# October 2014 Monthly Maths

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October 2014 Monthly Maths

Monthly Maths I s s u e Why solve problems? Although problem solving is a phrase currently on the lips of many maths teachers, people have been solving problems since the beginning of man, albeit by trial and error. 4 1 Problem solving in mathematics education How and when does problem solving fit into mathematics education? In her nrich article Problem Solving and the New Curriculum author Lynne McClure asks “What's the point of doing maths?” McClure says: “What children should be doing is solving problems, their own as In his 1998 paper well as those posed by others. Because Problem Solving, the whole point of learning maths is to be Kevin Dunbar able solve problems. Learning those rules gives focuses on a and facts is of course important, but they are the tools with which we learn to do number of important issues in maths fluently, they aren’t maths itself.” problem solving research, along with an overview of developments in problem solving research. Dunbar describes two crucial features of problem solving: “First, a problem exists when a goal must be achieved and the solution is not immediately obvious. Second, problem solving often involves attempting different ways of solving the problem.” MEI's response to the draft primary national curriculum (July 2012) states that: “Incorporating a problem solving cycle into the national curriculum is not, of itself, a sufficient means of improving classroom teaching. Fundamental improvements in classroom practice, supported by appropriate professional development and resources, are needed.” Click here for the MEI Maths Item of the Month www.mei.org.uk O c t o b e r 2 0 1 4 In his MEI Conference 2014 session, Using problem solving to develop mathematical thinking in post GCSE students, Phil Chaffé suggests that the following benefits will be enjoyed by teachers and students: More engaged students Students with a better understanding of mathematics, who rise above the symbol manipulation Students who appreciate mathematics techniques as tools rather than endpoints Students who are better prepared for HE and careers More possibilities to spot mathematical talent Better results In the following pages we will examine the impact on maths education of people involved in problem solving: George Pólya, Derek Holton, Charles Lovitt, Edward de Bono and Marylin vos Savant. Curriculum Update GCSE Mathematics for first teaching 2015, AQA, Edexcel Pearson and OCR are all accredited. Teaching of the level 3 Certificate in Quantitative Methods (MEI) has started. Resources are freely available online DfE and Ofqual consultations about A level Mathematics and Further Mathematics have now closed. The outcome of the consultations is expected Oct/Nov. Disclaimer: This newsletter provides links to other Internet sites for the convenience of users. MEI is not responsible for the availability or content of these external sites, nor does MEI endorse or guarantee the products, services, or information described or offered at these other Internet sites. George Pólya Teaching resources Carol Knights has adapted resources from three problem -solving sessions delivered by MEI staff at the MEI Conference 2014: Clare Parsons Kevin Lord Phil Chaffé Carol’s resulting teaching and learning resource is included at the end of this newsletter. As usual, you can download this resource in its original file formats, from the MEI Monthly Maths web page. George Pólya George Pólya (1887 – 1985) was a Hungarian mathematician and professor of mathematics at ETH Zürich and at Stanford University. His work on heuristics (general problem solving strategies) and pedagogy has had lasting influence on mathematical education. In 1945 Pólya published the book How To Solve It, which sold over one million copies and has been translated into 17 languages. In this book he identifies four basic principles of problem solving: 1. 2. 3. 4. Understand the problem Devise a plan Carry out the plan Look back In the Scholastics Teachers Resources section, the 4 Steps to Problem Solving page details Pólya’s problem solving steps as adopted by Billstein, Libeskind and Lott in their book A Problem Solving Approach to Mathematics for Elementary School Teachers. The one-hour 1965 film Pólya Guessing starts with Pólya explaining his attitude to teaching: “Teaching is giving students the opportunity to discover things by themselves.” “First guess then prove.” “Finished mathematics consists of proofs…but mathematics in the making consists of guesses.” In the film Pólya demonstrates his “extraordinary ability to stimulate a group to guess intelligently, to make reasonable conjecture, a process which is essential to mathematical discovery”. He plays a guessing