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Problem-solving in mathematics (Book E)

This master may only be reproduced by the original purchaser for use with their class(es). The publisher prohibits the loaning or onselling of this master for the purposes of reproduction.

Published by R.I.C. Publications® 2008 Republished under licence by Prim-Ed Publishing 2009 Copyright© George Booker and Denise Bond 2007 ISBN 978-1-84654-186-5 PR–6034

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

Titles available in this series: Problem-solving in mathematics (Book A) Problem-solving in mathematics (Book B) Problem-solving in mathematics (Book C) Problem-solving in mathematics (Book D) Problem-solving in mathematics (Book E) Problem-solving in mathematics (Book F) Problem-solving in mathematics (Book G)

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FOREWORD

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ensure appropriate explanations. The use of the pages foster discussion among pupils and suggest ways in which problems can be extended. Related problems occur on one or more pages that extend the problem’s ideas, the solution processes and pupils’ understanding of the range of ways to come to terms with what problems are asking.

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Problem-solving does not come easily to most people, so learners need many experiences engaging with problems if they are to develop this crucial ability. As they grapple with problem, meaning and find solutions, pupils will learn a great deal about mathematics and mathematical reasoning; for instance, how to organise information to uncover meanings and allow connections among the various facets of a problem to become more apparent, leading to a focus on organising what needs to be done rather than simply looking to apply one or more strategies. In turn, this extended thinking will help pupils make informed choices about events that impact on 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 pupils from the early years to the end of primary school and into secondary school. The series of problems, discussions of ways to understand what is being asked and means of obtaining solutions have been built up to improve the problem-solving performance and persistence of all pupils. It is a fundamental belief of the authors that it is critical that pupils and teachers engage with a few complex problems over an extended period rather than spend a short time on many straightforward ‘problems’ or exercises. In particular, it is essential to allow pupils time to review and discuss what is required in the problem-solving process before moving to another and different problem. This book includes extensive ideas for extending problems and solution strategies to assist teachers in implementing this vital aspect of mathematics in their classrooms. Also, the problems have been constructed and selected over many years’ experience with pupils at all levels of mathematical talent and persistence, as well as in discussions with teachers in classrooms, professional learning and university settings.

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

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

Problem-solving in mathematics

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FOREWORD

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that is being learned as well as 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, twodimensional geometry is used at first but extended to more complex uses over the range of problems, then joined by interaction with three-dimensional 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 giving 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 successful teaching of problem-solving. A section on likely difficulties points to reasoning and content inadequacies that experience has shown may well impede pupils’ success. In this way, teachers can be on the look out for difficulties and be prepared to guide pupils past these potential pitfalls. The final section suggests extensions to the problems to enable teachers to provide several related experiences with problems of these kinds in order to build a rich array of experiences with particular solution methods; for example, the numbers, shapes or measurements in the original problems might change but leave the means to a solution essentially the same, or the context may change while the numbers, shapes or measurements remain the same. Then numbers, shapes or measurements and the context could be changed to see how the pupils handle situations that appear different but are essentially the same as those already met and solved. Other suggestions ask pupils to make and pose their own problems, investigate and present background to the problems or topics to the class, or consider solutions at a more general level (possibly involving verbal descriptions and eventually pictorial or symbolic arguments). In this way, not only are pupils’ ways of thinking extended but the problems written on one page are used to produce several more problems that utilise the same approach.

Mathematics and language The difficulty of the mathematics gradually increases over the series, largely in line with what is taught at the various year levels, although problem-solving both challenges at the point of the mathematics

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Problem-solving in mathematics

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

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FOREWORD

An approach to solving problems

Analyse

Try

the problem

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

Explore

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means to a solution

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an approach

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 other 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 to develop a full solution.

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

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

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

George Booker and Denise Bond

Problem-solving in mathematics

v


CONTENTS How far? . ...................................................................... 34

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

How much? ................................................................... 35

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

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

A note on calculator use ................................................ xx

Fitting tyres ................................................................... 37

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

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

Painting cubes . ............................................................... 3

Area . ............................................................................. 39

Viewing cubes . ............................................................... 4

Area and perimeter 1 .................................................... 40

Nets and cubes ............................................................... 5

Area and perimeter 2 .................................................... 41

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

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

Holiday flights ................................................................. 7

Beading ......................................................................... 43

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

Library ........................................................................... 44

Bookworms ..................................................................... 9

The commuter train . ..................................................... 45

Calculator patterns ....................................................... 10

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

Caramel toffees ............................................................ 11

Squares and area .......................................................... 47

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

Teacher notes . .............................................................. 48

The plant nursery .......................................................... 13

Balancing out ................................................................ 49

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

Christmas shopping ...................................................... 50

Animal World ................................................................ 15

Calendar calculations ................................................... 51

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

Teacher notes . ...............................................................52

Up and down ................................................................. 17

Distance travelled ..........................................................53

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

Taking time . .................................................................. 54

Puzzle scrolls ................................................................. 19

Puzzle scrolls ................................................................. 55

Scrapbooking ................................................................ 20

Teacher notes . .............................................................. 56

Changing coins . ............................................................ 21

Football records ............................................................ 57

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

Teacher notes . .............................................................. 58

Fruit farm . ..................................................................... 23

Building houses . ........................................................... 59

At the delicatessen ....................................................... 24

Teacher notes . .............................................................. 60

The sugar mill ............................................................... 25

Running races ............................................................... 61

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

Solutions ................................................................ 62–65

Numbers in columns ..................................................... 27

Isometric resource page ............................................... 66

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

0–99 board resource page ............................................ 67

Magic squares .............................................................. 29

4-digit number expander resource page (x 5) ............... 68

Magic shapes . .............................................................. 30

10 mm x 10 mm grid resource page ............................. 69

Sudoku .......................................................................... 31

15 mm x 15 mm grid resource page ............................. 70

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

Triangular grid resource page ....................................... 71

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The mango orchard ....................................................... 14

Tourists in Australia ...................................................... 33

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Problem-solving in mathematics

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INTRODUCTION

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

Problem-solving

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

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

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

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

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Problem-solving and mathematical thinking

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

A large number of tourists visited Uluru during 2007. There were twice as many visitors in 2007 than in 2003 and 6530 more visitors in 2007 than in 2006. If there were 298 460 visitors in 2003, how many were there in 2006?

For a pupil aged 7-10 years, sorting out the information to see how the number of visitors each year are linked is a considerable task and then there is a need to use Prim-Ed PublishingÂŽ

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Problem-solving in mathematics

vii


INTRODUCTION

590 390 tourists visited Uluru in 2006

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

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

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298460 x 2 – 6530 = 590390

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

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multiplication and subtraction with large numbers. For a pupil in later primary years, an ability to see how the problem is structured and familiarity with computation could lead them to use a calculator, key in the numbers and operation in an appropriate order and readily obtain the answer:

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?

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Problem-solving in mathematics

Making sense in mathematics Making sense of the mathematics being developed and used needs to 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 putting meanings, representations and relationships in mathematical ideas to the forefront of thinking about and dealing with mathematics. Making sensible interpretations of any results and determining which of several possibilities is more or equally likely is critical in problem-solving. Number sense, which involves being able to work with numbers comfortably and competently,

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INTRODUCTION

• understanding relationships among numbers • a capacity to calculate and estimate mentally

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• fluent processes for larger numbers and adaptive use of calculators

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

The following problem highlights the importance of these understandings.

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

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

3 1 7

3 1

tens

7

ones

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

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• appreciating the relative size of numbers

10 317 is 31.7

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Number sense requires:

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:

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

Spatial sense is equally important as information is frequently presented in visual formats that need to be interpreted and processed, while the use of diagrams is often essential in developing conceptual understanding across all aspects of mathematics. Using diagrams, placing information in tables or depicting a systematic way of dealing with the various possibilities in a problem assist in visualising what is happening. It can be a very powerful tool in coming to terms with the information in a problem and it provides insight into ways to proceed to a solution. Spatial sense involves: • a capacity to visualise shapes and their properties • determining relationships among shapes and their properties • linking two-dimensional and three-dimensional 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.

Problem-solving in mathematics

ix


INTRODUCTION The following problem shows how these understandings can be used.

Many dice are made in the shape of a cube with arrangements of dots on each square face so that the sum of the dots on opposite faces is always 7. An arrangement of squares that can be folded to make a cube is called a net of a cube.

A small sheet of paper has been folded in half and then cut along the fold to make two rectangles. The perimeter of each rectangle is 18 cm. What was the perimeter of the original square sheet of paper?

Which of these arrangements of squares forms a net for the dice?

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Greengrocers often stack fruit as a pyramid.

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How many oranges are in this stack?

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Reading the problem carefully and analysing the diagram shows that the length of the longer side of the rectangle is the same as one side of the square while the other side of the rectangle is half this length. Another way to obtain this insight is to make a square, fold it in half along the cutting line and then fold it again. This shows that the large square is made up of four smaller squares:

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Measurement sense is dependent on both number sense and spatial sense as 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.

Since each rectangle contains two small squares, the perimeter of the rectangle, 18 cm, is the same as 6 sides of the smaller square, so the side of the small square is 3 cm. The perimeter of the large square is made of 8 of these small sides, so is 24 cm. Similar thinking is used with arrangements of twodimensional and three-dimensional shapes and in visualising how they can fit together or be taken apart.

Measurement sense includes: • • • • •

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understanding how numeration and computation underpin measurement extending relationships from number understandings to the metric system appreciating the relative size of measurements a capacity to use calculators, mental or written processes for exact and approximate calculations an inclination to use understanding and facility with measurements in flexible ways.

Problem-solving in mathematics

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INTRODUCTION The following problem shows how these understandings can be used.

Data sense involves: • understanding how numeration and computation underpin the analysis of data

A city square has an area of 160 m2. Four small triangular garden beds are constructed from each corner to the midpoints of the sides of the square. What is the area of each garden bed?

• 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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• an inclination to use understanding and facility with number combinations and arrangements in flexible ways.

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The following problem shows how these understandings can be used.

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Reading the problem carefully shows that there are 4 garden beds and each of them takes up the same proportions of the whole square. A quick look at the area of the square shows that there will not be an exact number of metres along one side. Some further thinking will be needed to determine the area of each garden bed.

• presenting and interpreting data in tables and graphs

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If the midpoints of each side are drawn across the square, four smaller squares are formed and each garden bed takes up 41 of a small square. Four of the garden beds will have the same area of one small square. Since area of the small square is 41 the area of the large square, the area of one small square is 40 m2 and the area of each triangular garden bed is 10 m2.

An understanding of the problem situation given by a diagram has been integrated with spatial thinking and a capacity to calculate mentally with simple fractions 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. 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. Prim-Ed Publishing®

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You are allowed 3 scoops of ice-cream: 1 chocolate, 1 vanilla and 1 strawberry. How many different ways can the scoops be placed on a cone?

There are six possibilities for placing the scoops of icecream on a cone. Systematically treating the possible placements one at a time highlights how the use of a diagram can account for all possible arrangements. Patterning is another critical 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 so as to determine a likely solution. Being able to see patterns is also very helpful in getting a solution more immediately or understanding whether or not a solution is complete.

Problem-solving in mathematics

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INTRODUCTION of the problem itself—what is being asked, what information might be used, what answer might be likely and so on—so that a particular approach is used only after the intent of the problem is determined. Establishing the meaning of the problem before any plan is drawn up or work on a solution begins is critical. Pupils need to see that discussion about the problem’s meaning, and the ways of obtaining a solution, must take precedence over a focus on ‘the answer’. Using collaborative groups when problem-solving, rather than tasks assigned individually, is an approach that helps to develop this disposition.

A farmer had emus and alpacas in one paddock. When she counted, there were 38 heads and 100 legs. How many emus and how many alpacas are in the paddock? There are 38 emus and alpacas. Emus have 2 legs. Alpacas have 4 legs. Number of

Number of

alpacas

emus

4

34

84 – too few

8

30

92 – too few

10

28

96 – too few

12

26

100

Number of legs

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 onethird 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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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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There are 12 alpacas and 26 emus.

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

Building a problem-solving process

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

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Problem-solving in mathematics

By carefully reading the problem, 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. 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.

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INTRODUCTION

Another way to solve this 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: Total amount available to spend:

She spent half of her money on a dress.

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Ways that may come to mind during the analysis include:

Materials could also have been used to work with 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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The next step is to explore possible ways to solve the problem. If the analysis stage has been completed, 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 a pupil needs to be encouraged to select a method that make sense and appears achievable to him or her.

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.

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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.

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

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

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• 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. • Backtrack using the numbers – The sandals were one-third of what was left after the dress, so the €60.00 would be two-thirds of what was left. Together, these two amounts would match the cost of the dress. • Use a diagram to represent the information in the problem. • Think of a similar problem – For example, it is like the car race problem in that the relative portions (places) are known and the final result (money left, winning position) are given. 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. Prim-Ed Publishing®

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

€30

€30

Each of the six equal parts represents €30, so Peta took €180 to spend. Having tried an idea, an answer needs to be analysed in the light of the problem in case another solution is required. It is essential to compare an answer back 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 Problem-solving in mathematics

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INTRODUCTION

Managing a problem-solving programme

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Thinking about how 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 what the problem was really asking. Next, all possible ways to solve the problem were explored before one, or a combination of ways, was/were selected to try. Finally, once something was tried, it was important to check the solution in relation to the problem to see if the solution was 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:

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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 to the solution that has minimised and simplified the calculations.

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 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 any answer obtained can be justified. Learning to reflect on the whole process leads to the development of a deeper understanding of problem-solving, and time must be allowed for reflection and discussion to fully build mathematical thinking.

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way the process is cyclic and should the answer be unreasonable, then the process would need to begin again.

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Analyse the problem

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

Try a solution strategy

Explore means to a solution

A plan to manage problem-solving This model for problem-solving provides pupils with a means of talking about the steps they engage with 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—that they 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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Problem-solving in mathematics

Correct answers that fully solve a problem are always important, but developing a capacity to use an effective problemsolving process needs to be the highest priority. A pupil who has an answer should be encouraged to discuss his or her solution

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INTRODUCTION

Building a problem-solving process

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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.

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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 pupils are able to participate in the discussion of the problem. In this way, problem-solving as a way of thinking comes to the fore, rather than focusing on the answers to a series of problems that some pupils see as the main aim of their mathematical activities.

of moving on to another different problem in the way that one example or topic shifts to another in other parts of mathematics learning. This can help pupils to understand the significance of asking questions of a problem, as well as seeing how a way of thinking can be adapted to other related problems. Having pupils engage in this process of problem posing is another way of both assessing and bringing them to terms with the overall process of solving problems.

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with others who believe they have a solution, rather than tell his or her answer to another pupil or simply move on to another problem. In particular, explaining to others why he or she believes an answer is reasonable, as well as why it provides a solution, gets other pupils to focus on the entire problem-solving process rather than just quickly getting an answer.

