FET Phase Grade 11 • Study Guide
Mathematical Literacy CAPS IEB
Mathematical Literacy Study guide
CAPS aligned
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2 111 - E - M A L - S G 0 1
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Grade 11
Study Guide G11 ~ Mathematical Literacy
CONTENTS LESSON ELEMENTS.......................................................................................................... 5 PREFACE............................................................................................................................ 6 ASSESSMENT REQUIREMENTS ...................................................................................... 6 THE ASSESSMENT PROGRAMME ................................................................................... 7 YEAR PLAN ........................................................................................................................ 7 SUGGESTED TIME TO SPEND ON EACH UNIT ............................................................... 8 UNIT 1: REVIEW OF BASIC SKILLS ............................................................................... 10 Part 1: Rounding............................................................................................................ 10 Activity 1 .................................................................................................................. 11 Part 2: Rounding to the nearest 10, 100 and 1000 ........................................................ 11 Activity 2 .................................................................................................................. 12 Part 3: Scientific notation ............................................................................................... 12 Activity 3 .................................................................................................................. 13 Part 4: Percentages ....................................................................................................... 14 Activity 4 .................................................................................................................. 18 Part 5: Ratios ................................................................................................................. 19 Activity 5 .................................................................................................................. 20 Part 6: Rate ................................................................................................................... 20 Activity 6 .................................................................................................................. 21 Part 7: Proportion .......................................................................................................... 21 Activity 7 .................................................................................................................. 22 UNIT 2: PATTERNS, RELATIONSHIPS AND REPRESENTATIONS .............................. 23 Lesson 1: Graphs that tell a story .................................................................................. 23 Activity 8 .................................................................................................................. 25 Lesson 2: Patterns with constant relationships .............................................................. 26 Lesson 3: Linear relationships ....................................................................................... 29 Activity 9 .................................................................................................................. 30 Activity 10 ................................................................................................................ 31 Lesson 4: Indirect/inverse proportion ............................................................................. 32 Activity 11 ................................................................................................................ 33 Lesson 5: Step graph .................................................................................................... 34 Activity 12 ................................................................................................................ 36 Lesson 6: Relationships with a constant ratio ................................................................ 37 Activity 13 ................................................................................................................ 38 Lesson 7: Combination relationships ............................................................................. 39 Activity 14 ................................................................................................................ 40 Lesson 8: Two relationships on a set of axes ................................................................ 42 Activity 15 ................................................................................................................ 43
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Study Guide G11 ~ Mathematical Literacy
UNIT 3: MEASUREMENT – CONVERSIONS AND TIME................................................. 45 Lesson 9: Revision – Measurement .............................................................................. 45 Activity 16 ................................................................................................................ 47 Lesson 10: Time ............................................................................................................ 48 Activity 17a .............................................................................................................. 48 Activity 17b .............................................................................................................. 50 Lesson 11: Temperature................................................................................................ 50 Activity 18 ................................................................................................................ 51 Lesson 12: Timetables................................................................................................... 51 UNIT 4: FINANCES – FINANCIAL DOCUMENTS, TARIFF SYSTEMS AND BREAKEVEN ANALYSIS .............................................................................................................. 54 Revision activity 1.................................................................................................... 55 Lesson 13: Quotations ................................................................................................... 57 Activity 19 ................................................................................................................ 58 Lesson 14: Invoices ....................................................................................................... 60 Activity 20 ................................................................................................................ 61 Lesson 15: Budgets, and income and expenditure statements ..................................... 62 Lesson 16: Cost price and selling price ......................................................................... 63 Activity 21 ................................................................................................................ 64 Lesson 17: Break-even analysis .................................................................................... 65 Activity 22 ................................................................................................................ 67 Activity 23 ................................................................................................................ 68 Lesson 18: Tariff systems .............................................................................................. 69 Activity 24 ................................................................................................................ 70 Activity 25 ................................................................................................................ 72 UNIT 5: FINANCES – INTEREST, BANKING AND INFLATION ...................................... 74 Lesson 19: Hire purchase .............................................................................................. 74 Activity 26 ................................................................................................................ 75 Lesson 20: Loans .......................................................................................................... 75 Activity 27 ................................................................................................................ 76 Lesson 21: Interest ........................................................................................................ 77 Activity 28 ................................................................................................................ 81 Lesson 22: Banking ....................................................................................................... 82 Activity 29 ................................................................................................................ 85 Activity 30 ................................................................................................................ 87 Lesson 23: Inflation ....................................................................................................... 88 Activity 31 ................................................................................................................ 89 Activity 32 ................................................................................................................ 91 UNIT 6: MEASUREMENT – LENGTH, MASS, VOLUME AND TEMPERATURE ............ 92 Revision activity 2.................................................................................................... 92 Lesson 24: Length and distance .................................................................................... 93 Activity 33 ................................................................................................................ 93 © Optimi