game with a group of students, explaining that guessing is the important beginning of solving a problem, and that looking at a simpler version of a problem will help to solve a more complex problem. Although the film’s picture quality is quite dark, it is worth watching and listening to follow Pólya’s encouraging style of engagement with the group, as they work together towards a solution to the problem using reasonable guessing, then observation, analogy and generalisation – the process of induction*. Pólya advises the students that they should not hold back from guessing; however they should not believe their own guesses, but test them. They should recognise the difference between a fact and a guess. By taking the students through a simple problem Pólya is able to steer them gently towards a more complex problem. You can observe the students becoming increasingly engaged and questioning during this process. *For more information see Pólya’s 1953 article in A. Bogomolny, INDUCTION AND ANALOGY IN MATHEMATICS Preface from Interactive Mathematics Miscellany and Puzzles. Derek Holton Problem solving Derek Holton resources The New Zealand Maths website that hosts the 400 Problem lesson resource (see right) also includes Problem Solving Information. This provides practical guidance about how to implement problem solving in a maths programme (referenced to The New Zealand Curriculum but much of it can be related to other curricula) as well as some of the philosophical ideas behind problem solving. British-born and Australianeducated Derek Holton, former Professor of Pure Mathematics at the University of Otago, New Zealand, has a special interest in problem solving. He has written several books on the subject, including a 2013 book, More problem solving: the creative side of mathematics, published by the Mathematical Association. The abstract states that the “the underlying aim of this book is to show that mathematics is more than a collection of results; there is also a creative, peopleside to the subject”, and that the book will demonstrate how the ordinary can become extraordinary when viewed through a mathematical prism”. Now retired and living in Melbourne, Australia, Holton continues to write about the importance of problem solving in mathematics. In his article What? No Moses? (in his blog Del’s Disturbances on the Casio Edu Australia website), Holton tells us about mathematics:“ there was no one person who brought down the precepts of the subject from a mountain...But if there are any precepts in The Problem mathematics, at least at the macro level, Solving section of they are problems, people and proof.” the nzmaths website provides The Mathematics problem-solving Association publishes the lessons that cover magazine Mathematics in Levels 1 to 6 (ages School; in the March 2006 5-15) of the New issue there is an article Zealand by Holton where he describes and Curriculum. discusses ‘The 400 Problem’: “This problem really does contain experimenting, conjecturing, proving, extending and generalizing. All these are the mathematician’s tools in trade”. Although written with a focus on mathematics at university level, Holton’s article Mathematics: What? Why? How? (page 21 of the Community for Undergraduate Learning in the Mathematical Sciences Newsletter No.1, July 2010) includes an attempt to describe the structure of the creative process in mathematics. Holton says that as mathematics teachers “We shy away from setting questions that are, in some sense ‘open’, and we avoid ‘natural questions’ while we move down set paths through traditional courses.” He stresses that the second type of question should be used more frequently: “Solving problems is not just about solving problems that everyone knows how to solve”. In Holton’s experience with bright secondary students they really enjoy the more open questioning approach, the human side of mathematics, as opposed to a set of things to be learned. Experiencing problem solving in this way will also prepare them to tackle new problems in their future careers. Charles Lovitt nrich problem solving resources In Jennifer Piggott’s excellent article, A Problem Is a Problem for All That, she advises: Give learners space Value their differences Learn from what they do and help them to make connections Use the inherent richness of opportunities to highlight interesting mathematics nrich also offers this post-16 investigation : Maths Problem Twisty Logic. “Sometimes mathematical setups which appear to be straightforward can lead to circular of selfcontradictory or 'paradoxical' logic. Give your brain a workout by thinking about these scenarios”. Charles Lovitt Charles Lovitt has been involved in Mathematics Professional Development for many years. Now retired, he was the director of a host of initiatives and