20764_Problem_solving_Year_5_Patterns_and_algebra_Number_and_place_value

Your partner in education 

YEAR 5 

PROBLEM-SOLVING 

IN MATHEMATICS 

Patterns and algebra/ 

Number and place value 

Two-time winner of the Australian 

Primary Publisher of the Year Award

Problem-solving in mathematics 

(Book F) 

Published by R.I.C. Publications ® 2008 

Copyright © George Booker and 

Denise Bond 2008 

RIC–20764 

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FOREWORD 

Books A–G of Problem-solving in mathematics have been developed to provide a rich resource for teachers 

of students from the early years to the end of middle school and into secondary school. The series of problems, 

discussions of ways to understand what is being asked and means of obtaining solutions have been built up to 

improve the problem-solving performance and persistence of all students. It is a fundamental belief of the authors 

that it is critical that students and teachers engage with a few complex problems over an extended period rather than 

spend a short time on many straightforward ‘problems’ or exercises. In particular, it is essential to allow students 

time to review and discuss what is required in the problem-solving process before moving to another and different 

problem. This book includes extensive ideas for extending problems and solution strategies to assist teachers in 

implementing this vital aspect of mathematics in their classrooms. Also, the problems have been constructed and 

selected over many years’ experience with students at all levels of mathematical talent and persistence, as well as 

in discussions with teachers in classrooms, professional learning and university settings. 

Problem-solving does not come easily to most people, 

so learners need many experiences engaging with 

problems if they are to develop this crucial ability. As 

they grapple with problem, meaning and find solutions, 

students will learn a great deal about mathematics 

and mathematical reasoning; for instance, how to 

organise information to uncover meanings and allow 

connections among the various facets of a problem 

to become more apparent, leading to a focus on 

organising what needs to be done rather than simply 

looking to apply one or more strategies. In turn, this 

extended thinking will help students make informed 

choices about events that impact on their lives and to 

interpret and respond to the decisions made by others 

at school, in everyday life and in further study. 

Student and teacher pages 

The student pages present problems chosen with a 

particular problem-solving focus and draw on a range 

of mathematical understandings and processes. 

For each set of related problems, teacher notes and 

discussion are provided, as well as indications of 

how particular problems can be examined and solved. 

Answers to the more straightforward problems and 

detailed solutions to the more complex problems 

ensure appropriate explanations, the use of the 

pages, foster discussion among students and suggest 

ways in which problems can be extended. Related 

problems occur on one or more pages that extend the 

problem’s ideas, the solution processes and students’ 

understanding of the range of ways to come to terms 

with what problems are asking. 

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 students to find 

a solution together with a range of means that can 

be followed. More often, problems are grouped as a 

series of three interrelated pages where the level of 

complexity gradually increases, while the associated 

teacher page examines one or two of the problems in 

depth and highlights how the other problems might be 

solved in a similar manner. 

R.I.C. Publications ® www.ricpublications.com.au Problem-solving in mathematics 

iii

FOREWORD 

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 students’ success. In this way, teachers 

can be on the look out for difficulties and be prepared 

to guide students past these potential pitfalls. The 

final section suggests extensions to the problems to 

enable teachers to provide several related experiences 

with problems of these kinds in order to build a rich 

array of experiences with particular solution methods; 

for example, the numbers, shapes or measurements 

in the original problems might change but leave the 

means to a solution essentially the same, or the 

context may change while the numbers, shapes or 

measurements remain the same. Then numbers, 

shapes or measurements and the context could be 

changed to see how the students handle situations 

that appear different but are essentially the same 

as those already met and solved. Other suggestions 

ask students to make and pose their own problems, 

investigate and present background to the problems 

or topics to the class, or consider solutions at a more 

general level (possibly involving verbal descriptions 

and eventually pictorial or symbolic arguments). 

In this way, not only are students’ ways of thinking 

extended but the problems written on one page are 

used to produce several more problems that utilise 

the same approach. 

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 

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. 

The language in which the problems are expressed is 

relatively straightforward, although this too increases 

in complexity and length of expression across the books 

in terms of both the context in which the problems 

are set and the mathematical content that is required. 

It will always be a challenge for some students 

to ‘unpack’ the meaning from a worded problem, 

particularly as 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 

iv 

Problem-solving in mathematics www.ricpublications.com.au R.I.C. Publications ®

FOREWORD 

diminished or left out of the problems’ situations. One 

reason for the suggestion that students work in groups 

is to allow them to share and assist each other with 

the tasks of discerning meanings and ways to tackle 

the ideas in complex problems through discussion, 

rather than simply leaping into the first ideas that 

come to mind (leaving the full extent of the problem 

unrealised). 

An approach to solving problems 

Try 

an approach 

Explore 

means to a solution 

Analyse 

the problem 

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 students to 

successfully solve the many types of problems, but 

also to give them a repertoire of solution processes 

that they can consider and draw on when new 

situations are encountered. In turn, this allows them 

to explore one or 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. 

