Rich Maths Tasks at KS3 to Engage and Motivate Jon Stratford Head of Maths John Flamsteed School Derbyshire

Aims for the Session To develop an understanding of what constitutes a Rich Maths task To examine how the Key Processes in Maths can be addressed through working with Rich Tasks To explore some of the pedagogy associated with Rich Tasks To look at possible sources for Rich Tasks, including own website.

Rich Tasks: 1. Features ofFeatures of 2. What makes a task RICH ?What makes a task RICH ? 3. ConjecturesConjectures 4. Ben’s GameBen’s Game

Key Processes – 1. Making sense ofMaking sense of 2. Working with the Key ProcessesWorking with the Key Processes

Teaching Strategies 1. Working with Rich TasksWorking with Rich Tasks 2. Group workGroup work 3. Who sits whereWho sits where 4. Writing framesWriting frames 5. Metacognitive PlenariesMetacognitive Plenaries

Rich Tasks – 1. Types ofTypes of 2. MysteriesMysteries 3. PhotographsPhotographs 4. Riching it UpRiching it Up 5. Website Website

Rich Tasks Are accessible and extendable. Allow individuals to make decisions. Involve learners in testing, proving, explaining, reflecting and interpreting. Promote discussion and communication. Encourage originality and invention. Encourage “what if?” and “what if not?” questions. Are enjoyable and contain opportunity to surprise. A. Ahmed

The richness of mathematical tasks does NOT lie in the tasks themselves NOR does it lie in the format of interactions The richness of learners’ mathematical experience depends on opportunities to use and develop their own powers, opportunities to make significant mathematical choices and being in the presence of mathematical awareness John Mason (OU and Oxford University) What makes a task RICH ?

Conjectures – what comes next?

Ben’s Game Ben, Jack and Emma were playing a game with a box of 40 counters – they were not using them all. They each had a small pile of counters in front of them. All at the same time Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. They all passed on more than one counter. After this they all had the same number of counters. How many could each of them have started with? What if they decided to play again, this time passing a different fraction each? Reproduced courtesy of NRICH

Key Processes Representing Analysing - Use mathematical reasoning Analysing - Use appropriate mathematical proceduresAnalysing - Use appropriate mathematical procedures Interpreting and evaluating Communicating and reflecting

Key Processes - Ben’s Game Representing: Identify the mathematical aspects of a situation or problem. Choose between representations. Select mathematical information, methods and tools to use.Representing Analysing - Use mathematical reasoning: Explore the effects of varying values and look for invariance and covariance. Make and begin to justify conjectures and generalisations, considering special cases and counter-examplesAnalysing - Use mathematical reasoning Analysing - Use appropriate mathematical procedures: Calculate accurately, selecting mental methods or calculating devices as appropriate. Record methods, solutions and conclusions.Analysing - Use appropriate mathematical procedures Interpreting and evaluating: Form convincing arguments based on findings and make general statementsInterpreting and evaluating Communicating and reflecting: Communicate findings effectively. Engage in mathematical discussion of resultsCommunicating and reflecting

Representing 1.Identify the mathematical aspects of a situation or problem 2.Choose between representations 3.Simplify the situation or problem in order to represent it mathematically, using appropriate variables, symbols, diagrams and models 4.Select mathematical information, methods and tools to use.

Analysing - Use mathematical reasoning 1.Make connections within mathematics 2.Use knowledge of related problems 3.Visualise and work with dynamic images 4.Identify and classify patterns 5.Make and begin to justify conjectures and generalisations, considering special cases and counter-examples 6.Explore the effects of varying values and look for invariance and covariance 7.Take account of feedback and learn from mistakes 8.Work logically towards results and solutions, recognising the impact of constraints and assumptions 9.Appreciate that there are a number of different techniques that can be used to analyse a situation 10.Reason inductively and deduce.

Analysing - Use appropriate mathematical procedures 1.Make accurate mathematical diagrams, graphs and constructions on paper and on screen 2.Calculate accurately, selecting mental methods or calculating devices as appropriate 3.Manipulate numbers, algebraic expressions and equations and apply routine algorithms 4.Use accurate notation, including correct syntax when using ICT 5.Record methods, solutions and conclusions 6.Estimate, approximate and check working.

Interpreting and evaluating 1.Form convincing arguments based on findings and make general statements 2.Consider the assumptions made and the appropriateness and accuracy of results and conclusions 3.Be aware of the strength of empirical evidence and appreciate the difference between evidence and proof 4.Look at data to find patterns and exceptions 5.Relate findings to the original context, identifying whether they support or refute conjectures 6.Engage with someone else’s mathematical reasoning in the context of a problem or particular situation 7.Consider the effectiveness of alternative strategies

Communicating and reflecting 1.Communicate findings effectively 2.Engage in mathematical discussion of results 3.Consider the elegance and efficiency of alternative solutions 4.Look for equivalence in relation to both the different approaches to the problem and different problems with similar structures 5.Make connections between the current situation and outcomes, and situations and outcomes they have already encountered

Teaching Strategies Allow pupils time to understand and engage with the problem Offer strategic rather than technical hints Encourage pupils to consider alternative methods and approaches Encourage explanation Model thinking and powerful methods

Group Work Plan to offer the task in a form that will encourage collaboration Plan how you will arrange the room Plan how you will group pupils Plan how you will introduce the purpose of discussing Plan how you will establish ground rules Plan how you will end the discussion

My Classroom - table arrangements

Writing frames – sentence starters My task is…. I think that ….. One reason for thinking that ….. Another reason is ….. In addition to this …… This is why I think that……

Further sentence starters and joiners alternatively … similarly … on the other hand … because of this …this suggests that … it would seem that …. never the less … however …instead of … in comparison to… although …this could mean that … one possibility is that … in contrast …perhaps …unless … additionally … furthermore… Nonetheless … consequently …

Metacognitive Plenaries Working Collaboratively This is how we Tried to understand what was said Built on what others had said. Demanded good explanations. Challenged what was said. This is how we Treated opinions with respect. Shared responsibility. Reached agreement. This is how we Gave everyone in our group a chance to speak. Listened to what people said. Checked that everyone else listened.

Types of Rich Tasks Problems / Puzzles –NRICH –UKMT –ATM Card Sort Activities –Tarsia Jigsaws –Mysteries –Always, Sometimes, Never Photographs Practical –Bowland –Data Gathering –Design/Construction ‘Big’ Questions Riched Up Tasks

Mysteries – Where’s the Cup? Our local football team, the Denby Dynamos, need your help. The Ripley Rockets have stolen their League Cup. Your task is to use the clues provided to find out where the cup is hidden.

Photographs – where’s the maths?

An exercise on Ratio (extract from SMP Interact) …becomes…

A richer task ?