1.1 Understanding prime factors, LCM and HCF

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1 Chapter 1 Number Specification GCSE 010 N 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 skills Select the mathematical information to use FS Performance Level 1 Select mathematics in an organised way to fi nd solutions Links mathsfi le/shockwave/games/gridgame html ActiveTeach resources Multiples and factors quiz Ladder method interactive HCF and LCM interactive 11 Understanding prime factors, LCM and HCF Concepts and skills Identify factros, multiples and prime numbers from a list of numbers Find the prime factor decomposition of positive integers Find common factors and common mulitples of two or three numbers Find the HCF and LCM of two or three numbers Functional skills L1 multiply and divide whole numbers using a range of strategies Prior key knowledge, skills and concepts Students should already know their multiplication tables up to be able to find factors, multiples and prime numbers (NC) Starter Check that students understand the terms prime number, factor and multiple List the factors of 1 (1,,, 4, 6, 1) List the multiples of 6 between 10 and 40 (1, 18, 4, 0, 6) List the first ten prime numbers (,, 5, 7, 11, 1, 17, 19,, 9) Introduce the word common into some questions Find two common factors of 1 and 18 (1,,, 6) Find two common multiples of and 4 (1, 4, 6 etc) Main teaching and learning Tell students that they are going to find 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 10 can be broken down into its prime factors (see Example ) Discuss the fact that you can start with any two numbers that multiply to give 10 Draw a second factor tree for 10 starting with a different factor pair to show that the same result is reached Tell students that they are going to find 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 finding the HCF and LCM for two small numbers (eg 4 and 6) Show students how these can be found by making a list of the factors and first few multiples of 4 and 6 Discuss why this method would not be appropriate for large numbers (eg 40 and 80) Explain how writing large numbers as the product of prime factors can be used to find 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 find the HCF and LCM of three large numbers (eg 40, 00 and 40) Students might like to know that the HCF of two numbers must be a factor of the difference between them So the HCF of 10 and 50 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 eg and 6 (), 4 and 10 (), 6 and 1 (6) Ask for the LCM of pairs of small numbers eg and 6 (6), 4 and 10 (0), 6 and 1 (1) common factor common multiple factor tree highest common factor (HCF) M01_MSAH_TG_GCSE_08_C01indd 1/05/010 11:40

2 Section 11 Understanding prime factors, LCM and HCF Section 11 Understanding prime factors, LCM and HCF 1 Write down all the factors of each of the following numbers Write down the factors in pairs a 1 b 0 c 6 d 40 Use this factor tree to write 54 as a product of its prime factors Complete the following factor trees a 4 b 70 c Write each of the following numbers as the product of its prime factors How can factor trees be drawn before working through to find out how many factors there are? a 4 b 40 c 50 d 7 e 100 Remember to draw factor trees first 5 a Write out all the factors of i 4 ii 6 iii 10 iv 16 v 0 b Use part a to help you write down the highest common factor (HCF) of Factors are numbers that go exactly into the given number i 4 and 10 ii 6 and 16 iii 4 and 0 iv 16 and 0 v 10 and 16 6 a Write out the first ten multiples of i 4 ii 6 b Write out the first six multiples of i 10 ii 16 iii 0 A D C C Section 11 Understanding prime factors, LCM and HCF Multiples are the numbers in the times table of the given number c Use your answers to parts a and b to help you write down the lowest common multiple (LCM) of i 4 and 6 ii 6 and 10 iii 4 and 16 iv 16 and 0 v 6 and 0 7 a Use your answers to question to write 4, 70 and 84 as products of their prime factors i 4 ii 70 iii 84 b Find the HCF and LCM of 4 and 70 c Find the HCF and LCM of 70 and 84 8 Find the HCF and LCM of the following pairs of numbers a 60 and 84 b 70 and 105 c 7 and 96 d 84 and 96 B C lowest common multiple prime factor M01_MSAH_TG_GCSE_08_C01indd 1/05/010 11:41

