SSIP JULY - AUGUST 2020 MATHEMATICAL LITERACY PARTICIPANT GUIDE
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SSIP JULY – AUGUST 2020 MATHEMATICAL LITERACY PARTICIPANT GUIDE 1
© COPYRIGHT This work is protected by the Copyright Act 98 of 1978. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission in writing from Matthew Goniwe School of Leadership and Governance. Whilst every effort has been made to ensure that the information published in this work is accurate, Matthew Goniwe School of Leadership and Governance takes no responsibility for any loss or damage suffered by any person as a result of the reliance upon the information contained therein. 2
TABLE OF CONTENTS NO. PREAMBLE PAGE A Foreword 3 B Purpose 3 C SSIP Aims/goals 4 D SSIP Objectives 4 E Learning Assumed to be in place 4 F Target Audience 4 G Notional Hours 4 H Course Design and Assessment Strategy 5 I Course outline/Map 5 J Table of Icons 7 K Table of Acronyms and Abbreviations 7 L Term ATP 8 M Course Timetable 8 Module 1: Probability 9 Unit 1 Expression and Prediction of Probability 10 Unit 2 Representation and Evaluation of probability 16 Module 2: Data Handling (Representing, Interpreting and Analysing Data) 28 Unit 1 Representing, Interpreting and Analysing Data 29 Module 3: Models and Packaging 51 Unit 1 Packaging 50 Unit 2 Assembling 58 Module 4: Finance (Interest and Banking) 64 Unit 1 Interest 65 Unit 2 Banking (Buying a car) 70 Unit 3 Banking (Buying a house) 74 A. FOREWORD This document is the result of the Just in Time Secondary School Intervention Programme (JIT SSIP) which is an intervention programme for FET teachers in the Gauteng Department Education (GDE) in collaboration with Matthew Goniwe School of Leadership and Governance (MGSLG). B. PURPOSE The purpose of this programme is professional development of FET teachers who are currently teaching in the school FET phase of the education system. The programme is aligned to the strategic goals of the GDE which focuses on improving the teaching and learning practice in the most classrooms through capacitation of teachers on Content, Pedagogy, and Assessment and ICT integration. 3
C. SSIP AIMS/GOALS The SSIP programme aims at professional development for Grade 10-12 teachers in the application of effective teaching and reflective practice to improve learner performance on the identified Grade 12 examinable topics. The overall goal for SSIP is to provide teachers with professional expertise, tools and skills to spot student learning difficulties and decide on the course of action. SSIP came about as result of the diagnostic needs that are identified through the end of the year NSC examination student learning data. In response to this design and development of teaching resources are developed to train teachers on the learner needs. The four interconnected outcomes that drive the professional development activities for SSIP are: • Enhancing Teachers knowledge: deep understanding of subject matter knowledge and students’ ideas on the content • Enhancing quality teaching and assessment for learning: effective instructional approaches that teachers may use to ensure improved understanding by most learners. • Developing ICT integration skills: Use of ICT to improve teaching and learning • Building professional learning communities: allow teachers to start collaborating and form professional networks in non-formal settings in context of their schools D. SSIP OBJECTIVES By the end of the workshop teachers should be able to: • have mastered and understood all aspects related to Interest, Banking, Models, Probability and Graphs within the FET Mathematical Curriculum • utilise ICT integration and encourage interactive lessons in teaching and learning E. LEARNING ASSUMED TO BE IN PLACE Participants are qualified teachers with a qualification in teaching at NQF 4 or above and teaching Mathematical Literacy. F. TARGET AUDIENCE Teachers who were identified through the 2018 NSC results, diagnostics report, and needs analyses of the teacher in the Integrated Quality Management System (IQMS) who teaching Mathematical Literacy in target schools. The course is aimed at professional development to improve learner performance in Mathematics Literacy. 4
G. NOTIONAL HOURS: The time required to successful completion has been allocated as follows: Contact face to face session 17, 5 hours Pre Test 1 hour Day 2 : Content Practice and Demonstration 10,5 hours Day 3: Content Practice and Demonstration 5 hours Post Test 1 hour H. COURSE DESIGN AND ASSESSMENT STRATEGY • The envisaged course focuses on four modules with at least two units each viz. o Module 1: Probability o Module 2: Data Handling (Representing, Interpreting and Analysing Data) o Module 3: Packaging and Assembling o Module 4: Finance (Interest and Banking) • Teachers will be subjected to the variety of content knowledge, formative activities to consolidate the content learnt, Pre-test at the beginning of each workshop session to further verify content gaps and post-test in the last session. • Pre-Post Test data will be used to monitor what learning has taken place in the 3 days of the session and workshop activities will also be used to support participant on subject matter knowledge I. COURSE OUTLINE/ MAP Module 1: Probability Objectives/Outcomes Units At the end of this module you will Unit 1: be able to: Expression and Prediction of Probability • Use the Probability Scale At the end of this Unit, you should be able to: to solve problems in • Express probability as numbers, fractions, decimals simple events (dice/coin and words. games, lotteries, etc.) • Interpret and use the probability scale to determine • Calculate Probability using Theoretical and the probability of simple events (dice/coin, national Experimental Probability lotteries, insurance risk assessment, weather • Work with situations predictions, etc.). involving the probability • Calculate Probability using theoretical/experimental of compound events probability. Unit 2: Representation and Evaluation of probability At the end of this Unit, you should be able to: • Represent compound events on a Tree diagram and on a contingency/Two-way table. • Interpret and analyse a drawn tree diagram and a contingency/two-way table. 5
