The Essentials Mathematics SBA Handbook - Glendon Steele

THE ESSENTIALMATHEMATICS SBA HANDBOOK A GUIDE FOR CSEC EXAMINATION GLENDON STEELE DEDICATION This book is dedicated to my Caribbean Examinations Council (CXC) O’ Level Mathematics teacher, Mr. Lloyd Elliott, who made me fall in love with the great subject, Mathematics. 01 ACKNOWLEDGEMENT I would like to thank Mr. Steve Hope for proofreading the SBA projects presented in this book and offering invaluable advice and support. I would also like to thank and acknowledge the work of Miss Candice Bharat in proofreading the book and correcting the English Language. Thank you! This book could not have been completed without the support and dedication of Miss Yvette Best who spent many long hours researching, collecting and preparing material for the SBA projects presented in this book. Thanks! Thanks! Thanks! Lastly, I would like to thank Mr. Kumar Ramsingh for always encouraging me to write my ideas. Thanks for believing in me. 02 PREFACE I attended a Mathematics School Based Assessment (SBA) Workshop approximately one year ago. At the end of the workshop, a number of my colleagues were lost, confused and did not seem to understand what a Mathematics SBA Project should entail. At that moment, I started thinking of writing an SBA guide that can help the students and teachers at my school. A few months later an idea struck me, why not write a Mathematics SBA Project guide that can work for all teachers and students who are involved in the CSEC Mathematics SBA Project? At that moment, the concept of this book was conceived and that was the genesis of this publication. The Essential Mathematics SBA Handbook: A Guide for CSEC Examination, was designed specifically to provide assistance to teachers and students who are preparing for the NEW CSEC Mathematics Examination in May 2018 and beyond. It takes the reader through the nine (9) sections that comprise a CSEC Mathematics SBA Project and with the use of simple language, each section is explained in detail. A sample project is used to explain each section of a project. This is done in order that students see exactly what is required at each stage of a project. There are ten (10) fully worked projects included in the book. These projects were written as guides in order to demonstrate to students how each section of an SBA Project can be structured and written as a report. The projects were intentionally written in order to stimulate lively and meaningful classroom discussions between the student and teaching bodies. At the end of the book, twenty (20) different real-world scenarios are presented together with suggested project titles aimed to assist students with the choice of research topics for an SBA Project. The book was written in order to dismiss any confusion that teachers and students may experience with respect to writing a CSEC Mathematics SBA Project. It is the author’s hope that teachers and students critique the projects presented in this book, add and subtract where necessary, so that in the end students produce on their own, a well researched and written SBA Project that satisfies the requirements of the Caribbean Examination Council (CXC). Do not forget to have fun along the way. 03 GOODLUCK! TABLE OF CONTENTS Section.......................................................................................................Page Number 1. Dedication.................................................................................................................1 2. Acknowledgement ...................................................................................................2 3. Preface......................................................................................................................3 4. Table of Contents......................................................................................................4 5. How To Use This Book..............................................................................................5 6. Rationale...................................................................................................................6 7. Structure of a CSEC Mathematics SBA Project........................................................8 8. Presentation of 10 Worked Mathematics SBA Projects..........................................20 • Project One.........................................................................................................21 • Project Two..........................................................................................................37 • Project Three.......................................................................................................52 • Project Four.........................................................................................................68 • Project Five..........................................................................................................82 • Project Six...........................................................................................................93 • Project Seven....................................................................................................103 • Project Eight......................................................................................................116 • Project Nine.......................................................................................................130 • Project Ten.........................................................................................................147 9. 20 Ideas for a Mathematics SBA with Scenarios and Titles.................................166 10. References............................................................................................................176 04 HOW TO USE THIS BOOK This book was written as a guide; the contents are not cast in stone. It is the author’s wish that students and teachers go through each project with critical eyes, analyze each section and discuss ways in which each section can be improved or rewritten more effectively. It is suggested that teachers allocate two (2) periods per week over a set period of time in order to deal with SBA issues. During these periods, students should be encouraged to read and analyze different sections of the projects presented in the book. They can be asked to rewrite a section with the use of their own words. Teachers and students can discuss ways in which each project can be rewritten so that it falls within the stipulated 1000 word guideline set by the Caribbean Examinations Council (CXC). Students can be grouped; each group can take a project and discuss it among the individuals in that group. After studying the ten (10) projects, students should be encouraged to go to the end of the book where there are twenty (20) possible scenarios and titles from which they can choose in order to develop their own projects. Students should also be given the freedom to choose any real-world situation that they wish to investigate for an SBA Project. GOODLUCK! 05 RATIONALE FOR SBA “The great book of nature can only be understood by those who understand the language in which it is so written and that language is Mathematics.” GALILEO When Galileo wrote these profound words years ago, he was trying to say to us that nature was given to us with a manual to its operation. However, the manual is written in a scientific language called Mathematics which must be understood if we want to unlock the mysteries of nature. Mathematics is the language used to study the principles of nature. Since nature reveals itself to us through patterns, Mathematics is a language that studies all possible patterns. Patterns mean, any regularities which the brain can recognize. Hence, any study of Mathematics should involve the investigation of patterns in the real world. When regularities and patterns are studied, the language of Mathematics can help us to establish models which can then be used to solve many problems in the real world. However, traditionally, Mathematics is taught from textbooks. Students are then assessed by answering questions from textbooks exercises, worksheets and examination papers. When the subject is presented in this way, students are generally confused and see Mathematics as a SET of rules to be learnt by rote; they have no idea of how to apply Mathematics in real-life situations. As such, they ask the question, “Why do we pursue Mathematics?”. The School Based Assessment (SBA) Project is an excellent way to link Mathematics to real-world events and help students to apply the concepts that have been taught in the classroom. 06 Projects in Mathematics will help students to recognize that Mathematics is a cultural pursuit; it is a language that pervades every culture. Projects will help students gain the following skills: 1. read and think critically 2. detect flaws in arguments 3. construct viable arguments 4. identify risks 5. make logical decisions 6. test anecdotal evidence for reliability 7. proficiency in oral and written communication Hence, an SBA Project will help students to answer the question, “When am I ever going to use Mathematics?”. In recent years, there has been a growing recognition that students should learn Mathematics in such a way that they can see its relevance in the real world. The Mathematics SBA Project has now become an essential part of the CSEC Syllabus and constitutes 20% of the final CSEC Examination. The SBA Projects require the students to: 1. Select a problem for research from the real world. 2. Select the relevant Mathematics topic from any part of the syllabus in order to research and investigate the chosen problem. 3. Write a clearly explained, logical and sequenced project that satisfies certain guidelines stipulated by the Caribbean Examinations Council (CXC). 07 STRUCTURE OF A CSEC MATHEMATICS SBA PROJECT After you have chosen a Mathematics Project to research, the next step is to decide how the project should be presented. The objective of the Mathematics SBA Project is to seek an answer to a problem that was identified and to generate a written report on the findings of the investigation. This report must be prepared and presented in a particular order The CSEC Mathematics SBA Project consists of seven (7) main sections and must be presented in the following order: SECTION OF MARKS ALCOATED THE PROJECT BY C.X.C. 1. Title of Project 1 2. Introduction 4 3. Method of Data Collection 2 4. Presentation of Data 5 5. Analysis of Data 2 6. Discussion of Findings 2 7. Conclusion 2 Total 18 There are two (2) marks that are awarded for overall presentation. Therefore, the maximum mark that can be obtained for the Mathematics SBA Project is twenty (20). NB: There are two other sub-sections of the Mathematics SBA Project that must be presented at the end of the project. These are: 1. the Reference 2. the Appendix They are called sub-sections because marks are not awarded for these two sections. 08 Therefore, a Mathematics SBA Project has nine (9) different sections; seven (7) sections are awarded marks and two (2) sections are presented for completeness of the project however, no marks will be awarded for them. Before we collect data for a project, we must know for what purpose we are collecting the data; what is the problem that we wish to solve with the data? Hence, the first step of the project is to identify a research problem that we would like to answer. In this initial stage, the subject teacher should play an integral role in assisting the student to identify simple real-world problems that can be solved using basic Mathematical techniques. The nine (9) sections of a project are outlined and explained below. A sample project is used to explain each section of the project. GETTING STARTED (Looking for data to solve a simple real-world problem) SAMPLE PROJECT BEGINS: Let us suppose that a student wants to determine the length of time that his/her peers spend waiting in line for service at the school’s cafeteria during the lunch break. The student will have to collect quantitative data in order to investigate the problem. The project can be presented as follows: SECTION 1: PROJECT TITLE The project title is a clear and comprehensive statement that states the real-world problem which is going to be investigated or explained. It tells the reader what the project is about. For example, in this sample project the project title can read as follows: “An investigation in order to determine the length of time students spend waiting in line for service at the school’s cafeteria during the lunch break.” SECTION 2: INTRODUCTION An introduction is a clearly written paragraph that introduces the reader to the project. It should tell the reader why the project was chosen. In this section, the student must clearly state the objective of the project. The introduction must also include a Table of Contents for the project. The Table of Contents tells the reader where to find the different sections of the project. 09 Here, the student should identify the page number on which each one of the nine (9) sections of the project can be found. The Table of Contents should be placed after the heading “Introduction” but before the actual Introduction is written. For example, the Introduction for the sample SBA Project should be presented as follows: INTRODUCTION TABLE OF CONTENTS: Project Title..................................................................................................Page number Introduction ................................................................................................Page number Method of Data Collection ..........................................................................Page number Presentation of Data....................................................................................Page number Analysis of Data...........................................................................................Page number Discussion of Findings................................................................................Page number Conclusion...................................................................................................Page number References..................................................................................................Page number Appendix.....................................................................................................Page number “Many of my peers usually arrive late for classes after the lunch break. When teachers enquire about this habit, the excuse is oftentimes that the student spent a long period of time waiting in line for service at the school’s cafeteria. This excuse is not always accepted by the teachers and my peers are sometimes punished. Therefore, I have decided to do an investigation in order to determine if the waiting time for service is as long as is reported by students.” SECTION 3: METHOD OF DATA COLLECTION This is a detailed description that states exactly how and where the data for a project was collected. It is important to remember that the research is seeking to answer a problem that the student has identified. Hence, in order to help the student to answer the research problem, data must be carefully collected and studied. Data can be collected in a number of different ways. The particular method chosen for data collection will vary and depends on what type of data the student is looking for. The following are a few different methods that can be used to collect data: (a) THE QUESTIONNAIRE In this method, the student gives the respondents written questions on a question sheet. The questions are standardized which means that all the respondents must answer the same set of questions. This method is an inexpensive way to collect data from a large number of persons. It also allows for collecting data easily without revealing the respondent’s identity. 10 However, some disadvantages of using questionnaires are: • Respondents are sometimes not motivated to answer all the questions. • If some responses are confusing, they are usually difficult to clarify. • Sometimes questionnaires are not returned on time. (b) PERSONAL INTERVIEWS Here, the student can structure some questions that he or she wants to be answered. The student can then question respondents and record the answers on a sheet. In this method, the student has the flexibility to probe the respondents in order to clarify answers and ask follow up questions. However, since the student is face to face with the respondent, this may cause discomfort to the respondent and he or she may not answer truthfully. In addition, the personal nature of the interview may cause the student to ask subjective questions that could lead the respondent in a particular direction. (c) OBSERVATION Without conducting interviews the student can integrate into a population and quietly record data on it. This data can then be studied in order to make inferences about the population. Note, that in this method of data collection, the student will have no control over the group that is going to be observed. (d) EXAMINATION OF RECORDS The student can seek permission to peruse records, reports, statements and regulations in order to collect data on a particular group. However, the disadvantages of this method are: • It can take a long time to sort and collect data from reports and records. • The student may not understand the data from the records and arrive at faulty conclusions. VARIABLES Remember, that a research investigates the relationship between variables. A variable is any factor that has a quantity or quality that can change. For example: • A dependant variable is a variable that can change based on the impact of another variable. • An independent variable is a variable that affects a dependant variable which can cause it to contribute to a problem. For example, in this sample project: Service at the cafeteria is an independent variable that can impact on the waiting time for the service. Waiting time is the dependant variable. When collecting data for a project it is good practice to identify the different types of variables. 11 For example, in this sample project the method of data collection could read as follows: The method of data collection chosen was personal interviews. “A form class was selected from each year group from Forms 1 to 5. The names of all the students from each class were placed into five (5) separate boxes. Each box was shaken and then the names of twenty (20) students were randomly selected from each box. This created a sample size of one hundred (100) students for the purpose of the research.” The one hundred (100) students were then each asked the standardized question: “How long did you spend waiting in line for service at the school’s cafeteria during the lunch break?” The data was collected from Monday to Friday, over a one week period. Permission to survey the students during the period immediately after the lunch break was granted by my teacher. SECTION 4: PRESENTATION OF DATA This is a very important part of the project where, the student presents the data that he/ she has collected. Generally, readers are not keen on reading through long qualitative presentations of data and are more inclined to peruse diagrams since diagrams, usually summarize large amounts of data. Hence, after collecting and organizing qualitative data, the student must present the data using at least two carefully labeled diagrams. These can be statistical diagrams such as: • Frequency Tables • Bar Graphs • Pie Charts • Comparative Bar Graphs • Histograms • Frequency Polygons • Cumulative Frequency Curves. 