common factor common multiple factor tree highest common factor (HCF)

�2 1 Number

60

GCSE 2010N c Use the concepts and vocabulary of factor (divisor), multiple, common factor, Highest Common Factor (HCF), Least Common Multiple (LCM), prime number and prime factor decomposition

FS Process skillsSelect the mathematical information to use

FS PerformanceLevel 1 Select mathematics in an organised way to � nd solutions

Speci� cation 1.1 Understanding prime factors, LCM and HCF

Concepts and skills

• Identify factors, multiples and prime numbers from a list of numbers.

• Find the prime factor decomposition of positive integers.

• Find common factors and common multiples of two numbers.

• Find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of two numbers.

Functional skills

• L1 … multiply and divide whole numbers using a range of strategies.

Prior key knowledge, skills and conceptsStudents should already

• know their multiplication tables up to 10 × 10

• understand and use positive integers and negative integers, both as positions and translations on a number line (N b)

• be able to � nd factors, multiples and prime numbers (N c).

Starter

• Check that students understand the terms prime number, factor and multiple. List the factors of 12. (1, 2, 3, 4, 6, 12) List the multiples of 6 between 10 and 40. (12, 18, 24, 30, 36) List the fi rst ten prime numbers. (2, 3, 5, 7, 11, 13, 17, 19, 23, 29)

• Introduce the word ‘common’ into some questions.Find two common factors of 12 and 18. (1, 2, 3, 6) Find two common multiples of 3 and 4. (12, 24, 36, etc)

Main teaching and learning

• Tell students that they are going to � nd out how to write any positive whole number as a product of its prime factors. Check that students understand the meaning of the word ‘product’.

• Explain that this can be done by using a factor tree (or repeated division). Draw a factor tree to show how 120 can be broken down into its prime factors (see Example 2).

• Discuss the fact that you can start with any two numbers that multiply to give 120. Draw a second factor tree for 120 starting with a different factor pair to show that the same result is reached.

• Tell students that they are going to � nd the HCF and LCM of two numbers.

• Explain that there are different methods that can be used to do this, depending on the size of the numbers involved.

• Discuss the best method for � nding the HCF and LCM for two small numbers (e.g. 4 and 6). Show students how these can be found by making a list of the factors and � rst few multiples of 4 and 6.

• Discuss why this method would not be appropriate for large numbers (e.g. 240 and 280).

• Explain how writing large numbers as the product of prime factors can be used to � nd the LCM and HCF.

Common misconceptions

• Remind students to include the multiplication signs when writing a number as a product of its prime factors. (These are often incorrectly replaced by addition signs or commas.)

Enrichment

• Suggest that students use the Venn diagram method to � nd the HCF and LCM of three large numbers (e.g. 240, 300 and 420).

• Students might like to know that the HCF of two numbers must be a factor of the difference between them. So the HCF of 210 and 250 must be a factor of 40. They may like to explore this and consider why this is the case.

Plenary

• Ask for the HCF of pairs of small numbers e.g. 2 and 6 (2), 4 and 10 (2), 6 and 12 (6).

• Ask for the LCM of pairs of small numbers e.g. 2 and 6 (6), 4 and 10 (20), 6 and 12 (12).

Linkshttp://www.bbc.co.uk/education/maths� le/shockwave/games/gridgame.html

ActiveTeach resourcesMultiples and factors quizLadder method interactiveHCF and LCM interactive

Resources

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cube cube number cube root square square number square root

�2 1 Number

62

GCSE 2010N d (part) Use the terms square, positive… square root, cube and cube rootN e (part) Use index notation for squares, cubes …

FS Process skillsUse appropriate mathematical procedures

FS PerformanceLevel 1 Use appropriate checking procedures at each stage

Speci� cation 1.2 Understanding squares and cubes

Concepts and skills

• Recall integer squares from 2 × 2 up to 15 × 15 and the corresponding square roots.

• Recall the cubes of 2, 3, 4, 5 and 10.

• Use index notation for squares and cubes.

• Find the value of calculations which include indices.

Functional skills

• L1 … multiply … whole numbers using a range of strategies.

Prior key knowledge, skills and concepts

• Students should already know how to multiply and divide positive and negative integers.

Starter

• Ask students to work out the value of 1 × 1, 2 × 2, 3 × 3 up to 10 × 10 (1, 4, 9, 16, 25, 36, 49, 64, 81, 100) and then 1 × 1 × 1, 2 × 2 × 2 up to 5 × 5 × 5 (1, 8, 27, 64, 125). Identify these as the square numbers and cube numbers respectively.

Main teaching and learning

• Explain that 102 is a shorter way of writing 10 × 10 and that (–53) is a shorter way of writing –5 × –5 × –5.

• Discuss how square root is the inverse (opposite) of square, therefore 100 = 10

because 102 = 100. Likewise –1253

= –5 because (–53) = –125.

• Ask students if it is possible for them to tell you the square root of any number. Which numbers can you write down the square root for without a calculator? (The square numbers.)

• Discuss the fact that each positive number has both a positive and negative square root.

• Explain why this is the case, e.g. 5 × 5 = 25 and –5 × –5 = 25.

Common misconceptions

• Remind students that squaring a negative number always gives a positive number.

• Warn students of the very common error: 32 = 6.

Enrichment

• Students could investigate the patterns formed from 112, 1112, etc.

• Some numbers can be expressed as the difference of two squares, for example 42 – 32 = 7, 32 – 12 = 8. Which numbers cannot be expressed as the difference of two squares? (2, 6, 10, 14, 18, …)

• How many squares are there on a standard chess board? (204 squares)

Plenary

• Ask students to give the values of, for example, 62 (36), 53 (125), 64 (8), 643

(4).