Module 2: Data Handling Graphs Objectives/Outcomes Units At the end of this module you will Unit 1: Data Handling Graphs be able to: At the end of this Unit, you should be able to: • Represent data • Read values directly from the values provided on the • Interpret and analyse data graphs. • Draw specified graph from a given table of data. • Represent, interpret and analyse graphs. Module 3 : Packaging and Assembling Objectives/Outcomes Units At the end of this module you will Unit 1: Packaging be able to: At the end of this Unit, you should be able to: • Investigate, by making use of • Investigate, by making use of calculations the most calculations the most cost cost effective and most appropriate way to package effective and most cans/boxes for optimal use of space. appropriate way to package Unit 2: Assembling cans/boxes for optimal use of At the end of this Unit, you should be able to: space. • Use instructions/assembly diagrams containing words or • Complete a task presented in pictures found in manuals in order to complete the task the instructions and explain presented in the instructions. what the Instructions mean • Explain what the instructions mean and represents using or represent using everyday everyday language. language. Module 4 : Finance ( Interest and Banking) Objectives/Outcomes Units At the end of this module you will Unit 1: Interest be able to: At the end of this Unit, you should be able to: • Perform calculations involving • Differentiate between simple and compound interest simple and compound interest. • Perform calculations involving simple or compound • Investigate hire purchase interest without using the formula. agreements. Unit 2: Banking • Investigate investments where At the end of this Unit, you should be able to: a fixed deposit is made every • Investigate the appropriate agreement to choose month. when buying items on hire purchase. • Perform calculations involving • Choose the best plan when buying a car. residual and balloon loans. • Use the factor table to calculate the monthly • Convert from rand to forex and payment of a home loan. from forex to rand. 6
J. TABLE OF ICONS TO BE USED IN THIS MANUAL Discussion Group ACTIVITY Individual ACTIVITY Study Tips Notes Ice Breaker Time Tools K. TABLE OF ACRONYMS AND ABBREVIATIONS Acronym Definition ATP Annual Teaching Plan CAPS Curriculum and Assessment Policy Statement ICT Information and Communication Technology LP Lesson Plan FG Facilitator’s Guide NPPPPR National Policy Pertaining to Programme and Promotion requirements PG Participant Guide FS Fact Sheet PPT PowerPoint Presentation TPACK Technological, Pedagogical, Content and Knowledge TS Training Session 7
L. TERM ANNUAL TEACHING PLAN DATE CONTENT CONTEXT 08/06 – 19/06 Interest and Banking • student/personal loans Hire Purchase, Loans, Exchange rate, • buying a car/house/furniture/etc Banking, Investment and Inflation • Planning trips to other countries • Cost of living comparisons • Stokvel, Annuities, Pension Fund Insurance Policy, Funeral Plans and CPI 17/08 – 28/08 Packaging and Assembling • Boxes for packaging fruits juices, tiles, • Packaging containers cool drinks etc. • 2D and 3D Models • Office spaces, venue preparations, building models etc. 09/03 – 13/03 Data Handling Graphs • National and Provincial Health • Box and whisker, Bar, Histogram, statistics, Education statistics, Road scatter plot, Pie and line graphs accident stats, Population statistics etc. 03/08 – 14/07 Probability • National lotteries, Gambling, Risk • Probability of simple events assessment and Newspaper articles • Compound events M. COURSE TIMETABLE Day 1 TIME ACTIVITY 15:30 – 16:30 Arrival 16:00 – 17:00 Plenary Session: Opening and Welcome 17:00 – 18:00 TS 1: Pre Test 18:00 – 19:30 Supper DAY 2 TIME ACTIVITY 06:30 – 08:00 Breakfast 08:00 – 10:30 TS 2: Module 1 – Unit 1 10:30 – 11:00 Tea Break 11:00 – 13:00 TS 3: Module 1 – Unit 2 13:00 – 14:00 Lunch 14:00 – 15:30 TS 4: Module 2 – Unit 1 15:30 – 17:00 TS 5: Module 3 – Unit 1 17:00 – 18:30 TS 6: Module 4 – Unit 1 19:00 – 20:30 Supper DAY 3 TIME ACTIVITY 06:30 – 08:00 Breakfast 08:00 – 10:30 TS 7: Module 4 – Unit 2 10:30 – 11:00 Tea Break 11:00 – 12:00 TS 8: Module 4 – Unit 3 12:00 – 13:00 TS 11: Post Test Closing Session and Remarks: 13:00 – 14:00 Lunch 8
MODULE ONE: PROBABILITY INTRODUCTION In this module you will look at the concept of Probability with specific focus on: • Language of Probability • Expressions of Probability • Probability scale • Practical and Theoretical Probability • Compound events OVERVIEW In this topic, you will work with complex projects in familiar and unfamiliar contexts SPECIFIC OBJECTIVES At the end of this module you will be able to: • Define Probability • Work with situations involving Probability. • Express Probability in different forms. • Recognize the difference between terms used in Probability. • Draw and analyze Contingency/Two-way tables and Tree diagrams. TERMINOLOGY Probability Numerical measure of the likelihood or chance that an event will happen or not happen. Event An activity. Outcome The result of a trial. Trial The act of letting an event happen or not happen e.g. action of tossing a coin. Chance A possibility of something happening. Possible Outcome The chance that the event will happen. Impossible Outcome No chance of the event happening. Certain Something is definitely going to happen. Likely Chance of something happening is greater than the chance of it not happening. Unlikely Chance of something happening is less than the chance of it not happening. Even(fifty-fifty) Chance of something happening or not happening is the same. Prediction Statement describing the chance of an outcome to happen based on given information. Tree diagram A Diagram using branches to display all the possible outcomes of the events. Two-way table Also known as Contingency table, represents all possible outcomes of two trials taking place together. Experiment Series of trials. 9
