2017 mathematical science changes

METROPOLE CENTRAL EDUCATION DISTRICT NAME OF CENTRE: CITY OF CAPE TOWN MATHEMATICS AND MATHEMATICAL SCIENCE LEARNER PORTFOLIO LEVEL: 4 YEAR: 2019 LEARNER NAME : ______________________ EDUCATOR NAME : ______________________ CENTRE MANAGER: C METH Compile by Compiled by R Boer : Ricardo Boer Mathematics and Mathematical Science Content                              Introduction ………………………………………………………………… .2 Integers……………………………………………………………………… 5 Bodmas……………………………………………………………………… 8 Add terms…………………………………………………………………… 12 Different terms ……………………………………………………………… 15 Exponents…………………………………………………………………… 20 Multiply Exponents…………………………………………………………… 25 Divide Exponents………………………………………………………………28 Highest Common Factor……………………………………………………… 33 Factorise ………………………………………………………………………. 36 Lowest Common Factor…………………………………………………… …40 Solve x……………………………………………………………………………44 Solve the variables ……………………………………………………………..48 Speed and Time……………………………………………………… 55 Number patterns …………………………………………………………… 65 Set Notation ………………………………………………………………… 73 Angles………………………………………………………………………… 90 Theorem of Pythagoras ……………………………………………………… 95 Measurement ………………………………………………………………… 98 Find the area ………………………………………………………… 101 Find the surface……………………………………………………… 113 Find the volumes……………………………………………………… 118 Find the volumes of the prisms ……………………………………… 122 Mode ,Medium and Mean ………………………………………………… Question paper 1…………………………………………………………… 132 Question paper 2…………………………………………………………… 143 Question paper 3…………………………………………………………… 149 Glossary ………………………………………………………………………155 Reference ……………………………………………………………………166 Compile by Ricardo Boer Page 1 Mathematics and Mathematical Science Introduction Understand the language of Mathematics is very important. If you don’t know the language, you will not be able to:   Read and understand Mathematics. Solve mathematical problems. grouping of numbers the following point is very important in Mathematics. What is an equation? It is identified when an = sign is part of a sentences, e.g. 2+2=4. It is very important tool in the mathematical world. A bit of history: Robert Recorde, a Welshman, first proposed the use of the = symbol in 1540 because he became tired of writing “is equal to” over and over again. Sign = + X ÷ ≠ < > ∴ Definition Equal to Plus/Add Minus/ less then Multiply/ times Divide/grouping Is not equal to Is smaller than Is greater than And there for Types of numbers       Even numbers:2,4,6,8,10,……..;divisible by 2 Uneven numbers:1,3,5,7,9,……..; not divisible by 2 Prime numbers:1,3,5,7,11,……….;only divisible by 1 and the number itself without a remainder. Natural numbers(N)1,2,3,4,5,6,7,……..; Integers (whole numbers)-3;-2;-1, 0, 1, 2, 3… Rational numbers include fractions and are the results of problems involving division of integers. Compile by Ricardo Boer Page 2 Mathematics and Mathematical Science   Irrational numbers. The square roots of prime numbers, and π will be irrational. These are the numbers which cannot be written in a finite or repetitive decimal. Real numbers (R) include rational and irrational numbers. 𝑴𝒖𝒍𝒕𝒊𝒑𝒍𝒊𝒄𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒔𝒊𝒈𝒏𝒔.     +𝑥+= + −𝑥−= + −𝑥+= − +𝑥−= − Remember:   Multiplication of like signs results in a positive answer. Multiplication of unlike signs results in a negative answer. Compile by Ricardo Boer Page 3 Mathematics and Mathematical Science SECTION 1 This module covers the following Unit Standards, Specific Outcomes and Assessment Criterions: SAQA US ID: 7447(6 credits) = 1𝑐𝑟𝑒𝑑𝑖𝑡 × 10ℎ𝑜𝑢𝑟𝑠 = 60 𝑛𝑜𝑡𝑖𝑎𝑛𝑎𝑙 ℎ𝑜𝑢𝑟𝑠 (Working with numbers in various contexts So 1: Express and interpret a range of contexts using mathematical symbols AC: 1, 2, 3 So 2: Solve a range of everyday problems using estimation and calculations Ac: 1-5 So 4: Perform operations on simple and complex numerical expressions Ac 1-2 So 5: Describe and compare counting systems from different cultures Ac 1-3 So 6: Critically analyse the development of the base ten number system. Ac 1-4 Compile by Ricardo Boer Page 4 Mathematics and Mathematical Science Date: ___________________ Level 4 Duration: 1 hour _________________________________________________________________ Activity 1 Us 7447 Working with numbers in various contexts So 1 Express and interpret a range of contexts using mathematical symbols Ac 1-2 For these pairs of numbers, fill in =; <; >:     38______-28 -7______7 2_______(-4+6) -3427____1984 (1) (1) (1) (1) Add the following integers:     (-9) + 10=_____ 2 +(-9)=______ 1569-(-2781)=______ 88-(+88) =__________ (2) (2) (2) (2) Give the additive inverse of the following numbers:     -65 0 100 1412 (1) (1) (1) (1) Complete the following,  John owes Edgars R 2050, 00 and pays R1005, 64 on his account. How much does he still owns? (2) ____________________________________________________________ ____________________________________________________________ _________________________________________________________ Compile by Ricardo Boer Page 5 Mathematics and Mathematical Science  Jane borrows R84, 12 then R341, 94 then R12, 34. How much debt does she have? 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by Ricardo Boer Page 7 Mathematics and Mathematical Science Date: ___________________ Level 4 Duration: 1 hour _________________________________________________________________ __ Activity 2 Us: 7447: Working with numbers in various contexts So: 4 AC 1-2 Example:      2+3-6x5÷2 =2+3-30÷2 =2+3-15 =5-15 =-10 Activity 1 1. 3-6+9÷5 2. 4x6-2+3÷5 (2) 3. 6÷2-2+3x5 4. 12x5-8÷3+6 5. 8÷2x4+2-3 6. 2 +4x3÷6-2 7. 8-4+6x2-3 8. 3 x2 ÷4+6-4 9. (-2)x(-3) 10. 3 x(-4) (2) (2) (2) (2) (2) (2) (2) (2) (2) Total: 20/________ Compile by Ricardo Boer Page 8 Mathematics and Mathematical Science Answer sheet for Activity 1: __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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by a number, the number is called the coefficient. For example, in 3a x a.3 is the coefficient and a variable or symbol. The meaning is 3 of a, 𝑜𝑟 𝑎 + 𝑎 + 𝑎. 4𝑥 = 4 𝑥 ᵡ Terms Terms are separated by + and/or – in an equation or expression. For example: 𝑎 +𝑏 2 𝑡𝑒𝑟𝑚𝑠 (𝑏𝑖𝑛𝑜𝑚𝑖𝑎𝑙) 𝑎 + 𝑏−𝑐 3 𝑡𝑒𝑟𝑚𝑠 (𝑡𝑟𝑖𝑛𝑜𝑚𝑖𝑎𝑙) 2𝑎 + 4𝑏 − 8𝑐 + 𝑑 𝑚𝑜𝑟𝑒 𝑡𝑒𝑟𝑚𝑠 (𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙) 2𝑎 + 4𝑏 = 8𝑐 3𝑡𝑒𝑟𝑚𝑠 𝑖𝑛 𝑎𝑛 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛. Instructions Do the following exercise in black or blue ink. Show all the calculations. Look at the example that was given to you in this regards. Example 1: 6𝑎 + 2𝑎 − 4𝑎 = (6 + 2 − 4)𝑎 = 4𝑎 Example 2 3𝑎 + 3𝑏 + 2𝑎𝑏 − 6𝑎 + 𝑎𝑏 = 3𝑏 + 3𝑎 − 6𝑎 + 2𝑎𝑏 + 𝑎𝑏 = 3𝑏 + (3 − 6)𝑎 + (2 + 1)(𝑎𝑏) = 3𝑏 − 3𝑎 + 3𝑎𝑏 Compile by Ricardo Boer Page 11 Mathematics and Mathematical Science Level: 4 Duration: 60 min Date: Activity 3 __________________________________________________________________________________________________ Us: 7447: Working with numbers in various contexts So: 4 AC: 1 -3. 1. 2. 3. 4. 5. 6. 7. 8. 3𝑎 + 2𝑎𝑏 − 6𝑎 + 2𝑏 − 𝑎𝑏 2𝑥𝑦 + 3𝑥 − 2𝑦 − 3𝑥𝑦 + 𝑥 𝑎 − 2 + 3𝑎 − 6 2𝑎𝑏 + 𝑏 − 4𝑎𝑏 − 5𝑏 3𝑝 + 2𝑞 − 4𝑝𝑞 − 2𝑞𝑝 2𝑎 + 3𝑎𝑏; 2𝑎 − 2𝑎𝑏 𝑎 − 2; 2𝑏 + 6 2𝑥𝑦 + 3𝑥 − 𝑦; 4𝑥𝑦 − 2𝑥 − 6 (3) (3) (2) (3) (2) (3) (2) ( 2) Total: 20/_________ Compile by Ricardo Boer Page 12 Mathematics and Mathematical Science Answer sheet for Activity __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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__________________________________________________________________________________________________ Activity 4 Us: 7447: Working with numbers in various contexts So: 3, 4 1. 