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Questions need to encourage pupils to explore possible means to a solution and try one or more of them, rather than point to a particular procedure. It can also assist pupils to see how to progress their thinking, rather than get in a loop where the same steps are repeated over and over. However, 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 pupils to become frustrated with the task and think that it is beyond them. Pupils need to experience the challenge of problem-solving and gain pleasure from working through the process that leads to a full solution. Taking time to listen to pupils as they try out their ideas, without comment or without directing them to a particular strategy, is also important. Listening provides a sense of how pupils’ 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 a difficulty with a particular problem mean that the problems that follow will also be as challenging.

Analyse • As pupils read a problem, the need to first read for the meaning of the problem can be stressed. This may require reading more than once and can be helped by asking pupils to state in their own words what the problem is asking them to do. • Further reading will be needed to sort out which information is needed and whether some is not needed or if other information needs to be gathered from the problem’s context (e.g. data presented within the illustration or table accompanying the problem), or whether the pupils’ mathematical understandings need to be used to find other relationships among the information. As the form of the problems becomes more complex, this thinking will be extended to incorporate further ways of dealing with the information; for example, measurement units, fractions and larger numbers might need to be renamed to the same mathematical form.

A teacher may also need to extend or adapt a given problem to ensure the problem-solving process is understood and can be used in other situations, instead

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INTRODUCTION • 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. • 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.

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

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

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Try a solution strategy

Try

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A fuller model to manage problem-solving can gradually emerge:

• 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?

• 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.

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

Expanding the problem-solving process

• 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

Explore means to a solution

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

Problem-solving in mathematics

Analyse • 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.

The role of calculators When calculators are used, pupils devote less time to basic calculations, providing time that might be needed to either explore a solution or find an answer to a problem. In this way, attention is shifted from computation, which the calculator can do, to thinking about the problem and its solution—work that the calculator cannot do. It also allows more problems (and more realistic problems) to be addressed in problemsolving sessions. In these situations, a calculator serves as a tool rather than a crutch, requiring pupils to think through the problem’s solution in order to

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INTRODUCTION 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.

Choosing, adapting and extending problems

language and mathematics. Within a problem, the processes that need to be used may be more or less obvious, the information that is required for a solution may be too much or too little, and strategic thinking may be needed in order to come to terms with what the problem is asking. Level

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

complex expression, complex mathematics

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increasing difficulty with problem’s expression and mathematics required

strategic thinking

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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.

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

processes processes too much too little obvious less obvious information information

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

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’ may have simple language and more difficult mathematics or more difficult language and simple mathematics; while a third-level has yet more difficult Prim-Ed Publishing®

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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 pupils may need to reverse what initially seemed to be required. (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 needs to be gathered and applied to the problem in order to obtain a solution. These problems do not contain first-hand all the necessary information required to find a solution but do contain a means to obtain the required information. The problem’s setting, the pupil’s mathematical understanding or the problem’s wording need to be searched for the additional material.

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INTRODUCTION

Assessing problem-solving

How? • Analyse • Explore • Try

Observations based on this analysis have led to a categorisation of many of the possible difficulties that pupils experience with problem-solving as a whole, rather than the misconceptions they may have with particular problems.

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Assessment of problem-solving requires careful and close observation of pupils working in a problemsolving setting. These observations can reveal the range of problem forms and the level of complexity in the expression and underlying mathematics that a pupil is able to confidently deal with. Further analysis of these observations can show to what extent the

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

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

pupil is able to analyse the question, explore ways to a solution, select one or more methods to try and then analyse any results obtained. It is the combination of two fundamental aspects—the types of problem that can be solved and the manner in which solutions are carried out—that will give a measure of a pupil’s developing problem solving abilities, rather than a one-off test in which some problems are solved and others are not.

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(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 as to determine a means to a solution.

Likely causes

Pupil is unable to make any attempt at a solution.

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These often involve inappropriate attempts at a solution based on little understanding of the problem.

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

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

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

Pupil uses an inappropriate operation.

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

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

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

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INTRODUCTION

• no attempt to assess the reasonableness of an answer

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

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

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

A final comment

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A major cause of possible difficulties is the lack of a well-developed plan of attack, leading pupils to focus on the surface level of problems. In such cases, pupils: • locate and manipulate numbers with little or no thought as to 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 • use the first available word cue to suggest the operation that might be needed.

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

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A NOTE ON CALCULATOR USE Many of the problems in this series demand the use of a number of consecutive calculations, often adding, subtracting, multiplying or dividing the same amount in order to complete entries in a table or see a pattern. This demands (or will build) a certain amount of sophisticated use of the memory and constant functions of a simple calculator.

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5. Using the memory keys M+, M– and MR will also simplify calculations. A result can be calculated and added to memory (M+). Then a second result can be calculated and added to (M+) or subtracted from (M–) the result in the memory. Pressing MR will display the result. Often this will need to be performed for several examples as they are entered onto a table or patterns are explored directly. Clearing the memory after each completed calculation is essential! A number of calculations may also need to be made before addition, subtraction, multiplication or division with a given number. That number can be placed in memory and used each time without needing to be re-keyed.

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2. To subtract a number such as 5 repeatedly, it is sufficient on most calculators to enter an initial number (e.g. 92) then press – 5 = = = = to subtract 5 over and over. • 92, 87, 82, 77, 72, … • To subtract 5 from a range of numbers, enter the first number (e.g. 92) then press – 5 = 95 – 5 = 90, 68 = gives 63, 43 = gives 38, 72 = gives 67, … • These are the answers when 5 is subtracted from each number.

4. To divide by a number such as 4 repeatedly, enter a number (e.g. 128). • Then press ÷ 4 = = = = to divide each result by 4. • 32, 8 , 2, 0.5, … • These are the answers when the given number is divided by 4.

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1. To add a number such as 9 repeatedly, it is sufficient on most calculators to enter an initial number (e.g. 30) then press + 9 = = = = to add 9 over and over. • 30, 39, 48, 57, 66, … • To add 9 to a range of numbers, enter the first number (e.g. 30) then press + 9 = 30 + 9 = 39, 7 = gives 16, 3 = gives 12, 21 = gives 30, … • These are the answers when 9 is added to each number.

6. The % key can be used to find percentage increases and decreases directly.

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3. To multiply a number such as 10 repeatedly, most calculators now reverse the order in which the numbers are entered. Enter 10 x, then press an initial number (e.g. 15) = = = = to multiply by 10 over and over. • 10, 150, 1500, 15 000, 150 000, … • This also allows squaring of numbers: 4 x = gives 16. • Continuing to press = gives more powers: • 4 x = = gives 64, 4 x = = = gives 256; 4 x = = = = gives 1024 and so on. • To multiply a range of numbers by 10, enter 10 x then the first number (e.g. 90) and = • 10 x 90 = 900, 45 = gives 450, 21 = gives 210, 162 = gives 1620, … • These are the answers when each number is multiplied by ON AC % 10. OFF

900

• To increase or decrease a number by a certain per cent (e.g. 20%), simply key the number and press + 20% or – 20% to get the answer: • 80 + 20% gives 96 (not 100) – 20% of 80 is 16, 80 + 16 is 96. • 90 – 20% gives 72 (not 70) – 20% of 90 is 18, 90 – 18 is 72.

7. While the square root key can be used directly, finding other roots is best done by a ‘try and adjust’ approach using the multiplication constant described above (in point 3).

CALCULATOR BSCX56

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Pupil worksheets and teachers notes

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Problem-solving in mathematics

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TEACHER NOTES

Curriculum links

Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Represent mathematical ideas and processes in different modes: pictorial and diagrammatic • Explore and investigate mathematical patterns and relationships Shape and space • Solve and complete practical tasks and problems involving 3-D shapes

Materials

Page 5 The activities on this page ask pupils to visualise how a two-dimensional representation can be folded to make a cube. Some pupils may need to cut out the nets from grid paper and fold them to see which form cubes. The dice at the top of the page shows 6, 3 and 2 dots, so the opposite sides would have 1, 4 and 5 dots. (It is important to realise that the opposing numbers on a dice always add to seven.) Seeing which dots would occur on the blank squares requires pupils to visualise how the squares fold and which faces would be opposite to each other. Again, some pupils may need to make the nets to see or check results. There are several ways the digits could be arranged on the blank die, with the opposite sides adding to seven. Writing letters on the net to match the given views extends this thinking.

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dice, cubes, grid paper to draw views of the shapes and nets of cubes, digital camera to photograph the shapes

dimensional shapes in terms of the component twodimensional forms. When viewed from above and below, the outline will appear the same. The views from opposite sides will be mirror images of each other. Understanding this can be difficult and pupils need to be encouraged to be systematic in constructing the shapes. When a pupil makes a shape for another pupil, rather than draw the different views, he or she may be able to take digital photos of each and then print them for other pupils to build from.

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To use spatial visualisation and logical reasoning to solve problems.

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Problem-solving

Focus

Discussion

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These pages explore arrangements and dissections of three-dimensional shapes in order to determine how particular outcomes are formed. Spatial and logical thinking and organisation are required as pupils investigate all likely arrangements to ensure the final shapes match the given criteria or visualise a shape in terms of its component parts.

Page 3 This page requires an ability to visualise the arrangements of cubes stacked in several layers. Some pupils will see layers from the bottom or top, while others will see slices going across the shape. They then need to use this information to determine where paint is applied to the complete shape and visualise the effect on the individual cubes. Some pupils may need to build the structure and manipulate the blocks in order to see what can happen. Encourage systematic analysis when they are working out the results. Only certain faces of the cubes are able to be painted and the number of cubes counted must equal the number of cubes in the shape.

Possible difficulties

• Unable to visualise the shapes from the twodimensional representations • Only considers the cubes that can be readily seen on the outside of the shapes • Unable to keep track of the possibilities in any of the investigations • Unable to visualise the opposite sides from the net of the cube

Extension

• Pupils make stacks using different arrangements of cubes and ask other pupils to investigate what would happen if all of the outside faces were painted. • Have pupils write their own name (if it has six letters) or a six-letter word on a cube and then challenge a partner to draw a net of the cube. • Ask pupils to write a six-letter word on a net and then challenge a partner to draw them on a representation of the cube.

Page 4 This investigation encourages pupils to see three-

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PAINTING CUBES Some cubes were joined together to make this shape. After they were joined together, the shape was painted yellow on all of its sides. When it was taken apart, some faces of the cubes were painted yellow and some weren’t. 1. How many cubes are there in the shape?

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. Make a table to show how many cubes have 2 0, 1, 2, 3, 4 or 5 faces painted.

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3. Have you included all of the cubes?

A wooden cube is painted so that two opposite sides are green, another two opposite sides are yellow and the last two sides are red. The cube is then cut into 64 small cubes.

4. How many of the smaller cubes will have one face painted green, one face painted yellow and one face painted red? 5. (a) How many of these cubes will have only two faces painted? (b) How many are red and green? 6. (a) How many of the cubes will have only one face painted? (b) How many are yellow? 7. How many of the cubes would have no paint? 8. Are there any cubes painted in other ways?

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VIEWING CUBES Draw what this shape would look like viewed from above. Would it look any different if viewed from below? 1. (a) Draw it from each of the sides in the space provided.

Bottom

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(b) How do the views compare?

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Get some cubes and make the shape.

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2. How many cubes did you need?

These are the views of an arrangement of cubes from the top, front and side. Top

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Side

Get some cubes and make the shape. 3. How many cubes did you need? 4. Make and draw a shape of your own. Ask a friend to construct the shape using your drawings. Compare the result to your drawings.

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NETS AND CUBES Most dice are made in the shape of a cube, with the dots on each square face arranged so the sum of the dots on opposite faces is always seven. An arrangement of squares that can be folded to make a cube is called the net of the cube. 1. Which of the arrangements of squares form nets for dice? Circle the ones that form a correct net and draw the missing dots on the blank squares.

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2. Put digits on the squares on the nets to make dice. Make sure the sum of the digits on the opposite faces is seven.

3. Here are three views of a dice on which Lauren wrote one letter of her name on each face. Can you draw a net to show how the letters would be arranged on the squares?

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Problem-solving in mathematics

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TEACHER NOTES To solve problems involving time and make decisions based on particular criteria.

Curriculum links

Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Devise and use mental strategies and procedures for carrying out mathematical tasks Measures • Read and interpret simple timetables • Solve and complete problems involving times

Similar thinking can be used for the flights that are too early. For example, catching the Lion Airlines plane at 09:50 would get to you to the hotel at around midday, which is about two hours too early. As such, the flights at 07:45, 08:10 and 09:15 are also too early and can be excluded. Pupils need to think in terms of 24-hour time for the flight information but the before and after times of 2 pm and 6 pm are in 12-hour time. Pupils need to be able to convert the times to 24-hour time. The information regarding the taxi and the waiting time at the airport is additional information which is not needed. Some pupils may try to include this in their calculations. The bus leaves the airport every 30 minutes (15 minutes and 45 minutes past each hour).

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This page explores reading for information from a number of sources (information about the plane, the timetable, and the shuttle bus) and using it to find solutions. The problems involve thinking about and working with time. Decisions about time being ‘too early’ or ‘too late’ are considered rather than an exact time.

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Time between the plane arriving and bus leaving also needs to be considered. If the plane arrives at 10:10 it would not be possible to catch the 10:15 bus as you would need to leave the plane, wait for luggage and then find the bus and you would need longer than 5 minutes to do this.

Materials

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Page 7 Pupils read the information on the page and use it to solve a number of questions. Pupils who are not familiar with 24hour time can complete the activity by using a conversion table. The worksheet can be used to introduce the concept of 24-hour time. Pupils need to read and sort information from a number of sources and fit this information against set criteria.

Unfamiliarity with a timetable Confusion with 24-hour time Not excluding the taxi and wait time information Thinking that an exact flight is needed rather than flights that are neither too early or too late

Extension

• Use the information and timetable with other criteria; for example, if you need to be in Barcelona for a lunchtime meeting what flights can you take?

There are a number of flights that fit the criteria, with other flights arriving at Barcelona being either ‘too early’ or ‘too late’. Once a flight is deemed to be too late, then others that are later still can be automatically excluded; for example, the Lion Airlines plane at 16:15 does not arrive at Barcelona until after dinner. This then also rules out the flights at 17:30 and 18:00.