12 The following is an example of the Presentation of Data for the sample project: The frequency table and the cumulative frequency table show the waiting time in minutes that the one hundred (100) students spent waiting in line at the cafeteria. Frequency Table Waiting time in minutes No. of students (f) 1-5 2 6 - 10 10 11 - 15 17 16 - 20 16 21 - 25 25 26 - 30 28 31 - 35 2 Cumulative Frequency Table Cummulative frequency Waiting time in minutes No. of students (f) (cf) 1-5 2 2 6 - 10 10 12 11 - 15 17 29 16 - 20 16 45 21 - 25 25 70 26 - 30 28 98 31 - 35 2 100 The students must also present the appropriate Mathematics concepts used to arrive at the research results. All mathematical results must be presented accurately using appropriate formulae. Here, the following results can be calculated and presented: • mathematical mean • mode • range • area • volume • profit 13 • loss • Hire Purchase price For example, all calculations for this sample project are presented as follows: 1. Calculation of the mean waiting time. Waiting time Number of Class midpoint f ×x in minutes students f x 1-5 2 3 6 6 - 10 10 8 80 11 - 15 17 13 221 16 - 20 16 18 288 21 - 25 25 23 575 26 - 30 28 28 784 31 - 35 2 33 66 ∑f = 100 2020 ∑f × x The average or mean = (the sum of f × x ) = (∑fx) = 2020 (the sum of f) (∑f) 100 Note: Σ is the Greek letter sigma. It is used here to replace the phrase “the sum of”. The Cumulative Frequency Curve is used to determine the number of students who waited for more than a certain number of minutes for service at the school’s cafeteria. I determined the number of students who waited for more than twenty (20) minutes by drawing a vertical line from twenty (20) minutes on the horizontal axis to meet the curve at point A as shown page 15. A horizontal line was then drawn from A to meet the vertical axis at 42. Hence, 100 - 42 = 58 students from the sample waited for more than twenty (20) minutes for service. The median waiting time was determined by drawing a horizontal line from 50 on the vertical axis to meet the curve at point B as shown. A vertical line was then drawn to meet the horizontal axis at twenty two (22) minutes. Hence, the median waiting time is twenty two (22) minutes. 14 Cumulative frequency curve for data y (NUMBER OF STUDENTS Scale: 2cm ≡ 10 students on vertical axis, 2cm ≡ 5 minutes on horizontal axis 100 90 80 70 60 50 B 40 A 30 20 10 0.5 5.5 10.5 15.5 20.5 25.5 30.5 35.5 15 x (WAITING TIME IN MINUTES) SECTION 5: ANALYSIS OF DATA In this section, the meaning of the data presented must be discussed in a clear and logical manner. The analysis must tell the reader what he or she must get from the data presented. It must say how the findings are related to the project objectives. For example, in this sample project, the analysis can read as follows: “The mean waiting time of 20.2 minutes and the median waiting time of 22 minutes suggest that a large number of students from the sample waited for more than twenty (20) minutes. This is supported by the Cumulative Frequency Curve where fifty eight (58) students waited for more than twenty (20) minutes. This also demonstrates that if a student is chosen at random from the sample, the probability that he/she will wait for more than twenty (20) minutes is 58 .” 100 SECTION 6: DISCUSSION OF FINDINGS: When discussing the findings, the student should clearly state the findings obtained from the analysis of the data and discuss how they are related to the purpose of the project that was stated in the introduction. The discussion of the findings should always be linked to the title and the introduction of the project. Hence, in writing the discussion, the student should always return to the project title and introduction for guidance. For example, in this sample project, the discussion of the findings can read as follows: “The data revealed that the mean waiting time for service at the school’s cafeteria is 20.2 minutes and the median waiting time is twenty two (22) minutes. Fifty eight (58) students waited in line for more than twenty (20) minutes daily. The lunch break is one hour long and students said that, it takes an average of twenty (20) minutes to consume their lunch. This leaves students with 19.8 minutes to prepare themselves for classes after lunch. This should give students enough time to arrive to class on time after the lunch break.” 16 SECTION 7: CONCLUSION The conclusion is a clearly written and concise paragraph that is linked to the purpose of the project. It offers the student a final chance to demonstrate that he/she has a complete understanding of the project’s objectives. When writing the conclusion, the student should return to the introduction of the project and look at the purpose for which the research was conducted. The conclusion should discuss and answer the purpose of the project. For example, in this sample project the conclusion can read as follows: “The research shows that the students’ claim of a long waiting time for service at the cafeteria during the lunch break is reasonable. However, an average waiting time of 20.2 minutes still leaves them with enough time for early arrival to classes for the afternoon session.” After the seven (7) main sections of the project are presented, there are two other sections that must be presented for completeness. However, marks will not be awarded for these two sections. These two sections are: 1. Reference 2. Appendix SECTION 8: REFERENCES A reference is the penultimate section of the project. It contains a list of all the sources such as: books, newspapers, magazines and websites which the student would have used in order to assist in his/her research for the project. For example, the references for the sample project are listed below. Greer, A & Layne, C.E, Certificate Mathematics: A Revision Course for the Caribbean. 2nd Ed. www.mathisfun.com www.purplemath.com www.home.auvanta.com SECTION 9: APPENDIX An Appendix is the section at the end of the project that contains supporting documents that were used by the student in the writing of the project. These supporting documents are essential for the completeness of the project but might be considered too long and cumbersome for the reader to peruse. Raw data, tables, charts, graphs, questionnaires and definitions are some of the things that can be included in the appendix. If you have more than one appendix, make sure that each one begins on a new page and is titled; Appendix A, Appendix B, Appendix C etc. 17 For example, the Appendix for the sample project should be presented as follows: APPENDIX Raw data showing the time that the one hundred (100) students spent waiting for service at the cafeteria. 31, 34, 16, 20, 2, 10, 5, 16, 17, 11, 12, 15, 14, 8, 7, 10, 21, 25, 27, 29, 28, 27, 16, 11, 15, 13, 21, 24, 22, 13, 14, 15, 29, 30, 27, 13, 15, 16, 19, 18, 26, 29, 30, 7, 10, 6, 8, 8, 9, 10, 12, 14, 15, 12, 11, 16, 17, 15, 20, 14, 15, 16, 20, 22, 23, 24, 25, 21, 23, 22, 24, 25, 21, 25, 24, 21, 22, 23, 24, 25, 22, 23, 24, 25, 30, 29, 26, 27, 25, 30, 27, 30, 29, 28, 29, 26, 27, 26, 30, 29. // PRESENTATION OF SAMPLE PROJECT ENDS // ROLE OF THE TEACHER: The basic role of the teacher is to: • Assist students with the selection of topics to be investigated for projects. • Provide meaningful guidance throughout the duration of the project. The teacher will set deadlines for students to complete each part of the project. The teacher must ensure that students adhere to required deadlines. • Allow the students to produce a number of drafts of the chosen project. The teacher should read the drafts, offer feedback and give guidance throughout the cycle of the project. It should be noted that, the teacher’s role is that of a facilitator and as such he/she should only offer guidance rather than give solutions to the problem being investigated in the project. • Dedicate at least one period per week in order to facilitate discussions on issues relating to the project. • Discuss students’ problems with the Head of Department in order to arrive at solutions. 18 ROLE OF THE STUDENT The basic role of the student is to: • Work closely with the teacher in order to select an SBA Project research topic. • Adhere to all deadlines set by the teacher. • Seek advice from the teacher on all the different parts of the project. • Read newspapers, journals, magazines and books in order to get information on the topic chosen for investigation. The internet is also an invaluable tool in the preparation of a project. • Produce drafts of the project for the teacher’s perusal before the final project is submitted. This will allow the teacher to give meaningful and timely feedback to students before the completion of projects and final submission. A WORD OF CAUTION • A Mathematics SBA Project is meant to test certain skills of a particular candidate. Therefore, when doing an SBA project, one must make every effort to ensure that one’s original work is produced and presented. If the student is using another person’s work, then credit must be given to the producer of the original work. Students may work in small groups when collecting data for a project. However, each student is expected to write his or her own analysis and conclusion of data collected. • The Mathematics SBA Project has a limit of one thousand (1,000) words and every effort must be made to stay within the stipulated limit. Failure to do so will result in a deduction of marks from the student’s final SBA Project mark. In order to avoid exceeding the word limit one should put extra materials such as charts, tables, raw data, pictures and definitions in an appendix. • When writing a project, do not use the actual names of businesses, schools, people and government institutions. • When working on projects ensure that work is continually backed up and saved. This will help students to easily replicate work in the event of theft or a disaster. It will also help in the submission of the final Mathematics SBA project to CXC as this will be presented via soft copy. • Two marks are awarded for the overall presentation of the project. Hence, every effort must be made in order to ensure that the information presented in the project is grammatically correct and must be written clearly and logically. The student can ask an English teacher to assist him or her with proofreading the Mathematics SBA Project before its final submission. 19 PRESENTATION OF 10 WORKED MATHEMATICS SBA PROJECTS 20 P R OJ E C T 1 SECTION 01 PROJECT TITLE An investigation into the waiting time for service at a health centre in Trinidad and Tobago. 21 PR O JECT 1 SECTION 02 INTRODUCTION TABLE OF CONTENTS 1. Title.......................................................................................... Page Number 2. Introduction.............................................................................. Page Number 3. Method of Data Collection....................................................... Page Number 4. Presentation of Data................................................................. Page Number 5. Analysis of Data....................................................................... Page Number 6. Discussion of Findings............................................................. Page Number 7. Conclusion............................................................................... Page Number 8. References............................................................................... Page Number 9. Appendix................................................................................. Page Number People complain via the media (radio, talk shows, nightly news, newspaper articles, etc.) about the long time they must wait for service at government institutions such as health centres, emergency rooms at hospitals, and immigration offices. Waiting a long time for service can be frustrating, inconvenient and may cause even the most tolerant person to become annoyed, anxious or even lose emotional control. Their account of long waiting time may, however, be regarded as either unreliable or hearsay. The purpose of this project is to use statistical techniques to investigate the claim that people wait a long time for service at government institutions. A health centre was chosen because most of the complaints were of tardy health care delivery. 22 P R OJ E C T 1 SECTION 03 METHOD OF DATA COLLECTION A health centre was visited from Monday to Friday between the hours of 7:00 am to 4:00 p.m. The data captured were that of the first thirty persons who entered for service each day. The registering clerk issued a consecutive number to each patient as he or she was seated. A method of data collection called non-participant observation was used. The data were generated by directly observing and recording the activities of each day as follows: • days of the week that persons arrived for service • consecutive numbers of 30 persons who entered the room for service • arrival times of each person • the times at which they were called to receive service. There was no interaction with the persons being observed. (See Appendix for raw data) 23 PR O JECT 1 SECTION 04 PRESENTATION OF DATA After the data were collected for the five days, the raw data were then organized using five frequency distribution tables. The tables consist of six class intervals each having a width of 50 minutes. MONDAY TUESDAY WEDNESDAY Waiting time Number of Waiting time Number of Waiting time Number of in minutes patients (f) in minutes patients (f) in minutes patients (f) 100 - 149 0 100 - 149 1 100 - 149 0 150 - 199 3 150 - 199 9 150 - 199 13 200 - 249 5 200 - 249 10 200 - 249 8 250 - 299 5 250 - 299 10 250 - 299 4 300 - 349 17 300 - 349 0 300 - 349 5 350 - 399 0 350 - 399 0 350 - 399 0 TOTAL 30 TOTAL 30 TOTAL 30 THURSDAY FRIDAY Waiting time Number of Waiting time Number of in minutes patients (f) in minutes patients (f) 100 - 149 7 100 - 149 1 150 - 199 9 150 - 199 4 200 - 249 4 200 - 249 8 250 - 299 8 250 - 299 7 300 - 349 2 300 - 349 8 350 - 399 0 350 - 399 2 TOTAL 30 TOTAL 30 24 Bar graph showing average waiting time per day in hours (2 cm ≡ 1 hour on vertical axis ) 5 4.5 4 AVERAGE WAITING TIME IN HOURSE 3.5 3 2.5 2 1.5 1 0.5 0 MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY The mean or average waiting time will now be calculated for each day. MONDAY Waiting time in Number of Class midpoint f ×x minutes patients (f) (x) 150 - 199 3 174.5 523.5 200 - 249 5 224.5 1122.5 250 - 299 5 274.5 1372.5 300 - 349 17 324.5 5516.5 TOTAL 30 TOTAL 8535 The average or mean = (the sum of f × x ) = (∑fx) = 8535 (the sum of f) (∑f) 30 = 284.5 minutes = 4.74 hours 25 TUESDAY Waiting time in Number of Class midpoint f ×x minutes patients (f) (x) 100 - 149 1 124.5 124.5 150 - 199 9 174.5 1570.5 200 - 249 10 224.5 2245.0 250 - 299 10 274.5 2745.0 TOTAL 30 TOTAL 6685 The average or mean = (the sum of f × x ) = (∑fx) = 6685 (the sum of f) (∑f) 30 = 222.83 minutes = 3.71 hours WEDNESDAY Waiting time in Number of Class midpoint f ×x minutes patients (f) (x) 150 - 199 13 174.5 2268.5 200 - 249 8 224.5 1796 250 - 299 4 274.5 1098 300 - 349 5 324.5 1622.5 TOTAL 30 TOTAL 6785 The average or mean = (the sum of f × x ) = (∑fx) = 6785 (the sum of f) (∑f) 30 = 226.17 minutes = 3.77 hours 26 THURSDAY Waiting time in Number of Class midpoint f ×x minutes patients (f) (x) 100 - 149 7 124.5 871.5 150 - 199 9 174.5 1570.5 200 - 249 4 224.5 898 250 - 299 8 274.5 2196 300 - 349 2 324.5 649 TOTAL 30 TOTAL 6185 The average or mean = (the sum of f × x ) = (∑fx) = 6185 (the sum of f) (∑f) 30 = 206.2 minutes = 3.44 hours FRIDAY Waiting time in Number of Class midpoint f ×x minutes patients (f) (x) 100 - 149 1 124.5 124.5 150 - 199 4 174.5 698 200 - 249 8 224.5 1796 250 - 299 7 274.5 1921.5 300 - 349 8 324.5 2596 350 - 399 2 374.5 749 TOTAL 30 TOTAL 7885 The average or mean = (the sum of f × x ) = (∑ fx) = 7885 (the sum of f) (∑ f) 30 = 262.8 minutes = 4.38 hours Σ is the Greek letter sigma. Note: 27 It is used here to replace the phrase “the sum of” PR O JECT 1 SECTION 05 ANALYSIS OF DATA The data collected from this research revealed: 1. The average waiting time on Monday was 4.74 hours The average waiting time on Tuesday was 3.71 hours The average waiting time on Wednesday was 3.77 hours The average waiting time on Thursday was 3.44 hours The average waiting time on Friday was 4.38 hours 2. . he bargraph shows that the average waiting time on Tuesday, Wednesday and T Thursday were approximately the same (i.e. about 3.6 hours). Patients waited much longer on Monday and Friday for service. On both of these days they waited approximately an hour longer than the other three days. On any given day they waited an average of at least three hours for service. 3. The average daily waiting time = (4.74+3.71+3.77+3.44+4.38) . 5 . = 20.04 . 5 . = 4.008 hours ≈ 4 hours This suggests that if all the patients were to wait for the same number of hours for service each day, that waiting time will be 4 hours. 28 P R OJ E C T 1 SECTION 06 DISCUSSION OF FINDINGS The data was collected using personal observation over a one week period revealed that, on average, patients waited for approximately four hours before being attended to at this particular health centre. 29 PR O JECT 1 SECTION 07 CONCLUSION Based on this research, the public’s perception that persons wait a long time for service at government institutions, namely health centres, is true. At this institution in particular, persons waited an average of 4 hours daily for service. This supports the anecdotal claim that people spend a long time waiting for service. This situation is untenable and requires urgent attention by those in authority. 