networks in Australia and beyond, including RIME (Reality in Mathematics Education). In his keynote presentation for an Australian Association of Mathematics Teachers Virtual Conferences, Investigations as a central focus for a Mathematics curriculum, Charles Lovitt pointed out “the unfortunate perception that one aspect of the problem solving picture is delivered through games and puzzles and therefore is relegated to the periphery or margins of mathematics”. He suggests that the term ‘Problem Solving’ might better be replaced by the term ‘Investigation Process’, as “The word (problem solving) has become so blurred that we have no common shared agreement on what it means”. Lovitt cites Derek Holton’s article What Mathematicians Do — and why it is important in the classroom (Item 6: Best of Set, ACER, Melb. 1994), where he listed the investigative process by which mathematicians create knowledge and solve problems. In his presentation Lovitt paraphrases the stages of this process (see also diagram on previous page), which could also provide easy to follow guidelines for both constructing and assessing curriculum: 1. Find an interesting (meaningful/ worthwhile) problem. 2. Informally explore, unstructured ‘play’ which generates data. 3. From patterns in the data, create hypotheses, conjectures, theories. 4. Invoke problem solving strategies to prove or disprove any theories. 5. Apply any basic skills I know as part of this proof process. 6. Extend and generalise the problem – what else can I learn from it? 7. Publish (or perish). 8. Go back to step 1. The basic skills mentioned in Stage 5 would include mathematical skills such as algorithms, graphing techniques, algebraic modelling, solution methods for equations, etc. How can we equip our students with the problem solving strategies mentioned in Stage 4? Lovitt suggests that a separate lesson could be devoted to developing thinking skills to create a toolbox of thinking strategies. In the next pages we look at Edward de Bono, a proponent of the teaching of thinking as a subject in schools. As a young teacher in the early 1980s I taught some of de Bono’s thinking skills in a lesson a week to my tutor group. This was as part of a wide humanities-based curriculum, but the thinking skills could be applied to any curricular subject, including mathematics. Edward de Bono Two types of thinking According to de Bono, thinking can be divided into two types: Vertical thinking: the traditionalhistorical method that uses the processes of logic Lateral (or ‘creative’) thinking: a relatively new type of thinking that complements analytical and critical thinking Lateral thinking seeks to solve problems by apparently illogical means; it is a learnable set of skills (a ‘tool’) that uses a process and willingness to look at things in a different way, using insight and creativity. Edward de Bono Edward de Bono was born in Malta in 1933. He developed the concept and tools of lateral thinking, making his work practical and available to everyone, from five years olds to adults. In his Problem Solving article on the thinking-approaches website, de Bono suggests that the traditional method of problem solving “is that you analyse the problem, identify the cause and then proceed to remove the cause. The cause of the problem is removed so the problem is solved.” However, this does not work in all cases, for example, where there is more than one cause for the problem, or where the cause cannot be found or cannot be removed. He says that analysis and argument are not enough; we need to develop the habits of constructive thinking: “The whole thrust of education is towards analysis…Everything should yield to analysis in our traditional methods of thinking. Very little emphasis is given to creativity.” In his 1970 book Lateral Thinking, de Bono says of the two types of thinking: “Lateral thinking is not a substitute for vertical thinking. Both are required. They are complementary. Lateral thinking is generative. Vertical thinking is selective.” He uses this analogy to explain: “Lateral thinking is like the reverse gear in a car. One would never try to drive along in reverse gear the whole time. On the other hand one needs to have it and to know how to use it for manoeuvrability and to get out of a blind alley.” De Bono stresses that lateral thinking isn’t something that occurs by chance and that the technique needs to be taught, preferably as a defined lesson, rather than along with another subject: “The best way to acquire skill in lateral thinking is to acquire skill in the use of a collection of tools which are all used to bring about the same effect.” After discussing attitudes towards lateral thinking and its use, de Bono describes in detail in the book, section by section, the different processes, including background