Division

Sharing or Grouping What does each mean? Model each with some counters for 8  2 What’s the same? What’s different? Ask participants to carry out the calculation with counters, both as a grouping and a sharing model

Multiplication Division Repeated Addition Scaling I have not filled in the blanks as I think they should have the discussion. It is an opportunity to think more deeply about the structures of division. The argument is that multiple addition links to grouping – multiple subtraction Scaling links to sharing – division can be seen as a scaling down process – finding a fraction of the number Pupils should work with a range of materials and contexts in which multiplication and division relate to grouping and sharing discrete and continuous quantities, relating these to fractions and measures (e.g. 40 ÷ 2 = 20, 20 is a half of 40). So dividing by 2 is the same as finding a half, dividing by 3 is the same as finding a third, dividing by 5 is the same as finding a fifth of a number. The operation of division relates to finding a fraction of a number. The sharing structure exemplifies this as 12 ÷ 4 means that 12 sweets shared between 4 people will result in them receiving a quarter each. The structure of division as sharing needs to be developed and not left at 1 for you, 1 for you. 1 for you.......... The new curriculum mentions scaling several times, particularly in the context of problem solving How do grouping and sharing relate to repeated addition and scaling?

An image for 56  7 Either: How many 7s can I see? (grouping) Or: If I put these into 7 groups how many in each group? (sharing) The key point here is that, by arranging the groups you remove into an array, grouping and sharing can be seen as ‘two sides of the same coin’ and not mutually exclusive. It relates to 8 x7

An image for 56  7 5 6 7 8 5 6 7 8 The array is an image for division too The key point here is that, by arranging the groups you remove into an array, grouping and sharing can be seen as ‘two sides of the same coin’ and not mutually exclusive. It relates to 8 x7

Sharers Groupers Share your counters out equally into piles according to the divisor then count how many in each pile. Put your multilink into groups of the divisor and count the number of groups 18÷3 24÷12 21÷3 20÷4

Video 2 https://www.ncetm.org.uk/resources/43589

Sharing and grouping in pairs What strategies are used to support the pupils in working collaboratively? How are the pupils supported in recognising that sharing and grouping give the same answer? The two structures are required and should be clearly taught. Consider how the context will determine which way children think about division.

363 ÷ 3 = 1 2 1 3 6 3 3 Division with no exchange or “re group” and no remainder Highlight that you are going to talk through division as grouping Emphasise that when we make a group of 3 in the hundreds column it is one group of 3 but the counters are hundreds and we can only call it one group of 3, within the context of it being in the hundreds column. The digit is multiplied by the column heading, so the one has a value of 100 make reference back to the place value in session 1 Now model as Sharing Talking through, in the first column, the counters shared between 3 means that they have 1 counter each Talk this through for each of the columns Identify that children need to see the model as grouping in order to more fluently use their tables.

844 ÷ 4 = 2 1 1 8 4 4 4 Division with no exchange or “re group” and no remainder Highlight that you are going to talk through division as grouping Emphasise that when we make a group of 3 in the hundreds column it is one group of 3 but the counters are hundreds and we can only call it one group of 3, within the context of it being in the hundreds column. The digit is multiplied by the column heading, so the one has a value of 100 make reference back to the place value in session 1 Now model as Sharing Talking through, in the first column, the counters shared between 3 means that they have 1 counter each Talk this through for each of the columns Identify that children need to see the model as grouping in order to more fluently use their tables.

Video 3 https://www.ncetm.org.uk/resources/43589

Division Vocabulary Consider the high expectations in terms of using the correct vocabulary. The teacher says “When children have the key vocabulary they can speak in full sentences about their learning and they learn more” Do you agree with this statement? Consider the strategy of learning the correct vocabulary alongside the procedure.

Division Vocabulary Things to notice: Children are given the opportunity to hypothesise and predict what the words might mean. The teacher says “I’m going to write this in a different way but it means the same thing.” The children are given the opportunity to transfer their understanding from one representation to another.

364 ÷ 3 = 1 2 rem 1 3 6 4 3 If there are some left over, children are quite happy with this and you could introduce the language of ‘remainder’ together with some way of recording this. N.B. There is the possibility, at a later stage, to exchanged the one for tens 0.1s. Might want to mention this here(?)

345 ÷ 3 = 1 1 5 3 4 5 3 1

Video 7 https://www.ncetm.org.uk/resources/43589

Using place value counters and recording division Notice how the children work in pairs where one manipulates and the other records. They then swap roles. This has proved to be an effective strategy to develop both reasoning and fluency in the use of the formal written method.

Using place value counters and recording division “... Using the place value counters ... Not only do they understand the process of the division algorithm but they also understand the concept” Notice how a thorough understanding of the physical process of re-grouping and exchange enables the children to use the language of re-grouping when talking about the algorithm.

Task Explore some division calculations using the different manipulatives. How well do the manipulatives help you to solve the calculation problems? How well do the manipulatives help to move pupils towards written methods? Reflect on your own practice about how a written method for division can be taught. You will need to determine how much time could be given to this. Participants will be ready to explore division using the resources at this stage so allow approx 20-30 minutes for exploration and discussion. You may wish to provide some examples of subtraction problems for the participants to try out using the diennes or place value disks. Use the questions above to help structure the discussion. In C1 this was a rich discussion which also looked at the various different subtraction problems that pupils might encounter. Draw the discussion back together with some closing thoughts.

Video 8 https://www.ncetm.org.uk/resources/43589

Problems involving division The three aims of the curriculum are to develop fluency, reason mathematically and solve problems. What aspects of these aims are present in the pupils’ discussions?

Problems involving division An important aspect of problem solving is making a sensible a interpretation of the solution. How do these problems provide the opportunity for pupils to develop this skill?