Mathematical Modelling A Collection of Classroom Tasks
An Institute of
Published by Alston Publishing House Pte Ltd 745 Lorong 5 Toa Payoh, #01-07, Singapore 319455 enquiry@alstonpublishinghouse.com www.alstonpublishinghouse.com Š 2012 Alston Publishing House Pte Ltd All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the copyright owner. First published 2012 ISBN 978-981-4370-04-2 Publisher: Sim Wee Chee Editorial Team:Varsha Primalani, Cheryl Lee Design Team: Melissa Lee, Eric Sto. Domingo Printed in Singapore
PREFACE Problem-solving is central to mathematics learning in the Singapore Mathematics curriculum. Mathematical modelling is gaining prominence in Singapore mathematics classrooms in recent years because modelling tasks are perceived to offer exciting platforms for problem-solving involving openended, multi-faceted and real-world interpretations. Through modelling tasks, students are exposed to different ways of mathematical representations and reasoning as they use mathematical lenses to appreciate the world around them. This book is written for teachers and students who are relatively new to mathematical modelling in the Singapore primary and secondary classrooms. The book has a mainly pedagogical focus. It starts with an introduction to what modelling tasks entail, some key features of these tasks and suggestions of design principles to elicit model development in students. The introduction chapter also includes a preliminary facilitation structure recommended for teachers who are new to using mathematical modelling in their classrooms. Such a facilitation structure tailors to a more student-centred approach in learning and at the same time, builds the teacher’s repertoire in scaffolding learning during open-ended tasks. In addition, the book offers a chapter illustrating how student learning and model development can be evaluated for further mathematical reasoning processes within real-world problem-solving. The book features eight simple yet enriching tasks which draw connections between school-based mathematics and real-world experiences. Each task is discussed in detail with prompting questions for students as well as teachers’ notes for facilitation according to the recommended facilitation structure. Mathematical content, skills and processes aligned with the syllabus are proposed for each task. Resources and pedagogical ideas for extension are included for further development of student thinking. Teachers can download the task sheets (without teachers’ notes) and samples of students’ work from the Outreach from any one of the following websites: www.alstonpublishinghouse.com www.myalston.com We hope this book will be useful to teachers. Dawn Ng Kit Ee and Lee Ngan Hoe
PREFACE
iii
ACKNOWLEDGEMENTS In June 2010, the Mathematics and Mathematics Education Academic Group of the National Institute of Education, Singapore, conducted a Mathematical Modelling Outreach over three days involving primary and secondary schools. Samples of students’ work were showcased in the Second Lee Peng Yee Symposium held immediately after the Mathematical Modelling Outreach. The modelling tasks recorded in this book were crafted and implemented by the following contributors during the Mathematical Modelling Outreach.
From the Mathematics and Mathematics Education Academic Group, National Institute of Education Dawn Ng Kit Ee Chan Chun Ming Eric Cheng Lu Pien Jaguthsing Dindyal Ho Weng Kin Yeo Kai Kow Joseph
Lee Ngan Hoe Cheang Wai Kwong Chua Kwee Gek Ho Foo Him Soon Wan Mei Amanda Zhao Dong Sheng
From Charles Stuart University, Australia Ho Siew Yin From Bachelor of Arts/Science (Education) July 2007 Intake, National Institute of Education Chiok Hwee Fen Goh Shu Yi Priscilla Hsieh Josie Lee Peiyi Pearly Ng Chun Hui Nur Liyana Binte Saine Phua Wang Yu Tan Guo Dong Thong Jia Sui Jessica
Goh Hui Qian Stella Hoon Hui Qi Nicola Kong May Hua Maybelline Neo Jie Qi Nur Azlin Binte Junari Ong Tze-Min Gracelyn Ramesh S/O Kunasgaran Tan Xiao Jing Tok Wei Ling
ACKNOWLEDGEMENTS
v
