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Embedding mathematical reasoning and problem solving into every day maths Session 5 : Roadmap to Mastery

Sealed solution : nrich.maths.org/1177

3 Aims: Fluency, reasoning & problem solving become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions

Complete this diagram so that the numbers in each row and in each column add up to 140. Can you make one of your won to add up to 250? Bottom Back Side What’s the largest open box that can be made by folding a piece of card 20cm by 30cm? 8 girls share 6 bars of chocolate equally. 12 boys share 9 bars equally. Clare says each girl gets more as there are fewer of them. Rob says each boy gets more as they have more chocolate to share. Explain why they are both wrong. Fluency, reasoning or problem solving?

Implications for learning activities How embedded are the three aims in everyday practice? What distinctions are made between Fluency, Reasoning & Problem Solving in day to day planning? What steps may be considered to develop this aspect?

Using previously mastered skills but developing reasoning & problem solving skills? Choosing activities that develop the reasoning & problem solving toolkit? Providing activities that require a wider range of skills

NRICH reasoning article “Reasoning is needed when logical thinking is required including…. Following a logical argument. Encountering a new challenge Selecting important information Recognising information to be found Choosing from a range of strategies Evaluating a solution Presenting a logical argument”

Money bags 1116

Problem solving skills …. Working systematically Trial and improvement Logical reasoning Spotting patterns Visualising Working backwards Conjecturing

Fermi questions My personal favourite…. “How many people can fit onto a soccer pitch?”

Developing & practising skills to be mastered in a problem solving and/or reasoning context? Which of these statements are correct? A square is a rectangle. A rectangle is a square. A rectangle is a parallelogram. A rhombus is a parallelogram. Explain your reasoning. 6 cm 4 cmIf the area of the shape is 36cm 2 what’s the max/min perimeter?

Card sort of…? Devise a set of cards that can be arranged to demonstrate the proof that angles in any triangle add up to 180 degrees Create a set of questions to illustrate the logically progression from 3 x 5 = 15to 150 ÷ 0.3 = 500

Providing independence to go deeper? Providing a positive role model by looking for links between different aspects of the maths curriculum and also demonstrating good reasoning in logical steps of progress within each topic Developing curiosity but encouraging mathematical rigour from the outset Poly Plugs

nrich Communicating reasoning … “.. in a succinct, elegant and mathematical way. Here strategies that we use in Literacy will prove helpful to us. It is helpful to model the communication ourselves (both as articulating our own thought processes and also staging conversations with other adults in the room), give sentence starters to help children construct their argument and also give time in lessons to improving children’s expression of their reasoning processes.”

“Children need to understand how to refine their sentences and chains of reasoning, and how to use appropriate mathematical language. A working wall is a great place for examples of superb reasoning that children can refer to when improving their own chain of reasoning to become more succinct and elegant. They also could collect good examples in the back of their mathematics books with annotations to explain what makes them ‘good’.”

Here are some possible sentence starters: I think this because... If this is true then... I know that the next one is... because... This can’t work because... When I tried ……. I noticed that... The pattern looks like... All the numbers begin with... Because ….. then I think …. This won’t work because..

“You might want to develop a checklist for children to use when considering their reasoning communication or that of others. It will need to be age/development/knowledge appropriate. It could contain the following: How clear is the reasoning? Can I follow the argument? How logical is the reasoning? Does it form a chain of reasoning? Is it a complete or partial chain? Does the argument/explanation use reasoning language, such as ‘because’? How succinct is the reasoning? Are the sentences short and to the point?”

1048 The large rectangle is divided into a series of smaller quadrilaterals and triangles. Each of the shapes is a fractional part of the large rectangle. Can you untangle what fractional part is represented by each of the ten numbered shapes?

Fraction fascination 5061 I drew this picture by drawing a line from the top right corner of a square to the midpoint of the opposite sides. Then I joined these two midpoints with another line. What fraction of the area of the square is each of these triangles?

Fraction fascination continued I then drew another picture. How has this been made using the first square? What is the shape that has been created in the middle of this larger square? What fraction of the larger square does this shape take up?

Heads and Feet 924 On a farm there were some hens and some sheep. Altogether there were 8 heads and 22 feet. How many hens were there?

Achi 1182 A game for two players Each player starts with 4 counters each. Take it in turns to place a counter on an empty circle until all the counters are on the board. Then take it in turns to slide one of your counters along a line into an empty circle. The winner is the first player to get three counters in a straight line.