20769_Problem_solving_Year_5_Number_and_place_value_Using_units_of_measurement_2
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YEAR 5<br />
PROBLEM-SOLVING<br />
IN MATHEMATICS<br />
<strong>Number</strong> <strong>and</strong> <strong>place</strong> <strong>value</strong>/<br />
<strong>Using</strong> <strong>units</strong> <strong>of</strong> <strong>measurement</strong> – 2<br />
Two-time winner <strong>of</strong> the Australian<br />
Primary Publisher <strong>of</strong> the <strong>Year</strong> Award
<strong>Problem</strong>-<strong>solving</strong> in mathematics<br />
(Book F)<br />
Published by R.I.C. Publications ® 2008<br />
Copyright © George Booker <strong>and</strong><br />
Denise Bond 2008<br />
RIC–<strong>20769</strong><br />
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FOREWORD<br />
Books A–G <strong>of</strong> <strong>Problem</strong>-<strong>solving</strong> in mathematics have been developed to provide a rich resource for teachers<br />
<strong>of</strong> students from the early years to the end <strong>of</strong> middle school <strong>and</strong> into secondary school. The series <strong>of</strong> problems,<br />
discussions <strong>of</strong> ways to underst<strong>and</strong> what is being asked <strong>and</strong> means <strong>of</strong> obtaining solutions have been built up to<br />
improve the problem-<strong>solving</strong> performance <strong>and</strong> persistence <strong>of</strong> all students. It is a fundamental belief <strong>of</strong> the authors<br />
that it is critical that students <strong>and</strong> teachers engage with a few complex problems over an extended period rather than<br />
spend a short time on many straightforward ‘problems’ or exercises. In particular, it is essential to allow students<br />
time to review <strong>and</strong> discuss what is required in the problem-<strong>solving</strong> process before moving to another <strong>and</strong> different<br />
problem. This book includes extensive ideas for extending problems <strong>and</strong> solution strategies to assist teachers in<br />
implementing this vital aspect <strong>of</strong> mathematics in their classrooms. Also, the problems have been constructed <strong>and</strong><br />
selected over many years’ experience with students at all levels <strong>of</strong> mathematical talent <strong>and</strong> persistence, as well as<br />
in discussions with teachers in classrooms, pr<strong>of</strong>essional learning <strong>and</strong> university settings.<br />
<strong>Problem</strong>-<strong>solving</strong> does not come easily to most people,<br />
so learners need many experiences engaging with<br />
problems if they are to develop this crucial ability. As<br />
they grapple with problem, meaning <strong>and</strong> find solutions,<br />
students will learn a great deal about mathematics<br />
<strong>and</strong> mathematical reasoning; for instance, how to<br />
organise information to uncover meanings <strong>and</strong> allow<br />
connections among the various facets <strong>of</strong> a problem<br />
to become more apparent, leading to a focus on<br />
organising what needs to be done rather than simply<br />
looking to apply one or more strategies. In turn, this<br />
extended thinking will help students make informed<br />
choices about events that impact on their lives <strong>and</strong> to<br />
interpret <strong>and</strong> respond to the decisions made by others<br />
at school, in everyday life <strong>and</strong> in further study.<br />
Student <strong>and</strong> teacher pages<br />
The student pages present problems chosen with a<br />
particular problem-<strong>solving</strong> focus <strong>and</strong> draw on a range<br />
<strong>of</strong> mathematical underst<strong>and</strong>ings <strong>and</strong> processes.<br />
For each set <strong>of</strong> related problems, teacher notes <strong>and</strong><br />
discussion are provided, as well as indications <strong>of</strong><br />
how particular problems can be examined <strong>and</strong> solved.<br />
Answers to the more straightforward problems <strong>and</strong><br />
detailed solutions to the more complex problems<br />
ensure appropriate explanations, the use <strong>of</strong> the<br />
pages, foster discussion among students <strong>and</strong> suggest<br />
ways in which problems can be extended. Related<br />
problems occur on one or more pages that extend the<br />
problem’s ideas, the solution processes <strong>and</strong> students’<br />
underst<strong>and</strong>ing <strong>of</strong> the range <strong>of</strong> ways to come to terms<br />
with what problems are asking.<br />
At the top <strong>of</strong> each teacher page, there is a statement<br />