Advisors In addition, we would like to thank the following esteemed researchers in the field of mathematical modelling for their valuable comments about the tasks during planning and implementation: Associate Professor Ang Keng Cheng (Head/Mathematics and Mathematics Education Academic Group, National Institute of Education, Singapore) Professor Lyn D. English (Queensland University of Technology, Australia) Dr Vincent Geiger (Australian Catholic University, Australia) Professor Toshikazu Ikeda (Yokohama National University, Japan) Professor Gabriele Kaiser (University of Hamburg, Germany) Associate Professor Gloria Stillman (Australian Catholic University, Australia) Support from Curriculum Planning and Development Division, Ministry of Education, Singapore Mr Soh Cheow Kian (Assistant Director, Curriculum Planning and Development Division, Sciences Branch, Math Unit, Ministry of Education, Singapore) Ms Gayatri Balakrishnan (Curriculum Specialist, Curriculum Planning and Development Division, Sciences Branch, Math Unit, Ministry of Education, Singapore) Participating Schools We would also like to thank these schools for their participation in the Mathematical Modelling Outreach. Singapore Primary Schools
Singapore Secondary Schools
Anchor Green Primary School
Boon Lay Secondary School
CHIJ St Nicholas Girls’ School (Primary)
Bukit Panjang Government High School
Da Qiao Primary School
Bukit View Secondary School
Henry Park Primary School
Catholic High School
Kheng Cheng School
CHIJ Katong Convent School
Kong Hwa School
Commonwealth Secondary School
Marsiling Primary School
Dunman High School
Nan Hua Primary School
Fuchun Secondary School
Park View Primary School
Hillgrove Secondary School
Princess Elizabeth Primary School
Ngee Ann Secondary School
Rulang Primary School
Northbrooks Secondary School
St Anthony’s Primary School
Paya Lebar Methodist Girls’ School
West View Primary School
Raffles Girls’ School River Valley High School Shuqun Secondary School Woodlands Ring Secondary School Overseas Schools
Stamford International School, Bandung, Indonesia
vi
ACKNOWLEDGEMENTS
A.B. Paterson College, Brisbane, Australia
CONTENTS Introduction viii
Primary Tasks Task 1 Dream Home Task 2 The Unsinkable Titanic
1 12
Primary/Secondary Task Task 3 Designing a Tent
22
Secondary Tasks Task 4 The Best Paper Plane
31
Task 5 Mobile Phone Plan
41
Task 6 Plane Punctuality
51
Task 7
61
My Hometown
Task 8 Designing a CafĂŠ
70
Evaluation of Modelling Efforts
80
References and Resources
81
INTRODUCTION What is Mathematical Modelling? Mathematical modelling is one way to help students make connections between school mathematics and real-world problems. In this book, we define mathematical modelling as a process of representing or describing real-world problems mathematically so as to understand or find solutions to the problems (Ang, 2009).
Mathematical Modelling Process There are many conceptual frameworks illustrating the mathematical modelling process. Most of them use four key components: Real-World Problem, Mathematical Problem, Mathematical Solution, and Real-World Solution. These components are included in the conceptual framework of the mathematical modelling process by the Curriculum Planning and Development Division (CPDD, 2012), Singapore Ministry of Education. A real-world problem can be interdisciplinary. Very often, we may need to use an integrated approach comprising multi-faceted interpretations, content knowledge and skills drawn from several disciplines for problem-solving in the real world. However, in mathematical modelling, we choose to place mathematical lenses in the foreground to plan solutions to a real-world problem by developing a mathematical model or improving on an existing model. A mathematical model leads the way for a possible mathematical approach to solve the real-world problem. This model is established on assumptions and parameters set by the person who proposed the model. We call this person a modeller. According to CPDD (2012), a mathematical model is a mathematical representation or idealisation of a real-world situation. It can be as complicated as a system of equations or as simple as a geometrical figure. A mathematical model can be used for at least three different purposes as a solution to the real-world problem: to describe, explain or predict. The model can be generated in at least three different ways: from empirical data, simulation or use of equations representing relationships between variables crucial to the real-world context. Ang (2009) provides useful examples on the various types of models. Nonetheless, any mathematical model should encompass many characteristics of the real-world conditions the problem is situated in. This is because we would like the model to address the problem as adequately and appropriately as possible in order for the solution proposed by the model to be useful in the real world.
