CAMBRIDGE IGCSE™ CORE AND EXTENDED MATHEMATICS:

Main teaching ideas

1 Prime factorisation (20–25 minutes)

TEACHER’S RESOURCE

Learning intention: Learners will make use of prime factorisation to solve problems.

Description and purpose: In this activity learners will find the prime factorisation of numbers using a prime factor tree and use it to find the highest common factor (HCF) and the lowest common multiple (LCM) in order to solve problems.

Demonstrate to learners how to find prime factors using a prime factor tree as shown in Worked Example 5 in the Coursebook and show how they can use these to find the HCF and LCM of the numbers.

An alternative method is to use a Venn diagram to show the HCF and LCM visually. For example, the diagram shows the prime factors of 20 and 42. The HCF is equal to the product of factors in the intersection (2) and the LCM is the product of all the factors shown in the Venn diagram (2 × 2 × 3 × 5 × 7 = 420).

Show learners that they can also write a number as a product of prime factors using index notation. For example, 168 = 2 × 2 × 3 × 7 = 23 × 31 × 71. (Note that the power of 1 is usually not written.)

Present the following problems and ask learners to solve them using what they know about prime factorisation.

a If a positive integer, n, is divisible by both 225 and 216, and n = 2x × 3y × 5z, where x, y and z are positive integers, what are the least values of x, y and z?

b What is the smallest possible integer n such that the product 84 × n is a perfect square?

Answers: a 3, 3, 2 b 21

Differentiation ideas:

Support: Ask the following questions to guide learners in their thinking:

• What does it mean that n is divisible by 225 and divisible by 216?

• What are the prime factorisations of 225 and 216, and how may they help you find the solution?

• What does the phrase ‘least values’ remind you of?

Challenge: Ask learners to come up with a proof that they are indeed the least values. They may use proof by contradiction, i.e., assume that there are smaller integers that are the smallest values, and show that it is a false assumption. Alternatively, learners may show and argue visually using a Venn diagram.

2 Power patterns (25–30 minutes)

Learning intention: Learners will explore patterns in the units (ones) digit of numbers raised to a power and use the pattern to find the units digit when the number is raised to a high power.

Description and purpose: The purpose of this activity is for learners to explore patterns in the units digit of numbers when they are raised to a power. Ask learners to start with the number 1 and find its value when it is raised to different powers (1, 2, 3 …). They should then repeat this starting with each of the numbers from 2 to 10. Ask them to see if they notice any patterns in the units digit each time. For example, the pattern for the exponents of the number 1 is always 1, for 2, it is 2, 4, 8, 6, 2, 4, 8, 6, … and continues in cycle of 4.

CAMBRIDGE IGCSE™ CORE AND EXTENDED MATHEMATICS: TEACHER’S RESOURCE

Complete the first two rows of this table and ask learners to complete the remaining rows. (Answers are shown in red.)

Learners may also notice a pattern for the ‘Length of cycle’ column: 1, 4, 4, 2, 1, … . Now, ask learners to use what they have found to help them determine the units digit for the following numbers, and explain their reasoning.

a 271 b 3123 c 1999 d 12857 × 429 e 1111 + 1313 + 1515 + 1717

Answers:

Differentiation ideas:

Support: Work through one or two problems together with the learners, to guide them step-by-step paying particular attention to the thinking process. Remind them that they need only to consider the units digit.

Challenge: Ask learners to explain why numbers raised to powers follow the particular patterns and also if they notice a pattern with the lengths of cycle (1, 4, 4, 2, 1 and repeats 1, 4, 4, 2, 1) and why this is so.

Plenary ideas

1 (Dis)order of operations (5–10 minutes)

Resources: A social media thread showing the problem of ambiguity when it comes to order of operations.

Description and purpose: The purpose of this activity is for learners to think about situations where there may be some ambiguity when using the order of operations and what they need to do.

Present this arithmetic problem to learners and ask them to solve it: 6 ÷ 2(1 + 2)