Primary Problem-solving in Mathematics: Book A - Ages 5-6 by Teacher Superstore

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Titles available in this series:

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

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FOREWORD

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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, students 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 students 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.

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Books A–G of Problem-solving in mathematics have been developed to provide a rich resource for teachers of students from the early years to the end of middle 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 students. It is a fundamental belief of the authors that it is critical that students 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 students 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 students at all levels of mathematical talent and persistence, as well as in discussions with teachers in classrooms and professional learning and university settings. 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 students’ understanding of the range of ways to come to terms with what the problems are asking.

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Student and teacher pages

The student 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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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 students 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.

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Problem-solving in mathematics

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FOREWORD

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 students’ success. In this way, teachers can be on the lookout for difficulties and be prepared to guide students 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 students handle situations that appear different but are essentially the same as those already met and solved.

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Mathematics and language

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

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 students 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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Other suggestions ask students 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 students’ ways of thinking extended but the problems written on one page are used to produce several more problems that utilise the same approach.

Problem-solving in mathematics

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FOREWORD

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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An approach to solving problems

Analyse

Try

the problem

an approach

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diminished or left out of the problems’ situations. One reason for the suggestion that students 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).

Not only is this model for the problem-solving process helpful in solving problems, it also provides a basis for students 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 students 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:

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 students to

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getting the answer is only the beginning rather than the end … an ability to communicate thinking is equally important.

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We wish all teachers and students who use these books success in fostering engagement with problemsolving and building a greater capacity to come to terms with and solve mathematical problems at all levels.

George Booker and Denise Bond

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Problem-solving in mathematics

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

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

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

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

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

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

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

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

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

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

Washing day ................................................................. 44 © R. I . C.Pu bl i c at i ons Dry-cleaning ................................................................... 45 Sorting clothes .............................................................. 14 •f orr evi ew u r p o.s esonl y• 46 Teacher notes .............................................................. Clown hats .................................................................... 15 p

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

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

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

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

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INTRODUCTION Problem-solving and mathematical thinking

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

Problem-solving lies at the heart of mathematics. New mathematical concepts and processes have always grown out of problem situations and students’ problem-solving capabilities develop from the very beginning of mathematics learning. A need to solve a problem can motivate students 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.

Problem-solving

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By learning problem-solving in mathematics, students 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.

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, students 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 student 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, students 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 students to become both able and willing to engage with and solve mathematical problems.

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For a student in Year 1, sorting out the information about being on the lily pad and being in the water may take some consideration and require counters to represent

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Problem-solving in mathematics

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INTRODUCTION the numbers and find the answer. For 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.

However, many students 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 students 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 students 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, students 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.

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 with mathematics. Making sensible interpretations of any results and determining which of several possibilities is more or equally likely is critical in problem-solving.

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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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Making sense in mathematics

Problem-solving in mathematics

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 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.

3 1 7

3 1

tens

ones

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.

• understanding relationships among numbers • appreciating the relative size of numbers • a capacity to calculate and estimate mentally • fluent processes for larger numbers and adaptive use of calculators • an inclination to use understanding and facility with numeration and computation in flexible ways.

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Number sense requires:

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

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.

these understandings.

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Spatial sense involves:

• a capacity to visualise shapes and their properties

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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 R.I.C. Publications®

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• 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

INTRODUCTION

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

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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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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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• understanding how numeration and computation underpin measurement

• extending relationships from number understanding to the metric system

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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.

• 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?

Problem-solving in mathematics

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INTRODUCTION Data sense involves: A snail crawls 3 m 15 cm around a square garden. At which corner will it stop?

• appreciating the relative likelihood of outcomes • a capacity to use calculators or mental and written processes for exact and approximate calculations

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• 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.

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• understanding how numeration and computation underpin the analysis of data

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?

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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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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.

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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.

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.

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Problem-solving in mathematics

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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Building a problem-solving process

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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 students 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 student 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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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.

