Division in all its aspects Dr Jenni Back Host Schools Project Lead

Key thresholds in mathematical development in arithmetic KS1 entry: conservation and counting KS2 entry: addition/subtraction, number bonds to 20, place value KS3 entry: multiplication/division, multiplication tables KS4 entry: proportional reasoning

The issues in years 3 and 4 Securing conservation and counting Additive reasoning, fluent recall and application of number bonds to 20, understanding place value Developing ideas of multiplication and division Intuitive understanding of proportional reasoning

The Big Ideas in arithmetic in Y3 & 4 Doing and undoing The number system and place value Securing and deepening understanding of addition and subtraction Building repertoire of known facts Developing understanding of multiplication and division

Some questions to explore Imagine a party scenario. Look at the question and think about the meaning of division that it relates to Consider the resources that might support an investigation of the problem Be ready to share your ideas with others

Party scenarios How many apricots will we each have if 1kg is shared between 12 of us? There are roughly 30 apricots in one kilogram. Sharing by counting out and cutting There are 40 sandwiches and 12 people at the party. How many will each guest be able to have? What shall we do with the spares? Sharing by counting out and cutting Here is a cake, we need to share it between the 12 people at the party. How do we do it? Sharing by cutting into congruent shapes    

Party Scenarios continued There are 5 pizzas. How can we share them between the 12 people? Sharing by cutting successively into congruent shapes We have 3 litres of juice for a party of 12 people. How many 120ml glasses will it fill? How many glasses each will that give everyone? Sharing by pouring liquids I am making invitations for a party and want them to be 10cm by 12cm. How many will I be able to cut out of each A4 sheet of card that I have? How many sheets of card will I need to make 12 invitations? Sharing by fitting into a shape

Party Scenarios continued I am making bunting to hang across the room. How many flags will I need to make to reach across the diagonal of the hall measuring 6m by 7m? You will need to think about the size of the flags, their shape, the tape or string they are fixed to and the gaps between them. To put them across both diagonals and along all four of the sides, how many more will I need? Finding out how many lengths x there are in y I have a bag of 30 balloons to decorate the hall. How many groups of 4 can I make? Grouping in 4s

Party scenarios continued If 12 people come to the party, how can I divide them into teams? What would be the best way to do this? Grouping in 2s, 3s, 4s etc or using multiplication facts If 12 people come to the party and they all go bowling by car from the hall, how many cars will be needed if each car can take 5 people? How many cars will be needed if each car has to have an adult to drive it in addition to the party guests? Successive subtraction or grouping and rounding up

Division: what does it mean? Children need to make sense of it Crucial to focus on communicating ideas How would you explain it? Sharing Grouping Inverse of multiplication Successive subtraction What other concepts is it dependent on? Addition, subtraction and multiplication

How can we support children’s understanding of it? Easy access to a range of appropriate and useful representations Support them in applying their ideas to new problems and eventually new concepts

Carousel of activities Division as the inverse of multiplication Exploring Factors – Maths Out Loud 5 lesson 15 Considering remainders Susie the Snake & Maisie the Mouse – Mathematical Challenges Able pupils at Key Stages 1 & 2 Division as repeated subtraction Dan the Dragon Slayer - Beam’s Big Book of Word Problems for Years 5 & 6 Unit 22 Division as sharing Party Time - Beam’s Big Book of Word Problems for Years 5 & 6 Unit 21 Division as grouping Cinderella -Beam’s Big Book of Word Problems for Years 3 & 4 Unit 22

Focus of engagement with activities Try the activity together – do the mathematics Identify the potential learning and progression of learning through the task Consider support and extension Identify the practical resources that might support the learning Develop an outline lesson plan using the task for your Y3 & 4 pupils Be prepared to feedback and share

Developing fluency in division calculations Building on know facts and inverses – mental methods and jottings Written methods Chunking up or down The ‘bus stop’ method for short division Using Dienes as a supporting resource The ultimate goal of long division Division of three-digit numbers – Maths Out Loud 6 lesson 15

Next Steps Where do you plan to take the work we have done today? Try some activity within the next week Feedback on the NCETM website about what you have done Share your experiences with your colleagues in school

Thank you for listening Dr Jenni Back jenni.back@ncetm.org.uk Telephone 07850100074