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.
Choose a topic to see its subtopics.
Chapter 1
Rational and Irrational Numbers, Exponents and Radicals
You have counted with 1, 2, 3, and you have used fractions and decimals. Are there numbers that are neither? Yes. The length of the diagonal of a square with a side of 1 is one of them, and no fraction can write it. In this chapter we sort the numbers into families, we meet the irrational numbers, and we place every real number on the number line. Then we learn how to add, multiply and rationalise them, and how to work with fractional powers. We use Python to see the digits.
2 courses
Chapter 2
Divisibility, HCF and LCM, Rational and Irrational Numbers
Every whole number above 1 is built from primes in exactly one way. This one fact finds the HCF and the LCM of any numbers, and it proves that a number such as the square root of 2 can never be written as a fraction. In this chapter you break numbers into primes, solve problems about bells, tiles and buses with the HCF and the LCM, and learn how to prove that a number is irrational. Python checks your work and finds patterns in thousands of numbers.
2 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 6
The Cartesian Plane, Distance, Midpoint and Section Formulae
How does a map tell you exactly where a place is? How does a game know where a character stands on the screen? In both cases, two numbers fix a position. In this chapter we learn how two number lines make a plane, how to name and plot any point on it, and then how to find the distance between two points, the midpoint of a line and the point that divides a line in a given ratio. We use Python to plot points and to check our calculations.
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 1Contains Standard content
Graphs of Functions, Composite and Inverse Functions, Relations and Functions, Functional Equations, Sets and Intervals, Functions as Models of Data, Counting Functions
A taxi fare depends on the distance. The height of a ball depends on the time. A temperature in kelvin depends on a temperature in celsius. In each case, one number decides another, and mathematics has one word for this: a function. In this chapter we start from sets and relations, and we build the idea of a function with care. Then we draw functions, move and stretch their graphs, combine them, and undo them with the inverse. Python draws the graphs and checks our answers.
4 courses
Chapter 2Contains Standard content
Compound and Multiple Angles, Angles and Radian Measure, Trigonometric Functions and Graphs, Trigonometric Identities, Trigonometric Equations
A Ferris wheel turns, a tide rises and falls, a guitar string shakes. Each motion repeats itself, and the trigonometric functions describe all of them. In this chapter you measure angles in radians, extend sine, cosine and tangent to angles of any size with the unit circle, draw their graphs, and prove the formulas for the sum and the difference of two angles. Python draws the graphs and checks each identity at thousands of angles.
4 courses