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Problem-solving in mathematics (Book A) Published by R.I.C. Publications® 2008 Republished under licence by Prim-Ed Publishing 2009 Copyright© George Booker and Denise Bond 2007 ISBN 978-1-84654-182-7 PR–6030

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Copyright Notice Blackline masters or copy masters are published and sold with a limited copyright. This copyright allows publishers to provide teachers and schools with a wide range of learning activities without copyright being breached. This limited copyright allows the purchaser to make sufficient copies for use within their own education institution. The copyright is not transferable, nor can it be onsold. Following these instructions is not essential but will ensure that you, as the purchaser, have evidence of legal ownership to the copyright if inspection occurs.

Titles available in this series:

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Problem-solving in mathematics (Book A) Problem-solving in mathematics (Book B) Problem-solving in mathematics (Book C) Problem-solving in mathematics (Book D) Problem-solving in mathematics (Book E) Problem-solving in mathematics (Book F) Problem-solving in mathematics (Book G)

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FOREWORD

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Books A–G of Problem-solving in mathematics have been developed to provide a rich resource for teachers of pupils from the early years to the end of primary school and into secondary school. The series of problems, discussions of ways to understand what is being asked and means of obtaining solutions have been built up to improve the problem-solving performance and persistence of all pupils. It is a fundamental belief of the authors that it is critical that pupils and teachers engage with a few complex problems over an extended period rather than spend a short time on many straightforward ‘problems’ or exercises. In particular, it is essential to allow pupils time to review and discuss what is required in the problem-solving process before moving to another and different problem. This book includes extensive ideas for extending problems and solution strategies to assist teachers in implementing this vital aspect of mathematics in their classrooms. Also, the problems have been constructed and selected over many years’ experience with pupils at all levels of mathematical talent and persistence, as well as in discussions with teachers in classrooms and professional learning and university settings.

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Problem-solving does not come easily to most people, so learners need many experiences engaging with problems if they are to develop this crucial ability. As they grapple with problem meaning and find solutions, pupils will learn a great deal about mathematics and mathematical reasoning—for instance, how to organise information to uncover meanings and allow connections among the various facets of a problem to become more apparent, leading to a focus on organising what needs to be done rather than simply looking to apply one or more strategies. In turn, this extended thinking will help pupils make informed choices about events that affect their lives and to interpret and respond to the decisions made by others at school, in everyday life and in further study.

Pupil and teacher pages

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The pupil pages present problems chosen with a particular problem-solving focus and draw on a range of mathematical understandings and processes. For each set of related problems, teacher notes and discussion are provided, as well as indications of how particular problems can be examined and solved. Answers to the more straightforward problems and detailed solutions to the more complex problems

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ensure appropriate explanations, and suggest ways in which problems can be extended. Related problems occur on one or more pages that extend the problem’s ideas, the solution processes and pupils’ understanding of the range of ways to come to terms with what the problems are asking. At the top of each teacher page, a statement highlights the particular thinking that the problems will demand, together with an indication of the mathematics that might be needed and a list of materials that can be used in seeking a solution. A particular focus for the page or set of three pages of problems then expands on these aspects. Each book is organised so that when a problem requires complicated strategic thinking, two or three problems occur on one page (supported by a teacher page with detailed discussion) to encourage pupils to find a solution together with a range of means that can be followed. More often, problems are grouped as a series of three interrelated pages where the level of complexity gradually increases, while the associated teacher page examines one or two of the problems in depth and highlights how the other problems might be solved in a similar manner.

Problem-solving in mathematics

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FOREWORD

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the various year levels, although problem-solving both challenges at the point of the mathematics that is being learned and provides insights and motivation for what might be learned next. For example, the computation required gradually builds from additive thinking, using addition and subtraction separately and together, to multiplicative thinking, where multiplication and division are connected conceptions. More complex interactions of these operations build up over the series as the operations are used to both come to terms with problems’ meanings and to achieve solutions. Similarly, two-dimensional geometry is used at first but extended to more complex uses over the range of problems, then joined by interaction with threedimensional ideas. Measurement, including chance and data, also extends over the series from length to perimeter, and from area to surface area and volume, drawing on the relationships among these concepts to organise solutions as well as give an understanding of the metric system. Time concepts range from interpreting timetables using 12-hour and 24-hour clocks, while investigations related to mass rely on both the concept itself and practical measurements.

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Each teacher page concludes with two further aspects critical to the successful teaching of problem-solving. A section on likely difficulties points to reasoning and content inadequacies that experience has shown may well impede pupils’ success. In this way, teachers can be on the lookout for difficulties and be prepared to guide pupils past these potential pitfalls. The final section suggests extensions to the problems to enable teachers to provide several related experiences with problems of these kinds in order to build a rich array of experiences with particular solution methods; for example, the numbers, shapes or measurements in the original problems might change but leave the means to a solution essentially the same, or the context may change while the numbers, shapes or measurements remain the same. Then numbers, shapes or measurements and the context could be changed to see how the pupils handle situations that appear different but are essentially the same as those already met and solved.

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Other suggestions ask pupils to make and pose their own problems, investigate and present background to the problems or topics to the class, or consider solutions at a more general level (possibly involving verbal descriptions and eventually pictorial or symbolic arguments). In this way, not only are pupils’ ways of thinking extended but the problems written on one page are used to produce several more problems that utilise the same approach.

Mathematics and language The difficulty of the mathematics gradually increases over the series, largely in line with what is taught at

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Problem-solving in mathematics

The language in which the problems are expressed is relatively straightforward, although this too increases in complexity and length of expression across the books in terms of both the context in which the problems are set and the mathematical content that is required. It will always be a challenge for some pupils to ‘unpack’ the meaning from a worded problem, particularly as the problems’ context, information and meanings expand. This ability is fundamental to the nature of mathematical problem-solving and needs to be built up with time and experiences rather than be

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FOREWORD

Analyse

Try

the problem

an approach

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Explore

means to a solution

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The careful, gradual development of an ability to analyse problems for meaning, organising information to make it meaningful and to make the connections among the problems more meaningful in order to suggest a way forward to a solution is fundamental to the approach taken with this series, from the first book to the last. At first, materials are used explicitly to aid these meanings and connections; however, in time they give way to diagrams, tables and symbols as understanding and experience of solving complex, engaging problems increases. As the problem forms expand, the range of methods to solve problems is carefully extended, not only to allow pupils to

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Not only is this model for the problem-solving process helpful in solving problems, it also provides a basis for pupils to discuss their progress and solutions and determine whether or not they have fully answered a question. At the same time, it guides teachers’ questions of pupils and provides a means of seeing underlying mathematical difficulties and ways in which problems can be adapted to suit particular needs and extensions. Above all, it provides a common framework for discussions between a teacher and group or whole class to focus on the problem-solving process rather than simply on the solution of particular problems. Indeed, as Alan Schoenfeld, in Steen L. (Ed) Mathematics and democracy (2001), states so well, in problem-solving:

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An approach to solving problems

successfully solve the many types of problems, but also to give them a repertoire of solution processes that they can consider and draw on when new situations are encountered. In turn, this allows them to explore one or another of these approaches to see whether each might furnish a likely result. In this way, when they try a particular method to solve a new problem, experience and analysis of the particular situation assists them in developing a full solution.

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diminished or left out of the problems’ situations. One reason for the suggestion that pupils work in groups is to allow them to share and assist each other with the tasks of discerning meanings and ways to tackle the ideas in complex problems through discussion, rather than simply leaping into the first ideas that come to mind (leaving the full extent of the problem unrealised).

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getting the answer is only the beginning rather than the end … an ability to communicate thinking is equally important. We wish all teachers and pupils who use these books success in fostering engagement with problem-solving and building a greater capacity to come to terms with and solve mathematical problems at all levels.

George Booker and Denise Bond

Problem-solving in mathematics

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CONTENTS Ordinal patterns ............................................................ 30

Contents .......................................................................... vi

Car race ......................................................................... 31

Introduction ........................................................... vii – xix

Teacher notes . .............................................................. 32

Teacher notes.................................................................... 2

Making sandwiches ...................................................... 33

Block time ....................................................................... 3

Sandwich choices ......................................................... 34

Block patterns ................................................................. 4

More choices ................................................................ 35

More blocks .................................................................... 5

Teacher notes . .............................................................. 36

Teacher notes . ................................................................ 6

At the toyshop . ............................................................. 37

Block Street . ................................................................... 7

Teacher notes . .............................................................. 38

Teacher notes . ................................................................ 8

Ice-creams 1 . ................................................................ 39

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Foreword .................................................................. iii – v

Ice-creams 2 . ................................................................ 40

How many? 2 ................................................................ 10

Chocolates .................................................................... 41

Animal pets ................................................................... 11

Teacher notes . .............................................................. 42

Teacher notes . .............................................................. 12

Hanging out the washing................................................ 43

Animal antics ................................................................ 13

Washing day ................................................................. 44

Sorting clothes .............................................................. 14

Dry-cleaning ................................................................... 45

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How many? 1 .................................................................. 9

Teacher notes . .............................................................. 46

Teacher notes . .............................................................. 16

Using squares ............................................................... 47

What’s my number? ...................................................... 17

Using 5 squares ............................................................ 48

Teacher notes . .............................................................. 18

Growing shapes ............................................................ 49

Missing numbers .......................................................... 19

Teacher notes . .............................................................. 50

Number sense ............................................................... 20

How long? ..................................................................... 51

Number stories ............................................................. 21

Solutions ................................................................. 52–54

Teacher notes . .............................................................. 22

Tangram resource page ................................................ 55

Number problems 1 ...................................................... 23

10 mm x 10 mm grid resource page ............................. 56

Number problems 2 ...................................................... 24

15 mm x 15 mm grid resource page ............................. 57

Number problems 3 ...................................................... 25

Triangular grid resource page ....................................... 58

Teacher notes . .............................................................. 26

Triangular isometric resource page .............................. 59

Toy animals ................................................................... 27

Square isometric resource page ................................... 60

Teacher notes . .............................................................. 28

4-digit number expander resource page (x 5) ................ 61

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Clown hats .................................................................... 15

Patterns ......................................................................... 29

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Problem-solving in mathematics

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INTRODUCTION

NCTM-Principles and standards for school mathematics (2000, p. 52)

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Problem-solving lies at the heart of mathematics. New mathematical concepts and processes have always grown out of problem situations and pupils’ problem-solving capabilities develop from the very beginning of mathematics learning. A need to solve a problem can motivate pupils to acquire new ways of thinking as well as come to terms with concepts and processes that might not have been adequately learned when first introduced. Even those who can calculate efficiently and accurately are ill prepared for a world where new and adaptable ways of thinking are essential if they are unable to identify which information or processes are needed.

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By learning problem-solving in mathematics, pupils should acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that will serve them well outside the mathematics classroom. In everyday life and in the workplace, being a good problem solver can lead to great advantages.

Well-chosen problems encourage deeper exploration of mathematical ideas, build persistence and highlight the need to understand thinking strategies, properties and relationships. They also reveal the central role of sense making in mathematical thinking—not only to evaluate the need for assessing the reasonableness of an answer or solution, but also the need to consider the interrelationships among the information provided with a problem situation. This may take the form of number sense, allowing numbers to be represented in various ways and operations to be interconnected; through spatial sense that allows the visualisation of a problem in both its parts and whole; to a sense of measurement across length, area, volume and chance and data.

