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1 Number Foundation Student Book reference What is covered in this section Units 1, 17 F Unit 1 Units 4, 18 1.1 Calculations Unit 18 E Objectives Units 1, 18 PR O O 1.1 Calculations • Estimate answers to calculations. • Write an inequality to represent an error interval. • Use a calculator for complex calculations. 1.2 Factors and multiples • Write a number as a product of its prime factors. • Use prime factor decomposition to find the HCF and LCM of two numbers. 1.3 Fractions • Add, subtract, multiply and divide mixed numbers. 1.4 Indices, powers and roots • Simplify expressions involving indices, roots and surds. • Understand and use zero and negative indices. 1.5 Standard form • Write very small and very large numbers in standard form. • Calculate with numbers in standard form. 1.6 Mixed exercise • Consolidate your learning with more practice. Key point 1 PL • Estimate answers to calculations. • Write an inequality to represent an error interval. • Use a calculator for complex calculations. SA M Measurements given to the nearest whole unit may be inaccurate by up to one half of a unit below and one half of a unit above. For example, the range of possible values for a length given as 3 cm to the nearest cm is 2.5 cm < length , 3.5 cm. Key point 2 To estimate the answer to a calculation, you can round every number to 1 significant figure (s.f.). c 9.1 Warm up 1 Round these numbers to 1 s.f. a 456 b 24 d 7.58 2 Estimate the answer to these calculations by first rounding each value to 1 s.f. __________ 1.7 + 12.2 a ________ b √ 84.37 + 18.6 5.9 3 Write an inequality to show the possible values for Q3 hint a 680 (rounded to the nearest 10) b 1.27 (rounded to 2 d.p.) c 17.2 (rounded to 3 s.f.) d 200 (rounded to 2 s.f.) <n, 1 4 A machine produces rolls of paper. Each roll is 20 m to the nearest centimetre. a What is the minimum possible length of paper in the roll? b Write an inequality to show the possible lengths of paper, l, in a roll. 5 Shampoo bottles contain 240 ml of shampoo. There is an error of ±5% in the volume of shampoo in the bottle. Work out the minimum and maximum possible volumes of shampoo in the bottle. 6 Problem-solving A machine fills packs of crisps with 120 g of crisps. There is an error of ±3% in the mass of the crisps in the packs. The crisps label says, ‘Minimum contents 115 g’. Is this true? Show how you get your answer.. Q6 A common error is to write ‘Yes’ or ‘No’ and to not show calculations to explain why. F 7 Tom calculates that 892.17 ÷ 18.4 = 85.8 (1 d.p.). Estimate the answer to his calculation to show he is wrong. O 8 Use estimation to see which of these calculations is wrong and which could be correct. __________ 3.2 2 + 9.17 2.17 2 + 57.2 _________ a = 3.98 (3 s.f.) = 4.99(3 s.f.)b __________ 6.19 − 2.3 3.9 2 5436 − 82 2 9.8 3 × 3.7 c __________ = 71.5(3 s.f.) = 149(3 s.f.)d ________2 5.36 + 12.97 1.8 × 5.2 O √ 9 Use a calculator to work out the correct answer to the calculations in Q8 that were wrong. Round to 3 s.f. PR Q9 A common error is to round to 3 d.p. instead of 3 s.f. 10 Use a calculator to work these out. Round your answers to the number of s.f. given. _________ __________ 4.1 2 + 2.15 3 __________ (2 3.8 2 24 − 3.29 s.f.)b √ (3 s.f.) √ 11.7 + 3.2 −3) + 2 × 4 × 3 3 + √ ( 5 + 13.2 __________________ (3 s.f.)d (4 s.f.) √ ________ 19.2 6 a 2 3 _________ ______________ 2 3 2 3 Q10 hint Estimate the answers first, in order to check your calculations. PL c E _________ 1.2 Factors and multiples Objectives SA M • Write a number as a product of its prime factors. • Use prime factor decomposition to find the HCF and LCM of two numbers. Key point 3 Warm up All numbers can be written as a product of prime factors. This is called prime factor decomposition. 