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Looking at a problem and working through what is needed to solve it will shed light on the problemsolving process.

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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. Students 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 problemsolving, rather than tasks assigned individually, is an approach that helps to develop this disposition.

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.

Problem-solving in mathematics

The next step is to explore possible ways to solve the problem. If the analysis stage has been completed,

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INTRODUCTION then ways in which the problem might be solved will emerge. 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 students need to be encouraged to select a method that makes sense and appears achievable.

Total amount available to spend:

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Ways that may come to mind during the analysis include:

• 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.

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 student work backwards through the problem from when Peta had $60.00 left.

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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:

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.

• Use a diagram to represent the information in the problem.

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• 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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$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.

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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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• 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.

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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

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INTRODUCTION to the solution that has minimised and simplified the calculations.

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Analyse the problem

A plan to manage problem-solving

Managing a problem-solving program Teaching problem-solving differs from many other aspects of mathematics in that collaborative work can be more productive than individual work. Students 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 student can access. Thus, time has to be allowed for discussion and exploration rather than insisting that students 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.

Explore means to a solution

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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.

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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 student 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 student 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 students to focus on the entire problemsolving process rather than just quickly getting an answer.

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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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INTRODUCTION become frustrated with the task and think that it is beyond them. Students 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 students 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 students’ 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.

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Questions must encourage students 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 students 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 too few may cause students to

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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 problemsolving should then be taken further, as particular groups discuss their solutions with the whole class and all students 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.

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 students 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 students 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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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. Analyse

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• As students 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 students to state in their own words what the problem is asking them to do. • 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 students’ 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.

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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• 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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Expanding the problem-solving process • 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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• 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.

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.

Analyse the problem

Try a solution strategy

Problem-solving in mathematics

• 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 • 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. Analyse

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 students asking themselves further questions about the situation and pose problems of their own as the significance of the problem’s structure is uncovered.

The role of calculators When calculators are used, students 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 students 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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• 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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should engage the interests of the students and also be able to be solved in more than one way.

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.

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Choosing, adapting and extending problems

When problems are selected, they need to be examined to see if students already have an understanding of the underlying mathematics required and that the problem’s expression can be meaningfully read by the group of students who will be attempting the solution—though not necessarily by all students 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

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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, simnple mathematics simple expression, more complex mathematics

complex expression, complex mathematics

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INTRODUCTION The varying levels of problem structure and expression (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 students 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 student is able to confidently deal with. Further analysis of these observations can show to what extent the student 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 student’s developing problem-solving abilities, rather than a one-off test in which some problems are solved and others are not.

(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. (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 student’s mathematical understanding or the problem’s wording need to be searched for the additional material.

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(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 students may need to reverse what initially seemed to be required.

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Assessing problem-solving

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

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A major cause of possible difficulties is the lack of a well-developed plan of attack, leading students to focus on the surface level of problems. In such cases, students:

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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 program. 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 students.

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categorisation of many of the possible difficulties that students 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.

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(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.

How? • Analyse • Explore • Try

Problem-solving in mathematics

• 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 Likely causes

Student is unable to make any attempt at a solution.

• • • •

Student 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

Students uses an inappropriate operation.

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

Student 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

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

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

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are discussed with students, 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.

answer • little perseverance if an answer is not obtained using the first approach tried

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• 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

A final comment

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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, students 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 student 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. R.I.C. Publications®

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TEACHER NOTES Problem-solving To reason logically, and to identify, create and describe patterns. Materials Blocks such as Unifix™ cubes, counters, teddies or koalas in different colours Focus To explore making patterns, changing patterns and using patterns and numbers. To have students analyse what makes a pattern and make predictions based on their experiences.

