have only eight weeks instead of hundreds of years, the course will not quite prove everything there is to know about planar geometry. Instead, we will cover all the concepts which may already be familiar to students, but in a manner consistent with the Greek spirit. Students will be expected to prove all their assertions, and will be given problems that look obvious, but in fact take some work to solve.

At first, we will build geometric intuition with compass and ruler constructions. Students will then examine the familiar topics of circles, triangles, and area in more detail. Towards the end, the course takes on a slightly different flavour, as students are introduced to combinatorial geometry, studying properties of configurations of points and lines in the plane. The final lesson will give a somewhat more historical overview of the Greeks and how they approached geometry.

Since we have only eight weeks instead of hundreds of years, the course will not quite prove everything there is to know about planar geometry. Instead, we will cover all the concepts which may already be familiar to students, but in a manner consistent with the Greek spirit. Students will be expected to prove all their assertions, and will be given problems that look obvious, but in fact take some work to solve.

At first, we will build geometric intuition with compass and ruler constructions. Students will then examine the familiar topics of circles, triangles, and area in more detail. Towards the end, the course takes on a slightly different flavour, as students are introduced to combinatorial geometry, studying properties of configurations of points and lines in the plane. The final lesson will give a somewhat more historical overview of the Greeks and how they approached geometry.

Topics

Course 5-A

Prerequisites

This course takes on a combinatorial flavour, hence combinatorial concepts from previous junior and intermediate courses will be very helpful.

About the content

The course begins with various methods for counting how many objects there are that have a certain property. Of particular note is the idea of first showing that there are as many objects as there are elements in some other set, and then counting the number of elements in this other set because it is much easier to count the number of elements in the second set. This idea is often used in higher mathematics, so it is important to be familiar with it.

We will also introduce probability at an elementary level, since uncertainty is baked into our daily lives, and have a good grasp of basic concepts paves the way to applying the ideas of probability to the world around us in a meaningful way.

Another important idea in higher mathematics is that of continuity. In this we will define discrete continuity, so that students have some intuition of what it means and how to solve problems with discrete continuity, preparing them for later courses in analysis.

The course ends with more lessons on analysing games and configurations on a grid, and a final lesson on how drawing pictures can simplify the problem at hand greatly, proving that a picture really can be 'worth a thousand words'.

We will also introduce probability at an elementary level, since uncertainty is baked into our daily lives, and have a good grasp of basic concepts paves the way to applying the ideas of probability to the world around us in a meaningful way.

Another important idea in higher mathematics is that of continuity. In this we will define discrete continuity, so that students have some intuition of what it means and how to solve problems with discrete continuity, preparing them for later courses in analysis.

The course ends with more lessons on analysing games and configurations on a grid, and a final lesson on how drawing pictures can simplify the problem at hand greatly, proving that a picture really can be 'worth a thousand words'.

Topics

Course 5-B

Prerequisites

This course requires understanding the material from the junior level courses, in particular the number theory: familiarity with divisibility and the greatest common divisor will be assumed. The other crucial skill will be dexterity with algebraic manipulation: there will be many variables and unknowns, so being able to work with symbols without being fazed is a must. At very least, students should be totally comfortable with the level of algebraic manipulation required for school.

About the content

We will start by giving students some more algebra training, as this will be indispensable for the following lessons. We will also give some harder problems about divisibility and gcd, ideas that students who attended the junior courses should already be familiar with.

The next lessons will cover key concepts from number theory, in particular the fundamental theorem of arithmetic. Students will also learn how to compare numbers in creative ways: even if they do not know the exact value of a numerical expression, and it is not feasible to evaluate it by hand, they can often meaningfully compare it to other expressions.

The course continues with modular arithmetic and Fermat's Little Theorem, workhorses of both Olympiad and professional mathematics. At the end, students will see a proof of the fundamental theorem of arithmetic, the first 'big' theorem proven in the Kvanta course, which is demanding, but extremely rewarding when students finally understand it.

