1. Primary National Strategy Mathematics 3 plus 2 day course: Session 3

2. © Crown copyright 2003 Primary National Strategy Slide 3.1 Objectives To analyse test questions on division as an aid to assessment To consider the errors that pupils may make with division and the implications for teaching

3. © Crown copyright 2003 Primary National Strategy Slide 3.2 Discussion point 1 What are the things that pupils need to know and be able to do before they move on to division calculations that extend beyond ‘tables facts’?

4. © Crown copyright 2003 Primary National Strategy Slide 3.3 Division questions at level 2 involve: interpretation of words such as: how many?, half, pair, left over, divided exactly by simple word problems involving grouping of two-digit numbers into 2s, 4s, 5s, 6s or 10s, sometimes with a remainder, and sometimes involving rounding up money problems involving division by 10p or 20p (coin values), sometimes with a remainder recognition of multiples of 10

5. © Crown copyright 2003 Primary National Strategy Slide 3.4 Division questions at level 3 involve: ‘missing number’ questions involving inverses, with the ‘box’ in different positions word problems involving: –division of two- and three-digit numbers by a single digit or multiple of 10, sometimes with remainder, sometimes involving rounding up –mixed units of money (e.g. £4 ÷ 40p) –in KS2 calculator paper, divisors like 20 and 25 ‘I am thinking of a number’ problems recognition of multiples of 3 and 5

6. © Crown copyright 2003 Primary National Strategy Slide 3.5 Division questions at level 4 involve: mental division of two- and three-digit numbers short division without calculator (e.g. 847 ÷ 7) more complex ‘missing number’ questions word problems involving: –division of two- and three-digit numbers, including rounding up from a decimal answer on a calculator –harder mixed units of money (e.g. £12.30 ÷ 15p) –simple direct proportion ‘I am thinking of a number’ problems

7. © Crown copyright 2003 Primary National Strategy Slide 3.6 Division questions at level 5 involve: in the mental paper, division of decimals by 10, simple direct proportion recognition of conventional short division layout ‘missing number’ problems (big numbers, decimals) word problems involving: –more reading and interpretation –division of four- and five-digit numbers, with rounding up from decimal answer on calculator –mixed units (e.g. 10 m ÷ 9.2 cm, 3 kg ÷ 60 g) –direct proportion ‘I am thinking of a number’ problems

8. © Crown copyright 2003 Primary National Strategy Slide 3.7 Criteria for calculation methods A calculation method for division should be: –reliable (the pupil gets the right answer) –appropriate (it is suitable for the type of calculation and the available tools: mental, written, calculator) –efficient (it is not too time-consuming) –checkable

9. An inefficient and inappropriate method Slide 3.8 A tent holds 6 children. How many tents are needed to hold 70 children? (Paper A) Leah achieved level 4 in the test.

10. An inefficient and inappropriate method Slide 3.9 568.1 ÷  = 24.7 (Paper B) Gemma achieved level 4 in the test. She divided by 5, 6, 8, 16 and 20 before trying 23.

11. © Crown copyright 2003 Primary National Strategy Slide 3.10 TASK: Pupils’ errors with division Look at each error and consider its nature. Is it caused by: –a careless slip (e.g. a division fact recalled incorrectly)? –basic misunderstanding of place value? –incorrect or inappropriate application of a method? –another reason? Annotate each example with your analysis.

12. © Crown copyright 2003 Primary National Strategy Slide 3.11 TASK: Pupils’ errors with division Take one of the errors. What could you do about it as a teacher: –to avoid the error happening in the first place? –to model and explain the correct approach? If you have time, repeat with another error.

13. © Crown copyright 2003 Primary National Strategy Slide 3.12 Summary Analyse each error and think about its possible cause – don’t simply re-teach the method Draw careless slips to a pupil’s attention and encourage the pupil to learn arithmetical facts ‘by heart’ Tackle misunderstanding of place value by the use of place value boards, multibase blocks and calculators to emphasise that when multiplying and dividing a number by a power of 10 the digits move to the left or right (see the Framework, section 6, pages 6 and 7)

14. © Crown copyright 2003 Primary National Strategy Slide 3.13 Summary Misapplication of a written method of division may be caused by: –lack of understanding of partitioning and the principle of the distributive law of division – more work on informal recording of mental methods of division is needed –being moved on too quickly, without a thorough grasp of all the necessary prerequisite skills If remainders are misinterpreted, model the process on a number line