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red blue yellow

yellow blue red

blue yellow red

red yellow blue

yellow red blue

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green

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blue yellow

red

yellow

green

red

blue

red

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yellow green

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green yellow

Page 4 The six different arrangements now need to be continued to form a pattern using more blocks in the same repeating order. The students 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 students to problem-solve how to make a pattern and what to do if they don’t have enough blocks.

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

green

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

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Discussion Page 3 The blocks can be lined up in six ways. For example, a student might put the same block first and then find two ways by swapping the other two blocks. Then the student 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:

blue

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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Page 5 Pattern-making and numbers are combined in these activities. When lining up their blocks in groups of two and three, some students 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 students 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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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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Draw the different groups of two you made.

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How many different groups can you make? R.I.C. Publications®

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Problem-solving in mathematics

BLOCK PATTERNS

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

Did you have enough to make a pattern or did you have to get some more ©R . I . Cblocks? .Publ i cat i ons

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

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How many different patterns can you make using the blocks?

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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?

r o e t s Bo r e p o u k How many groups did you make? S Guess and then count them.

Can you line them up in groups of 3? How many groups did you make?

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Can you line up your blocks in groups of 2?

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

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TEACHER NOTES Problem-solving To reason logically and use patterns to represent and solve problems. Materials Blocks such as Unifix™ cubes, Multilink™ cubes or other counters (For each group, 10 in total: 5 red, 3 blue and 2 green.)

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1 blue apartment

1 green house

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1 blue house

1 red house

1 green house

1 red apartment block

1 green house

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1 blue house

1 green house

1 red house

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1 red apartment block

1 red house

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• Take a digital photo of each block street and ask the pairs of students to write about their street, listing how many houses and apartments there are.

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Discussion Page 7 Individually or in pairs, students 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

Extension • Use the ‘Block Street’ blackline masters and make up other criteria for students to make streets.

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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 students should be encourage to explore and try a number of different possibilities.

Possible difficulties • Only making houses • Mixing the colours in the apartments • Not realising that a different order gives a different solution

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house = 1 block

Put blocks on the street to show:

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BLOCKT STREET

. te apartment = 3 blocks of the same colour

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Problem-solving in mathematics

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.

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TEACHER NOTES 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. Students 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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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.

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

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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.

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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 comparison • Combining the information about bugs and beetles • Not determining which information to use first when graphing

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being counted by Dillon. The questions ask the students 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.

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 students 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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HOW MANY? 1 Mary ate 9 grapes. Peter ate 5 grapes. Danny ate 7 grapes. Who ate the most grapes?

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Wendy has 9 marbles.

Adam has 16 marbles.

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Who has more?

Wendy or Simon?

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Problem-solving in mathematics

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

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r o e t s Bo r e p o u k How many times did Josh skip? S

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. te o c On Monday, Dillon counted 24 bugs and 4 beetles. . che e r o t On Tuesday, he counted 16 bugs and 9 beetles. r s super 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? 10

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ANIMAL PETS Sonia’s class has collected information about 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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Number of pets

9 8 7

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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 students have dogs as pets. • There is one fewer budgie than guinea pigs. cats

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guinea pigs

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TEACHER NOTES Problem-solving To analyse and sort data according to one or more criteria. 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 students to find solutions. Students should be encourage to explore and try a number of different possibilities.

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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. Students would need to be able to explain and justify their solutions.

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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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, eggs or live births, and living in colonies or not are possibilities that could be used.

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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

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Page 15 There are 12 different ways the hats can be placed on the pairs of clowns. Students 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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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?

r o e t s Bo r e p o u k How many ways can you sort them? S spider

spider

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Now choose another way to sort them.

spider

butterfly

dragonfly

grasshopper

bee

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butterfly

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bee

ladybird

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lizard

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Problem-solving in mathematics

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

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

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Now sort the clothes another way. 14

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

Hat A

r o e t s Hat B C r e oo p u k S Hat B

Hat D

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Problem-solving in mathematics

TEACHER NOTES 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 students to disregard numbers that are not possible rather than simply look for ones that are likely to work.

