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Our curriculum

What is studied in Axiom Maths Circles?

Axiom Maths Circles is a taught enrichment programme. It is not an acceleration programme: in their circles, students do not encounter content from a later Key Stage in the National Curriculum (or equivalent).

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Areas of study

The areas of study in the Axiom Maths curriculum are chosen to build on what students will have already learnt at school, but also to allow them to progress systematically through a network of related ideas in areas which develop deep mathematical significance: for example, combinatorics and geometry. Students increase and improve their factual knowledge and procedural fluency as they do so. The focus of a circle session is the underlying mathematical structure of the content.

Topics are loosely organised within strands, so that, once an idea has been encountered, the concept is reinforced and its use practised when that strand is revisited in one or more subsequent Blocks. Within strands, specific content has been chosen because it provides plenty of opportunities for students to reason mathematically, in an increasingly ambitious way, but with only a manageable level of technical apparatus and a limited dependence on where students have reached in the school curriculum.

Principles

The long-term sequencing of the Axiom Maths curriculum and the specific task design in each session enable mentors to apply the research-validated principles of effective pedagogy that can also be seen in the best teaching in UK classrooms. These principles include:

  • Tasks should develop understanding

    Tasks should allow students to develop their conceptual understanding of the ideas and work out for themselves how to apply them, not just to practise methods they have been taught.

  • Learning needs to be broken down into steps

    These steps need to be sufficiently small to be manageable but also appropriately challenging.

  • Problem-solving is not a generic skill

    Learning to solve problems in mathematics involves learning different skills in different topic areas.

  • Practice needs to be purposeful

    Variation between questions should be designed to draw students’ attention to the current idea. Tasks should include those which illustrate where a technique is not applicable as well as its range of applications. Student practice should be guided, so students learn appropriately from what they did right and what they did wrong.

  • Ideas need reinforcement

    Practice should be spaced, so that new ideas are regularly reinforced until they are secure. Some tasks should revisit ideas introduced in previous sessions; some should look ahead to ideas to be met in later sessions.

  • Learning should not be easy

    Tasks should be designed not to make learning a skill as easy as possible, but instead to put in place ‘desirable difficulties’ which force deeper processing. Allowing students to experience productive struggle is essential: mentors should not always provide immediate support even when students seem unable to make progress.

  • Students should experience creative mathematics

    Students should be expected to generate mathematics for themselves: this can range from finding their own examples and non-examples, through writing questions with given answers, to finding their own strategies for unfamiliar problems.

Parity

The parity of an integer is its oddness or evenness. The mathematicians learn results such as that “a product can only be odd if all its factors are odd” and use these to prove that some outcomes of a process are impossible because the parity of a variable is constant, or alternates. They use parity problems as an accessible context for learning to make formal arguments using known results, and to establish the difference between not ruled out and possible. This also allows exemplification of the idea that identifying and ignoring irrelevant aspects of a problem can be an important part of solving it. This strand is closely entwined with that of Games, because parity is an important example of an invariant. It only appears as a strand in the early blocks but the ideas later develop into colouring problems in the Network stand and the idea of remainders on division by integers other than two in the Digits, remainders and divisibility strand. In Computer Science, parity checking is often used as a simple error detection protocol and the theory of elementary particles exhibits symmetries which have a parity and combine like even and odd numbers.

Example question

Write down the whole numbers from 1 to 1001 and place a + or – sign in front of each. Can you choose the signs so that the sum of the signed numbers is 2024?

The key to this problem is to ignore the specific numbers and ask instead “Can the sum of 500 even integers and 501 odd integers come to an even total?” This means that you are using just the parity of the numbers involved. Axiom mathematicians establish, in this strand, the general result that a sum which includes an odd number of odd integers must be odd, so the answer is “No, you can’t”.

Games

In the two-player mathematical games we focus on in this strand, players take turns to change the game state, and compete to be the one who achieves the winning state. The mathematicians analyse these games by comparing possible strategies; by drawing, or imagining, game trees; by using symmetry; by identifying invariants or monovariants, such as parity, and by working backwards from the goal to identify good and bad states for them and their opponent to face. A number of the games they analyse also depend on knowledge of other mathematical domains and some of them turn out to be pseudo-games, where one of the players is bound to win – understanding why this is helps the mathematicians understand the constraints imposed by the structure of the domain in which the game is set.

Example question

Ten counters are placed on a table and two players take turns to remove either one or four counters. Who wins?