CONTENT You will study this module through the following units Unit 1 Experimental and Theoretical probability Unit 2 Probability of Compound events UNIT 1: EXPERIMENTAL AND THEORETICAL PROBABILITY INTRODUCTION In this unit you will look at • the concept of expressions of Probability, • working with the probability scale and • calculating probability using theoretical and experimental probability. LEARNING OUTCOME At the end of this Unit, you should be able to: • Express probability as numbers, fractions, decimals and words. • Interpret and use the probability scale to determine the probability of simple events (dice/coin, national lotteries, insurance risk assessment, weather predictions, etc.). • Calculate Probability using theoretical/experimental probability. LESSON NOTES • The Probability of an event happening is expressed as a number. It can be written as: ✓ A fraction ✓ A decimal fraction ✓ A percentage 6 2 • Probability must always be written in its simplest form e.g. 18 = 6 • The Probability of events can be expressed using a Probability scale. ✓ The scale ranges from 0% - 100% OR 0-1. ✓ 0 stands for Probability of an event that certainly will not happen. ✓ 1 OR 100% stands for Probability of an event that will certainly happen. 10
The following is a visual representation of the Probability scale: Example 1 State the Probability of the following events using the Probability scale: 1. Flipping a coin to land on tails. Proposed solution 1 2. The sun setting in the evening. 1. Even or 50/50 3. Pigs flying. chance. 2. Certain 3. Impossible Theoretical Probability • Uses known known facts about the Probability or the situation. • No experiments are conducted, but knowledge about the situation is gained by using logical reasoning or known formula. • Formula that is used to calculate Probability of an event happening: Theoretical Probability = Example 2: A jar contains 7 red balls and Proposed Solution 12 blue balls. Determine the probability of picking a red 7 ball. P(red ball) = 19 11
EXPERIMENTAL PROBABILITY Found by repeating the event many times (trial) and then calculating the Probability based on the outcomes of these trials. Example 3: Proposed Solution A dice is thrown and the outcome is noted. A dice has 1, 2, 3, 4, 5, 6 on it. Determine the Probability of throwing a: 1 1. P(Three)= 6 1. Three 3 1 2. Odd number 2. P (Odd number) = 6 = 2 3 1 3. Even number 3. P (Even number) = 6 = 2 Example 4: A standard deck of cards is made up of 13 spades, 13 hearts,13 diamonds and 13 clubs as indicated below. 1. What is the Probability that Poppy will pick up a Queen of Spades?” 2. Nikiwe decides to pick a card from the deck, while Poppy is holding the Queen of Spades in her hand. What is the Probability, as a decimal that Nikiwe will pick a Queen from the deck? Solution: 1. P (Queen Spades) 1 = 52 3 2. P(Queen) = 51 = 0,06 12
Activity 1.1.1 Group Discussion (30 Minutes) Instructions • Participants should form groups of 4 – 5 • Refer to the questions provided • These questions are intended to advance skills that can be used in the classroom and possible ways in which this section can be taught • Task 1: Answer all the questions • Report Back and Discussion • Resources: Pen and Calculator Answer the following questions: 1.1. Lihle's favourite band is performing at an open-air arena. It is predicted that it is most unlikely that it will rain on the night of the performance. Choose ONE of the values below that best describes this probability: (2) , , % , % 1.2. A nurse is carrying a rectangular medicine box. The medicine box contains FOUR identical smaller boxes. EACH small box contains four different types of pills in cylindrical containers which are labelled A, B, K and U, as shown below. Determine (as a Percentage) the probability of randomly selecting a type U container from the medicine box. (3) 1.3. Peter is in the front of a queue to buy a chocolate for his mother and randomly chooses a chocolate. There are 50 chocolates left that are wrapped as follows: • 17 silver • 20 gold • 13 red Determine the probability that Peter will choose a chocolate wrapped in: 1.3.1 Red foil (2) 1.3.2 Green foil (2) 13
1.4. A company installed computers at a computer centre in October 2019. The manager used a bank account to pay the employees' wages for the project. Below is a calendar for October 2019. Use the calendar above to answer the following question. Determine the probability of randomly selecting a workday(Monday-Friday) in October 2019 (3) with a date that is an even number. [12] Activity 1.1.2 (30 Minutes) Instructions • Refer to the questions provided • These questions are intended to advance skills that can be used in the classroom and possible ways in which this section can be taught • Task 1: Answer all the questions • Report Back and Discussion • Resources: Note Pad, Pen and Calculator The TWO pie charts below show why and how people in South Africa travel. 14
1. Study the TWO pie charts above and answer the questions that follow. a) Calculate the percentage of people whose reason for travel is sport. b) Which mode of transport is used by most people? c) Determine the probability (written as a fraction in its simplest form) of randomly selecting a person whose mode of transportation is travelling by bus. 2. A group of tourists visited the Kgalagadi Transfrontier Park. The layout plan of the Twee Rivieren Camp is given above. This camp offers two types of accommodation: • Camping facilities (for tents) • Cottages The visitors booked a drive activity. Determine the probability that the activity booked was NOT a night drive. 3. Mrs Zulu owns a large national cleaning company that cleans office blocks in various cities. The number of employees are as follows: • 11 handymen. • 272 cleaners (men and women in the ratio 1: 3). • 12 supervisors (in the same gender ratio as the cleaners). • 11 drivers. 15
a) Calculate the probability of randomly selecting a female cleaner as the employee of the month, rounded off to THREE decimal places. b) Explain why the chance of randomly selecting a male supervisor as the employee of the month is most unlikely. 4. Human blood is classified into eight main blood groups. The SANBS is regularly appealing to eligible people to donate blood. TABLE 1 below shows the distribution of the eight different blood groups in the South African population. Table 2 shows the comptibility between the blood group of the donor and the possible recipient. a) Write down the Probability of randomly selecting a South African who is clasified in the O blood group. b) Identify the blood group of a recipient that has a 100% Probability of being able to receive blood from any donor blood group. c) Verify with a reason whether it is most likely for an − blood group recipient to be able to receive from any donor blood group. 16