The next lessons will cover key concepts from number theory, in particular the fundamental theorem of arithmetic. Students will also learn how to compare numbers in creative ways: even if they do not know the exact value of a numerical expression, and it is not feasible to evaluate it by hand, they can often meaningfully compare it to other expressions.

The course continues with modular arithmetic and Fermat's Little Theorem, workhorses of both Olympiad and professional mathematics. At the end, students will see a proof of the fundamental theorem of arithmetic, the first 'big' theorem proven in the Kvanta course, which is demanding, but extremely rewarding when students finally understand it.

Topics

Course 6-A

Prerequisites

Many concepts from the junior and 4-A courses will be further developed in this course, so it is essential that students are on top of the material covered in those courses. The difficulty of problems will also have increased substantially, so mathematical maturity is a must. Maturity in general will also be helpful, as students will be expected to struggle for longer with the problems before solving them.

About the content

The first half of the course will revisit the extremal principle, parity, and invariants. Mathematical induction will also be revisited, this time in full formality and expecting students to be able to handle the abstract statements instead of saying et cetera. This is a key topic for further studies in mathematics, so it is critical that students get this right.

Also featured is the pigeonhole principle: if there are 10 pigeonholes and 11 pigeons in those pigeonholes, then some pigeonhole must have at least 2 pigeons. This apparently simple idea will be fleshed out in detail since, as students will find out, some fiendishly hard problems have a clever application of this as the main step.

The next two courses will be a further lesson in algebra, and how it can be used to prove combinatorial statements. The course finishes by exploring some of the many fascinating properties of Pascal's triangle.

Also featured is the pigeonhole principle: if there are 10 pigeonholes and 11 pigeons in those pigeonholes, then some pigeonhole must have at least 2 pigeons. This apparently simple idea will be fleshed out in detail since, as students will find out, some fiendishly hard problems have a clever application of this as the main step.

The next two courses will be a further lesson in algebra, and how it can be used to prove combinatorial statements. The course finishes by exploring some of the many fascinating properties of Pascal's triangle.

Topics

Course 6-B

Prerequisites

This course has relatively few technical prerequisites: a working understanding of probability and being generally comfortable with problem solving and abstract mathematics should be enough. Ideally, students should be able to process new, difficult concepts, and be curious about the endless mathematical wonders that are just at their fingertips.

About the content

Facility with the material covered in the past five intermediate courses will make a student a strong candidate for their national mathematics competitions. In this course, we present a number of topics not particularly related to Olympiads which should nonetheless be of interest to the budding mathematician, giving a first taste of what awaits students who persevere with mathematics to university.

The first three lessons deal with games and information: how does one make use of the information at hand, whether complete or not, to formulate the best possible strategy? We will also introduce conditional probability, which is at the heart of statistics: how does one make use of*new* information?

Next, we explore the basics of set theory. Certain logical paradoxes arise due to a poorly defined set theory, and the work of professional logicians who sort these out begins in understanding set theory: what is a set?

We then move on to geometry in 3 dimensions, and of particular interest are the Platonic solids: these display a high degree of symmetry. What can we say about them? How many are there? And we finish by talking about paradoxes in probability: there are problems where depending on how to approach them you will get different answers. So we will discuss them.

The first three lessons deal with games and information: how does one make use of the information at hand, whether complete or not, to formulate the best possible strategy? We will also introduce conditional probability, which is at the heart of statistics: how does one make use of

Next, we explore the basics of set theory. Certain logical paradoxes arise due to a poorly defined set theory, and the work of professional logicians who sort these out begins in understanding set theory: what is a set?

We then move on to geometry in 3 dimensions, and of particular interest are the Platonic solids: these display a high degree of symmetry. What can we say about them? How many are there? And we finish by talking about paradoxes in probability: there are problems where depending on how to approach them you will get different answers. So we will discuss them.

Topics