Facilitating the Mathematical Modelling Process Singapore Ministry of Education highlights formulating, solving, interpreting and reflecting as four important elements which describe the activities students will engage in during the modelling process as they progress through the cycle starting from the Real-World Problem. In this book, we suggest how the teacher can facilitate a modelling task using six phases (Figure 1): discuss, plan, experiment, verify, present and reflect. These phases provide a loose structure to help teachers activate the four elements in the Ministry’s conceptual framework. The conceptual framework on the modelling process provided by the Ministry of Education (CPDD, 2012) as well as the six-phase facilitation framework presented in Figure 1 suggest that the modelling process is non-linear. We will briefly explain how the six phases can be realised along with the four elements next. The section that follows after will exemplify these ideas using a real-world problem from this book.
viii
INTRODUCTION
Phase 1: Discuss
Phase 2: Plan
Phase 3: Experiment
Phase 4: Verify
Phase 5: Present
Phase 6: Reflect Figure 1: A suggested six-phase facilitation framework for the modelling process (Ng, 2011)
We encourage teachers to allow at least the first four phases depicted in Figure 1 to be cyclical when facilitating students’ modelling processes. Student modellers normally work in groups. In Phase 1, time should be provided for groups to discuss and negotiate meanings so as to establish an initial understanding of the real-world problem. Group members need to seek consensus in the mathematical interpretations of the real-world problem, define key variables, set goals, list assumptions and decide on the parameters to work within. In Phase 2, the group makes plans for model development such as choosing the variables to focus on, noting the information needed to quantify the chosen variables and framing the computational approach. Both Phases 1 and 2 provide platforms for groups to embark in the element of ‘formulating’ the mathematical problem from the given real-world problem. To progress further, there is a need to narrow down the scope of the real-world problem by crafting a mathematical problem statement with a clear goal. Nonetheless, the open-ended nature of a real-world problem may result in groups formulating mathematical problems with differing goals. At times, such goals may be difficult to achieve given resource and time constraints. Hence, the teacher may like to guide beginning student modellers in the selection of goals during problem formulation. Phase 3 is referred to as experiment because plans are implemented by the collection of data through experiments, computer simulations, data bases and spreadsheets or other means such as surveys, interviews and internet search. The data collected should help develop a preliminary version of a mathematical model together with a cohesive set of arguments in answer to the problem formulated. Here, modellers activate the element of ‘solving’ as appropriate mathematical methods and tools are used to craft a mathematical solution. Next, in Phase 4, the preliminary model is verified within the context INTRODUCTION
ix
and requirements of the real-world problem. The element of ‘interpreting’ is evoked during this phase when modellers check to see if the model is an adequate and appropriate representation of the realworld problem. The potential usefulness of the associated solution is evaluated too. The presentation of the model and its arguments in Phase 5 can be carried out in both written and oral forms. This is an important phase where modellers explain and defend their chosen approach against the background of the real-world problem they started off with. Lastly, Phase 6 requires modellers to reflect on their modelling processes and critically examine their models for limitations and scope of applications in other contexts so that improvements on the models can be suggested. This phase coincides with the element of ‘reflecting’. During facilitation, it is crucial that student modellers be allowed opportunities to engage in the cyclical movements between the phases suggested above, particularly Phases 1 to 4. This encourages model refinement towards more sophisticated mathematical representations. We will now look at how the phases and elements can be exemplified in the following problem.
A Real-World Problem According to the 2010 census, there are about 539,356 permanent residents (PRs) in Singapore. Permanent residents are entitled to most of the rights and duties of citizens with the exception of voting at the general elections. The following problem was crafted within this context.