2. 3. 4. 5. 6. 7. 𝑎 − 3 + 𝑏 + 4 + 𝑐 − 12 𝑎 − 2 − 2𝑏 + 2 2𝑏 + 𝑏 − 3𝑏 + 2 3𝑥𝑦 + 2𝑥 − 𝑦 − 𝑥 − 3𝑦 + 2𝑥𝑦 2 + 4𝑥 − 6𝑦 − 4 − 2𝑥 + 3𝑦 (2𝑥 + 3𝑦) + (8𝑥 − 2𝑦) (2𝑥² + 3𝑥 − 2) − (𝑥² + 2𝑥 + 1) (3) (3) (3) (3) (3) (2) (3) Total: 20/_________ Compile by Ricardo Boer Page 15 Mathematics and Mathematical Science Answer sheet for Activity __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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__________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ Compile by Ricardo Boer Page 17 Mathematics and Mathematical Science “I don’t regret the things I’ve done, I regret the things I didn’t do when I had the chance.” – Unknown “Challenges are what make life interesting and overcoming them is what makes life meaningful.” – Joshua J. Marine “It’s hard to wait around for something you know might never happens; but it’s harder to give up when you know it’s everything you want.” – Unknown Compile by Ricardo Boer Page 18 Mathematics and Mathematical Science Multiplication of variables It takes too long to write 𝑎 × 𝑎 × 𝑎 × 𝑎. it is however necessary in Mathematics so 𝑎4 shorter notation was developed. 𝑎 × 𝑎 × 𝑎 × 𝑎 Is written as 𝑎4 Example:     2 𝑥 2² = 2𝑥(2 𝑥2) = 2³ 𝑏 𝑥 𝑏 = 𝑏² 𝑏 𝑥 𝑏 𝑥 𝑏 = 𝑏³ 𝑎² 𝑥 𝑎² = 𝑎4 Compile by Ricardo Boer Page 19 Mathematics and Mathematical Science Level: 4 Duration: 60 min Date: _________________________________________________________________________________________________ Activity 5 US: 7447: Working with numbers in various contexts So 5 Ac 1-3 Please simplify the following sums: 1. 2𝑥 𝑥 3𝑥³ 2. 2 𝑥 𝑏 𝑥 𝑏² 3. 2 𝑥 𝑎 𝑥 3 𝑥 𝑎² 4. 3𝑎 𝑥(4𝑎)² 5. 2𝑎²(4𝑎³) 6. (−2𝑎)(4𝑎𝑐) 7. (−𝑎²)(−4𝑎𝑏) 8. (−2𝑎𝑏)(−3𝑎²𝑏𝑐) 9. (3𝑎)²(2𝑏)²3𝑎𝑏 10. 3𝑥² 𝑥 5𝑥² (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) Total: 20/__________ Compile by Ricardo Boer Page 20 Mathematics and Mathematical Science Answer Sheet __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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__________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ Compile by Ricardo Boer Page 22 Mathematics and Mathematical Science THE FOUR MAIN OPERATIONS IN ALGEBRA The pocket calculator Objectives and overview On completion of this section, you should be able to:          Do calculations including, the four basic operations, and extraction of roots and involution with the aid of a calculator. Use the memory keys in calculations. Use the special keys e.g. 𝐼𝑁𝑉, ( ), 𝐸𝑋𝑃, 𝜋, Multiply binomials Multiply trinomials. Division of terms Division by a monomial Division by a binomial Long division. Compile by Ricardo Boer Page 23 Mathematics and Mathematical Science Example: Calculate the following with the aid of a calculator: 1) 2) 3) 4) 5) 24 (2,4)5 (3,6)2,4 3 √16 √126 4 Solutions: 1) 2) 3) 4) 24 =16 (2,4)5=79,626 ( 3,6)2,4=21,633 3 √16= (16)13=2,520 4 5) √126=3,350 Press2. 𝑦 𝑥 4=on the Calculator Press 2.4 𝑦 𝑥 5= on the Calculator. Press 3.6 𝑦 𝑥 2,4= on the Calculator Press 16 𝑦 𝑥 3𝑥1 = on the Calculator 1 Press 126 𝑦 𝑥 4 𝑥 Example: 1) 2) 3) 4) x + (2 + 𝑥) = 𝑥 + 2 + 𝑥 = 𝟐𝒙 + 𝟐 𝑥 − (2 + 𝑥) = 𝑥 – 2 – 𝑥 = −𝟐 3𝑎 − 2(2𝑎 – 6) = 3𝑎 − 4𝑎 + 12 = −𝒂 + 𝟏𝟐 2(2𝑎 + 3) − 3(𝑎 − 6) = 4𝑎 + 6 − 3𝑎 + 18 = 𝒂 + 𝟐𝟒 Compile by Ricardo Boer Page 24 Mathematics and Mathematical Science Level: 4. Duration: 60 min Date: _________________________________________________________________________________________________ Activity: 6 Us7447 So 1, 2, 4 1. 2. 3. 4. 5. 6. 7. 8. 9. 