Learning Paths
Each learning path is one chapter: a guided route through a topic, one small course at a time. Pick a class to see its chapters in order.
Chapter 3Contains Standard content
Complex Numbers, Quadratic Equations
The equation x squared plus 1 equals 0 has no real solution. Mathematicians did not give up: they invented a number i whose square is minus one, and a whole new world of numbers opened up. In this chapter you calculate with complex numbers, picture them as points on the Argand plane, find their modulus and conjugate, and solve every quadratic equation, even those with a negative discriminant. Python has complex numbers built in, so you can check every step.
3 courses
Chapter 4Contains Standard content
Permutations and Combinations
How many number plates can a state issue? In how many ways can a team of five be chosen from twelve players? Listing every case would take forever, so mathematicians count cleverly. In this chapter you learn the fundamental principle of counting, factorials, permutations when order matters and combinations when it does not. Python lists small cases so you can check every formula before you trust it.
4 courses
Chapter 5Contains Standard content
Sequences and Series
A loan that grows by the same percentage each year, a ball that bounces back to two-thirds of its height, a pattern of tiles that adds three more each row: these are sequences, and adding their terms gives a series. In this chapter you revisit arithmetic progressions, meet geometric progressions and their sums, and prove that the arithmetic mean is never less than the geometric mean. Python lists terms and partial sums so you can check every formula.
3 courses
Chapter 7Contains Standard content
Binomial Theorem
What is the value of 1.01 to the power 10? Expanding it by hand means ten multiplications. The binomial theorem gives the whole answer at once, term by term. In this chapter you meet Pascal's triangle, write the expansion of any power of a sum, find a single term without writing the rest, and use the coefficients to answer questions on divisibility and on large numbers. Python expands powers and checks every formula.
3 courses
Chapter 3Contains Standard content
Polynomials, Algebraic Identities and Factorisation
The cost of a taxi, the height of a thrown ball and the area of a garden can all be written with expressions such as x² + 3x + 2. These expressions are called polynomials. In this chapter we learn what they are, how to find the numbers that make them zero, and how to split them into smaller factors. We also learn the identities that make long multiplications short. We use Python to calculate values and to check our factors.
3 courses
Chapter 4Contains Standard content
Quadratic Equations, Polynomials
A ball thrown up comes down again, and a garden of a given area has a length that you can find. Both lead to an equation with a squared term: a quadratic equation. In this chapter you learn to recognise such equations, to solve them by factorising, by completing the square and with the quadratic formula, and to tell from the discriminant how many roots there are. Python solves and checks equations by the hundred.
3 courses
Chapter 5
Sequences and Series
A taxi fare that rises by the same amount every kilometre, the seats in rows that grow by two, a savings plan that adds the same sum each month: each is a list of numbers with a common difference. In this chapter you learn to spot an arithmetic progression, find any of its terms, and add up many terms at once with a formula. Python lists the terms and checks your formulas in a moment.
2 courses
Chapter 7
Linear Equations in Two Variables, Pairs of Linear Equations, Straight Lines
A pen and a notebook together cost 10 dollars. How much does each one cost? There is not one answer. There are many. In this chapter we write such a puzzle as an equation with two unknowns, and we draw all its answers as a straight line. Then we put two equations together, and we find the one answer that fits both. We use Python to draw the lines and to check our work.
2 courses
Chapter 8
Pairs of Linear Equations
Two taxi companies charge a fixed fee plus a rate for each kilometre. For how many kilometres do they cost the same? Two linear equations answer a question like this. In this chapter you draw a pair of equations as two lines, decide when they meet, never meet or coincide, and solve them by substitution, elimination and cross-multiplication. Then you turn word problems about ages, speeds and fractions into pairs of equations. Python solves any pair and checks the answer.
2 courses