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Study Guide G11 ~ Mathematical Literacy
Lesson 25: Mass (Weight) ............................................................................................. 94 Activity 34 ................................................................................................................ 94 Activity 35 ................................................................................................................ 97 Lesson 26: Volume ........................................................................................................ 99 Activity 36 .............................................................................................................. 100 Activity 37 .............................................................................................................. 100 Lesson 27: Temperature.............................................................................................. 101 Activity 38 .............................................................................................................. 101 UNIT 7: MAPWORK ........................................................................................................ 102 Revision activity 3.................................................................................................. 103 Lesson 28: National road maps ................................................................................... 105 Activity 39 .............................................................................................................. 107 Lesson 29: Strip maps ................................................................................................. 108 Activity 40 .............................................................................................................. 109 Lesson 30: Street maps ............................................................................................... 109 Activity 41 .............................................................................................................. 113 Lesson 31: Indexes ..................................................................................................... 114 Lesson 32: Residential or housing estate maps .......................................................... 115 Activity 42 .............................................................................................................. 116 Lesson 33: Scales ....................................................................................................... 117 Activity 43 .............................................................................................................. 118 Lesson 34: Profile and elevation maps ........................................................................ 120 Application activity 1 .............................................................................................. 121 UNIT 8: MEASUREMENT – PERIMETER/CIRCUMFERENCE, AREA AND VOLUME . 123 Lesson 35: Perimeter .................................................................................................. 125 Activity 44 .............................................................................................................. 127 Lesson 36: Area and total surface area ....................................................................... 128 Activity 45 .............................................................................................................. 132 Lesson 37: Volume ...................................................................................................... 133 Activity 46 .............................................................................................................. 135 UNIT 9: FLOOR AND ELEVATION PLANS, MODELS AND SCALE DRAWINGS ....... 139 Lesson 38: Floor plans ................................................................................................ 139 Activity 47 .............................................................................................................. 142 Activity 48 .............................................................................................................. 143 Lesson 39: Elevations.................................................................................................. 144 Activity 49 .............................................................................................................. 146 Lesson 40: Models and instructions ............................................................................ 147 Activity 50 .............................................................................................................. 148 Lesson 41: Scale drawings .......................................................................................... 148
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Study Guide G11 ~ Mathematical Literacy
UNIT 10: FINANCES – VAT AND UNEMPLOYMENT INSURANCE FUND (UIF) ......... 151 Lesson 42: Value added tax (VAT) .............................................................................. 151 Activity 51 .............................................................................................................. 152 Lesson 43: Unemployment insurance fund (UIF) ........................................................ 152 Activity 52 .............................................................................................................. 153 UNIT 11: PROBABILITY ................................................................................................. 154 Lesson 44: Probability scale ........................................................................................ 154 Lesson 45: Expression of probability ........................................................................... 155 Activity 53 .............................................................................................................. 156 Activity 54 .............................................................................................................. 158 Activity 55 .............................................................................................................. 160 Lesson 46: Prediction of probability ............................................................................. 161 Activity 56 .............................................................................................................. 162 UNIT 12: FINANCES – EXCHANGE RATES.................................................................. 164 Lesson 47: Conversion of currency values .................................................................. 164 Activity 57 .............................................................................................................. 165 Lesson 48: Buying and selling currency ...................................................................... 166 Activity 58 .............................................................................................................. 166 UNIT 13: DATA HANDLING ........................................................................................... 167 Lesson 49: Interpreting statistics ................................................................................. 167 Activity 59 .............................................................................................................. 170 Activity 60 .............................................................................................................. 170 Activity 61 .............................................................................................................. 176 Activity 62 .............................................................................................................. 177 Activity 63 .............................................................................................................. 183 BIBLIOGRAPHY ............................................................................................................. 187 ADDENDUM A: TERMINOLOGY.................................................................................... 192 ADDENDUM B: FORMULAE .......................................................................................... 196
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Study Guide G11 ~ Mathematical Literacy
Lesson elements LEARNING OBJECTIVES
What you should know at the end of the lesson. Taken from CAPS.