2 + (𝑥 − 1) 𝑥 − (1 + 𝑥) 2𝑎 – (1 + 2𝑎 – 𝑏) 2𝑎 − 3(2 – 𝑎 + 4𝑏) 𝑥 – (𝑥 + 𝑦) + 2(𝑥 − 𝑦) 𝑎 + 2𝑏 – 𝑐 – (𝑎 − 3𝑏 + 2𝑐 3𝑎 + 2{𝑎 − 3(2𝑎 + 4)} 4 − {2𝑎 + 3[𝑎 − (𝑎 − 4)]} 2𝑎 + 2{𝑎 − 3[2𝑎 − (2 − 𝑎)]} (2) (2) (2) (2) (2) (2) (2) (2) (4) Total: 20/_________ Compile by Ricardo Boer Page 25 Mathematics and Mathematical Science Answer Sheet for Activity __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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Duration: 60 min Date: __________________________________________________________________________________________________ Activity 7 US 7447 So: 4 AC1-2 Example; Division by Symbols 1) a³÷a= 2) 𝑎 𝑎4 𝑎³ =a² 𝑎 4 = a÷ 𝑎 =𝑎1−4 = 𝑎 −3 3) 9𝑎2÷ 3a= 39𝑎2−1 =3a Activity 1) 2) 3) 4) 5) 6) 𝑥4 ÷ 𝑥 8𝑥 2 ÷ 2𝑥 3𝑥 ÷ 9𝑥 2 𝑑𝑖𝑣𝑖𝑑𝑒 4𝑥 3 + 8𝑥 2 + 2𝑥 − 6 𝑏𝑦 2𝑥 𝑑𝑖𝑣𝑖𝑑𝑒 3 − 𝑥 + 3𝑥 2 + 9𝑥 3 by 3x 𝑑𝑖𝑣𝑖𝑑𝑒 𝑥 2 − 4 𝑏𝑦 𝑥. (2) (2) (2) (6) (6) (2) Total: 20/_________ Compile by Ricardo Boer Page 28 Mathematics and Mathematical Science Answer Sheet: ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ ______________________________________________________________________________________________ Compile by Ricardo Boer Page 29 Mathematics and Mathematical Science Factorisation Objectives and overview On completion of this section, you should be able to: 1. Factorise polynomials with common factors. 2. Determine the highest common factor.( HCF) 3. Regroup terms that have common factor and factorise them. General Products are built up of factors. 6=3× 2 ∴ 3 𝑖𝑠 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 6 𝑎𝑛𝑑 2 𝑖𝑠 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 6 2ab = 2 × 𝑎 × 𝑏 ∴ 2 𝑖𝑠 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 2𝑎𝑏, 𝑎 𝑖𝑠 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 2𝑎𝑏 𝑎𝑛𝑑 𝑏 𝑖𝑠 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 2𝑎𝑏 Prime factors Prime factors are the smallest factors. 3 and 4 are factors of 12 because 12=3× 4. 𝑡ℎ𝑒 𝑝𝑟𝑖𝑚𝑒 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑎𝑟𝑒 3 × 2 × 2, 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 12 = 3 × 2 ×× 2. Resolving into factors Resolving into factors means to calculate the factors of a number expression. It also means to write it as a product. Compile by Ricardo Boer Page 30 Mathematics and Mathematical Science Examples: Resolve the following into prime factors: 16 Solutions: 2 2 2 16 8 4 2 ∴ 16 = 2 × 2 × 2 × 2 81𝑎2 𝑏 Solutions: 3 3 3 81 27 9 3 ∴ 81 = 3 × 3 × 3 × 3 ∴ 81𝑎2 𝑏 = 3 × 3 × 3 × 3 × 𝑎 × 𝑎 × 𝑏 Common factors In every term of an expression can be divided by the same number or symbol, this number or symbol is called a common factor. For example: 3 is a common factor of 3𝑥 + 3𝑦. Resolving into factors can be done as follow: ∴ 3𝑥 + 3𝑦 = 3(𝑥 + 𝑦) Compile by Ricardo Boer Page 31 Mathematics and Mathematical Science 20𝑥𝑦 2 +12𝑥 2 ∴2×2×5×𝑥×𝑦×𝑦+2×2×3×𝑥×𝑥 = 4𝑥{5𝑦 2 +3𝑥} = 44𝑥 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑜𝑚𝑚𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟. 38𝑎2 𝑏 + 57𝑎𝑏 2 = 2 × 19 × 𝑎2 𝑏 + 3 × 19 × 𝑎𝑏 2 = 19𝑎𝑏 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑜𝑚𝑚𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟. The highest common factor (HCF) The common factor in the above-mentioned example must be highest (biggest). If it is not then the expression is not completely factorised. The method to find the highest common factor is the same as the method used in the above problems. Example: Calculate the HCF of 24𝑥 2 𝑦 𝑎𝑛𝑑 6𝑥𝑦 2 Solution: 24𝑥 2 𝑦 = 2 × 2 × 2 × 3 × 𝑥 × 𝑥 × 𝑦 ∴ 𝐻𝐶𝐹 = 2 × 3 × 𝑥 × 𝑦 = 6𝑥𝑦 Compile by Ricardo Boer Page 32 Mathematics and Mathematical Science Level: 4. Duration: 60 min Date: __________________________________________________________________________________________________ Activity 8 Us 7447 So 4 AC1 -2 Calculate the HCF of: 1) 3𝑎𝑏 𝑎𝑛𝑑 12 𝑏𝑐 2) 128 𝑎𝑛𝑑 16 3) 147 𝑎𝑛𝑑 21 4) 105,30,45 5) 30,18 𝑎𝑛𝑑 54 6) 25𝑎2 𝑏 2 𝑐 2 𝑎𝑛𝑑 50 𝑎𝑏𝑐 7) 34𝑎𝑏 2 and 8𝑎2 𝑐 8) 14𝑥 2 𝑦, 49𝑥𝑦 2 𝑎𝑛𝑑 21𝑥𝑦 9) 124𝑎𝑐, 81𝑎𝑏 𝑎𝑛𝑑 𝑎2 𝑏 2 𝑐 2 10)22𝑥 3 , 33𝑥 2 and 44𝑥 (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) Total: 20/___________ Compile by Ricardo Boer Page 33 Mathematics and Mathematical Science Answer sheet __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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Duration: 60 min Date: __________________________________________________________________________________________________ Activity 9 Us 7447 So 4 AC1- 3 Resolve into factors: 1. 