3 Chapter 1 Number GCSE 010 N d (part) Use the terms square, positive square root, cube and cube root N e (part) Use index notation for squares, cubes FS Process skills Use appropriate mathematical procedures Specification FS Performance Level 1 Use appropriate checking procedures at each stage 1 Understanding squares and cubes Concepts and skills Recall integer squares from up to and the corresponding square roots Recall the cubes of,, 4, 5 and 10 Use index notation for squares and cubes 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,, up to (1, 4, 9, 16, 5, 6, 49, 64, 81, 100) and then 1 1 1, up to (1, 8, 7, 64, 15) Identify these as the square numbers and cube numbers respectively Main teaching and learning Explain that 10 is a shorter way of writing and that ( 5 ) is a shorter way of writing Discuss how square root is the inverse (opposite) of square, therefore 100 = 10 because 10 = 100 Likewise 15 = 5 because ( 5 ) = 15 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, eg 5 5 = 5 and 5 5 = 5 Common misconceptions Remind students that squaring a negative number always gives a positive number Warn students of the very common error: = 6 Enrichment Students could investigate the patterns formed from 11, 111 etc Some numbers can be expressed as the difference of two squares, for example 4 = 7, 1 = 8 Which numbers cannot be expressed as the difference of two squares? (, 6, 10, 14, 18, ) How many squares are there on a standard chess board? (04 squares) Plenary Ask students to give the values of, for example, 6 (6), 5 (15), 64 (8), 64 (4) 4 cube cube number cube root square square number square root M01_MSAH_TG_GCSE_08_C01indd 4 1/05/010 11:41

4 Section 1 Understanding squares and cubes Section 1 Understanding squares and cubes Section 1 Understanding squares and cubes 1 Work out c 4 d 49 a b 4 c 5 d 7 e 10 Lizzy says that to work out 6 you 6 so the answer is 1 Lizzy is wrong Explain why do e 9 7 Write down Work out a 1 b 64 a b c 4 d 5 e 10 Here is a list of 8 numbers From the numbers in the list write down a all the numbers that are square numbers b all the numbers that are cube numbers Work out 5 a ( ) b ( ) c ( 6) d ( 8) e ( ) Write down 6 a 5 b 100 c 7 d 1000 e 8 out 8 Work a 5 + b 8 c 4 10 d 9 16 e 100 f g 5 6 h 81 + ( 1) 5A 5B Section 1 Understanding squares and cubes i j 4 ( 6) 9 Work out a 1 b 8 5 c d e 5 1 f 4 8 g 1000 h ( ) 4 i ( 5) 4 j C 5 M01_MSAH_TG_GCSE_08_C01indd 5 1/05/010 11:41

5 Chapter 1 Number GCSE 010 N q (part) Understand and use number operations and the relationships between them, including hierarchy of operations FS Process skills Use appropriate mathematical procedures FS Performance Level 1 Apply mathematics in an organised way to fi nd solutions Questions for plenary Variety of calculators ActiveTeach resources Squaring quiz BIDMAS animation The audience video Specification 1 Understanding order of operations Concepts and skills Multiply and divide numbers using he 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 concepts Students should already know how to add, subtract, multiply and divide positive and negative integers (N a) Understand and use positive number 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 + 5 on the calculator they have been given Ask for the answers from the calculators You should get the answers 16 (incorrect) and 1 (correct) Discuss why the calculators (which are always correct!) are giving two different answers Try some other calculations, eg 0 14 (1), + 4 (14) Main teaching and learning Tell students that they are going to find 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 remember to use BIDMAS; this must be worked out as 9 = 18 When left with just addition and subtraction then you must work from left to right, eg = 4 + = ( cannot be worked out as 6 1) Enrichment Using just four 4s and any arithmetic operations, how many of the positive integers can you make? For example, = 4; = Plenary Have some pre-prepared questions on the board and ask students to work these out For example, (15), 4 (8 ) (8) 6 BIDMAS operation power, powers value M01_MSAH_TG_GCSE_08_C01indd 6 1/05/010 11:41

6 Section 1 Understanding order of operations Section 1 Understanding order of operations 1 Work out a b 4 + c 0 4 d Work out a b 0 4 Remember Brackets Indices Divide Multiply Add Subtract c d Work out a 5 (6 + ) b (18 1) 6 c (9 ) (8 + ) d (9 5) (4 + ) 4 Work out a 7 (6 ) b (7 + ) (16 9) c d 45 (8 ) e 8 4 f g h (0 1) 5 Work out In c remember to square the 4 before you multiply by a 5 + b (5 + ) c 4 d A D Section 1 Understanding order of operations 6 Work out a 5 (10 7) b c d e f (1 9) g h Work out a b c d e f ( ) 8 g 1 5 ( ) h B D 7 M01_MSAH_TG_GCSE_08_C01indd 7 1/05/010 11:41

