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2 2 Fractions Specification GCSE 2010 N a (part) Add, subtract… any number N h Understand equivalent fractions, simplifying a fraction by cancelling all common factors N i Add and subtract fractions FS Process skills Recognise that a situation has aspects that can be represented using mathematics FS Performance Level 2 Apply a range of mathematics to find solutions Resources ActiveTeach resources Simplifying fractions quiz Adding fractions 2 interactive 2 2.1 Adding and subtracting fractions and mixed numbers Concepts and skills • • • Add, subtract … fractions. Find equivalent fractions. Add and subtract fractions. Functional skills • L2 Carry out calculations with numbers of any size in practical contexts. Prior key knowledge, skills and concepts Students should already know how to • add and subtract integers • • • • find the LCM of two numbers write a fraction in its simplest form (N h) change from mixed numbers to improper fractions and vice versa (N h) order fractions (N b). Starter • Ask students to find the LCM of pairs of numbers, e.g. 3 and 4 (12), 2 and 6 (6), 4 and 6 (12), 8 and 10 (40). • Discuss this question with students. Can you tell me all the fractions that are equivalent 1 1 to 2 ? (No – there are an infinite number.) Ask them for other fractions equivalent to 2 . Main teaching and learning • • Tell students that they are going to learn how to add and subtract fractions. • Discuss how you could add 32 + 41 . Ask students for ideas. Drawing a diagram to illustrate each fraction on a rectangular grid, using 3 columns and 4 rows, is one way to move into a discussion of common denominators. • • Explain the subtraction of fractions in the same way, using diagrams where necessary. Explain that, to add fractions, the denominators must be the same. Diagrams are a useful way to show that, for example, 25 + 51 = 53 . Discuss adding and subtracting mixed numbers. Encourage students to deal with the integer parts first and then the fraction parts. Common misconceptions • When adding fractions you do not ‘add the top numbers’ and ‘add the bottom numbers’. This is a very common error. • When adding fractions that have a common denominator add only the numerators; the denominator remains the same. Enrichment • Students could consider the circumstances under which the LCM of two numbers is not the product of the two numbers (when they have factor(s) in common). Plenary • 68 equivalent fractions 7 Students could be asked to find different fraction sums that lead to an answer of 8 11++33, ,11––11, ,55++11,etc etc 22 88 88 88 44 (( mixed number )) 2 2 Fractions Specification GCSE 2010 N a (part) … multiply… any number N o (part) Interpret fractions … as operators FS Process skills Recognise that a situation has aspects that can be represented using mathematics Use appropriate mathematical procedures FS Performance Level 2 Apply a range of mathematics to find solutions 2 2.2 Multiplying fractions and mixed numbers Concepts and skills • • • … multiply… fractions…. Multiply… by any number between 0 and 1. Find a fraction of a quantity. Functional skills • L2 Carry out calculations with numbers of any size in practical contexts … Prior key knowledge, skills and concepts Students should already know how to • multiply integers • change between mixed numbers and improper fractions. Starter Resources ActiveTeach resources Adding fractions quiz Multiplying a fraction 2 animation • Ask students to change some mixed numbers to improper fractions and vice versa. For 23 3 7 1 2 10 , 3 2 3 . example 2 53 135 , 4 29 389 , 10 () () ( ) ( ) Main teaching and learning • • Tell students that they are going to learn to multiply both fractions and mixed numbers. • Discuss the fact that if the final answer needs simplifying, then it is likely that the simplifying could have been done before the multiplication. • • Ask students how they would multiply mixed numbers. Explain that to multiply fractions you multiply the numerators and multiply the denominators. Explain that mixed numbers need to be changed to improper fractions before multiplication can take place. Common misconceptions • When multiplying fractions students often multiply the whole numbers together and then multiply the fractions together. Enrichment • More able students could practise multiplying three mixed numbers together. Plenary • 70 improper fraction Practise multiplying simple fractions together mentally. For example, 71 × 25 (352 ). 2 2 Fractions 2 2.3 Dividing fractions and mixed numbers Specification GCSE 2010 N a (part) … divide any number FS Process skills Recognise that a situation has aspects that can be represented using mathematics Use appropriate mathematical procedures FS Performance Level 2 Apply a range of mathematics to find solutions Concepts and skills • • … divide… fractions…. … divide by any number between 0 and 1. Functional skills • L2 Carry out calculations with numbers of any size in practical contexts … Prior key knowledge, skills and concepts Students should already know how to • multiply integers and fractions • change between mixed numbers and improper fractions. Starter • Remind students that the reciprocal of a whole number is 1 divided by the number, so 1 the reciprocal of 4 is 4 , and to find the reciprocal of a fraction you turn the fraction upside down. • Ask students to give you the reciprocal of various fractions and integers. For example 1 1 2 3 5 4 5, 5 2, 2 , 3 2 , 4 5 . Resources ActiveTeach resources Dividing integers quiz Dividing a fraction 2 animation () () () () Main teaching and learning • Tell students that they are going to learn how to divide by a fraction and by mixed numbers. • Discuss the connection between multiplication and division, e.g. multiplying by 2 (the reciprocal of 2) is the same as dividing by 2. • Explain that to divide by a fraction you multiply by the reciprocal of the fraction. Illustrate this by a worked example (e.g. Example 10). • Ask students how this could be extended to mixed numbers. First change the mixed numbers to improper fractions and then proceed in the same way as for dividing by fractions. • Encourage students to cancel before division and to give their final answers in their simplest form, using mixed numbers where appropriate. 1 Common misconceptions • • It is the second fraction that must be ‘turned upside down’, not the first one. Fractions must be kept in the correct order and division is not commutative. Enrichment • Students could be given problems that contain a mixture of division and multiplication. For example, 121 × 2 23 ÷ 131 or calculations involving fractions and BIDMAS. Plenary • 72 inverted Use a mixture of division and multiplication of fraction questions on the board to ensure that students can differentiate between the two methods. 2 2 Fractions 2 2.4 Fraction problems Specification GCSE 2010 N a Add, subtract, multiply and divide any number N o Interpret fractions, decimals and percentages as operators FS Process skills Recognise that a situation has aspects that can be represented using mathematics Use appropriate mathematical procedures FS Performance Level 2 Apply a range of mathematics to find solutions Concepts and skills • • ActiveTeach resources Multiplication and division quiz Fraction and percentage finder interactive RP KC Fractions knowledge check RP PS Fractions problem solving Find a fraction of a quantity. Functional skills • L2 Carry out calculations with numbers of any size in practical contexts … Prior key knowledge, skills and concepts Students should already be able to • multiply and divide by an integer • • add, subtract, multiply and divide fractions solve word problems. Starter • Resources Add, subtract, multiply and divide… fractions…. Use a number of word problems using integer values to ensure that students know when to use the different arithmetic operations. Main teaching and learning • Ask students for examples in real life where fractions are used. For example, sales in shops, interest rates. • • Tell students that they are going to learn about the use of fractions in solving problems. • How would you find 14 of 20? (20 ÷ 4 = 5). How could you use this result to find (5 × 3 = 15). 3 4 of 20? Discuss the need to be able to work with fractions in problems. Use the examples in the Student Book to illustrate problems using fractions. Common misconceptions • When working with questions involving a mix of units students need to think carefully about their final answer. • Students use the wrong arithmetical operation. Plenary • 74 Give students a mixture of simple fraction calculations to underpin the different techniques needed for adding, subtracting, multiplying and dividing fractions and mixed numbers.