Division

1. st mary's / OPF / 7/6/08 sourced from 1 Division

2. 2 st mary's / OPF / 7/6/08 sourced from The problems with division Division 2 Ask teachers to have a go at these calculations by using the traditional ‘guzinter’ format as shown. After they have done this explain the problems with this method: 1. The method doesn’t work for these numbers – e.g. 6 ‘goes into’ 1 won’t go, so cross out the one and put it with the units/ones to make 18. You are back where you started! The same happens with the second example. You end up with ’24 goes into 202’, again exactly where you started. If children do not have another approach to division they are now stuck. 2. The importance of place value and the importance of referring to the real value of digits has been constantly emphasised to children. However in this method no reference to place value is made (for the second calculation on the slide you say ’24 goes in to 2, won’t go’, yet the 2 is really 200 and we know that 24 will go into 200). It therefore becomes a very difficult method to explain. 3. The method reverses the order of the numbers in the calculation compared to the horizontal notation (202 ? 24)Division 2 Ask teachers to have a go at these calculations by using the traditional ‘guzinter’ format as shown. After they have done this explain the problems with this method: 1. The method doesn’t work for these numbers – e.g. 6 ‘goes into’ 1 won’t go, so cross out the one and put it with the units/ones to make 18. You are back where you started! The same happens with the second example. You end up with ’24 goes into 202’, again exactly where you started. If children do not have another approach to division they are now stuck. 2. The importance of place value and the importance of referring to the real value of digits has been constantly emphasised to children. However in this method no reference to place value is made (for the second calculation on the slide you say ’24 goes in to 2, won’t go’, yet the 2 is really 200 and we know that 24 will go into 200). It therefore becomes a very difficult method to explain. 3. The method reverses the order of the numbers in the calculation compared to the horizontal notation (202 ? 24)

3. 3 st mary's / OPF / 7/6/08 sourced from What is division? Division 3 Ask everyone to quickly draw an illustration of the calculation and then compare the different ways of recording. It is common for most people to automatically ‘share’ (share the 12 objects 3 ways). It is less common for ‘grouping’ to be illustrated (by showing how many threes are in 12). The next two slides will demonstrate the difference between grouping and sharing.Division 3 Ask everyone to quickly draw an illustration of the calculation and then compare the different ways of recording. It is common for most people to automatically ‘share’ (share the 12 objects 3 ways). It is less common for ‘grouping’ to be illustrated (by showing how many threes are in 12). The next two slides will demonstrate the difference between grouping and sharing.

4. 4 st mary's / OPF / 7/6/08 sourced from Division 4 (animated slide) This slide illustrates how 12 objects can be practically shared. Division 4 (animated slide) This slide illustrates how 12 objects can be practically shared.

5. 5 st mary's / OPF / 7/6/08 sourced from Division 5 (animated slide) In this slide the same calculated is completed by considering how many groups of three there are in 12. Division 5 (animated slide) In this slide the same calculated is completed by considering how many groups of three there are in 12.

6. 6 st mary's / OPF / 7/6/08 sourced from Division 6 It is important that the full range of vocabulary is used. The understanding division pages in the Framework (section 5 page 49 and section 6 pages54-55) provide a useful reference for the variety of ways in which questions can be phrased. It may be helpful to reflect on which words are most commonly used by teachers/children and which are underused. There is often an overemphasis on the word ‘share.’ Therefore when referring to the mathematical symbol (?) it is useful to read it as ‘divided by’. The other language will depend on the way the calculation is tackled – it is very important that the vocabulary matches to the image being shown (you would not want to be demonstrating grouping and use language such as ‘share equally’).Division 6 It is important that the full range of vocabulary is used. The understanding division pages in the Framework (section 5 page 49 and section 6 pages54-55) provide a useful reference for the variety of ways in which questions can be phrased. It may be helpful to reflect on which words are most commonly used by teachers/children and which are underused. There is often an overemphasis on the word ‘share.’ Therefore when referring to the mathematical symbol (?) it is useful to read it as ‘divided by’. The other language will depend on the way the calculation is tackled – it is very important that the vocabulary matches to the image being shown (you would not want to be demonstrating grouping and use language such as ‘share equally’).