IMPORTANT TERMINOLOGY
New terminology to improve understanding of the subject as part of the lesson.
IMPORTANT
A summary or explanation of the main concepts of a lesson.
ACTIVITY
Formative assessment to test your progress and knowledge at the end of each lesson.
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Study Guide G11 ~ Mathematical Literacy
Preface Recommended books Any additional book may be used with this study guide. It is always a good idea to refer to other textbooks to develop a broader perspective on the subject. • •
The Answer Series: Grade 11 Mathematical Literacy 3 in 1 Mathematical Literacy for the Classroom Grade 11 Learner’s Book
Assessment requirements Note that there are constant references to TL1, TL2, TL3 and TL4 throughout the facilitator guide. These are the thinking levels required to answer the specific question asked. The thinking levels represent the following skills Thinking level 1 Knowing Thinking level 2 Applying routine procedures in familiar contexts Thinking level 3 Applying multi-step procedures in a variety of contexts Thinking level 4 Reasoning and reflecting
• • • •
When tasks, investigations and especially tests and examinations are set, the guidelines below are used to allocate marks to a specific thinking level. Mark distribution according to the thinking levels Percentage of marks allocated
Thinking levels Thinking level 1: Knowing
30% (±5%)
Thinking level 2: Applying routine procedures in familiar contexts Thinking level 3: Applying multi-step procedures in a variety of contexts Thinking level 4: Reasoning and reflecting
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30% (±5%) 20% (±5%) 20% (±5%)
Study Guide G11 ~ Mathematical Literacy
The assessment programme Refer to Impaq’s online platform for assessment tasks, examinations and the assessment plan.
Year plan Unit
Topic
Term
1
Review of basic skills
2
Patterns, relationships and representations
3
Measurement: Conversions and time
4
Finances: Financial documents, tariff systems and break-even analysis
5
Finances: Interest, banking and inflation
6
Measurement: Length, mass, volume and temperature
7
Mapwork
8
Measurement: Perimeter, area and volume
9
Floor and elevation plans, models and scale drawings
10
Finances: VAT and unemployment insurance fund (UIF)
11
Probability
12
Finances: Exchange rates
13
Data handling
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Term 1 January – March
Term 2 April – June
Term 3 July – September
Term 4 October – November
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Study Guide G11 ~ Mathematical Literacy
Suggested time to spend on each unit (according to CAPS) Week
Term 1
1
2
3
4
5
6
7
8
9
6
7
8
9
Contexts focusing on Patterns, relationships and representations Topics
Assessment
Contexts focusing on Measurement (conversions and time) Contexts focusing on Finance (financial documents, tariff systems and break-even analysis) Assignment/investigation Control test (covering measurement and finance, integrated with numbers and patterns concepts) Week
Term 2
1
2
3
4
5
Contexts focusing on Finance (interest, banking, inflation) Topics
Contexts focusing on Measurement (length, mass, volume, temperature) Contexts focusing on Maps, plans and other representations of the physical world (scale and mapwork) Revision
Assessment
Assignment/investigation June examination (2 papers, 1½ hours each, 75 marks each; covering Finance, Measurement and Maps, integrated with Numbers and Patterns concepts) Week
Term 3
1
2
3
4
5
6
7
8
9
8
9
Contexts focusing on Measurement (perimeter, area and volume) Topics
Contexts focusing on Maps, plans and other representations of the physical world (models and plans) Contexts focusing on Finance (taxation) Contexts focusing on Probability
Assessment
Assignment/investigation Control test (covering Measurement, Models and plans, Finance and Probability, integrated with Numbers and Patterns concepts) Week
Term 4
1
2
3
4
5
6
Contexts focusing on Finance (exchange rates) Topics
Contexts focusing on Data handling Revision
Assessment
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Assignment/investigation November examination (2 papers, 2 hours each, 100 marks each; covering all topics in the curriculum)
8
7
Study Guide G11 ~ Mathematical Literacy
Time allocation per topic serves as a guideline only and it can be adjusted to your own pace. Bear in mind that you must first complete the relevant lessons before you will be allowed to take a test or the relevant examination. You need to spend 4,5 hours per week on Mathematical Literacy. Take note that this time allocation per week excludes all activities, assessments and examinations; it gives an indication only of the time that must be spent on theoretical aspects. If you tend to work more slowly, the necessary adjustments must be made to ensure that you still master all the work in time. Proposed instructional time per week: Subject
Time allocation per week (hours)
Home Language
4,5
First Additional Language
4,5
Mathematical Literacy
4,5
Life Orientation
2
Three optional subjects: Optional subject 1 Optional subject 2 Optional subject 3
4 4 4
Total
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27,5
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Study Guide G11 ~ Mathematical Literacy
Unit
1
UNIT 1: REVIEW OF BASIC SKILLS Learning objectives At the end of the unit, you must be able to: • apply basic mathematical skills • understand rounding, scientific notation, percentages, ratios and proportion, and do calculations
Part 1: Rounding When we use rounding, we need to be aware of the context of the problem. Context is the circumstances in which something occurs. This will determine whether we round up or round down as well as to how many decimal places we need to round up or down. Important When we round numbers, we cannot simply say the original number is equal to the rounded number. For this reason, we don’t use the =, but the ≈.