material, theory and nature of the process being discussed in that section, followed by practical formats for trying out and using the process under discussion – “actual involvement” is vital, he says. However, de Bono stresses that “what is supplied is supplied more as an example than as anything else. Anyone who is teaching lateral thinking…must supplement the material offered here with his own material.” He suggests different types of materials that could be collected for this purpose. De Bono stated that: “Some people with high IQs turn out to be relatively ineffective thinkers and others with much more humble IQs are more effective.” Six Thinking Hats Using Six De Bono defined thinking as: The Thinking Hats in operating skill with which intelligence acts upon experience.” He felt that the classroom In this nine-minute video Kim Wells, one of three de Bono master trainers in education, describes what de Bono's thinking hats are and how they can be used as a learning and thinking strategy. In her article Dr Edward de Bono’s six thinking hats and numeracy (Australian Mathematics Primary Classroom (3) 2006), Anne Patterson, a teacher and lecturer in Victoria, applies the teaching approach of “thinking hats” to mathematics education. there is enough individuality in thinking styles and sufficient difference between individuals to suggest that thinking may be a skill that can be developed. With this in mind, de Bono designed the CoRT Thinking Lessons for schools; these lessons have been in use since 1970. De Bono summarises the CoRT Cognitive Research Trust) aims as follows: 1. To acknowledge thinking as a skill. 2. To develop the skill of practical thinking. 3. To encourage students to look objectively at their own thinking and the thinking of others. Edward de Bono explains the importance of and need for thinking tools in a ten-minute video introduction to a lecture that has been uploaded in six parts. This includes a demonstration of the process of addition, and how we can rearrange things in our mind so that we deal with them more simply and more effectively. His ‘Six Thinking Hats’ method is also mentioned. The ‘Six Thinking Hats’ framework is widely used across the world from primary school to board level for any sort of discussion or debate, as an alternative to traditional argument. The ‘parallel’ nature of this method, where everyone is thinking in the same direction, from the same perspective, at the same time, “enables each person's unique point of view to be included and considered.” This thinking tool is designed to help people think clearly and thoroughly by directing their thinking attention in one direction at a time. Each metaphorical 'Thinking Hat' is a different colour that represents a different style of thinking. If you look at a problem with the 'Six Thinking Hats' technique, then you will solve it using all approaches. The White Hat calls for information known or needed. The Red Hat signifies feelings, hunches and intuition. The Black Hat is judgment -the devil's advocate or why something may not work. The Yellow Hat symbolizes brightness and optimism. The Green Hat focuses on creativity: the possibilities, alternatives and new ideas. The Blue Hat is used to manage the thinking process. This four-minute video explains the Six Thinking Hats method further. Marilyn vos Savant The problem The teaser was based on one of the games in the US show Let's Make a Deal! hosted 1963-1991 by Monty Hall. Let's Make a Deal unexpectedly spawned a mathematical conundrum dubbed the Three Door Puzzle. Yet this wasn’t a new problem to some mathematicians, who called it the Monty Hall Problem. An earlier version of the problem, the Three Prisoner Problem, was analysed in 1959 by Martin Gardner in his Mathematical Games column in the journal Scientific American, noting that "in no other branch of mathematics is it so easy for experts to blunder as in probability theory." Marilyn vos Savant A good example of using thinking hats to use different perspectives to consider a problem is that of Marilyn vos Savant and the Monty Hall Problem. In September 1990, vos Savant, puzzle columnist for the U.S. magazine Parade, was sent a probability teaser by a reader. Its publication in her "Ask Marilyn" column together with her solution has produced much debate amongst mathematicians and laymen ever since. “Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, ‘Do you want to pick door No. 2?’ Is it to your advantage to switch your choice?” Marilyn vos Savant replied: “Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?” She came under vehement criticism from mathematicians for her reply. In a follow-up column vos Savant called on school teachers to show the problem to classes, and published the results of more than 1,000 school experiments. Nearly 100% found it pays to switch. Despite this, the Monty Hall Problem continues to be a much-debated topic – we have found a few links that we hope will help to model the problem: The Dr Math forum illustrates the debate well, with several people posting their explanations and solutions to the problem. The Virtual Laboratories in Probability and Statistics project provides free, high quality, interactive, web-based resources for students and teachers of probability and statistics. The Games of Chance section includes the Monty Hall Problem. Alan Davies and Oxford Mathematics Professor Marcus Du Sautoy test out the Monty Hall problem in this YouTube video in response to the comments in this video. The Monty Hall problem was also featured in Mark Haddon’s novel The Curious Incident of the Dog in the Night-time; the SparkNotes literature study guide to the book (see Analysis: Chapters 97-101) sums it up thus: “In essence, Christopher shows that intuition, which says is what people use in life to make decisions, can lead a person to the wrong answer. A problem that appears straightforward turns out to be not straightforward at all.” Further Reading Teachers’ resources and ideas shared The NCETM weekly Twitter #mathscpdchat on 1 July 2014 discussed different ways to promote problem solving in maths and useful resources. The chronological account of the chat is available on the NCETM website, with some of the ideas highlighted. Several links to relevant books are also provided. A search in TES Connect reveals a wide array of problem solving resources shared by teachers for use in the secondary mathematics classroom. Further Reading Laura E. Hardin’s 2002 paper Problem Solving Concepts and Theories provides an overview of educational research on problem solving. Hardin considers problem solving in the context of behavioural, cognitive, and informationprocessing pedagogy, concluding that “both content knowledge and general problem-solving skill are necessary for expert problem solving to occur.” Why Is Teaching With Problem Solving Important to Student Learning? In their NCTM research brief authors Jinfa Cai and Frank Lester provide some directions and useful suggestions, for both teachers and curriculum writers, on teaching with problem solving based on research . (Judith Reed Quander, Series Editor. National Council of Teachers of Mathematics Research Brief. 2010.) Fostering Mathematical Thinking and Problem Solving: The Teacher’s Role Nicole R. Rigelman’s article includes considerations for teachers who want to foster their students’ mathematical thinking and problem solving. (Teaching Children Mathematics. February 2007. The National Council of Teachers of Mathematics, Inc.) Kevin Niall Dunbar (Professor of Human Development and Quantitative Methodology at the University of Maryland College Park) has written several papers on scientific thinking heuristics, including PROBLEM SOLVING (1998. In W. Bechtel, & G. Graham (Eds.). A companion to Cognitive Science. London, England: Blackwell, pp 289-298.); Problem Solving and Reasoning (2006. Kevin Dunbar & Jonathan Fugelsang. To appear in E.E. Smith & S. Kosslyn. An introduction to cognitive psychology. Chapter 10). In her article Mathematics Through Problem Solving, Margaret Taplin (Institute of Sathya Sai Education, Hong Kong) asks What Is A 'Problem-Solving Approach'?, and examines The Role of Problem Solving in Teaching Mathematics as a Process. MATHEMATICAL PROBLEM SOLVING by James W. Wilson, Maria L. Fernandez, and Nelda Hadaway (Wilson, P. S. (Ed.) (1993). Research Ideas for the Classroom: High School Mathematics. New York: MacMillan. Chapter 4). The authors review and discuss the research on how students in secondary schools can develop the ability to solve a wide variety of complex problems: PISA 2012 results: Creative Problem Solving (OECD Publishing). This volume presents an assessment of student performance in creative problem solving, which measures students’ capacity to respond to non-routine situations in order to achieve their potential as constructive and reflective citizens. LeMaPS: Lessons for Mathematical Problem Solving (University of Nottingham School of Education; Centre for Research in Mathematics Education. Principal Investigators: Geoff Wake; CoInvestigators: Malcolm Swan, Colin Foster). “This Nuffield funded project seeks proof-of-concept of new and sustainable models of partnerships that support professional learning in secondary school mathematics with the involvement of Higher Education. The focus is on improving students’ problemsolving capabilities in mathematics. “The project will build on the outcomes of a Bowland Maths funded pilot that explored the use of Japanese lesson study principles to consider the teaching of mathematical problem solving.” MEI and FMSP Problem