7. 7 st mary's / OPF / 7/6/08 sourced from Introducing formal division Division 8 The first division objectives appear in Year 2 and children are expected to have experience of grouping and sharing. Research has shown that children have an innate ability to share and this has been identified in very young children. By the time division is taught children will already have a good understanding of sharing fairly (it is probably something they have done with sweets/toys etc in everyday life). They probably will not have an understanding of grouping and so this is where emphasis needs to be placed. Some Key Stage 1 teachers have introduced division as grouping and when children have got a grasp of this concept have then looked at sharing.Division 8 The first division objectives appear in Year 2 and children are expected to have experience of grouping and sharing. Research has shown that children have an innate ability to share and this has been identified in very young children. By the time division is taught children will already have a good understanding of sharing fairly (it is probably something they have done with sweets/toys etc in everyday life). They probably will not have an understanding of grouping and so this is where emphasis needs to be placed. Some Key Stage 1 teachers have introduced division as grouping and when children have got a grasp of this concept have then looked at sharing.

8. 8 st mary's / OPF / 7/6/08 sourced from Division 9 (animated slide) Sharing is much easier to do practically than on paper. With practical apparatus children could count out 18 objects and share them one by one into 3 piles. However recording this process is not as easy. Often children try to use a drawing like the one in the slide. Even with the benefit of the straight lines produced by the computer you can see how complex the diagram becomes with relatively low numbers. Division 9 (animated slide) Sharing is much easier to do practically than on paper. With practical apparatus children could count out 18 objects and share them one by one into 3 piles. However recording this process is not as easy. Often children try to use a drawing like the one in the slide. Even with the benefit of the straight lines produced by the computer you can see how complex the diagram becomes with relatively low numbers.

9. 9 st mary's / OPF / 7/6/08 sourced from 18 ? 3 = Division 11 (animated slide) The illustration of grouping is much clearer. This slide shows how the same 18 objects as shown in the previous slide can be grouped. The groups are very clear and easily counted. Obviously we would not want children to rely on drawings such as this for large numbers so it is also helpful to relate the image to equal jumps on a number line. It would be useful to show the ‘Grouping’ Interactive Teaching Program here. If you have not yet got a copy it is available to download free from the following website: www.numeracy.org.uk (Username: Y1to3 Password: smethwick) Division 11 (animated slide) The illustration of grouping is much clearer. This slide shows how the same 18 objects as shown in the previous slide can be grouped. The groups are very clear and easily counted. Obviously we would not want children to rely on drawings such as this for large numbers so it is also helpful to relate the image to equal jumps on a number line. It would be useful to show the ‘Grouping’ Interactive Teaching Program here. If you have not yet got a copy it is available to download free from the following website: www.numeracy.org.uk (Username: Y1to3 Password: smethwick)

10. 10 st mary's / OPF / 7/6/08 sourced from Modelling grouping on beadstrings Division 12 Beadstrings can also be used to practically group. If you have got 100 beadstrings in your school you might want to try a calculation such as 54 ? 6. The colours clearly show when a group of 6 has crossed a tens boundary. 20 and 100 Beadstrings are available from www.beadstring.com Large bead frames are available from Taskmaster (0116 270 4286) and Autopress (0870 240 3565)Division 12 Beadstrings can also be used to practically group. If you have got 100 beadstrings in your school you might want to try a calculation such as 54 ? 6. The colours clearly show when a group of 6 has crossed a tens boundary. 20 and 100 Beadstrings are available from www.beadstring.com Large bead frames are available from Taskmaster (0116 270 4286) and Autopress (0870 240 3565)