Not only is this model for the problem-solving process 

helpful in solving problems, it also provides a basis for 

students to discuss their progress and solutions and 

determine whether or not they have fully answered 

a question. At the same time, it guides teacher 

questions of students and provides a means of seeing 

underlying mathematical difficulties and ways in 

which problems can be adapted to suit particular 

needs and extensions. Above all, it provides a common 

framework for discussions between a teacher and 

group or whole class to focus on the problem-solving 

process rather than simply on the solution of particular 

problems. Indeed, as Alan Schoenfeld, in Steen L (Ed) 

Mathematics and democracy (2001), states so well, in 

problem-solving: 

getting the answer is only the beginning rather than 

the end … an ability to communicate thinking is 

equally important. 

We wish all teachers and students who use these 

books success in fostering engagement with problemsolving 

and building a greater capacity to come to 

terms with and solve mathematical problems at all 

levels. 

George Booker and Denise Bond 


v

CONTENTS 

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

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

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

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

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

Coloured cubes .............................................................. 3 

Growing cubes ............................................................... 4 

Viewing cubes ................................................................ 5 

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

Market days ................................................................... 7 

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

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

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

Puzzle scrolls 1.............................................................. 11 

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

The seedling nursery .................................................... 13 

The tropical fruit orchard ............................................. 14 

Animal Safari Park ...................................................... 15 

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

Desert adventures ........................................................ 17 

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

Abstract art .................................................................. 19 

Time taken .................................................................... 20 


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

The school’s records ..................................................... 23 

The town’s centenary ................................................... 24 

Keeping records ........................................................... 25 

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

Number patterns .......................................................... 27 

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

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

Sudoku ......................................................................... 30 

Alphametic puzzles ...................................................... 31 

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

At the shops ................................................................. 33 

At the delicatessen ...................................................... 34 

The sugar mill .............................................................. 35 

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

The fish market ............................................................ 37 

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

Designing shapes ......................................................... 39 

Different designs ......................................................... 40 

Using designs ............................................................... 41 

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

How many? ................................................................... 43 

How far? ....................................................................... 44 

How much? ................................................................... 45 

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

Training runs ................................................................. 47 

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

Balancing business ...................................................... 49 

Calendar calculations .................................................. 50 


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

Tall buildings ................................................................ 53 

Good sports .................................................................. 54 


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

Tank water .................................................................... 57 

Square-deal nursery ..................................................... 58 

Salad days .................................................................... 59 

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

Shelley Beach .............................................................. 61 

Solutions .................................................................62–68 

Isometric resource page .............................................. 69 

0–99 board resource page ........................................... 70 

4-digit number expander resource page (x 5) ............. 71 

10 mm x 10 mm grid resource page ............................ 72 

15 mm x 15 mm grid resource page ............................ 73 

Triangular grid resource page ...................................... 74 

vi 


TEACHER NOTES 

Problem-solving 

To use strategic thinking to solve problems 

Materials 

grid paper, counters in several different colours 

Focus 

These pages explore more complex problems in which 

the most difficult step is to find a way of coming to terms 

with the problem and what the question is asking. Using 

materials to assist is one way this can be done. Another is 

to use a diagram to assist in thinking backwards or making 

trials and adjusting to find a solution that matches all of 

the conditions. 

Discussion 

Page 19 

The first problem can be solved by using coloured 

counters on a grid or colouring the squares to see what is 

happening. The number of possibilities can then be seen 

directly or patterns can be sought: 

1 2 3 4 5 6 7 8 9 10 

The flashing light problem can be solved by considering 

multiples of 2, 7 and 5. It is similar to the cafe problem 

on page 50 and can be solved using a table in a similar 

manner. Recognising similarities in problems even though 

the context is different is a critical aspect of problem 

solving. This also gives the number of times the lights 

flashed after dusk but the initial flashes also need to be 

included. A diagram will help see what is happening in 

the third problem. 

Page 21 

The puzzle scrolls contain a number of different problems 

all involving strategic thinking to find possible solutions. In 

most cases, students will find tables, lists and diagrams are 

needed to manage the data while exploring the different 

possibilities. Problem 4 has more than one solution. 

Possible difficulties 

• Not using a diagram or table to come to terms with 

the problem conditions 

• Unable to see how to connect the time cycled to the 

distance travelled 

• Considering only some aspects of the puzzle scrolls 

Extension 

• Write problems based on those in the puzzle scrolls. 

• Use different speeds and times for the problems on 

page 20. 

Analysis of the patterns shows that the squares represent 

the factors of the number of the column in which they 

occur. Determining the factors of each number from 1 to 

50 shows that 48 has the most factors in a column. The 

other questions are solved by considering prime numbers, 

squares of prime a number and systematically examining 

the pairs of factors in each number. 

Page 20 

For the first problem, consider the distances covered each 

hour (a table or list would help). A difference of 25 km 

is needed to allow for the 1 hour of travel and 1 hour of 

rest. After 5 hours, cycling at 10 km per or covers 50 km 

and cycling at 15 km per hour covers 75 km. Travelling one 

more hour at 10 km per hour would give 60 km, which 

travelling 1 less hour at 15 km per hour would also give 60 

km. He would need to travel for 5 hours at 12 km per hour. 