30 P R OJ E C T 1 SECTION 08 REFERENCES Toolsie, R.; Mathematics: A Complete Course with CXC Questions. Vol. 2 www.compass.port.ac.uk www.yourarticlelibrary.com 31 PR O JECT 1 SECTION 09 APPENDIX MONDAY Patient No. Arrival Time for Service Time Called for Service Time Waited for Service 1 7:00 a.m. 9:30 a.m. 2 hours 30 minutes 2 7:00 a.m. 9:41 a.m. 2 hours 41 minutes 3 7:03 a.m. 10:00 a.m. 2 hours 57 minutes 4 7:05 a.m. 10:20 a.m. 3 hours 24 minutes 5 7:10 a.m. 10:40 a.m. 3 hours 30 minutes 6 7:15 a.m. 10:50 a.m. 3 hours 35 minutes 7 7:18 a.m. 11:02 a.m. 3 hours 44 minutes 8 7:20 a.m. 11:20 a.m. 4 hours 0 minutes 9 7:23 a.m. 11:35 a.m. 4 hours 12 minutes 10 7:28 a.m. 11:44 a.m. 4 hours 16 minutes 11 7:30 a.m. 11:58 a.m. 4 hours 28 minutes 12 7:38 a.m. 12:15 p.m. 4 hours 37 minutes 13 7:42 a.m. 12:38 p.m. 4 hours 56 minutes 14 8:00 a.m. 1:00 p.m. 5 hours 0 minutes 15 8:10 a.m. 1:15 p.m. 5 hours 5 minutes 16 8:12 a.m. 1:30 p.m. 5 hours 18 minutes 17 8:30 a.m. 1:38 p.m. 5 hours 8 minutes 18 8:33 a.m. 1:45 p.m. 5 hours 12 minutes 19 8:38 a.m. 1:50 p.m. 5 hours 12 minutes 20 8:40 a.m. 2:00 p.m. 5 hours 20 minutes 21 8:45 a.m. 2:12 p.m. 5 hours 27 minutes 22 8:56 a.m. 2:20 p.m. 5 hours 24 minutes 23 9:01 a.m. 2:25 p.m. 5 hours 24 minutes 24 9:15 a.m. 2:35 p.m. 5 hours 10 minutes 25 9:28 a.m. 2:38 p.m. 5 hours 12 minutes 26 9:30 a.m. 2:45 p.m. 5 hours 15 minutes 27 9:36 a.m. 2:48 p.m. 5 hours 12 minutes 28 9:39 a.m. 2:50 p.m. 5 hours 11 minutes 29 9:44 a.m. 2:55 p.m. 5 hours 11 minutes 30 9:45 a.m. 3:00 p.m. 5 hours 15 minutes 32 TUESDAY Patient No. Arrival Time for Service Time Called for Service Time Waited for Service 1 7:00 a.m. 9:20 a.m. 2 hours 20 minutes 2 7:10 a.m. 9:40 a.m. 2 hours 30 minutes 3 7:11 a.m. 9:48 a.m. 2 hours 37 minutes 4 7:18 a.m. 10:01 a.m. 2 hours 43 minutes 5 7:25 a.m. 10:15 a.m. 2 hours 50 minutes 6 7:29 a.m. 10:21 a.m. 2 hours 51 minutes 7 7:30 a.m. 10:30 a.m. 3 hours 0 minutes 8 7:35 a.m. 10:39 a.m. 3 hours 4 minutes 9 7:39 a.m. 10:50 a.m. 3 hours 1 minutes 10 7:43 a.m. 11:01 a.m. 3 hours 18 minutes 11 7:49 a.m. 11:15 a.m. 3 hours 26 minutes 12 7:53 a.m. 11:20 p.m. 3 hours 27 minutes 13 7:59 a.m. 11:28 a.m. 3 hours 29 minutes 14 8:05 a.m. 11:35 a.m. 3 hours 30 minutes 15 8:08 a.m. 11:43 a.m. 3 hours 35 minutes 16 8:15 a.m. 11:49 a.m. 3 hours 34 minutes 17 8:18 a.m. 11:53 a.m. 3 hours 35 minutes 18 8:23 a.m. 12:20 p.m. 3 hours 57 minutes 19 8:29 a.m. 12:35 p.m. 4 hours 6 minutes 20 8:30 a.m. 12:40 p.m. 4 hours 5 minutes 21 8:39 a.m. 12:49 p.m. 4 hours 10 minutes 22 8:42 a.m. 12:53 p.m. 4 hours 11 minutes 23 8:50 a.m. 1:10 p.m. 4 hours 20 minutes 24 8:55 a.m. 1:15 p.m. 4 hours 20 minutes 25 9:01 a.m. 1:30 p.m. 4 hours 29 minutes 26 9:08 a.m. 1:38 p.m. 4 hours 20 minutes 27 9:11 a.m. 1:44 p.m. 4 hours 33 minutes 28 9:15 a.m. 1:55 p.m. 4 hours 40 minutes 29 9:18 a.m. 2:10 p.m. 4 hours 52 minutes 30 9:33 a.m. 2:15 p.m. 4 hours 42 minutes 33 WEDNESDAY Patient No. Arrival Time for Service Time Called for Service Time Waited for Service 1 7:00 a.m. 9:05 a.m. 2 hours 5 minutes 2 7:00 a.m. 9:25 a.m. 2 hours 25 minutes 3 7:00 a.m. 9:28 a.m. 2 hours 28 minutes 4 7:05 a.m. 9:39 a.m. 2 hours 34 minutes 5 7:12 a.m. 9:45 a.m. 2 hours 33 minutes 6 7:14 a.m. 9:55 a.m. 2 hours 41 minutes 7 7:15 a.m. 10:05 a.m. 2 hours 50 minutes 8 7:18 a.m. 10:15 a.m. 2 hours 57 minutes 9 7:23 a.m. 10:31 a.m. 3 hours 8 minutes 10 7:30 a.m. 10:39 a.m. 3 hours 9 minutes 11 7:33 a.m. 10:50 a.m. 3 hours 17 minutes 12 7:39 a.m. 10:55 a.m. 3 hours 16 minutes 13 7:41 a.m. 11:00 a.m. 3 hours 19 minutes 14 7:50 a.m. 11:11 a.m. 3 hours 21 minutes 15 7:54 a.m. 11:20 a.m. 3 hours 26 minutes 16 8:00 a.m. 11:29 a.m. 3 hours 29 minutes 17 8:05 a.m. 11:40 a.m. 3 hours 35 minutes 18 8:08 a.m. 11:49 a.m. 3 hours 41 minutes 19 8:12 a.m. 11:55 a.m. 3 hours 43 minutes 20 8:18 a.m. 12:20 p.m. 4 hours 2 minutes 21 8:23 a.m. 12:30 p.m. 4 hours 7 minutes 22 8:29 a.m. 12:45 p.m. 4 hours 16 minutes 23 8:33 a.m. 1:10 p.m. 4 hours 37 minutes 24 8:39 a.m. 1:30 p.m. 4 hours 51 minutes 25 8:44 a.m. 1:39 p.m. 4 hours 55 minutes 26 8:48 a.m. 1:48 p.m. 5 hours 0 minutes 27 8:55 a.m. 2:00 p.m. 5 hours 5 minutes 28 8:58 a.m. 2:15 p.m. 5 hours 17 minutes 29 9:01 a.m. 2:25 p.m. 5 hours 14 minutes 30 9:05 a.m. 2:30 p.m. 5 hours 25 minutes 34 THURSDAY Patient No. Arrival Time for Service Time called for Service Time waited for Service 1 7:00 a.m. 9:03 a.m. 2 hours 3 minutes 2 7:00 a.m. 9:10 a.m. 2 hours 10 minutes 3 7:05 a.m. 9:20 a.m. 2 hours 15 minutes 4 7:10 a.m. 9:30 a.m. 2 hours 20 minutes 5 7:12 a.m. 9:38 a.m. 2 hours 26 minutes 6 7:15 a.m. 9:50 a.m. 2 hours 35 minutes 7 7:30 a.m. 9:55 a.m. 2 hours 25 minutes 8 7:35 a.m. 10:15 a.m. 2 hours 35 minutes 9 7:39 a.m. 10:20 a.m. 2 hours 41 minutes 10 8:00 a.m. 10:28 a.m. 2 hours 28 minutes 11 8:05 a.m. 10:35 a.m. 2 hours 30 minutes 12 8:08 a.m. 10:49 a.m. 2 hours 41 minutes 13 8:11 a.m. 11:55 a.m. 2 hours 44 minutes 14 8:13 a.m. 11:11 a.m. 2 hours 58 minutes 15 8:25 a.m. 11:20 a.m. 2 hours 55 minutes 16 8:39 a.m. 11:30 a.m. 2 hours 51 minutes 17 8:50 a.m. 12:10 a.m. 3 hours 20 minutes 18 8:55 a.m. 12:25 a.m. 3 hours 30 minutes 19 9:00 a.m. 1:00 p.m. 4 hours 0 minutes 20 9:08 a.m. 1:10 p.m. 4 hours 2 minutes 21 9:10 a.m. 1:29 p.m. 4 hours 19 minutes 22 9:15 a.m. 1:30 p.m. 4 hours 14 minutes 23 9:21 a.m. 1:39 p.m. 4 hours 18 minutes 24 9:23 a.m. 2:00 p.m. 4 hours 37 minutes 25 9:25 a.m. 2:11 p.m. 4 hours 46 minutes 26 9:28 a.m. 2:20 p.m. 5 hours 52 minutes 27 9:30 a.m. 2:25 p.m. 5 hours 55 minutes 28 9:31 a.m. 2:29 p.m. 5 hours 58 minutes 29 9:34 a.m. 2:35 p.m. 5 hours 1 minutes 30 9:35 a.m. 2:41 p.m. 5 hours 6 minutes 35 FRIDAY Patient No. Arrival Time for Service Time called for Service Time waited for Service 1 7:00 a.m. 9:15 a.m. 2 hours 15 minutes 2 7:00 a.m. 9:30 a.m. 2 hours 30 minutes 3 7:00 a.m. 9:35 a.m. 2 hours 35 minutes 4 7:05 a.m. 10:00 a.m. 2 hours 55 minutes 5 7:08 a.m. 10:35 a.m. 3 hours 17 minutes 6 7:12 a.m. 10:44 a.m. 3 hours 32 minutes 7 7:19 a.m. 10:55 a.m. 3 hours 36 minutes 8 7:25 a.m. 11:30 a.m. 4 hours 5 minutes 9 7:30 a.m. 11:35 a.m. 4 hours 5 minutes 10 7:35 a.m. 11:38 a.m. 4 hours 3 minutes 11 7:39 a.m. 11:44 a.m. 4 hours 5 minutes 12 7:44 a.m. 11:49 a.m. 4 hours 5 minutes 13 7:48 a.m. 11:55 a.m. 4 hours 6 minutes 14 7:55 a.m. 12:08 p.m. 4 hours 13 minutes 15 8:00 a.m. 12:20 p.m. 4 hours 20 minutes 16 8:05 a.m. 12:35 p.m. 4 hours 20 minutes 17 8:10 a.m. 12:49 p.m. 4 hours 39 minutes 18 8:18 a.m. 1:00 p.m. 4 hours 42 minutes 19 8:25 a.m. 1:20 p.m. 4 hours 55 minutes 20 8:30 a.m. 1:28 p.m. 4 hours 58 minutes 21 8:33 a.m. 1:39 p.m. 5 hours 6 minutes 22 8:40 a.m. 1:45 p.m. 5 hours 5 minutes 23 8:45 a.m. 2:00 p.m. 5 hours 15 minutes 24 8:49 a.m. 2:20 p.m. 5 hours 31 minutes 25 8:52 a.m. 2:29 p.m. 5 hours 37 minutes 26 8:55 a.m. 2:38 p.m. 5 hours 43 minutes 27 8:59 a.m. 2:45 p.m. 5 hours 46 minutes 28 9:01 a.m. 2:50 p.m. 5 hours 49 minutes 29 9:03 a.m. 3:00 p.m. 5 hours 57 minutes 30 9:05 a.m. 3:10 p.m. 6 hours 6 minutes 36 P R OJ E C T 2 SECTION 01 PROJECT TITLE An investigation into various Hire Purchase plans from four different stores in Trinidad and Tobago. 37 PRO JEC T 2 SECTION 01 INTRODUCTION TABLE OF CONTENTS 1. Title.......................................................................................... Page Number 2. Introduction.............................................................................. Page Number 3. Method of Data Collection....................................................... Page Number 4. Presentation of Data................................................................. Page Number 5. Analysis of Data....................................................................... Page Number 6. Discussion of Findings............................................................. Page Number 7. Conclusion............................................................................... Page Number 8. References............................................................................... Page Number 9. Appendix................................................................................. Page Number At one time or another everyone purchases something from a store. Sometimes payment may be in full at the time of purchase referred to as cash. However, the buyer may choose to enter into an arrangement whereby payments can be made in equal monthly instalments. This arrangement and its process is referred to as Hire Purchase. For example: my parents are planning to purchase a living room set for Christmas but they do not have enough money to pay cash for the item. As such they must enter into a hire purchase arrangement with the store. Having studied consumer arithmetic in school, I have decided to use my knowledge of Hire Purchase to assist them in investigating various Hire Purchase plans. The purpose of this project is to investigate various Hire Purchase plans in order to determine how these plans are calculated and their cost effectiveness. 38 P R OJ E C T 2 SECTION 03 METHOD OF DATA COLLECTION Four (4) stores were visited. Store A – East Trinidad Store B – West Trinidad Store C – North Trinidad Store D – South Trinidad An interview with the supervisor of each store provided the data regarding the cash price and the various Hire Purchase plans on the same living room set in his/her respective store. The data collected are as follows: (i) Cash Price (ii) Hire Purchase Plans: (a) 3 months for cash (b) 1 year at regular price (c) down payment and 24 equal monthly instalments (d) no down payment and 24 monthly instalments 39 PRO JEC T 2 SECTION 04 PRESENTATION OF DATA STORE A Cost of living Type of Plan Method of Payment Plan room set ($) Cash one-off full payment 12,309.60 Regular Price Cost of living Type of Plan Method of Payment Plan ($) room set ($) Hire Purchase 3 months for cash 15,387.00 16,156.35 Plan a Hire Purchase 1 year at regular price 15,387.00 18,464.40 Plan b Hire Purchase Down payment and 24 equal 15,387.00 21,541.80 Plan c monthly instalments Hire Purchase No down payment and 24 monthly 15,387.00 21,541.80 Plan d instalments STORE B Cost of living Type of Plan Method of Payment Plan room set ($) Cash one-off full payment 12,800.00 Regular Price Cost of living Type of Plan Method of Payment Plan ($) room set ($) Hire Purchase 3 months for cash 16,000.00 16,800.00 Plan a Hire Purchase 1 year at regular price 16,000.00 19,200.00 Plan b Hire Purchase Down payment and 24 equal 16,000.00 22,400.00 Plan c monthly instalments Hire Purchase No down payment and 24 monthly 16,000.00 22,400.00 Plan d instalments 40 STORE C Cost of living Type of Plan Method of Payment Plan room set ($) Cash one-off full payment 12,309.60 Regular Price Cost of living Type of Plan Method of Payment Plan ($) room set ($) Hire Purchase 3 months for cash 14,800.00 15,540.00 Plan a Hire Purchase 1 year at regular price 14,800.00 17,760.00 Plan b Hire Purchase Down payment and 24 equal 14,800.00 20,720.00 Plan c monthly instalments Hire Purchase No down payment and 24 monthly 14,800.00 20,720.00 Plan d instalments STORE D Cost of living Type of Plan Method of Payment Plan room set ($) Cash one-off full payment 11,200.00 Regular Price Cost of living Type of Plan Method of Payment Plan ($) room set ($) Hire Purchase 3 months for cash 14,000.00 14,700.00 Plan a Hire Purchase 1 year at regular price 14,000.00 16,800.00 Plan b Hire Purchase Down payment and 24 equal 14,000.00 19,600.00 Plan c monthly instalments Hire Purchase No down payment and 24 monthly 14,000.00 19,600.00 Plan d instalments 41 The graph shows how the cash price compares to the different Hire Purchase plans at store A. 20 18 16 PRICE IN $ × 1,000 14 12 10 08 06 04 02 a b c d TYPE OF HIRE PURCHASE PLAN CP HP PRICE All four stores offer the same payment structure under cash and Hire Purchase, as such, the data from one store will be used to demonstrate the general principle of Hire Purchase plans. Please note that the regular price is the price that is used to calculate the different Hire Purchase plans. The regular price is obtained by marking up the cash price by 25%. In order to calculate the different Hire Purchase plans, for every 3 months, 5 % interest is added to the regular price. (e.g.) 3 months for cash: 5% interest is added to the regular price. 6 months for cash: 10% interest is added to the regular price. 12 months for cash: 20% interest is added to the regular price. 24 months for cash: 40% interest is added to the regular price. 42 Using Data From Store A Calculation of cost of living room set under two different plans Plan # 1 Cash: a one-off full payment of the cash price at the time of purchase. CASH PRICE = $12,309.60 Plan #2 HIRE PURCHASE: There were four Hire Purchase plans offered: a, b, c, d Plan a: 3 Months for Cash This consists of the regular price plus 5% interest of the regular price. This amount must be paid in 3 months. Regular price = $15,387.00 Interest = 5% of regular price = 5 × $15,387 = $769.35 100 ∴ Cost of Living Room set = Regular Price + Interest = $15,387 + $769.35 = $16,156.35 Number of monthly instalments = 3 ∴ Amount of each monthly instalment = Cost of set 3 = $15,156.35 3 = $5,385.45 Therefore, if the living room set is purchased using this plan, my parents will pay $16,156.35 in 3 equal monthly instalments of $5,385.45. 43 Plan b: One Year at Regular Price This consists of the regular price plus 20% interest of the regular price. This amount must be paid in 12 months. Regular price = $15,387.00 Interest = 20% of regular price = 20 × $15,387 = $3,077.40 100 ∴ Cost of living room set = Regular Price + Interest = $15,387 + $3,077.40 = $18,464.40 Number of monthly instalments = 12 ∴ Amount of each monthly instalment = Cost of set 12 = $18,464.40 12 = $1,538.70 Therefore, if the living room set is purchased using this plan, my parents will pay $18,464.40 in 12 equal monthly instalments of $1,538.70. Plan c: Down Payment and 24 Monthly Instalments Regular price = $15,387.00 Interest = 40% of regular price = 40 × $15,387 = $6,154.80 100 ∴ Cost of living room set = Regular Price + Interest = $15,387 + $6,154.80 = $21,541.80 Down payment = 25% of cost of set = 25 × 21,541.8 0 100 = $5,385.45 Balance to be paid in monthly instalment = cost of set – down payment = $21,541.80 – $5,385.45 = $16,156.35 Number of monthly instalments = 24 ∴ Amount of each monthly instalment = Balance 24 = $21,541.80 24 = $673.20 Therefore, if the living room set is purchased using this plan, my parents will pay $21,541.80 in 24 equal monthly instalments of $673.20 and an initial down payment of $5,385.45. 44 Plan d: No Down Payment and 24 Monthly Instalments Regular price = $15,387.00 Interest = 40% of regular price = 40 × $15,387 = $6,154.80 100 ∴ Cost of living room set = Regular Price + Interest = $15,387 + $6,154.80 = $21,541.80 Number of monthly instalments = 24 ∴ Amount of each monthly instalment = Cost of set 24 = 21,541.80 24 = $897.60 Therefore, if the living room set is purchased using this plan, my parents will pay $21,541.80 in 24 equal monthly instalments of $897.60. 45 PRO JEC T 2 SECTION 05 ANALYSIS OF DATA Each Hire Purchase plan will now be analysed and compared to the cash price for cost effectiveness. Hire Purchase Plan a Hire Purchase price of living room set = $16,156.35 Cash price = $12,309.60 Difference in price = ($16,156.35 – $12,309.60) = $ 3,846.75 Therefore, under Plan a the consumer pays $ 3,846.75 more than the cash price. This increase in price as a percentage = 3,846.75 × 100% 12,309.60 = 31.25% Hire Purchase Plan b Hire Purchase price of living room set = $18,464.40 Cash price = $12,309.60 Difference in price = ($18,464.40– $12,309.60) = $ 6,154.80 Therefore, under Plan b the consumer pay $ 6,154.80 more than the cash price. This increase in price as a percentage = 6,154.80 × 100% 12,309.60 = 50% Hire Purchase Plan c Hire Purchase price of living room set = $ 21,541.40 Cash price = $12,309.60 Difference in price = ($21,541.40 – $12,309.60) = $ 9,231.80 Therefore, under Plan c the consumer pay $ 9,231.80 more than the cash price. This increase in price as a percentage = 9,231.80 × 100% 12,309.60 = 75% 46 Hire Purchase Plan d Hire Purchase price of living room set = $21,541.80 Cash price = $12,309.60 Difference in price = ($21,541.40 – $12,309.60) = $ 9,231.80 Therefore, under Plan d the consumer pay $ 9,231.80 more than the cash price. This increase in price as a percentage = 9,231.80 × 100% 12,309.60 = 75% NB: To qualify for a Hire Purchase plan, my parents will be subjected to a Means Test. (See Appendix) 47 PRO JEC T 2 SECTION 06 DISCUSSION OF FINDINGS From the data collected, it is seen that: 1. all stores have the same payment structure as stated in the presentation of data. 2. all Hire Purchase prices originated from the cash price. 3. purchasing the living room set under the cash plan is the cheapest option. 4. for each method of Hire Purchase the increase in interest varies incrementally from 31% up to 75%. 5. there is a correlation between the length of time of repayment, the amount of monthly instalments, and the interest charged under each plan. (e.g. the longer my parents take to repay, the smaller the monthly instalments and the higher the interest) 6. The only true cash price is a one-off full payment made at the time of purchasing the item. All other prices are derived from the regular price, which is 25% more than the cash price. (e.g. 3 months for cash is a term used by stores to mislead consumers into believing that they are paying cash for the item). There are many payment plans available to my parents in purchasing the living room set via Hire Purchase. However, all these plans come with extra charges because hire purchase is a form of borrowing and there is a cost attached to borrowing. By using consumer arithmetic, I was able to compute the percentage increase on the cash price of the living room set through the four different Hire Purchase plans. Using these calculations my parents can determine beforehand the advantages and disadvantages of each plan before entering into a Hire Purchase agreement. The main advantage is that it allows my parents to make use of the living room set whilst still paying for it. On the other hand, my parents may be faced with the disadvantage of paying exorbitant prices for the living room set. In some cases almost double the cash price. 48 P R OJ E C T 2 SECTION 07 CONCLUSION The investigation of the different Hire Purchase payment plans reveal the following calculations: 1. Cost of the living room set under Plan a: 3 months for cash is $16,156.35, which represents a 31% increase in interest on the cash price. 2. Cost of the living room set under Plan b: 1 year at regular price is $18,464.40 which represents a 50% increase in interest on the cash price. 3. Cost of the living room set under Plan c: down payment and 24 monthly instalments is $21,541.40 which represents a 70% increase in interest on the cash price. 4. Cost of the living room set under Plan d: 24 monthly instalments is $21,541.40 which represents a 70% increase in interest on the cash price. Based on the above, the most cost effective plan is Plan a: 3 months for cash. 49 PRO JEC T 2 SECTION 08 REFERENCES Greer, A. & Layne, C. E. (1991) Certificate Mathematics: A Revision Course for the Caribbean. Toolsie, R, (1999). A Complete Course with CXC Questions. www.everythingmaths.co.za www.wikiEducator.org 50 P R OJ E C T 2 SECTION 09 APPENDIX Terms used in this project Cash Price : One full payment at the time of purchase Hire Purchase/Regular price : the Hire Purchase price is also referred to as the Regular Price. It consists of 25% increase on the cash price Down payment : The first payment the buyer makes on the item to begin the Hire Purchase process. Instalments : Weekly or monthly payments within the purchase plan Hire Purchase Agreement : An agreement between the buyer and the store outlining the terms and conditions of the payment plan. It also outlines measures to be taken regarding non-payment on the buyer’s part. Means Test : This is used to determine whether the buyer will qualify for a particular Hire Purchase plan. Requirements for Means Test are: Job letter and payslip Identification Card Utility bill Two (2) references Bank Statement for last 6 months if self-employed. 