UNIT 2: PROBABILITY OF COMPOUND EVENTS INTRODUCTION • • Sometimes in Probability we have more than one event taking place. • To calculate Probability in these cases, we need to determine all the possible outcomes. • In this unit, we use tree diagrams and two-way (contingency) tables to determine all the other outcomes accurately. LEARNING OUTCOME At the end of this Unit, you should be able to: • Represent compound events on a Tree diagram and on a contingency/Two-way table. • Interpret and analyse a drawn tree diagram and a contingency/two-way table. • Use the drawn tree diagram and a contingency/two-way table to complete a task. LESSON NOTES TREE DIAGRAMS • A tree diagram is shaped like a tree. • It has branches that represents the possible outcomes. • A tree diagram is a visual way of expressing the outcomes, particularly when there are more than two events. Example 1 a) Draw a tree diagram to represent the possible outcomes of tossing a coin two times. b) State the number of possible outcomes in the tree diagram. c) Determine the probability of getting the following: I. 1 heads a 1 tails in any order. II. 2 heads. Proposed Solution a) b) TT; TH; HT; HH = 4 outcomes c) 2 1 I. P(H, T in any order)= 4=2 1 II. P(2 Heads)= 4 17
Example 2 AK Bazaar soccer team still have two matches to play before they qualify to take part in the Premier Soccer league. There are three possible outcomes for each match: win (W), lose (L) or draw (D). A tree diagram is drawn to work out the possible outcomes for the two matches. Look at the tree diagram and answer the questions below: a) Complete the tree diagram above to show all the possible outcomes of the two matches. b) Explain the meaning of the outcomes: WD and DW. c) What is the Probability of winning both matches? d) What is the Probability of winning only one match? e) What is the Probability of drawing at least one of the matches? Proposed Solution a). b) WD: Win 1st match and Draw second match. DW: Draw 1st match and win 2nd match. 1 c) P(WW)=9 4 d) P(W)= 9 5 e) P(D) and P(DD)= 9 18
TWO-WAY TABLES (CONTINGENCY TABLES) • A two-way table is also called a contingency table. • A two-way table that shows all the possible outcomes of two trials taking place together. • Has the outcomes of one event across a row in a table and the other outcomes are found running down a column. Example 3 Siya spins two spinners at the same time. Spinner 1 has divisions A,B,C and D. Spinner 2 is labelled 1,2, and 3. Siya draws up a contingency table to show the possible outcomes of his experiment Spinner1 A B C D Spinner 2 1 A1 B1 C1 D1 2 A2 B2 C2 D2 3 A3 B3 C3 D3 a) Write down the number of outcomes in this experiment. b) What is the probability of scoring a 2 and any letter? c) What is the Probability of scoring a D and a 1? Proposed Solution a. There are 12 outcomes. 4 b. P(2 and any letter)= 12 = 1 3 1 c. P(D and a 1)= 12 19
Example 4 The table below shows the results of a survey conducted to determine how learners of Bophelong Secondary get to their school on Monday and Friday. Mode Cycle Car Taxi Walk Total Day Monday 12 42 20 100 Friday 8 50 122 Total 50 92 60 a) Complete the table above by filling in the missing values. b) Determine the probability that a learner, taken at random: I. Cycled to school. II. Did not walk to school Friday. Solutions Example 4 a). 41 c) P(Did not walk)122 = 61 Mode Cycle Car Taxi Walk Total b) P(Cycling)222 = 111 10 Day 82 20 Monday 12 26 42 20 100 Friday 8 24 50 40 122 Total 20 50 92 60 222 Notes--------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- 20
Activity 1.2.1 Group Discussion (30 Minutes) Instructions • Participants should form groups of 4 – 5 • Refer to the questions provided • These questions are intended to advance skills that can be used in the classroom and possible ways in which this section can be taught • Task 1: Answer all the questions • Report Back and Discussion • Resources: Pen and Calculator 1. Thabo cannot decide what to wear to a party. He has three pairs of pants- a grey pair, a black pair and a blue pair of jeans. He puts his pants on his bed, closes his eyes and chooses a pair of pants to wear. He has four shirts-a white shirt, red shirt , black shirt and green shirt. He pulls a shirt from his drawer without looking. 1.1. Complete the two-way table below to show the different possible combinations of shirts and pants that Thabo has to choose from and answer the questions that follow. (4) Shirts White shirt Red shirt(RS) Black shirt(BS) Green (WS) shirt(GS) Pants Grey pants(G) Black pants (B) Blue jeans (BJ) BJ;RS 1.1.1. How many compound outcomes are there in total? (2) 1.1.2. What is the probability that Thabo will be wearing his Green shirt (3) 1.1.3. What is the probability that Thabo will NOT be wearing something black? (3) 2. At Alex High School the basic boys' uniform consists of a pair of pants and a shirt with the option of wearing a tie. The pants may be either long or short, and the shirt may be either long-sleeved or short-sleeved. They are allowed to wear any combination of these three items of clothing when they are on a trip. 2.1. Complete the tree diagram below to illustrate all the possible combinations of these three items of clothing that the boys may wear on a trip. (7) KEYS: CODE EXPLANATION LP LONG PANTS SP SHORT PANTS LS LONG SLEEVE SS SHORT SLEEVE T TIE NT NO TIE 21
2.2. When the boys are at school, they are only allowed to wear ONE of the following combinations of the uniform: • Long pants with long-sleeved shirt and a tie. • Short pants with a short-sleeved shirt and no tie. If one of the boys in the bus were randomly selected, use the completed tree diagram above to determine the probability (in a decimal form) that he would be wearing on of these two combinations. (2) 22