This is an example of the type of game we focus on in this strand: players take turns to choose how to change the game state—here, the number of counters on the table, and compete to be the one who achieves the winning state—here, leaving no counters. Chance is not involved. A game of this sort can be represented by a tree diagram, where players take it in turns to decide which branch to follow. Axiom mathematicians establish who will win by considering good and bad positions to face and working backwards from the winning state: it’s good to face 1 or 4 counters; so bad to face 2 or 5; so good to face 3 or 6 or 9; so bad to face 7 or 10. The first player will lose.

Combinatorics

Counting might seem trivial, indeed for most of us it’s where our journey in mathematics begins, but when you’re asked to count the number of patterns satisfying some constraints you’re going to need some clever ways of counting. In this strand, students learn how to list objects systematically in a variety of contexts, and to enumerate a large number of possibilities without listing them. They learn the vocabulary to explain their system, and learn to use representations that help explain their choices, for instance tree diagrams, Venn diagrams and two-way tables. They learn the idea of conditioning on the values of a particular variable to break up a large problem into a series of smaller ones, and begin to think recursively, so that they recognise the smaller problems can be broken up further by similar conditioning. The mathematicians also meet the idea of overcounting by a constant factor and use this to develop ways of enumerating the number of selections of objects from a set. This culminates in the appearance of Pingala’s triangle in the counting of routes through a maze, and the surprising links between these problems.

Example question

For a counter moving from square to adjacent square on a 5 × 5 grid, how many shortest paths are there between opposite corners? How many ways are there to divide a group of 8 children into two teams of four? Why do these questions have the same answer?

Both of them are equal to the number of ways of colouring a list of eight things so that four of them are red and the other four blue: in the first case, a shortest path consists of eight steps, of which four need to be vertical (red) and four horizontal (blue); in the second case, the colouring divides the list of children into a red team and a blue team. There are 70 different ways to make this colouring.

Networks

In the networks strand, our mathematicians get to explore something really quite different from mathematics in the national curriculum. Networks are key to re-representing a whole raft of complicated mathematical and real-life structures. It’s also highly current—there are mathematicians all over the world working right now to solve yet unsolved problems in network theory.

A problem can be represented as a network problem whenever the structure is a set of objects and a relationship which holds between some pairs of objects but not others. This could be physical things like computers, pairs of which are or are not connected by a cable in a network, or towns, pairs of which are or are not connected by roads, but could be non-physical things like actors, pairs of whom are or are not connected by the relationship “has been in a movie with” or the objects can be mathematical, for instance positive integers pairs of which are or are not in the relationship “sum to a prime”. The objects are represented as points called nodes, and the relationship by lines between the pairs for which the relationship holds, called edges.

In our maths circles, mathematicians learn about features that might or might not be present in a network, such as a “Hamiltonian path”—a path through the network which visits every node in the network exactly once, and they interpret what that might mean in the context of the problem, e.g. whether its possible to visit every town exactly once for a given arrangement of roads between them. This gives opportunities to allow the mathematicians to distinguish between showing that such paths are possible (by exhibiting one) and showing that none is impossible, by identifying an obstruction to any attempt to construct one and justifying their claim that this will definitely rule out its existence.

Example question

Can you write all the whole numbers from 1 to 20 in a line, so that the sum of any two adjacent numbers is an exact square? What if you include 0 as well?

This problem hinges on whether or not each pair of numbers is related by summing to an exact square—so it is a good example of a problem which can be represented on a network: let each number be a node and connect each pair of numbers which sum to a square with an edge. The question is now “is their a path through the network which visits each node once and only once?”

Digits, remainders and divisibility

In this strand the mathematicians work on what might loosely be termed ‘number theory’ and consider a variety of problems involving positive integers. Initially, they look at cryptarithmetic problems to develop the idea of arithmetic with the ones digits of numbers and particularly that only some digits can be the ones digit of a square number. They move on to look at factors and primes, especially highest common factors and lowest common multiples; here they will use the ideas they have met in school lessons, but have time to tackle problems of significantly greater depth than they are likely to meet at KS3. Similarly they will prove the divisibility rules for 3, 4, 9 and 11, which they may well have met previously, and see how to use them in less obvious ways—for instance recognising that a number is not a square, by showing that it is a multiple of 3, but not of 9. Later they meet the idea of arithmetic in different bases, including binary, and relate the ones digits of numbers in base 4 to the remainder when dividing a number by 4 – and specifically that squares are multiples of 4 or are one more than a multiple of four.