Extension • Students 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 students to try and solve. • Students could start with a one-digit number and then try it with a two-digit number.

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 students may work down the list of conditions, while others students 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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Some students will use the information provided to discard numbers until only the correct number remains. Other students may prefer to try each number in turn against all of the criteria until only one number suits all of the conditions.

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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

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WHAT’S MY NUMBER? 1

My number is:

r o e t s Bo r e ok • greater than u 4p S • an odd number

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• between 3 and 8

• greater than 5. My number is

©WHAT’S R. I . C.MY Pub l i cat i ons NUMBER? 2 •f orr evi ew pur posesonl y•

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My number is:

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. t0 and 7 o • between e c . c e r • lower than 6 h er o t s super • an odd number • greater than 4. My number is

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Problem-solving in mathematics

TEACHER NOTES Problem-solving To interpret and organise information in a series of interrelated problem statements. Focus These pages explore ways to put given numbers into problem situations so that the resulting story makes sense. Students 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 students think of their own numbers to fit problem situations.

r o e t s Bo r e p ok u S

Page 20 Reading the stories and working out the relationships among the number of stickers and counters is needed to solve the problem. Students need to select numbers based on the information given. The questions do not ask for the numbers in order, so students 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.

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

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Discussion Page 19 The first two problems involve students reading the story and putting numbers into the story. Students 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 students to think of their own numbers so the story makes sense.

Teac he r

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

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Page 21 These problems requires the students 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.

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

Jason is younger than Rose.

r o e t s Bo r e p years old. ok u S years old.

Rose is

Put in the numbers so the story makes sense.

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Jason is

Mason is older than Will. Will is older than Bella.

Mason is

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years old.

Bella is

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o c . Mason is olderc than Will. Will is older than Bella. e her r o t s s r u e p years old. Mason is

Now use your own numbers so the story makes sense.

Will is Bella is

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Problem-solving in mathematics

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

Jason has one more sticker than Rose.

r o e t s Bo r e p ok Harry has one more sticker than Bree. u S stickers. Jason has

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Rose has one more sticker than Harry.

stickers.

Bree has

Harry has

stickers.

Rose has

stickers.

©so Rthe . I . C .Pmakes ubl i c at i ons Put in the numbers story sense.

•f orr evi ew pur posesonl y•

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

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

The yellow box has

counters.

The green box has

counters.

The blue box has

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.

r o e t s Bo r e p ok u S stickers. Rose Harry has

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stickers than Jason. She has

has fewer stickers than Harry.

stickers.

Rose has

The red box© hasR one more the blue box. . I . C .Pcounter ubl i cthan at i o ns

The blue box one more counter than green box. •f o rhas r ev i ew pu r pos esthe on l y•

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The red box has

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counters.

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

o c counters. The green box has . che e r o t r s sup counters. The blue box has er The yellow box has

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counters.

Problem-solving in mathematics

TEACHER NOTES Problem-solving

To analyse and use information in addition problems.

Possible difficulties

Materials

• 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

Focus

Extension

Counters or blocks These pages explore word problems that require addition. Students 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.

Discussion

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.

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

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

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Page 24 These problems involves 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, students need to read the problem, identify that they need to add to solve it and then select the numbers to be added.

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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.

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These are simple problems. Students need to read the problem, identify that they need to add to solve it and then correctly select the numbers to be added.

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NUMBER PROBLEMS 1

r o e t s Bo r e p ok u S

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

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3 frogs on the lily pad. © R. I . C.Pub i caint i on s 4l frogs the water. many frogs altogether? •f orr evi ew puHow r po ses on l y•

o c . clake. e 9 ducks on the her r o t s super 4 geese in the sky. 5 ducks on the bank. How many ducks altogether?

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Problem-solving in mathematics

NUMBER PROBLEMS 2

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

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m . u

The parrot lost 8 yellow feathers and 9 blue © R. I . C.Pub l i cat i ons feathers. How many feathers did• •f orr evi ew pu r po ses onl y the parrot lose?

