Leicestershire Numeracy Team 2003 Division Division 1 This presentation was written to support you when discussing division at your school. It could form a staff meeting to look at the progression in the development of division concepts from Key Stage 1 to Key Stage 2. It follows the progression in division within the Leicestershire Recommendations for Pencil & Paper Procedures. It places particular emphasis on division as grouping and develops the idea of chunking based on repeated subtraction. Leicestershire Numeracy Team 2003

The problems with division Try these: 6 18 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) 24 202 Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 What is division? How would you illustrate this division to a child? What would you draw and what language would you use? 12  3 = 4 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. Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 Skills in Early Division 12  3 = 4 Sharing There are three children and 12 cakes. How many can they each have, if I share them out equally? (Sharing 12 things equally into 3 piles. How many in each) Division 4 (animated slide) This slide illustrates how 12 objects can be practically shared. Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 Skills in Early Division 12  3 = 4 Grouping There are 12 cakes. How many children can have three each? (How many threes are there is 12?) Division 5 (animated slide) In this slide the same calculated is completed by considering how many groups of three there are in 12. Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 Language and division Since the  sign represents both the sharing and grouping aspects of division, encourage the children to read this as ‘divided by’ rather than ‘shared by’. 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’). Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 6000  1000 = Would you group or share for this calculation? Division 7 Ask teachers to briefly discuss this. It would be very inefficient to share 6000 by 1000. To answer this question most people would think about how many 1000s there are in 6000s (grouping). Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 Introducing division In Year 2 children are encouraged to understand the operation of division as: sharing equally grouping or repeated subtraction e.g. How many tens are in 60? 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. Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 18  3 = 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. Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 Sharing Supports an understanding of halving and the 1 to 1 correspondence between objects. Requires little knowledge or skill beyond counting. Becomes more difficult to visualise as the divisor increases. Is inefficient. Division 10 Sharing has limitations. It is useful for halving and when looking at fractions. However as the size of numbers increase it can quickly become inefficient. Leicestershire Numeracy Team 2003

Division and number lines 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) 0 3 6 9 12 15 18 Leicestershire Numeracy Team 2003

Modelling division on beadstrings 20  4 = 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) Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 20  4 = Division 13 Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 20  4 = Division 14 Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 20  4 = Division 15 Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 20  4 = Division 16 Leicestershire Numeracy Team 2003

Key Stage 1 - Calculations Encourage children to use jottings, as well, to check answers to calculations that have been reached by mental methods Division 17 This was a question from the 2002 National test paper. Feedback from QCA has indicated that when children record they are now more likely to use grouping. The slide shows two examples of this. A sharing drawing would have been nightmare! Q29 1 2c 2b 2a 3 All 4% 12% 27% 61% 31% Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 Grouping Links to counting in equal steps on a number line. Requires knowledge of subtraction facts (repeated subtraction) and addition facts (counting up). Is more efficient than sharing as the divisor increases. Provides a firmer basis on which to build children’s 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. Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 Introducing division In Year 3 and 4 children also need to know that: dividing a whole number by 1 leaves the number unchanged: e.g. 12  1 =12 16  2 does not equal 2  16 division reverses multiplication (the inverse) – this allows them to solve division calculations by using multiplication strategies (18  3 by counting the hops of 3 to 18) there will be remainders for some division calculations (to be expressed as whole-number remainders). 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. Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 How many eights in 48?                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. Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 Continuing division In Year 4 children need to begin to : relate division and fractions use a written method for division (chunking). 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. Leicestershire Numeracy Team 2003

 2 3 the number to be divided the divisor 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. Leicestershire Numeracy Team 2003

2  3 the number to be divided the divisor Division 23 Leicestershire Numeracy Team 2003

2  3 the number to be divided the divisor Division 24 Leicestershire Numeracy Team 2003

Teaching chunking - partitioning 72  5 Partition 72 in to a convenient multiple of 5 + the rest 72 = 50 + 22 Divide each part 50 ÷ 5 = 10 22 ÷ 5 = 4 rem 2 Recombine the parts Answer: 14 remainder 2 Division 26 This shows an example of partitioning. Here the 72 is partitioned into 10x the divisor and the rest e..g 72 = 50 + 22 Each part is then divided by 5 using knowledge of tables and if necessary place value These are then recombined to give the answer In a word problem this would then be put into the appropriate context Leicestershire Numeracy Team 2003

Teaching chunking - number line 72 ÷ 5 = Grouping - How many 5’s are there in 72? Adding groups of 5 5 x 10 or 10 groups of 5 5 x 4 or 4 groups of 5 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. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 72 Leicestershire Numeracy Team 2003

Teaching chunking - vertical 5 x 1 = 5 5 x 2 = 10 5 x 5 = 25 5 x 10 = 50 72  5 = 72 50 (5 x 10) 22 20 (5 x 4) 2 Answer: 14 remainder 2 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. Leicestershire Numeracy Team 2003

Using calculators for repeated subtraction The constant function To calculate 72  5 using repeated subtraction Press 5 - - = then press 72 Division 27 Calculators can be used to help demonstrate the effect of repeated subtraction by using its constant function. This converts the calculator into a counting machine – in this example one that counts backwards in fives. On your calculator press 5 and then the subtract key twice (a small k should appear on the calculator display – this shows the constant function is in operation) then =. Now press 72. From now on every time you press the equals key 5 will be subtracted from 72. You can count how many times 5 can be subtracted from 72 (the calculator will continue counting back in fives into negative numbers to you will need to discuss with children when to stop!) Leicestershire Numeracy Team 2003

Teaching chunking - larger numbers 256  7 256 = 210 + 46 210 ÷ 7 = 30 46 ÷ 7 = 6 remainder 4 7 x 1 = 7 7 x 2 = 14 7 x 5 = 35 7 x 10 = 70 256  7 = 256 210 (7 x 30) 46 42 (7 x 6) 4 or 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. Answer: 36 remainder 4 Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 Continuing division In Year 5 and 6 children also need to understand: that a number cannot be divided by zero how a quotient can be expressed as a fraction and as a decimal fraction how to interpret the display when dividing with a calculator. 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. Leicestershire Numeracy Team 2003

185 people go to the school concert. They pay £1.35 each. How much ticket money is collected? Programmes cost 15p each. Selling programmes raises £12.30 How many programmes are sold? £ Division 30 This is a question from the calculator paper in the 2002 National tests. Ask teachers to quickly answer this question. It may be worth discussing what is meant by ‘show your method’ – basically this means write down the keys you would press on your calculator to answer the question. Share solutions. Then look at the following slide. Show your method you may get a mark. Leicestershire Numeracy Team 2003

Leicestershire Numeracy Team 2003 Division 31 This shows a child’s response to the question. This was a common answer. Children need to ensure that the answer displayed on their calculator makes sense in the context of the question (0.82 programmes could not have been sold). They also need to appreciate when questions involve mixed units. This would enable them to calculate 12.3  0.15 = 82. Leicestershire Numeracy Team 2003

Solve these word problems To make a box pieces of wood 135mm long have to be cut from a 2.5m length. How many lengths of wood can be cut? Train fares cost £14.50. I have £52. How many people can I take on the journey? 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). Leicestershire Numeracy Team 2003