51 PR O JEC T 3 SECTION 01 PROJECT TITLE An investigation into teachers’ punctuality and regularity at a government school. 52 P R OJ E C T 3 SECTION 02 INTRODUCTION TABLE OF CONTENTS 1. Title.......................................................................................... Page Number 2. Introduction.............................................................................. Page Number 3. Method of Data Collection....................................................... Page Number 4. Presentation of Data................................................................. Page Number 5. Analysis of Data....................................................................... Page Number 6. Discussion of Findings............................................................. Page Number 7. Conclusion............................................................................... Page Number 8. References............................................................................... Page Number 9. Appendix................................................................................. Page Number Stakeholders such as PTA, leaders within the Ministry of Education and students themselves, complain about habitual tardiness and absenteeism on the part of teachers within the government school system. The classroom teacher is also a major stakeholder in the education system. The success of the delivery of the curriculum and discipline are dependent on the teacher being in the classroom, as such, teacher punctuality and regularity are major factors within the system. The claim of teacher absenteeism and habitual tardiness is usually derived from anecdotal evidence rather than data driven claims. The purpose of this project is to examine the records of teacher punctuality and regularity for one month at a government secondary school in order to determine whether the lates and absences are as high as the anecdotal reports suggest. 53 PR O JEC T 3 SECTION 03 METHOD OF DATA COLLECTION Permission was granted by the school’s principal to peruse the Teachers’ Attendance Register weekly for a period of one month in order to extract data on regularity and punctuality for that specific month. The sample size consisted of 45 teachers. 54 P R OJ E C T 3 SECTION 04 PRESENTATION OF DATA After the data were collected, they were organized using a table, comparative bar charts and frequency tables as follows: The table shows the number of teachers who were late and absent for the particular month Week Day Late Absent Present Monday 10 11 34 Tuesday 12 14 31 Week 1 Wednesday 14 08 37 Thursday 17 14 31 Friday 14 14 31 Monday 11 03 42 Tuesday 13 07 38 Week 2 Wednesday 16 09 36 Thursday 16 06 39 Friday 12 17 28 Monday 09 12 33 Tuesday 16 11 34 Week 3 Wednesday 10 09 36 Thursday 12 03 42 Friday 11 14 31 Monday 12 12 33 Tuesday 12 08 37 Week 4 Wednesday 09 09 36 Thursday 08 02 43 Friday 10 20 25 Table One 55 Bar graph showing number of times late and absent in Week One 20 NUMBER OF TIMES LATE/ABSENT 18 16 14 12 10 08 06 04 02 MON TUES WED THUR FRI DAYS OF THE WEEK LATE ABSENT Bar graph showing number of times late and absent in Week Two 20 NUMBER OF TIMES LATE/ABSENT 18 16 14 12 10 08 06 04 02 MON TUES WED THUR FRI DAYS OF THE WEEK LATE ABSENT 56 Bar graph showing number of times late and absent in Week Three 20 NUMBER OF TIMES LATE/ABSENT 18 16 14 12 10 08 06 04 02 MON TUES WED THUR FRI DAYS OF THE WEEK LATE ABSENT Bar graph showing number of times late and absent in Week Four 20 NUMBER OF TIMES LATE/ABSENT 18 16 14 12 10 08 06 04 02 MON TUES WED THUR FRI DAYS OF THE WEEK LATE ABSENT 57 Raw data showing the number of minutes the 45 teachers were late in a month 3, 10, 8, 7, 12, 16, 3, 10, 1, 20, 39, 2, 14, 21, 8, 9, 43, 49, 50, 52, 40, 1, 5, 6, 12, 17, 58, 60, 55, 45, 48, 41, 45, 35, 33, 39, 25, 22, 30, 24, 30, 21, 30, 22, 30 Table Two: Frequency table showing how the raw data were organised Time in minutes Number of Teachers 1 – 10 13 11 – 20 06 21 – 30 10 31 – 40 05 41 – 50 07 51 – 60 04 Table Three: Table showing the calculation of the mean number of minutes late by teachers in a month Time in Minutes Number of Teachers (f) Midpoint x f×x 1 – 10 13 5.5 71.5 11 – 20 06 15.5 93 21 – 30 10 25.5 255 31 – 40 05 35.5 177.5 41 – 50 07 45.5 318.5 51 – 60 4 55.5 222 ∑ f = 45 ∑fx = 1,137.5 The sum of f = 45 The sum of fx = 1,137.0 The mean x̄̄ = ∑fx = 1137.5 = 25.3 minutes ∑f 45 58 Scale: 1cm ≡1 teacher on the vertical axis 2cm ≡10 minutes on the horizontal axis 14 13 12 11 10 9 NUMBER OF TEACHERS 8 7 6 5 4 3 2 1 0 0.5 10.5 20.5 30.5 40.5 50.5 60.5 NUMBER OF MINUTES LATE Histogram showing the number of minutes that the 45 teachers have been late for the month. 59 Table Four: The table shows a summary of absences and late in percentages Week One Total no. of No. of No. of Percentage Percentage teachers teachers teachers of teachers of teachers on roll, absent on late on day, absent on late on day Day A day, y day. y ×100% x x A ×100% A Monday 45 11 10 24 22 Tuesday 45 14 12 31 27 Wednesday 45 8 14 18 31 Thursday 45 14 17 31 38 Friday 45 14 14 31 31 Week Two Total no. of No. of No. of Percentage Percentage teachers on teachers teachers of teachers of teachers roll, absent on late on day, absent on late on day Day A day, y day. y x x A ×100% A ×100% Monday 45 3 11 7 24 Tuesday 45 7 13 16 29 Wednesday 45 9 16 20 36 Thursday 45 6 16 13 36 Friday 45 17 12 38 31 60 Week Three Total no. of No. of No. of Percentage Percentage teachers on teachers teachers of teachers of teachers roll, absent on late on day, absent on late on day Day A day, y day. y ×100% x x A ×100% A Monday 45 12 9 27 20 Tuesday 45 13 16 29 36 Wednesday 45 9 10 20 22 Thursday 45 3 12 7 27 Friday 45 14 11 31 24 Week Four Total no. of No. of No. of Percentage Percentage teachers on teachers teachers of teachers of teachers roll, absent on late on day, absent on late on day Day A day, y day. y ×100% x x A ×100% A Monday 45 12 12 27 27 Tuesday 45 8 12 18 27 Wednesday 45 9 9 20 20 Thursday 45 2 8 04 18 Friday 45 20 10 44 22 61 Week One Average percentage of teachers late : 22+27+31+38+31 = 29.8% 5 Average percentage of teachers absent : 24+31+18+31+31 = 27% 5 Week Two Average percentage of teachers late : 24+29+26+26+31 = 31.2% 5 Average percentage of teachers absent : 7+16+20+13+38 = 18.8% 5 Week Three Average percentage of teachers late : 20+36+22+27+24 = 25.8% 5 Average percentage of teachers absent : 27+29+20+7+31 = 22.8% 5 Week Four Average percentage of teachers late : 27+27+20+18+22 = 22.8% 5 Average percentage of teachers absent : 27+18+20+4+44 = 22.6% 5 62 P R OJ E C T 3 SECTION 05 ANALYSIS OF DATA Table Four shows that from a staff of 45 teachers: 1. In Week One, approximately 27% of the teaching staff was absent daily and 30% of teachers were late daily 2. In Week Two, approximately 19% of the teaching staff was absent daily and 31% of teachers were late daily 3. In Week Three, approximately 23% of the teaching staff was absent daily and 26% of teachers were late daily 4. In Week Four, approximately 23% of the teaching staff was absent daily and 23% of teachers were late daily The average percentage of teachers absent for the four weeks is: 27 + 19 + 23 + 23 = 92 ≈ 23% 4 4 The average percentage of teachers late for the four week is: 30 + 31 + 26 + 23 = 110 ≈ 28% 4 4 Therefore, the data revealed that for the month under consideration: 1. 23% of teacher were absent on a daily basis 2. 28% of teacher were late on a daily basis Table Two and the histogram show that 19 teachers were late for 1 to 20 minutes for the month, whereas 26 teachers were late for 21 to 60 minutes for the month. 63 The four comparative bar graphs revealed that in any given week, more teachers were late than absent. This is seen in: Bargraph (1) – where 2 out of the 5 days, more teachers were late than absent and on one day the same number was late as absent Bargraph (2) – where 4 out of 5 days, more teachers were late than absent Bargraph (3) – where 3 out of 5 days, more teachers were late than absent. Bargraph (4) – where 2 out of 5 days, more teachers were late than absent and for 2 days the same number was late as absent. The graphs also show that teachers were more likely to be absent on a Friday than any other day. This is evident in the fact that on the Friday of: (a) Week One – 14 teachers were absent (b) Week Two – 17 teachers were absent (c) Week Three – 14 teachers were absent (d) Week Four – 20 teachers were absent Table Three gives the mean number of minutes late by teachers in the month to be 25.3 minutes. This means that if all the teachers were to be absent for the same number of minutes every day that number will be 25.3 minutes. 64 P R OJ E C T 3 SECTION 06 DISCUSSION OF FINDINGS The data showed that on average, 23% of teachers are absent everyday which is about 10 teachers from a staff of 45. Also, an average of 28% of teachers are late every day for an average of 25.3 minutes per day. This computes to 506 minutes of teaching time per month or about 17 periods per month. There are 8 (35 minute) periods in a teaching day at the school, therefore 25.3 minutes per day late, translate to 2 days absences per month in late minutes per teacher. 65 PR O JEC T 3 SECTION 07 CONCLUSION The findings revealed that 23% of the teachers were late every day and 28% were absent everyday over the period of one month. This suggests, that on average, out of a staff of 45 teachers, approximately 10 of them are late ever day and 13 are absent every day. This supports the anecdotal claim that at this particular school, teachers are tardy in getting to school and the absenteeism rate is high. 66 P R OJ E C T 3 SECTION 08 REFERENCES Greer, A. & Layne, C.E. Certificate in Mathematics: A Revision Course for the Caribbean. 2nd Edition. www.eduflow.wordpress.com/tardiness www.theastofed.com 67 PR O JEC T 4 SECTION 01 PROJECT TITLE An investigation in order to determine if the School Feeding Programme increases punctuality and regularity at a particular school. 68 P R OJ E C T 4 SECTION 02 INTRODUCTION TABLE OF CONTENTS 1. Title.......................................................................................... Page Number 2. Introduction.............................................................................. Page Number 3. Method of Data Collection....................................................... Page Number 4. Presentation of Data................................................................. Page Number 5. Analysis of Data....................................................................... Page Number 6. Discussion of Findings............................................................. Page Number 7. Conclusion............................................................................... Page Number 8. References............................................................................... Page Number 9. Appendix................................................................................. Page Number The School Feeding Programme was introduced in secondary schools as a safety net, aimed at assisting children of low socio-economic households. Its main focus is to: • provide nutritious meals for children • increase enrolment • promote academic success • reduce absenteeism and improve regularity and punctuality in schools. The purpose of this project is to determine whether the School Feeding Programme is fulfilling one of its mandates of reducing absenteeism and improving punctuality in schools. 69 PR O JEC T 4 SECTION 03 METHOD OF DATA COLLECTION The data was collected via a questionnaire. The questionnaire consisted of eight (8) pre-set standardized questions, in order to allow the same information to be generated from each student. The questions were asked in the same order. The sample consisted of 100 students, 20 of whom were chosen from each form level. There are 4 classes at each form level where the students were all asked if they wanted to take part in a survey. From the group of students who gave a positive response, five students were randomly chosen from each form class. The survey was conducted during the lunch break after students would have consumed the breakfast and lunch. (See appendix for questionnaire used in survey) It took four (4) weeks to collect the data from the sample size. 70 P R OJ E C T 4 SECTION 04 PRESENTATION OF DATA After the data was collected, it was organized and presented using tables, comparative bar charts and pie charts as follows. 1. The table below shows the number of students participated in the school feeding programme Form Number Partaking Number Not Partaking 1 18 2 2 15 5 3 14 6 4 12 8 5 10 10 69 31 Table 1 The comparative bar chart compares students who participated and who did not participate in the School Feeding Programme 20 NUMBER OF TIMES LATE/ABSENT 18 16 14 12 10 08 06 04 02 1 2 3 4 5 FORM CLASS 71 Figure 1 YES NO 2. The table below shows the number of students by form who arrived early for breakfast and participated in the School Feeding Programme Early for Not early for Form Total breakfast breakfast 1 18 2 20 2 15 5 20 3 14 6 20 4 12 8 20 5 10 10 20 69 31 100 Table 2 3. The table below shows the number of students who had breakfast everyday by form Number Having Number Not Having Form Total Breakfast Everyday Breakfast Everyday 1 20 0 20 2 20 0 20 3 20 0 20 4 20 0 20 5 20 0 20 100 0 100 Table 3 4. The table below shows the number of students who obtained breakfast from the School Feeding Programme, home and the cafeteria by form. Where breakfast is obtained School Feeding Form Home Cafeteria Total Programme 1 1 18 1 20 2 3 15 2 20 3 2 14 4 20 4 2 12 6 20 5 3 10 7 20 11 69 20 100 Table 4 72 The pie chart shows the proportion of students who obtained breakfast from the School Feeding Programme, home and cafeteria. Home Cafeteria School Feeding Programme 5. The table below shows the number of students who attended school regularly for lunch from the School Feeding Programme Attend School Do Not Attend Form for Lunch School for Lunch 1 18 2 2 15 5 3 14 6 4 12 8 5 10 10 69 31 Table 5 73 6. The table below shows the number of students who had lunch every day by form. Number Having Number Not Having Form Lunch Every Day Lunch Every Day 1 20 0 2 20 0 3 20 0 4 20 0 5 20 0 100 0 Table 6 7. The table shows the number of students who obtain lunch from the School Feeding Programme Where lunch is obtained School Feeding Form Home Cafeteria Total Programme 1 1 18 1 20 2 3 15 2 20 3 2 14 4 20 4 2 12 6 20 5 3 10 7 20 11 69 20 100 74 The pie chart shows the proportion of students who obtained lunch from the School Feeding Programme, home and cafeteria. Home Cafeteria School Feeding Programme 75 PR O JEC T 4 SECTION 05 ANALYSIS OF DATA Table 1 shows that out of the 100 students who were surveyed, 69 were participants of the School Feeding Programme, which means that 69% of the sample were recipients of meals from the School Feeding Programme. Table 1 and the comparative bar chart (Figure 1) show that the students in the lower forms (1, 2 and 3) were more likely to participate in the School Feeding Programme than children from the upper forms. This is supported by the following calculations: From Table I: (i) the number of lower form students who participated in the School Feeding Programme was (18 + 15 + 14) = 47. The total number of lower form students in the survey was 60. Hence, the percentage of lower form students who participated in the School Feeding Programme was 47 × 100% = 78.3% 60 (ii) the number of upper form students who participated in the School Feeding Programme (12 + 10) = 22. The total number of upper form students in the survey was 40. Therefore, the percentage of upper form students who participated in the programme was 22 × 100% = 55% 40 Table 3 and 6 show that 100% of the students surveyed had breakfast and lunch every day. However, tables 2 and 4 show that 69% got breakfast and lunch from the School Feeding Programme; whilst 11% got from home and 20% from the cafeteria. This is illustrated in Figure 2 and 3. In constructing the pie charts, angles were calculated to represent the sectors: - School Feeding Programme - home - cafeteria This was done as follows: School Feeding Programme 69 Home 11 Cafeteria 20 76 100 Angle representing School Feeding Programme = 69 × 360° = 248.4° 100 Angle representing Home = 11 × 360° = 39.6° 100 Angle representing Cafeteria = 20 × 360° = 72° 100 From the 100 students who were surveyed 69 came early for breakfast and attended school regularly for lunch from the school feeding programme. Therefore, if a student is chosen at random from the sample, the probability of that child arriving early for breakfast and regularly for lunch is 69% or 0.69 77 PR O JEC T 4 SECTION 06 DISCUSSION OF FINDINGS Although all of the students surveyed had breakfast and lunch every day, the majority of them (69%) came early and regularly to school in order to obtain their meals from the school feeding programme. The students from forms 1, 2, and 3 expressed greater interest in arriving early and regularly to participate in the School Feeding Programme. 78 P R OJ E C T 4 SECTION 07 CONCLUSION Given the high proportion of students who arrive early and regularly for the School Feeding Programme, one can conclude that the School Feeding Programme has a positive impact on the lives of students which resulted in an increase in punctuality and regularity. 79 PR O JEC T 4 SECTION 08 REFERENCES www.wikihow.com www.ttconnect.gov.tt www.guardian.co.tt 80 P R OJ E C T 4 SECTION 09 APPENDIX QUESTIONNAIRE FOR SURVEY 1. Please tick one response only Form Class  1  2  3  4  5 2. Do you partake in the school feeding programme? .  Yes  No 3. Do you arrive early to receive breakfast from the school feeding programme? .  Yes  No 4. Do you have breakfast every day? .  Yes  No 5. If the answer to question 4 is yes, then where do you get breakfast? .  Home  School  Cafeteria 6. Do you attend school regularly to receive lunch? .  Yes  No 7. Do you have lunch every day? .  Yes  No 8. If the answer to question 7 is yes, then where do you get lunch? .  Home  School  Cafeteria 81 PRO JEC T 5 SECTION 01 PROJECT TITLE An investigation to determine the number of birds that must be purchased in order for a small business to realize a maximum profit. 