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Problem-solving in mathematics

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HOLIDAY FLIGHTS Imagine you have decided to travel from Dublin to Barcelona for a holiday. Look at the following information and plan your holiday. As most hotels have a 2 pm check-in time, you want to arrive after 2 pm. However, you also want to arrive before dinner at 6 pm. TAXI INFORMATION from home to Dublin Airport

• waiting time

25 mins

• travel time

45 mins

• flight time to Barcelona 1 hour 50 mins Lion Airlines

LA47

LA51

LA68

14:30

15:10

16:15

LA74

LA32 09:50

18:00

LA23

Flight numbers

08:10

17:30

Departure times

g ST74

ST68

15:00

13:50

1 hour

m

• wait time at airport

in

ST51

ST47 12:40

09:15

Vi ew

ST32

ST23 07:45

Departure times

Flight numbers

Sa

Star Airlines

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PLANE INFORMATION

BUS INFORMATION from Barcelona Airport to hotel • hotel bus leaves airport every 30 mins (15 and 45 mins past each hour) • travel time to hotel

15 mins

1. I can catch the following flights: 2. These flights are too early: 3. These flights are too late:

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Problem-solving in mathematics

7


TEACHER NOTES

Curriculum links

Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Devise and use mental strategies and procedures for carrying out mathematical tasks Number • Explore and identify place value in numbers • Solve word problems involving addition and subtraction • Solve and complete problems involving multiplication and division of whole numbers

Materials

calculator, number expanders

Focus

Sa

These pages explore concepts of place value and number sense. The relationships among numbers and place value is analysed and pupils are encouraged to not only find possibilities but to also disregard numbers and combinations that are not possible. Place value and number sense are needed rather than addition or multiplication.

digit number (345 345), it is at first surprising that it can be divided exactly by 13. The result also divides exactly by 11 and this result divides exactly by 7 to give the initial three-digit number. Reading the number as three hundred and forty-five thousand, three hundred and forty-five gives a clue—the original number is multiplied by 1000 and added to itself or multiplied by 1001. The product of 13 x 11 x 7 is 1001, so the divisions have undone the effect of the multiplying. Problem 2 requires pupils to notice a pattern: That the product of the two increased and decreased numbers is the square of the original number, subtracted the square of the number that was originally added and subtracted. Using the calculator’s memory to store the square and then subtract the product of the two changed numbers allows this pattern to be readily seen. It could be expressed as: (number + increase) x (number – increase) = number2 – increase2. This leads to a powerful and well-known algebraic identity. Page 11 In this investigation pupils need to read and interpret the information and use it to find combinations that match specific criteria. Pupils need to think of possible combinations as well as discarding combinations that don’t work. In Problem 1, each person needs to receive 15 caramel toffees. Possible combinations are boxes 9, 5, 1; or 8, 4, 3; or 7, 6, 2. Problem 2 has many possible combinations. The two children can be given boxes that total to 18 or 20 caramel toffees and each total has several possible combinations; for example, they could have boxes 2, 3, 5, 8 and 1, 4, 6, 7 (if totalling to 18); or 1, 4, 6, 9 and 2, 3, 7, 8 (if totalling to 20). The last question asks for a total of 13 toffees, so possible combinations include 9, 3, 1 and 7, 4, 2 or 2, 3, 8 and 1, 5, 7.

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To read, interpret and analyse information.

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Problem-solving

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Discussion

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Page 9 This investigation requires pupils to read and interpret information while thinking in terms of place value. As the book has 10 chapters of equal length, an understanding of place value can be used to solve each problem. Given that the book has 1380 pages, we know from place value that this number has 138 tens so each chapter must be 138 pages long. This information can be used to calculate the various problems. Before starting, pupils may find it helpful to draw up a table outlining the starting page of each chapter. Each chapter has 138 pages and Chapter 1 starts on page 1, so Chapter 2 must start on page 139 and so on. The page number that Wanda has read to is given in the beginning and this information is needed to answer some of the problems. For example, in order to determine how many pages Wanda has read past the middle of the book, it is necessary to keep in mind that there are 1380 pages in the book and she has read up to page 759. Similarly, this information is again needed to determine how many pages need to be read to finish the book. Page 10 These problems require pupils to use their calculator to generate some results in order to think about the underlying numeration and multiplication patterns. For Problem 1, each time a three-digit number, such as 345, is changed to a six

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Problem-solving in mathematics

Possible difficulties

• Poor understanding of place value • Wanting to add, subtract or multiply rather than using place value or number sense • Not using all of the criteria • Not using place value to solve the questions • Not taking into consideration the starting page of each chapter

Extension

• Change the criteria involving the number of pages in a book and the page number read to and explore the problems again based on the new criteria. • Pupils could think up their own calculator problems and write the criteria to match. • Work out the different possibilities for two children to have three boxes, each with different totals.

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BOOKWORMS Wanda’s book starts on page 1 and has 1380 pages. There are 10 chapters of equal length in the book and she has read up to page 759. 2. Wanda’s favourite page is 409. What chapter is it in?

3. Wanda reads 10 pages each morning and 10 pages each night. How long has she been reading the book?

4. Wanda’s favourite chapter is Chapter 5. What pages are in Chapter 5?

Sa

m

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1. How many pages are in each chapter?

6. How many pages has Wanda read past the middle of the book?

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in

g

5. The most exciting part was from page 619 to page 694. What chapters are these pages in?

7. Wanda’s friend is also reading this book. She has read eight pages of Chapter 6. What page is she up to?

8. How many more pages does Wanda’s friend need to read to finish the book?

9. How many more pages does Wanda need to read to finish the book?

10. If Wanda reads 10 pages per day, how long will it take her to read the book?

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Problem-solving in mathematics

9


CALCULATOR PATTERNS Use your calculator to help you think about what happens. Increasing and dividing 1. Enter a three-digit number (e.g. 345). (b) Try again with another threedigit number. What happens?

• Divide this number by 13.

• Divide the result by 11.

• Divide this result by seven.

(c) Why does this happen?

(a) What number is this?

pl e

• Repeat the three digits to make a six-digit number (e.g. 345 345).

Squaring and multiplying

What do you notice?

in

g

Sa

m

2. Enter a two-digit number (e.g. 78), square it and store the result in your calculator’s memory. • Multiply the numbers that are one more and one less than your original twodigit number (e.g. 79 x 77). • Compare the answer to the square of your original number. • Try some other two-digit numbers.

Vi ew

3. Use another two-digit number, but now multiply the numbers that are two more and two less. What pattern can you see?

4. Now try another two-digit number, but multiply the numbers that are three more and three less. (a) What is the pattern now? (b) What do you think will happen if you try four more and four less? (c) What about five more and five less? 5. Describe a pattern that will explain what happens whenever you multiply two numbers, each increased and decreased by the same amount from an original number.

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Problem-solving in mathematics

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CARAMEL TOFFEES Trevor is having a party. He has wrapped an amount of caramel toffees in nine boxes. The first box (Box 1) has 1 caramel toffee, the second box (Box 2) has 2 caramel toffees, the third box (Box 3) has 3 caramel toffees and so on.

Box 1

Box 2

Box 3

Box 4

Box 5

Box 6

Box 7

Box 8 Box 9

1. How many caramel toffees has Trevor wrapped?

pl e

Trevor wants to give the same number of caramel toffees to each person.

(b)

(c)

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g

Sa

(a)

m

2. If he gives three children 3 boxes each and an equal amount of toffees to each child, which boxes could he give?

Vi ew

3. If he gives two children 4 boxes and an equal amount of toffees each, which boxes can he give? (a)

(b)

4. If he gives two children 3 boxes each, which boxes could he give so each child receives 13 caramel toffees?

(a)

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(b)

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Problem-solving in mathematics

11


TEACHER NOTES

Curriculum links

Materials

base 10 materials, place value chart or a calculator

Focus

Page 15 The problems require more than one step and involve a number of operations, including multiplication. The wording has been kept simple to assist with the problem-solving process. Some problems have additional information, often involving weight, which is not needed and some problems may not have an ‘exact answer’; for example, Question 1 involves 52 boxes of fish. If the dolphins eat eight boxes of fish a day, then there is enough fish for six full days, with four boxes left over. Pupils could discuss their solution in terms of ‘six days’ or even ‘six and a half days’. Pupils should be encouraged to explore and try different ways of arriving at a solution.

Sa

Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Devise and use mental strategies and procedures for carrying out mathematical tasks Number • Solve word problems involving addition and subtraction • Solve and complete problems involving multiplication and division of whole numbers Measures • Solve and complete problems involving units of money

Page 14 These investigations involve information about a mango orchard with 12 hectares of mango trees. The information given in the beginning statement (about the number of hectares of mangoes) is needed to answer the subsequent questions. Using the original information as a basis, the numbers change to meet new criteria as mangoes are picked, sorted and packed into trays. As with the investigations on the previous page, some solutions may not necessarily be exact; for example, in Problem 1, 273 trees are watered each day and so it will take three days for 819 trees to be watered, leaving just 117 trees to be watered on the fourth day. Some pupils may discuss their solution in terms of ‘three and a bit days’ while others may say ‘four days’. Some questions require pupils to keep track of new information and use it to answer the subsequent questions.

pl e

To analyse and calculate information in word problems.

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Problem-solving

Vi ew

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g

The pages explore word problems that mostly require addition or subtraction. Pupils need to determine what the problem is asking and, in many cases, carry out more than one calculation in order to find a solution. Analysis of the problems reveals that some problems contain additional information that is not needed. Materials can, if necessary, be used to assist with the calculation as these problems are about reading for information and determining what the problem is asking rather than computation or basic facts.

Discussion

Page 13 Pupils need to carefully read each problem to determine what the problem is asking. In most cases each problem has more than one step and involves multiplication. The third investigation involves the concept of profit. Pupils need to have an understanding that ‘profit’ is the money left after all costs have been calculated. The fourth problem involves pots being fertilised and plants being watered; however, the solution only involves plants being watered and the information about pots and fertiliser is not required.

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Problem-solving in mathematics

Possible difficulties

• Inability to identify the need to add, subtract or multiply • Confusion over the need to carry out more than one step to arrive at a solution • Using all of the numbers listed in the problems rather than just the numbers needed • Difficulty with the concept of profit

Extension

• Discuss how problems can have more than one answer depending on different interpretations. • Pupils could write their own problems and give them to other pupils to solve. • Explore how many of these problems could be solved using the repeated addition technique on the calculator; i.e. enter 273 + 273 and then hit the ‘equal’ key—keep hitting the key and you will keep adding 273 without having to continue entering the number.

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pl e

THE PLANT NURSERY

Sa

m

1. During the week, the plant nursery planted 45 rows of flower seedlings and 18 rows of palm seedlings. There are 18 palm seedlings or 56 flower seedlings in each row. How many seedlings were planted?

Vi ew

in

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2. The nursery sold 28 bags of bark chips on Monday, 23 bags on Tuesday, 34 bags on Wednesday and 29 on Thursday. If each bag costs €27 to buy, how much money did the nursery make from selling bark chips?

3. If the nursery buys each bag of bark chips for €12, how much profit did the nursery make from selling the bark chips?

4. During the morning, 46 pots were fertilised and 45 rows of plants were watered. During the afternoon, another 56 rows of plants were watered and 36 pots were fertilised. If there are 65 plants in each row, how many plants were watered? 5. The nursery uses large punnets that hold 9 seedlings and small punnets that hold 6 seedlings. How many seedlings were planted if 236 large punnets were used?

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Problem-solving in mathematics

13


THE MANGO ORCHARD

Sa

m

pl e

The tropical fruit farm has 12 hectares of mangoes, eight hectares of bananas and three hectares of lychees.

Vi ew

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1. Each hectare of mangoes has 78 trees and is watered on a rotational basis. Three and a half hectares are watered each day before sunrise. How many trees are watered each day and how many days will it take for all of the trees to be watered?

The mango crop is ready to harvest. Each tree has over 80 mangoes. 2. How many mangoes are ready for sorting and packing into trays after 6 hectares have been picked?

After harvesting, the mangoes are sent to the packing shed for sorting and are packed into trays of 16. Some damaged and blemished mangoes are rejected and are not packed into trays. About seven mangoes per 100 are rejected. 3. How many trays of mangoes were packed after the six hectares were picked? 4. How many trays of mangoes were packed after all of the mangoes were picked?

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Problem-solving in mathematics

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ANIMAL WORLD

1. The dolphins at Animal World eat 8 boxes of fish a day. Each box weighs 2 kg and costs €9.50. How many days does €494 worth of fish last?

m

Sa

2. The car park has 9 sections for cars to park in. Each section holds up to 230 cars. If sections one to six are completely full and sections seven to nine are half full, how many cars are in the car park?

in

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3. The giraffes eat 7 bales of hay over two days. Each bale of hay costs €12.00. How much would it cost to feed the giraffes during the month of September?

4. The tropical birds in the walk-through aviary eat 14 sacks of seed a week. Each sack weighs 40 kg and costs €58.00. How many sacks are used over one year? 5. The area to see the dolphin show has 18 rows of seats, with 26 seats in each row. If there are 247 adults and 59 children already seated, how many more people can watch the show? 6. The seals eat 2 cartons of fish a week. Each carton weighs 12 kg, contains six boxes of fish and costs €57.00. How many kilograms of fish do the seals eat over a year?

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Problem-solving in mathematics

15


TEACHER NOTES

Curriculum links

For Problem 1, the notional ‘middle’ of the ladder has to be taken for granted and the effects of moving up and down the ladder are measured against this position. This means that in effect the fireman has moved up 12 rungs and then 14 rungs to get to the roof. There are 26 rungs from the middle to the top so there must also be 26 rungs from the middle to the bottom; therefore, the ladder has 53 rungs. Similar reasoning provides the path of the ball, remembering that it only drops the full distance once, then each other distance twice until the last distance, which is also only measured once—6.75 m in total. The lift and airplane problems are solved in the same manner.

Possible difficulties

• Unable to use a diagram or materials to come to terms with the problems • Not considering the movements relative to an initial position to obtain an appropriate starting point; e.g. Thinking there are 52 rungs rather than 53 • Not realising that a ball will bounce both up and down before getting to its new level • Simply working on the basis of calculations with the numbers in the problem to obtain incorrect answers

Sa

Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Devise and use mental strategies and procedures for carrying out mathematical tasks Number • Solve word problems involving addition and subtraction • Solve and complete problems involving multiplication and division of whole numbers Measures • Solve and complete problems involving the addition, subtraction, multiplication and division of units of length

in the introduction (pages xiv – xvi). In the problems, the information needs to be carefully analysed to determine the distance at the beginning rather than at the end of a situation. Some pupils may think of using counters to represent the changing positions as the situation is worked through either forwards or backwards. Using a diagram or ‘try and adjust’ are other alternatives to find a solution.

pl e

To use logical reasoning and spatial visualisation to solve problems.

m

Problem-solving

Materials

g

counters, grid paper, calculator

in

Focus

Vi ew

This page explores problems based on a conceptual understanding of height, along with an awareness that it is the initial position that is required, not the end point. One way of solving the problems is by backtracking from the answers. Using counters on a vertical grid or line can be useful as they allow the individual movements up and down to be considered while the intent of the problem is kept in mind. The use of a diagram or calculator are other ways to manipulate the data in the order in which it needs to be used.