that highlights the particular thinking that the<br />
problems will dem<strong>and</strong>, together with an indication<br />
<strong>of</strong> the mathematics that might be needed <strong>and</strong> a list<br />
<strong>of</strong> materials that could be used in seeking a solution.<br />
A particular focus for the page or set <strong>of</strong> three pages<br />
<strong>of</strong> problems then exp<strong>and</strong>s on these aspects. Each<br />
book is organised so that when a problem requires<br />
complicated strategic thinking, two or three problems<br />
occur on one page (supported by a teacher page with<br />
detailed discussion) to encourage students to find<br />
a solution together with a range <strong>of</strong> means that can<br />
be followed. More <strong>of</strong>ten, problems are grouped as a<br />
series <strong>of</strong> three interrelated pages where the level <strong>of</strong><br />
complexity gradually increases, while the associated<br />
teacher page examines one or two <strong>of</strong> the problems in<br />
depth <strong>and</strong> highlights how the other problems might be<br />
solved in a similar manner.<br />
R.I.C. Publications ® www.ricpublications.com.au <strong>Problem</strong>-<strong>solving</strong> in mathematics<br />
iii
FOREWORD<br />
Each teacher page concludes with two further aspects<br />
critical to successful teaching <strong>of</strong> problem-<strong>solving</strong>. A<br />
section on likely difficulties points to reasoning <strong>and</strong><br />
content inadequacies that experience has shown may<br />
well impede students’ success. In this way, teachers<br />
can be on the look out for difficulties <strong>and</strong> be prepared<br />
to guide students past these potential pitfalls. The<br />
final section suggests extensions to the problems to<br />
enable teachers to provide several related experiences<br />
with problems <strong>of</strong> these kinds in order to build a rich<br />
array <strong>of</strong> experiences with particular solution methods;<br />
for example, the numbers, shapes or <strong>measurement</strong>s<br />
in the original problems might change but leave the<br />
means to a solution essentially the same, or the<br />
context may change while the numbers, shapes or<br />
<strong>measurement</strong>s remain the same. Then numbers,<br />
shapes or <strong>measurement</strong>s <strong>and</strong> the context could be<br />
changed to see how the students h<strong>and</strong>le situations<br />
that appear different but are essentially the same<br />
as those already met <strong>and</strong> solved. Other suggestions<br />
ask students to make <strong>and</strong> pose their own problems,<br />
investigate <strong>and</strong> present background to the problems<br />
or topics to the class, or consider solutions at a more<br />
general level (possibly involving verbal descriptions<br />
<strong>and</strong> eventually pictorial or symbolic arguments).<br />
In this way, not only are students’ ways <strong>of</strong> thinking<br />
extended but the problems written on one page are<br />
used to produce several more problems that utilise<br />
the same approach.<br />
Mathematics <strong>and</strong> language<br />
The difficulty <strong>of</strong> the mathematics gradually increases<br />
over the series, largely in line with what is taught<br />
at the various year levels, although problem-<strong>solving</strong><br />
both challenges at the point <strong>of</strong> the mathematics<br />
that is being learned as well as provides insights<br />
<strong>and</strong> motivation for what might be learned next. For<br />
example, the computation required gradually builds<br />
from additive thinking, using addition <strong>and</strong> subtraction<br />
separately <strong>and</strong> together, to multiplicative thinking,<br />
where multiplication <strong>and</strong> division are connected<br />
conceptions. More complex interactions <strong>of</strong> these<br />
operations build up over the series as the operations<br />
are used to both come to terms with problems’<br />
meanings <strong>and</strong> to achieve solutions. Similarly, twodimensional<br />
geometry is used at first but extended<br />
to more complex uses over the range <strong>of</strong> problems,<br />
then joined by interaction with three-dimensional<br />
ideas. Measurement, including chance <strong>and</strong> data, also<br />
extends over the series from length to perimeter, <strong>and</strong><br />
from area to surface area <strong>and</strong> volume, drawing on<br />
the relationships among these concepts to organise<br />
solutions as well as giving an underst<strong>and</strong>ing <strong>of</strong> the<br />