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Problem-solving and mathematical thinking

A problem is a task or situation for which there is no immediate or obvious solution, so that problemsolving refers to the processes used when engaging with this task. When problem-solving, pupils engage with situations for which a solution strategy is not immediately obvious, drawing on their understanding of concepts and processes they have already met, and will often develop new understandings and ways of thinking as they move towards a solution. It follows that a task that is a problem for one pupil may not be a problem for another and that a situation that is a problem at one level will only be an exercise or routine application of a known means to a solution at a later time.

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On the other hand, pupils who can analyse the meaning of problems, explore means to a solution and carry out a plan to solve mathematical problems have acquired deeper and more useful knowledge than simply being able to complete calculations, name shapes, use formulas to make measurements or determine measures of chance and data. It is critical that mathematics teaching focuses on enabling all pupils to become both able and willing to engage with and solve mathematical problems.

Problem-solving

For a pupil aged 5–6 years, sorting out the information about being on the lily pad and being in the water may take some consideration and require counters to represent the numbers and find the answer. For Prim-Ed Publishing®

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Problem-solving in mathematics

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INTRODUCTION

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However, many pupils feel inadequate when they encounter problem-solving questions. They seem to have no idea of how to go about finding a solution and are unable to draw on the competencies they have learned in number, space and measurement. Often these difficulties stem from underdeveloped concepts for the operations, spatial thinking and measurement processes. They may also involve an underdeveloped capacity to read problems for meaning and a tendency to be led astray by the wording or numbers in a problem situation. Their approach may then simply be to try a series of guesses or calculations rather than consider using a diagram or materials to come to terms with what the problem is asking and using a systematic approach to organise the information given and required in the task. It is this ability to analyse problems that is the key to problem-solving, enabling decisions to be made about which mathematical processes to use, which information is needed and which ways of proceeding are likely to lead to a solution.

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As the world in which we live becomes ever more complex, the level of mathematical thinking and problem-solving needed in life and in the workplace has increased considerably. Those who understand and can use the mathematics they have learned will have opportunities opened to them that those who do not develop these ways of thinking will not. To enable pupils to thrive in this changing world, attitudes and ways of knowing that enable them to deal with new or unfamiliar tasks are now as essential as the procedures that have always been used to handle familiar operations readily and efficiently. Such an attitude needs to develop from the beginning of mathematics learning as pupils form beliefs about meaning, the notion of taking control over the activities they engage with and the results they obtain, and as they build an inclination to try different approaches. In other words, pupils need to see mathematics as a way of thinking rather than a means of providing answers to be judged right or wrong by a teacher, textbook or some other external authority. They must be led to focus on means of solving problems rather than on particular answers so that they understand the need to determine the meaning of a problem before beginning to work on a solution.

of him. When another 6 cars passed him, there were now 9 ahead of him. If he is to win, he needs to pass all 9 cars. The 4 and 6 implied in the problem were not used at all! Rather, a diagram or the use of materials is needed first to interpret the situation and then see how a solution can be obtained.

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children in the middle primary years, understanding of the addition concept and knowledge of the addition facts would lead them immediately to think about the sum of 3 and 4 and come up with the solution of 7 frogs.

In a car race, Jordan started in fourth place. During the race, he was passed by six cars. How many cars does he need to pass to win the race?

In order to solve this problem, it is not enough to simply use the numbers that are given. Rather, an analysis of the race situation is needed first to see that when Jordan started, there were 3 cars ahead

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Problem-solving in mathematics

Making sense in mathematics Making sense of the mathematics being developed and used must be seen as the central concern of learning. This is important, not only in coming to terms with problems and means to solutions, but also in terms of bringing meaning, representations and relationships in mathematical ideas to the forefront of thinking about and dealing with mathematics. Making sensible interpretations of any results and determining which of several possibilities is more or equally likely is critical in problem-solving. Number sense, which involves being able to work with numbers comfortably and competently, is important in many aspects of problem-solving, in making judgments, interpreting information and communicating ways of thinking. It is based on a full understanding of numeration concepts such

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INTRODUCTION

Number sense requires: • understanding relationships among numbers • a capacity to calculate and estimate mentally

• fluent processes for larger numbers and adaptive use of calculators

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• an inclination to use understanding and facility with numeration and computation in flexible ways.

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The following problem highlights the importance of these understandings.

There were 317 people at the New Year’s Eve party on 31 December. If each table could seat 5 couples, how many tables were needed?

Reading the problem carefully shows that each table seats five couples or 10 people. At first glance, this problem might be solved using division; however, this would result in a decimal fraction, which is not useful in dealing with people seated at tables: 10 317 is 31.7 Prim-Ed Publishing®

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3 1 7

3 1

tens

7

ones

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This provides for all the people at the party and analysis of the number 317 shows that there have to be at least 32 tables for everyone to have a seat and allow partygoers to move around and sit with others during the evening. Understanding how to rename a number has provided a direct solution without any need for computation. It highlights how coming to terms with a problem and integrating this with number sense provides a means of solving the problem more directly and allows an appreciation of what the solution might mean.

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• appreciating the relative size of numbers

In contrast, a full understanding of numbers allows 317 to be renamed as 31 tens and 7 ones:

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as zero, place value and the renaming of numbers in equivalent forms, so that 207 can be seen as 20 tens and 7 ones as well as 2 hundreds and 7 ones (or that 52, 2.5 and 2 12 are all names for the same fraction amount). Automatic, accurate access to basic facts also underpins number sense, not as an end in itself, but rather as a means of combining with numeration concepts to allow manageable mental strategies and fluent processes for larger numbers. Well-understood concepts for the operations are essential in allowing relationships within a problem to be revealed and taken into account when framing a solution.

Spatial sense is equally important, as information is frequently presented in visual formats that need to be interpreted and processed, while the use of diagrams is often essential in developing conceptual understanding across all aspects of mathematics. Using diagrams, placing information in tables or depicting a systematic way of dealing with the various possibilities in a problem assist in visualising what is happening. It can be a very powerful tool in coming to terms with the information in a problem, and it provides insight into ways to proceed to a solution. Spatial sense involves: • a capacity to visualise shapes and their properties • determining relationships among shapes and their properties • linking two-dimensional and threedimensional representations • presenting and interpreting information in tables and lists • an inclination to use diagrams and models to visualise problem situations and applications in flexible ways. The following problem shows how these understandings can be used. Problem-solving in mathematics

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INTRODUCTION

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Measurement sense is dependent on both number sense and spatial sense, since attributes that are one‑, two- or three-dimensional are quantified to provide both exact and approximate measures and allow comparison. Many measurements use aspects of space (length, area, volume), while others use numbers on a scale (time, mass, temperature). Money can be viewed as a measure of value and uses numbers more directly, while practical activities such as map reading and determining angles require a sense of direction as well as gauging measurement. The coordination of the thinking for number and space, along with an understanding of how the metric system builds on place value, zero and renaming, is critical in both building measurement understanding and using it to come to terms with and solve many practical problems and applications.

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Reading the problem carefully shows that only two spaces in the box can be used each time and that no use of the spaces can be duplicated. A systematic approach, placing one chocolate in a fixed position and varying the other spaces will provide a solution; however, care will be needed to see that the same placement has not already occurred:

How many cubes are needed to make this shape?

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Cathy has 2 chocolates and 1 box. In how many different ways can she place the chocolates in the box?

There are six possible arrangements. The placement of objects on the diagram has provided a solution, highlighting how coming to terms with a problem and integrating this with spatial sense allows a systematic analysis of all the possibilities.

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Similar thinking is used with arrangements of twodimensional and three-dimensional shapes and in visualising how they can fit together or be taken apart.

Measurement sense includes: • understanding how numeration and computation underpin measurement • extending relationships from number understanding to the metric system • appreciating the relative size of measurements • a capacity to use calculators, mental or written processes for exact and approximate calculations • an inclination to use understanding and facility with measurements in flexible ways. The following problem shows how these understandings can be used.

Which of these shapes can be made using all of the tangram pieces?

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Problem-solving in mathematics

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INTRODUCTION Data sense involves:

• a capacity to use calculators or mental and written processes for exact and approximate calculations • presenting and interpreting data in tables and graphs • an inclination to use understanding and facility with number combinations and arrangements in flexible ways.

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Carefully reading the problem shows that the snail will travel 45 cm as it moves along each side of the square. In order to come to terms with what is needed, 3 m 15 cm needs to be renamed as 315 cm. The distances the snail travels along each side can then be totalled until 315 cm is reached. It can also be inferred that it will travel along some sides more than once as the distance around the outside of the square is 180 cm. At this point, the snail will be back at A. Travelling a further 45 cm will take it to B, a distance of 225 cm. At C it will have travelled 270 cm and it will have travelled 315 cm (or 3 m 15 cm) when it reaches D for the second time.

• appreciating the relative likelihood of outcomes

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At which corner will it stop?

• understanding how numeration and computation underpin the analysis of data

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A snail crawls 3 m 15 cm around a square garden.

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By using an understanding of the problem situation, a diagram has been integrated with a knowledge of metres and centimetres and a capacity to calculate mentally using addition and multiplication to provide an appropriate solution. Both spatial sense and number sense have been used to understand the problem and suggest a means to a solution.

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Data sense is an outgrowth of measurement sense and refers to an understanding of the way number sense, spatial sense and a sense of measurement work together to deal with situations where patterns need to be discerned among data or when likely outcomes need to be analysed. This can occur among frequencies in data or possibilities in chance.

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The following problem shows how these understandings can be used. You are allowed 3 scoops of ice-cream: 1 chocolate, 1 vanilla and 1 strawberry. How many different ways can the scoops be placed on a cone?

There are six possibilities for placing the scoops of icecream on a cone. Systematically treating the possible placements one at a time highlights how the use of a diagram can account for all possible arrangements. This problem also shows how patterning is another aspect of sense-making in mathematics. Often a problem calls on discerning a pattern in the placement of materials, the numbers involved in the situation or the possible arrangements of data or outcomes to determine a likely solution. Being able to see patterns is also very helpful in getting an immediate solution or understanding whether or not a solution is complete. Allied to patterning are notions of symmetry, repetition and extending ideas to more general cases. All of these aspects of mathematical sense-making are critical to developing the thinking on which problemsolving depends, as well as solving problems per se.

Problem-solving in mathematics

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INTRODUCTION As more experience in solving problems is gained, an ability to see patterns in what is occurring will also allow solutions to be obtained more directly and help in seeing the relationship between a new problem and one that has been solved previously. It is this ability to relate problem types, even when the context appears to be quite different, that often distinguishes a good problem solver from one who is more hesitant.

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On Saturday, Peta went to the shopping centre to buy a new outfit to wear at her friend’s birthday party. She spent half of her money on a dress and then one-third of what she had left on a pair of sandals. After her purchases, she had £60.00 left in her purse. How much money did she have to start with?

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While the teaching of problem-solving has often centred on the use of particular strategies that could apply to various classes of problems, many pupils are unable to access and use these strategies to solve problems outside of the teaching situations in which they were introduced. Rather than acquire a process for solving problems, they may attempt to memorise a set of procedures and view mathematics as a set of learned rules where success follows the use of the right procedure to the numbers given in the problem. Any use of strategies may be based on familiarity, personal preference or recent exposure rather than through a consideration of the problem to be solved. A pupil may even feel it is sufficient to have only one strategy and that the strategy should work all of the time—and if it doesn’t, then the problem can’t be solved.