≈ means almost equal to OR more or less the same (rounded OR estimated)
EXAMPLES
Round the following numbers to the number of decimal places indicated in brackets: 1.1
8,25632 (2 decimal places) Solution: ≈ 8,26
Note: Look at the third decimal. A number less than 4 makes no difference to the second decimal. If the third decimal is 5 or higher, the second decimal should then be greater, therefore the 5 changed to a 6. 1.2
1.3
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16,89432 (3 decimal places) Solution: ≈ 16,894 428,899 (2 decimal places) Solution: ≈ 428,90
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Study Guide G11 ~ Mathematical Literacy
Unit
1
Activity 1 Round the numbers to TWO decimal places: 1. 4,3568 2. 18,3578 3. 1,9999 4. 45,4386 5. 834,8765 6. 29,99124 7. 492,6324 8. 9,3334 9. 16,448 10. 74,97553
Part 2: Rounding to the nearest 10, 100 and 1 000 Use place values to help you when rounding numbers. For example, if you must round numbers to the nearest 10, you must look at the Units column. Thousands Th
Hundreds H
Tens T
Units U
Rounding to the nearest 10 Numbers smaller than 5 in the Units column are rounded to the previous ten. Numbers larger than 5 in the Units column are rounded to the next ten. Rounding to the nearest 100 Numbers smaller than 5 in the Tens column are rounded to the previous 100. Numbers larger than 5 in the Tens column are rounded to the next 100. Rounding to the nearest 1 000 Numbers smaller than 5 in the Hundreds column are rounded to the previous 1 000. Numbers larger than 5 in the Hundreds column are rounded to the next 1 000. EXAMPLES 2.1
Round the numbers to the nearest 10: 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5
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75 ≈ 80 17 ≈ 20 163 ≈ 160 689 ≈ 690 3≈0
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Study Guide G11 ~ Mathematical Literacy
2.2
1
Round the numbers to the nearest 100: 2.2.1 2.2.2 2.2.3
2.3
Unit
75 ≈ 100 2 082 ≈ 2 100 43 ≈ 0
Round the numbers to the nearest 1 000: 2.3.1 2.3.2 2.3.3
899 ≈ 1 000 3 956 ≈ 4 000 282 ≈ 0
Activity 2
Round the numbers to the nearest 10, 100 or 1 000: 1. 2. 3. 4. 5.
48 (nearest 10) 245 (nearest 100) 2 367 (nearest 1 000) 4 356 (nearest 10, 100 and 1 000) 3 835 (nearest 10, 100 and 1 000)
Part 3: Scientific notation Important One million: 1 000 000 (6 zeros) One billion: 1 000 000 000 (9 zeros) One trillion: 1 000 000 000 000 (12 zeros) We use scientific notation to write very large or very small numbers in a way that makes it easy to read. The number is written using a number between 1 and 10, which is then multiplied by a power of 10. For large numbers, put a decimal comma after the first digit so the number is between 1 and 10. Count the number of places from the decimal comma to the end of the number – this will tell you what power of 10 to use. Remove any zeros and write the number multiplied by 10 to the power of 𝑥𝑥 (𝑥𝑥 being the number of places from the decimal comma to the end of the number). For very small numbers, count the number of places from the decimal comma until you have a number between 1 and 10. Remove any zeros in front of the decimal comma and write the number multiplied by 10 to the power of negative 𝑥𝑥. © Optimi
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Study Guide G11 ~ Mathematical Literacy
Unit
EXAMPLES Write 81 500 in scientific notation.