The Problem: According to the Urban Redevelopment Authority, more than 80 percent of the 4000 Singaporeans, permanent residents and foreigners who participated in a lifestyle survey conducted in 2010 found Singapore to be a great place to live, work and play in. When interviewed, people shared what they liked about the town they live in, however new or old. The editor of Good Living, a lifestyle magazine, is featuring an article to showcase the most suitable town for a family of new permanent residents. The editor has invited your team to submit a proposal for the town you would recommend. This problem is relevant to the current concerns of Singaporeans and PRs alike. Although the problem has provided a clear goal, it is deliberately worded in a general, open-ended way to invite multiple interpretations at the beginning of the modelling process. Prompting questions for student groups to respond to with respect to this problem are provided in Task 7 (My Hometown) of this book. Notes for teachers are also provided in that task.
Phase 1: Discuss Teacher may like to encourage students to • individually read and think about the given real-world problem • record questions to clarify the problem and main ideas (e.g. Whose needs are to be met for town selection?) • highlight key words or phrases for further definitions (e.g. permanent residents, town, suitability) Teacher may conduct a whole class discussion to generate ideas on the above once students have had time to think through the problem individually.
x
INTRODUCTION
Next, students can get into groups for discussions to formulate the mathematical problem from the realworld problem as part of the modelling process. Encourage student groups to • articulate their mathematical goal clearly as they transform the given real-world problem into a mathematical problem (e.g. to design a system of evaluation criteria to mathematically determine the most suitable town for a family of new permanent residents using statistical measures) • consider the profile and needs of the PR family • make assumptions about the profile of the PR family For example, the family - comprise both parents and at least two school-going children, - does not own a car, - does not have special needs, - intends to stay in the same town for at least 5 years, and - has a total household income of $8000 (comparable to a middle-income family in Singapore based on statistics of average monthly household income in 2011) • propose all variables affecting the selection of town Teacher may conduct a whole class discussion again to • classify variables as important to the model, to be discarded or controlled so as to narrow the scope of work • cite the information needed for important variables • list the assumptions to be made in relation to the controlled variables • set parameters for model development An example: Variable
Housing prices
Important
Yes
Information Needed
Information Sources
Assumptions
• Prices of 3–5 room HDB flats • Prices of condominiums • Rental amounts
Websites from • Urban Redevelopment Authority • Singapore Property search
• Housing costs in most towns will remain stable for the next 3 years
Parameters
Considered: Median housing prices and rentals for 4-room HDB flats and comparable condominium units with similar floor area (99-year lease) — using costs for the current 6-month period Not considered: Landed or freehold properties
Transport convenience
Yes
• Types of public transport available • Location of public transport stations • Public transport travel route options and costs involved
Websites from • Singapore Mass Rapid Transit • Singapore Bus Service
• Public transport charges will not be revised within the next 3 years
Considered: Total transport cost per month; bus and MRT fares included; using costs for the current 6-month period Not considered: Taxi charges, parking charges or other transport costs due to non-routine activities
INTRODUCTION
xi
Variable
Availability of amenities
Important
Yes
Information Needed
Information Sources
Availability of • Schools • Childcare facilities • Market • Hospital • Community centre • Library • Entertainment centre • Sports stadium
Websites from the various town portals
Assumptions
Parameters
Considered: Basic amenities Not considered: Other services like counselling and tuition centres
Location of town with respect to the city
No/ Controlled
-
-
• Most towns are easily accessible to the city since the transport system in Singapore is efficient
Considered: Towns near to workplace or famous schools
Country of origin of PR family
No/ Discarded
-
-
-
-
Student groups continue formulating the mathematical problem as they discuss • how they can quantify or measure the suitability of a town by converting the variables into evaluation criteria Examples: - A maximum of 30% of total family income can be taken up for rental or monthly cash home loan repayment - Transport cost for the whole family should not involve more than 10% of total income
Phase 2: Plan Student groups continue formulating their mathematical problem as they • decide what data would be relevant to the focused variables and strengthen mathematical arguments • suggest the mathematics needed for making robust arguments in the evaluation criteria for suitability of town (e.g. consider the advantages and limitations of using mean housing prices, critical choice of graphical representations of useful data) • list the calculations needed (e.g. use of weighted averages, ranking of towns based on mean, median, mode of data sets) • plan how they can find out more about the profile and needs of a PR family • allocate responsibilities to each group member • recommend a timeline for model development