11. 11 st mary's / OPF / 7/6/08 sourced from 20 ? 4 = Division 13Division 13

12. 12 st mary's / OPF / 7/6/08 sourced from Division 14Division 14

13. 13 st mary's / OPF / 7/6/08 sourced from Division 15Division 15

14. 14 st mary's / OPF / 7/6/08 sourced from Division 16Division 16

15. 15 st mary's / OPF / 7/6/08 sourced from Grouping higher order skill than sharing Provides a firmer basis on which to build children’s understanding of division. 6000 ? 1000 = Division 18 The slide summarises why an emphasis should be placed on grouping. It links to number lines and the use of number facts, recording is more efficient and as children progress towards chunking it becomes a better base on which to build understanding of division. Division 18 The slide summarises why an emphasis should be placed on grouping. It links to number lines and the use of number facts, recording is more efficient and as children progress towards chunking it becomes a better base on which to build understanding of division.

16. 16 st mary's / OPF / 7/6/08 sourced from Division 19 While children need to consider grouping and sharing in Year 3 and 4 they also need to understand the effect of dividing by 1; the fact that unlike multiplication division is not commutative, that multiplication facts can help with division equations and that some calculations will result in a remainder.Division 19 While children need to consider grouping and sharing in Year 3 and 4 they also need to understand the effect of dividing by 1; the fact that unlike multiplication division is not commutative, that multiplication facts can help with division equations and that some calculations will result in a remainder.

17. 17 st mary's / OPF / 7/6/08 sourced from Division 20 (animated slide) Children in Years 2 and 3 are encouraged to use arrays to help them understand multiplication. These are useful images to discuss in relation to division. The slide shows how the array can be used to discuss how many groups of 8 there are in 48. (This image could also demonstrate 48 shared between 6.) Arrays are a good image from which to write ‘fact families’ – this array shows the following facts; 6 x 8 = 48, 8 x 6 = 48, 48 ? 8 = 6 and 48 ? 6 = 8.Division 20 (animated slide) Children in Years 2 and 3 are encouraged to use arrays to help them understand multiplication. These are useful images to discuss in relation to division. The slide shows how the array can be used to discuss how many groups of 8 there are in 48. (This image could also demonstrate 48 shared between 6.) Arrays are a good image from which to write ‘fact families’ – this array shows the following facts; 6 x 8 = 48, 8 x 6 = 48, 48 ? 8 = 6 and 48 ? 6 = 8.

18. 18 st mary's / OPF / 7/6/08 sourced from Division 21 Year 4 children are also expected to start looking at pencil and paper procedures for division (chunking) and to relate relate division to fractions.Division 21 Year 4 children are also expected to start looking at pencil and paper procedures for division (chunking) and to relate relate division to fractions.

19. 19 st mary's / OPF / 7/6/08 sourced from Division 22 (animated slide) It is useful to discuss the meaning of the different parts of a fraction with children. This animation shows how the fraction bar is linked to the division symbol. It helps explain why you key in 2 ? 3 on a calculator when you want to convert two thirds into a decimal. Division 22 (animated slide) It is useful to discuss the meaning of the different parts of a fraction with children. This animation shows how the fraction bar is linked to the division symbol. It helps explain why you key in 2 ? 3 on a calculator when you want to convert two thirds into a decimal.