•
81500 4
3
2
1
= 8,15 × 10
4
Write 0,0043 in scientific notation.
•
0, 0 0 4 3 1
2
3
= 4,3 × 10-3
Decimal notation 34 000 000 734 000 000 000 0,000 001 4 0,000 000 000 03 674 965,38
Activity 3
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Scientific notation 3,4 × 107 7,34 × 1011 1,4 × 10-6 3,0 × 10-11 6,7496538 × 105
1.
Write the numbers in scientific notation: 1.1 2 700 000 1.2 4 500 1.3 0,000 000 56 1.4 0,000 18
2.
Write the following as an ordinary number: 2.1 6,4 × 105 2.2 7,23 × 109 2.3 4,3 × 10-6 2.4 7,5 × 10-9 13
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Study Guide G11 ~ Mathematical Literacy
Unit
1
Part 4: Percentages A percentage is a way of expressing the parts of a whole (the numerator) as a value out of 100. It is always written with the % symbol directly after the number. A percentage can also be written as a fraction. The denominator will always be 100. The term ‘per cent’ derives from the Latin word per centum, which means ‘per hundred’ or ‘for every hundred’. Percentages can be expressed as a common fraction or as a decimal number, 83
e.g. 83% = 100 OR = 0,83.
4.1 Calculation of percentage To calculate a percentage of an amount, write the per cent as a fraction and multiply by the amount. per cent × amount 100 EXAMPLES 4.1.1
Calculate 25% of R5 000,00. Note: The word ‘of’ means multiplication. 25
100
4.1.2
× R5 000 = R1 250,00
Sequence on calculator: (25 ÷ 100) × 5 000 = 28
Thandi gets 40 in a Mathematical Literacy test.
What is her percentage for the test? 28 40
× 100 = 70%
Sequence on calculator: (28 ÷ 40) × 100 =
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Study Guide G11 ~ Mathematical Literacy
Unit
1
4.2 Determining new values for given percentages Sometimes we need to calculate the new price of an item or product after it has been increased by a certain percentage, e.g. VAT (Value added tax) which is levied on the original price. EXAMPLES 4.2.1
A packet of biscuits costs R15,30 without VAT. How much will you pay after VAT is added? (In South Africa, the VAT rate is currently 15%) 15
100
× R16,00 = R2,40
OR
R16,00 × 115% = R18,40
R16,00 + R2,40 = R18,40
You will pay R18,40 after VAT is added. 4.2.2
A vendor increases his prices by 8%. If the original price is R45,99, what will the price be after the increase? 8
100
× R45,99 = R3,68
OR
R45,99 × 108% = R49,67
R45,99 + R3,68 = R49,67 The new price is R49,67. 4.2.3
A shoe shop has a sale with 25% off all shoes. What will you pay for a pair of shoes that was originally priced at R645,00?
25
100
× R645,00 = R163,50
OR
R645,00 – R163,50 = R490,50
100% – 25% = 75% 75% × R645,00 = R490,50
You will now pay R490,50 for the pair of shoes. © Optimi
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Study Guide G11 ~ Mathematical Literacy
4.2.4
Unit
1
The petrol price decreases with 2,5%. How much will motorists now pay for a litre of petrol if the price was R14,40/ℓ? 2,5
100
OR
× R14,40 = R0,36
100% – 2,5% = 97,5%
R14,40 – R0,36 = R14,04 Motorists will now pay R14,04/ℓ.
97,5% × R14,40 = R14,04
4.3 Determining the initial value when new values and percentages are given Sometimes we need to calculate the original price of an item or product before it has been decreased by a certain percentage, e.g. discount on products during a sale. Important You must determine the original value – the previous value, before it was increased or decreased. EXAMPLES 4.3.1
Dishwashing liquid costs R23,10, VAT included. VAT is 15%. What was the dishwashing liquid’s price before VAT was added? P: price without VAT P × 115% = R23,10
P=
23,10
115%
(115% = 100% + 15%) OR
R23,10 ÷ 1,15
P = R20,09 The dishwashing liquid’s original price was R20,09. 4.3.2
A butchery increases their prices by 9%, which means that biltong now costs R195,10/kg. What was the original price before the increase? P: original price P × 109% = R195,10 P = 195,10 ÷ 109% = R178,99 The biltong’s original price was R178,99.