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Building a problem-solving process

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By reading the problem carefully, it can be determined that Peta had an original amount of money to spend. She spent some on a dress and some on shoes and then had £60.00 left. All of the information required to solve the problem is available and no further information is needed. The question at the end asks how much money did she start with, but really the problem is how much did she spend on the dress and then on the sandals.

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In contrast, observation of successful problem-solvers shows that their success depends more on an analysis of the problem itself—what is being asked, what information might be used, what answer might be likely and so on—so that a particular approach is used only after the intent of the problem is determined. Establishing the meaning of the problem before any plan is drawn up or work on a solution begins is critical. Pupils need to see that discussion about the problem’s meaning, and the ways of obtaining a solution, must take precedence over a focus on the answer. Using collaborative groups when problem-solving, rather than tasks assigned individually, is an approach that helps to develop this disposition. Looking at a problem and working through what is needed to solve it will shed light on the problemsolving process.

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Problem-solving in mathematics

The discussion of this problem has served to identify the key element within the problem-solving process; it is necessary to analyse the problem to unfold its meanings and discover what needs to be considered. What the problem is asking is rarely found in the question in the problem statement. Instead, it is necessary to look below the surface level of the problem and come to terms with the problem’s structure. Reading the problem aloud, thinking of previous problems and other similar problems, selecting important information from the problem that may be useful, and discussion of the problem’s meaning are all essential. The next step is to explore possible ways to solve the problem. If the analysis stage has been completed,

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INTRODUCTION

Ways that may come to mind during the analysis include:

She spent half of her money on a dress.

She then spent one-third of what she had left on sandals, which has minimised and simplified the calculations.

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• Materials – Base 10 materials could be used to represent the money spent and to help the pupil work backwards through the problem from when Peta had £60.00 left.

Total amount available to spend:

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It is here that strategies, and how they might be useful to solving a problem, can arise. However, most problems can be solved in a variety of ways, using different approaches, and pupils need to be encouraged to select a method that makes sense and appears achievable.

Another way to solve the problem is with a diagram. If we use a rectangle to represent how much money Peta took with her, we can show by shading how much she spent on a dress and sandals:

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then ways in which the problem might be solved will emerge.

• Try and adjust – Select an amount that Peta might have taken shopping, try it in the context of the question, examine the resulting amounts, and then adjust them, if necessary, until £60.00 is the result.

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• Backtrack using the numbers – The sandals were one-third of what was left after the dress, so the £60.00 would be two-thirds of what was left. Together, these two amounts would match the cost of the dress.

• Use a diagram to represent the information in the problem. • Think of a similar problem – For example, it is like the car race problem in that the relative portions (places) are known and the final result (money left, winning position) are given.

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Now one of the possible means to a solution can be selected to try. Backtracking shows that £60 was twothirds of what she had left, so the sandals (which are one-third of what she had left) must have cost £30. Together, these are half of what Peta took, which is also the cost of the dress. As the dress cost £90, Peta took £180 to spend. Materials could also have been used with which to work backwards: 6 tens represent the £60 left, so the sandals would cost 3 tens and the dress 9 tens—she took 18 tens or £180 shopping.

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At this point she had £60 left, so the twounshaded parts must be worth £60 or £30 per part—which has again minimised and simplified the calculations.

£30

£30

Each of the six equal parts represents £30, so Peta took £180 to spend. Having tried an idea, an answer needs to be analysed in the light of the problem in case another solution is required. It is essential to compare an answer to the original analysis of the problem to determine whether the solution obtained is reasonable and answers the problem. It will also raise the question as to whether other answers exist and even whether there might be other solution strategies. In this way the process is cyclic and should the answer be unreasonable, then the process would need to begin again. We believe that Peta took £180 to shop with. She spent half (or £90) on a dress, leaving £90. She spent one-third of the £90 on sandals (£30), leaving £60. Looking again at the problem, we see that this is correct and the diagram has provided a direct means

Problem-solving in mathematics

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INTRODUCTION

Analyse the problem

Try a solution strategy

Explore means to a solution

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Managing a problem-solving programme

Teaching problem-solving differs from many other aspects of mathematics in that collaborative work can be more productive than individual work. Pupils who may be tempted to quickly give up when working on their own can be encouraged to see ways of proceeding when discussing a problem in a group; therefore building greater confidence in their capacity to solve problems and learning the value of persisting with a problem in order to tease out what is required. What is discussed with their peers is more likely to be recalled when other problems are met, while the observations made in the group increase the range of approaches that a pupil can access. Thus, time has to be allowed for discussion and exploration rather than insisting that pupils spend ‘time on task’ as for routine activities.

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Thinking about the various ways this problem was solved highlights the key elements within the problem-solving process. When starting the process, it is necessary to analyse the problem to unfold its layers, discover its structure and understand what the problem is really asking. Next, all possible ways to solve the problem are explored before one, or a combination of ways, are selected to try. Finally, once something is tried, it is important to check the solution in relation to the problem to see if the solution is reasonable. This process highlights the cyclic nature of problem-solving and brings to the fore the importance of understanding the problem (and its structure) before proceeding. This process can be summarised as:

solving another problem at a later stage. It allows the thinking to be carried over to the new situation in a way that simply trying to think of the strategy used often fails to reveal. Analysing problems in this way also highlights that a problem is not solved until the answer obtained can be justified. Learning to reflect on the whole process leads to the development of a deeper understanding of problem-solving, and time must be allowed for reflection and discussion to fully build mathematical thinking.

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to the solution that has minimised and simplified the calculations.

A plan to manage problem-solving

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This model for problem-solving provides pupils with a means of talking about the steps they take whenever they have a problem to solve: Discussing how they initially analysed the problem, explored various ways that might provide a solution, and then tried one or more possible solution paths to obtain a solution— which they then analysed for completeness and sense-making—reinforces the very methods that will give them success on future problems. This process brings to the fore the importance of understanding the problem and its structure before proceeding.

Further, returning to an analysis of any answers and solution strategies highlights the importance of reflecting on what has been done. Taking time to reflect on any plans drawn up, processes followed and strategies used brings out the significance of coming to terms with the nature of the problem, as well as the value and applicability of particular approaches that might be used with other problems. Thinking of how a related problem was solved is often the key to

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Problem-solving in mathematics

Correct answers that fully solve a problem are always important, but developing a capacity to use an effective problem-solving process needs to be the highest priority. A pupil who has an answer should be encouraged to discuss his or her solution with others who believe they have a solution, rather than tell his or her answer to another pupil or simply move on to another problem. In particular, explaining to others why he or she believes an answer is reasonable, as well as why it provides a solution, gets other pupils to focus on the entire problemsolving process rather than just quickly getting an answer.

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INTRODUCTION

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too few may cause pupils to become frustrated with the task and think that it is beyond them. Pupils need to experience the challenge of problem-solving and gain pleasure from working through the process that leads to a full solution. Taking time to listen to pupils as they try out their ideas, without comment or without directing them to a particular strategy, is also important. Listening provides a sense of how pupilsâ€™ problem-solving is developing, as assessing this aspect of mathematics can be difficult. After all, solving one problem will not necessarily lead to success on the next problem, nor will difficulty with a particular problem mean that the problems that follow will also be as challenging. A teacher also may need to extend or adapt a given problem to ensure the problem-solving process is understood and can be used in other situations, instead of moving on to a different problem in another area of mathematics learning. This can help pupils to understand the significance of asking questions of a problem, as well as seeing how a way of thinking can be adapted to other related problems. Having pupils engage in this process of problem posing is another way of both assessing them and bringing them to terms with the overall process of solving problems.

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Expressing an answer in a sentence that relates to the question stated in the problem also encourages reflection on what was done and ensures that the focus is on solving the problem rather than providing an answer. These aspects of the teaching of problem-solving should then be taken further, as particular groups discuss their solutions with the whole class and all pupils are able to participate in the discussion of the problem. In this way, problem-solving as a way of thinking comes to the fore, rather than focusing on the answers as the main aim of their mathematical activities.

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Questions must encourage pupils to explore possible means to a solution and try one or more of them, rather than point to a particular procedure. It can also help pupils to see how to progress in their thinking, rather than get into a loop where the same steps are repeated over and over. While having too many questions that focus on the way to a solution may end up removing the problem-solving aspect from the question, having

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Problem-solving in mathematics

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INTRODUCTION Building a problem-solving process The cyclical model Analyse–Explore–Try provides a very helpful means of organising and discussing possible solutions. However, care must be taken that it is not seen simply as a procedure to be memorised and then applied in a routine manner to every new problem. Rather, it needs to be carefully developed over a range of different problems, highlighting the components that are developed with each new problem.

Explore

• When a problem is being explored, some problems will require the use of materials to think through the whole of the problem’s context. Others will demand the use of diagrams to show what is needed. Another will show how systematic analysis of the situation using a sequence of diagrams, on a list or table, is helpful. As these ways of thinking about the problem are understood, they can be included in the cycle of steps.

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• Further reading will be needed to sort out which information is needed and whether some is not needed or if other information needs to be gathered from the problem’s context (e.g. data presented within the illustration or table accompanying the problem), or whether the pupils’ mathematical understandings need to be used to find other relationships among the information. As the form of the problems becomes more complex, this thinking will be extended to incorporate further ways of dealing with the information; for example, measurement units, fractions and larger numbers might need to be renamed to the same mathematical form.

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• As pupils read a problem, the need to first read for the meaning of the problem can be stressed. This may require reading more than once and can be helped by asking pupils to state in their own words what the problem is asking them to do.

• Developing a capacity to see ‘through’ the problem’s expression—or context to see similarities between new problems and others that might already have been met—is a critical way of building expertise in coming to terms with and solving problems.

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Analyse

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Try

• Thinking about any processes that might be needed and the order in which they are used, as well as the type of answer that could occur, should also be developed in the context of new levels of problem structure. Expanding the problem-solving process

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• Put the solution back into the problem. • Does the answer make sense? • Does it solve the problem? • Is it the only answer? • Could there be another way?

• Use materials or a model. • Use a calculator. • Use pencil and paper. • Look for a pattern.

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Analyse the problem

Try a solution strategy

Problem-solving in mathematics

• Many pupils often try to guess a result. This can even be encouraged by talking about ‘guess and check’ as a means to solve problems. Changing to ‘try and adjust’ is more helpful in building a way of thinking and can lead to a very powerful way of finding solutions.

• Read carefully. • What is the problem asking? • What is the meaning of the information? Is it all needed? Is there too little? Too much? • Which operations will be needed and in what order? • What sort of answer is likely? • Have I seen a problem like this before?

Explore means to a solution

• Use a diagram or materials. • Work backwards or backtrack. • Put the information into a table. • Try and adjust.

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INTRODUCTION

• The point in the cycle where an answer is assessed for reasonableness (e.g. whether it provides a solution, is only one of several solutions or whether there may be another way to solve the problem) also needs to be brought to the fore as different problems are met.

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The role of calculators

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When calculators are used, pupils devote less time to basic calculations, providing time that might be needed to either explore a solution or find an answer to a problem. In this way, attention is shifted from computation, which the calculator can do, to thinking about the problem and its solution—work that the calculator cannot do. It also allows more problems (and more realistic problems) to be addressed in problemsolving sessions. In these situations, a calculator serves as a tool rather than a crutch, requiring pupils to think through the problem’s solution in order to know how to use the calculator appropriately. It also underpins the need to make sense of the steps along the way and any answers that result, as keying incorrect numbers, operations or order of operations quickly leads to results that are not appropriate.