2𝑎𝑐 − 4𝑎𝑑 + +𝑏𝑐 − 2𝑏𝑑 2. 3𝑥 2 − 9𝑥𝑦 + 𝑥𝑦 − 3𝑦 2 3. 𝑎𝑥 + 𝑎𝑦 + 𝑏𝑐+𝑏𝑦 4. 𝑎𝑥 + 𝑏𝑥 − 𝑎𝑦 − 𝑏𝑦 5. 6𝑎𝑥 − 4𝑎𝑦 + 3𝑏𝑐 − 2𝑏𝑐 6. 𝑏 + 𝑐 + 2𝑎𝑏 + 2𝑎𝑐 7. 2𝑎𝑐 + 4𝑎𝑑 + 3𝑏𝑐 + 6𝑏𝑑 8. 𝑎𝑐 2 +2𝑎𝑑 −ae+𝑏𝑐 2 + 2𝑏𝑑 − 𝑏𝑒 9. 6𝑦 + 4𝑧 − 2𝑥 + 3𝑣𝑦 + 2𝑣𝑧 − 𝑣𝑥 10. 3𝑎𝑥 − 6𝑏𝑥 + 6𝑎𝑦 − 12𝑏𝑦 (2) ( 2) (2) (2) (2) (2) (2) (2) (2) (2) Total: 20/____________ Compile by Ricardo Boer Page 36 Mathematics and Mathematical Science Answer Sheet for the Activity: __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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__________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ Compile by Ricardo Boer Page 38 Mathematics and Mathematical Science Fractions Objectives and overview: On completion of this section in the workbook, you should be able to do the following:     Multiply and divide fractions Add and subtract fractions Simplify fractions Determine the lowest common multiple.(LCM) Multiplication and division of fractions. In 2 3 , 2 is called the numerator and 3 the denominator. Another way of writing 3 is 3 1 4 1 3 1 2÷ 3 0𝑟 2 3. Therefore 5 can be written as 4 5 and 4 = 3(4) If fractions are multiplied, the numerators are multiplied and the denominators are multiplied. For example: 2 3 1 ×5 = 2×1 3𝑥5 2 = 15 If fractions are divided the divisor is inverted and then multiplied For example: 2 3 1 2 5 ÷5 =3×1= 10 3 {This is so because ÷ 𝑖𝑠 𝑡ℎ𝑒 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 ×} Addition and subtraction of fractions Only similar fractions can be added or subtracted. That means that sixths can be added to sixths etc. Example: 1 6 2 3 +6 = 6 Compile by Ricardo Boer Page 39 Mathematics and Mathematical Science Level: 4. Duration: 60 min Date: __________________________________________________________________________________________________ Activity 10 Us: 7447 So: 1, 2, 3. Calculate the LCM of 1. 2. 3. 4. (2) (2) (2) (2) 6 𝑎𝑛𝑑 15 3,12 𝑎𝑛𝑑 18 3,4 𝑎𝑛𝑑 6 2𝑎𝑏 𝑎𝑛𝑑 8𝑎2 𝑏 2 5. 32𝑥𝑦 2 and 16𝑦𝑥 6. 18𝑎2 12𝑎2 𝑏𝑐, 42𝑎2 𝑏 2 𝑐 2 (2) (2) Simplify 1. 2. 3. 4. 2 𝑠 1 +7 3 (2) 5 (2) + 12 3 3 4 2 3 7 (2) − 12 1 5 + 2 − 12 (2) Total: 20/__________ Compile by Ricardo Boer Page 40 Mathematics and Mathematical Science Answer Sheet: __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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__________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ ________________________________________________________________________________________________ Compile by Ricardo Boer Page 42 Mathematics and Mathematical Science Equations, word problems and manipulation of formulae Objectives and overview On completion of this section, you should be able to:   Change the subject of a given formula. Determine the value of a new subject by substituting the values of the known quantities. Changing the subject: A formula is nothing more than a relationship between quantities. As we saw earlier, equations are handy ways of writing problems. These equations make it possible to solve mathematical problems. Example Calculate the number that must be added to 16 to give an answer of 25. Solution Let the number be 𝑥 ∴ 𝑥 + 16 = 25 ∴ 𝑥 + 16 − 16 = 25 − 16 {Subtract 16 on both sides} ∴𝑥=9 Notice that if something is added to 𝑎𝑛𝑑 𝑤𝑒 𝑚𝑢𝑠𝑡 𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥 𝑡ℎ𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛, 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛, 𝑖𝑠 𝑎𝑝𝑝𝑙𝑖𝑒𝑑. 