20. 20 st mary's / OPF / 7/6/08 sourced from Division 23 Division 23

21. 21 st mary's / OPF / 7/6/08 sourced from Division 24 Division 24

22. 22 st mary's / OPF / 7/6/08 sourced from Division 25 (animated slide) Grouping is the concept used in chunking. As illustrated in the slide the question 72 ? 5 can be thought of as ‘how many fives are there in 72?’ One way of working this out would be to start at 72 and add 5 as many times as possible. Division is associated with repeated addition here but you could also use subtracting five as many times as possible. This, however, is time-consuming and inefficient. Instead of adding individual fives it is possible to add ‘chunks’ of five. In the above example ten chunks of five have been added then another 4 chunks of 5 leaving a remainder of 2. This means there are 14 chunks of 5 in 72 with 2 left over, so the answer is 14 remainder 2.Division 25 (animated slide) Grouping is the concept used in chunking. As illustrated in the slide the question 72 ? 5 can be thought of as ‘how many fives are there in 72?’ One way of working this out would be to start at 72 and add 5 as many times as possible. Division is associated with repeated addition here but you could also use subtracting five as many times as possible. This, however, is time-consuming and inefficient. Instead of adding individual fives it is possible to add ‘chunks’ of five. In the above example ten chunks of five have been added then another 4 chunks of 5 leaving a remainder of 2. This means there are 14 chunks of 5 in 72 with 2 left over, so the answer is 14 remainder 2.

23. 23 st mary's / OPF / 7/6/08 sourced from Teaching chunking – vertical Division 26 This shows the same example but written vertically. This is the written method referred to as ‘chunking’. Children often find it helpful to record some key multiplication facts by the side of the calculation as a useful reference point. The most useful facts are generally 1 times, 2 times, 5 times and 10 times the divisor. Initially it is helpful to pick a dividend (the number to be divided) that is more than 10x (but less than 20x) the divisor.Division 26 This shows the same example but written vertically. This is the written method referred to as ‘chunking’. Children often find it helpful to record some key multiplication facts by the side of the calculation as a useful reference point. The most useful facts are generally 1 times, 2 times, 5 times and 10 times the divisor. Initially it is helpful to pick a dividend (the number to be divided) that is more than 10x (but less than 20x) the divisor.

24. 24 st mary's / OPF / 7/6/08 sourced from Division 28 Another example of chunking but with larger numbers Both the partitioning method and vertical methods are shown. Again the multiplication facts can help provide a useful starting point.Division 28 Another example of chunking but with larger numbers Both the partitioning method and vertical methods are shown. Again the multiplication facts can help provide a useful starting point.

25. 25 st mary's / OPF / 7/6/08 sourced from Continuing division Division 29 Finally in Years 5 and 6 children need to appreciate that a number cannot be divided by 0 (you cannot answer the question ‘How many zeros are there in 9?’). They also need to have experience of expressing remainders as fractions and where appropriate as decimals. Year 5 and 6 children also need to experience using calculators as calculating aids.Division 29 Finally in Years 5 and 6 children need to appreciate that a number cannot be divided by 0 (you cannot answer the question ‘How many zeros are there in 9?’). They also need to have experience of expressing remainders as fractions and where appropriate as decimals. Year 5 and 6 children also need to experience using calculators as calculating aids.

26. 26 st mary's / OPF / 7/6/08 sourced from Standard Short division - in late year 4 early year 5 e.g. 256 divided by 7 0 3 6 r 4/7 = 7 22546

27. 27 st mary's / OPF / 7/6/08 sourced from LONG DIVISION by CHUNKING 2645 divide by 23

28. 28 st mary's / OPF / 7/6/08 sourced from Year 6 - Word problems Division 32 Finally two more questions to solve using the calculator. The first question involves quite complex mixed units and the need to interpret the display and make sense of the decimals. The second example again requires the ability to interpret the answer and decide whether to round up or down (it is also possible to use the constant function to answer this).Division 32 Finally two more questions to solve using the calculator. The first question involves quite complex mixed units and the need to interpret the display and make sense of the decimals. The second example again requires the ability to interpret the answer and decide whether to round up or down (it is also possible to use the constant function to answer this).

29. 29 st mary's / OPF / 7/6/08 sourced from Other Issues Calculators - place value, rounding checking, lessons on practical use of. Not ‘normally’ used in KS 1 Homework - not always written; sometimes practical; possibly preparation for future lessons; problem solving; generally follow up. Games Keeping You Informed -curriculum guide/ calculations booklet ??????????? ANY QUESTIONS ??????