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Choosing, adapting and extending problems

When problems are selected, they need to be examined to see if pupils already have an understanding of the underlying mathematics required and that the problem’s expression can be meaningfully read by the group of pupils who will be attempting the solution— though not necessarily by all pupils in the group. The problem itself should be neither too easy (so that it is just an exercise, repeating something readily done before), nor too difficult (thus beyond the capabilities of most or all in the group). A problem should engage

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As a problem and its solution is reviewed, posing similar questions—where the numbers, shapes or measurements are changed—focuses attention back on what was entailed in analysing the problem and in exploring the means to a solution. Extending these processes to more complex situations shows how the particular approach can be extended to other situations and how patterns can be analysed to obtain more general methods or results. It also highlights the importance of a systematic approach when conceiving and discussing a solution and can lead to pupils asking themselves further questions about the situation and pose problems of their own as the significance of the problem’s structure is uncovered.

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Analyse

the interests of the pupils and also be able to be solved in more than one way.

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• When materials, a diagram or table have been used, another means to a solution is to look for a pattern in the results. When these have revealed what is needed to try for a solution, it may also be reasonable to use pencil and paper or a calculator.

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Problem structure and expression When analysing a problem, it is also possible to discern critical aspects of the problem’s form and relate this to an appropriate level of mathematics and problem expression when choosing or extending problems. A problem of first-level complexity uses simple mathematics and simple language. A secondlevel problem may have simple language and more difficult mathematics or more difficult language and simple mathematics, while a third-level problem has yet more difficult language and mathematics. Within a problem, the processes that must be used may be more or less obvious, the information that is required for a solution may be too much or too little, and strategic thinking may be needed in order to come to terms with what the problem is asking. Level

processes obvious

processes less obvious

too much information

too little information

strategic thinking

simple expression, simple mathematics increasing difficulty with problem’s expression and mathematics required

more complex expression, simple mathematics simple expression, more complex mathematics

complex expression, complex mathematics

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INTRODUCTION Assessing problem-solving

(ii) The processes required are not immediately obvious, as these problems contain all the information necessary to find a solution but demand further analysis to sort out what is wanted and pupils may need to reverse what initially seemed to be required.

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(iii) The problem contains more information than is needed for a solution, as these problems contain not only all the information needed to find a solution but also additional information in the form of times, numbers, shapes or measurements.

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(i) The processes to be used are relatively obvious, as these problems are comparatively straightforward and contain all the information necessary to find a solution.

Assessment of problem-solving requires careful and close observation of pupils working in a problemsolving setting. These observations can reveal the range of problem forms and the level of complexity in the expression and underlying mathematics that a pupil is able to confidently deal with. Further analysis of these observations can show to what extent the pupil is able to analyse the question, explore ways to a solution, select one or more methods to try and then analyse any results obtained. It is the combination of two fundamental aspects—the types of problem that can be solved and the manner in which solutions are carried out—that will give a measure of a pupil’s developing problem-solving abilities, rather than a one-off test in which some problems are solved and others are not.

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The varying levels of problem structure and expression

How? • Analyse • Explore • Try

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(iv) Further information must be gathered and applied to the problem in order to obtain a solution. These problems do not contain all the information necessary to find a solution but do contain a means to obtain the required information. The problem’s setting, the pupil’s mathematical understanding or the problem’s wording need to be searched for the additional material.

What? • Problem form • Problem expression Assessment informs: • Mathematics required

(v) Strategic thinking is required to analyse the question in order to determine a solution strategy. Deeper analysis, often aided by the use of diagrams or tables, is needed to come to terms with what the problem is asking so a means to a solution can be determined.

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This analysis of the nature of problems can also serve as a means of evaluating the provision of problems within a mathematics programme. In particular, it can lead to the development of a full range of problems, ensuring they are included across all problem forms, with the mathematics and expression suited to the level of the pupils.

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Observations based on this analysis have led to a categorisation of many of the possible difficulties that pupils experience with problem-solving as a whole, rather than the misconceptions they may have about particular problems.These often involve inappropriate attempts at a solution based on little understanding of the problem. A major cause of possible difficulties is the lack of a well-developed plan of attack, leading pupils to focus on the surface level of problems. In such cases, pupils:

• locate and manipulate numbers with little or no thought about their relevance to the problem • try a succession of different operations if the first ones attempted do not yield a (likely) result • focus on keywords for an indication of what might be done without considering their significance within the problem as a whole • read problems quickly and cursorily to locate the numbers to be used

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INTRODUCTION Problem

Likely causes

Pupil is unable to make any attempt at a solution.

• • • •

Pupil has no means of linking the situation to the implicit mathematical meaning.

• needs to create diagram or use materials • needs to consider separate parts of question and then bring parts together

Pupil uses an inappropriate operation.

• misled by word cues or numbers • has underdeveloped concepts • uses rote procedures rather than real understanding

Pupil is unable to translate a problem into a more familiar process.

• cannot see interactions between operations • lack of understanding means he/she unable to reverse situations • data may need to be used in an order not evident in the problem statement or in an order contrary to that in which it is presented

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not interested feels overwhelmed cannot think of how to start to answer question needs to reconsider complexity of steps and information

• use the first available word cue to suggest the operation that might be needed.

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Other possible difficulties result from a focus on being quick, which leads to:

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• no attempt to assess the reasonableness of an answer

• little perseverance if an answer is not obtained using the first approach tried • not being able to access strategies to which they have been introduced.

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When the approaches to problem processing developed in this series are followed and the specific suggestions for solving particular problems or types of

problems are discussed with pupils, these difficulties can be minimised, if not entirely avoided. Analysing the problem before starting leads to an understanding of the problem’s meanings. The cycle of steps within the model means that nothing is tried before the intent of the problem is clear and the means to a solution have been considered. Focusing on a problem’s meaning and discussing what needs to be done builds perseverance. Making sense of the steps that must be followed and any answers that result are central to the problem-solving process. These difficulties are unlikely to occur among those who have built up an understanding of this way of thinking.

A final comment If an approach to problem-solving can be built up using the ideas developed here and the problems in the investigations on the pages that follow, pupils will develop a way of thinking about and with mathematics that will allow them to readily solve problems and generalise from what they already know to understand new mathematical ideas. They will engage with these emerging mathematical conceptions from their very beginnings, be prepared to debate and discuss their own ideas, and develop attitudes that will allow them to tackle new problems and topics. Mathematics can then be a subject that is readily engaged with and can become one in which the pupil feels in control, instead of one in which many rules devoid of meaning have to be memorised and applied at the right time. This early enthusiasm for learning and the ability to think mathematically will lead to a search for meaning in new situations and processes that will allow mathematical ideas to be used across a range of applications in school and everyday life.

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Problem-solving in mathematics

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TEACHER NOTES

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Page 5 Pattern-making and numbers are combined in these activities. When lining up their blocks in groups of two and three, some pupils will have blocks left over. This encourages discussion about who can line up their blocks with none left over and who have some left over. The last question involves pupils looking at their blocks to see if they can make a pattern. They should have enough blocks to make some sort of pattern.

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Curriculum links England (Year 1) • Using and applying: Describe simple patterns and relationships. Northern Ireland (Foundation) • Patterns and relationships: Copy, continue and create patterns. • Progression: Solve practical problems. Scotland (Early) • Patterns and relationships: Copy, continue and create own patterns. Wales (Foundation) • Skills: Develop a variety of mathematical approaches and strategies. • Skills: Recognise patterns through practical activities.

Page 4 The six different arrangements now need to be continued to form a pattern using more blocks in the same repeating order. The pupils are not told how many blocks to get and some may not have enough blocks to make a pattern and may need to get more to complete their pattern. This requires pupils to problem-solve how to make a pattern and what to do if they don’t have enough blocks.

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Problem-solving To reason logically, and to identify, create and describe patterns.

Materials Blocks such as Unifix™ cubes, counters, teddies or plastic animals in different colours

Extension • Pupils can be encouraged to make and describe more complex patterns of their own.

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Focus To explore making patterns, changing patterns and using patterns and numbers. To have pupils analyse what makes a pattern and make predictions based on their experiences.

Possible difficulties • Indiscriminately moving blocks around • Inability to keep track of what they are trying. • Difficulty in repeating a consistent pattern. • Content to find only one or two possibilities.

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Discussion Page 3 The blocks can be lined up in six ways. For example, a pupil might put the same block first and then find two ways by swapping the other two blocks. Then the pupil changes the colour of the first block and swaps the other two colours again. This can be done a third time and gives the following six arrangements:

red blue yellow

yellow blue red

blue yellow red

red yellow blue

yellow red blue

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blue red yellow

The next activity involves taking four blocks in different colours and lining them up in different groups of two. Possible arrangements are:

blue

2

red

red

blue

blue yellow

red

blue

red

green

yellow blue

green

blue

yellow

yellow

green

red

green

yellow green

red

green yellow

Problem-solving in mathematics

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BLOCK TIME Take 3 blocks, each a different colour. How many ways can you line them up?

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Draw a picture of all the ways they can be lined up.

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Take 4 blocks, each a different colour.

How many ways can you line them up in groups of two?

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Draw the different groups of two you made.

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Problem-solving in mathematics

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BLOCK PATTERNS

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Choose some blocks in 3 different colours and make a pattern. Draw your pattern.

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Did you have enough to make a pattern or did you have to get some more blocks?

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Think about other patterns you can make using the same blocks. Draw the different patterns.

How many different patterns can you make using the blocks?

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Problem-solving in mathematics

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MORE BLOCKS How many blocks can you hold in one hand? Guess and then count them. How many blocks can you hold in two hands?

How many groups did you make?

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Can you line them up in groups of 3?

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Can you line up your blocks in groups of 2?

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Guess and then count them.

How many groups did you make?

Look at your blocks and see if they make a pattern.

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Draw your pattern below.

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Problem-solving in mathematics

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TEACHER NOTES

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Problem-solving in mathematics

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1 red house

1 blue house

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Focus To analyse problems with a three-dimensional aspect. This extends the previous work with coloured blocks on page 3. There are a number of solutions to this problem, and pupils should be encouraged to explore and try a number of different possibilities.

1 red house

Materials Blocks such as Unifix™ cubes, Multilink™ cubes or other counters (For each group, 10 in total: 5 red, 3 blue and 2 green.)

1 blue house

1 blue apartment

1 blue house

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1 red house

1 red apartment block

1 green house

1 red apartment block

1 green house

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Curriculum links England (Year 1) • Using and applying: Solve problems involving counting, use diagrams to solve problems and describe simple patterns and relationships. Northern Ireland (Foundation) • Patterns and relationships: Explore pattern in number. • Progression: Solve practical problems. Scotland (Early) • Patterns and relationships: Create own patterns. Wales (Foundation) • Skills: Develop a variety of mathematical approaches and strategies. • Skills: Recognise patterns through practical activities.

Discussion Page 7 Individually or in pairs, pupils use the 10 blocks and arrange them on their street according to the street map. For example, the five red people could either be five houses or one apartment block and two houses, while the green people can be only houses. Possible arrangements could include:

1 red house

Problem-solving To reason logically and use patterns to represent and solve problems.