Phase 3: Experiment Student groups implement the plans made and embark on solving the mathematical problem they crafted earlier. They should aim to come up with a preliminary mathematical model along with a cohesive set of written mathematical arguments with reasoning based on relevant data. xii
INTRODUCTION
An example of a Preliminary Ranking System Model: The town with lowest total score will be recommended to the PR family. Towns
Housing Prices (HDB)
Housing Prices (Condominium Unit)
Transport Cost
Transport Convenience
Availability of Amenities
Total Score
Clementi
3
3
1
1
1
9
Woodlands
2
2
3
2
2
11
Punggol
1
1
3
3
6
14
Ranks for Housing Prices (median housing prices of 4-room HDB flats and comparable condominium units in each town over the current 6-month period used): 1 – less than 10% of total family income; 2 – between 10% and 20%; 3 – between 20% and 30% of total family income spent on housing loan or rental installments. Ranks for Transport Cost (mean total transport cost over the current 6 month period used): 1 – less than 5% of total family income; 2 – between 5% and 8% of total family income; 3 – between 8% and 10% of total family income spent on routine transport needs Ranks for Transport Convenience: 1 – Town is well connected to other places; 2 – Town is still developing public transport services; 3 – Town is still establishing basic public transport network Ranks for Availability of Amenities: 1 – fulfilling all 8 of the listed amenities; 9 – fulfilling none of the listed amenities This table is to be accompanied by cost calculations (e.g. percentages, fractions) and statistical information.
Phase 4: Verify Student groups can work on verifying their preliminary model within the context of the real-world problem, checking how robust their model is based on the following: • Alignment of their interpretations of the real-world problem and the model developed (i.e. Did the group answer the problem within the assumptions made from the various interpretations?) • How representative their model is in terms of meeting the requirements of the real-world problem (i.e. adequacy and appropriateness of the model) • Justification of the mathematical approaches used in terms of appropriateness, accuracy and reasonableness • How the model compares with other possible models within the parameters set by the group (i.e. how confident the group is about their model in addressing the requirements of the problem and why) Such verification processes activate the element of interpreting, particularly linking the mathematical solution to the real-world solution as indicated in the Ministry’s conceptual framework. This phase highlights the potential of modelling tasks in sensitising students’ awareness of the importance of interpretations, assumptions and parameters in a real-world problem and their impact on the level of sophistication and usefulness of the solution. If time permits in class, it is ideal that student groups are encouraged to engage in another cycle of model development through Phases 1 to 4 if they find their model lacking in many aspects.
Phase 5: Present Teacher may help student groups to organise their written and oral presentations of their final models. Students need to experience using mathematical language to express and defend their reasoning clearly and logically. They should also provide documentation of their modelling process through the phases so that their thinking can be better appreciated and understood by the targeted audience. One way of organising their report is to present it according to the phases. Every group member should be involved in preparing the report as part of a coherent set of arguments in answer to the real-world problem as
INTRODUCTION
xiii
part of their learning experience. Members of the group may answer questions from the audience during their presentation.
Phase 6: Reflect Teacher can guide students to do individual and group reflections on their modelling journeys by getting them to think through the following: • How successful the group was in achieving the goal in the real-world problem • How applicable the group’s model is in other similar contexts (i.e. the usefulness and limitations of the model) • Suggestions of possible further improvements to the model for a more adequate and appropriate response to the real-world problem (e.g. re-examining or replacing some of the assumptions and parameters set) It may be more fruitful for students to embark on their reflections after having thought through their peers’ models with respect to their own. The six-phase facilitation framework for the modelling process presented in Figure 1 is used for all modelling tasks presented in this book. However, readers may be interested in another useful framework from the New South Wales Curriculum K-12 Directorate (2011) which appears to incorporate the modelling process and facilitation stages. The reference to this conceptual framework is provided in the last section of the book.