18 


ABSTRACT ART 

Justin made a grid with 50 rows and 50 columns. He 

asked his friends to help him colour squares on the 

grid to make an abstract design for his art class. He 

started and coloured all of the squares in the first 

row. 

His first friend coloured every second square on the 

second row. The next friend coloured every third 

square on the third row and so on, until every row 

had some squares coloured. 

1 1 1 1 1 

2 2 

3 

4 

5 

1. When the design is complete, which column would have the most squares coloured? 

Some columns would only have 2 squares coloured. 

2. Which columns would they be? 

Other columns would have only 3 squares coloured. 

3. Which columns would they be? 

4. (a) Would any other column(s) have an odd number of squares coloured? 

(b) Why do most columns have an even number of squares coloured? 

Justin’s teacher was very impressed with his design and decided to make a similar 

project for the whole class. On the wall of the art classroom, he drew a grid that had 

150 rows and 150 columns. The students then took it in turns to come to the wall and 

colour the squares following the same pattern. 

5. Which column(s) would have the largest number of squares coloured in the class project? 


19

TIME TAKEN 

1. A cyclist needs to meet his friend for coffee 

at midday. He knows that if he travels at 

15 km per hour, he will arrive an hour early, 

and if his speed is 10 km per hour, he will 

be an hour late. At what speed should he 

travel in order to arrive on time? 

2. At the entrance to the town, the council 

installed three coloured lights to draw 

travellers’ attention to the town’s 40 km 

per hour night time speed limit. The blue 

light flashes every 2 minutes, the red light 

flashes every 3.5 minutes and the green 

light flashes 5 minutes. When the lights 

come on at dusk, they all flash together. 

How long will it be before they next flash 

together? 

3. When Ernesto got his driving licence, he agreed to collect his sister, Ana, from the 

train station on the way home from work in exchange for being allowed to use the 

family car on the weekend. He usually picked her up from the station at 6:30 pm, 

but today she got an earlier train and arrived an hour earlier. Ana started walking 

home until Ernesto saw her and was able to drive her home. If they arrived home 

24 minutes earlier than usual, how long had Ana walking before Ernesto picked her 

up? 

20 


PUZZLE SCROLLS 2 

1. $38 000 is divided among 4 

sisters with each one getting 

$1000 more than their younger 

sibling. 

How much will the eldest sister 

get? 

2. A rectangular table is twice as 

long as it is wide. If it was 2 m 

shorter and 2 m wider it would be 

a square. 

What size is the table? 

3. How many more cubes would 

you need to make this object 8 

steps high? 

4. I spent $10 on stamps at the post 

office and bought 50c and 70c 

stamps. 

How many of each stamp did I 

buy? 

5. How often in a 12 hour period 

does the sum of the digits on a 

digital clock equal 8? 

: 

01 43 

GIGA-BLASTER 

ON 

OFF 

6. A triangle has a perimeter of 

60 cm. Two of its sides are 

equal in length and the third 

side is 9 cm longer than the 

equal-length sides. 

What is the length of the third 

side? 


21

SOLUTIONS 

Note: Many solutions are written statements rather than just numbers. This is to encourage teachers and 

students to solve problems in this way. 

ABSTRACT ART ......................................................... page 19 

1. Use counters or colour the squares on a grid: 

1 2 3 4 5 6 7 8 9 

This shows the factors of the number of the column in 

which they occur. 

Checking the factors of each number 1–50 shows that 48 

has the most entries in a column. 

2. columns whose numbers are prime numbers 

3. 4, 9, 25, 49 (the square numbers) 

4. (a) 1 

(b) Factors occur in pairs except for the square numbers 

5. 144 (it has the largest number of factors of the numbers 

1–150.) 

TIME TAKEN ............................................................. page 20 

1. 12 km per hour 

The difference in the distance travelled must be 25 km 

(the sum of 1 hour @ 10 km and 1 hour @ 15 km/hr) 

hours 10 km/hr 15 km/hr difference 

1 10 15 5 

2 20 30 10 

3 30 45 15 

4 40 60 20 

5 50 75 25 

This occurs after 5 hours – the distance must be 60 km 

and he must travel at 12 km/hr 

2. 1 hour 10 minutes later 

3. If Ernesto got home 24 minutes early, he drove for 36 

minutes. He drove 18 minutes before meeting Ana. Ana 

walked for 42 minutes. 

PUZZLE SCROLLS 2 ................................................... page 21 

1. $1000 

2. 8 m x 4 m 

3. 36 cubes 

4. 13 x 50c and 5 x 70c or 6 x 50c and 10 x 70c 

5. 57 times 

6. 26 cm 


63

10 mm x 10 mm GRID RESOURCE PAGE 

72 


15 mm x 15 mm GRID RESOURCE PAGE 


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20764_Problem_solving_Year_5_Patterns_and_algebra_Number_and_place_value

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