1 Write as a product of powers 2×5×3×5×2×2 2 Find a the HCF of 12 and 32 b the LCM of 6 and 10. 3 a Copy and complete these factor trees. b Write 84 and 105 as products of their prime factors. Q3 hint Write the products of factors using index notation, smallest factor first. 2 Q1 hint 2 × 3 × 5 84 2 105 5 4 What is the prime factor decomposition of 72? 5 Copy and complete this Venn diagram to find the HCF and LCM of 84 and 72. Prime factors of 84 Prime factors of 72 Q5 A common error is to give the LCM answer for the HCF or the HCF for the LCM. 2 Q7 hint F 6 Find the HCF and LCM of 28 and 70c 132 and 60 a 105 and 84b 7 Two numbers have HCF 8 and LCM 120. What are the two numbers? O 8 8 Problem-solving Find two numbers with LCM 70 and HCF 5. ×8× = 120 O 9 A = 22 × 5 × 7 B = 3 × 52 × 11 2 C = 2 × 13 D = 2 × 32 × 72 a Find the HCF of i A and D ii A and C iii B and D b Which two numbers have LCM i 1820 ii 42 900 iii8820 c Which two numbers have no common factors? PR Q9 hint Draw Venn diagrams. Q10 A common error is to not realise that this is an LCM question. PL E 10 Problem-solving Number 6 buses stop at the railway station every 9 minutes. Number 15 buses stop at the railway station every 12 minutes. At 10 am a number 6 bus and a number 15 bus both stop at the station. When will a number 6 bus and a number 15 bus next stop at the station together? 1.3 Fractions SA M Objective • Add, subtract, multiply and divide mixed numbers. Key point 4 To add, subtract, multiply or divide mixed numbers, you can write them as improper fractions first. 2 Cancel then multiply a _56 × _37 6 c __ 12 × _23 b d _47 × _38 3 _ 79 × __ 14 3 Work out a 2 _15 + 1 _25 b 3 _17 + 2 _35 3 _1 c 5_ 14 + 1 _49 d 4 __ 10 + 1 3 d _ 23 ÷ _38 3 × 5 __ 1 ⟋ 5 × 3 ______ Q2 hint _____ = = 2 6 × 7 ⟋ 6 × 7 Warm up 1 Work out a _56 + _35 b _49 − _16 c _ 34 × _57 Give your answers as mixed numbers where appropriate. Q3 hint Write your answers as mixed numbers. 3 4 Work out 7 a 2 _ 35 + 1 __ 3 _ 58 + 4 _34 10 b 11 _5 c 5_ 67 + 2 _23 d 1 __ 12 + 3 9 Q4 A common error when adding or subtracting fractions is to add/subtract the numerators and denominators. 5 Work out 8 _2 a 5 _49 − 1 _15 b 3 _34 − 1 _58 c 6 _67 − 3 _13 d 4 __ 11 − 3 5 6 Work out 5 _2 a 4_ 14 − 1 _35 b 3 _17 − 2 _14 c 5 _23 − 2 _79 d 4 __ 12 − 2 3 7 Work out a 3_ 15 × _23 b _25 × 2 _14 c _ 57 × 3 _12 d _34 × 2 _25 F O Q8 A common error when dividing fractions is to forget to ‘flip’ the fraction you are dividing by, when you change the calculation to multiplication. O 8 Work out a 2 _14 ÷ _15 b 1 _12 ÷ _47 c 3_ 23 ÷ _56 d 1 _78 ÷ _58 Q7 A common error is to not to change to improper fractions first. PR 9 Work out a 2 _12 × 1 _16 b 1 _23 × 1 _18 c 1 _17 × 3 _34 d 3 _15 × 2 _12 10 Work out a 2 _34 ÷ 2 _13 b 1 _45 ÷ 1 _34 c 2 _14 ÷ 3_ 35 d 2 _ 23 ÷ 1 _79 1 58 kg 11 212 kg PL E 11 Problem-solving Karl has these bags of sand. He needs 8 kg of sand to make cement. Does he have enough? Show working to explain. SA M 12 Problem-solving Ailsa has 8 _35 metres of wire. a How many 1 _ 12 metre lengths can she cut from it? b How much is left over? 1.4 Indices, powers and roots Objectives • Simplify expressions involving indices, roots and surds. • Understand and use zero and negative indices. Key point 5 Any number (or term) raised to the power 0 is equal to 1. Any number (or term) raised to the power –1 is the reciprocal of the number. To raise a power of a number (or term) to another power, multiply the indices. Key point 6 3 −2is the same as (3 −1) 2. To find a negative power, find the reciprocal and then raise the value to the positive power. 4 9 310 kg