Example question

A repunit is a positive integer such as 11 or 111111, all of whose digits are 1s. Which repunits are square numbers?

Well, 1 obviously is and 11 and 111 aren’t. All the repunits with four or more digits can be partitioned into 1…100 + 11, but 1…100 is divisible by 4 and 11 is three more than a multiple of 4. Aha! All square numbers are multiples of 4, or one more than a multiple of 4 (a result the mathematicians prove in this strand), so 1 is the only square number.

Algebra

The mathematicians will all be studying algebra in their school lessons, so we concentrate in this strand on parts of the topic which are less emphasised in KS4 exams and their teachers therefore may not have had time to cover in the depth which would be desirable for progression to A-level and beyond. The strand stresses thinking about algebraic expressions as having inputs and outputs – as representing functions, in more formal terms. The mathematicians start by using algebraic expressions to represent general claims, such as that the first n odd integers have sum n^2 and move on to think about how to construct appropriate proofs and arguments in a geometrical context. There are sessions in this strand giving the mathematicians the opportunity to practise setting up linear equations from more complex information than they might meet in their school lessons and using the strategies of substitution and elimination in simultaneous equations in more than two variables. Later, the mathematicians look at the use of algebra to prove general claims, culminating in a proof that every Pythagorean triple includes a multiple of 5.

Example question

On holiday, if you join two of the three excursions on offer you pay £24, £30 or £40, depending which you choose; what do the three excursions cost? The areas of the faces of a cuboid are 24 cm2, 30 cm2 and 40 cm2; what are the dimensions of the cuboid?

An algebraic approach to the first could lead to a + b = 24, b + c = 30, c + a + 40, which are simultaneous equations not quite like those usually tackled in school but susceptible to the same techniques—in this case elimination by, for instance, adding the first two and subtracting the third. The second might give ab = 24, bc = 30, ca = 40, which look very different, but can be solved by analogy: multiply the first two and divide by the third.

Investigation

The purpose of this strand is to offer the mathematicians more extended problems, one or two of them occupying a whole session. The intention is that they should develop their problem-solving behaviours as a result of this: trying something when they don’t know whether it will be successful, co-operating with others in a problem which requires several linked steps for its solution, organising and articulating their approaches, being persistent when faced with difficulties and monitoring their progress towards the goal.

Example question

The differences between the four integers 0, 1 , 3, 5 include 1 – 0 = 1, 3 – 1 = 2 (and 5 – 3 = 2), 3 – 0 = 3, 5 – 1 = 4 and 5 – 0 = 5: all the integers from 1 to 5 occur as a difference of two of them. Can you find four integers, including 0 and 7, whose differences include all the integers from 1 to 7? Can you find five integers, including 0 and 9, whose differences include all the integers from 1 to 9? Can you find five integers, including 0 and 10, whose differences include all the integers from 1 to 10?

Four integers A, B, C, and D only have six gaps A-B, A-C, A-D, B-C, B-D and C-D, so they cannot include all the integers from 1 to 7. Five integers have ten gaps but, although it is possible to find five integers, including 0 and 9, whose differences include all the integers from 1 to 9 (and repeat one of the differences), it is not possible to have all ten gaps different from each other.

Logic

These sessions are not part of a content strand, but offer opportunities for the mathematicians to look at a particular type of reasoning. The first session in this strand gives examples of mathematical questions that may have many, one or no solutions. Later in the strand, the mathematicians look at arguments from contradiction, in the context of problems of the ‘pigeonhole principle’ type and at Venn diagrams and their connection to logical arguments.

Example question

There are 744 pupils in a school. Convince me that that there are three pupils who share a birthday. How many pupils would there have to be in a school before you could be sure that at least four of them all shared a birthday?

If there were no more than two pupils sharing any birthday, then there would be at most 2 × 366 = 732 pupils in the school, but there are 744, so there must be at least three pupils sharing at least one birthday. To use this argument to show that at least four shared a birthday, you would need 3 × 366 + 1 = 1099 pupils. The if … then … but … so … structure indicates that an argument by contradiction is being used to justify a claim.

Geometry

This strand is under development, as we work out how to enable both our online and in-person circles to have a similarly rich experience when exploring this topic. We currently look at the geometry of transformations, at the geometry of polygons, and at the use of algebra in mensuration problems, but we hope to take the transformation work further and expand the mathematicians’ experience of parallel line geometry, triangle geometry, the classification of quadrilaterals and congruence.