Mathematical Modelling Problems vs Applications Tasks Problem-solving is at the heart of the Singapore Mathematics Curriculum Framework (Figure 2). Teachers are encouraged to provide opportunities for students to engage in a variety of problem-solving activities including non-routine real-world problems. The framework emphasises five inter-related components: Concepts, Skills, Processes, Metacognition and Attitude. Applications and modelling problems are highlighted in the ‘Processes’ component of the framework because these enhance connections between school mathematics and the real world. Our curriculum suggests that such problems promote understanding of key mathematical concepts and methods during the development of mathematical competencies. Although both applications and modelling problems are situated in real-world contexts, the latter is unique in one main way. A modelling problem always spring board from a real-world situation (Stillman, Brown, and Galbraith, 2008). Modellers need to first deal with the context of the problem (i.e. make interpretations, work with the ambiguities involved, draw connections, identify assumptions and parameters) before deciding what mathematical concepts and skills are appropriate for a mathematical solution to the problem. In other words, the context of the real-world problem governs the choice of mathematics during the modelling process (reality ➔ mathematics). In contrast, other problems situated in real-world contexts (e.g. applications tasks) usually start with the teacher determining what mathematics they would like the students to apply after teaching-learning activities and then subsequently looking for appropriate realworld contexts to showcase the use of the chosen mathematics (mathematics ➔ reality).
xiv
INTRODUCTION
tti
Skills
Numerical calculation Algebraic manipulation Spatial visualisation Data analysis Measurement Use of mathematical tools Estimation
og
nit
Mathematical Mathematical Problem Solving Problem-Solving
ion
Proc
A
tac
Monitoring of one’s own thinking Self-regulation of learning
s
s
e tud
Me
esse
Beliefs Interest Appreciation Confidence Perseverance
Reasoning, communication and connections Applications and modelling Thinking skills and heuristics
Concepts Numerical Algebraic Geometric Statistical Probabilistic Analytical
Figure 2: Singapore Mathematics Curriculum Framework (CPDD, 2012)
Designing a Modelling Task for Students We suggest some guidelines for crafting modelling tasks. Some of these are adapted from Galbraith (2006): • Meaningfulness — the problem should relate to the real-world experiences of students • Mathematically Open — the problem should allow for various mathematical interpretations • Mathematically Rich — students should be able to transform the given general real-world situation into a mathematically specific problem • Mathematically Feasible — there exists at least one approach to the solution process and students can make use of the mathematics available to them, make assumptions and collate relevant data • Mathematically Solvable — there exists at least one solution to the problem and this is attainable by the students • Mathematically Verifiable — it is possible to engage in a series of checks for mathematical accuracy and appropriateness of solution with respect to the real-world setting
About this Book A total of eight modelling problems are presented in this book. The suggested guidelines above have been used in crafting the problems. Each problem has been implemented with school students using the facilitation structure proposed in Figure 1. More reports on students’ work on these tasks can be found in Lee and Ng (in press). The eight tasks comprise of two at primary level, five at secondary level and the remaining one at both primary and secondary levels. Record sheets for each task are included and these are designed according to the six phases of facilitation with detailed notes for teachers. A proposed generic evaluation rubric for the tasks is presented on page 80 of this book. Student task sheets (without teacher’s notes) can be downloaded from any one of the following websites: www.alstonpublishinghouse.com www.myalston.com INTRODUCTION
xv
1
DREAM HOME Suitable for: Primary levels
TASK SHEET
A standard 5-room flat in Singapore comprises three or four bedrooms and a living room. Other features include two washrooms/toilets, a kitchen, and a storeroom. An interior design company, Noble Construction Designers, is working with one of the major furniture companies to furnish the master bedroom of a 5-room flat. They are hiring architects to design and draw the floor plan for the flat and to propose how the master bedroom can be furnished within the given budget. Your team is invited to work out the layout design and furnishing ideas. The drawing should be properly labelled, stating the various parts of the flat and its real-life dimensions. Furnishings for the master bedroom, including the furniture items, should also be drawn accordingly.