Activity 1.2.2 (30 Minutes) Instructions • Refer to the questions provided • These questions are intended to advance skills that can be used in the classroom and possible ways in which this section can be taught • Task 1: Answer all the questions • Report Back and Discussion • Resources: Pen and Calculator 1. Mary wants ice cream. The ice cream shop has the tree diagrams in the window to show customers possible combination of ice cream they are selling. Study the tree diagram below and answer the questions that follow: 1.1. How many outcomes are there in total? (2) 1.2. How many combinations can Mary get with vanilla ice cream? (2) 1.3. How many combinations can Mary get if the ice cream shop runs out of nuts? (2) 1.4. How many combinations can she get with chocolate and strawberries? (2) 23
2. The accelerated schools infrastructure delivery initiative (ASIDI) of the Department of Education was created to renovate schols so that they are conducive to learning. TABLE 1 below shows a summary of the completed renovated projects per province. PROBLEMS AT THE SCHOOL PROVINCE INAPPROPRIATE WATER SANITATION ELECTRICAL TOTAL STRUCTURES NUMBER OF PROJECTS EC 145 271 171 202 789 FS 21 53 12 50 136 GP 0 0 14 2 16 KZN 1 206 103 58 368 LP 3 104 88 5 200 MP 5 36 38 45 124 NW 2 3 10 0 15 NC 1 5 13 2 21 WC 25 3 19 8 55 TOTAL 203 681 468 372 1724 Use the information above to answer the questions that follow. 2.1. Write down, in simplified form, the ratio of water proects to sanitation projects in Limpopo. (3) 2.2. Determine, as a decimal, the probability of randmly selecting a renovation project that successfully completed inappropriate structures. (3) 2.3. Determine, to the nearest percentage, the probability of randomly selecting a completed project at a school in KZN that did NOT have electric repairs. (4) [15] 24
Activity 1.2.3 (30 Minutes) Instructions Refer to the questions provided • These questions are intended to advance skills that can be used in the classroom and possible ways in which this section can be taught • Task 1: Answer all the questions • Report Back and Discussion • Resources: Pen and Calculator 1.1. Katleho Secondary School kept records of all the learners that arrived late for school. The Mathematical Literacy teacher noticed that the late arrival is influenced by the occurrence of rain. The tree diagram below was drawn to show the outcomes and probability of late arrivals when the chance of rain is 25%. 0,2 late (L) R,L 0,05 ...…. rain (R) 0,25 0,8 (a) 0,2 not late (N) (b) 0,075 0,75 0,1 late (L) ...…. no rain (D) 0,9 D,N 0,675 not late (N) ...…. [Adapted fromSASAMS 2019] Study the tree diagram and answer the questions that follow. 1.1.1 Write down the percentage of learners who arrive late if it does not rain. (2) ...…. 1.1.2 Write down the missing outcomes (a) and (b). (4) 1.1.3 Write down the probability (as a simplified common fraction) of randomly selecting a learner who arrived late for school on a rainy day. (2) 1.1.4 If the school has 1 562 learners, determine how many learners will not be late if the chance for rain is 25%. (3) 25
2. All the members of the Math Lit club at Cape High are in Grades 10, 11 or 12. The number of learners belonging to the club is given in the table below: Grade 10 Grade 11 Grade 12 TOTAL Girls 33 77 0 110 Boys 132 0 60 192 TOTAL 165 77 60 302 Use the table above to determine the probability of randomly choosing a member who is : 2.1. A boy in grade 12. (2) 2.2. A learner who is NOT in grade 10. (3) 3. The principals of Lebohang Secondary School and Jet Secondary School compared their schools' 2019 matric results. The comparisons of the passes are shown in the table below. TABLE 4: Comparison of 2019 matric passes for the two schools NUMBER OF LEARNERS PER TYPE PERCENTAGE NAME OF OF MATRIC PASSES PASS SCHOOL Degree Diploma Certificate Lebohang Sec 4 5 20 96,67 Jet Secondary 65 44 25 87,58 3.1. How many learners of Lebohang Secondary failed their matric examination in (4) 2019 3.2. If a learner from Jet Secondary School is chosen randomly from those who passed, what is the probability that this learner qualified for degree studies? (4) Give the answer as a percentage, rounded off to ONE decimal place. REFERENCE USED 1. Solutions for all Mathematical Literacy 2. Exam fever Mathematical Literacy 3. Previous Grade 12 NSC Question Papers and Memo 4. Curriculum and Assessment Policy System (CAPS) 5. 2020 Annual Teaching Plan (ATP) 26
SUMMARY In this Module, Participants learned: • The importance of Teaching of the drawing probability scale line in all the three FET grades, explanation and interpretation of the probability scales. • Use the Probability Scale to solve problems in simple events (dice/coin games, lotteries, etc.) • The importance of Teaching of the drawing probability scale line in all the three FET grades, explanation and interpretation of the probability scales. • Use the Probability Scale to solve problems in simple events (dice/coin games, lotteries, etc.) • Calculate Probability using Theoretical and Experimental Probability • Working with situations involving the probability of compound events REFLECTION • More emphasis should be placed on probability irrespective of the fact that according to CAPS should only cover 5% of the question paper. • A careful reading of tables is a crucial skill in solving mathematical problems. Teachers should give learners enough opportunities, during contact time, to practise and develop this skill. • Teachers must incorporate large numbers in their lessons, across all topics in Mathematical Literacy. • It is advisable that every assessment (formal or informal) task should involve a problem on big numbers so that learners can familiarise themselves with them. • Teachers should encourage learners to write a glossary of the different terms’ meanings, at the back of their books, as they complete each topic. • Teachers should encourage candidates to use the LOLT always during the lessons. Scenarios should be discussed and critically analysed during lessons to give learners the opportunity to think critically and develop analytical and problem-solving skills. END OF MODULE 1 27