82 P R OJ E C T 5 SECTION 02 INTRODUCTION TABLE OF CONTENTS 1. Title.......................................................................................... Page Number 2. Introduction.............................................................................. Page Number 3. Method of Data Collection....................................................... Page Number 4. Presentation of Data................................................................. Page Number 5. Analysis of Data....................................................................... Page Number 6. Discussion of Findings............................................................. Page Number 7. Conclusion............................................................................... Page Number 8. References............................................................................... Page Number 9. Appendix................................................................................. Page Number My dad has decided to start a small business to supplement his monthly income. He plans to purchase chickens and ducks weekly to retail at a profit. After hearing dad’s plan, I decided to use a branch of Mathematics called linear programming to help him model his business plan. Linear programming is a mathematical technique that is used to maximize profit or minimize loss within a specified system, given all the constraints in the system. This technique will allow dad to predict the exact number of birds he should purchase weekly in order to maximize profit on his investment, without violating certain constraints. While trying to establish the business, dad may encounter limitations and restrictions. These limitations and restrictions are called constraints. 83 PRO JEC T 5 SECTION 03 METHOD OF DATA COLLECTION 1. I visited a poultry farm and collected data as follows: (i) wholesale prices on chickens and ducks (ii) the minimum number of birds that must be purchased in order to obtain the wholesale price. (iii) the suggested prices at which the birds can be retailed. (iv) the maximum number of birds that can be stored in the fowl run at home. 2. I also got dad’s weekly budget for the purchase of birds. 84 P R OJ E C T 5 SECTION 04 PRESENTATION OF DATA The data collected shows, that in order to obtain wholesale and retail prices, dad must satisfy the following four constraints: 1. He must buy at least 10 ducks; this means that the minimum amount of ducks that he can purchase is 10 2. He must buy at least 20 chickens; this means that he must buy 20 or more chickens 3. He has storage space for only 73 birds; this means that he cannot buy more than 73 birds. 4. He only has $3,000.00 to spend on birds. Ducks cost $80.00 each and chickens cost $30.00 each. The data also shows that at the end of this venture, dad is expected to make a profit of $70.00 on each duck and $30.00 on each chicken. In linear programming, we use inequalities to represent real world constraints which can then be represented graphically by regions. For dad’s business model, each constraint will give rise to an inequality. In order to write inequalities, we must introduce variables. For this model, I will use two variables, x and y. I will let x represent the number of ducks and y represent the number of chickens that dad wants to purchase. 1. Dad must buy at least 10 ducks Then x must be greater than or equal to 10 This is written as x ≥ 10 2. He must buy at least 20 chickens Then y must be greater than or equal to 20 This is written as y ≥ 20 3. He has storage for only 73 birds Then the total number of birds must be less than or equal to 73 This is written as x + y ≤ 73 85 4. The cost of x ducks at $80.00 each is $80 x The cost of y chickens at $30.00 each is $30 y He has only $3,000.00 to spend on birds Then the total amount of money to be spent must be less than or equal to $3000.00 This is written as 80x + 30y ≤ $3,000.00 Each of the above inequalities will represent a region on a graph sheet. To identify regions, we must have boundaries. The equality part of each inequality will be used as a boundary line for each inequality. 1. For the region x ≥ 10; the straight line x = 10 will be drawn as a boundary line 2. For the region y ≥ 20; the straight line y = 20 will be drawn as the boundary line. 3. For the region x + y ≤ 73; the straight line x + y = 73 will be drawn as the boundary line 4. For the region 80x + 30y ≤ 3000; the straight line 80x + 30y = 3000 will be drawn as a boundary line To draw the line x + y = 73 ; the two points (0, 73) and (73, 0) will be used x 0 73 y 73 0 Before the line 80x + 30y = 3000 is drawn, the equation will be simplified by dividing by 10. The equation is now 8x + 3y = 300; and the two points (0, 100) and (30, 20) will be used to draw the boundary line. x 0 30 y 100 20 The graph below represents the system of the four inequalities. The shaded region S represents dad’s degree of freedom. In this region he is free to purchase a combination of ducks and chickens without violating any of the four constraints. 86 Shaded region S represents dad’s degree of freedom y Scale: 2cm ≡10 ducks on horizontal, 2cm ≡10 chickens on vertical 100 x = 10 90 80 y (NUMBER OF CHICKENS) 70 B (10, 63) 60 C (16, 57) 50 40 30 s D (30, 20) 20 A (10, 20) y = 20 x + y = 73 10 8x + 3 y = 300 x 10 20 30 40 50 60 70 x (NUMBER OF DUCKS) 87 PRO JEC T 5 SECTION 05 ANALYSIS OF DATA Dad is expected to make a profit of $70.00 on each duck and $30.00 on each chicken. x ducks at $70.00 each will yield a profit of $70 x and y chickens at $30 each will yield a profit of $30 y. Therefore the profit equation for dad’s business model is written as: P = 70x + 30y The aim is to determine the number of ducks and chickens that dad should purchase to achieve the maximum profit. Now, the four inequalities gave rise to a polygon with the vertices A (10, 20); B (10, 63); C (16, 57); D (30, 20) as shown on the graph. The fundamental theorem of linear programming states that one of the four vertices will give values of x and y that will maximize the equation P = 70x + 30y Testing for the maximum profit we get: 1. At A (10, 20); where x = 10, y = 20 P = 70 × 10 + 30 × 20 = 700 + 600 = $1,300.00 This means that if dad purchases 10 ducks and 30 chickens, he will make a profit of $1,300.00 per week. 2. At B (10, 63); where x = 10, y = 63 P = 70 × 10 + 30 × 63 = 700 + 1,890 = $2,590.00 This means that if dad purchases 10 ducks and 63 chickens, he will make a profit of $2,590.00 per week. 3. At C (16, 57); where x = 16, y = 57 P = 70 × 16 + 30 × 57 = 1,120 + 1,710 = $2,830.00 This means that if dad purchases 16 ducks and 57 chickens, he will make a profit of $2,830.00 per week. 88 4. At C (30, 20); where x = 30, y = 20, P = 70 × 30 + 30 × 20 = 2,100 + 600 = $2,700.00 This means that if dad purchases 30 ducks and 20 chickens, he will make a profit of $2,700.00 per week. Therefore, in order for dad to obtain a maximum profit of $2,830, he must purchase 16 ducks and 57 chickens each week. 89 PRO JEC T 5 SECTION 05 DISCUSSION OF FINDINGS This simple linear programming model, gives dad a guide so he can make an informed decision when purchasing the birds to retail at a profit. It helps him to avoid unnecessary speculations and estimates based on guesswork. For example, based on his degree of freedom, dad knows that: 1. if he purchases 10 ducks and 20 chickens, he will realize a profit of $1,300.00 per week. 2. if he purchases 10 ducks and 63 chickens, he will realize a profit of $2,590.00 per week. 3. if he purchases 16 ducks and 57 chickens, he will realize a profit of $2,830.00 per week. 4. if he purchases 30 ducks and 20 chickens, he will realize a profit of $2,700.00 per week. Hence, dad now knows the exact number of ducks and chickens he must purchase weekly within the given constraints in order to maximize profit on his investment. 90 P R OJ E C T 5 SECTION 06 CONCLUSION Based on the findings, it can be concluded that dad must purchase 16 ducks and 57 chickens, a total of 73 birds in order to generate a maximum profit of $2,830.00 per week. 91 PRO JEC T 5 SECTION 08 REFERENCES Greer, A. & Layne, C. E., (1980); Certificate Mathematics: A Revision Course for the Caribbean. Toolsie, R.; Mathematics: A Complete Course with CXC Questions. Vol. 2 www.math.ucla.edu www.mathswork.com 92 P R OJ E C T 6 SECTION 01 PROJECT TITLE An investigation to determine the maximum area of a vegetable garden that can be constructed from a given plot of land. 93 PR O JEC T 6 SECTION 02 INTRODUCTION TABLE OF CONTENTS 1. Title.......................................................................................... Page Number 2. Introduction.............................................................................. Page Number 3. Method of Data Collection....................................................... Page Number 4. Presentation of Data................................................................. Page Number 5. Analysis of Data....................................................................... Page Number 6. Discussion of Findings............................................................. Page Number 7. Conclusion............................................................................... Page Number 8. References............................................................................... Page Number 9. Appendix................................................................................. Page Number To deal with rising prices, my parents and I have decided to create a vegetable garden at the back of our home. Since there are three (3) dogs in the yard, we would like to enclose the garden with a wire fence. Dad has a piece of wire in storage which he wants to use to fence the garden. We also want to give the garden a rectangular shape, with a 1 metre footpath around. This will allow for easy access, watering and fertilizing. After studying quadratic equations in school, I have decided to use this principle to help my parents maximize the use of the land space for the construction of the garden. 94 PR O JEC T 6 SECTION 03 METHOD OF DATA COLLECTION Dad and I went to the back of the house and used a measuring tape where we measured the dimensions of the available land. We also measured the length of the wire that dad had in storage. The land had a rough rectangular shape, therefore, we measured the length and width as accurately as possible. 95 PR O JEC T 6 SECTION 06 PRESENTATION OF DATA The data collected revealed that: (i) The length of the available land is approximately 29.7 metres. (ii) The width of the land is approximately 21.9 metres. (iii) The area of the land is 29.7 metres × 21.9 metres (i.e. approximately 650 sq. metres). (iv) The length of the wire to be used is 100 metres long. y metres 1 metre footpath x metres Vegetable Garden x metres y metres The diagram above is a model of the plot of land to be fenced with 100 metres of fencing wire. Let the length of the plot of land to be fenced be y metres. Let the width of the plot of land to be fenced be x metres. Length of wire to enclose the plot is 100 metres. Hence, perimeter (P) of the plot is 100 metres. ∴ 2x + 2y = 100 or x + y = 50 => y = 50 – x (Equation 1) 96 Since there is a 1 metre foot path around the garden; the Width of the garden = (x – 2) metres Length of the garden = (y – 2) metres Substitute y = 50 – x from equation 1 into the length to get: Length of Garden = (50 – x – 2) metres = (48 – x) metres Area of rectangular vegetable garden is length multiply by width ∴ Area = (48 – x) (x – 2) = 48x – 96 – x2 + 2x = 50x – 96 – x2 For the relation A = 50x – 96 – x2, a table of values of possible widths, x, was tabulated, a graph of A vs x was drawn for the domain values 5 ≤ x ≤ 45 x 5 10 15 20 25 30 35 40 45 A 129 304 429 504 529 504 429 304 129 See Appendix for calculation of the range (A) The graph below shows the relationship between the area (A) of the land to be enclosed and the width of the land (x) 600 500 Area in square metres (m2) 400 300 200 100 0 0 5 10 15 20 25 30 35 40 45 50 Length in metres (m) Figure 1: Scale used: 1cm ≡ 5 metres on horizontal axis, 1cm ≡ 100 square metres on vertical axis Note: The length of the available land is 29.7 m. Therefore, choosing domain values (x) that are greater than 29 metres will serve no practical purpose in the project. These values were included strictly 97 for the theroretical purpose of obtaining a complete parabolic shape on the graph. PR O JEC T 6 SECTION 05 ANALYSIS OF DATA From the graph: 1. The maximum value of A is 529 square metres 2. The maximum area occurs when x = 25 metres The length of the land to be fenced to give the maximum area of the vegetable garden is y = 50 – x = 50 – 25 = 25 metres. The width of the land to be fenced to give the maximum area of the vegetable garden is: x = 25 m (from graph) The maximum area of the vegetable garden is 529 metres2 (from graph) Verifying the graphical results by calculation we get: (i) width of vegetable garden = (x – 2) metres = (25 – 2) metres = 23 metres (ii) length of vegetable garden = (y – 2) metres = (25 – 2) metres = 23 metres Area of vegetable garden = length × width = (y – 2) (x – 2) = (23 × 23) metres2 = 529 metres2 The vegetable garden can be constructed using the above dimensions since the area of 529 metres2 is within the area of the available land, which is 650 metres2. Also the area of the plot of land to be fenced which will contain the vegetable garden and the one metre footpath is (25 × 25) metres2 = 625 metres2 This too is within the area of the available land to be used for construction. 98 P R OJ E C T 6 SECTION 06 DISCUSSION OF FINDINGS We can easily construct a vegetable garden by enclosing an area of land at the back of the house without considering any rigorous analysis. However, this simple algebraic model puts us in a position to make more informed choices in terms of the dimensions to be used. For example, the model shows that: 1. The maximum area that can be enclosed from the given land space is a perfect square of 625 square metres with dimensions 25 metres by 25 metres. 2. The area of the vegetable garden to be enclosed with a one metre pathway around it is a perfect square of 529 square metres with dimensions 23 metres by 23 metres. 3. This construction will still leave a reasonable amount of land space around the enclosed area. 4. The land space remaining after construction of fence is (650 – 625) square metres = 25 square metres. This space will allow for the dogs to play outdoors. 99 PR O JEC T 6 SECTION 07 CONCLUSION The findings reveal that the maximum area of a vegetable garden that can be set up from an available piece of land of area 650 square metres, is 529 square metres. To allow for the footpath of one metre around the garden, an area of 625 m2 of land must be fenced with a 100 metre long fence. 25 metres 1m outer fence 25 metres 1m Vegetable Garden 1m 25 metres footpath 1m 25 metres MODEL OF COMPLETED CONSTRUCTION 100 P R OJ E C T 6 SECTION 08 REFERENCES Clarke, L. Harwood, Additional Pure Mathematics, 3rd Edition Heng, H. H.; Talbert, J. F., Additional Mathematics, 6th Edition www.uncw.edu/alegbra www.varsitytutors.com 101 PR O JEC T 6 SECTION 09 APPENDIX CALCULATION OF RANGE A = 50x – 96 – x2 x 5 10 15 20 25 30 35 40 45 A 129 304 429 504 529 504 429 304 129 When x = 5m A = 50 × 5 – 96 – 52 = 250 – 96 – 25 = 129 m2 When x = 10m A = 50 × 10 – 96 – 102 = 500 – 96 – 100 = 304 m2 When x = 15m A = 50 × 15 – 96 – 152 = 750 – 96 – 225 = 429 m2 When x = 20m A = 50 × 20 – 96 – 202 = 1000 – 96 – 400 = 504 m2 When x = 25m A = 50 × 25 – 96 – 252 = 1250 – 96 – 625 = 529 m2 When x = 30m A = 50 × 30 – 96 – 302 = 1500 – 96 – 900 = 504 m2 When x = 35m A = 50 × 35 – 96 – 352 = 1750 – 96 – 1225 = 429 m2 102 P R OJ E C T 7 SECTION 01 PROJECT TITLE Creating the most cost effective budget to complete a room measuring 14’ x 12’ x 10’. 103 PRO JEC T 7 SECTION 02 INTRODUCTION TABLE OF CONTENTS 1. Title.......................................................................................... Page Number 2. Introduction.............................................................................. Page Number 3. Method of Data Collection....................................................... Page Number 4. Presentation of Data................................................................. Page Number 5. Analysis of Data....................................................................... Page Number 6. Discussion of Findings............................................................. Page Number 7. Conclusion............................................................................... Page Number 8. References............................................................................... Page Number 9. Appendix................................................................................. Page Number My parents have just completed the construction of a bedroom measuring 14’ x 12’ x 10’. With a budget of $5,000.00, they are considering the following: 1. installing ceramic floor tiles 2. painting the walls yellow, doors blue 3. installing a ceiling of either gypsum or P.V.C. panels With my knowledge of consumer arithmetic and measurement, I have decided to assist them with this venture. The purpose of this project is to determine the most cost effective estimate to complete the room. 104 P R OJ E C T 7 SECTION 03 METHOD OF DATA COLLECTION Four (4) different hardware stores A, B, C, and D were visited and prices were obtained on the material to be used. I also obtained prices on labour costs from two workmen named 1 and 2 in the project. The labour cost quoted from the workmen included the cost of labour for framing and installing the gypsum or PVC ceiling. 