. tebags of toffee for the fete. Later, he made Bert made 16 21 o c . bags of fudge. How many bags did he make toe sell at the ch r er o fete? st super

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NUMBER PROBLEMS 3

r o e t s Bo r e p ok u S

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

The gardener planted 15 apple trees and 27 pear trees. How many holes © R. I . C.Pub l i c at i o nsdid he need to dig?

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•f orr evi ew pur posesonl y•

o c The shopping c bag had a hole . e r and 27 oranges h felle out. Then 18 o t r s super apples fell out. How many pieces of fruit fell out of the shopping bag?

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Problem-solving in mathematics

TEACHER NOTES Problem-solving

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

Materials

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

Focus

Teac he r

Extension

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

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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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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. Students 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.

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

• Confusion with the $ (dollar) 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

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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.

Discussion

Possible difficulties

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TOY ANIMALS

r o e t s Bo r e p ok u S cat $2

cow $2

sheep $3

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chicken $1

goat $3

horse $4

pig $3

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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?

. tebought a goat, a dog and a pig. o Daniel has c . How much didc he spend? e her r o t Choose 3 s r u Mike has $8. He wantss to buy 3 toy animals. pe different things he can buy with his money.

If you had $10, what would you buy?

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Problem-solving in mathematics

TEACHER NOTES Problem-solving

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

Focus

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

Discussion

• 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

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

Extension

• Have students make patterns in which three blocks are specified. • In pairs, have one student decide the criterion for a pattern while the other student makes the pattern. • In pairs, play a game where one student forms a pattern behind a barrier that stops the other from seeing the pattern. The first student then describes the pattern, using colours and ordinal numbers, to the other student. The second student 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 students use a table to find the results of their car races.

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Page 29 Some students 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 students may be able to think of a pattern in their mind for the second problem and then arrange the blocks accordingly.

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Possible difficulties

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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 students have difficulty organising the data, have them use colour 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.

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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.

r o e t s Bo r e p the 14th block be? ok What colour u would S Make a pattern that starts with 2 blue blocks. Draw your pattern.

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What colour would the 10th block be?

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Problem-solving in mathematics

ORDINAL PATTERNS

r o e t s Bo r e p ok u S

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

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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. © Rthe . I . C .block Pub i ca t i o n7th s block is Make a pattern where 4th isl red and the green. •f orr evi ew pur posesonl y• Make and draw other patterns where the 4th block is red and the 7th block is green.

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CAR RACE

FINISH

Use the clues to colour the cars.

r o e t s Bo r e p o u k The yellow car is placed second. S 37

The 6th car is black.

The red car is in front of the yellow car.

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

©R I . C Pub l i c at i o s Use the drawing to . fill in . which place each carn is positioned. •f orr vi e w pur posesonl y• e 4th

5th

3rd

6th

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Draw your own car race and write clues to match.

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Problem-solving in mathematics

TEACHER NOTES Problem-solving

then investigated.

To organise data and make predictions.

Materials

Some students may need materials to assist them.

Focus

Discussion

• 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

r o e t s Bo r e p ok u S

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

• 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. • Students could list the sandwiches they would choose to make and take to a picnic.

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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.

Possible difficulties

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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 students 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 students think about their experiences and make a prediction, which is

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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. Students 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.

r o e t s Bo r e p ok u S

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We have white bread.

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

FILLING

white

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How many sandwiches can they make?

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white

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

. t BREAD e

white white

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FILLINGS

white

How many sandwiches can they make?

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Problem-solving in mathematics

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

We have white bread and brown bread.

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WHITE

BROWN

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r o e t s Bo r e p ok u Write all the different sandwiches they can make using S one filling.

How many sandwiches can they make?

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

BROWN

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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.

We have ham, cheese and tomato.