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Study Guide G11 ~ Mathematical Literacy
4.3.3
Unit
1
A computer shop has a sale with all items at 25% discount. If a printer’s selling price is currently at R1 538,40, what was the original price before the discount? P: original price (100% – 25% = 75%) P × 75% = R1 538,40 P = R1 538,40 ÷ 75% = R2 051,20 The printer’s original price was R2 051,20.
4.4 Percentage increase and decrease We can use the following formula to determine percentage increase or decrease:
Percentage increase/decrease =
current value − previous value previous value
× 100
EXAMPLES 4.4.1
4.4.2
In 2019, a school had 1 538 learners and in 2020, there were 1 645 learners. By what percentage has the enrolment increased? Percentage increase =
𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜 𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯 − 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯
Percentage increase =
𝟏𝟏 𝟔𝟔𝟔𝟔𝟔𝟔 − 𝟏𝟏 𝟓𝟓𝟓𝟓𝟓𝟓
𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯
𝟏𝟏 𝟓𝟓𝟓𝟓𝟓𝟓
× 𝟏𝟏𝟏𝟏𝟏𝟏
× 𝟏𝟏𝟏𝟏𝟏𝟏
= 6,96% ≈ 7%
In 2018, a glass factory produced 796 faulty glasses and in 2019, only 675. By what percentage did the faulty glasses decrease? Percentage decrease =
𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜 𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯 − 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯
Percentage decrease =
𝟔𝟔𝟔𝟔𝟔𝟔 − 𝟕𝟕𝟕𝟕𝟕𝟕
𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯
𝟕𝟕𝟕𝟕𝟕𝟕
= -15,20%
× 𝟏𝟏𝟏𝟏𝟏𝟏
× 𝟏𝟏𝟏𝟏𝟏𝟏
The negative value indicates a decrease. In this example, the percentage of faulty glasses decreased by 15,20%.
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Study Guide G11 ~ Mathematical Literacy
Unit
1
Activity 4 1.
Calculate: 1.1 45% of R2 500 1.2 33⅓% of R8 500
2.
Thabo gets 75 in a Life Sciences test. What is his percentage?
38
Round your answer to the nearest percentage. 3.
A shop assistant forgot to add VAT when she calculated the price of several items. The prices of the following items must be recalculated: • A pair of sneakers at R189,99 • A pair of jeans at R539,99 • A T-shirt at R125,90 • A scarf at R85,99 Calculate the correct new price of each item. VAT is 15%.
4.
Peter buys a second-hand bakkie at R185 000, 15% VAT included. What would the bakkie have cost before VAT was added?
5.
The petrol price increases by 2,3% and motorists now pay R16,30/ℓ. How much did motorists pay for a litre of petrol before the increase?
6.
A Blu-ray player that previously cost R1 235,00 now costs R855,00. Calculate the percentage discount to the nearest whole number.
7.
What percentage is 36 minutes of an hour?
8.
After an inflation increase of 8,5%, a bag of groceries now costs R685,00. What was the cost of the same bag of groceries before the increase?
9.
A company has 840 employees, of which 195 are female. What percentage of the employees are male? Round your answer to the nearest whole number.
10. Mandla earns R18 845,00 per month. He previously earned R16 750,00. The average salary increase at his company was 10%. Show by calculation that Mandla’s new salary is in line with the company average.
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Study Guide G11 ~ Mathematical Literacy
Unit
1
Part 5: Ratios A ratio is the comparison of values that can be measured in the same unit. You need to ensure that the units are the same because values with different units cannot be expressed as a ratio. •
2
Notation: 2 to 5 OR 2 : 5 OR 5 (We read it as two to five)
•
Always write ratios in its simplest form, e.g. 10 : 25 → 2 : 5 (divide each number by 5)
•
Increase R12,50 in the ratio 3 : 5 5
•
3
× R12,50 = R20,83
(numerator highest value)
× R12,50 = R7,50
(numerator lowest value)
Decrease R12,50 in the ratio 3 : 5 3 5
Ratios mostly consist of two terms, but they can also have more than two terms. The order of the terms in a ratio is important. A ratio of 1 : 6 is NOT the same as a ratio of 6 : 1. DISTRIBUTION OF RATIOS a)
R8 000 is divided between Anne, William and Susan in the ratio 2 : 3 : 5. How much will each receive? Determine the total by adding up 2 + 3 + 5 = 10. 2
Anne: 10 × R8 000 = R1 600 3
William: 10 × R8 000 = R2 400 5
b)
Susan: 10 × R8 000 = R4 000
A piece of ribbon of 30 m is divided in the ratio 1 : 2 : 3. How long is each piece? 1
Part 1: 6 × 30 m = 5 m 2
Part 2: 6 × 30 m = 10 m 3
Part 3: 6 × 30 m = 15 m © Optimi
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Study Guide G11 ~ Mathematical Literacy
Unit
1
Activity 5 1.