MODULE TWO: DATA HANDLING (REPRESENTING, INTEPRETING AND ANALYSING DATA) INTRODUCTION In this module you will look at the concept of representing, interpreting and analysing data of the following graphs: • Bar graph • Histogram • Pie chart • Line graph • Box and whisker plot • Scatter plot OVERVIEW In this topic, you will work with complex projects in familiar and unfamiliar contexts SPECIFIC OBJECTIVES At the end of this module you will be able to: • Represent different data on statistical graphs • Represent and analyse graphs • identify and describe trends • identify and describe sources of bias • answer questions under investigation • identify and describe any misleading representations and data summaries TERMINOLOGY TYPE OF GRAPH DESCRIPTION Bar graph Graph represented in the form of bars representing discreet data Histogram Graph represented in the form of bars representing continuous data Pie chart It is a circular graphical representation of relative portions of data in the form of slices. Line graph It is a type of graph that displays data in the form of co-ordinates joined by line segments. Box and whisker It is a graphical representation of the minimum value, quartile 1, quartile 2, quartile plot 3 and maximum value. Scatter Plot It is the type of graph that shows the relationship between two sets of data. CONTENT You will study this module through the following unit(s): Data Handling Graphs 28
UNIT 1: REPRESENTING, INTERPRETING AND ANALYSING DATA INTRODUCTION In this unit you will look at the concept of Graphs (Box and Whisker, Pie chart, Line, Bar, Histogram, Scatter plot, etc.) with specific focus on representing, interpreting and analysing data. LEARNING OUTCOME At the end of this Unit, you should be able to: • Read values directly from the values provided on the graphs. • Draw specified graph from a given table of data. • Represent, interpret and analyse graphs. LESSON NOTES STATISTICAL CYCLE • REPRESENTATION AND INTERPRETATION OF GRAPHS The following should be considered when working with graphs: The type of graph Graph title, labels and axes Dependant and independent variables Relationship between variables Trends represented in the graph Reasons for changes in the graph Conclusion about the graph 29
• BAR GRAPH It is a graph representing discrete data which is categorised and represented by bars with spaces in between. Discrete data is the quantitative data that can be counted and has a finite number of possible values. It represents dependent and independent variable. Each bar represents a specific category. Bar graph can be represented in the following forms: • VERTICAL BAR GRAPH Number of cases is the DATE TOTAL POSITIVE CASES RECOVERIES dependent variable 28-05-20 25 937 13 451 Date recorded is the 21-05-20 19137 8950 independent variable 14-05-20 12739 5676 First bar representing positive 07-05-20 8232 3153 cases recorded on different 30-04-20 5647 2073 days and 2nd bar representing 23-04-20 3953 1473 recoveries Notes:----------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------- 30
• HORIZONTAL BAR GRAPH Number of reported covid-19 cases per province 14-05-20 Date recorded 21-05-20 28-05-20 0 2000 4000 6000 8000 10000 12000 14000 Number of cases KWAZULU NATAL EASTERN CAPE GAUTENG WESTERN CAPE Number of cases is the dependent variable Date recorded is the independent variable Each bar representing a different province • VERTICAL BAR STACKS Number of reported covid-19 cases per province 35000 30000 Number of cases 25000 20000 15000 10000 5000 0 WESTERN CAPE GAUTENG EASTERN CAPE KWAZULU NATAL Provinces 28-05-20 21-05-20 14-05-20 Number of cases is the dependent variable Province recorded is the independent variable. Bars stacked vertically to represent data collected on different days. 31
• HORIZONTAL BAR STACKS Number of reported covid-19 cases per province KWAZULU NATAL EASTERN CAPE Provinces GAUTENG WESTERN CAPE 0 5000 10000 15000 20000 25000 30000 35000 Number of cases 28-05-20 21-05-20 14-05-20 Number of cases is the dependent variable Province is the independent variable Bars stacked horizontally to represent data collected on different days • HISTOGRAM It is a graphical representation of a continuous data using bars. It represents the frequency of continuous data The data should be first grouped into interval It represents dependent and independent variable. If the frequency is zero, then there will be no bar plotted There are no gaps between bars 32
• LINE GRAPH • It is a type of graph that displays data in the form of co-ordinates joined by line segments. It compares two variables; plotted along an axis; vertical and horizontal axis. These represent dependent and independent variable. Independent variable is a variable that is not affected by other factors (e.g. Age); and it is a plotted on a horizontal axis. Dependent variable is a variable that is changed by other factors; (e.g. No of pregnant girls travelled over a period of time). In this case, distance depends on the time taken to travel. It is plotted on the vertical axis. • PIE CHART It is a circular graphical representation of relative portions of data in the form of slices. To represent the data in a pie chart, each slice is represented as a proportion of 3600 and represented as a percentage. The total percentage of all slices of a pie chart is 100% If 1 portion is unknown, then the unknown portion should be subtracted from 100% 33
NUMBER OF POSITIVE CASES AT 5 TOP INFECTED PROVINCES ON THE 28TH OF MAY 2020 PROVINCE POSITIVE CASES WESTERN CAPE 12888 GAUTENG 3167 EASTERN CAPE 3047 KWAZULU NATAL 2186 FREE STATE 221 PROVINCES WESTERN EASTERN KWAZULU GAUTENG FREE CAPE CAPE NATAL STATE RECOVERIES 9157 1700 1180 2019 123 34
• BOX AND WHISKER PLOT It is a graphical representation of quartiles, minimum and maximum values The graph has 4 segments (quarters) Each segment has 25% of data If the segment is bigger, it implies that data is spread apart If the segment is small, it implies that values are close to each other 25% 25% 25% 25% It is a graphical representation of the minimum value, quartile 1, quartile 2, quartile 3 and maximum value. Each segment represents 25% The whiskers represent minimum and maximum values The boxes represent quartile 1, quartile 2 and quartile 3 The data that lies below the lower quartile (Q1) is 25% The data that lies falls below the median (Q2) is 50% The data that lies below the upper quartile (Q3) is 75% The data that lies above the lower quartile (Q1) is 75% The data that lies above the median (Q2) is 50% The data that lies above the upper quartile (Q3) is 25 N.B learners are not expected to draw the box and whisker plot 35