Solving Resources Links GCSE Problem Solving Resources Problem Solving Resources Problem Solving CPD for teaching problem solving in GCSE Maths or in the Sixth Form STEP/AEA/MAT support - The FMSP will be running several national programmes of CPD to support teachers helping students to develop problem solving skills and to prepare for examinations, also Student and Teacher Problem Solving Conferences where students and teachers will look at various aspects of problem solving skills. STEP/AEA/MAT Year 12 Problem Solving Summer School - A series of five live online workshops to help year 12 students develop their problem solving skills in pure mathematics. MEI curriculum development, resources and professional development Links to web pages about problem solving in mathematics education, with links to publications and resources. delivered by Stella Dudzic: IQM: Modelling and estimation IQM: Probability and risk IQM: Financial problem solving IQM: Statistical problem solving Realistic Mathematics Education (RME) London Schools’ Excellence Fund CPD: Integrating Mathematical Problem Solving (IMPs) Introducing problem-solving into the Key Stage 4 curriculum Integrating Mathematical Problem Solving (IMPS) resources - Free of charge, designed to help teachers of mathematics and teachers of other subjects at A level to teach relevant aspects of mathematics and statistics, showing how they are used in solving real problems. Critical Maths: a mathematics-based thinking curriculum for Level 3 Critical Maths resources - Designed for post-16 students at level 3; especially useful for Core Maths classes. The resources enable students to think about real problems using mathematics. Many start by engaging the students in giving an initial opinion and then encourage them to think more deeply and to evaluate their initial thoughts. Quantitative Methods OCR Level 3 Certificate in Quantitative Methods resources - Can be subscribed to free of charge by centres, thanks to sponsorship from OCR. Teaching Introduction to Quantitative Methods CPD A series of MEI Conference 2014 session Developing mathematical thinking in post 16 students LSEF Problem Solving Conference Resources - Resources from the first London Schools’ Excellence Fund Mathematical Problem Solving Conference that took place on Wednesday 9th July 2014 at Birkbeck, University of London. STEP and AEA support - MEI provides real-time online tutorials and teaching sessions in STEP and AEA Mathematics. Students can access live interactive tuition at a time and location to suit them through an online learning platform. Problem Solving and STEP – An MEI Conference 2014 session delivered by FMSP Area Coordinators Martin Bamber & Abi Bown The Further Mathematics Support Programme (FMSP) supports the development of problem solving in mathematics at both GCSE and A level with professional development courses for teachers, events and activities for students and resources for use in schools and colleges. See left column for links. Spot the Pattern On the next slide is a grid and on each subsequent slide there are 4 pieces of information. Can you work out how the grid should be coloured in? Spot the Pattern There are 4 red squares (arranged in a square) in the middle of the design. There are 7 red squares in the bottom right hand quarter of the design. There is one square of each colour in the top row of the design (the rest are blank). No blue square is directly next to a yellow square. Spot the Pattern There are 9 blank squares (arranged in a square) in the bottom right hand corner of the design. There is one square of each colour in the first column of the design (the rest are blank). The top left corner to the bottom right corner is a line of reflection symmetry. The blue square in the top row has two blank squares between it and the yellow square in the top right hand corner. Spot the Pattern The yellow squares are only on the top right to bottom left diagonal. There are 5 more red squares than blue squares. The design has one line of reflection symmetry. There are 6 blue squares in the top left hand quarter of the design Spot the Pattern There are 6 blue squares in the top left hand quarter of the design There are 13 red squares in the design. Some of the squares are not coloured in. The design does not have rotation symmetry. Spot the Pattern The top left corner to the bottom right corner diagonal has 5 red squares on it (the rest are blank) The design uses 3 different colours. The top right corner to bottom left corner diagonal has red and yellow squares only in the ratio 1:3. The ratio of yellow squares to blue squares is 3:4. Algebra with cards and paper 7cm b 7cm 5 cm a Three rectangular business cards are shown. What is the perimeter of each? a Algebra with cards and paper 7cm 5 cm 5 cm+7cm+5 cm+7cm = 24 cm a a cm+7cm+a cm+7cm = 14+2a cm 7cm b a a cm+ b cm+ a cm+ b cm=2a + 2b cm Algebra with cards and paper Using this business card, three arrangements of 2 cards are shown below. What is the perimeter of each? Algebra with cards and paper 2a+4b 4a+2b 2a+4b Which is smaller: 2a +4b or 4a+2b? Algebra with cards and paper Putting 2 cards together ‘edge to edge’, what other perimeters can you find? How might you write an expression for the perimeter of these arrangements? Algebra with cards and paper Can you describe how to arrange the two cards to obtain: • the maximum perimeter? • the minimum perimeter? Can you explain how you know these are the maximum and minimum values? How many different arrangements can you find for 3 cards? Algebra with cards and paper Explore the maximum and minimum perimeters for: • 3 cards • 4 cards • 5 cards • … • n cards Can you come up with general algebraic expressions for the maximum and minimum perimeters for n cards? Algebra with cards and paper Does using a different sized rectangular card affect the arrangements that give the minimum perimeter? Does it affect the algebraic value of the minimum perimeter? Perimeter of Rectangular Rings Putting 4 cards together, it is possible to make a ring as shown. Write down an expression for: • the perimeter of the outer rectangle of the ring; • the perimeter of the inner rectangle of the ring; • the total perimeter. Can you simplify your expressions? Perimeter of Rectangular Rings outer rectangle of the ring: 4a + 4b inner rectangle of the ring: 4b – 4a the total perimeter: 8b Perimeter of Rectangular Rings Putting 6 cards together there are 2 possible rings. Find them and write expressions for: • the perimeter of the outer rectangle of the ring; • the perimeter of the inner rectangle of the ring; • the total perimeter. Simplify the expressions where possible Perimeter of Rectangular Rings outer rectangle of the ring: 4a + 6b inner rectangle of the ring: 6b – 4a the total perimeter: 12b Perimeter of Rectangular Rings outer rectangle of the ring: 4a + 6b inner rectangle of the ring: 6b – 4a the total perimeter: 12b Perimeter of Rectangular Rings Explore for different numbers of cards. What do you notice each time? Can you explain why? Monty Hall Problem Monty Hall was a U.S. game show host in the 1970s. His show provides us with a probability problem. Contestants on the show would either win a car… …or a goat. Monty Hall Problem Monty presents the contestant with a choice of 3 doors. Behind one of them is a car, behind the other two are goats. Green! Monty Hall Problem Having chosen a door, Monty shows her what is behind one of the other doors – he knows where the car is and always shows her a goat. Monty Hall Problem He now asks her whether she wants to stick with the green door or switch to the pink one. Monty Hall Problem Should she stick with the door she chose first or switch? What’s your initial instinct? Try it out several times with a partner to see what happens. Do you win more times if you stick or switch? Monty Hall Problem Let’s look at the problem mathematically. Supposing the car is behind the green door. Fill in the table on the following slide and decide whether it’s generally better to stick or switch. Monty Hall Problem Door chosen by you: Behind it is a.. Door Monty would then show you: Stick, and Switch and you win you win a… a… Monty Hall Problem Door chosen by you: Behind it is a.. Door Monty would then show you: or Stick, and you win a… Switch and you win a… Monty Hall Problem So if you switch, you can expect to win a car 2 out of 3 times, whereas if you stick you would only win the car 1 out of 3 times. This problem is famous for puzzling mathematicians during the last century and illustrates that although probability questions can seem confusing and even counter-intuitive sometimes, using a logical approach helps to unravel them. Teacher notes In this edition, 4 short activities from the MEI conference are used. All four activities could be used with a wide range of students, although they are in approximate order of ‘age appropriateness’. The full session PowerPoint and materials can be downloaded from the conference page ‘Spot the Pattern’ problem by Phil Chaffé was in Session B ‘Algebra with cards and paper’ by Kevin Lord was in Session C ‘Perimeter of Rectangular Rings’ is from the same session. ‘Introduction to Probability (S1)’ by Clare Parsons was in session I Teacher notes: Problem Solving, Phil Chaffé During his session, Phil led teachers to consider what is meant by ‘problem solving’, why it is important, how we might enable students to improve their problem solving skills and he also looked at a range of resources and their sources. One of his problems is presented here which requires students to discern a pattern detailed by a