Discussion

Extension

• Change the numbers in the problems but leave the questions the same. • Change the problems’ contexts but leave the numbers the same. • Have pupils make up problems like these of their own and challenge others to solve them using diagrams or materials.

Page 17 These problems can be used to highlight the power of the ‘Analyse-Explore-Try’ model of problem-solving that has evolved over the varied number, spatial and measurement situations posed in this book. This is discussed in detail

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Problem-solving in mathematics

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UP AND DOWN 1. A firefighter stood on the middle rung of an extended ladder, spraying water onto a burning apartment block. As he succeeded in dampening down the fire, he climbed seven rungs. A sudden flare-up sent him down 12 rungs. After it died down, he moved up 17 rungs. When the fire was put out, he climbed the remaining 14 rungs to the top of the ladder and got onto the roof. How many rungs did the extended ladder have?

Sa

m

pl e

2. Roberta drops a ball from the terrace to the patio 2.4 m below. The ball then bounces up and down until her friend, Shen, catches it at a height of 15 cm from the ground. It is a bouncy ball, at each bounce travelling half the distance of the previous bounce. How far did the ball travel from the time Roberta dropped it until it was caught by Shen?

g

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3. You enter the lift at a certain floor of a tall building. Then the lift moves up 16 floors, down 19 floors, and up 11 floors. You are now at Floor 31. At which floor did you enter the lift?

4. A plane is flying at cruising altitude when turbulence occurs and the pilot flies to a new altitude 2300 m higher to get above the rough weather. Then a lightning storm hits and she descends 4800 m to avoid it. To be safe, when the storm has finished she ascends 3100 m to be at a height of 11 200 m. What is the usual cruising altitude for the plane?

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Problem-solving in mathematics

17


TEACHER NOTES

Curriculum links

Number of pages

4 x number of pages + 7

5 x 1 less than number of pages

4

23

15

8

39

35

12

55

55

There are 12 pages and 55 photos.

Page 21 Problem 1 can be solved by using coins or counters to keep track of what is happening. Another way is to put the information in a table where what is occurring is readily seen:

Sa

Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Record the processes and results of work using a variety of methods • Devise and use mental strategies and procedures for carrying out mathematical tasks Number • Solve word problems involving addition and subtraction • Solve and complete problems involving multiplication and division of whole numbers Algebra • Explore, recognise and record patterns in number

Page 20 These problems require knowledge of division with remainders, multiples and factors. There are many ways they can be solved—using counters to see what is happening, ‘try and adjust’ using different numbers, placing the attempts in a table to organise an approach, and using some form of simple algebraic thinking. Analysing Problem 1 shows that 4 x number of pages + 7 must be the same as 5 x 1 less than the number of pages:

pl e

To use strategic thinking to solve problems.

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Problem-solving

Materials

10c

counters in several different colours, coins

10c

10c

€1

Focus

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Discussion

Page 19 The puzzle scrolls contain a number of different problems all requiring strategic thinking to find possible solutions. In most cases, pupils will find tables, lists and diagrams are needed to manage the data while exploring the different possibilities. In Problem 2, combinations involving four digits that add to 19 need to be considered remembering that 0 can be used in a pin number. Pupils also need to consider that a number can be used more than once; for example, 4, 4, 4 and 7 could be used.

18

10c

€1

Problem-solving in mathematics

10c €1

€2

This page explores other complex problems in which the most difficult step is to find a way of coming to terms with the problem and what each question is asking. Using materials to explore the situation is one way this can be done. Another is to use a diagram to assist in thinking backwards or trialling and adjusting to find a solution that matches all of the conditions.

10c

10c

10c

€2 50c

10c

€1

10c

10c

€1 €2

€1 €2

50c 20c

10c

50c 20c

He would have €9.20 and three 10c coins would not change. This approach will readily solve the second problem as well. The pattern is that the coins in a prime number position (2, 3, 5, 7, 11, 13, …) only change twice; and only those in square number positions will show heads (1, 4, 9, 16, … ).

Possible difficulties

• Considering only some aspects of the puzzle scrolls • Not taking into consideration the remainders when considering multiples • Not being able to keep track of the changes in the coins

Extension

• Write problems based on the puzzle scrolls. • Provide other combinations of pictures and pages for the scrapbook problem. • Extend the coin problems to start with larger numbers of coins and look for patterns.

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PUZZLE SCROLLS 2. My pin number has four digits which when added totals 19. It could be 1, 3, 6 and 9.

If the amount of water it stored doubled each day, how long did it take to be half full?

What else could it be?

27

36

in

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63

48

54

m

24

4. A maths club starts with one member, who then recruits two new members the following week. Each new member then does the same each following week.

Sa

3. (a) What two-digit number is 3 times the sum of its digits?

pl e

1. A new dam was built and it took 48 days for it to fill with water.

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(b) Can you find any that are four times the sum?

How many members are there after 5 weeks?

5. An atlas has 6 pages of maps of Australia. The sum of these pages is 237.

6. What 2 whole numbers, neither of which contain a zero, have a product of 1 000 000?

What are the page numbers?

How many can you find?

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Problem-solving in mathematics

19


SCRAPBOOKING

Sa

m

pl e

1. Layla is organising photos of her family to put in her scrapbook. If she puts 4 photos on each page, 7 photos will not be able to be included in her scrapbook. If she puts 5 photos on each page, 1 page will be empty. How many photos does she have to put in her scrapbook and how many pages does the scrapbook have?

Vi ew

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2. When he saw what his sister was organising, Sammy decided to put pictures of his favourite cars in a scrapbook like Layla’s. He had some large and some smaller pictures, so he decided to put 4 pictures on each page. However, this meant there were 3 pages without any pictures. When he changed the arrangement to 3 pictures on a page, there were 5 pictures that he had no space for. How many pictures did he have and how many pages were in his scrapbook?

3. Sammy noticed that when the football players in his club were being organised to sit on benches for the end-of-season photo, if 5 players were seated on each bench, 4 players did not get a seat. When they squashed up a bit and fitted 6 players on each bench, they now had enough room for the coach and assistant coach to sit down. How many football players are there in Sammy’s club? 4. What is the smallest number that when it is divided by 3, there is a remainder of 1; when it is divided by 4, there is a remainder of 2; when it is divided by 5, there is a remainder of 3; and when it is divided by 6, there is a remainder of 4?

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Problem-solving in mathematics

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CHANGING COINS John put out a row of twelve 10c coins. He asked his mother to exchange every second coin for a €1 coin, his father to exchange every third coin for a €2 coin, his brother to exchange every fourth coin for a 50c piece and his sister to exchange every fifth coin for a 20c piece. 1. (a) How much money did he have after all the exchanges were made? (b) How many of the original 10c coins were not changed? (c) Were any coins changed three times? (d) Which coins were changed twice?

Once?

Sa

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His sister, Jane, set out sixteen 10c coins tails up in a row. Then she changed the coins by turning them over so that tails became heads or heads became tails.

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The first time she turned over every coin, the next time every second coin, the next time every third coin and so on until she had turned the 16th coin five times.

2. (a) After she made all the changes, which coins displayed heads? (b) Which coins only changed once?

Twice?

(c) If she had put out more coins and turned them, what would be the next coin to show heads? (d) What other coins would only be changed twice? (e) Can you describe a pattern for the changes?

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Problem-solving in mathematics

21


TEACHER NOTES

Curriculum links

Materials

place value chart, calculator

g

Focus

Vi ew

in

These pages explore word problems that require a number of operations, including division. The wording has been kept fairly simple to help with the problem-solving process. Pupils need to determine what the problem is asking and in many cases carry out more than one equation in order to find solutions. If necessary, materials can be used to assist with the calculation as these problems are about reading for information and determining what the problem is asking rather than computation or basic facts.

Discussion

Page 23 These problems require more than one step and involve a number of operations, including division. The wording has been kept fairly simple to assist with the problem-solving process. In many cases, the problem involves more than one fruit and pupils need to read the problem carefully to determine what is needed to find a solution; for example, in Problem 4 pupils need to consider apricots and pears as well as trays and boxes in order to find a solution.

22

Page 25 These investigations involve weight; in particular, tonnes. In most cases more than one calculation is needed to find a solution. The first question results in a calculation of 763.3 bins. This is an example of where the answer does not result in a whole number and would need to be thought of as either 763 full bins and 1 bin which is not full or 764 bins. Similar thinking is required in Problem 6—37 full loads and one incomplete load are required to transport all of the sugar. As three trucks are used, there would be 12 trips using all three trucks and one trip needing only two trucks.

Sa

Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Devise and use mental strategies and procedures for carrying out mathematical tasks Number • Solve word problems involving addition and subtraction • Solve and complete problems involving multiplication and division of whole numbers Measures • Solve and complete problems involving the addition, subtraction, multiplication and division of units of weight

Page 24 These investigations involve more than one step and use the concepts of grams and kilograms (weight). The concept of profit and repackaging is explored in many of the problems. Pupils may find it helpful to draw a diagram to visualise what is happening in order to find a solution; for example in the first question, pupils could draw and label the cost of a large container followed by 10 small tubs. In this way they can visualise the actual process of repackaging and the cost involved in buying the large container and selling the smaller tubs. Problem 2 requires pupils to think in terms of how many containers are needed, instead of an exact amount. Four containers are needed to make 34 tubs, but not all of the olives will have been used as another three tubs could be made from the last container.

pl e

To analyse and use information in word problems.

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Problem-solving

Problem-solving in mathematics

Possible difficulties

• Inability to identify the need to add, subtract, multiply or divide • Confusion over the need to carry out more than one calculation to arrive at a solution • Using all of the numbers listed in the problems rather than just the numbers needed • Not thinking in terms of the problem and writing solutions such as ‘763.3 bins’ • Difficulty with the concept of ‘tonnes’

Extension

• Pupils could write their own problems and give them to other pupils to solve.

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pl e

FRUIT FARM

m

1. An amount of pears, peaches and apricots have been picked and are ready for packing into trays. There are 406 pears, 739 peaches and 615 apricots. If each tray holds nine peaches, how many trays are needed?

Sa

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2. A box of pears holds 4 trays and each tray holds 12 large pears. If there are 1152 pears to be packed, how many boxes and trays are needed?

in

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3. (a) The fruit pickers picked 2340 medium-sized mangoes over 6 days, which were then packed into 52 boxes. How many mangoes are in each box?

(b) If each box holds 3 trays, how many mangoes are in each tray? 4. There are 35 boxes of apricots and 68 trays of pears ready for the market. Each box has 48 apricots and each tray has 12 pears. How many pears and apricots are packed? 5. The fruit pickers picked 487 peaches in the morning and 364 peaches in the afternoon. If they worked for 6 hours, approximately how many peaches did they pick per hour?

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Problem-solving in mathematics

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AT THE DELICATESSEN OPEN SIX DAYS A WEEK (CLOSED MONDAYS)

FRESHLY SLICED

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1. The delicatessen buys 2 kg containers of sun-dried tomatoes, which they repackage and sell in 200 g tubs for €6. If each container costs them €19, how much profit do they make on each container?

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2. About 34 tubs are sold each week. How many containers does the delicatessen need to buy each week so that there are enough tubs to sell?

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3. How many containers would they need during the month of March?

4. The deli sells marinated green olives in 400 g tubs and loose by the kilogram. The tubs sell for €8.50 and the loose olives sell for €18.50 kg. Which is the better value way to buy 1.6 kg of olives?

5. The deli buys its loose olives for €12.30 kg. During the summer months, about 13 kg of loose olives are sold each week and during the winter months about 9 kg are sold. What is the difference in weekly profit between the summer and winter months? 6. Eighteen boxes of bacon were delivered. Each box contains 12 packets of bacon. Six boxes were unpacked in the morning and 8 boxes in the afternoon. How many packets have been unpacked?

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THE SUGAR MILL

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1. During the crushing season, the sugar cane farm transported 4580 tonnes of cane by train to the sugar mill. A train usually hauls between 135 and 145 bins. If each bin holds 6 tonnes of cut sugar cane, how many bins of cane were transported to the sugar mill?

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2. If each train is able to haul a maximum of 145 bins, how many train trips to the mill were there during the season?

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3. During one week, trains transported cane from four different farms. If 19 370 tonnes of cane was transported, how many bins and trains were used?

4. During the harvesting season, one farm transported 6790 tonnes of sugar cane to be crushed. If the harvesting season is from June to December, approximately how much cane was harvested per month? 5. It takes approximately 8 tonnes of cut sugar cane to produce one tonne of raw sugar. How much raw sugar would be produced from 8972 tonnes of cut sugar cane? 6. During the crushing season, raw sugar from the mill is transported to the port by trucks, each of which can carry 7 tonnes. How many trips would be needed to transport 790 tonnes of raw sugar if 3 trucks were used?

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Problem-solving in mathematics

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TEACHER NOTES

Curriculum links

Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Devise and use mental strategies and procedures for carrying out mathematical tasks Number • Solve word problems involving addition and subtraction • Solve and complete problems involving multiplication and division of whole numbers Algebra • Explore, recognise and record patterns in number

The second problem is based on multiples of 7. Dividing a number by 7 will give a remainder from 1 to 6 (a multiple of 7 will always be in Column B). 1 will be in Column A, 2 in Column C, 3 in Column E and so on. 1000 is in Column D. Prime numbers are important as they provide insight into all numbers and are the basis for security codes for credit cards, bank accounts and building entry cards. This arrangement of numbers does not include one as one is not prime (its only factor is one). 2, 3, 5 and 7 are prime numbers, while 4 and 6 also have 2 as a factor and are not prime numbers. Any number in the column under 2 will have 2 as a factor, any number in the column under 3 will have 3 as a factor, any number under 4 will have 2 as a factor and any number under 5 will have 2 and 3 as factors.

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Finding a number is just a matter of determining how the patterns can be used. After a multiple of 8, the numbers start from Column A again and distribute themselves as the digits in the first two rows. A number which is a multiple of eight and 1 more is in Column A, 2 more is under B, 3 more is under C and so on. Dividing a number by eight provides the remainder and the column is readily found. 1000 is in Column A and 1001 will also be in Column A.

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To use patterns and logical reasoning to determine numbers in a table.

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The prime numbers can only be in the columns headed by 5 and 7. However, not all of the numbers in these columns are prime; for example, 25 occurs in the column under 7 and 35 occurs in the column under 5. This means that a prime number must be one more or one less than a multiple of 6—but not all numbers one more or one less than a multiple of 6 will be a prime number.

Discussion

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This page explores pupils’ understanding of numbers in order to discern patterns that allow larger numbers to be determined without laboriously writing or counting all of the numbers up to the point asked for. It also highlights the value of using factors and multiples when thinking about numbers.