Possible difficulties • Only making houses • Mixing the colours in the apartments • Not realising that a different order gives a different solution Extension • Use the ‘Block Street’ copymaster and make up other criteria for pupils to make streets. • Take a digital photo of each block street and ask the pairs of pupils to write about their street, listing how many houses and apartments there are.

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Problem-solving in mathematics

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apartment = 3 blocks of the same colour

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Use the same blocks to make a different-looking street.

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10 people live on Block Street: 5 red people, 3 blue people and 2 green people. Make your street with houses and apartments to show where the people live.

house = 1 block

BLOCK STREET

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Put blocks on the street to show:

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Possible difficulties • An inability to compare numbers • Combining all numbers present, rather than seeing that some information is not needed • Using the cue ‘more’ to find an answer by adding rather than comparing • Combining the information about bugs and beetles • Not determining which information to use first when graphing

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Curriculum links England (Year 1) • Using and applying: Use numbers to solve problems and answer questions by sorting information. • Counting and understanding number: Compare and order numbers. Northern Ireland (Foundation) • Patterns and relationships: Compare sets and understand the terms ‘more than’ and ‘less than’. • Progression: Solve practical problems. Scotland (Early) • Number processes: Use numbers to describe order. Wales (Foundation) • Skills: Develop a variety of mathematical approaches and strategies. • Number: Compare and order numbers.

Page 11 The information in the graph is analysed to sort out which of the entries on the graph belong to which animal. Understanding comparison enables pupils to work out which label goes with the largest number (dogs) and smallest number (fish). The statement that the number of cats added to the number of fish is the same as the number of dogs is not needed, but it does show if the columns have been labelled correctly. This leaves only guinea pigs and budgies. There must be seven guinea pigs and six budgies to match the criteria of there being one less budgie than guinea pigs.

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Problem-solving To identify and use information in a problem.

Focus These pages explore the reading and interpretation of information to solve problems involving numeration. No addition or subtraction is needed. Pupils analyse the problem to locate the required information, decide what information is not needed and then use comparison, rather than addition or subtraction, to obtain solutions.

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Discussion Page 9 These problems state information that has to be analysed in order to answer a series of questions. The first problems involve reading and identifying the information needed. The next problem requires several comparisons among the number of marbles the children have.

Extension • In pairs, have the pupils use the problem structure on page 9 to write problems of their own to give to other pupils to solve.

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The last problem is about the red cars, with the blue cars included only as additional information. Although Ashley counted the largest number of cars, she, in fact, counted more blue cars than red cars, so Rob counted the largest number of red cars. Page 10 Analysis of the first problem reveals that Kim skipped more times than Josh, because she skipped as many times as he did and then a further six times.

The second problem involves the two types of insects being counted by Dillon. The questions ask the pupils to sort out on what day of the week a certain number of bugs were counted and when a number of beetles were counted and to understand that at no time are the numbers of bugs and beetles combined.

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Problem-solving in mathematics

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HOW MANY? 1 Mary ate 9 grapes. Peter ate 5 grapes. Danny ate 7 grapes.

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How many grapes did Peter eat?

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Who ate the most grapes?

Wendy has 9 marbles. Adam has 16 marbles.

Who has the most marbles? Who has more? Wendy or Simon?

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Simon has 7 marbles.

Lucy counted 7 red cars. Ashley counted 6 red cars and 10 blue cars. Rob counted 9 red cars. Who counted the most red cars? How many red cars did Ashley count?

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Problem-solving in mathematics

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HOW MANY? 2 Josh and Kim were skipping. Josh skipped 9 times. Kim skipped 6 more times.

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How many times did Josh skip?

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Who skipped the most?

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On Monday, Dillon counted 24 bugs and 4 beetles. On Tuesday, he counted 16 bugs and 9 beetles. The next day, he counted 19 bugs and 7 beetles. On which day did he count the most bugs? How many beetles did he count on Wednesday? On which day did he count the most beetles?

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Problem-solving in mathematics

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ANIMAL PETS Sonia’s class has collected information about pets.

Animal pets

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11 10 9 8 7 6 5

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Number of pets

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Class members keep five different kinds of animals as pets. Below is a graph showing how many there are of each pet. The names of the types of animals have not been put on the graph. Cut out the animals’ names and glue them onto the graph. Use the information below to help you.

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4 3 2 1 0

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Clues: • There are 3 fewer cats than dogs. • The smallest number of pets kept are fish. • The number of cats added to the number of fish is the same as the number of dogs. • Most pupils have dogs as pets. • There is one fewer budgie than guinea pigs. cats

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fish

budgies

guinea pigs

✁ dogs

Problem-solving in mathematics

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TEACHER NOTES

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Curriculum links England (Year 1) • Using and applying: Answer questions by sorting information or objects. • Using and applying: Describe ways of solving problems and explain choices. Northern Ireland (Foundation) • Sorting: Sort collections of materials and talk about the arrangements. • Progression: Solve practical problems. Scotland (Early) • 2-D shapes and 3-D objects: Investigate and sort objects. Wales (Foundation) • Skills: Develop a variety of mathematical approaches and strategies. • Handling data: Sort and classify sets of objects.

Page 15 There are 12 different ways the hats can be placed on

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Problem-solving To analyse and sort data according to one or more criteria.

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Focus These pages explore sets of objects to make decisions about how and what criteria can be used to organise them. Identifying, analysing and writing about the specific criteria to sort objects is needed for pupils to find solutions. Pupils should be encouraged to explore and try a number of different possibilities.

the pairs of clowns. Pupils might keep the hat on the first clown the same and change the hat on the second clown. This activity extends the previous work with the blocks (pages 2–5). Putting a hat on a clown, rather than just lining up hats, means that having Hat A and Hat B on one pair is different from having Hat B and Hat A on another pair. In the first example, Hat A is on the first clown, and in the second example, Hat B is on the first clown.

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Discussion Page 13 There are a number of ways the animals can be sorted. Criteria such as the number of legs, flying or not flying, insects or not insects, and living in colonies or not, are possibilities that could be used.

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Page 14 Again, there are a number of ways the clothes can be sorted. One possibility would be size, so the small things (such as underwear and socks) could go in the first drawer, the larger, bulky things in the bottom drawer, and the shirt and T-shirts in the other drawers. Pupils would need to be able to explain and justify their solutions.

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Problem-solving in mathematics

Possible difficulties • Needing help to think of how the objects can be sorted • Sorting according to only one criteria • Placing the clothes into drawers but not according to any criteria • Using two sets of clowns and drawing each of the four hats only once Extension • Make an A3-sized display of the animals and write how they have been sorted. • Discuss whether some of the clothes would be best hung in a cupboard rather than using the chest of drawers. • Look at a bedroom cupboard and sort what could be hung up and what could go into the drawers. • Explore what would happen if there were three clowns with four types of hats.

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ANIMAL ANTICS Frances looked for and found lots of animals in her garden. Help her to sort and display them. Cut out the pictures and sort them into groups. How did you sort them?

spider

butterfly

butterfly

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spider

spider

butterfly

dragonfly

dragonfly

grasshopper

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spider

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How many ways can you sort them?

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Now choose another way to sort them.

grasshopper

bee

bee

bee

bee

ladybird

ladybird

ladybird

ant

ant

ant

ant

ant

ant

ant

ant

frog

frog

frog

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grasshopper

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Problem-solving in mathematics

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SORTING CLOTHES Nick has a new chest of drawers. He needs to put his socks, shorts, underwear, shirt, jumper, tracksuit trousers and T-shirts into the four drawers.

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Can you help him to sort out what clothes go into each drawer?

Now sort the clothes another way. 14

Problem-solving in mathematics

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CLOWN HATS The clowns have 4 types of hats. Draw 2 different hats on each pair of clowns.

Hat B

Hat C

Hat D

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Hat A

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Did all of the clowns get a hat? Prim-Ed Publishing速

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Problem-solving in mathematics

15

TEACHER NOTES

e

Extension • Pupils think of a number and make up criteria to match, using similar criteria of between, greater than and lower than. • The new problem is given to other pupils to try and solve. • Pupils could start with a one-digit number and then try it with a two-digit number.

Sa m

Curriculum links England (Year 1) • Using and applying: Use numbers to solve problems and answer questions by sorting information. • Counting and understanding number: Compare and order numbers. Northern Ireland (Foundation) • Understanding number: Explore the number that comes after, before, between a given number. • Progression: Solve practical problems. Scotland (Early) • Number processes: Explore numbers and use them to describe order. Wales (Foundation) • Skills: Develop a variety of mathematical approaches and strategies. • Number: Compare and order numbers.

Possible difficulties • Selecting a number only based on the first criteria • Not using all of the criteria • Selecting a number only based on the last criteria

pl

Problem-solving To read, interpret and analyse information.

Focus This page explores relationships among numbers and uses this analysis to find a number that matches specific criteria. This process encourages pupils to disregard numbers that are not possible rather than simply look for ones that are likely to work.

ew in

g

Some pupils will use the information provided to discard numbers until only the correct number remains. Other pupils may prefer to try each number in turn against all of the criteria until only one number suits all of the conditions.

Vi

Discussion Page 17 The criteria listed allow numbers to not only be selected but also to be ruled out; for example, when a number has to be between three and eight, then two and nine can be ruled out. Some pupils may work down the list of conditions, while others pupils might read all the conditions and then decide where to start.

One way in which problem-solving differs from work with computation, measurement and other direct applications is that finding what is not likely is often more important than simply using a known method to obtain a definite result.

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WHAT’S MY NUMBER? 1

7

2

5

9

3

My number is:

e

• between 3 and 8

pl

• greater than 4 • an odd number

Sa m

• greater than 5. My number is

.

g

WHAT’S MY NUMBER? 2

8

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3

6

9

5

My number is:

Vi

• between 0 and 7 • lower than 6 • an odd number • greater than 4. My number is

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.

Problem-solving in mathematics

17

TEACHER NOTES

e

Possible difficulties • Putting the numbers in the correct order in the story • Not considering the sense of the story. • Inability to see that there can be more than one answer. • Not taking into account the context of the story when selecting numbers. • Difficulty thinking up their own numbers for a story for it to make sense.

Sa m

Curriculum Links England (Year 1) • Using and applying: Use numbers to solve problems and answer questions by sorting information. • Counting and understanding number: Compare and order numbers. Northern Ireland (Foundation) • Understanding number: Explore the number that comes after, before, between a given number. • Progression: Solve practical problems. Scotland (Early) • Number processes: Explore numbers and use them to describe order. Wales (Foundation) • Skills: Develop a variety of mathematical approaches and strategies. • Number: Compare and order numbers.

Page 21 These problems require the pupils to think of their own numbers; however, this time there is no model of what makes sense, as in the problem on page 19. There is considerable leeway as to what numbers could be used for the story to make sense. Often it is what does not make sense as opposed to what does.

pl

Problem-solving To interpret and organise information in a series of interrelated problem statements.

g

Focus These pages explore ways to put given numbers into problem situations so that the resulting story makes sense. Pupils need to read the stories carefully to work out which number goes where. The numbers are not listed in the order they are used in the story. This thinking is then extended so that pupils think of their own numbers to fit problem situations.

Extension • Pupils can be encouraged to write further stories of their own and swap them with others in the class to solve. • Pupils could write stories with and without numbers.

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Discussion Page 19 The first two problems involve pupils reading the story and putting numbers into the story. Pupils need to make decisions about who is the older child and who is the younger based on the story and put the numbers in accordingly. The last problem uses the story structure of the previous problem and requires pupils to think of their own numbers so the story makes sense.