Write the following ratios in its simplest form: 1.1 54 : 81 1.2 5 days : 5 weeks 1.3 2,5 ℓ : 500 mℓ
2.
The following ratio is given on a bottle of cold drink concentrate: ‘mix 1 : 4 with water’. 2.1 What does it mean? 2.2 How much concentrate and how much water should you add to mix 1 ℓ of cold drink for you and some friends?
3.
Divide R4 500 between Kate, Megan and Sophie in the ratio 5 : 8 : 12.
4.
Increase R168,50 in the ratio 2 : 3.
5.
Decrease 520 kg in the ratio 9 : 7.
Part 6: Rate Rate is the relationship between two values of different units. Unlike ratios where we work with the same unit, rate works with different units. Think of the exchange rate, in which the values of different currencies are compared with one another, for example, the value of the rand against that of the dollar. More examples include km/h; R/kg; m/s and kg/m2. EXAMPLES 6.1 The exchange rate is R18,45 for one euro (€). If you have €1 500, how many rands do you have? €1 500 × R18,45 = R27 675
6.2 If you have R100 000, how many euros do you have? R100 000 ÷ R18,45 = €5 420,05 6.3 A water tank with a capacity of 100 ℓ leaks at a rate of 1,4 ℓ/min. How long will it take before there is 64 ℓ water left in the tank? Give your answer in minutes and seconds. 100 – 64 = 36 ℓ 36 ℓ ÷ 1,4 = 25,71 minutes 25 minutes and (0,71 × 60) 25 minutes 42 seconds © Optimi
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Study Guide G11 ~ Mathematical Literacy
Unit
Activity 6 1.
£1 (British pound) → R19,65 1.1 How many rands are equivalent to £20 000? 1.2 If you had R55 000, how many British pounds (£) do you have?
2.
5 kg of potatoes costs R45,89. How much will 1 kg of potatoes cost?
Part 7: Proportion Proportion means that two ratios are of equal value, e.g. 1 : 2 = 2 : 4. Direct proportion As one quantity increases, another quantity increases at the same rate. The relationship between the two quantities remains constant. Direct proportion is represented by a straight line on a graph which increases with the same slope (gradient).
Indirect/inverse proportion As one quantity increases, another quantity decreases. We can draw a graph to show indirect or inverse proportion. A curve on this graph will show the increase in one quantity and the decrease in another quantity.
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Study Guide G11 ~ Mathematical Literacy
1
Unit
Activity 7 1.
Bananas cost R15,99/kg. Complete the table by filling in the missing values:
Weight in kg
1
5
10
1.3
20
1.5
30
Cost in rand
R15,99
1.1
1.2
R239,85
1.4
R399,75
1.6
2.
Draw a line graph on a labelled set of axes to graphically represent the table.
3.
What type of proportion is represented here? Motivate your answer.
4.
A group of learners want to hire a bus to take them to Gold Reef City. It costs R3 000 to rent the bus. The amount that each learner should contribute, depends on how many people will be on the bus. 4.1 Complete the table to indicate the amount per person.
Number of people on bus Amount per person
1
2
3
4
5
10
50
60
R3 000
R1 500
4.1.1
4.1.2
4.1.3
R300
4.1.4
R50
4.2 Draw a line graph to graphically represent the table. 4.3 What type of proportion is represented here? Motivate your answer.