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒: +𝑟𝑒𝑠𝑢𝑙𝑡𝑠 𝑖𝑛 −, −𝑖𝑛 +,÷ 𝑖𝑛 ×,× 𝑖𝑛 ÷, 𝑝𝑜𝑤𝑒𝑟𝑠 𝑖𝑛 𝑟𝑜𝑜𝑡𝑠 𝑎𝑛𝑑 𝑟𝑜𝑜𝑡𝑠 𝑖𝑛 𝑝𝑜𝑤𝑒𝑟𝑠. Example: Solve for 𝑥 𝑖𝑓 𝑥 + 𝑎 − 4 = 6 Solution 𝑥+𝑎−4=6 ∴𝑥+𝑎−4−𝑎+4=6−𝑎 +4 ∴ 𝑥 = 10 − 𝑎 Compile by Ricardo Boer Page 43 Mathematics and Mathematical Science Level: 4. Duration: 60 min Date: __________________________________________________________________________________________________ Activity 11 Us: 7447 So 1, 2, 3 Solve for 𝒙 𝒊𝒇: 1) 𝑥 + 4 = 12 2) 𝑥 − 6 = 2 3) 3𝑥 = 9 𝑥 4) 3 = 2 (2) (2) (2) (2) 5) 𝑥 2 = 16 6) 3𝑥 − 6 = 2 𝑦 7) 𝑥 = 4 (2) (2) (2) 8) 𝑥𝑦 = 4 (2) 9) 2 3 𝑥+7=2 10)√2𝑥 = 4 (2) (2) Total: 20/____________ Compile by Ricardo Boer Page 44 Mathematics and Mathematical Science ANSWER SHEET __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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__________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ Compile by Ricardo Boer Page 46 Mathematics and Mathematical Science Technical Formulae A technical formula is nothing more than an equation. The subject of the equation is the single symbol standing to the left of the = sign. 2𝑦𝑟 e.g. 𝑡 𝑖𝑛 𝑡 = 𝑞+2 𝑦𝑝𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 ℎ𝑒𝑟𝑒. 2 Example Make 𝑢 𝑡ℎ𝑒 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑜𝑓 𝑣 = 𝑢 + 𝑎𝑡 Solution: 𝑣 = 𝑢 + 𝑎𝑡 ∴ 𝑣 − 𝑎𝑡 = 𝑢 + 𝑎𝑡 − 𝑎𝑡 {Subtract 𝑎𝑡 on both sides} ∴ v−𝑎𝑡 = 𝑢 ∴ 𝑢 = 𝑣 − 𝑎𝑡 Express 𝑡 in terms of the other quantities in 𝑅 = 𝑟(𝑖 + 𝑎𝑡) Solution 𝑅 = 𝑟(𝑖 + 𝑎𝑡) 𝑅 𝑟 ∴ 𝑟 = 𝑟(𝑖 + 𝑎𝑡) 𝑅 ∴ 𝑅 = 𝑖 + 𝑎𝑡 Compile by Ricardo Boer Page 47 Mathematics and Mathematical Science Level: 4. Duration: 60 min Date: __________________________________________________________________________________________________ Activity 12 Us: 7447 So: 1, 2, 3 Make the symbol in brackets the subject of the formula: 1) 𝑉 = 𝑙𝑏ℎ(ℎ) 2) 𝐴 = 4𝜋𝑟 2 (𝑟) (4) (4) 4 3) 𝑉 = 3 𝜋𝑟 3 (r) (4) 4) 𝑥 = 4ℎ𝐷 − 4ℎ (𝐷) 2 2 𝑥2 5) 𝐷 = ℎ + 4ℎ(𝑥) (4) (4) Total: 20/_________ Compile by Ricardo Boer Page 48 Mathematics and Mathematical Science ANSWER SHEET __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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__________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ Compile by Ricardo Boer Page 50 Mathematics and Mathematical Science SECTION 2 This module covers the following Unit Standards, Specific Outcomes and Assessment Criterions: SAQA US ID: 7450(2 credits) = 1𝑐𝑟𝑒𝑑𝑖𝑡 × 10ℎ𝑜𝑢𝑟𝑠 = 20 𝑛𝑜𝑡𝑖𝑎𝑛𝑎𝑙 ℎ𝑜𝑢𝑟𝑠 (Work with measurements in a variety of contexts So 1: Apply relationships between common quantities in various contexts. AC: 1, 2, 3 So 2: Use measuring instruments to measure and calculate quantities in various contexts Ac: 1-3 Compile by Ricardo Boer Page 51 Mathematics and Mathematical Science Speed, distance, time and revolutions. Objectives and overview On completion of this section, you should be able to solve problems on speed, distance, time and revolution. Speed Speed can be measured in kilometres per hour; in SI units it is written as km/h. it can also be written as÷ ℎ𝑜𝑢𝑟. Speed is also measured in metres per second, that is 𝑚/𝑠 or 𝑚. 𝑠 −1. It can also be written as 𝑚 ÷ 𝑠 Therefore speed= 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑖𝑚𝑒 𝑠 (𝑉 = 𝑡 ) Example Calculate the average speed of a car if it covers 315km in 3 hours. Solution: Speed = = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑖𝑚𝑒 𝑠 (𝑉 = 𝑡 ) 315 3 = 105𝑘𝑚/ℎ Distance Distance is usually measured in kilometres (km) or metres (m). If distance is to measure in terms of speed and time the relationship can be seen from the description of speed. 