Vi

Page 20 Reading the stories and working out the relationships among the number of stickers and counters is needed to solve the problem. Pupils need to select numbers based on the information given. The questions do not ask for the numbers in order, so pupils are required to read and scan to find a solution. The last problem has two possible answers. This introduces the understanding that there may be more than one answer to a problem.

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MISSING NUMBERS Put in the numbers so the story makes sense.

7

9

Jason is younger than Rose.

e

years old.

Rose is

years old.

pl

Jason is

Sa m

Put in the numbers so the story makes sense.

15

19

13

Mason is older than Will. Will is older than Bella. years old.

g

Mason is

years old.

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Will is

years old.

Bella is

Vi

Now use your own numbers so the story makes sense. Mason is older than Will. Will is older than Bella. Mason is

years old.

Will is

years old.

Bella is

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Problem-solving in mathematics

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NUMBER SENSE Put in the numbers so the story makes sense.

27

29

26

28

Jason has one more sticker than Rose.

e

Rose has one more sticker than Harry.

pl

Harry has one more sticker than Bree. stickers.

Jason has

stickers.

Sa m

Bree has Harry has

stickers.

Rose has

stickers.

41

42

ew in

44

g

Put in the numbers so the story makes sense.

45

43

The red box has one more counter than the blue box. The blue box has one more counter than the green box.

Vi

The green box has one more counter than the yellow box. The red box has

counters.

The yellow box has

counters.

The green box has

counters.

The blue box has

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Problem-solving in mathematics

counters.

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NUMBER STORIES Make up your own numbers to go in the stories below. Jason has

stickers. Rose has more stickers.

pl

e

stickers than Jason. She has

Sa m

stickers. Rose Harry has has fewer stickers than Harry. Rose has

stickers.

The red box has one more counter than the blue box.

g

The blue box has one more counter than the green box.

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The green box has one more counter than the yellow box.

The red box has

counters. counters.

The green box has

counters.

Vi

The yellow box has

The blue box has

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Problem-solving in mathematics

21

TEACHER NOTES Problem-solving

added.

Possible difficulties

• Inability to identify the need to add to find a solution. • The need to add two different items to get a solution. • Adding all of the numbers written rather than just the numbers required.

Extension

• Using the problems on page 23 as a model, have pupils write simple problems for other pupils to solve.

Sa m

England (Year 1) • Using and applying: Solve problems involving addition and answer questions by sorting information. Northern Ireland (Foundation) • Understanding number: Carry out simple mental calculations. • Progression: Carry out mental calculations and solve practical problems. Scotland (Early) • Addition: Count on to help understand addition. Wales (Foundation) • Skills: Develop a variety of mathematical approaches and strategies. • Number: Calculate in a variety of ways.

e

Curriculum links

Page 25 Addition of two unlike items is again used in these problems but this time with larger numbers. The idea of holes being dug to plant the trees is more complex than just asking how many trees have been planted. The last problem involves more reading then the previous problems.

pl

To analyse and use information in addition problems.

Materials

Counters or blocks

Focus

g

These pages explore word problems that require addition. Pupils need to determine what the problem is asking in order to find a solution. Analysis of the problems reveals that more information may be needed.

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Counters or blocks can be used to assist with these problems. Pupils are not simply concerned with basic facts but about reading for information and determining what the problems are asking.

Discussion

Vi

Page 23 Each problem involves adding like items together—e.g. bugs—and has the clearly identified question of ‘How many altogether?’ at the end, which suggests use of addition. The last problem has additional information about geese, which is not needed. These are simple problems. Pupils need to read the problem, identify that they need to add to solve it and then correctly select the numbers to be added.

Page 24 These problems involve the addition of two unlike items to get a total—e.g. boys and girls, to get the number of children. This is more complex than adding two like items together. Accordingly, the language has been kept fairly simple in each problem. Again, pupils need to read the problem, identify that they need to add to solve it and then select the numbers to be

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pl 3 frogs on the lily pad. 4 frogs in the water. How many frogs altogether?

Vi

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g

Sa m

4 bugs on a bush. 6 bugs on another bush. How many bugs altogether?

e

NUMBER PROBLEMS 1

9 ducks on the lake. 4 geese in the sky. 5 ducks on the bank. How many ducks altogether?

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Problem-solving in mathematics

23

NUMBER PROBLEMS 2

Sa m

pl

e

7 girls riding bikes. 5 boys on the grass. How many children altogether?

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g

The parrot lost 8 yellow feathers and 9 blue feathers. How many feathers did the parrot lose?

Vi

Bert made 16 bags of toffee for the market. Later, he made 21 bags of fudge. How many bags did he make to sell at the market?

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NUMBER PROBLEMS 3

The gardener planted 15 apple trees and 27 pear trees. How many holes did he need to dig?

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pl

e

14 glasses of orange juice. 13 glasses of apple juice. How many drinks altogether?

Vi

The shopping bag had a hole and 27 oranges fell out. Then 18 apples fell out. How many pieces of fruit fell out of the shopping bag?

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25

TEACHER NOTES Problem-solving

Discussion

Curriculum links

Page 27 Pupils read the items for sale and note how much each costs. Pupils who are not familiar with money can still do the activity with a calculator. This investigation involves the pupils obtaining information not only from the question but also from another source—the pictures.

e

They need to remember what they are buying and then work out how much it is, and in some cases add on to compare amounts, to see if they have enough money. The last two questions have a number of possible solutions. Pupils might choose three items they would like and then add and compare only to discover they don’t have enough money, while others may just choose the three cheapest items. Either way, they need to compare money amounts and make decisions accordingly.

Sa m

England (Year 1) • Using and applying: Solve problems involving addition in the context of money. Northern Ireland (Foundation) • Understanding money: Use number skills in shopping activities. • Progression: Carry out mental calculations and solve practical problems. Scotland (Early) • Addition: Count on to help understand addition. • Money: Develop an awareness of how money is used. Wales (Foundation) • Skills: Develop a variety of mathematical approaches and strategies. • Number: Calculate in a variety of ways. • Measures and money: Find money totals.

pl

To solve problems involving money and to make decisions based on particular criteria.

Possible difficulties

Some pupils may need counters, play money or a calculator.

• Confusion with the £ (pound) symbol. • The concept of ‘enough money’ as opposed to an ‘exact amount’. • Not buying different things when necessary. • Thinking the exact amount of £10 has to be spent as opposed to not spending all that is available.

Focus

Extension

• Make a list of all the different possibilities of how pupils could spend their £10. • In pairs, have pupils write other questions about the toy animals.

g

Materials

ew in

This page explores the concepts of reading for information, obtaining information from another source (the picture) and using both to find solutions. The problems are about using money, making decisions based on money and comparing amounts of money, rather than addition or mental facts.

Vi

Solutions can be obtained using materials and comparison of amounts. The item amounts have been kept small to assist with the problem-solving. Counters, blocks, play money or a calculator can be used if needed. This investigation lends itself to using a calculator and could be used to introduce this tool or to extend work previously completed on a calculator.

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TOY ANIMALS

cat £2

sheep £3

Sa m

pl

e

chicken £1

goat £3

g

cow £2

dog £2

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horse £4

pig £3

Nancy has £4. Can she buy a chicken and a cat? Maria has £5. Does she have enough to buy the cow and the sheep?

Vi

Daniel has bought a goat, a dog and a pig. How much did he spend? Mike has £8. He wants to buy 3 toy animals. Choose 3 different things he can buy with his money.

If you had £10, what would you buy?

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Problem-solving in mathematics

27

TEACHER NOTES

Curriculum links

Possible difficulties

• Moving blocks around indiscriminately • Focusing on the particular positions; e.g. forming a line where the fourth block is red and the seventh block is blue, but without a pattern in between—blocks are just in a line and no pattern is evident • Difficulty in repeating a consistent pattern

Sa m

England (Year 1) • Using and applying: Describe simple patterns and relationships. Northern Ireland (Foundation) • Understanding number: Explore ordinal number. • Patterns and relationships: Copy, continue and create patterns. • Progression: Solve practical problems. Scotland (Early) • Patterns and relationships: Copy, continue and create own patterns. Wales (Foundation) • Skills: Develop a variety of mathematical approaches and strategies. • Skills: Investigate repeating patterns and relationships and make simple predictions.

Page 31 The car race problem provides a very engaging way of consolidating the use of patterns and the manner in which ordinal numbers are used to describe them. If pupils have difficulty organising the data, have them use coloured cubes to represent the cars and then transfer the results to the drawn cars. They may also need to do this first when planning their own race.

e

To investigate patterns and order and make predictions based on these.

pl

Problem-solving

Focus

• Have pupils make patterns in which three blocks are specified. • In pairs, have one pupil decide the criterion for a pattern while the other pupil makes the pattern. • In pairs, play a game where one pupil forms a pattern behind a barrier that stops the other from seeing the pattern. The first pupil then describes the pattern, using colours and ordinal numbers, to the other pupil. The second pupil forms this pattern, as he/she understands it to be, on his/her side of the barrier. When the second pattern has been completed, the barrier is removed and the two patterns are compared. • Have pupils use a table to find the results of their car races.

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g

These pages build and extend the earlier activities of using coloured blocks, and introduce ordinal numbers as an aspect of forming and describing patterns. Pupils are required to manipulate items (both real and drawn) to fit particular criteria and determined patterns, including how they relate to ordinal place.

Extension

Discussion

Page 29 Some pupils may have to put additional blocks out and count them to find their solutions, while others may be able to predict the block’s colour from the pattern they can see. Similarly, some pupils may be able to think of a pattern in their mind for the second problem and then arrange the blocks accordingly.

Vi

Page 30 These activities require pupils to work backwards to form different patterns. Ask pupils to check the blocks other classmates have arranged. Pupils have to see if the arrangements do show a pattern and then come up with a way of describing the pattern.

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PATTERNS Take 3 blue, 3 red and 3 yellow blocks. Line them up like this: blue, red, yellow, blue, red, yellow. Continue the pattern.

pl

What colour would the 14th block be?

e

What colour would the 10th block be?

Vi

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g

Draw your pattern.

Sa m

Make a pattern that starts with 2 blue blocks.

What colour is the 8th block? What colour would the 16th block be? What about the 24th block? Prim-Ed Publishing速

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Problem-solving in mathematics

29

ORDINAL PATTERNS

Sa m

pl

e

Make a pattern where the 6th block is green. Make and draw other patterns where the 6th block is green.

Vi

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g

How many patterns did you make? Make a pattern where the 9th block is red. Make five patterns where the 9th block is red. Make a pattern where the 8th and 9th blocks are red. Make a pattern where the 4th block is red and the 7th block is green. Make and draw other patterns where the 4th block is red and the 7th block is green.

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Problem-solving in mathematics

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CAR RACE

37

The yellow car is placed second.

Sa m

The 6th car is black.

45

e

37

pl

FINISH

Use the clues to colour the cars.

The red car is in front of the yellow car.

The orange car is between the black car and the blue car. The green car is behind the yellow car. 1st

g

Use the drawing to fill in which place each car is positioned. 4th

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2nd

5th

3rd

6th

Vi

Draw your own car race and write clues to match.

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Problem-solving in mathematics

31

TEACHER NOTES

Curriculum links

Materials

Some pupils may need materials to assist them.