7 Chapter 1 Number Specification GCSE 010 N q Understand and use number operations and the relationships between them, including inverse operations and hierarchy of operations N v (part) Use calculators effectively and efficiently FS Process skills Decide on the methods, operations and tools, including ICT, to use in a situation Use appropriate mathematical procedures FS Performance Level Use appropriate checking procedures and evaluate their effectiveness at each stage 14 Using a calculator Concepts and skills Understand reciprocal as multiplicative inverse, knowing that any non-zero number multiplied by its reciprocal is 1 (and that zero has no reciprocal because division by zero is not defined) Find reciprocals Understand that the inverse operation of raising a positive number to a power n is raising the result of this operation tht power 1 n Understand and use unit fractions as multiplicative inverses Use a calculator effectively and efficiently Know how to enter complex calculations Understand, and interpret, the calculator display Understand that premature rounding can cause problems when undertaking calculations with more than one step Functional skills L Carry out calculations with numbers of any size to a given number of decimal places Calculators Prior key knowledge, skills and concepts Students should be able to use a calculator to enter numbers and carry out the four arithmetical operations Starter Ask students to use their calculators to work out the following 45 (6005), 67 ( ) Ask students how they used their calculators to work out these sums Main teaching and learning Tell students that they are going to learn how to use some of the buttons on their scientific calculators Ask students to work out various calculations using their calculators and discuss the key sequences Discuss the different ways that calculations such as (17965 ) can be 1 evaluated correctly Common misconceptions The calculator does not always give the correct answer it depends what was entered and how it was entered Encourage students to work out approximations to calculations before using their calculators Remind students to take care when using a calculator to work out a fraction that has a calculation in the numerator and/or denominator Plenary Give students various sums to be evaluated using their calculators 8 inverse operations reciprocal M01_MSAH_TG_GCSE_08_C01indd 8 1/05/010 11:41

8 Section 14 Using a calculator Section 14 Using a calculator 1 Work out Remember, to work out powers on your calculator, use the y x or x or ^ key a b 1 c ( 51) d 5 e 15 4 Work out a 59 b 6 69 c 961 d e 601 Work out a ( + 89) 5 b + 15 c (194 78) d Work out, giving your answers correct to one decimal place a 17 7 b A Section 14 Using a calculator c 84 1 d Work out, giving your answers correct to three significant figures Take care when working out sums that involve fractions Work out the top and bottom of the fraction separately or use brackets a b c d Work out, giving your answers correct to three significant figures a b c d Work out, giving your answers correct to three significant figures a b c d ( ) B D C 9 M01_MSAH_TG_GCSE_08_C01indd 9 1/05/010 11:41

9 Chapter 1 Number GCSE 010 N f (part) Use index laws for multiplication and division of integer powers FS Process skills Use appropriate mathematical procedures FS Performance Level 1 Use appropriate checking procedures at each stage ActiveTeach resources RP Number knowledge check RP Multiples problem solving Specification Follow up 51 Using zero and negative powers 5 Working with fractional indices 15 Understanding the index laws Concepts and skills 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 concepts Students should be able to work out the square of a number and the cube of a number understand index notation eg know that 4 = Starter Write on the board and ask students what this means ( ) Ask them for another way to write (6 4 ) 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 4 and Ask students to write out 4 in full ( ) and explain that this could be written as 6 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 (7 ) (7 7 7 ) Discuss how this can be simplified to 7 ++ which can be written as 7 or 7 6 Encourage students to give a general rule for expressions of the form (a m ) n (multiply the powers) Common misconceptions A common error is to work out 4 as 4 rather than Questions that ask students to Simplify or Write as a power of 4 want the answer to be given as 4 9 ; they are not asking the student to work out 4 9 Students should be encouraged to show their working ie writing 4 6+ before the final 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 is Students often multiply rather than add the powers in 6 5 and divide rather than subtract the powers in 8 4 Students often multiply the numbers as well as adding the powers, ie to give the answer to 4 7 as 9 11 instead of 11 Students often think that 4 is the same as 4 0 rather than 4 1 Plenary Ask students to simplify expressions such as (7 11 ), (8 10 ), (6 4 ) (6 8 ) 10 index number laws of indices M01_MSAH_TG_GCSE_08_C01indd 10 1/05/010 11:41

10 Section 15 Understanding the index laws Section 15 Understanding the index laws Section 15 Understanding the index laws 1 Work out 6 Write as a power of 6 (a m ) n = a mn C a 4 b 9 a (6 ) 4 b (6 4 ) c 6 d 10 4 e 5 Write in index form c (6 8 ) 5 d (6 9 ) a b e (6 ) c d Write as a power of 7 a m a n = a m +n C 7 Write as a power of a single number a b a b (9 4 ) c 7 7 d c d e e 5 (5 4 ) 5 a m a n = a m n 4 Write as a power of 5 a b Write as a power of 5 a b B c d e Write as a power of 5 a 9 b 7 c 10 4 d 4 e c 6 d e A 11B Section 15 Understanding the index laws 9 Work out a 4 5 b 8 6 B 9 4 c d 4 10 e C 11 M01_MSAH_TG_GCSE_08_C01indd 11 1/05/010 11:41