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑇𝑖𝑚𝑒 = 𝑠𝑝𝑒𝑒𝑑 Compile by Ricardo Boer Page 52 Mathematics and Mathematical Science Conversion between different units of speed and velocity.  The SI unit for speed and velocity is 𝑚. 𝑠 −1 but we usually measure the speed of cars and other means of transport in 𝑘𝑚. ℎ−1  Convert 𝑘𝑚. ℎ−1 to 𝑚. 𝑠 −1 as follows: o 1km=1000m o 1hour=1×60× 60 = 3600𝑠 o ∴ 1𝑘𝑚 1ℎ = 1×1𝑜𝑜𝑜𝑚 1×3600𝑠 o = 0,278 𝑚. 𝑠 −1  Convert 𝑚. 𝑠 −1 to 𝑘𝑚. ℎ−1 as follows: 1𝑚 1𝑠 1 1 = 1000 𝑘𝑚 = 1000 × 3600 1 3600 = 1000 = 3,6𝑘𝑚. ℎ−1 Example: A cyclist travels 400m in a straight line, turns travels 700m in the opposite direction. He cycles the distance in 100s. Calculate the  The average speed Solution: Distance travelled: 𝑑 = 400 + 700 = 1100𝑚 Average speed:𝑣 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑡𝑖𝑚𝑒 1100𝑚 100𝑠 So he travel at the following speed 𝑉 = 11𝑚. 𝑠 Acceleration If the velocity of a moving object changes, i.e. if the magnitude increase or decrease, or if the direction changes, we say that the object accelerates. Acceleration = Compile by Ricardo Boer 𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒗𝒆𝒍𝒐𝒄𝒊𝒕𝒚 𝒕𝒊𝒎𝒆 𝒆𝒍𝒂𝒑𝒔𝒆𝒅 = ∆𝒗 𝛁𝒕 Page 53 Mathematics and Mathematical Science = 𝑓𝑖𝑛𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 − 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 ∆𝑡 ∴𝑎= 𝑉𝑓 − 𝑉𝑖 ∆𝑡 Acceleration is the rate of change of velocity  Acceleration is a vector quantity and the direction of the acceleration has to be indicated  The symbol for acceleration is a.  The SI unit is meters per second second = 𝒎.𝒔 −𝟏 𝒔 Metres per second squared (m.s -2 ) Compile by Ricardo Boer Page 54 Mathematics and Mathematical Science Level: 4. Duration: 60 min Date: __________________________________________________________________________________________________ Activity 1 Us: 7450 So: 1 Ac 1-3 Instructions Please answer all the questions. Show all the calculations. Write in black or blue ink. 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__________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ Compile by Ricardo Boer Page 57 Mathematics and Mathematical Science “One of the most important keys to Success is having the discipline to do what you know you should do, even when you don’t feel like doing it.” - Unknown “Good things come to those who wait… greater things come to those who get off their ass and do anything to make it happen.” – Unknown “Happiness cannot be traveled to, owned, earned, or worn. It is the spiritual experience of living every minute with love, grace & gratitude.” – Denis Waitley Compile by Ricardo Boer Page 58 Mathematics and Mathematical Science SECTION 3 This module covers the following Unit Standards, Specific Outcomes and Assessment Criterions: SAQA US ID: 7448(4 credits) = 1𝑐𝑟𝑒𝑑𝑖𝑡 × 10ℎ𝑜𝑢𝑟𝑠 = 40 𝑛𝑜𝑡𝑖𝑎𝑛𝑎𝑙 ℎ𝑜𝑢𝑟𝑠 (Work with patterns in various contexts) So 1: Recognise, identify and describe patterns in various contexts AC: 1, 2, 3, 4 So 2: Complete, extent and generate patterns in a variety of contexts Ac: 1-3 So 3: Devise processes for a general rule Ac 1-3 So 4: Represent patterns using different generalised mathematical forms Ac 1-4 So 5: Use general rules to generate patterns. Ac 1-2 Compile by Ricardo Boer Page 59 Mathematics and Mathematical Science NUMBER PATTERNS In earlier grades you saw patterns in the form of pictures and numbers. In this chapter, we learn more about the mathematics of patterns. Patterns are repetitive sequences and can be found in nature, shapes, events, sets of numbers and almost everywhere you care to look. For example, seeds in a sunflower, snowflakes, geometric designs on quilts or tiles, the number sequence. Numbers can have interesting patterns. Here we examine some types of patterns and how they are formed.  