Focus

Possible difficulties

• Not using a table or list to manage the data. • Inability to see that a ham and cheese sandwich is the same as a cheese and ham sandwich.

Extension

Sa m

England (Year 1) • Using and applying: Answer questions by sorting information. • Handling data: Answer questions by recording information in lists. Northern Ireland (Foundation) • Progression: Sort materials, recording the outcomes. • Progression: Solve practical problems. Scotland (Early) • Data and analysis: Gather information and organize and display findings. Wales (Foundation) • Skills: Develop a variety of mathematical approaches and strategies. • Handling data: Represent data using simple tables.

Page 35 In this scenario, there is now the choice of three types of bread (white, brown or multi-grain) with three different fillings. Again pupils may be able to predict how many sandwiches can be made and what will happen if two fillings are used, based on experience with the previous problems. The questions about more, fewer or the same number of sandwiches is designed to have pupils think about their experiences and make a prediction, which is then investigated.

e

To organise data and make predictions

pl

Problem-solving

• Make a class table showing the possible combinations using one type of bread, two types and three types with the three fillings. • Explore what would happen with four fillings and the possible combinations. • Pupils could list the sandwiches they would choose to make and take to a picnic.

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g

These pages explore the various ways sandwiches can be made using two different types of bread and one of three fillings. The variations are then recorded. These explorations are then extended to include a larger number of bread types and fillings. In order to manage the possibilities, it is important to organise the data in a list or table or to use materials.

Discussion

Vi

Page 33 There are three possible white bread sandwiches—ham, cheese or tomato. The question about using two fillings is designed to make pupils think about whether there will be more sandwiches, fewer sandwiches or the same number of sandwiches. In this case it is the same number.

Page 34 This problem builds on the problem on page 33 and adds a second choice of bread. Now sandwiches can be made using brown or white bread with the three different fillings. Pupils may be able to predict how many sandwiches can be made and what will happen if two filling are used, based on their experience with the previous problem.

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MAKING SANDWICHES Zac and Rachel are making sandwiches for a picnic. They have decided to make each sandwich different. We have ham, cheese and tomato.

Sa m

pl

e

We have white bread.

Write all the different sandwiches they can make if they use one filling in each sandwich. BREAD

FILLING

white

g

white

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white

How many sandwiches can they make?

Vi

Write all the different ways they can make sandwiches if they use two fillings. BREAD

FILLINGS

white

white

white

How many sandwiches can they make?

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Problem-solving in mathematics

33

SANDWICH CHOICES Zac and Rachel are making sandwiches for a picnic. They decide to make each sandwich different. We have ham, cheese and tomato.

pl

e

We have white bread and brown bread.

Sa m

Write all the different sandwiches they can make using one filling. WHITE

BROWN

g

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How many sandwiches can they make? Write all the different ways they can make sandwiches if they use two fillings.

Vi

WHITE

BROWN

What if they use all three fillings? Altogether, Zac and Rachel can make sandwiches using all three fillings.

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Problem-solving in mathematics

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MORE CHOICES Zac and Rachel are making sandwiches for a picnic. They have decided to make each sandwich different. We have white, brown and multigrain bread.

pl

e

We have ham, cheese and tomato.

WHITE

Sa m

Write all the different sandwiches they can make if they use one filling. BROWN

g

MULTI-GRAIN

ew in

How many sandwiches can they make? What happens if they use two fillings? Will they have more, fewer or the same number of sandwiches? Write all the different ways they can make sandwiches if they use two fillings.

Vi

WHITE

BROWN

MULTI-GRAIN

How many sandwiches can they make?

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Problem-solving in mathematics

35

TEACHER NOTES Problem-solving

Discussion

Curriculum links

Page 37 Pupils read the items for sale and note how much each one costs. Pupils who are not familiar with money can still do the activity with a calculator. This investigation involves pupils’ reading for information but also getting information from another source—the illustration.

Materials

Some pupils may need counters, play money or a calculator.

e

Possible difficulties

• Confusion with the £ (pound) symbol. • The concept of ‘enough money’ as opposed to an ‘exact amount’. • Not buying different things when necessary. • Thinking the exact amount of £8, £7 or £5 has to be spent as opposed to not spending all that is available.

Extension

g

Focus

They need to remember what they are buying and then work out how much it is—and in some cases add and in others compare amounts—to see if they have enough money. The last three questions have a number of possible solutions. Pupils might choose three items they would like and then add and compare only to discover they don’t have enough money, while others may choose the three cheapest items. Either way, they need to compare money amounts and make decisions accordingly.

Sa m

England (Year 1) • Using and applying: Solve problems involving addition in the context of money. Northern Ireland (Foundation) • Understanding money: Use number skills in shopping activities. • Progression: Carry out mental calculations and solve practical problems. Scotland (Early) • Addition: Count on to help understand addition. • Money: Develop an awareness of how money is used. Wales (Foundation) • Skills: Develop a variety of mathematical approaches and strategies. • Number: Calculate in a variety of ways. • Measures and money: Find money totals.

pl

To solve problems involving money and to make decisions based on particular criteria.

• Make a list of all the different possibilities for the last three questions. • In pairs, ask pupils to write other questions about the fruit shop and give them to another pair to solve.

Vi

ew in

This page explores reading for information, obtaining information from another source (a drawing) and using it to find solutions. The problems are about using money, making decisions based on money and comparing amounts of money, rather than addition or mental facts. Solutions can be obtained by using materials and comparing amounts. The item amounts have been kept small to assist with the problem-solving. Counters, blocks, play money or a calculator can be used if needed. This investigation builds on the earlier ‘Toy animals’ investigation on page 27. It has the added dimension of a number of items for a particular price while the ‘Toy animals’ investigation involved only one item for a particular price.

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Problem-solving in mathematics

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AT THE TOYSHOP £1

£3

Sa m

£1

pl

e

£2

£2

£4

Kelly has £6. Can she buy a koala and a teddy bear?

g

Mandy has £3. Does she have enough to buy a helicopter and a

ew in

boat?

Davey bought a whistle, a koala and a car. How much did he spend?

Vi

Mark has £8. Choose 3 different things he can buy with his money.

Jane spent £7. List what she could have bought.

What would you buy if you had £5 to spend?

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Problem-solving in mathematics

37

TEACHER NOTES

Curriculum links

Materials

coloured pencils; counting materials, if needed

Focus

Possible difficulties

• Colouring all of the cones or boxes, whether or not they are all needed. • Not organising the data and just randomly positioning any combination. • Using the same combination more than once.

Extension

Sa m

England (Year 1) • Using and applying: Answer questions by sorting information. • Handling data: Answer questions by presenting outcomes using pictures. Northern Ireland (Foundation) • Progression: Sort materials, recording the outcomes. • Progression: Solve practical problems. Scotland (Early) • Data and analysis: Gather information and organize and display findings. Wales (Foundation) • Skills: Develop a variety of mathematical approaches and strategies. • Handling data: Represent data using pictures.

Page 41 This problem is based on a similar idea, exploring the various positions in which an item can be placed in a box. There are six possible ways to position the two chocolates in the box. Some pupils might think that, because there are two chocolates, the solution will be the same as the two scoops of ice-cream. As with the previous problems, there are more boxes than required.

e

To organise data and make predictions.

pl

Problem-solving

• Revisit the problem on page 39 and explore the possibilities of using any combination of flavours, rather than only one scoop of each flavour; e.g. two chocolate and one vanilla.

ew in

g

These pages explore the recording of different ways ice-creams and chocolates can be organised according to various criteria. In each investigation there are more cones or boxes than needed. This requires the pupils to carefully analyse their solutions and to begin to be able to justify their responses.

Discussion

Vi

Page 39 There are six possible ways the scoops can be placed on the cones. In this activity eight cones are drawn on the page and some pupils may simply repeat a previous combination in order to fill all the cones. The question states that pupils choose one of each flavour; therefore each cone needs one chocolate scoop, one vanilla scoop and one strawberry scoop.

Page 40 Pupils can choose two scoops of ice-cream from three possibilities. This problem does not state that they must choose different flavours, so there are nine possibilities, since they could have two scoops of the same flavour if they wished. Again, there are more cones on the page than needed and some pupils may simply repeat a previous combination in order to fill all the cones.

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Problem-solving in mathematics

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ICE-CREAMS 1

e

You are allowed 3 scoops of ice-cream: 1 chocolate, 1 vanilla and 1 strawberry.

Vi

ew in

g

Sa m

pl

Colour the scoops to show the different ways each ice-cream flavour could be placed on the cone.

Did you need all of the cones? Prim-Ed Publishing速

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Problem-solving in mathematics

39

ICE-CREAMS 2

pl

e

You are allowed 2 scoops of ice-cream. You can choose from vanilla, chocolate and strawberry.

Vi

ew in

g

Sa m

Colour the scoops to show the different ways the ice-cream flavours could be placed on the cone.

Did you need all of the cones? 40

Problem-solving in mathematics

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CHOCOLATES Cathyâ€™s box has spaces for 4 chocolates.

Sa m

pl

e

There are 4 ways she can put 1 chocolate into the box:

Cathy has 2 chocolates and 1 box.

Vi

ew in

g

Draw the different ways she can place the chocolates in the box. The first one has been done for you.

Did you use all the boxes? What if there were 3 chocolates? There are Prim-Ed PublishingÂŽ

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ways.

Problem-solving in mathematics

41

TEACHER NOTES

Curriculum links

Materials

Drawing materials, including coloured pencils

Focus

Page 45 Using the road map, pupils track a path from one point to another without backtracking along the way. Each route can be used only once. Some pupils may need to use different coloured pencils to help keep track of the different paths. There are at least 10 possibilities. Discuss the various paths available by using an enlarged version of the picture to show all of the different possibilities.

Sa m

England (Year 1) • Using and applying: Display results using pictures. • Handling data: Answer questions by presenting outcomes using pictures. Northern Ireland (Foundation) • Patterns and relationships: Explore pattern in number. • Progression: Solve practical problems. Scotland (Early) • Data and analysis: Organize and display findings. Wales (Foundation) • Skills: Develop a variety of mathematical approaches and strategies. • Skills: Investigate patterns and relationships and make simple predictions.

Page 44 This investigation continues from the previous page. Now the towels are joined together so that, rather than four pegs for two towels, there are three pegs for two towels. In the case of three pegs for every two towels, pupils could draw the towels and then arrange the pegs to count them. The same diagram could be used to solve the problem for 10 towels instead of pupils having to draw a new diagram. Again, some pupils may be able to predict the number of pegs needed by using the previous activity and will only need to draw a new diagram to confirm this.

e

To use diagrams, make predictions and reason logically.

pl

Problem-solving

• Putting two pegs on each towel when there should be three pegs per two towels. • Taking only the most direct route and not thinking about all the possibilities.

Extension

• Make a table listing how many towels and pegs are used when there are two pegs per towel and how many are used when there are three pegs per two towels. • Use several enlarged pictures to show all of the different paths.

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These pages explore how many pegs are needed to hang towels in various combinations, as well as exploring the different paths between two points. Pupils are required to make predictions and use diagrams to gather the information needed to find solutions. The problems could be completed using multiplication; however, they can also be done using a diagram and counting, using addition with counters, calculating doubles or with a calculator.