metric system. Time concepts range from interpreting<br />
timetables using 12-hour <strong>and</strong> 24-hour clocks while<br />
investigations related to mass rely on both the concept<br />
itself <strong>and</strong> practical <strong>measurement</strong>s.<br />
The language in which the problems are expressed is<br />
relatively straightforward, although this too increases<br />
in complexity <strong>and</strong> length <strong>of</strong> expression across the books<br />
in terms <strong>of</strong> both the context in which the problems<br />
are set <strong>and</strong> the mathematical content that is required.<br />
It will always be a challenge for some students<br />
to ‘unpack’ the meaning from a worded problem,<br />
particularly as problems’ context, information <strong>and</strong><br />
meanings exp<strong>and</strong>. This ability is fundamental to the<br />
nature <strong>of</strong> mathematical problem-<strong>solving</strong> <strong>and</strong> needs to<br />
be built up with time <strong>and</strong> experiences rather than be<br />
iv<br />
<strong>Problem</strong>-<strong>solving</strong> in mathematics www.ricpublications.com.au R.I.C. Publications ®
FOREWORD<br />
diminished or left out <strong>of</strong> the problems’ situations. One<br />
reason for the suggestion that students work in groups<br />
is to allow them to share <strong>and</strong> assist each other with<br />
the tasks <strong>of</strong> discerning meanings <strong>and</strong> ways to tackle<br />
the ideas in complex problems through discussion,<br />
rather than simply leaping into the first ideas that<br />
come to mind (leaving the full extent <strong>of</strong> the problem<br />
unrealised).<br />
An approach to <strong>solving</strong> problems<br />
Try<br />
an approach<br />
Explore<br />
means to a solution<br />
Analyse<br />
the problem<br />
The careful, gradual development <strong>of</strong> an ability to<br />
analyse problems for meaning, organising information<br />
to make it meaningful <strong>and</strong> to make the connections<br />
among them more meaningful in order to suggest<br />
a way forward to a solution is fundamental to the<br />
approach taken with this series, from the first book<br />
to the last. At first, materials are used explicitly to<br />
aid these meanings <strong>and</strong> connections; however, in<br />
time they give way to diagrams, tables <strong>and</strong> symbols<br />
as underst<strong>and</strong>ing <strong>and</strong> experience <strong>of</strong> <strong>solving</strong> complex,<br />
engaging problems increases. As the problem forms<br />
exp<strong>and</strong>, the range <strong>of</strong> methods to solve problems<br />
is carefully extended, not only to allow students to<br />
successfully solve the many types <strong>of</strong> problems, but<br />
also to give them a repertoire <strong>of</strong> solution processes<br />
that they can consider <strong>and</strong> draw on when new<br />
situations are encountered. In turn, this allows them<br />
to explore one or other <strong>of</strong> these approaches to see<br />
whether each might furnish a likely result. In this way,<br />
when they try a particular method to solve a new<br />
problem, experience <strong>and</strong> analysis <strong>of</strong> the particular<br />
situation assists them to develop a full solution.<br />
Not only is this model for the problem-<strong>solving</strong> process<br />
helpful in <strong>solving</strong> problems, it also provides a basis for<br />
students to discuss their progress <strong>and</strong> solutions <strong>and</strong><br />
determine whether or not they have fully answered<br />
a question. At the same time, it guides teacher<br />
questions <strong>of</strong> students <strong>and</strong> provides a means <strong>of</strong> seeing<br />
underlying mathematical difficulties <strong>and</strong> ways in<br />
which problems can be adapted to suit particular<br />
needs <strong>and</strong> extensions. Above all, it provides a common<br />
framework for discussions between a teacher <strong>and</strong><br />
group or whole class to focus on the problem-<strong>solving</strong><br />
process rather than simply on the solution <strong>of</strong> particular<br />
problems. Indeed, as Alan Schoenfeld, in Steen L (Ed)<br />
Mathematics <strong>and</strong> democracy (2001), states so well, in<br />
problem-<strong>solving</strong>:<br />
getting the answer is only the beginning rather than<br />
the end … an ability to communicate thinking is<br />
equally important.<br />
We wish all teachers <strong>and</strong> students who use these<br />
books success in fostering engagement with problem<strong>solving</strong><br />