Teac he r

WHITE

BROWN

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r o e t s Bo r e psandwiches they can make okif they use one u Write all the different S filling. MULTI-GRAIN

How many sandwiches can they make?

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fewer or the same number of sandwiches?

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What happens if they use two fillings? Will they have more, Write all the different ways they can make sandwiches if they use two fillings.

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MULTI-GRAIN

How many sandwiches can they make? R.I.C. Publications®

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Problem-solving in mathematics

TEACHER NOTES Problem-solving

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

Materials

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

Focus

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

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Teac he r

Extension

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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. Students 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.

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

• Confusion with the $ (dollar) 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

r o e t s Bo r e p ok u S

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.

Discussion

Possible difficulties

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AT THE TOYSHOP

r o e t s Bo r e p ok u S $1

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Teac he r

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

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

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spend?

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

o c . che e r o t r s bought. Jane spent $7. List whats she could have up er What would you buy if you had $5 to spend?

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Problem-solving in mathematics

TEACHER NOTES Problem-solving

To organise data and make predictions.

Materials

coloured pencils; counting materials, if needed

Focus

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 students to carefully analyse their solutions and to begin to be able to justify their responses.

• 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

r o e t s Bo r e p ok u S

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 students may simply repeat a previous combination in order to fill all the cones. The question states that students choose one of each flavour; therefore each cone needs one chocolate scoop, one vanilla scoop and one strawberry scoop.

• 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.

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Teac he r

Discussion

Possible difficulties

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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 students 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.

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Page 40 Students 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 students may simply repeat a previous combination in order to fill all the cones.

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

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r o e t s Bo r e p oeach ice-cream Colour the scoops to show the different ways k u flavour couldS be placed on the cone.

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Problem-solving in mathematics

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

Teac he r

flavours could be placed on the cone.

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r o e t s Bo r e p ok u S Colour the scoops to show the different ways the ice-cream

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Did you need all of the cones? 40

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CHOCOLATES Cathy’s box has spaces for 4 chocolates.

r o e t s Bo r e p ok u S

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Teac he r

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

Cathy has 2 chocolates and 1 box.

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box. The first one has been done for you.

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ways.

Problem-solving in mathematics

TEACHER NOTES Problem-solving

To use diagrams, make predications and reason logically.

Materials

Drawing materials, including coloured pencils

Focus

• 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.

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Discussion

Page 43 Students 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 students 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 students 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.

• 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. Students 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

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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, students 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 students having to draw a new diagram. Again, some students 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.

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Page 45 Using the road map, students track a path from one point to another without backtracking along the way. Each route can be used only once. Some students may need to use different coloured pencils to help keep track of the different paths. There are at least 10 possibilities, but it is unlikely that one student alone will see all 10 paths, since the students have been asked to draw only four paths. Discuss the various paths available by using an enlarged version of the picture to show all of the different possibilities.

Problem-solving in mathematics

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HANGING OUT THE WASHING Mr Smith is hanging out his towels to dry.

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r o e t s Bo r e p ok u S He uses 2 pegs for each towel.

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

WASHING DAY

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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? Draw the towels and pegs.

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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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What about 10 towels? Draw the towels and pegs.

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Problem-solving in mathematics

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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.

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Mr Smith’s home

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Problem-solving in mathematics

TEACHER NOTES Problem-solving

To visualise relationships among two-dimensional shapes.

Materials

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

Focus

• 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

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Discussion

Page 47 A number of shapes can be made using the four squares, all of which are commonly called tetrominoes. Some students 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.

Extension

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

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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 students investigate all possible arrangements and extensions. Being able to visualise patterning of this form will assist students in solving many other problems, including number, measurement, and chance and data, as well as other spatial situations.

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

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Page 48 This activity extends from the previous problem. Again, some students 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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Problem-solving in mathematics

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USING SQUARES This shape is a square

. Find 4 squares.

r o e t s B4o r What other shapes can you make using squares? e p ok Here is one. u S How many other shapes can you make? Draw each one.

