Number Theory Lesson PlanIGCSE Maths 0580 · for teachers, tutors and self-study

Write the three sets on the board one at a time. Give students 30 seconds per set, then ask two or three for their answer and their reason. Accept any well-argued answer, then steer to the intended one. Close by asking: "Is 1 a prime number?" and take a quick vote — it sets up the main misconception for later.

Write the odd one out and a one-line reason in your exercise book, then hold it up or tell a partner when asked (mini whiteboards work too, if your class has them).

Build a nested diagram on the board: natural numbers inside integers inside rational numbers, with irrational numbers in a separate region next to the rationals. Add one example to each region as you define it (use the key vocabulary). Then model Worked example 1: Selecting numbers by type from a list — choosing a prime, a cube, an irrational number and a number that is both square and cube from a list. Think aloud: for each number, say which test you apply.

Point out the exam trap in part (d): students find a square or a cube and stop. Numbers that are both are sixth powers: $1, 64, 729, \dots$

Copy the diagram and add one more example of your own to each region. Follow the worked example and note the test used for each type of number.

Ask: "Is $0.333\dots$ rational?" Most students say no because it never ends. Show $0.\dot{3} = \dfrac{1}{3}$ to settle it: every recurring decimal is rational. Then model Worked example 3: Converting a recurring decimal to a fraction, which turns $0.\dot{2}\dot{7}$ to $\dfrac{3}{11}$:

• Let $x = 0.\dot{2}\dot{7}$.
• Two digits recur, so multiply by $100$: $100x = 27.\dot{2}\dot{7}$.
• Subtract: $99x = 27$, so $x = \dfrac{27}{99} = \dfrac{3}{11}$.

Then do one together ("we do"): $0.\dot{4}$. Finally, for roots, use the rule: $\sqrt{n}$ is rational only when $n$ is a perfect square, so $\sqrt{49} = 7$ is rational and $\sqrt{50}$ is irrational.

Follow the method, then convert $0.\dot{4}$ with the teacher, writing every line. Explain to a partner why $\sqrt{50}$ is irrational.

Read each question aloud (or write it on the board); students write the answer in their exercise books and hold them up on your signal. With mini whiteboards, use those instead. For a quick check without writing, ask for hand signals: thumbs up if they think $\sqrt{50}$ is rational, thumbs down if irrational. Use the answers to decide who needs the support tasks in the next stage. Question 4 is the stretch question — praise a correct method even if the final number is wrong.

Answer each question in your exercise book and hold it up when asked.

Students work through the three tiers in order and move on when they are confident. Students who struggled in the guided practice start on the support task (see Support and stretch). Circulate and check recurring-decimal working: the subtraction step is where marks are lost. Early finishers move to the Practice Exam Questions for this subtopic.

Work through the Core, Extended and Challenge questions, showing full working. Then continue with the Practice Exam Questions.

Students answer the three exit-ticket questions on paper and hand them in. Sort the tickets into three piles (all correct / one wrong / two or more wrong) to plan the start of the next lesson. Finish by returning to the starter: "Is 1 prime?" — the class should now answer with the reason.

Answer the three questions on your own, then check them against the success criteria at the top of the lesson.