Rich Maths Tasks at KS3 to

Engage and Motivate

Jon StratfordHead of MathsJohn Flamsteed SchoolDerbyshirejondstratford@yahoo.com

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 of2. What makes a task RICH ?3. Conjectures4. Ben’s Game

Key Processes – 1. Making sense of2. Working with the Key Processes

Teaching Strategies1. Working with Rich Tasks2. Group work3. Who sits where4. Writing frames5. Metacognitive Plenaries

Rich Tasks – 1. Types of2. Mysteries3. Photographs4. Riching it Up5. 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 The richness of mathematical tasks does NOT lie in the tasks themselvesNOT lie in the tasks themselves

NOR does it lie in the format of interactionsNOR does it lie in the format of interactions

The richness of learners’ mathematical The richness of learners’ mathematical experience depends on opportunities to use experience depends on opportunities to use and develop their own powers, and develop their own powers, opportunities to make significant opportunities to make significant mathematical choices and being in the mathematical choices and being in the presence of mathematical awarenesspresence 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 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.

• 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-examples

• Analysing - Use appropriate mathematical procedures: Calculate accurately, selecting mental methods or calculating devices as appropriate. Record methods, solutions and conclusions.

• Interpreting and evaluating: Form convincing arguments based on findings and make general statements

• Communicating and reflecting: Communicate findings effectively. Engage in mathematical discussion of results

Representing

1. Identify the mathematical aspects of a situation or problem

2. Choose between representations3. 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 mathematics2. Use knowledge of related problems3. Visualise and work with dynamic images4. Identify and classify patterns5. Make and begin to justify conjectures and generalisations,

considering special cases and counter-examples6. Explore the effects of varying values and look for

invariance and covariance7. Take account of feedback and learn from mistakes8. Work logically towards results and solutions, recognising

the impact of constraints and assumptions9. Appreciate that there are a number of different

techniques that can be used to analyse a situation10. Reason inductively and deduce.

Analysing - Use appropriate mathematical

procedures 1. Make accurate mathematical diagrams, graphs and

constructions on paper and on screen2. Calculate accurately, selecting mental methods or

calculating devices as appropriate3. Manipulate numbers, algebraic expressions and equations

and apply routine algorithms4. Use accurate notation, including correct syntax when

using ICT5. Record methods, solutions and conclusions6. 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 exceptions5. Relate findings to the original context, identifying whether

they support or refute conjectures6. Engage with someone else’s mathematical reasoning in

the context of a problem or particular situation7. Consider the effectiveness of alternative strategies

Communicating and reflecting

1. Communicate findings effectively2. Engage in mathematical discussion of results3. Consider the elegance and efficiency of alternative

solutions4. 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 joinersalternatively … 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 weTried to understand what

was saidBuilt on what others had

said.Demanded good

explanations.Challenged what was said.

This is how weTreated opinions with respect.Shared responsibility.Reached agreement.

This is how weGave 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

http://emmaths.jfcs.org.uk/

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 ?