• SCATTER PLOT It is the type of graph that shows the relationship between two sets of data. It shows one variable on the horizontal axis and another one on the vertical axis. The pattern of the scatter points will show whether there is correlation between the variables or not. The closer the points, the stronger the correlation. Take Note: Learners are not expected to do line of best fit Positive correlation [From left to right] When one variable increases, the other one increases too’ 6 5 4 3 Series1 2 1 0 3.28 3.3 3.32 3.34 3.36 3.38 3.4 3.42 3.44 3.46 EASTERN CAPE ACTIVE CASES VERSUS RECOVERIES 2000 1800 1600 1400 ACTIVE CASES 1200 1000 800 600 400 200 0 0 200 400 600 800 1000 1200 1400 1600 1800 RECOVERIES 36
Negative correlation [from right to left] When one variable increases, the other one decreases ACHIEVED BACHELOR 100000 90000 80000 70000 60000 50000 40000 30000 20000 10000 0 0 5 10 15 20 25 30 35 No correlation There is no relationship between the variables, the pattern is irregular 3.9 3.8 3.7 3.6 3.5 Seri… 3.4 3.3 3.2 0 2 4 6 8 10 12 14 37
• MISLEADING GRAPHS Data represented is not obvious but needs further analysis Data is usually represented in a different format from the usual one. The following graphs represent the data on the table below DATE TOTAL POSITIVE CASES 28-05-20 25 937 21-05-20 19137 14-05-20 12739 07-05-20 8232 30-04-20 5647 23-04-20 3953 Example 1: TOTAL POSITIVE CASES 25 937 100% 80% 60% 40% 20% 0% 21-05-20 14-05-20 7/5/2020 30-04-20 23-04-20 TOTAL POSITIVE CASES 25 937 25000 20000 15000 10000 5000 0 21-05-20 14-05-20 7/5/2020 30-04-20 23-04-20 The 2 graphs are representing the same information, 1 graph represented in percentage form and 1 represented in actual numbers. 38
Example 2: 28-05-20 WESTERN CAPE GAUTENG EASTERN CAPE KWAZULU NATAL FREE STATE LIMPOPO NORTH WEST MPUMALANGA NORTHERN CAPE UNALLOCATED TOTAL PROVINCE WC GP EC KZN FS LP NW MP NC POSITIVE 16 3167 3047 3040 221 141 138 106 48 CASES 893 • Smallest values do not appear on the graph Notes:----------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------- 39
ACTIVITY 2.1.1 (Q2.1 – 2.3) Instructions • You should form groups of 4 – 5 • Duration: 15 minutes • Refer to the questions provided • Task 1: Answer all the questions Task 2: Prepare a marking grid for the activities below Task 3: explain mark allocation for the solutions • These questions are intended to prompt you to consolidate the unit and possible ways in which this section can be taught • Report Back and Discussion • Resources: Training manual, Note pad, Pen and Calculator 2.1. The table below shows the statistics on Medical aid scheme coverage from 2008 to 2017. During this period, the number of individuals who were covered by a medical aid scheme increased from 8,1 million to 9,5 million. Use the table below to answer the questions that follow. 2.1.1. State why the data for people covered by medical aid is regarded as discrete. (2) 2.1.2. Name the type of data collecting instrument used to collect this data. (2) 2.1.3. Write down in words, the number of people not covered by a medical aid scheme in 2011. (2) 2.1.4. Calculate the median number of the unspecified people. (3) 2.1.5. Write down the ratio of the total population to the number of people covered by a medical aid scheme in 2008, in the form … : 1 (3) 2.1.6. Determine the probability (as a percentage) of randomly selecting the number of people who do not know about medical aid scheme coverage in 2013. (3) 2.1.7. Draw a broken line graph on the ANSWER SHEET to represent the percentage of people covered by a medical aid scheme from 2008 - 2017. (5) [20] 40
TABLE 1 : Medical aid coverage in thousands from 2008 – 2017 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 Number of people 8 057 8 502 8 967 8 312 9 157 9 608 9 470 9 307 9 447 9 475 covered by a medical aid scheme Number of people not 41 26 41 28 41 60 43 01 42 81 43 30 43 94 45 06 45 46 covered by a 6 4 6 3 9 0 6 5 646 654 medical aid scheme Subtotal 49 49 78 50 57 51 32 51 97 52 90 53 41 54 37 55 09 56 323 6 3 5 6 8 6 2 3 129 Percentage of people covered by a 16,3 17,1 17,7 16,2 17,5 18,2 17,7 17,1 17,1 16,9 medical aid scheme Number of people who do not know 101 19 23 0 58 36 46 71 53 24 about medical aid scheme coverage Unspecified 56 347 254 249 291 161 451 308 474 369 people Total population 49 48 50 15 50 85 51 57 52 32 53 10 53 91 54 75 55 62 56 0 2 0 4 5 5 2 1 0 522 [Adapted from www.statssa.gov.za] 41
2.2. The graphs below show two types of graphs. The pie chart shows the percentages of different perfume sales for March to May. The incomplete bar graph shows ONLY the number of Fantasy perfumes sold for March and April. A total of 12 000 bottles of perfume was sold during this period 2.2.1. Calculate the number of bottles of Fantasy perfume during period. (5) 2.2.2. Pat claimed that more Fantasy perfumes were sold in May than in March. Verify his claim. (3) 42
PERCENTAGE OF PERFUMES SOLD FROM MARCH - MAY BLACK OPIUM OLYMPEA 19% 23% BUTTERFLY BLACK XS 18% ANGEL 8% FANTASY D% D represent unknown percentages of perfumes indicated in the pie Number of perfumes sold in a term 800 700 Number of Perfumes Sold 600 500 400 689 681 300 550 200 100 0 March April May term in months 43
2.3. Regal plans to tour the country over holidays. He then investigated temperature changes to help him decide on the suitable time to visit different areas. The table below shows the average temperatures recorded for 11 cities. 2.3.1. Why are temperatures regarded as average temperatures? (2) 2.3.2. Identify the town and season on the table with lowest average temperature. (2) 2.3.3. A line graph representing the summer maximum temperatures has been drawn, on the same system of axis, draw the graph representing summer minimum temperatures. (5) MAXIMUM AND MINIMUM SUMMER AND WINTER TEMPERATURES CITIES SUMMER WINTER MAXIMUM MINIMUM MAXIMUM MINIMUM Bloemfontein 31 15 17 -2 Cape Town 26 16 18 7 Durban 28 21 23 11 East London 26 18 21 10 Johannesburg 26 15 17 4 Kimberley 33 18 19 3 Polokwane 28 17 20 4 Port Elizabeth 25 18 20 9 Pretoria 29 18 20 5 Richards Bay 29 21 23 12 Upington 36 20 21 4 Min and max summer temperatures of 11 cities 25 20 Temperatures in C 15 10 5 0 Cities SUMMER MINIMUM 44