Page 27 There are many ways to solve the problems, all involving the concept of factors and multiples; together with searching for patterns in the table of values. The first thing to notice is how the numbers increase from left to right in one column then right to left in the next. For Problem 1, this means considering multiples of 4 by looking across a row or multiples of 8 to describe the digits under a particular letter. Other pupils may consider multiples of 20, as the whole table repeats itself in blocks of 20 (except that one switches from Column A to D for each block of 20). This may lead other pupils to focus on blocks of 40. All of these ways of thinking will show that 101 is in Column D.

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Problem-solving in mathematics

Possible difficulties

• Thinking that writing out all of the numbers is the only way to be sure of a solution • Only considering the ones place for the pattern • Unable to verbalise a mathematical description of how the numbers are placed in columns or where the prime numbers are found

Extension

• For Problems 1 and 2, describe a pattern for the row where a number occurs. • Investigate the topic of prime numbers further. • Find some background information about Eratosthenes and other Greek mathematicians who were interested in numbers, such as Pythagoras.

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NUMBERS IN COLUMNS The counting numbers are arranged in four columns: A, B, C and D.

A

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1. (a) In which column will 101 appear?

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(b) Can you describe a pattern for the columns to help you find where any number will be?

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The arrangement of counting numbers was changed again to seven columns:

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(b) In which column will 1000 appear?

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2. (a) Describe a pattern for the columns to help you find where numbers are.

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(c) In which column will 1001 appear?

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A prime number is a number that has only two factors, one and itself. They are very important in understanding numbers.

A famous Greek mathematician, Eratosthenes, arranged the counting numbers in columns to see if he could discover a pattern for finding all the prime numbers. This was his arrangement: 1

2 8 14 20

3 9 15 21

3. (a) Circle the prime numbers.

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4 10 16 22

5 11 17 ‌

6 12 18

7 13 19

(b) Can you find a pattern?

Problem-solving in mathematics

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TEACHER NOTES Problem-solving

To identify and use pupils’ understanding of number.

Curriculum links

Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Explore and investigate mathematical patterns and relationships Algebra • Explore, recognise and record patterns in number

Page 31 This page explores sudoku. The word ‘sudoku’ roughly means ‘digits must only occur once’. In this case, 4-by-4 and 6-by-6 grids are used. In Problem 1, every row, column and mini-grid must contain one of each of the digits 1 to 4 while the second set uses the digits 1 to 6. No addition or basic facts are involved and pupils need to use logical reasoning to find solutions.

counters, blocks or a calculator

Focus

Possible difficulties

• Considering only rows or columns rather than rows, columns and diagonals

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These pages explore problems involving number sense, magic squares and logic. Analysis of the problems to locate given information is necessary to find the magic number or the arrangement of numbers. Counters, blocks or a calculator can be used to assist as the focus of the problems is on the concepts of number sense and number logic rather than solving basic facts.

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Page 30 In these questions, the concepts of magic squares has been extended from squares to different shapes, where each line or diagonal adds to the same number. The first set of shapes uses the digits 1 to 6 in various combinations to make different magic numbers. Problem 2 uses the digits 1 to 9. Unlike the first examples, pupils are already given the magic numbers for each slope and they need to organise the digits accordingly. The last shape has no starting numbers and pupils may need to use the ‘try and adjust’ strategy in order to find a solution.

Extension

Discussion

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• Investigate other magic squares, magic numbers and magic shapes. • Explore sudoku games in magazines, newspapers and on the Internet that involve 6-by-6 grids and 9-by-9 grids.

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Page 29 This investigation involves the concept of magic squares. Simple 3-by-3 magic squares have been used to help pupils to come to terms with the idea that each row, column and diagonal adds to a magic number. This concept is further explored as pupils investigate 4-by-4 magic squares. The last magic square has a number of points of interest aside from just being a magic square—when turned upside down it is still a magic square and although the numbers change, the magic number stays the same!

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MAGIC SQUARES Magic squares have numbers that all add to the same total. All of the rows, columns and diagonals add to the same total. 1. This magic square has a magic number of

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2. Complete the magic squares below. Remember, all rows, columns and diagonals must add to the same number. (c)

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Magic number

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3. Now complete these 4-by-4 magic squares.

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Look at this unusual magic square! Find the magic number for this magic square. Check the totals for each row, column and diagonal. 68 4. (a) The magic number is

89

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88

Now turn it upside down and check again.

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(b) What do the numbers add to now?

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Problem-solving in mathematics

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MAGIC SHAPES 1. Using the numbers 1 to 6, make each line add to the same total. Write each shape’s magic number. (a)

(b)

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2. Try these shapes, using the digits 1 to 9. The magic number is written under each shape. (b)

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SUDOKU Sudoku puzzles use numbers and to solve them you must use logic to work out where the numbers go. Every row, column and mini-grid must contain one of each of the numbers 1, 2, 3 and 4; for example: 2

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The completed sudoku has the numbers 1, 2, 3 and 4 in every row, column and mini-grid.

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1. Complete each sudoku, using the digits 1 to 4.

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2. Complete each sudoku, using the digits 1 to 6. (a)

2 5

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TEACHER NOTES

Curriculum links

Materials

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These pages explore the concepts of averages, distance and payments and involve pupils carefully reading interrelated statements within problem situations. Pupils need to read the stories carefully in order to take into consideration a number of different criteria. Tables and lists can be used to help manage the various criteria.

Discussion

Page 33 These investigations consist of a series of interrelated statements that need to be carefully understood in order to find a solution. The use of a table or list can be very helpful to manage the data. For example, in Problem 1, a table listing the years 2003 to 2007 can be used as a starting point. The problem states how many visitors there were in 2003 and this information can be used to work out the number of visitors in 2007—twice as many. That information can in turn be used to work out the numbers for the other years. A similar table or list can be used for the other problems. The last problem contains additional information which is not needed to find a solution.

32

Problem 4 contains information about a morning stop and lunch that needs to be factored into the amount of timespent driving. If they leave at 8:30 in the morning, stop for 30 minutes during the morning and then have lunch at 12:30 pm, they have driven for three and a half hours, not four hours. The information about the distance from Paris to Marseille is additional information that is not needed to find a solution.

Page 35 Careful reading of each problem is required to determine what each problem is asking. With Problem 1 it is necessary to take into consideration whether or not it is a leap year, as you would need to divide by either 365 days or 366 days. With Problem 3, one way to solve the problem is to find a common multiple, such as 24. For every 24 peaches he sells he makes €3 profit. To have made €45 profit he needs to have sold 15 sets of six peaches. With Problem 4 it is necessary to understand that 39 is three-quarters and so one-quarter is 13—so four-quarters would be 52 litres. The last problem requires pupils to calculate how many 250 gram packs can be made from 10 kg of cheese and then to use this information to solve how much profit is made over a month.

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The third investigation deals with the concept of distance travelled over a month. Discussion could centre on how this would vary from month to month. A table could be constructed to show the distance travelled for each month of the year.

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Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Devise and use mental strategies and procedures for carrying out mathematical tasks Number • Solve word problems involving addition and subtraction • Solve and complete problems involving multiplication and division of whole numbers Measures • Solve and complete problems involving the addition, subtraction, multiplication and division of units of length, weight, capacity, time and money

Page 34 These problems explore the concept of average distance travelled over a period of time. In many cases the solution is not necessarily exact but rather an approximate time or distance; for example, Problem 1 states that Gemma swims 100 m in ‘about 2 minutes’. This is not an exact time and would vary from lap to lap, so the solution of how far she has swum would again be an approximate distance.

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To interpret and organise information contained in a series of interrelated statements and to use logical thinking to find solutions.

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Possible difficulties

• Not using a table or list to manage the data • Not understanding ‘average’ • Confusion when dealing with approximate times and distances

Extension

• Construct a table to show the running distance and how it varies from month to month. • Write other problems, using the same form of complex reasoning for other pupils to solve. www.prim-ed.com

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TOURISTS IN AUSTRALIA 1. The Olgas, in the Northern Territory, had a large number of tourists during 2007. There were twice as many visitors in 2007 than in 2003. There were 89 530 more visitors in 2007 than in 2006 and 106 219 less visitors in 2004 than 2006. If there were 208 460 visitors in 2003, how many were there in 2006?

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2. During September, 258 693 tourists visited Sydney. May had twice as many visitors as January, but 6328 less than July. July had 8269 more visitors than September. How many visitors were there in January?

3. About 210 000 people visit Kakadu National Park each year. Visitor numbers are greatest during the dry season months of June to September, and lowest during the wet season months of December to February. During July, there were 33 843 visitors. June had 24 450 more visitors than January and February had 26 730 less than July. January had 25 470 less visitors than August and August had 18 390 more visitors than February. How many visitors were there in June?

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Problem-solving in mathematics

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HOW FAR?

1. Gemma swims Mondays, Wednesdays and Fridays in a 50-m pool and Tuesdays and Thursdays in a 25-m pool. She can swim 100 m in about two minutes. She usually swims for one hour in the 50-m pool and one hour and 45 minutes in the 25-m pool. Approximately how far does she swim each week?

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2. Lena caught the bus from Rome, in Italy to Munich, in Germany. The bus left at 7.30 am and, due to traffic, averaged 49 km for the first 2 hours. Once on the motorway, the average speed increased to 93 km per hour. The bus stopped for lunch at 12.15 pm. How far had Lena travelled?

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3. Brenda trains each day for the marathon. During the week, she runs 14 km in the morning and 11 km in the afternoon. On the weekend, she runs 32 km on Saturday and 29 km on Sunday. How far does she run in August?

4. Avril and her sister drove from Paris to Marseille, a distance of 809 km. They left at 8.30 in the morning and had lunch at 12.30. They stopped for 30 mins during the morning and one hour for lunch. They averaged 87 km per hour before lunch and 93 km per hour after lunch. How far had they driven by 4.00 pm?

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HOW MUCH? 1. The city council charges each householder yearly. In Kath’s last council bill she was charged €124.75 per quarter for water usage and €106.50 per quarter for rubbish collection. How much does she pay per day for water usage and rubbish collection?

2. Bev bought 54.2 kg of lavender honey and 39.4 kg of heather honey. When she got home she mixed them together and stored the honey in 78 jars. How much honey is in each jar?

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3. Malik bought peaches from the market at 8 for €3. He then sold them at his fruit shop for 6 for €3. Over the weekend he made a profit of €45 selling his peaches. How many peaches did he sell?

4. Scott’s car was out of petrol. He pushed it to the service station and pumped 39 litres of petrol into the tank. As he drove away, he noticed that the tank was still one-quarter empty. How much petrol does a full tank hold?

5. Greta’s cheese shop sells 100 kg of cheddar cheese each month. She buys the cheese in 10 kg blocks for €96.50. She then cuts the cheese into smaller 250 g packs that she sells for €4.50 each. How much profit does she make selling cheddar cheese each month?

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Problem-solving in mathematics

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TEACHER NOTES

Curriculum links

Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Devise and use mental strategies and procedures for carrying out mathematical tasks Number • Solve word problems involving addition and subtraction • Solve and complete problems involving multiplication and division of whole numbers Measures • Solve and complete problems involving the addition, subtraction, multiplication and division of units of money

Materials

Motorcycles (2 tyres) 119 117 115 113 111 109 107 105 103 101

Cars (4 tyres) 0 1 2 3 4 5 6 7 8 9

Number of vehicles 119 118 117 116 115 114 113 112 111 110

Number of tyres 238 238 238 238 238 238 238 238 238 238

Problems 3 and 4 require careful reading to determine how the information needs to be used. Some pupils may simply try to subtract the smaller number from the larger and obtain an answer of €1000 for Problem 1, which can not be correct. The expression ‘more than’ is critical—if the trailer tyres cost €1000, the tractor tyres would cost €10 000 and the total cost would be €11 000, which is too much. Subtraction gives the difference between the two prices, while the actual amounts need to match both the total and the difference. A table can be used to keep track of the possibilities until a match is found. Logical reasoning about the different amounts is another way of determining the solutions.

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would not be possible to have only cars). If pupils put the numbers in a table and work through all of the possible combinations (with the amount of tyres remaining constant and there being more than 100 vehicles), this will provide all of the possible combinations of motorcycles and cars:

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To use strategic thinking to solve problems.

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Discussion

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This page explores problems that may have several answers and further analysis of the data is needed to determine whether this is the case or if there is a unique solution. A process of ‘try and adjust’ could be used; however, logical reasoning of the possibilities and the use of a table or diagram is more productive. These ways of thinking provide a means to solve other complex problems.

Page 37 There are several ways these problems can be solved. The first problem states that cars need four tyres and trucks need ten tyres. Since 92 vehicles were given new tyres, they could not all be cars because then only 368 tyres would be needed. The other 282 tyres must have been put on the trucks. Since a truck has six more tyres than a car, there would be 27 trucks and 45 cars (a total of 650 tyres). Another way would be to put multiples of two or four into a table and systematically check and adjust results until a solution is reached. Problem 2 can be solved in a similar way by using a table. However, the second condition in this problem does not give a particular number, only the requirements that there are more than 100 vehicles and an odd number of motorcycles, so there are several possibilities. If there were only motorcycles, there would be 119 of them (it

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Problem-solving in mathematics

Possible difficulties

• Not using a table or diagram to manage the data • Not considering all of the possible answers when only information about the tyres is given for the cars and motorcycles—there are 19 possibilities! • Only considering one condition when there are two to be considered

Extension

• Discuss the various methods available to solve the problems. Include those discussed above. Ask pupils to solve each problem using a different method to that they used or tried first. Encourage them to use a table to organise their approach. • Change the conditions for the first problem; e.g. 650 motorcycle and truck tyres, 650 motorcycle and car tyres, different numbers of vehicles or tyres. • With the second problem, use an odd number of trucks rather than motorcycles. • Ask the pupils to make problems, based on Problems 3 and 4, for other pupils in the class to answer. www.prim-ed.com

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FITTING TYRES 1. A tyre company replaces tyres on motorcycles, cars and trucks. Motorbikes have two wheels, cars have four wheels and trucks have ten wheels. On Tuesday, they replaced 650 car and truck tyres on a total of 92 vehicles. How many cars and how many trucks did they fit new tyres to?

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2. On Saturday, the tyre company is only open in the morning and works on cars and motorcycles but not trucks. At the end of the morning, the supervisor noticed that 238 tyres had been used, that more than 100 vehicles had been given new tyres and that the number of motorcycles that had received new tyres was an odd number. How many cars and how many motorcycles could there have been?

3. Purchasing tyres for a tractor and its trailer costs €10 000. If the tractor tyres cost €9000 more than the trailer tyres, what is the cost of the trailer tyres?

4. The tyres for a truck and its trailer costs €100 000. If the truck tyres cost €25 000 more than the trailer tyres, what is the cost of the truck tyres?

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Problem-solving in mathematics

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TEACHER NOTES Problem-solving

To use spatial visualisation and measurement to solve problems.