105 PRO JEC T 7 SECTION 04 PRESENTATION OF DATA In this project, 4 estimates were considered for comparison. After the raw data was collected from the 4 different hardware stores and the 2 workmen, it was organized and presented in tables as follows: Estimate from Hardware A Quantity Description Unit Price $ Amount $ 100 16” x 16” ceramic floor tiles 20.00 2,000.00 4 bags thinset 40.00 160.00 6 lbs grout 20.00 120.00 1 pk tile spaces 60.00 60.00 1 gal concrete primer 180.00 180.00 1 gal water based paint 250.00 250.00 ¼ gal wood paint 70.00 70.00 45 pieces 24” x 24” gypsum ceiling tiles 30.00 1,350.00 4,190.00 VAT @ 12.5% 523.75 Total 4,713.75 Estimate from Hardware B Quantity Description Unit Price $ Amount $ 100 16” x 16” ceramic floor tiles 18.00 1,800.00 4 bags thinset 45.00 180.00 6 lbs grout 26.00 104.00 1 pk tile spaces 70.00 70.00 1 gal concrete primer 150.00 150.00 1 gal water based paint 230.00 230.00 ¼ gal wood paint 90.00 90.00 45 pieces 24” x 24” gypsum ceiling tiles 35.00 1,575.00 4,199.00 VAT @ 12.5% 524.88 Total 4,723.88 106 Estimate from Hardware C Quantity Description Unit Price $ Amount $ 100 16” x 16” ceramic floor tiles 17.00 1,700.00 4 bags thinset 40.00 160.00 6 lbs grout 30.00 180.00 1 pk tile spaces 65.00 65.00 1 gal concrete primer 120.00 120.00 1 gal water based paint 200.00 200.00 ¼ gal wood paint 105.00 105.00 15 lengths 10” P.V.C. panels @ 19 ½ ft. 70.00 1,050.00 3,580.00 VAT @ 12.5% 447.50 Total 4,027.50 Estimate from Hardware D Quantity Description Unit Price $ Amount $ 100 16” x 16” ceramic floor tiles 17.00 1,700.00 4 bags thinset 45.00 180.00 6 lbs grout 20.00 120.00 1 pk tile spaces 30.00 30.00 1 gal concrete primer 120.00 120.00 1 gal water base paint 150.00 150.00 ¼ gal wood paint 50.00 50.00 15 lengths 10” P.V.C. panels @ 19 ½ ft. 50.00 750.00 3,100.00 VAT @ 12.5% 387.50 Total 3,487.50 107 Labour Cost: Two workmen who were selected to do the work gave the following labour cost: Labour cost from Workman 1 = $2,000.00 Labour cost from Workman 2 = $1,500.00 Summary of Estimates and Labour Costs Estimate Hardware A $4,713.75 Hardware A $4,713.75 Labour Workman #1 $2,000.00 Labour Workman #2 $1,500.00 Total $6,713.75 $6,213.75 Estimate Hardware B $4,723.88 Hardware B $4,723.88 Labour Workman #1 $2,000.00 Labour Workman #2 $1,500.00 Total $6,723.88 $6,223.88 Estimate Hardware C $4,027.50 Hardware C $4,027.50 Labour Workman #1 $2,000.00 Labour Workman #2 $1,500.00 Total $6,027.50 $5,523.50 Estimate Hardware D $3,375.00 Hardware D $3,487.50 Labour Workman #1 $2,000.00 Labour Workman #2 $1,500.00 Total $5,375.00 $4,987.50 Please note that Hardware A and Hardware B had no PVC ceiling on sale. Therefore, PVC ceiling was not seen in the budgets for Hardware A and Hardware B. Hardware C and Hardware D had no gypsum on sale. Therefore, gypsum tiles were not seen in the budgets for Hardware C and Hardware D. In order to create the 4 estimates, the following calculations were performed. In performing the calculations the following conversions were made; (i) inches to feet – 1 feet = 12 inches ∴ to convert inches to feet we divide by 12 (ii) ounces to pounds – 1 pound = 16 ounces ∴ to convert ounces to pounds we divide by 16 108 The Floor Item needed to tile the floor are: (i) tiles (ii) thinset (iii) grout (iv) tile spacers (i) Tiles One floor tile measures 16” x 16” which is 1.333 feet x 1.333 feet. Since the tiles have a square shape and the area of a square is side x side, it follows that the area of 1 tile is 1.333 feet x 1.333 feet = 1.8 square feet. Since the floor has a rectangular shape of 14 feet x 12 feet and the area of a rectangle is length x width, it follows that the area of the floor to be tiled is 14 feet x 12 feet = 168 square feet. To tile an area of 168 square feet using 1.8 square feet tiles, the number of tiles needed is 168 square feet = 94 tiles approximately. 1.8 square feet An extra 6 tiles are to be bought in order to make provision for breakage. ∴ Total number of tiles to be bought is 100. Thinset Thinset is the material used to plaster the floor before laying the tiles. The information on the package states that 1 bag covers a surface area of 40 sq. ft. ∴ The total number of bags that covers 168 sq. ft. of floor surface is 168 = 4 bags 40 ∴ The total number of bags of thinset to be bought is 4 bags Grout Grout is the material used to seal the spaces between each tile. The information on the package states that 1 ounce of grout seals approximately one 16” x 16” tile. ∴ The amount of grout needed is 1 ounce by 94 tiles = 94 ounces = 5.9 lbs ∴ The amount of grout to be bought is 6 pounds Tile Spacers Tile spacers are used to create the even space between each tile. Tile spacers are sold in packages of 100 per pack. ∴ The amount of spacers needed for 94 tiles is 1 pack Summary of items needed for tiling the floor are as follows: 4 bags of thinset 6 lbs of grout 109 1 pack of spacers Ceiling The space to be sealed is flat and rectangular in shape. Length of ceiling = 14 ft. Width of ceiling = 12 ft. ∴ Area of Ceiling = 14 ft. x 12 ft. = 168 sq. ft. The space is sealed with either gypsum ceiling tiles or PVC ceiling panels. (i) Gypsum Ceiling Tiles Area of 1 tile = 24” x 24” square inches = 2’ x 2’ sq. ft. = 4 sq. ft. ∴ No. of ceiling tiles required = 168 = 42 tiles 4 3 extra tiles are bought ∴ Total number of ceiling tiles to be bought = 42 + 3 = 45 (ii) PVC Ceiling Tiles 1 Length of 1 PVC panel = 19 2 ft. 10 Width of 1 PVC panel = 10 inches = 12 ft. Width of room = 12 ft. ∴ No. of PVC panel required = 12 = 12 × 12 = 144 = 14.4 10 10 10 12 Therefore, total number of panels to be bought = 15 110 Paint The walls are to be painted yellow and the door blue. The room has four walls, all of which are rectangular in shape. Two sides measure 14’ x 10’ each ∴ Total area of the two sides = (14’ x 10’) x 2 = 280 sq. ft. The other 2 sides measure 12’ x 10’ each ∴ Total area of the other 2 sides = (12’ x 10’) x 2 = 240 sq. ft. Therefore, the total area of wall space to be painted = 280 + 240 = 520 sq. ft. Door size = 54” x 24” sq. ft. = 4.5’ x 2’ sq. ft. = 9 sq. ft. Window size = 24” x 24” sq. inches = 2’ x 2’ sq. ft. = 4 sq. ft. ∴ Total area of window and door = (9 + 4) sq. ft. = 13 sq. ft. Therefore, the total area of wall space to be painted = (520 - 13) sq. ft. = 507 sq. ft. The information on the paint tin states that one gallon of primer and one gallon of paint each has a coverage of 420 to 520 sq. ft. ∴ Amount of paint and primer needed for walls are: 1 gallon of primer 1 gallon of yellow paint Door area = 9 sq. ft. ∴ 1 gallon of blue oil paint is needed to paint door. 4 111 PRO JEC T 7 SECTION 05 ANALYSIS OF DATA The data revealed the following; 1. estimate from Hardware B with labour cost from Workman 1 came in with the highest budget of $6,723.88 2. estimate from Hardware A with labour cost from Workman 1 gave the 2nd higher estimate of $6,713.75 3. estimate from Hardware B with labour cost from Workman 2 gave the 3rd highest estimate of $6,223.88 4. estimate from Hardware A with labour cost from Workman 2 gave the 4th highest estimate of $6,213.75 5. estimate from Hardware C with labour cost from Workman 1 gave the 5th highest estimate of $6,027.50 6. estimate from Hardware C with labour cost from Workman 1 gave the 6th highest estimate of $5,523.50 7. estimate from Hardware D with labour cost from Workman 1 gave the 7th highest estimate of $5,375.00 8. estimate from Hardware D with labour cost from Workman 2 gave the cheapest estimate of $4,987.50 To determine whether further saving was possible and which estimate is most cost effective, all 8 estimates were perused and the lowest price of each item was extracted from the different hardwares. Hardware C & D offered the cheapest tile Hardware C offered the cheapest thinset Hardware A & D offered the cheapest grout Hardware D offered the cheapest spaces Hardware C & D offered the cheapest primer Hardware D offered the cheapest water based paint, wood paint and PVC panels From the above observation, the estimate which offered the most cost effective budget was generated. Overall, Hardware D, offered the cheapest prices with the exception of Hardware C which offered the cheapest thinset. Therefore all materials are to be purchased from Hardware D, with the exception of thinset will be purchased from Hardware C. 112 From the above consideration the cheapest estimate is as follows: Hardware Item Unit Price $ Quantity Cost $ Cost $ D Ceramic tiles 17.00 100 1,700.00 - C Thinset 40.00 4 bags - 160.00 D Grout 20.00 6 lbs 120.00 - D Spacers 30.00 1 pk 30.00 - D Primer 120.00 1 gal 120.00 - D Water Paint 150.00 1 gal 150.00 - D Oil Paint 50.00 ¼ gal 50.00 - D PVC Panels 50.00 15 lengths 750.00 - Total without VAT 2,920.00 160.00 VAT @ 12.5% 365.00 20.00 Total with VAT 3,285.00 180.00 Total cost of material = Total cost from Hardware D + Total cost from Hardware C = $3,285.00 + $180.00 = $3,465.00 Labour from Workman 2 = $1,500.00 ∴ Total cost of estimate = $4,965.00 113 PRO JEC T 7 SECTION 06 CONCLUSION The most cost effective estimate therefore is the combination of prices from Hardware C and D and labour cost from Workman 2 which totals $4,965.00 and falls within the budget of $5,000.00. 114 P R OJ E C T 7 SECTION 07 REFERENCES www.wikihow.com www.selfbuildnewhomes.com 115 PRO JEC T 8 SECTION 01 PROJECT TITLE An investigation into fixed rate mortgages. 116 P R OJ E C T 8 SECTION 02 INTRODUCTION TABLE OF CONTENTS 1. Title.......................................................................................... Page Number 2. Introduction.............................................................................. Page Number 3. Method of Data Collection....................................................... Page Number 4. Presentation of Data................................................................. Page Number 5. Analysis of Data....................................................................... Page Number 6. Discussion of Findings............................................................. Page Number 7. Conclusion............................................................................... Page Number 8. References............................................................................... Page Number 9. Appendix................................................................................. Page Number My parents are considering taking a loan from a bank to purchase a house. They are employed as Clerks in the government service and have savings which will be used as a down-payment towards the purchase of the house. The purpose of this project is to show how monthly mortgage payments are calculated on fixed rate mortgages given different terms of repayment. 117 PRO JEC T 8 SECTION 03 METHOD OF DATA COLLECTION I sat with my parents and collected their personal data such as age, joint income, savings, cost of the house and other information which would aide in their mortgage calculation. I also visited a local bank and interviewed a mortgage Loans Officer who provided information on mortgages such as: type of mortgage, principal, interest rate, terms of repayment and other facts which were pertinent to calculating mortgages. 118 P R OJ E C T 8 SECTION 03 METHOD OF DATA COLLECTION The data collected from my parents are as follows: Age of both parents – 30 years Joint income – $23,000 per month Savings (down payment) – $50,000.00 Cost of the house – $500,000.00 My interview with the mortgage Loans Officer introduced me to the following facts about mortgage which must be known before a mortgage is calculated. 1. Mortgages are usually calculated at rates which stay the same for the lifetime of the mortgage, therefore, they are called fixed rate mortgages. These carry a low interest rate ranging between 3% and 10% because the loan is secured, giving the lender the redress to seize the property if the loan is not being repaid. 2. The age of the borrower is a major factor. The younger the borrower, the smaller the mortgage rate and the longer the lifespan of the loan. 3. The older the borrower, the larger the monthly payment over a shorter loan lifespan. 4. A down payment of at least 10% must be made 5. The monthly payment must end on or before, 30 years. This means that my parents have a maximum of 30 years in which to repay their mortgage. Therefore, I have decided to calculate 3 mortgage plan options with 3 different terms of repayment of 20 years, 25 years and the maximum 30 years. The following presentation gives a detailed outline of how the three different mortgage plans will be setup and calculated: The value of each monthly payment will be calculated using a standard compound interest formula. This formula has four variables. These variables are: 1. The Principal amount of money that my parents want to borrow. This is represented by the letter P in the formula. 2. The interest rate per month that the bank is charging my parents to borrow the Principal sum. This is represented by the letter r in the formula. 3. The number of months that my parents will take 119 to repay the Principal sum. This is represented by the letter n in the formula. 4. The amount of each monthly payment that my parents will have to repay. This represented by the letter M in the formula. Values for the three variables; principal (P), time (n) and rate (r) will be collected from the bank and substituted into the formula to obtain the fourth variable, the monthly payment (M). The compound interest formula that will be used in this project is: M=P { r (1+r)n (1+r)n - 1 } All calculations within the curly brackets must be done first to obtain a value. This value is then multiplied by P to obtain the value of M. The down payment of 10% of $500,000.00 will be calculated. This down payment will be subtracted from $500,000.00 to obtain the principal (P) that will be used to calculate the monthly payment (M) Down payment to be made by my parents is 10% of $500,000.00 ∴ Down payment = 10% of 500,000 = 10 × 500,000 = $50,000.00 100 The mortgage can be obtained under three different terms. (1) Term A: Principal (P) = $450,000 Time (n) = 30 years = (30 × 12) months = 360 months Rate (r) = 4.5% per year = 4.5 %= 0.375% per month 12 (2) Term B: Principal (P) = $450,000 Time (n) = 25 years = (25 × 12) months = 300 months Rate (r) = 7% per year = 7 % = 0.583% per month 12 (3) Term C: Principal (P) = $450,000 Time (n) = 20 years = (20 × 12) months = 240 months Rate (r) = 9.5% per year = 9.5 % = 0.792% per month 12 120 The following calculations were performed to determine the value of each monthly payment that my parents will have to make under the three different mortgage plans. Calculation of monthly payment under Term A P = $450,000 n = 360 months r = 0.375% = 0.375 = 0.00375 100 M =P { r (1+r)n (1+r)n - 1 } = 450000 { (0.00375 (1 + 0.00375)360 (1 + 0.00375)360 - 1 } = 450000 { (0.00375 × (3.84769805) 3.84769805 - 1 } = 450000 { 0.014428867 2.84769805 } = 450000 × 0.005066852 = $2,280.08 Therefore, under Term A: = the total cost of the house is: down payment + monthly payment × 360 = $50,000.00 + $2,280.08 × 360 = $50,000.00 + $820,828.80 = $870,828.80 Cost of loan = monthly payment x 360 – $450,000 = $2,280.08 × 360 – $450,000 = $820,828.80 – $450,000 = $370,828.80 121 Calculation of monthly payment under Term B P = $450,000 n = 300 months r = 0.583% = 0.583 = 0.00583 100 M =P { r (1+r)n (1+r)n - 1 } = 450000 { 0.00583 (1 + 0.00583)300 (1 + 0.00583)300 - 1 } = 450000 { 0.00583 × (5.719728815) 5.719728851 - 1 } = 450000 { 0.03346018 4.719728815 } = 450000 × 0.00706524 = $3,179.36 Therefore, under Term A: = the total cost of the house is: down payment + monthly payment × 300 = $50,000.00 + $3,179.36 × 300 = $50,000.00 + $953,808.00 = $1,003,808.00 Cost of loan = monthly payment x 360 – $450,000 = $2,280.08 × 300 – $450,000 = $953,808 – $450,000 = $503,808 122 Calculation of monthly payment under Term C P = $450,000 n = 240 months r = 0.792% = 0.792 = 0.00792 100 M =P { r (1+r)n (1+r)n - 1 } = 450000 { 0.00792 (1 + 0.00792)240 (1 + 0.00792)240 - 1 } = 450000 { 0.00792 ×(6.641330639) 6.641330639 - 1 } = 450000 { 0.052599338 5.641330639 } = 450000 × 0.009323923 = $4,195.77 Therefore, under Term A: = the total cost of the house is: down payment + monthly payment × 240 = $50,000.00 + $4,195.77 × 240 = $50,000.00 + $1,006,984.80 = $1,056,984.80 Cost of loan = monthly payment x 360 – $450,000 = $4,195.77 × 240 – $450,000 = $1,006,984.80 – $450,000 = $556,984.80 123 Bar graph showing the final cost of the house under the three different mortgage Terms. Scale: 1cm ≡ $100,000 on the vertical axis 12 10 8 Final cost of house (x $100,000) 6 4 2 0 Term A Term B Term C Mortgage Terms 124 P R OJ E C T 8 SECTION 05 ANALYSIS OF DATA Using Term A, it was found that my parents will have to pay $2,280.08 per month for 360 months (30 years). The final cost of the house will be $870,828.80 Using Term B, they will have to pay $3,179.36 per month for 300 months (25 years). The final cost of the house will be $1,003,808.00 Using Term C, they will have to pay $4,195.77 per month for 240 months (20 years). The final cost of the house will be $1,056,984.80 The bar graph on the previous page compares the final cost of the house under the three mortgage Terms A, B and C. 125 PRO JEC T 8 SECTION 06 DISCUSSION OF FINDINGS It is impossible for my parents to purchase a house that is valued at $500,000.00 without a mortgage and buying this house is likely to be the biggest investment they will ever make. Taking a mortgage gives them the flexibility of spreading the repayment of the house over periods of 20, 25, or 30 years. Therefore, monthly repayments will be more manageable for them. If they are desirous of paying the smallest monthly repayment with the smallest interest charge, they can take the option that offers the mortgage for 30 years. Here, the repayment will be $2,880.08 per month with an interest of 4.5% per annum. With this option the cost of the mortgage will be $370,828.80. Although the other two options offer a shorter repayment time, the cost of the mortgage is significantly greater. 126 P R OJ E C T 8 SECTION 07 CONCLUSION The mortgage calculations revealed that: 1. The final cost of the house using Plan A will be $870,828.80. The monthly repayment will be $2,880.08. The interest charge for the mortgage will be $370,828.80 2. The final cost of the house using Plan B will be $1,003,808.00. The monthly repayment will be $3,179.36. The interest charge for the mortgage will be $503,808.00 3. The final cost of the house using Plan C will be $1,056,984.00. The monthly repayment will be $4,195.77. The interest charge for the mortgage will be $556,984.00 127 PRO JEC T 8 SECTION 08 REFERENCES www.moneysupermarket.com www.edelmonfinancial.com www.weberawler.com 128 P R OJ E C T 8 SECTION 09 APPENDIX Terms used in this project: 1. Mortgage: a loan for the purpose of buying a property which is secured by the property 2. Fixed Rate Mortgage: a mortgage loan with an interest that does not change over the life of the loan 3. Down payment: an initial percentage payment made when the property is bought on credit 4. Principal: the initial amount borrowed after the down payment is made 5. Rate / Interest Rate: the price that the bank charges for the mortgage. This charge is usually expressed as a percentage 6. Time: the number of months it will take for the loan to be repaid 7. Monthly payment: the amount of each monthly instalment 8. Variable: a quantity that can change 9. Formula: an equation showing how the four variables, principal, rate, time and monthly payment are related 129 PR O JEC T 9 SECTION 01 PROJECT TITLE An investigation to determine if the Form Four Science group in a secondary school performs better in the area of Mathematics than other subject groups. 