ACTIVITY 2.1.2 (Q2.4 – 2.7) Instructions • You should form groups of 4 – 5 • Duration: 15 minutes • Refer to the questions provided • Task 1: Answer all the questions Task 2: Decide on marks to be allocated to these questions Task 3: explain mark allocation for the solutions • These questions are intended to prompt you to consolidate the unit and possible ways in which this section can be taught • Report Back and Discussion • Resources: Training manual, Note pad, Pen and Calculator 4.4. On the first of March 2020 the first case of Covid -19 was reported in South Africa and the 27th of March 2020, the first Covid-19 death was reported. Since then the death cases of covid-19 were recorded. The following graph above represent the death cases of covid-19 deaths recorded on the 7th of June 2020. Study the graph and Use it to answer the following questions 2.4.1. Calculate the range of the death cases. 2.4.2. Show that the median of the death cases is ---------------- 2.4.3. Determine the interquartile range of the set of data COVID-19 DEATH RATE PER AGE CATEGORY ON 7 JUNE 2020 300 290 280 270 260 250 240 230 220 210 200 190 NUMBER OF CASES 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 0-9 19-Oct 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90-99 AGES 2.5. Western Cape has recorded the highest number of Covid-19 cases. The active cases and infection cases for Western Cape were recorded for the whole month of May and the graph was drawn from the data collected. Study the graph below and answer the following questions: 45
12000 10000 8000 RECOVERY CASES 6000 4000 2000 0 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 INFECTIONS CASES 2.5.1. Write a suitable title for this graph 2.5.2. Which method was used to collect this set of data? 2.5.3 Explain the trend represented by this graph. 2.5.4. Estimate the maximum cases of infection cases represented on this graph. 2.6. Infection cases were recorded for the 2 provinces that had high infection rate for the month of May 2020. Note that: infection cases recorded is accumulative, the data is added to the recorded one. The data collected was summarized into 5 number summary. Study the table and graph below and answer the following questions. GAUTENG EASTERN CAPE MINIMUM 1507 691 Q1 1845.5 959 Q2 2210 1662 Q3 2703 2808 MAXIMUM 3583 3583 2.6.1. Which province shows the high rate of infection? Explain. 2.6.2. Write the median of Gauteng. 2.6.3. Determine the interquartile range of Eastern Cape. 2.6.4. What percentage of data lies above the upper quartile? 46
2.7. The data of the 4 provinces with highest reported cases was recorded and represented on the following graph. Study the graph and answer the following questions: KWAZULU NATAL EASTERN CAPE PROVINCES GAUTENG WESTERN CAPE 0 5000 10000 15000 20000 25000 30000 35000 RECORDED CASES 14-05-20 21-05-20 28-05-20 2.7.1. Which instrument was used to collect this data? 2.7.2. Is number of cases the dependent or independent variable? 2.7.3. How many cases were recorded at Western Cape on the 21st of May 2020? 2.7.4. Determine the estimated difference between the cases recorded on the 28th of May 2020 between Eastern Cape and Kwazulu-Natal? 47
RESOURCES FOR MODULE 3 • Mathematical Literacy NSC Exam 2018 Diagnostic Report • Statistics SA reports • DBE question papers • Provincial question papers SUMMARY In this module, you: • Represented and interpreted box and whisker plots, pie charts, bar, histogram, line graphs and scatter plot • Determined trends in the data and use these to answer questions and make predictions based on the data • Read and analyse information from the graph. REFLECTION • A careful reading of tables/graphs is a crucial skill in solving mathematical problems. Teachers should give learners enough opportunities, during contact time, to practise and develop this skill. • Teachers must incorporate large numbers in their lessons, across all topics in Mathematical Literacy. It is advisable that every assessment (formal or informal) task should involve a problem on big numbers so that learners can familiarise themselves with them. • Learners should be taught how a trend should be described. • Teachers should teach learners first to make sense of the information before attempting the questions. END OF MODULE 2 48
MODULE 3: PACKAGING AND ASSEMBLING INTRODUCTION In this module you will look at the concept of Models with specific focus on: • Packaging of material in boxes and tilling • Instructions and assembly diagrams OVERVIEW In this topic, you will work with complex projects in familiar and unfamiliar contexts SPECIFIC OBJECTIVES At the end of you will be able to work with 2-dimensional and 3-dimensional models in order to: • Investigate, by making use of calculations the most cost effective and most appropriate way to package cans/boxes for optimal use of space. • Understand different packaging arrangements. • Determine quantities of materials used for different packaging arrangements. TERMINOLOGY Model 2-Dimensional representation of a 3-Dimensional object. 2-D Models A model of a real object drawn to scale that shows only length and breadth of the real object. 3-D Models A model of a real object drawn to scale that shows length, width and height of the real object. Packaging Process of packing items into a limited space, like a box, suitcase, cupboard, etc. Packaging Models showing different ways in which items are arranged. arrangement Most appropriate way Packaging arrangement that uses the minimum amount of space of packaging to package the maximum number of items. Most cost-effective Packaging arrangement that uses the minimum amount of way of packaging material to package the maximum number of items. Scale Determines how many times smaller an object shown on a plan or map is that its actual size Number scale A number scale such as 1 : 50 000 means that 1 unit on the map represent 50 000 units in real life Scale drawing A diagram of a real-life object drawn in proportion. Floor plan Shows the design and dimensions of the inside of a building, from a top view. 49
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