DREAM HOME
1
Requirements given by Noble Construction Designers 1. The 5-room flat has a total floor area of 132 m2 and should comprise the following: • Four bedrooms • A living room • Two washrooms/toilets • A kitchen • A storeroom • A small balcony 2. The floor plan must be able to fit into the size of the flip chart given. 3. The floor plan drawing should represent the real-life dimensions of the 5-room flat. 4. The total cost of furnishing (flooring and furniture types) for the master bedroom should not exceed $3000. 5. You may use the catalogues provided to choose the furniture items.
2
Task 1
RECORD SHEET FOR GROUP WORK Part
1
Discuss
1. What is your team’s idea of an ‘ideal’ 5-room flat home? Notes For Teachers
Possible discussion points: • Students’ experiences staying in HDB 5-room flats • Various designs of 5-room flats • Definition of the ‘ideal’ home (e.g. area of bedrooms/living room/kitchen/storeroom/balcony, position of each room in relation to other parts of the flat, the types of furniture that the master bedroom should have)
2. What is your team’s goal? Notes For Teachers
Students may articulate their goal in a mathematical problem statement that incorporates their idea of what qualities the ‘ideal’ 5-room flat home should have. For example: The goal of the task is to determine the largest living room and lowest cost of furnishing the master bedroom.
DREAM HOME
3
3. What are the variables that will affect your team’s goal? Notes For Teachers
Discuss with students how dependent and independent variables can be identified according to students’ definition of the ‘ideal’ 5-room flat home. Examples of dependent variables: • Area of each of the rooms • Cost of furnishing the master bedroom Examples of independent variables: • Dimensions of each of the rooms • Cost of furniture needed to furnish the master bedroom • Cost of flooring for the master bedroom Encourage students to use a spreadsheet to calculate the cost of furnishing the master bedroom.
4. Which variables does your team intend to begin your investigation with? Explain your choice. Notes For Teachers
Encourage students to • focus on the relevant dependent and independent variables based on their problem statement, and • investigate how a particular independent variable affects a dependent variable by keeping all other variables constant.
4
Task 1
5. What additional information does your team need? Notes For Teachers
Students may want to find out how certain variables can be measured or quantified.
6. What assumptions has your team made? Notes For Teachers
Possible assumptions students may make: • The number of people staying in such a home • The normal lifestyle of the people staying in the master bedroom Encourage students to define the conditions for their investigation. For example: • The requirements given by Noble Construction Designers have been met. • All bedrooms should have a door and a window. • The washrooms/toilets should be equipped with a W.C. each.
DREAM HOME
5
Part
2
✓ ✓ ✓ ✓ ✓
Plan
1. Make a clear record of the things your team needs to do. Notes For Teachers
Suggest that students assign roles and duties to the members of their team. Students may sketch possible floor plans and make lists to show the possible ways of furnishing the master bedroom to test the identified variables by considering the following: • Choice of dimensions of rooms • Types and cost of furniture needed for the master bedroom • Types and cost of furnishing the flooring of the master bedroom • How to make appropriate calculations from the proposed dimensions, and type and cost of furnishing for the master bedroom to determine the impact on the variables to obtain the ideal design for the 5-room flat home • How to record and present the data
2. Sketch your team’s floor plan here. Label and show the dimensions of each clearly. You can use sheets of blank, grid or graph paper to help you. Include for each floor plan, a list of materials and the respective costs to furnish the master bedroom. Notes For Teachers
Encourage students to • consider different possible floor plans, and • come up with different lists of materials and their respective costs to furnish the master bedroom for each floor plan.