130 P R OJ E C T 9 SECTION 02 INTRODUCTION TABLE OF CONTENTS 1. Title.......................................................................................... Page Number 2. Introduction.............................................................................. Page Number 3. Method of Data Collection....................................................... Page Number 4. Presentation of Data................................................................. Page Number 5. Analysis of Data....................................................................... Page Number 6. Discussion of Findings............................................................. Page Number 7. Conclusion............................................................................... Page Number 8. References............................................................................... Page Number 9. Appendix................................................................................. Page Number Students usually say and eventually come to believe that generally, Science students perform much better in Mathematics than students in the other subject groups of Languages, Business and Technical. This claim is not usually supported by hard data or facts but is more of an anecdotal nature. After studying simple statistics and realizing that it can be used to investigate anecdotal claims using quantitative data, I have decided to use this branch of Mathematics to study the claim of my peers. The purpose of this research is to determine whether there is a relationship between the subject groupings and the performance in Mathematics at a particular school. 131 PR O JEC T 9 SECTION 03 METHOD OF DATA COLLECTION Data was collected from four (4) groups of students; Science, Languages, Business and Technical. Each group consisted of thirty (30) students. The names of 30 students from each group were placed in separate boxes, fifteen (15) of which were randomly selected to minimize bias. The marks of the fifteen (15) students from each group were extracted from the teacher’s mark-book. The examination was marked out of 100% with the pass mark being 50%. Each group was given two (2) hours to complete the examination. All four (4) groups wrote the same Mathematics examination. It should be noted that, the project was discussed with the Mathematics teachers of the four (4) different groups and permission was granted to peruse and extract data from their mark-books. 132 P R OJ E C T 9 SECTION 04 PRESENTION OF DATA After the raw data was collected from the four (4) groups, they were organized into one table as follows: Class Marks 4 Science 40, 46, 51, 66, 67, 75, 76, 77, 79, 79, 81, 85, 88, 95, 95 4 Language 23, 28, 40, 45, 45, 46, 47, 58, 58, 59, 60, 61, 62, 65, 66 4 Business 27, 32, 34, 39, 49, 51, 52, 54, 55, 55, 55, 56, 66, 70, 73 4 Technical 37, 39, 42, 43, 46, 46, 47, 51, 52, 55, 59, 62, 66, 70, 72 Summary of the results from the four groups No. of No. of % % % % Class Students Students 20 - 39 40 - 59 60 - 79 80 - 100 Passed Failed Science 13 2 0 3 7 5 Language 8 7 2 8 5 0 Business 10 5 4 8 3 0 Technical 8 7 2 9 4 0 133 After the mean for each group was calculated, they were displayed on a bar graph and a pie chart as follows. Bar graph showing averages of the 4 groups Sale: 1cm ≡ 10% on vertical axis 80 70 60 CLASS AVERAGE 50 40 30 20 10 SCIENCE LANGUAGE BUSINESS TECHNICAL FORM CLASS Pie chart comparing averages of the 4 groups TECHNICAL 84° SCIENCE 115° BUSINESS 81° LANGUAGE 81° 134 All calculations which were done for the Mean x, Standard Deviation, Range and the Pie Chart will be shown here. The marks from each class were used to calculate the mean (x̄) or average mark for each class as follows; The simple arithmetic mean (x̄) for a sample is calculated using the formula: Mean x̄ = Sum of all scores in the sample Number of scores in the sample ∴ Mean (x̄) for Science group = 40 + 46 + 51 + 66 + 67 + 75 + 76 + 77 + 79 + 79 + 81 + 85 + 88 + 95 + 95 15 = 1100 = 73.3 % 15 ∴ Mean (x̄) for Language group = 23 + 28 + 40 + 45 + 45 + 46 + 47 + 58 + 58 + 59 + 60 + 61 + 62 + 65 + 66 15 = 763 = 50.9 % 15 ∴ Mean (x̄) for Business group = 27 + 32 + 34 + 39 + 49 + 51 + 52 + 54 + 55 + 55 + 55 + 56 + 66 + 70 + 73 15 = 768 = 51.2 % 15 ∴ Mean (x̄) for Technical group = 37 + 39 +42 + 43 + 46 + 46 + 47 + 51 +52 + 55 + 59 + 62 + 66 + 70 + 72 15 = 787 = 52.5 % 15 The mean mark from each class was then used to calculate the Standard Deviation. The Standard Deviation for a simple group is calculated using the formula Standard Deviation = ∑(x - x̄)2 n Where x is each individual score x̄ is the group mean n is the number of scores in the sample Note: Σ is the Greek letter sigma It is used in Mathematics to indicate the sum of a set of data. Therefore: Σ (x - x̄)2 is interpreted as the sum of (x - x̄)2 135 1. Calculations of Standard Deviation for Science group Mean x̄ = 73.3% MARKS x x - x̄ (x - x̄)2 40 -33.3 1108.89 46 -27.3 745.29 51 -22.3 497.29 66 -7.3 53.29 67 -6.3 39.69 75 1.7 2.89 76 2.7 7.29 77 3.7 13.69 79 5.7 32.49 81 7.7 59.29 58 11.7 136.89 88 14.7 136.89 88 14.7 216.09 95 21.7 470.89 95 21.7 470.89 Σ (x - x̄)2 n = 15 = 3887 .35 Standard Deviation = ∑(x - x)̄ 2 n = 3887.35 15 = 259.15667 = 16.1% 136 1. Calculations of Standard Deviation for Language group Mean x̄ = 50.9% MARKS x x - x̄ (x - x̄)2 23 -27.9 778.41 28 -22.9 524.41 40 -10.9 118.81 45 -5.9 34.81 45 -5.9 34.81 46 -4.9 24.01 47 -3.9 15.21 58 7.1 50.41 58 7.1 50.41 59 8.1 65.61 60 9.1 82.81 61 10.1 102.01 62 11.1 123.21 65 14.1 198.81 66 15.1 228.01 Σ (x - x̄)2 n = 15 = 2431.75 Standard Deviation = ∑(x - x)̄ 2 n = 2431.75 15 = 162.116667 = 12.7% 137 1. Calculations of Standard Deviation for Business group Mean x̄ = 51.2% MARKS x x - x̄ (x - x̄)2 27 -24.2 585.64 32 -19.2 368.64 34 -17.2 295.84 39 -12.2 148.84 49 -2.2 4.84 51 -0.2 0.04 52 0.8 0.64 54 2.8 7.84 55 3.8 14.44 55 3.8 14.44 55 3.8 14.44 56 4.8 23.04 66 14.8 219.04 70 18.8 353.44 73 21.8 475.24 Σ (x - x̄)2 n = 15 = 2526.4 Standard Deviation = ∑(x - x)̄ 2 n = 2526.4 15 = 168.426667 = 12.98% 138 1. Calculations of Standard Deviation for Technical group Mean x̄ = 52.5% MARKS x x - x̄ (x - x̄)2 27 -15.5 240.25 32 -13.5 182.25 34 -10.5 110.25 39 -9.5 90.25 49 -6.5 42.25 51 -6.5 42.25 52 -5.5 30.25 54 -1.5 2.25 55 0.5 0.25 55 2.5 6.25 55 6.5 42.25 56 9.5 90.25 66 13.5 182.25 70 17.5 306.25 73 19.5 380.25 Σ (x - x̄)2 n = 15 = 1747.75 Standard Deviation = ∑(x - x)̄ 2 n = 1747.75 15 = 116.51667 = 10.79% 139 Calculations for the range of each group. The Range is the difference between the largest and smallest values in a sample. ∴ Range = Largest Value – Smallest Value 1. Largest value for Science group = 95% Smallest value for Science group = 40% Range for Science group = 95% – 40% = 55% 2. Largest value for Language group = 66% Smallest value for Language group = 23% Range for Language group = 66% - 23% = 43% 3. Largest value for Business group = 73% Smallest value for Business group = 27% Range for Business group = 73% - 27% = 46% 4. Largest value for Technical group = 72% Smallest value for Technical group = 37% Range for Technical group = 72% - 37% = 35% CLASS AVERAGE % Science 73 Language 51 Business 51 Technical 53 228 Calculations for angles to represent each group sector on the pie chart Angle to represent Science group = 73 × 360° ≈ 115° 228 Angle to represent Language group = 51 × 360° ≈ 81° 228 Angle to represent Business group = 51 × 360° ≈ 81° 228 Angle to represent Technical group = 53 × 360° ≈ 84° 228 140 P R OJ E C T 9 SECTION 05 ANALYSIS OF DATA The table gives a summary of important measures calculated in the project. MEAN x̄ STANDARD RANGE GROUP % DEVIATION % % Science 73.3 16.1 55 Language 50.7 12.7 43 Business 51.2 12.98 46 Technical 52.5 10.79 35 The data show that the mean for the four groups: Science, Language, Business and Technical are 73.3%, 50.9%, 51.2% and 52.5% respectively. The Standard Deviation are 16.1%, 12.7%, 12.8% and 10.79% respectively. The ranges are 55%, 43%, 46% and 35% respectively. The Mean of the Science group is approximately 20-23% higher than the mean of the other three groups. This suggests that the marks from the Science group is much better than those from the other groups. The Standard Deviation and Range are measures of spread. The Range describes how well the mean represents the group and the Standard Deviation shows by how much the students individual scores differ from the mean of their respective group. The Range for the Science group is 55%. This range is fairly large indicating that the Mean of 73.3% may not be as representative of the entire group. It suggests that there are a few students in the class who are excellent at Mathematics and there are a few students in the class who are weak at Mathematics. Since the range is large, the Standard Deviation will also be large, suggesting large spread away from the mean by a few students. 141 The data shows that the three groups: Language, Business, and Technical; are very close to the same mean, with the Technical group performing slightly better than the other two groups. The ranges and Standard Deviation of the three groups are less than the Science group. This suggests that there is less spread away from the mean in these three groups than that of the Science group. Since the means of the other groups are much less than that of the Science group, one can conclude that the performance of the Science students is better than that of the other three groups. The bar graph and the pie chart show that the means for the Language and Business group are approximately the same. The Technical group’s performance is slightly better than that of the Language and Business groups. The diagrams also indicate that the Science group’s mean mark is approximately 20 - 23% higher than that of the other three groups. 142 P R OJ E C T 9 SECTION 06 DISCUSSION OF FINDINGS The findings of the data revealed that 13 out of the 15 Science students passed the examination with 10 students obtaining between 75% and 95%. 8 out of the 15 Language students passed the examination with the highest score being 66% and 10 students scoring between 23% and 59%. 10 out of the 15 Business students passed the examination with 12 students scoring between 27% and 56%. 8 out of the 15 Technical students passed the examination with 11 students scoring between 37% and 59% and a highest score of 72%. The highest score in each of the three other groups was less than the scores of 10 students from the Science group. 143 PR O JEC T 9 SECTION 07 CONCLUSION The Science group with Mean, Range and Standard Deviation of 73.3%, 55% and 16.1% respectively suggest that, this grouping of students has an impact on the Mathematics performance at this particular school. 144 P R OJ E C T 9 SECTION 08 REFERENCES Toolsie, R, (2012). Additional Mathematics: A Complete Course for CSEC. www.quora.com www.mathgoodies.com www.mathisfun.com 145 PR O JEC T 9 SECTION 08 APPENDIX 1. Mathematics marks collected from the Science group: 95, 95, 85, 79, 75, 77, 79, 66, 76, 67, 88, 81, 46, 51, 40 2. Mathematics marks collected from the Language group: 59, 47, 65, 58, 23, 28, 45, 60, 58, 62, 46, 40, 45, 61, 66 3. Mathematics marks collected from the Business group: 32, 27, 56, 55, 51, 52, 34, 66, 49, 70, 54, 73, 55, 39, 55 4. Mathematics marks collected from the Technical group: 70, 42, 37, 46, 39, 62, 59, 51, 43, 72, 46, 55, 47, 52, 66 146 P R OJ E C T 1 0 SECTION 01 PROJECT TITLE An investigation into Pythagoras’ Theorem to determine its usefulness in the real world. 147 PRO JEC T 10 SECTION 02 INTRODUCTION TABLE OF CONTENTS 1. Title.......................................................................................... Page Number 2. Introduction.............................................................................. Page Number 3. Method of Data Collection....................................................... Page Number 4. Presentation of Data................................................................. Page Number 5. Analysis of Data....................................................................... Page Number 6. Discussion of Findings............................................................. Page Number 7. Conclusion............................................................................... Page Number 8. References............................................................................... Page Number 9. Appendix................................................................................. Page Number My fourth form class recently completed a study in geometry. Pythagoras’ Theorem was emphasized. However, many of my peers failed to understand its importance and usefulness in the real-world. After observing the confusion of my peers, I decided to investigate this theorem and its application in the real-world. The purpose of this study is to (i) research and (ii) demonstrate how the theorem can be used to solve real-world problems. 148 P R OJ E C T 1 0 SECTION 02 METHOD OF DATA COLLECTION I perused a number of websites in order to obtain historical information on Mathematician Pythagoras and his famous theorem. I also interviewed my Mathematics teacher to ascertain how Pythagoras’ Theorem can be applied in real-life situations. I used a measuring tape to measure the following objects: 1. a bedroom 2. a television set 3. a playing field 4. a school building 149 PRO JEC T 10 SECTION 04 PRESENTATION OF DATA Historical Perspective and Background of Pythagoras’ Theorem: Pythagoras (569 BC – 475 BC) was a Greek philosopher and mathematician. He made important contributions to Pure Mathematics and Geometry, and is credited for coining the word Mathematics. One of the most important contributions to geometry is a famous theorem which bears his name: Pythagoras’ Theorem. It is said that this theorem was known to the Egyptians for centuries before Pythagoras. However, Pythagoras was the first person to prove the theorem formerly. Pythagoras’Theorem states that: In any right-angled triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides. Where the hypotenuse is the side opposite the right angle. Pythagoras saw this result through geometric eyes and actually drew squares on the three sides of a right- angled triangle to represent the theorem. Figure 1: A Picture of Pythagoras’ Theorem c2 b2 b c a a2 Area of square on hypotenuse c is c2 Area of square on side a is a2 Area of square on side b is b2 The theorem is saying that the two smaller squares on sides a and b 150 can be combined to exactly fit on the big square on side c. The following presentation is an outline of how the theorem can be proven: c Area of square with side c is c2 c c c b Area of square with side b is b2 b b b a Area of square with side a is a2 a a a Method (1): practical proof: If the three square above are placed on the three sides of a right-angled triangle, then based on the Pythagorean Theorem: a2 + b2 = c2 This result can be established by constructing a right-angled triangle on graph paper as shown in figure 2. Let c = 5 cm, b = 4 cm and a = 3 cm We then draw squares on the three sides as shown. We then find the area of each square by counting all the little squares inside the 151 three squares. c b a Figure 2 The number of little squares inside the square on side c should be equal to the number of little square inside the square on the side a plus the number of little square inside the squares on the side b. 152 Method (2) Algebraic / Geometric proof It may not always be practical to draw triangles and count squares. Therefore, a general proof will be established using algebraic and geometric principles. Algebraic results used in proof: (a + b)2 = (a + b) (a + b) = a2 + 2ab +b2 Geometric results used in proof: (i) The sum of the three interior angles in any triangle is 180° x° x° + y° + z° =180° y° z° (ii) The sum of the adjacent angles on a straight line is equal to 180° x° + y° + z° =180° y° z° x° (iii) The area of a right angle triangle is = base × height 2 h A= b×h 2 b 153 x° x° x° x° c c c c b b b b y° y° y° y° a a a a 1. Four congruent triangles will be drawn with sides a, b, and c and interior angles 90°, x° and y° 2. The four triangles will be joined together to form a square of side (a + b) as shown below. b a x° y° The sum of the three angles in a a b triangle is 180° c c ∴ x + y + 90° = 180° y° x° => x + y= 90° ∴ the quadrilateral inside the big x° y° square of side (a + b) is also a square side c. c c b a y° x° a b The area of the big square with side (a + b) is equal to the sum of the areas of the four triangles plus the area of the square of side c. This will be used to prove the result: c2 = a2 + b2 154 The following is a presentation of two proofs of the Pythagorean Theorem Method 1 Practical Proof: counting squares. 5 4 10 3 9 15 2 8 14 1 20 1 2 3 4 7 13 19 6 25 12 5 6 7 8 18 24 11 c 17 = 23 5c b = 4cm m 9 10 11 12 16 22 13 14 15 16 21 a = 3cm 1 2 3 4 5 6 7 8 9 The number of little squares in the square of side 3 cm is 9 The number of little squares in the square of side 4 cm is 16 The number of little squares in the square of side 5 cm is 25 Hence: (i) area of square with side 5 cm is 25 cm2 (ii) area of square with side 4 cm is 16 cm2 (ii) area of square with side 3 cm is 9 cm2 This is so because each little square has an area of 1 cm2 When we add the areas of the two square on sides a and b we get 9 cm2 + 16 cm2 = 25 cm2. ∴ Area of square with side 5 cm = Area of square with side 3 cm + Area of square with side 4 cm OR c2 = a2 + b2 155 Method 2 Algebraic / Geometric Method b a x° y° a b Required to prove that a2 + b2 = c2 c c y° x° x° y° c c b a y° x° a b Proof: (i) The area of each triangle A = b × h = ab 2 2 ∴ The total area of the four triangles is 4 × ab = 2ab 2 (ii) The area of the interior squares with side c is c2 (iii) The area of the big square with side (a + b) is (a + b)2 = (a + b) (a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2 Combining (i), (ii) and (iii) to get: Area of square with side (a + b) = Area of four triangles + Area of Square with side c ∴ a2 + 2ab + b2 = 2ab + c2 => a2 + b2 + 2ab – 2ab = 2ab – 2ab + c2 => a2 + b2 = c2 Q.E.D.  156 P R OJ E C T 1 0 SECTION 05 ANALYSIS OF DATA For the given triangle, Pythagoras’ Theorem states that c2 = a2 + b2 c b a What does this mean in a real life context? How can Pythagoras’ Theorem help us to solve problems in the real world? If we look at the algebraic statements c2 = a2 + b2 closely, and take the square root of both sides we get c2 = a2 + b2 which gives: c = a2 + b2 Now c is a length, which suggests that Pythagoras’ Theorem can be used to determine lengths or distances. If the length c can be placed in a right angle triangle with two known sides, then Pythagoras’ Theorem can be used to find the length of c. 1. Bedroom Workmanship Approximately two (2) years ago my parents bought a house in a particular housing development. I have always heard residents complain about poor workmanship in the units. I decided to apply my knowledge of Pythagoras’ Theorem to test the basic alignment of a bedroom A c b B C a At a glance, my bedroom has a rectangular shape as shown above. The diagonal AB is the hypotenuse of triangle ABC if and only if c2 = a2 + b2. Also, if c2 = a2 + b2, then the angle ACB is a right angle and the room is lined up properly and is a perfect rectangular shape. 