series of snippets of information. There are 20 information cards and a blank grid for students to work on, which also appear on the PPT slides. This activity can be tackled in small groups with rules for collaborative work imposed to prevent some students from dominating and others from not participating, or it could be tackled in pairs or individually. Teacher notes: Problem Solving, Phil Chaffé Either present the whole class with the slides one at a time so that they have 4 bits of information to work with at a time (some to-ing and fro-ing may be necessary to check wording) or print the slides for groups to use. Extension ideas: • Ask if there are any pieces of information that are unnecessary • Ask students to come up with their own designs (on a 4x4 grid, perhaps) and describe them with only 8 pieces of information. Teacher notes: Algebra with cards and paper, Kevin Lord Kevin’s session began very simply and built up to more complex use of algebra. The activities can be used to support students with reasoning, justification and proof as well as use of algebra. Just two of his activities are used here; several more are available on the conference page. His activities all used business cards as a starting point, but identical rectangles of paper or card would work just as effectively. The first activity looks at the maximum and minimum areas for two or more cards put together, the second at perimeters of rings of cards. Teacher notes: Algebra with cards and paper, Kevin Lord Slide 13 Since b is the long edge, 4a + 2b will be smaller in value than 2a + 4b Slide 14 The maximum perimeter will be when the two cards are almost separate (as shown) so the limit is 4a+4b The minimum occurs when the cards have the long edges together: Teacher notes: Algebra with cards and paper, Kevin Lord Slide 16 When working with n cards, the maximum perimeter will be n(2a + 2b) ‘Taking out’ the longest edges by putting them together will always minimise the perimeter, reducing n(2a + 2b) by 2b each time, thus the minimum perimeter is n(2a + 2b) – (n-1)2b = 2na + 2b Explanation: Placing one card initially, the perimeter is 2a + 2b; another card makes the perimeter 2(2a+2b), but by placing a long edge of the second card against the first one this is reduced by a maximum of 2b, there being a ‘b’ ‘taken out’ on each side of the join. Adding a third card, the maximum perimeter is 3(2a +2b),this is reduced by a maximum of 2(2b) by ensuring that joins are made at 2 long edges. Teacher notes: Algebra with cards and paper, Kevin Lord Slide 17 The ratio of the sides of the rectangle affects the arrangements that are possible to obtain the minimum perimeter, but the algebraic value of the minimum perimeter is unchanged; being 6a+2b for 3 cards . Similar arrangements for 3 different rectangles are shown. Arrangement for minimum perimeter Arrangement for minimum perimeter Not an arrangement for minimum perimeter Teacher notes: Perimeter of Rectangular Rings, Kevin Lord Slides 19-24 When creating a ring of cards, if they are always placed so that the long edge is on the outside then the following occurs: Length a from 4 of the rectangles on the outside at the ‘corners’. Length b from each rectangle on the outside. Total outside perimeter is 4a + nb Teacher notes: Perimeter of Rectangular Rings, Kevin Lord Slides 19-24 When creating a ring of cards, if they are always placed so that the long edge is on the outside then the following occurs: Length b from n-4 of the rectangles on the inside’. Length b-a from 4 rectangles on the inside. Total inside perimeter is (n-4)b + 4(b-a) = nb - 4a Teacher notes: Perimeter of Rectangular Rings, Kevin Lord Slides 19-24 When creating a ring of cards, if they are always placed so that the long edge is on the outside then the following occurs: Total perimeter is inside + outside: nb - 4a + 4a + nb =2nb Teacher notes: Perimeter of Rectangular Rings, Kevin Lord Slides 19-24 When creating a ring of cards, if they are always placed so that the long edge is on the outside then the following occurs: An alternative way of getting to this result is to consider the perimeter of all the cards used: 2n(a+b) = 2na + 2nb At each ‘join’, 2a is lost from the perimeter. There are n such joins. Perimeter = 2na + 2nb – 2na Perimeter = 2nb Teacher notes: Introducing Probability (S1), Clare Parsons To begin her session, Clare used a short activity which highlighted the challenge that understanding probability presents. Not wishing to spoil the session for her with future TAM teachers, I have instead used the ‘Monty Hall’ problem, of which it was reminiscent. During the session Clare used several approaches to teaching probability which made solving problems much more straight-forward, retaining understanding and insight whilst giving a very helpful structure.