GCSE 2010N a (part) Add, subtract… any numberN h Understand equivalent fractions, simplifying a fraction by cancelling all common factorsN i Add and subtract fractions

FS Process skillsRecognise that a situation has aspects that can be represented using mathematics

FS PerformanceLevel 2 Apply a range of mathematics to � nd solutions

Speci� cation

ActiveTeach resourcesSimplifying fractions quizAdding fractions 2 interactive

Resources

equivalent fractions mixed number

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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 conceptsStudents should already know how to

• add and subtract integers

• � nd 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 � nd 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 to 1

2? (No – there are an in� nite number.) Ask them for other fractions equivalent to 12 .

Main teaching and learning

• Tell students that they are going to learn how to add and subtract fractions.

• Explain that, to add fractions, the denominators must be the same. Diagrams are a useful way to show that, for example, 25

15

35+ = .

• Discuss how you could add 2314+ . 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.

• Discuss adding and subtracting mixed numbers. Encourage students to deal with the integer parts � rst 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

• Students could be asked to � nd different fraction sums that lead to an answer of 78 1

238

18

58

141+ – + etc, ,( ), 1

238

18

58

141+ – + etc, ,( )

2

GCSE 2010N a (part) … multiply… any numberN o (part) Interpret fractions … as operators

FS Process skillsRecognise that a situation has aspects that can be represented using mathematicsUse appropriate mathematical procedures

FS PerformanceLevel 2 Apply a range of mathematics to � nd solutions

Speci� cation

ActiveTeach resourcesAdding fractions quizMultiplying a fraction 2 animation

Resources

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 conceptsStudents should already know how to

• multiply integers

• change between mixed numbers and improper fractions.

Starter

• Ask students to change some mixed numbers to improper fractions and vice versa. For example 2 4 2 23

5135

29

389

2310

310

73

13( ) ( ) ( ) ( ), , , .

Main teaching and learning

• Tell students that they are going to learn to multiply both fractions and mixed numbers.

• Explain that to multiply fractions you multiply the numerators and multiply the denominators.

• Discuss the fact that if the � nal 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 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

• Practise multiplying simple fractions together mentally. For example, 1725

235×× ( ).

improper fraction

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GCSE 2010N a (part) … divide any number

FS Process skillsRecognise that a situation has aspects that can be represented using mathematicsUse appropriate mathematical procedures

FS PerformanceLevel 2 Apply a range of mathematics to � nd solutions

Speci� cation

ActiveTeach resourcesDividing integers quizDividing a fraction 2 animation

Resources

2.3 Dividing fractions and mixed numbers

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 conceptsStudents 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 the reciprocal of 4 is 1

4 , and to � nd 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 5, 1

5( ) 2, 1

2( ), 23 32( ), 5

4 45( ).

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 12

(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 � nal answers in their simplest form, using mixed numbers where appropriate.

Common misconceptions

• It is the second fraction that must be ‘turned upside down’, not the � rst 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, 1 2 11

223

13×× ÷ or calculations involving fractions and BIDMAS.

Plenary

• Use a mixture of division and multiplication of fraction questions on the board to ensure that students can differentiate between the two methods.

2

GCSE 2010N a Add, subtract, multiply and divide any numberN o Interpret fractions, decimals and percentages as operators

FS Process skillsRecognise that a situation has aspects that can be represented using mathematicsUse appropriate mathematical procedures

FS PerformanceLevel 2 Apply a range of mathematics to � nd solutions

Speci� cation

ActiveTeach resourcesMultiplication and division quizFraction and percentage � nder interactiveRP KC Fractions knowledge checkRP PS Fractions problem solving

Resources

2.4 Fraction problems

Concepts and skills

• Add, subtract, multiply and divide… fractions….

• 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 conceptsStudents should already be able to

• multiply and divide by an integer

• add, subtract, multiply and divide fractions

• solve word problems.

Starter

• 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 fi nd 14 of 20? (20 ÷ 4 = 5). How could you use this result to fi nd 34 of 20?

(5 × 3 = 15).

• 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 � nal answer.

• Students use the wrong arithmetical operation.

Plenary

• Give students a mixture of simple fraction calculations to underpin the different techniques needed for adding, subtracting, multiplying and dividing fractions and mixed numbers.

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