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Study Guide G11 ~ Mathematical Literacy
Unit
2
UNIT 2: PATTERNS, RELATIONSHIPS AND REPRESENTATIONS Learning objectives At the end of the unit, you must be able to: • analyse graphs that tell a story • understand, draw and analyse: o Fixed relationships o Linear relationships o Indirect/inverse proportions o Step graphs o Constant relationships o Combination graphs
Lesson 1: Graphs that tell a story Graphs found in financial magazines or in the business supplement of newspapers tell a story. In Mathematical Literacy, each problem or question also starts with a story. You must analyse and solve these stories. It is easier to understand a picture or representation than a list of numbers. A graph is a mathematical representation of the relationship between two values, such as distance and time. The advantage of a graph is that you can see the full picture or problem at a glance. In this lesson, we will look at the messages conveyed by graphs. EXAMPLE Ester went shopping. She visited various shops and then went home. The distance she walked is represented by the adjacent graph:
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Study Guide G11 ~ Mathematical Literacy
Unit
Distance that Ester walked 600
2
Shop 3
Distance (metres)
500
400 Shop 2
300
200 Shop 1
100
0
0
5
10
15
20
25
30
35
40
Time (minutes)
1.
Write the independent and dependent variables. Independent variable: time in minutes Dependent variable: distance in metres
2.
What is the maximum value on the graph? 600 metres
3.
How much time did Ester spend in the shops altogether? Shop 1: 5 minutes Shop 2: 15 minutes Shop 3: 5 minutes Total time: 25 minutes
4.
Calculate the speed at which Ester walked home. Speed = distance ÷ time Speed = 600 metres ÷ 10 minutes = 60 m/min
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45
50
55
60
Study Guide G11 ~ Mathematical Literacy
2
Unit
Activity 8 Study the graph and answer the questions. This graph represents the time it takes to fill a bath with a certain amount of water and then draining the bath. y
Volume of water (litres)
20
B
15
A
C
10
5
0
0
2
4
6
8
10
12
14
16
18
20
x
Time (minutes)
1. 2. 3. 4. 5. 6.
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Give the graph a suitable heading. Name the independent and dependent variables. How much time did the person spend in the bath? What happens during A, B and C? If the capacity of a standard bath is 80 ℓ, what percentage of the bath is filled with water? Explain why the slope (gradient) of the graph differs at A and C.
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Study Guide G11 ~ Mathematical Literacy
Unit
2
Lesson 2: Patterns with constant relationships VARIABLES Variables are values that can change. The independent variable is the value that is shown on the 𝑥𝑥-axis. It is independent from the dependent variable and is not affected by other variables. For example, time is always an independent variable. It changes as time passes, without it being affected by something else or being dependent on something. The dependent variable is the value that is shown on the 𝑦𝑦-axis. It is determined by the independent variable. TYPES OF VARIABLES Discrete values consist of whole numbers, for example, the number of people standing in a queue, the number of apples in a bag, days of the week. These are values that are not used as fractions in context. Discrete values are indicated by points (dots) which can be connected by dotted lines.
Continuous values consist of fractions (common or decimal fractions), for example, length, weight, age. Continuous values are indicated by solid lines because the values between the points are also included.
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Study Guide G11 ~ Mathematical Literacy
Unit
2
Increasing relationships exist when the independent and dependent variables both increase. 𝑦𝑦
𝑦𝑦 = 𝑎𝑎𝑥𝑥 𝑏𝑏 (𝑏𝑏 > 1)
𝑥𝑥
Decreasing relationships exist when the independent variable increases and the dependent variable decreases. 𝑓𝑓(𝑥𝑥)
𝑓𝑓 𝑥𝑥
Constant or fixed relationships occur when the dependent variable (𝑦𝑦) remains constant and the independent variable (𝑥𝑥) increases. 𝑓𝑓(𝑥𝑥) 𝑓𝑓
𝑥𝑥
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Study Guide G11 ~ Mathematical Literacy
2
Unit
EXAMPLE The Naidoo family wants to go on holiday in Cape Town. They receive two quotations to rent apartments for 8 people. • •
Apartment 1: R1 500,00 per day Apartment 2: R1 800,00 per day
We can represent the above information with a: • table • formula • graphical representation Table Number of people
0
1
2
3
4
5
6
7
8
Apartment 1
1 500
1 500
1 500
1 500
1 500
1 500
1 500
1 500
1 500
Apartment 2
1 800
1 800
1 800
1 800
1 800
1 800
1 800
1 800
1 800
Formula Apartment 1 = 1 500 Apartment 2 = 1 800 Graphical representation
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