Curriculum links

4 requires the visualisation of the smaller triangles within the overlapping large triangle and the hexagon shape of the overlap that make up the garden and pond to see the area and perimeter of the shapes.

Problems 2 and 3 can be solved in a similar way by determining what part the triangles are of the total area. Rather than using complex mathematics, pupils can calculate the area of the triangles once they know the area of the entire shape—in this case 160 m2. Because the triangles span the midpoints of the squares, we know that they are a certain fraction of the entire area; for example, in Problem 2, the garden bed is one-eighth of the entire area or 20 m2. Using the same thinking as Problem 1, the gardens form 1 quarter of the square or 40 m2.

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These pages explore pupils’ ideas of area and perimeter, using their knowledge of squares and triangles to visualise shapes and to determine the areas of smaller shapes or lengths from which the squares and triangles are composed. Spatial and logical thinking, as well as numerical reasoning and organisation, are involved as pupils investigate the interrelationship of area and perimeter.

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Page 41 This page extends the thinking about areas. For Problem 1, trying to calculate the area of the path alone would be tedious and require a level of mathematics beyond the capacity of most of the pupils at this age. Instead, it is better to think of the park as being a square with a path and four triangular garden beds. From the information provided, we know that the length of the long side of each triangle is 12 metres. These 4 gardens can be put together to form a square with an area of 12 m x 12 m = 144 m2. Each side of the park is 18 m in length so the park has an area of 324 m2. Subtracting the area of the garden beds from the size of the park shows that the area of the paths is 180 m2.

Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems Measures • Understand and measure the perimeter of regular 2-D shapes • Solve and complete practical tasks and problems involving length

Discussion

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Page 39 Pupils need to be able to visualise the way the shape in Problem 1 is made from three squares, each of which is made of four smaller squares. In this way, the shape can be seen as the sum of two large squares and two small squares; or one large square and two shapes made of three small squares; or three large squares with an area of half a large square (or two small squares) removed. Since the length of each side of the large square is 12 cm, the area of a large square is 144 cm2 and the area of a small square is 36 cm2. For Problem 2, half of the large rectangle is made of two smaller rectangles, each of which contains two small triangles. This means that the smaller triangle must be seen as half of the area of the larger triangle and one-eighth of the area of the whole shape. For the third problem, each small square has an area of 100 cm2 and the new area is found by extending the pattern to find the number of small squares. Page 40 For Problem 1, pupils need to be able to visualise how the shapes are made of squares and how the perimeter of the final shape uses only some of the sides of the squares, each of which has an area of 16 cm2 and length of 4 cm. Problem

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Problem-solving in mathematics

Possible difficulties

• Uncertain of area and perimeter • May confuse area and perimeter because of a reliance on rules in place of understanding • Does not understand that sides in a square or equilateral triangle are of equal length • Unable to visualise the smaller shapes within the large shapes • Cannot keep track of the number of sides that need to be used to determine perimeter

Extension

• Have pupils investigate shapes made from overlapping equilateral triangles in the same way as those made from overlapping squares. • Ask pupils to create their own shapes made up of squares or equilateral triangles to determine areas and perimeters. • Investigate parks made up of paths and triangularshaped lawns or garden beds with different dimensions for the sides of the garden, path or triangular areas. • Investigate paths and garden beds within different shaped parks.

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AREA Area is the amount of surface there is of a shape.

This shape is made from three overlapping squares that are all the same size and which all have sides of 12 cm in length. 1. What is the area of the shape?

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This rectangle has an area of 144 cm2. The larger triangle had been drawn by connecting the mid-points of three of the sides of the rectangle. 2. What is the area of one of the smaller triangles?

This shape is made from 10 small squares and has a height of 40 cm. The shape is continued until it has a height of 120 cm. 3. What is the area of the new shape?

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AREA AND PERIMETER 1 Perimeter is the distance around the boundary of a shape.

1. This shape is made from eight small squares. It has an area of 128 cm2. What is the perimeter of the shape?

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2. Nine square-shaped areas were used to form a terrace around a large goldfish pond. If the perimeter of the terrace is 160 m, what is the area that has to be tiled to form the terrace?

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3. This garden was made from 17 square pieces of turf, each made of a different type of native grass. If the distance around the boundary of the garden is 180 m, what is the area of the garden?

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4. (a) A star-shaped garden was made by overlapping two equilateral triangles. If the perimeter of each triangle is 7 m 20 cm, what is the perimeter of the garden?

(b) The gardener has decided to turn the centre of the garden (where the two equilateral triangles overlap) into a lily pond. What will be the shape of the pond and what is the length of the pond wall he needs to make?

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AREA AND PERIMETER 2 1. A small park in the city is the shape of a square, with two diagonal paths and four triangular garden beds. On each border of the park, the length of the garden bed is 12 m, while the distance from one corner of the park to the garden bed is 3 m. (a) What is the area of each of the four garden beds? (b) What is the area of the paths?

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2. The city council plans to put a memorial garden in another of the city’s squares. They decide to construct the garden in one corner of the square and leave the rest of the square as lawn for children to play on. The long edge of the garden bed will extend from the midpoints of two of the adjacent sides of the squares. If the area of the square is 160 m2, what will be the area of the garden?

3. Some local people protested at the small size of the garden. They suggest an alternative with four small triangular garden beds from the corners to the midpoint of the sides of the square. (a) What is the area of the garden now proposed? (b) How much less lawn is there as compared to the first proposal?

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TEACHER NOTES To analyse and use information in word problems.

Curriculum links

Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Devise and use mental strategies and procedures for carrying out mathematical tasks Number • Explore and identify place value in numbers • Solve word problems involving addition and subtraction • Solve and complete problems involving multiplication and division of whole numbers

Page 45 Pupils need to carefully read each problem to determine what the question is asking. The problems focus around trains, carriages, people and seats. In some problems, people are getting on and getting off, and in others it is necessary to determine how many people are sitting or standing in a carriage. Each problem requires more than one step and two or more operations are needed to find a solution. There are a number of ways to find a solution and pupils should be encouraged to explore and try different possibilities of arriving at a solution.

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Page 44 These investigations involve various stories about a library, with a number of interrelated questions arising from the written information. The situation begins with a set number of books, magazines and CD-ROMs. Using this information as a basis, the numbers change to meet new criteria where amounts of books, magazines and CDROMs are borrowed, returned, shelved and dusted. Pupils are required to keep track of the new information and use it to answer the subsequent questions. Computation is not always needed to find a solution; for example, Problem 1 states that each shelf holds 100 books, and since there are 29 572 books in the library, 296 shelves are needed. No division is necessary, as use of place value tells us that 29 572 has 295 hundreds, so 296 shelves must be needed. Similarly with Problem 2, 79 shelves being dusted means 70 hundreds so 7900 books have been dusted.

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The pupils explore word problems that require addition, subtraction, multiplication and place value understanding. The wording is more complex than in previous problems and involve a combination of the three operations. Pupils need to determine what the problem is asking and in many cases carry out more than one calculation in order to find solutions. Materials can be used to assist with the calculation, if necessary, as these problems are about reading for information and determining what the problem is asking rather than computation or basic facts.

Page 43 These problems mostly involve multiplication, with some addition and subtraction. Each problem is fairly straightforward, with all of the information needed to find a solution readily available. Each problem requires more than one calculation and some include additional information which is not needed to find a solution. Pupils may find it helpful to draw a diagram in order to work out what is happening in the problem and to determine what needs to be multiplied to find a solution.

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Possible difficulties

• Inability to identify the need to add, subtract, multiply or divide • Not using place value to solve the problems • Confusion over the need to carry out more than one calculation to arrive at a solution • Unable to understand the concept of ‘capacity’

Extension

• Pupils could write their own problems and give them to other pupils to solve.

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BEADING 1. Jackie has 19 bags of coloured beads, with 65 beads in each bag. She makes 4 necklaces. If she uses 183 beads to make each necklace, how many beads does she have left?

2. Grant has 28 bags of beads. He then buys 17 more bags so that he has enough beads to make 12 bracelets. If each bracelet requires 27 beads and each bag contains 32 beads, how many beads does he have once he has made the bracelets?

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3. At the bead factory, Marissa bought bags of beads for herself and her friends. She gives 16 bags to her friend, Elly, and 38 bags to another friend, Sarah. If she bought 85 bags of beads and each bag holds 36 beads, how many beads did she give away?

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4. Neil has 34 large bags of beads and 27 small bags of beads. The large bags hold 25 beads and the small bags contain 15 beads. He uses 24 beads to make a necklace and 16 beads to make a bracelet. If he makes 5 necklaces and eight bracelets, how many beads does he now have?

5. Ava has 19 bags of beads, with 28 beads in each bag. She uses 36 beads to make two necklaces and 54 beads to make six bracelets. She wants to make five more sets of the same amount of jewellery. Does she have enough beads?

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LIBRARY The library contains books, magazines and stories on CD-ROM. It has a total stock of 29 572 books, 341 magazines and 273 CD-ROMs. 1. In the library, each set of shelves holds 100 books. How many sets of shelves are needed to hold all of the books? 2. The cleaner has dusted 79 sets of shelves. How many books have been dusted?

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3. At the beginning of the week, the library computer showed that there were 12 841 books currently in the library. On Monday, 473 were borrowed and 168 returned. On Tuesday, 312 were borrowed and 489 returned, and on Wednesday, 654 books were borrowed and 536 returned. How many books were in the library at closing time on Wednesday?

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4. Two hundred large boxes of books, fifty-three small boxes of books, eight boxes of magazines and fourteen boxes of CD-ROMs are delivered to the library. If each large box of books holds 48 books and each small box holds 36 books, how many books will the library now have?

5. If each box of magazines holds 16 magazines and each box of CD-ROMs holds 32 CD-ROMs, how many magazines and CD-ROMs will the library now have? 6. In order to fit in all of the new books, magazines and CD-ROMs, 317 old and damaged books, 57 out-of-date magazines and 29 CD-ROMs were removed from the collection. How many books, CD-ROMs and magazines are now in the collection?

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THE COMMUTER TRAIN

1. During peak hour, a daily commuter train has 18 carriages. Each carriage has 47 seats and can hold 65 people. How many people can travel on the train and how many of them can sit?

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2. There were 1104 people on the train. At the first station, 208 people got on and 394 people got off. At the next station, 243 people got on and 172 people got off. How many people are now on the train?

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3. At the end of the line, there are 41 people in the first carriage, 52 in the second carriage, 39 in the third carriage and 47 in the fourth carriage. How many more people are needed to fill the train to capacity?

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4. On the weekend the train only has 11 carriages. After the fourth stop there are 527 people on the train. If each seat is taken, how many people are not sitting down?

5. At the first station, 143 people got on and 238 people got off. At the next station, 154 people got on and 236 people got off. There are now 491 people on the train. How many people were on the train to begin with? 6. On Tuesday, the train breaks down and all of the passengers have to be taken by bus to the next station. If there are 14 carriages on the train and each carriage is full to capacity, how many buses are needed to take the passengers to the next station if each bus holds 72 people?

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TEACHER NOTES Problem-solving

To use spatial visualisation and logical reasoning to solve problems.

Curriculum links

These relationships can now be used to solve Problem 7. If the area of the original tangram is now calculated to be 144 cm2, the area of half the tangram is 72 cm2. This means that the area of a large triangle is 36 cm2 and the area of the square is half of this (18 cm2). Alternatively, half of the tangram is formed by the two small triangles, square, trapezium and medium-sized triangle. The area of half the whole tangram, 72 cm2, is four times the area of the small square, so the small square is 18 cm2. For Problem 8, since the area of a large triangle is the same as twice the area of the medium-sized triangle and the area of the medium-sized triangle is the same as the area of the trapezium, 8 cm2, the area of a large triangle must be 16 cm2.

Materials

Possible difficulties

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Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems Measures • Solve and complete practical tasks and problems involving length

Focus

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This page explores arrangements and dissections of twodimensional shapes to explore the relationships among the area of the individual pieces and the whole. Spatial as well as logical thinking and organisation are required as pupils investigate all likely arrangements of the pieces to visualise the area of each shape in terms of its component parts.

• Unable to visualise the tangram parts in terms of each other • Try to work with the numbers to calculate areas rather than visualise the part/whole relationships that allow the areas to be determined directly from each other

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• Ask what the area of the other parts and the original square would be if the area of the trapezium was 15 cm2, the area of the large triangle is 28 cm2, or the area of the original square was 16 cm2. • Have pupils make up their own values for particular areas and challenge others to find the areas of each part and the whole square. • Ask pupils to express the areas of each of the parts as sums of the area of the small triangle. • Challenge the pupils to express each part as a fraction of the original square. • Another challenge! If a square has a diagonal of length 8 cm, what is the area of the square? This builds on Problem 1, page 41.

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Page 47 Pupils will need access to a tangram set in order to manipulate the component pieces and see the relationship of parts to wholes. At first the problem may seem impossible, as the square cannot simply be placed inside the large triangle to directly determine its area. Instead, the problem needs to be broken down into a series of smaller problems that together will answer the initial question.

By placing the two small triangles on top of the small square, pupils can see that the area of the small triangle is half that of the small square; i.e. an area of 5 cm2. The two triangles can also be arranged to form the trapezium and the medium-sized triangle, so each of these has an area of 10 cm2. When combined, the two small triangles, square, trapezium and medium triangle have an area of 40 cm2. This is also the area of half of the original tangram, so the area of one large triangle must be 20 cm2. Some pupils may immediately see that the size of a large triangle is the same as the square and the two small triangles (or that of two squares).

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Hint: Use the four large triangles from two tangram sets to think about dissecting and rearranging the area of the original square.

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SQUARES AND AREA The area of the small square in this tangram is 10 cm2.

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1. What is the area of one of the small triangles? 2. What is the area of the trapezium?

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3. What is the area of the medium-sized triangle?

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4. When the two small triangles, square, trapezium and medium-sized triangle are placed together, what is the combined area?

5. How much of the tangram do they form? 6. (a) What is the area of one of the large triangles? (b) How did you work out the answer? 7. If the original tangram had an area of 144 cm2, what would be the area of the square? 8. If the trapezium has an area of 8 cm2, what would be the area of one large triangle?

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TEACHER NOTES

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Sara originally spent 34 of the amount of money she had. She then spent 13 of what was left on her work colleagues (not one-third of what she originally had). The money she spent on dinner (Suzie, Sharon and herself—€72), together with what she had left (€18), is shown in the two unshaded parts of the diagram. This means that each part must be €45. This would mean that she originally took 12 x €45 or €540 to spend. It also shows that she spent 9 x €45 or €405 on her family and (for Problem 2) she could have saved €121.50 if she had shopped in November. The last problem can be solved by using a similar diagram.