<strong>and</strong> building a greater capacity to come to<br />
terms with <strong>and</strong> solve mathematical problems at all<br />
levels.<br />
George Booker <strong>and</strong> Denise Bond<br />
R.I.C. Publications ® www.ricpublications.com.au <strong>Problem</strong>-<strong>solving</strong> in mathematics<br />
v
CONTENTS<br />
Foreword .................................................................. iii – v<br />
Contents .......................................................................... vi<br />
Introduction ........................................................... vii – xix<br />
A note on calculator use ................................................ xx<br />
Teacher notes ................................................................. 2<br />
Coloured cubes .............................................................. 3<br />
Growing cubes ............................................................... 4<br />
Viewing cubes ................................................................ 5<br />
Teacher notes ................................................................. 6<br />
Market days ................................................................... 7<br />
Teacher notes ................................................................. 8<br />
Bookworms .................................................................... 9<br />
Calculator patterns ...................................................... 10<br />
Puzzle scrolls 1.............................................................. 11<br />
Teacher notes ............................................................... 12<br />
The seedling nursery .................................................... 13<br />
The tropical fruit orchard ............................................. 14<br />
Animal Safari Park ...................................................... 15<br />
Teacher notes ............................................................... 16<br />
Desert adventures ........................................................ 17<br />
Teacher notes ............................................................... 18<br />
Abstract art .................................................................. 19<br />
Time taken .................................................................... 20<br />
Puzzle scrolls 2.............................................................. 21<br />
Teacher notes ............................................................... 22<br />
The school’s records ..................................................... 23<br />
The town’s centenary ................................................... 24<br />
Keeping records ........................................................... 25<br />
Teacher notes ............................................................... 26<br />
<strong>Number</strong> patterns .......................................................... 27<br />
Teacher notes ............................................................... 28<br />
Magic squares ............................................................. 29<br />
Sudoku ......................................................................... 30<br />
Alphametic puzzles ...................................................... 31<br />
Teacher notes ............................................................... 32<br />
At the shops ................................................................. 33<br />
At the delicatessen ...................................................... 34<br />
The sugar mill .............................................................. 35<br />
Teacher notes ............................................................... 36<br />
The fish market ............................................................ 37<br />
Teacher notes ............................................................... 38<br />
Designing shapes ......................................................... 39<br />
Different designs ......................................................... 40<br />
<strong>Using</strong> designs ............................................................... 41<br />
Teacher notes ............................................................... 42<br />
How many? ................................................................... 43<br />
How far? ....................................................................... 44<br />
How much? ................................................................... 45<br />
Teacher notes ............................................................... 46<br />
Training runs ................................................................. 47<br />
Teacher notes ............................................................... 48<br />