Possible difficulties

Discussion

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Page 43 Pupils have to look at the diagram of the towels and pegs. Using the diagram, they need to figure out how many pegs will be needed if there are five towels to hang. Some pupils may be able to look at the diagram and use it to solve the problem, while others may need to draw the five towels and count the pegs. This information can then be used to figure out how many pegs are needed for nine towels. Some pupils may be able to predict the number of pegs from the previous activity and will only need to draw a new diagram to confirm this.

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HANGING OUT THE WASHING

He uses 2 pegs for each towel.

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Mr Smith is hanging out his towels to dry.

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How many pegs does he use if he hangs out 5 towels? Draw the towels and pegs.

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What about for 9 towels? Draw the towels and pegs.

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Problem-solving in mathematics

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WASHING DAY

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Mr Smith has noticed that many of his pegs are broken and that he does not have enough pegs to use for each towel. He decides to join the towels together and use 3 pegs for 2 towels.

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Draw the towels and pegs.

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He wants to hang out 7 towels. How many pegs would he need if he was to peg the corners together, using only 3 pegs for 2 towels?

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What about 10 towels? Draw the towels and pegs.

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DRY-CLEANING Mr Smith has to walk to get to the drycleaners. He likes to walk a different way each time.

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He can only walk along each road once on his way to the drycleaners. Using a different colour each time, draw the different ways he can walk to the drycleaners.

Mr Smith’s home

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TEACHER NOTES

Curriculum links

Possible difficulties

• Not joining squares along a side. • Making only one or two possible shapes. • Inability to visualise the pattern needed to grow a shape. • Only rotating or flipping existing shapes, to create duplicates.

Extension

• Investigate other shape patterns that ‘grow’ using different numbers of squares or shapes.

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England (Year 1) • Using and applying: Answer questions by sorting shapes and displaying results using pictures. • Using and applying: Describe simple patterns and relationships involving shapes. Northern Ireland (Foundation) • Shape and space: Create patterns with 2-D shapes. • Progression: Solve practical problems. Scotland (Early) • Properties of 2-D shapes: Investigate shapes and be creative with them. Wales (Foundation) • Skills: Develop a variety of mathematical approaches and strategies. • Skills: Recognise patterns, sequences and relationships through practical activities.

Page 49 Pupils will need to physically manipulate squares or draw squares on grid paper to see how the pattern ‘grows’. Encourage pupils to make predictions and give a verbal description of their findings.

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To visualise relationships among two-dimensional shapes.

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Materials

Square tiles, and grid paper or cut-out squares and shapes

Focus

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These pages explore arrangements of squares and other shapes. Spatial thinking, as well as logical thinking and organisation, is involved as pupils investigate all possible arrangements and extensions. Being able to visualise patterning of this form will assist pupils in solving many other problems, including number, measurement, and chance and data, as well as other spatial situations.

Discussion

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Page 47 A number of shapes can be made using the four squares, all of which are commonly called tetrominoes. Some pupils may need to physically manipulate the squares in order to find the combinations and analyse whether the shapes are the same or different. This might involve rotating or flipping.

Page 48 This activity extends from the previous problem. Again, some pupils may need to physically manipulate the squares in order to find the combinations and analyse whether shapes are the same or different. This might involve rotating or flipping. Shapes made with five squares are called pentominoes.

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USING SQUARES This shape is a square

. Find 4 squares.

Make a larger square using 4

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What other shapes can you make using 4 squares? Here is one.

How many other shapes can you make?

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Draw each one.

How many shapes did you make?

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USING 5 SQUARES This shape is a square.

Find 5 squares.

What shapes can you make using 5

?

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Here is one.

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Draw each shape.

How many shapes did you make? 48

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GROWING SHAPES Make each shape. How many squares are in each shape? Shape 1

Shape 3

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Look for a pattern.

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Shape 2

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Make Shape 4. Draw it.

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How many squares are in it? Make Shape 5. Draw it.

How many squares are in it?

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TEACHER NOTES Discussion

Curriculum links

Possible difficulties

• Figuring out how far was really travelled when moving forward and back

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England (Year 1) • Using and applying: Solve problems involving addition and subtraction and answer questions by sorting information. Northern Ireland (Foundation) • Understanding number: Carry out simple mental calculations. • Progression: Carry out mental calculations and solve practical problems. Scotland (Early) • Addition and subtraction: Count on and back to help understand addition and subtraction. Wales (Foundation) • Skills: Develop a variety of mathematical approaches and strategies. • Skills: Interpret solutions to calculations within the context of the given problem.

Page 51 The first problem involves the character climbing three metres and then slipping down one metre so that, each day, the distance travelled is two metres. As the tree is eight metres high, it will take exactly four days to reach the top. The next problem involves a distance travelled of three metres, as the snail climbs five metres but also slips down two metres. As the rock wall is 16 metres high, it will take six days of travel, with not all of the last day needed to complete the climb. On the sixth day, only one metre of travel is needed to reach the top, so there will be time to spare.

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To use visualisation to understand measurement.

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Problem-solving

Materials

Drawing materials, counters

Focus

Extension

• Make a table showing how far each animal travelled each day.

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This page explores the concept of reading and interpreting information to solve problems involving distance travelled. Analysis of the problems reveals that the distance entails movement back and forth; for example, climbing forward three metres and then slipping back one metre is an actual travelling distance of two metres. Some pupils many need to draw a diagram or act out each problem to fully grasp the concept.

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HOW LONG? Cathy Caterpillar wants to climb to the top of an 8-metre high tree.

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How long will it take her to reach the top of the tree?

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Each day she climbs forward 3 metres, but slips back 1 metre overnight.

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Suzy Snail is climbing a steep 16-metre high rock wall. Each day she climbs forward 5 metres but slips back 2 metres overnight. How long will it take her to get to the top of the rock wall?

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SOLUTIONS Note: Many solutions are written statements rather than simply numbers. This is to encourage teachers and students to solve problems in this way. BLOCK TIME ....................................................... page 3 YBR YRB

BYR BRY

RY BG GB

RG YB GR

ANIMAL ANTICS . ............................................ page 13 Answers will vary.

SORTING CLOTHES ......................................... page 14 Answers will vary. BR YR GY

CLOWN HATS ................................................... page 15

BLOCK PATTERNS ............................................. page 4

1. Answers will vary; for example: R B G R R R B R 2.–4. Answers will vary.

1. 2.

AB BA CA DA Yes

AC BC CB DB

AD BD CD DC

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6 RYB RBY 12 RB BY YG 12

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1. 2. 3. 4. 5.

WHAT’S MY NUMBER? ................................... page 17

MORE BLOCKS ................................................... page 5

MISSING NUMBERS ....................................... page 19

Answers will vary.

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G B

B R

BLOCK STREET ................................................... page 7 Teacher check Mary 5 Adam Wendy Rob 6

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1. 2. 3. 4. 5. 6.

HOW MANY? 2 ................................................. page 10 Kim 9 Monday 7 Tuesday

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1. 2. 3. 4. 5.

ANIMAL PETS .................................................. page 11

NUMBER SENSE .............................................. page 20 1. (a) 29 (b) 26 (c) 27 2. (a) 45 or 44 (b) 42 or 41 (c) 43 or 42 (d) 44 or 43

(d) 28

NUMBER STORIES .......................................... page 21 Answers will vary.

NUMBER PROBLEMS 1 .................................. page 23 1. 10 bugs 2. 7 frogs 3. 14 ducks

NUMBER PROBLEMS 2 .................................. page 24 1. 12 children 2. 17 feathers 3. 37 bags

1. 27 drinks 2. 42 holes 3. 45 pieces of fruit

11 10 9 8 7 6 5 4

dogs

0

fish

1

cats

2

budgies

3

guinea pigs

Number of pets

(c) 13

NUMBER PROBLEMS 3 .................................. page 25

Animal pets

12

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1. (a) 9 (b) 7 2. (a) 19 (b) 15 3. Answers will vary.

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HOW MANY? 1................................................... page 9

1. 7 2. 5

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SOLUTIONS Note: Many solutions are written statements rather than simply numbers. This is to encourage teachers and students to solve problems in this way.

Yes, they cost £3 Yes, they cost £5 £8 Answers will vary; for example, chicken, cat, sheep. Answers will vary

PATTERNS ......................................................... page 29 1. (a) blue (b) red 2. Teacher check

ORDINAL PATTERNS . ..................................... page 30 CAR RACE . ........................................................ page 31 1. 1st – red 4th – blue 2. Teacher check

2nd – yellow 5th – orange

3rd – green 6th – black

MAKING SANDWICHES ................................. page 33

1. There are 6 variations 2. No

SANDWICH CHOICES ..................................... page 34

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1. white bread – ham, cheese, tomato brown bread – ham, cheese, tomato 2. 6 3. white bread – ham & cheese, ham & tomato, cheese & tomato brown bread – ham & cheese, ham & tomato, cheese & tomato 4. 2

MORE CHOICES . .............................................. page 35 1. 2. 3.

white bread – ham, cheese, tomato brown bread – ham, cheese, tomato multi-grain – ham, cheese, tomato 9 same

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Yes, they cost £4. No, they cost £4. £7 Answers will vary, e.g. whistle, boat, car Answers will vary, e.g. whistle, boat, koala, helicopter Answers will vary, e.g. teddy, helicopter

ICE–CREAMS 1................................................. page 39

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ham, cheese, tomato 3 ham & cheese ham & tomato cheese & tomato 3

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1. 2. 3. 4.

AT THE TOYSHOP ............................................ page 37

1. 2. 3. 4. 5. 6.

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Answers will vary

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1. 2. 3. 4. 5.

4. white bread – ham & cheese, ham & tomato, cheese and tomato brown bread – ham & cheese, ham & tomato, cheese and tomato multi-grain bread – ham & cheese, ham & tomato, cheese and tomato 5. 9

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TOY ANIMALS .................................................. page 27

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ICE–CREAMS 2................................................. page 40 1. There are 9 variations 2. No

CHOCOLATES . .................................................. page 41 1.

• •

•

•

•

•

•

• • • •

• 2. No, there are only 6 ways. 3. There are 4 ways:

• •

•

•

• •

• •

•

•

• •

HANGING OUT THE WASHING ..................... page 43 1. 10 pegs 2. 18 pegs

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SOLUTIONS Note: Many solutions are written statements rather than simply numbers. This is to encourage teachers and students to solve problems in this way. WASHING DAY . ............................................... page 44 1. 8 pegs 2. 11 pegs

DRY-CLEANING ................................................ page 45 Teacher check

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USING SQUARES ............................................. page 47

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2. 4 (Any other answers would be only one of the original shapes reflected or rotated.)

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1.

USING 5 SQUARES .......................................... page 48

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2. 11 are possible (Others are only the same shapes rotated or reflected.)

GROWING SHAPES ......................................... page 49 1. 2.

Shape 1 – 4 Shape 2 – 7 Shape 3 – 10 13 16

HOW LONG? ..................................................... page 51 1. 4 days 2. 6 days (or 5 days and 8 hours)

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TANGRAM RESOURCE PAGE

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10 mm x 10 mm GRID RESOURCE PAGE

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15 mm x 15 mm GRID RESOURCE PAGE

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TRIANGULAR GRID RESOURCE PAGE

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TRIANGULAR ISOMETRIC RESOURCE PAGE

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SQUARE ISOMETRIC RESOURCE PAGE

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ones tens

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thousands

thousands

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ones

4-DIGIT NUMBER EXPANDER RESOURCE PAGE (x 5)

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