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Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Devise and use mental strategies and procedures for carrying out mathematical tasks Number • Solve word problems involving addition and subtraction • Solve and complete problems involving multiplication and division of whole numbers • Solve and complete problems involving fractions Measures • Solve and complete problems involving dates • Solve and complete problems involving the addition, subtraction, multiplication and division of units of money

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Curriculum links

backwards and carrying out fractions calculations, but this may be difficult and time consuming for pupils. Similarly, a ‘try and adjust’ approach would be a long process, although counters could be used to make this approach more manageable. It is much easier to use a diagram to organise the data; for example, with Problem 1:

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To use logical reasoning and measurement to solve problems involving a balance, money or a calendar.

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These pages explore different ways of coming to terms with problem situations and analysing the possibilities that make up the whole solution. Logical reasoning, as well as an understanding of measurement concepts as money and a calendar, is required. Diagrams can be used to organise, sort and explore the data.

Discussion

Page 49 The problems on this page are essentially solved by ruling out different possibilities one at a time. Pupils are encouraged to look for an efficient means of solving the problems rather than the tedious task of comparing the weights of the coins one at a time. In this way, they can come to terms with the idea of creating three distinct groupings of the coins or gold bars to quickly determine which groups contain the lighter or heavier object and then repeat the process until they can identify which coin is lighter. Page 50 It is possible to solve these investigations by working

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Page 51 This page explores pupils’ understanding of the days in a week and how they are arranged in a calendar. For Problem 1, while it might seem that the variable days in a month might affect the result, all that is really being asked is to divide 100 by 7 and to find the remainder to see how many weekdays later the birthday will be. The other problems use a similar understanding of the cyclical nature of the days of the week. An actual calendar, counters or a time line will help with the second problem. Problem 3 calls on an understanding of multiples, while the last problem requires a careful elimination of the possibilities, so a table may help.

Problem-solving in mathematics

Possible difficulties

• Thinking that all of the €600 was taken to spend rather than just ‘some money’

Extension

• Seek out other famous problems dealing with counterfeit coins or ‘fool’s gold’ by using a balance to work out the false items. • Continue the Christmas shopping problems by changing the amounts spent and left over (e.g. the cost of coffee rather than dinner) or determine the discounted price if each purchase had been made in November. • Have pupils make up their own problems based on the calendar and days in a week.

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BALANCING OUT 1. Seven gold coins all look the same, but you know that one is not made of pure gold and is lighter than the rest. How do you find the lighter coin? You can use a balance to help, but you can only use it twice.

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2. (a) You are given 24 silver coins and told that 23 of the coins are the same weight and one is heavier than the rest. How many times would you have to use the balance to find the heavy coin?

(b) What is the least amount of times you can use the balance to find the heavier coin?

3. An unscrupulous worker at the mint stole some gold by hollowing out one gold bar during a delivery of 60 bars. How could his suspicious manager be able to find the underweight gold bar, and use the balance the least amount of times?

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1. Sara received €600 for her holiday pay. On 23 December, she took some money to spend on Christmas shopping and spent 43 of the amount on her family and 1 3 of what she had left on her colleagues at work. Afterwards, she met with her girlfriends, Suzie and Sharon, and bought dinner for everyone at a cost of €24 each. If she had €18 left, how much money did she originally have to shop with?

2. If Sara had completed her Christmas shopping in November, she could have bought the same presents for 30% less. How much would she have saved on the presents for her family? 3. Sharon also did her Christmas shopping on 23 December. After she spent 1 1 5 of the money she had on her brother, she spent 2 of what she had left on her mother and 41 of her remaining money on her father. As she only had €15 left, she was glad that Sara paid for her dinner. How much money did Sharon originally take to spend?

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CALENDAR CALCULATIONS 1. Today is my birthday. My friend’s birthday is exactly 100 days after mine. If my birthday is on a Wednesday this year, what day of the week will she have her birthday?

2. What day of the week was yesterday if 4 days before the day after tomorrow was Monday?

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3. Three friends visit the gym on different days: the first boy once every 3 days, his girlfriend once every 4 days and the second boy once every 5 days. They were last together at the gym on a Friday. In how many days will they again be together at the gym and what day of the week will it be?

4. June has 30 days. One year, June had exactly four Mondays. On which days of the week could June 30 not have occurred that year?

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TEACHER NOTES

Curriculum links

Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems Measures • Understand and measure the perimeter of regular 2-D shapes • Solve and complete practical tasks and problems involving length and time

Materials

counters in different colours, clocks

Focus

Page 54 Each of these problems requires a good understanding of time, the hours and minutes in a day and how time is shown on a clock. Access to a clock with moveable hands will help pupils understand what happens when time is gained or lost on a clock, as well as the way an analogue clock shows time over a 24 hour period. Using a list or table will help pupils keep track of the changes and possibilities. Page 55 The puzzle scrolls contain a number of different problems, all involving strategic thinking to find possible solutions. In most cases, pupils will find diagrams and lists are needed to manage the data while exploring the different possibilities. Concepts of space and measurement are explored in each scroll and pupils may use a number of different ways to find possible solutions, including the ‘try and adjust’ strategy.

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These pages explore different ways of visualising the problems and analyse different possible solutions. Logical reasoning is needed, as well as an understanding of measurement (length, perimeter, area and direction). For each question, materials, diagrams or tables can be used to organise, sort and explore the data.

pupils need to understand that the path traces a rectangle because the directions are at right angles to each other and show that the final point is one side of the rectangle or 750 m from her starting point.

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To use spatial visualisation, logical reasoning and measurement to solve problems.

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Problem-solving objective

Possible difficulties

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• Unable to visualise the paths taken by the person watering the garden, cyclists or balloonist • Using only 12-hour and not 24-hour time in considering the correct time • Thinking that the hands on a clock always line up on the hour • Including 12 o’clock as a time when the hands line up on a clock • Unable to draw or interpret diagrams to see the relationships among perimeter, side-length and area

Page 53 Using a diagram or counters to model the first two problems will help pupils understand the different pieces of information that the problems present. For Problem 1, Kevin needs to walk as far as the second, fourth, sixth (and so on) tree and back to the start. However, he can finish watering at the last tree and does not need to return to the tap. (Next time he can do the same in reverse, starting from the last tree.) Clearly, the answer to the second problem is not that the cyclists returned at the same time! A diagram or use of counters sorts out the difficulty posed by the problem. In the last problem, the information about ascent and descent of the balloon is not relevant. Rather,

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Extension

• The problem involving watering trees from a bucket could include more trees, less water per tree and hence more trees per bucket. • The cyclists could go from the clubhouse to the bay, then go back to the clubhouse and finally return to the bay. • Pupils could write more complex problems involving balloons; clocks which lose or gain time; and area, side length and perimeters for a pentagon, hexagon or other polygon.

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DISTANCE TRAVELLED 1. Kevin has planted 16 small trees in a row to provide a hedge along the side of his property. He planted them 3 m apart, with the first tree next to a garden tap. Because of a drought, he is only allowed to water them using a single bucket filled at the tap. If the bucket only contains enough water for 2 trees, how far does he need to walk to water all the trees?

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2. Two cyclists set out from the clubhouse, cycled to the bay and returned to the clubhouse without stopping. The first cyclist always travelled at the same speed, but the second cyclist started at half the speed and returned at twice the speed. Which cyclist returned first?

3. An adventurer made a flight in her balloon. First she ascended 300 metres. Then she flew 1500 metres towards the north-west, descended 100 metres and flew 750 metres to the north-east. After that, she flew 1500 metres towards the south-east and descended 100 metres. How far was she from her starting point?

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TAKING TIME 1. Judy bought a clock for €10 from the flea market. It was a bargain, but it does not keep accurate time and gains one minute every hour. If the clock showed the correct time when she bought it, when will it next show the correct time?

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2. Grandma’s clock is so old a key is used to wind it up to keep time. Unfortunately, the clockwork spring has stretched with time and it now loses six minutes every hour. When I last visited Grandma, I set it to the correct time of 10.30 am. What will be the correct time when her clock shows 3 pm?

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3. A fast clock gains one minute per hour and a slow clock loses two minutes per hour. At a certain time, both clocks are set to the correct time. Less than 24 hours later, the fast clock registers 9 o’clock at the same moment that the slow clock registers 8 o’clock. What is the correct time at that moment?

4. How many times a day do the hands of an analogue clock lie directly opposite each other?

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PUZZLE SCROLLS 1. Divide the face of a clock into three parts by using two lines. Make the numbers in each part add to the same total. 11

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4. An isosceles triangle has two sides of equal length. If the base is half the length of one of the other two sides and the perimeter is 1 m 35 cm, how long is the base?

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3. The area and perimeter of a small square are numerically equal. What is the length of one side of the square?

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5. A man built a fence around an octagonal garden. On each side there are seven posts. How many post holes did he have to dig for the posts?

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6. If you double the length of each side of a square, do you double the area of the square?

Problem-solving in mathematics

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TEACHER NOTES Problem-solving

To organise data and use an understanding of numbers to solve problems.

adjusted. One way to do this is to focus on the number of digits on a page, the number of pages with that number of digits and the number of stickers this requires: number of digits number of pages on a page

Curriculum links

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The problem comes down to knowing how many two-digit and three-digit numbers are used. Pupils who try to use a computational approach often get incorrect answers of 89 two-digit numbers by reasoning 99 minus 10; or 90 twodigit numbers by using 100 minus 10. Instead, knowing that the last two-digit number is 99 and that there are 9 one-digit numbers gives the correct result of 90, so that 180 stickers would be used for two-digit numbers. Similar reasoning gives 900 three-digit numbers (then 9000 fourdigit numbers, 90 000 five-digit numbers and so on). Since 435 stickers were used, some of the pages must need three stickers. 189 stickers were used for the one- and two-digit numbers, so 246 stickers were used on threedigit pages. That means there were 82 three-digit pages (246 ÷ 3) and the book has 181 pages.

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Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Record the processes and results of work using a variety of methods • Devise and use mental strategies and procedures for carrying out mathematical tasks Number • Solve word problems involving addition and subtraction • Solve and complete problems involving multiplication and division of whole numbers

number of stickers used

calculator

Focus

Applying this way of thinking to the other two problems shows that the page number is 102 when 198 stickers are used; and 60 stickers for each digit requires 600 stickers altogether, which is not enough to number 240 pages.

Discussion

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This page explores problems that require an ability to carefully analyse the relationships among the data and use an understanding of numbers to organise the information to keep track of the possible answers. Writing the various interrelated aspects in a table is a very helpful way of approaching the problem and provides a systematic way of dealing with a range of problems that have several overlapping conditions.

Page 57 The problems require a clear understanding of how pages in a book are numbered. Each side of one sheet of paper that makes up a book is one page. This may lead some pupils to divide the total number of stickers/digits that are used by two. This could not be correct as it assumes that each page would be numbered with a 1-digit number and there are clearly more than nine pages. There will be pages that need two stickers as they are numbered with two-digit numbers and there are also some that use threedigit numbers and need three stickers. Some form of organised list or table is needed to keep track of all the information or attempts that are tried and

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Problem-solving in mathematics

Possible difficulties

• Not understanding how pages in a book are numbered • Not reasoning correctly about the pages with two or three stickers per page • Confusing the page numbers with the number of stickers to provide an answer of 246

Extension

Extend the problem to other situations where pages are numbered; for example: • After writing a new novel, the author asked her computer to place page numbers on each page. The last page number was 798. How many digits were needed? • Mary’s computer crashed when only part way through numbering the pages on her autobiography. It had used 3000 digits: Will there be any pages with fourdigit numbers? What was the last page it numbered? (52 pages more than page 999 or page 1051.) • Ask pupils to write similar problems for books by their favourite authors and give them to each other to solve. www.prim-ed.com

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FOOTBALL RECORDS Jeremy plays for the district football team. He was asked to take photos, using the district’s digital camera, to record the times and dates of practice sessions and matches. He also collected articles and stories about the team from magazines and newspapers and, along with photos, put them into a scrapbook.

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Jeremy did not want this to happen again, so he decided to number all the pages of the scrapbook. He bought a box of stickers with one of the digits from 0 to 9 on each sticker and used them to number each page of the scrapbook, starting with 1.

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Unfortunately, on the way to football practice, he dropped the scrapbook and all of the pages fell out. Luckily, by noting the date of the photos, articles and stories, he was able to put the pages back in order.

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When Jeremy had finished, he noticed he had used 435 stickers.

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1. How many pages are in the scrapbook?

2. How many pages had he numbered by the time he had used 198 stickers?

3. Would he have enough stickers to number 240 pages if he had 60 stickers for each of the digits from 0 to 9?

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TEACHER NOTES Problem-solving

To analyse information and use proportional reasoning to solve problems.

Base 10 materials could also be used in an array to show how 3 by 28 can be rearranged as 4 by 21:

Curriculum links

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The second problem is similar, but this time some thought about what is involved in the problem would suggest that more time is needed. Care needs to be taken that this group of workers only works five days per week. The team expected to need 140 days, so four plumbers would need 35 days each and the job would take seven weeks altogether (not five weeks). Again, materials could be used or proportional reasoning needed to calculate that it would take five-quarters as long to work the 35 days needed. The last problem is similar, but now there are more pieces of information to keep in mind. It has to take less time, so there must be more workers. Originally it was expected that there would be 7 x 5 x 6 days of work (210 days). If the workers agreed to work every day for three weeks, each would work for 21 days and 10 workers would be needed.

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Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Devise and use mental strategies and procedures for carrying out mathematical tasks Number • Solve word problems involving addition and subtraction • Solve and complete problems involving multiplication and division of whole numbers Measures • Solve and complete problems involving the addition, subtraction, multiplication and division of units of time

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This page explores problems that require an ability to carefully analyse the relationships among the data and use an understanding of proportional reasoning to suggest possible answers. Several different pieces of information need to be kept in mind—the total number of days worked, the number of people working and the number of days they work each week—to determine when more or less time would be needed and the number of days and weeks that each would amount to. Page 59 The problems require a careful analysis of all of the information presented, as well as a sense of proportion to see that the first task would be completed in less time while the second job would take more time. While a week has seven days, a working week varies according to the type of work and the expectations of the workers. In Question 1, each of the three tilers works for four weeks and four days (28 days). This means the job requires 3 x 28 or 84 days work. If four tilers were working, each would work for 21 days and the job would take three weeks and three days and not three weeks if a week is considered as seven days rather than the six days worked. Some pupils may also be able to reason that when a team of three tilers is replaced by a team of four tilers, this would take three-quarters of the time to give 21 days. Counters or

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Problem-solving in mathematics

Possible difficulties

• Not taking note of the number of days in each working week • Thinking that the number of days worked needs to be divided by the workers and then multiplied to get a larger number of days for the first problem and a smaller number of days for the second problem

Extension

Extend the problem to other situations where work crew numbers vary: • What would happen if one of the tilers for the first job was ill and could not work? • What would happen if the plumber in the second problem was only sick for two weeks? (Four plumbers for two weeks would give 40 working days a week. The remaining days requires 20 days for each of the five plumbers or four weeks work. The job would now take six weeks to complete.) • For the last problem, if the workers only agreed to work six days per week, how many workers would be needed to finish the job in time? (11 workers would not be enough. 12 workers would be needed to finish the job in time and the last day would not be a full day’s work.) www.prim-ed.com

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BUILDING HOUSES 1. A big housing development is rapidly nearing completion. A team of 3 tilers took 4 weeks and 4 days to tile the roofs of one street of 10 houses. How long would it take a team of 4 tilers to do the same job if each tiler in both teams worked at the same rate? (The tilers work six days a week.)