Balancing business ...................................................... 49<br />
Calendar calculations .................................................. 50<br />
Puzzle scrolls 3.............................................................. 51<br />
Teacher notes ............................................................... 52<br />
Tall buildings ................................................................ 53<br />
Good sports .................................................................. 54<br />
Puzzle scrolls 4.............................................................. 55<br />
Teacher notes ............................................................... 56<br />
Tank water .................................................................... 57<br />
Square-deal nursery ..................................................... 58<br />
Salad days .................................................................... 59<br />
Teacher notes ............................................................... 60<br />
Shelley Beach .............................................................. 61<br />
Solutions .................................................................62–68<br />
Isometric resource page .............................................. 69<br />
0–99 board resource page ........................................... 70<br />
4-digit number exp<strong>and</strong>er resource page (x 5) ............. 71<br />
10 mm x 10 mm grid resource page ............................ 72<br />
15 mm x 15 mm grid resource page ............................ 73<br />
Triangular grid resource page ...................................... 74<br />
vi<br />
<strong>Problem</strong>-<strong>solving</strong> in mathematics www.ricpublications.com.au R.I.C. Publications ®
TEACHER NOTES<br />
<strong>Problem</strong>-<strong>solving</strong><br />
To use logical reasoning <strong>and</strong> <strong>measurement</strong> to solve<br />
problems<br />
Materials<br />
paper to draw diagrams or tables <strong>and</strong> record times,<br />
calculator<br />
Focus<br />
This page investigates distance <strong>and</strong> time expressed as<br />
fractions <strong>of</strong> the distance around a running track. Logical<br />
thinking <strong>and</strong> organisation are needed to see how the<br />
runners progress, keep track <strong>of</strong> their positions <strong>and</strong><br />
determine when they will coincide.<br />
Discussion<br />
Page 47<br />
Drawing a diagram to show the oval, the running track <strong>and</strong><br />
the relative positions <strong>of</strong> the runners may help students. For<br />
<strong>Problem</strong> 1, there are 24 markers (one every 10 m) around<br />
the 240 m track <strong>and</strong> these show how the relative jogging<br />
rates <strong>of</strong> 1-half, 1-fourth <strong>and</strong> 1-third can be determined. In<br />
one minute, Heather jogs 120 m, Hannah jogs 80 m <strong>and</strong><br />
Helen jogs 60 m. This information can be organised to<br />
keep track <strong>of</strong> how far each has jogged <strong>and</strong> when each<br />
reaches the starting line—i.e. has jogged 240 m. One way<br />
is to <strong>place</strong> the information in a table to show how far each<br />
girl has jogged after 1, 2, 3 etc. minutes:<br />
Time in<br />
minutes<br />
Heather Hannah Helen<br />
1 120 80 60<br />
2 start 160 120<br />
3 120 start 180<br />
4 start 80 start<br />
5 120 160 60<br />
After 4 minutes, all <strong>of</strong> the girls have reached the start at<br />
least once, but never all at the same time.<br />
Other ways to find a solution can also be used; for<br />
example, while the first solution focuses on the distance<br />
travelled, other students may prefer to work with time,<br />
using the pattern that Heather reaches the start every<br />
2 minutes, Hannah every 3 minutes, <strong>and</strong> Helen every 4<br />
minutes. Some students may quickly follow a pattern or<br />
seek a common multiple <strong>of</strong> 2, 3 <strong>and</strong> 4 which is 12.<br />
The second problem can be solved in a similar manner:<br />
Expressing the relative distances run as written fractions<br />
may require further thought for some students <strong>and</strong> this<br />
problem asks for the distance travelled rather than the<br />
time taken.<br />
Possible difficulties<br />
• Unable to visualise the distance each person jogs or<br />
the time they take<br />
• Unable to organise the information to keep track <strong>of</strong><br />
the various criteria<br />
• Does not coordinate the different times people take to<br />
get to the start<br />
• Unable to convert metres to kilometres <strong>and</strong> vice versa<br />
Extension<br />