2

BIDMAS operation power, powers value

�2 1 Number

64

1.3 Understanding order of operations

Concepts and skills

• Add, subtract, multiply and divide whole numbers, negative numbers, integers . . . .

• Find the value of calculations which include indices.

• Multiply and divide numbers using the commutative, associative, and distributive laws and factorisation where possible, or place value adjustments.

• Use brackets and the hierarchy of operations.

Functional skills

• L1 Add, subtract, multiply and divide whole numbers using a range of strategies.

Prior key knowledge, skills and conceptsStudents should already know how to

• add, subtract, multiply and divide positive and negative integers (N a)

• understand and use positive numbers and negative integers, both as positions and translations on a number line (N b).

Starter

• Give out a variety of different calculators. (The calculator function on less sophisticated mobile phones is useful here.)

• Ask students to work out 3 + 5 × 2 on the calculator they have been given. Ask for the answers from the calculators. You should get the answers 16 (incorrect) and 13 (correct).

• Discuss why the calculators (which are always correct!) are giving two different answers.

• Try some other calculations, e.g. 20 – 14 ÷ 2 (13), 2 × 3 + 4 × 2 (14)

Main teaching and learning

• Tell students that they are going to � nd out about the order in which arithmetic operations should be carried out.

• Explain to students why it is important that there is a standard order of operations. (So that we all arrive at the same answer.)

• Discuss the meaning of the letters in BIDMAS.

Common misconceptions

• When working out calculations such as 2 × 32 remember to use BIDMAS; this must be worked out as 2 × 9 = 18.

• When left with just addition and subtraction then you must work from left to right, e.g. 6 – 10 + 2 = –4 + 2 = –2 (6 – 10 + 2 cannot be worked out as 6 – 12).

Enrichment

• Using just four 4s and any arithmetic operations, how many of the positive integers can you make? For example, 4 × 4 + 4 + 4 = 24; 4 ÷ 4 + 4 ÷ 4 = 2.

Plenary

• Have some pre-prepared questions on the board and ask students to work these out. For example, 7 + 4 × 2 (15), 24 – (8 × 2) (8).

GCSE 2010N a Add, subtract, multiply and divide any numberN e Use index notation for squares, cubes and powers of 10N q (part) Understand and use number operations and the relationships between them, including … hierarchy of operations

FS Process skillsUse appropriate mathematical procedures

FS PerformanceLevel 1 Apply mathematics in an organised way to � nd solutions…

Speci� cation

ResourcesQuestions for plenaryVariety of calculators

ActiveTeach resourcesSquaring quizBIDMAS animation

Resources

2

index number laws of indices

�2 1 Number

66

1.4 Understanding the index laws

Concepts and skills

• Find the value of calculations which include indices.

• Use index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integer… powers.

Functional skills

• L1 … multiply and divide whole numbers using a range of strategies.

Prior key knowledge, skills and conceptsStudents should

• be able to work out the square of a number and the cube of a number

• understand index notation e.g. know that 34 = 3 × 3 × 3 × 3.

Starter

• Write 23 on the board and ask students what this means (2 × 2 × 2).

• Ask them for another way to write 6 × 6 × 6 × 6 (64).

• Have other similar examples written ready on the board to use for practice.

Main teaching and learning

• Tell students that they are going to learn the index laws, which will enable them to simplify expressions such as 32 × 34 and 65 ÷ 62.

• Ask students to write out 32 × 34 in full (3 × 3 × 3 × 3 × 3 × 3) and explain that this could be written as 36.

• Discuss other similar examples and encourage students to give a general rule for combining the powers to give a single power (add the powers).

• Ask students to investigate examples involving division and ask them to come up with a rule this time (subtract the powers).

• Ask students to explain the meaning of (72)3 (Answer: 72 × 72 × 72). Discuss how this can be simpli� ed to 72+2+2 which can be written as 72 × 3 or 76. Encourage students to give a general rule for expressions of the form (am)n (multiply the powers).

Common misconceptions

• A common error is to work out 34 as 3 × 4 rather than 3 × 3 × 3 × 3.

• Questions that ask students to ‘Simplify 46 × 43’ or ‘Write 46 × 43 as a power of 4’ want the answer to be given as 49; they are not asking the student to work out 49. Students should be encouraged to show their working, i.e. writing 46+3 before the � nal answer to show their method.

• Questions that use the word ‘evaluate’ or the phrase ‘work out’ do require an answer that is not in index form. For example, the answer to ‘Evaluate 22 × 23’ is 32.

• Students often multiply rather than add the powers in 26 × 25 and divide rather than subtract the powers in 28 ÷ 24.

• Students often multiply the numbers as well as adding the powers, i.e. to give the answer to 34 × 37 as 911 instead of 311.

• Students often think that 4 is the same as 40 rather than 41.

Plenary

• Ask students to simplify expressions such as 78 × 73 (711), 812 ÷ 82 (810), (64)2 (68).

GCSE 2010N e (part) … Use index notation… N f (part) Use index laws for multiplication and division of integer… powers

FS Process skillsUse appropriate mathematical procedures

FS PerformanceLevel 1 Use appropriate checking procedures at each stage

Speci� cation

ActiveTeach resourcesRP KC Number knowledge checkRP PS Multiples problem solving

Follow up5.1 Using zero and negative powers 5.3 Working with fractional indices

Resources

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