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Make a larger square using 4

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Problem-solving in mathematics

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.

Draw each shape.

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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

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Shape 3

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Shape 2

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Make Shape 4. Draw it.

o c . e Make Shape 5. c Draw it. her r o t s super How many squares are in it?

How many squares are in it?

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Problem-solving in mathematics

TEACHER NOTES Problem-solving

To use visualisation to understand measurement.

Materials Focus

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 students many need to draw a diagram or act out each problem to fully grasp the concept.

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

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

Extension

Drawing materials, counters

Discussion

Possible difficulties

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Problem-solving in mathematics

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HOW LONG? Cathy Caterpillar wants to climb to the top of an 8-metre high tree. Each day she climbs forward 3 metres, but slips back 1 metre overnight.

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r o e t s Bo r e pher to ok How long will it take u reach the topS of the tree?

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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.

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How long will it take her to get to the top of the rock wall?

Problem-solving in mathematics

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 6 RYB RBY 12 RB BY YG 12

YBR YRB

BYR BRY

RY BG GB

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

SORTING CLOTHES ......................................... page 14 Answers will vary.

r o e t s Bo r e p ok u S RG YB GR

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 wil vary.

1. 2.

AB BA CA DA Yes

AC BC CB DB

AD BD CD DC

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1. 2. 3. 4. 5.

G B

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

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

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

B R

Answers will vary.

1. 7 2. 5

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

© R. I . C.Pu bl i cat i ons NUMBER SENSE .............................................. page 20 HOW MANY? 1................................................... page 9 •f orr evi ew pur posesonl y• BLOCK STREET ................................................... page 7 Teacher check

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

Mary 5 Adam Wendy Rob 6

NUMBER PROBLEMS 1 .................................. page 23

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10 9 8 7 6 5 4

dogs

fish

1 0

cats

budgies

Guinea pigs

Number of pets

1. 12 children 2. 17 feathers 3. 37 bags

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

Animal pets

1. 10 bugs 2. 7 frogs 3. 14 ducks

NUMBER PROBLEMS 2 .................................. page 24

ANIMAL PETS .................................................. page 11 12

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HOW MANY? 2 ................................................. page 10 1. 2. 3. 4. 5.

(d) 28

NUMBER STORIES .......................................... page 21

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

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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. TOY ANIMALS .................................................. page 27 Yes, they cost $3 Yes, they cost $5 $8 Answers will vary; for example, chicken, cat, sheep. Answers will vary

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1. (a) blue (b) red 2. Teacher check

ORDINAL PATTERNS . ..................................... page 30 Answers will vary

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. 2. 3. 4.

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

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

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

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 © R. I . C.Publ i cat i ons •f orr evi ew pur p osesonl y• ICE–CREAMS 2................................................. page 40 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

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

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1. There are 9 variations 2. No

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

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

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PATTERNS ......................................................... page 29

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

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

Problem-solving in mathematics

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

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

USING 5 SQUARES .......................................... page 48 1.

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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)

Problem-solving in mathematics

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TANGRAM RESOURCE PAGE

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Problem-solving in mathematics

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10 mm x 10 mm GRID RESOURCE PAGE

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Problem-solving in mathematics

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15 mm x 15 mm GRID RESOURCE PAGE

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Problem-solving in mathematics

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TRIANGULAR GRID RESOURCE PAGE

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Problem-solving in mathematics

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TRIANGULAR ISOMETRIC RESOURCE PAGE

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Problem-solving in mathematics

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SQUARE ISOMETRIC RESOURCE PAGE

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ones tens

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tens

ones

4-DIGIT NUMBER EXPANDER RESOURCE PAGE (x 5)

hundreds

thousands R.I.C. Publications®

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Problem-solving in mathematics

thousands

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hundreds

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