• Find out how far each <strong>of</strong> the children would jog if they<br />
trained for one hour each day.<br />
• How far would they have jogged if they began<br />
training for one hour per day four weeks before the<br />
cross-country meet was scheduled?<br />
• If each girl wanted to jog for one minute <strong>and</strong> finish<br />
together at the original starting line, where would<br />
they each have to start running?<br />
• If each boy wanted to jog 2 km, how many minutes<br />
apart would they each finish?<br />
• Challenge students to come up with similar problems<br />
<strong>of</strong> their own involving other activities.<br />
46<br />
<strong>Problem</strong>-<strong>solving</strong> in mathematics www.ricpublications.com.au R.I.C. Publications ®
TRAINING RUNS<br />
To prepare for the school’s cross-country run, Heather <strong>and</strong> her two sisters, Hannah<br />
<strong>and</strong> Helen, decide to jog on the oval each morning. The running track around the oval<br />
is 240 m long, with markers to show the distance every 10 m <strong>and</strong> a large clock to keep<br />
track <strong>of</strong> how they are going. After one minute, Heather has jogged halfway around the<br />
track, Hannah has run 1-third <strong>of</strong> the way <strong>and</strong> Helen has jogged 1-fourth <strong>of</strong> the distance<br />
around the track.<br />
1. If they all continue jogging at the same rate, after how many minutes will they all cross<br />
the starting line at the same time?<br />
2. (a) Their cousins, Len, Liam <strong>and</strong> Lachlan, also have a cross-country race coming up<br />
<strong>and</strong> decide to follow a similar training schedule. The running track on their oval is<br />
300 m long, with markers to show the distance every 10 m. It also has a large<br />
clock. After 1 minute, Len has jogged 1 2 the track, Liam has jogged 1 5 <strong>of</strong> the way<br />
<strong>and</strong> Lachlan has jogged 1 3 <strong>of</strong> the distance. If they all continue jogging at the same<br />
rate, how many kilometres will each <strong>of</strong> the boys have jogged when they first cross<br />
the starting at the same time?<br />
(b) If they start jogging at 7:30 am, what will be the time when they first cross the<br />
starting line together?<br />
(c) What would be the next time they cross the starting line together <strong>and</strong> how far<br />
would each boy have jogged by then?<br />
R.I.C. Publications ® www.ricpublications.com.au <strong>Problem</strong>-<strong>solving</strong> in mathematics<br />
47
SOLUTIONS<br />
Note: Many solutions are written statements rather than just numbers. This is to encourage teachers <strong>and</strong><br />
students to solve problems in this way.<br />
TRAINING RUNS ....................................................... page 47<br />
1. In one minute, Heather jogs<br />
120 m, Hannah jogs 80 m<br />
<strong>and</strong> Helen jogs 60 m. They<br />
reach the starting line after<br />
jogging 240 m:<br />
The table can be continued until after 12 minutes they<br />
are all at the start together.<br />
Looking for a pattern in the table—Heather is at the start<br />
every 2 minutes, Hannah every 3 minutes, <strong>and</strong> Helen<br />
every 4 minutes—also shows that they will be at the<br />
start together after 12 minutes.<br />
2. (a) The second problem is solved in a similar manner.<br />
Expressing the relative distances run as written<br />
fractions may require further thought for some<br />
students <strong>and</strong> this problem asks for the distance<br />
travelled rather than the time taken. Students need<br />
to calculate the total distances travelled <strong>and</strong> then<br />
convert the metres to kilometres.<br />
Distance jogged Len Liam Lachlan<br />
1 150 60 200<br />
2 300 – start 120 400<br />
3 450 180 600 – start<br />
4 600 – start 240 800<br />
5 750 300 – start 1000<br />
6 900 – start 360 1200 – start<br />
The table can be continued until after 30 minutes<br />
they are all at the start together. Looking for a<br />
pattern in the table – Len is at the start every 2<br />
minutes, Liam every 5 minutes, <strong>and</strong> Lachlan every 3<br />
minutes – also shows that they will be at the start<br />
together after 30 minutes. Len will have made 15<br />
laps <strong>and</strong> jogged 4500 m or 4 km 500 m. Liam will<br />
have made 6 laps <strong>and</strong> jogged 1800m or 1 km 800 m.<br />
Lachlan will have made 10 laps <strong>and</strong> jogged 3000 m<br />
or 3 km.<br />
(b) 8:00<br />
(c) 8:30 Len 9 km, Liam 3.6 km, Lachlan 6 km<br />
R.I.C. Publications ® www.ricpublications.com.au <strong>Problem</strong>-<strong>solving</strong> in mathematics<br />
67