6
Task 1
Part
3
Experiment
1. Carry out the necessary calculations for the floor plans and list of furnishing for the master bedroom to decide on the ideal 5-room flat home. Record and present the calculations here clearly. Notes For Teachers
Students may show calculations relating to • the area of each of the rooms, • the total area of the flat, and • the total cost for furnishing the master bedroom. Encourage students to • record and present the data for each plan and accompanying list of items to furnish the master bedroom on separate sheets of paper, and • label each sheet for easy reference later.
2. Explain what other calculations your team needs to make using the data in order to determine the ‘ideal’ 5-room flat home. Notes For Teachers
Encourage students to develop mathematical arguments that will help them arrive at a decision for their ‘ideal’ 5-room flat home (e.g. tabulating a systematic listing). Students should justify their decisions based on the results of their calculations rather than by personal preference.
DREAM HOME
7
Part
4
✓ ✓ 5
Verify
1. For the choice of floor plan and furnishing for the master bedroom, show how it meets the requirements given by Noble Construction Designers and how it satisfies your team’s goal. 2. Draw the final floor plan on the flip chart provided, showing the real-life dimensions of the flat (in metres) on the flip chart (in exact measurement in centimetres). Label the various parts of the drawing to show the rooms and the furniture for the master bedroom clearly. 3. Print out a photo of the final floor plan that your team has drawn and paste it here. Notes For Teachers
Check that students verify that the requirements given by Noble Construction Designers are met, and the goal(s) set are satisfied.
8
Task 1
Part
5
Proposal
Present
1. Create a proposal to present your team’s plan to Noble Construction Designers. How would your team convince the company to accept its proposal? Notes For Teachers
Students should include the following points in their proposal: • Their definition of the ‘ideal’ 5-room flat home • Assumptions made • Conditions set • Variables investigated • Findings • Verification process • Recommendations and reasoning process [Arguments should be based on calculations and mathematical reasoning.]
DREAM HOME
9
Part
6
Reflect
1. How successful was your team in achieving its goal? Give reasons. Notes For Teachers
Students may want to review their definition of the ‘ideal’ 5-room flat home, their approach and their investigation procedures.
2. We can/cannot use the same experimental process on other situations because … Notes For Teachers
Students may consider the relevance of their approach (i.e. defining goals, selecting variables, conducting experiments, verifying) for other real-world situations, such as selecting the best location for a home in Singapore.
3. If we had the chance to do this all over again, we would have … Notes For Teachers
Encourage students to reflect on other approaches they could have tried.
10
Task 1
GENERAL NOTES FOR TEACHERS This task involves designing the ‘ideal’ 5-room flat home based on a quality/qualities that students deem most important in such a home (e.g. areas of bedrooms, cost of furnishing the master bedroom). Students will be able to relate to this task as they would likely have experience staying in HDB flats. They may make calculations on the various aspects of designing such a home. In the process, they will discover the relevance of school-based mathematical concepts and skills in their daily lives, especially the applications of measurement and geometry. In addition, the task emphasises the following: Thinking skills — spatial visualisation, analysing parts and wholes Mathematical concepts and skills — numerical, measurement, geometrical, estimation Representations of students’ mathematical arguments — use of tables and scaled drawings Some useful websites for this task: Sample floor plans from HDB — http://www.hdb.gov.sg/fi10/fi10221p.nsf/Attachment/AR2000/$file/P81-82HDB.pdf IKEA Singapore catalogue download — http://www.ikea.com/ms/en_SG/virtual_catalogue/online_catalogues.html Free download of floor plans drawing software — http://www.smartdraw.com/specials/ppc/draw-floor-plans.htm?id=11008 7&gclid=CPCf09vng68CFQ8b6wod3wgu1Q Some suggested materials for the task: These are to be placed at the front of the class. Students choose from the provided materials what they need for their own interpretation of the task. Rulers Measuring tapes Calculators Coloured paper Glue Scissors Flip charts Markers Furniture catalogues The following should preferably also be made available for students when attempting the task: PC with spreadsheet software and guide Internet access Printers
DREAM HOME
11