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2. The team of 5 plumbers had planned to complete the bathrooms, laundry and kitchen for 3 houses in 5 weeks and 3 days. Unfortunately, one of the plumbers fell sick. How long would it take the remaining four plumbers to complete the three houses?

(Plumbers work five days per week.)

3. The last job on the housing estate was to pave the driveways. The builders had expected to have 5 weeks to do the job, with a team of 7 men working 6 days per week. If there were only 3 weeks to do the job but the workers agreed to work 7 days a week, how many workers would be needed?

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TEACHER NOTES

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This page explores problems based on a conceptual understanding of averages. Many pupils move immediately to calculations to determine a total, then divide by the amount of numbers that were totalled. This approach works when the average is of a particular group of numbers or results, but these problems require a careful analysis of what is occurring before using a knowledge of the calculation to process the appropriate numbers.

Discussion

Page 61 Applying the ‘analyse-explore-try’ model of problemsolving to this problem requires that the information needs to be carefully analysed to determine what is known and what is needed. The most important information is just what gives an average—the total time taken divided by the number of races. This has to be given and interpreted by the person solving the problem and is not explicitly stated in the problem. For Problem 1, the initial average time at the end of seven races was 71.9. What is needed is to find the total time taken when she has run another

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The last problem again requires the use of averages—it is only a matter of unfolding the numbers and calculations to get an answer from the total time taken in the 14 races he would have run.

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This thinking now has to be applied in reverse for the second problem. Since 2 minutes 50 seconds is 170 seconds, Jacob has to have a total time less than 12 x 170 or 2040 seconds at the end of his final race. He has already run 11 races, so the time taken is 11 x 170.8 or 1878.8 seconds. For the final race he must run faster than 2040 – 1878.8 (or 161.2 seconds). His time for the last race must be 2 minutes 41 seconds. If pupils incorrectly use only the two numbers given in the problem and divide by two, they would need to calculate 2 x 170. This would imply that the new race needed to take no more than 169.2 seconds. But putting this information back into the problem would suggest a new average of 2 minutes 50.666 seconds—and that would not be low enough to qualify.

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Fourth Class Skills • Apply concepts and processes in a variety of contexts • Analyse problems and plan an approach to solving them • Select and apply a variety of strategies to complete tasks and solve problems • Devise and use mental strategies and procedures for carrying out mathematical tasks Number • Solve word problems involving addition and subtraction • Solve and complete problems involving multiplication and division • Solve problems involving decimals Measures • Solve and complete problems involving the addition, subtraction, multiplication and division of units of time

three races, then divide this by the total number of races she has run now. First pupils need to calculate the time over Jade’s first seven races (7 x 71.9 or 503.3 seconds) and then add the three additional times to reach a new total of 716.4 seconds. Dividing by 10 gives her new average as 71.64 seconds. If a pupil simply adds the four numbers given in the problem and divides the number by four, they would get an average of 71.25 seconds, which gives a false idea of her progress.

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To use logical reasoning and mathematical understanding to solve problems.

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Problem-solving

Problem-solving in mathematics

Possible difficulties

• Not understanding the concept of an average • Only using the numbers given in the problems • Not analysing the answers to see if they correctly solve the problem • Not attempting a solution because of the complexity of the steps or calculations

Extension

• Ask what time Jade needs to beat in the first problem if she needs a qualifying time of less than 72 seconds and has one more race to see if she can lower her average: Would she be able to equal a qualifying time of less than 70 seconds if she has two more races to run? • Have a discussion on the nature of averages. It is not just a calculation but a typical performance over the range of calculations. Pupils may wish to look into its origin in maritime law.

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RUNNING RACES

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1. Jade is a keen runner. After 7 competitions, her average time for running 400 metres was 71.9 seconds. In her next three competitions, she recorded times of 71.2, 71.1 and 70.8 seconds. What is her new average time?

2. Jade’s older brother, Jacob, also enjoys running competitions. He competes in 800 m events. In order to qualify for the district meeting, he needs to get his average time below 2 minutes 50 seconds. After 11 competitions, he has an average time of 170.8 seconds. How fast must he run in his next race to be below the time to qualify for the meeting? 3. Jacob has to run in two more races. In the next race, he runs 2 minutes 51 seconds. What would he need to run in the last 2 races to keep his average low enough to qualify?

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

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Yes, there are 21 cubes. 8 (a) 24 (b) 8 (a) 24 (b) 8 8 No, all 64 cubes are accounted for.

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NETS AND CUBES ...................................................... page 5 1. Pupils should have circled b, c, d and f. (d)

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2. Answers will vary. Possible answers:

3 1 2 6 5 4 3.

2 3 6 5 1 4

1 2 3 5 4 6

N R U L E A

HOLIDAY FLIGHTS ...................................................... page 7 1. ST47, ST51, ST68, LA47, LA51 2. ST23, ST32, LA23, LA32 3. ST74, LA68, LA74

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CALCULATOR PATTERNS ........................................ page 10 1. (a) The same three-digit number that you start with. (b) It is always the same number. (c) Teacher check 2. The answer is always 1 less than the square of the original number. 3. The answer is always 4 less. 4. (a) The answer is always 9 less. (b) 16 less (c) 25 less 5. It will always be the same two-digit number squared, subtracted by the square of the amount it was increased or decreased by.

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VIEWING CUBES . ....................................................... page 4 1. (a)

BOOKWORMS ............................................................. page 9 1. 138 pages in each chapter 2. Page 409 is in Chapter 3 3. She has been reading the book for 38 days 4. 553–690 5. Some of these pages are in Chapters 5 and some in 6 6. 69 pages 7. Her friend is up to page 698 8. Her friend needs to read 682 pages to finish the book 9. Wanda needs to read 621 pages 10. It will take her 63 days to read the book

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PAINTING CUBES ....................................................... page 3 1. 21 2.

Problem-solving in mathematics

CARAMEL TOFFEES .................................................. page 11 1. 45 2. Teacher check: There are many possible combinations. 3. Teacher check: There are many possible combinations. 4. Teacher check: There are many possible combinations. THE PLANT NURSERY . ............................................ page 13 1. 2844 seedlings were planted 2. €3078 was made selling bark chips 3. €1710 profit from selling bark chips 4. 6565 plants were watered 5. 2124 seedlings were planted THE MANGO ORCHARD . ......................................... page 14 1. 273 trees are watered each day. It will take 4 days. 2. 37 440 mangoes have been picked 3. 2176 trays were packed after 6 hectares 4. 4352 trays were packed altogether

ANIMAL WORLD ....................................................... page 15 1. 6 days (and there are enough for another 1/2 day) 2. 1725 cars are in the car park 3. It costs €1260 to feed the giraffes 4. 728 sacks of seed were used over one year 5. 162 people can watch the show 6. The seals eat 1248 kg of fish per year

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

SCRAPBOOKING ....................................................... page 20 1. 12 pages, 55 photos 2. 17 pages, 56 pictures 3. There are 34 players 4. 58

MAGIC SQUARES ..................................................... page 29 1. 27 2. 80 10 60 23 28 21 24 3 18

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CHANGING COINS . .................................................. page 21 1. (a) €9.20 (b) three (c) Yes, the twelfth coin. (d) Twice – 4th, 6th, 8th, 10th; Once – 2nd, 3rd, 5th, 9th 2. (a) 1st, 4th, 9th, 16th coins (b) Once – 1st coin; Twice – 2nd, 3rd, 5th, 7th, 11th, 13th coins (c) 25th coin (d) 17th, 19th, 23rd (e) Prime number positions only change twice and only square number positions will show heads.

NUMBERS IN COLUMNS ........................................ page 27 1. (a) D (b) Answers will vary (c) A 2. (a) Answers will vary (b) D 3. (a) Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19 (b) After 2 and 3, all prime numbers are 1 more or 1 less than a multiple of six.

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PUZZLE SCROLLS . .................................................... page 19 1. It will be half full after 47 days 2. Answers will vary. Some are: 1, 4, 5, 9; 5, 2, 3, 9; 1, 0, 9, 9; 2, 4, 7, 6 3. (a) 27 (b) 24, 36, 48 4. 81 members after 5 weeks 5. Page number 37, 38, 39, 40, 41, 42 6. 64 x 15 625; There are no others.

4. 970 tonnes per month 5. about 1122 (1121.5) tonnes of raw sugar 6. Two trucks would make 37 trips and one truck would make 38 trips.

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UP AND DOWN ......................................................... page 17 1. The extended ladder has 53 rungs 2. The ball travelled 6.75 m 3. You entered the lift at Floor 23 4. The usual cruising altitude is 10 600 m

FRUIT FARM . ............................................................. page 23 1. 82 trays and one peach left over 2. 24 boxes, 96 trays are needed 3. (a) 45 mangoes in each box (b) 15 mangoes in each tray 4. 1680 apricots and 816 pears are packed 5. about 142 peaches per hour AT THE DELICATESSEN ........................................... page 24 1. €41 profit on each container 2. 4 containers need to be bought 3. 18 containers are needed for March 4. Loose olives are the better buy 5. €24.80 more profit in summer 6. 168 packets had been unpacked THE SUGAR MILL ...................................................... page 25 1. 764 bins were transported 2. About 6 train trips 3. 3229 bins and 23 trains were used

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SOLUTIONS Note: Many solutions are written statements rather than just numbers. This is to encourage teachers and pupils to solve problems in this way. SUDOKU ..................................................................... page 31 1. 3 1 2 4 3 1 4 2 3 1 4 2 3

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BEADING .................................................................... page 43 1. 503 beads are left. 2. He has 1116 beads. 3. She gave away 1944 beads. 4. He has 1007 beads. 5. No, five sets require 450 beads and Ava only has 442 beads.

TOURISTS IN AUSTRALIA . ..................................... page 33 1. 327 390 visitors in 2006 2. 130 317 visitors in January 3. 24 483 visitors in June

LIBRARY ..................................................................... page 44 1. 296 sets of shelves 2. 7900 books 3. 12 595 books 4. The library has 41 080 books (11 508 added). 5. 469 magazines and 721 CD-ROMs. 6. 40 763 books, 412 magazines and 692 CD-ROMs.

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HOW FAR? . ................................................................ page 34 1. She swims 19 500 m or 19.5 km each week 2. Leila travelled approximately 354 km (353.73 km) 3. If August has five weekends and 21 days—830 km. If there are only four weekends she would run 819 km. 4. They had driven approximately 537 km

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AREA AND PERIMETER 2 ...................................... pages 41 1. (a) 36 m2 (b) 180 m2 2. 20 m2 3. (a) 40 m2 (b) 20 m2

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THE COMMUTER TRAIN . ........................................ page 45 1. 1170 can travel on the train and 846 can sit. 2. 989 people on the train 3. 991 people are needed to fill the train. 4. 10 people are not sitting. 5. 668 people were on the train. 6. 13 buses are needed.

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HOW MUCH? ............................................................. page 35 1. €2.53 per day 2. 1.2 kg in each jar 3. He sold 360 peaches. 4. 52 litres in a full tank 5. €785 profit per month

FITTING TYRES . ........................................................ page 37 1. 45 cars and 47 trucks 2. Answers will vary (see p. 36) 3. The trailer tyres cost €500. 4. The truck tyres cost €62 500. AREA ........................................................................... page 39 1. Area of shape is 360 cm2 2. Area of smaller triangle is 18 cm2 3. Area of new shape is 7800 cm2 AREA AND PERIMETER 1 ...................................... pages 40 1. Perimeter is 64 cm 2. Area to be tiled is 576 m2 3. Area of the garden is 425 m2 4. (a) Perimeter of the garden is 9 m 60 cm (b) A hexagon. The pond wall is 4.8 m long.

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Problem-solving in mathematics

SQUARES AND AREA . ............................................. page 47 1. Area of small triangle is 5 cm2 2. Area of the trapezium is 10 cm2 3. Area of the medium sized triangle is 10 cm2 4. The combined area is 40 cm2 5. They form half of the tangram 6. (a) The area of a large triangle is 20 cm2 (b) It is 14 of the whole square. 7. Area of the square would be 18 cm2 8. Area of a large triangle would be 16 cm2 BALANCING OUT ...................................................... page 49 1. Put three coins on each balance pan. If they balance, the remaining coin is lighter. If one pan is lighter, then further divide the three lighter pan’s coins on each pan. 2. (a) Answers will vary (b) four times is the least 3. Answers will vary: Four is the least number of times.

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SOLUTIONS Note: Many solutions are written statements rather than just numbers. This is to encourage teachers and pupils to solve problems in this way. CHRISTMAS SHOPPING . ........................................ page 50 1. She had €540 to shop 2. €121.50 would be saved 3. Sharon took €50 to spend

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TAKING TIME . ........................................................... page 54 1. 60 hours or 2 days, 12 hours later 2. 3:27 pm is the correct time 3. 8:40 is the correct time 4. The hands are opposite 23 times

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DISTANCE TRAVELLED ............................................ page 53 1. He walks 339 m to finish at the last tree. 2. First cyclist would return first 3. 750 m north-east of the starting point

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CALENDAR CALCULATIONS ................................... page 51 1. She has her birthday on Friday 2. Tuesday 3. They will be together again on Tuesday (in nine weeks time) 4. Monday or Tuesday

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PUZZLE SCROLLS........................................................ page 55 1.

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2. The area is half that of the large square, 2 m2

3. 4 cm 4. 27 cm 5. He needs 48 holes to have 1 hole at each corner and 5 more along each side 6. No, you increase it by four times. FOOTBALL RECORDS................................................. page 57 1. 181 pages in the scrapbook 2. He had numbered 102 pages 3. No, he needs 612 stickers. BUILDING HOUSES.................................................... page 59 1. three weeks, three days (21 days) for four tilers 2. seven weeks (35 days) for four plumbers 3. 10 workers would be needed. RUNNING RACES........................................................ page 61 1. Her new average time is 71.64 seconds 2. He needs to run his next race in a time of 2 minutes 41 seconds 3. He needs to run 1 minute 51 seconds in each of the last races

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6034IRE Primary Problem-solving in Maths Book E