157 By measurement, the length of a measured 14 ft and the length of b measured 12 ft. Hence, for perfect line-up of the bedroom the following must be true: c2 = a2 + b2. Using the measurements a2 + b2 = 142 + 122 = 196 + 144 = 340 square feet Taking the square root on both sides we get: a2 + b2 = 340 ≈ 18.4 ft Hence, if the bedroom was lined up correctly the length of the diagonal should be approximately 18.4 feet. By measurement, the diagonal of the room measured 17.5 feet. Therefore, based on Pythagoras’ Theorem, the angle ACB is not equal to 90° and my bedroom is not a perfect rectangular shape. Therefore, I can conclude that the workmanship was somewhat poor. 2. Television Screen Another real-world situation that is worthy of investigation is that of the size of a television screen. c 37” 66” The size of the family’s television set was said to be 75”. However, I measured the length of the television set and discovered that it is 66” in length and 37” in width. The television set has a rectangular shape, so I decided to apply Pythagoras’ Theorem to determine the length of the diagonal c of the television set. Using Pythagoras’ Theorem we get: c2 = 662 + 372 = 4356 + 1369 = 5725 square inches Therefore: c = 5725 = 75.7” Therefore, our television set is approximately 75” along its diagonal. Therefore, in general, the reported size of a television set is the length of its diagonal. 158 3. The Shortest Distance Between Two Points B 50 m C A 120 m There is a rectangular playing field behind the secondary school that I attend. My friend and I was wondering what is the shortest path to get from point A to point B as shown in the diagram. One path is to walk from A to C and then to B. Another path is to walk along a straight line from A to B. The length of the field is 120 m and its width is 50 m. Hence, the length of the path from A to C and then to B is (120 + 50) m = 170 m. Since the straight path from A to B is the diagonal of a rectangular shape, we can determine the straight path from A to B using Pythagoras’ Theorem. Based on the Pythagorean Theorem: AB2 = 1202 + 502 = 14400 + 2500 = 16900 m2 Therefore: AB = 16900 = 130 m Hence, the shortest path from A to B is the straight line from A to B along the diagonal of the rectangular shaped field. This also verifies that the shortest distance between two points is a straight line joining the two points. 159 4. Determine The Length Of The Ladder Needed To Place A Banner On A Wall space for banner 20 ft er dd la wall 15 ft The school is planning its annual Carnival activity. The Carnival Committee is planning to erect a banner 20 feet above the ground at the front of the building as shown above. There is a space of 15 feet between the front of the school and a wall on which the ladder could be placed in order to erect the banner. Pythagoras’ Theorem can be used to determine the length of ladder that is needed to erect the banner. Let the length of the ladder be c feet. Using Pythagoras’ Theorem we get: c2 = 152 + 202 = 225 + 400 = 625 square feet Therefore: c = 625 = 25” Therefore, a ladder of length 25 feet is needed to help the committee in erecting the banner. 160 P R OJ E C T 1 0 SECTION 06 DISCUSSION OF FINDINGS Based on the results, we see that Pythagoras’ Theorem is not just an abstract formula to be learnt by rote. It can be used to help determine lengths and establish the accuracy of measurement. In this project it was used to check the accuracy of the measurement of a bedroom, verify the size of a television set, determine the shortest distance between two points on a playing field and determine the length of the ladder that is needed for a particular job. In conducting this project a few challenges were encountered. Measuring the dimensions of the school’s playing field posed some problems since the measuring tape that was used was only 25 feet long. Two of my peers assisted in measuring and marking points along the dimensions of the field. Also, it was difficult to ascertain whether the field had square edges. Hence, it was assumed that the edges of the field formed 90° angles when they met. A spirit level was used to determine if the school’s walls and ground were both vertical and horizontal. 161 PRO JEC T 10 SECTION 07 CONCLUSION Pythagoras’ Theorem can be demonstrated practically by constructing a right-angle triangle on graph paper and counting the squares on the three sides of the triangle. Algebra and Geometry can be used to prove the theorem by constructing four congruent triangles around a square and setting up an algebraic identity. The theorem can be used in the real-world to investigate the following situations: 1. the accurate construction of a rectangular room, 2. the size of a television set, 3. the shortest distance between two points, 4. the length of the ladder needed to erect a banner on a vertical wall 162 P R OJ E C T 1 0 SECTION 08 REFERENCES www.brighthubeducation.com www.thefamouspeople.com www.geo.uiuc.edu www.storyofmathematics.com 163 PRO JEC T 10 SECTION 09 APPENDIX (1) Results used in this project A square is a quadrilateral with 4 equal sides. Each internal angle equal to 90°. Area of a square = side × side = side2 h Area of a right angle triangle = base × height 2 b a The sum of the 3 interior angles in a triangle is equal to 180° ∴ a° + b° + c° = 180° b c The sum of the adjacent angles on a straight line ∴ a° + b° + c° = 180° b a c 164 (2) Terms used in project: (i) Hypotenuse: The side opposite the right angle in a right-angle triangle A Hypotenuse C B (ii) Diagonal: A line joining two vertices of a polygon when those vertices are not on the same edge. AB is a diagonal B vertex vertex A (iii) Theorem: A mathematical statement that was proven to be true within a given system. (iv) Q.E.D.: Abbreviation for the Latin phrase “quod erat demonstrandum” translated into English means “that which was to be demonstrated” usually used at the end of a proof. (v) Horizontal Plane: flat smooth surface with no irregularities. 165 20 IDEAS FOR A MATHEMATICS SBA WITH SCENARIOS AND TITLES 166 Mathematical relations permeate the world, therefore, students have a wealth of options from which they can choose. Students should first look within their immediate environment and observe situations that may require investigation. For example: 1. The home is an excellent place to start looking for SBA Project ideas: a. Creating a budget or estimate for home renovations can give rise to an SBA Project. b. Purchasing major items for the home can also spark ideas for an SBA Project. 2. The school is another place where SBA Project ideas can be developed. The following are all great concepts for SBA Projects: a. absenteeism b. late coming c. improper uniform d. poor cafeteria service e. students’ academic performance f. classroom: Students should pay particular attention to teachers when they are presenting their lessons. Topics such as: measurements, statistics, geometry, relations and functions and consumer arithmetic are all excellent topics that can be used in order to investigate real-world problems. When a teacher is presenting a particular topic, students should ask the teacher to explain how the given topic can assist them with the SBA Projects. 3. The community is also a great place to get SBA Project ideas. Students should listen to complaints of poor service by customers. These complaints can then be investigated by using simple Mathematical techniques. Generally, students can look at any complaint or real-world problem which may be anecdotal in nature and can be investigated by using a simple Mathematical technique. The following is a list of twenty (20) scenarios and titles that can be used as research projects under different topics in Mathematics. These topics are statistics, consumer arithmetic, algebra, geometry and matrices. 167 STATISTICS SCENARIO 1 Students at the Form three (3) level write the National Certificate of Secondary Education (NCSE) Examination every year. However, the results of these examinations may not be seriously analyzed by school management to inform teaching strategies. A student can therefore do a research project based on the NCSE results and the title can read as follows: “ An investigation of the last four (4) years of NSEC results at a particular school.” SCENARIO 2 A student may want to know how his or her school performs in a particular subject area at the Caribbean Secondary Education Certificate (CSEC) Examination. The following title can be used to pursue a project: “A study of the last five (5) years of the CSEC English Language examination results at a particular school.” SCENARIO 3 A group may want to study the performance of lower form students at a school in a particular subject area. The following title can be used for a research project: “A student may want to study the performance of the lower school students of his or her school in a particular subject area. The following title can be used for a research project.” SCENARIO 4 Teachers often complain of Form 5 students being tardy and absent from school. This claim can be investigated in a project. The following title can set the general research theme for the project: “An investigation into the punctuality and regularity of Form 5 students at a secondary school.” 168 SCENARIO 5 The above situation can also be investigated along the following line: “An investigation in order to determine if the distance that students live from the school has an impact on their punctuality”. SCENARIO 6 Parents usually say that extracurricular activities affect the children’s academic performance. This claim can be investigated by using the following title as a general theme for the project: “An investigation of students’ academic performance at a particular school in order to determine if they are affected by extracurricular activities..’’ SCENARIO 7 The school’s management has serious problems with students altering the school uniform. A project can be done on the severity of the problem. The following title can be used as a guide for a project: “An investigation into the number of upper school students at a particular school who alter their school uniform.” SCENARIO 8 Students may choose to investigate the academic performance of the Form one students in a particular subject area at a school. The following title can be used to guide the project: “An investigation into the performance of Form 1 students in the area of Mathematics at a secondary school.” SCENARIO 9 There are always complaints of students not submitting homework assignments. This claim can be investigated by using the following title as a guide: “An investigation of the homework habits of the upper and lower forms in order to determine which group is more likely to submit homework assignments.” 169 CONSUMER ARITHMETIC SCENARIO 10 A student can choose to assist his or her parent with the family’s budget. A budget can be created by taking into account salaries and expenditure. The following title can be a great theme for the project: “Creating a family budget using consumer arithmetic techniques.” SCENARIO 11 The House Systems at schools usually use fundraisers to help with the purchase of uniforms and other items for Sports Day. A group of students can create a business plan for a particular house in order to assist with fundraising. They can list the items to be sold and the price at each items must retailed in order to generate a significant profit. The following title can be used to guide the project development: “Developing a business plan to raise funds for one of the houses that belong to a particular school’s house system.” SCENARIO 12 Suppose a student’s father wants to sell his motor vehicle after eight (8) years. The vehicle has depreciated by a given percentage every year for the last eight (8) years. The student can develop a project in order to determine the present value of the vehicle and other factors that may affect the final selling price. The title of the project can be stated as follows: “An investigation into the value of my father’s motor vehicle after eight (8) years.” 170 FUNCTIONS AND RELATIONS GRAPHS & LINEAR PROGRAMMING SCENARIO 13 Dad is a plumber. He charges a fixed fee of $200 per site visit. He also charges a rate of $100 per hour. A linear relation can be set up and used in order to predict the total fee that Dad can charge for spending x hours on a job site. The following title can guide the project: “Using a linear relation in order to determine the total amount of money that my father can charge for his plumbing services.” SCENARIO 14 1 The circumference (C) of any circle is 3 7 times its diameter (d). This is stated as 1 C= πd, where π = 3 7 . This implies that C is directly proportional to d. Hence, a graph of C vs d is a straight line. This concept can be used to determine the approximate value of π by using practical measurements. The title of the project can read as follows: “An investigation into the relationship between the circumference of a circle and its diameter in order to determine the value of pi (π)”. SCENARIO 15 Students usually find it difficult to manage their time for studies and other activities. This is so because, there are usually a number of constraints on their lives. Linear programming can assist students to determine how many hours of study that they need on a weekly basis in order to achieve maximum benefit given all the constraints on their lives. A title for this project can read as follows: “Using linear programming techniques to determine the maximum number of hours that a student should study each week in order to achieve maximum benefits, given all the constraints on his or her life.” 171 ALGEBRA SCENARIO 16 There is a piece of land at the back of the school that can be used by the Young Leaders Group to create a rectangular vegetable garden. The garden must be fenced and a footpath must be placed around it for access. A quadratic function can be created to model the garden and the method of completing the square can be used to determine the maximum area of land that can be used. The project title could read: “Using the method of completing the square to determine the maximum area of a vegetable garden.” SCENARIO 17 Students are often asked to factorize the algebraic expression a2 - b2. Theoretically, the result of this factorization is a2 - b2 = (a + b) (a - b). A group of students can arrange with the Woodwork Department or a tradesman to cut a piece of wood into a perfect square of a particular area. A smaller square can then be cut from the bigger square and the students can demonstrate the results a2 - b2= (a + b) (a - b) in a practical way. The title of a project can read as follows: “A study into the algebraic identity a2 - b2 = (a + b) (a - b) in order to physically demonstrate the result.” 172 GEOMETRY SCENARIO 18 A general formula (2n - 4) × 900 is usually given to students to determine the sum of the interior angles in a polygon. This formula is usually given without proof. A group of students can physically prove this result by cutting different polygons out of Bristol Board and relating each polygon to a triangle as shown below. Shape divided into Number Calcula- Sum of Shape triangles of sides tion angles 3 1 x 180° 180° 4 2 x 180° 360° 5 3 x 180° 540° 8 6 x 180° 1080° Students should be able to look at the pattern and formulate a general formula for the sum of the interior angles of any polygon. The title of the project can read: “An investigation into different polygons in order to determine a general formula for the sum of the interior angles of any polygon with the use of a practical method.” 173 MATRICES SCENARIO 19 My parents usually give away hampers every Christmas. This year they want to give out ten (10) hampers consisting of 1 litre milk cartons, 5 kilogram packs of rice and 2 litre bottles of cooking oil. Each hamper has ten (10) items. Milk costs $12.50, rice costs $30.00 and oil costs $45.00. Hamper Milk Rice Oil 1 5 3 2 2 4 5 1 3 6 1 3 4 2 7 1 5 3 4 3 6 1 6 3 7 4 2 4 8 3 3 4 9 1 7 2 10 3 4 3 Item Cost $ Milk 12.50 Rice 30.00 Oil 45.00 Students can create two matrices in order to determine: 1. the total value of each hamper 2. the total cost to create all 10 hampers The title of the project can be written as: “Using matrices in order to determine the total cost of production for ten (10) Christmas hampers.” 174 APPLICATION OF MATHEMATICS SCENARIO 20 Students always complain that Mathematics has no real world meaning to them. They always ask the questions; “Why am I doing this topic?” and “When will I use Mathematics in life?” This project gives students a chance to answer their own questions. Students can choose three careers and show how Mathematics can play a key role in these fields. The following title can be used as a guide for this project: “An investigation in order to demonstrate the importance of Mathematics in different careers.” 175 REFERENCES Clarke, L. Harwood, Additional Pure Mathematics, 3rd Edition Greer, A. & Layne, C. E. (1991) Certificate Mathematics: A Revision Course for the Caribbean. Heng, H. H.; Talbert, J. F., Additional Mathematics, 6th Edition Toolsie, R, (1999). A Complete Course with CXC Questions. www.brighthubeducation.com www.edelmonfinancial.com www.eduflow.wordpress.com/tardiness www.everythingmaths.co.za www.freepik.com www.geo.uiuc.edu www.guardian.co.tt www.math.ucla.edu www.mathgoodies.com www.mathisfun.com www.mathswork.com www.moneysupermarket.com www.quora.com www.selfbuildnewhomes.com www.storyofmathematics.com www.theastofed.com www.thefamouspeople.com www.ttconnect.gov.tt www.uncw.edu/alegbra www.varsitytutors.com www.weberawler.com www.wikiEducator.org www.wikihow.com 176 ABOUT THE AUTHOR Glendon Steele has been teaching Mathematics for the past twenty (20) years. He currently serves as Head of the Mathematics Department at St. James Secondary School, Trinidad and Tobago. He is a graduate of The University of the West Indies and holds a B.Sc. Mathematics (Hons) and a Post Graduate Diploma in Education- Teaching of Mathematics. He has also been a Caribbean Examinations Council Assistant Examiner for the past fifteen (15) years. Over the years he has been conscientiously preparing students for the annual Australian Mathematics Olympiad Competition. Glendon Steele is a consummate teacher in many regards and holds the distinguished accomplishment of effectively presenting the television series, ‘A Conceptual Approach to Learning Mathematics’, on The Information Channel 4 for approximately five years during which, emphasis was placed on learning Mathematics by building on a set of fundamental principles. Here, he emphasized that Mathematics should not be taught by presenting a set of techniques that students should memorize by rote but instead it should be taught by helping students to recognize that Mathematics is based on a set of fundamental concepts that form the basis of learning any topic in Mathematics. Mr. Steele’s philosophy is that Mathematics should be presented in a manner in which students can see, acknowledge and appreciate its relevance in the real world and eventually come to fall in love with the subject.