1; 4;7;10;13;16;19;………. There is difference of 3 between successive terms. The pattern is continued by adding 3 to the last term.  3;8;13;18;23;28;33;38;………… There is a difference of 5 between successive terms. The pattern is continued by adding 5 to the last term.  2 ;4;8;16;32;64;………….. The sequence has a factor of 2 between successive terms. The pattern is continued by multiplying the last term with 2. Example: You and 3 friends decide to study for maths and are sitting together at a square table. A few minutes later 2 other friends arrive and would like to sit at your table. You move another table next to yours so that 6 people can sit at the table. Another 2 friends also want to join the group, so you take a third table and add it to the exiting tables. Now 8 people can sit together. Examine how the number of people sitting is related to the number of tables. Is there a pattern? Compile by Ricardo Boer Page 60 Mathematics and Mathematical Science Solution Step 1: Make a table to see if a pattern forms Number of tables,𝒏 1 2 3 4 ⋮ 𝒏 Number of people seated 4= 𝟒 𝟒+𝟐=𝟔 𝟒+𝟐+𝟐=𝟖 𝟒 + 𝟐 + 𝟐 + 𝟐 = 𝟏𝟎 ⋮ 𝟒 + 𝟐 + 𝟐 + 𝟐 + ⋯+ 𝟐 Step 2: Describe the pattern. We can see that for 3 tables we can seat 8 people, for 4 tables we can seat 10 people and so on. We started out with 4 people and added two times. So for each table added, the number of people increased by 2. DESCRIBING SEQUENCES To describe terms in a number pattern we use the following notation: The 1st term of a sequence is 𝑇1 The 4th term of a sequence is 𝑇4 The 10th term of a sequence is 𝑇10 Compile by Ricardo Boer Page 61 Mathematics and Mathematical Science The general term is often expressed as the 𝑛𝑡ℎ term and is written as 𝑇𝑛. A sequence does not have to follow a pattern but when it does, we can write down the general formula to calculate any term. For example, consider the following linear sequence: 1;3;5;7;9;…….. The 𝑛𝑡ℎ term is given by the general formula 𝑇𝑛=2𝑛−1 You can check this by substituting values into the formula: 𝑇1=2(1)−1=1 𝑇2=2(2)−1=3 𝑇3=2(3)−1=5 𝑇4=2(4)−1=7 𝑇5=2(5)−1=9 If we find the relationship between the position of a term and its value, we can describe the pattern and find any term in the sequence. Common difference In some sequences, there is a constant difference between any two successive terms. This is called a common difference. Definition: Common difference The common difference is the difference between any term and the term before it and is denoted by 𝑑. For example, consider the sequence 10;7;4;1;…. To calculate the common difference, we find the difference between any term and the previous term: 𝒅 = 𝑻𝟐−𝑻𝟏 𝟕 − 𝟏𝟎 = −𝟑 Compile by Ricardo Boer Page 62 Mathematics and Mathematical Science Let us check another two terms: 𝒅 = 𝑻𝟒−𝑻𝟑 = 𝟏 − 𝟒 = −𝟑 𝑊𝑒 𝑠𝑒𝑒 𝑡ℎ𝑎𝑡 𝑑 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. 𝒊𝒎𝒑𝒐𝒓𝒕𝒂𝒏𝒕: 𝒅 = 𝑻𝟐−𝑻𝟏, Not 𝑻𝟏−𝑻𝟐 You and 3 friends decide to study for maths and are sitting together at a square table. A few minutes later 2 other friends arrive and would like to sit at your table. You move another table next to yours so that 6 people can sit at the table. Another 2 friends also want to join the group, so you take a third table and add it to the exiting tables. Now 8 people can sit together. 1. Find the expression for the number of people seated at 𝑛 𝑡𝑎𝑏𝑙𝑒𝑠. 2. Use the general formula to 3. How many tables are needed to seat 20 people? Solution Step1: Make a table to see the pattern. Number of tables,𝒏 Number of people seated 1 4= 𝟒 2 𝟒+𝟐=𝟔 3 𝟒+𝟐+𝟐=𝟖 4 𝟒 + 𝟐 + 𝟐 + 𝟐 = 𝟏𝟎 ⋮ ⋮ 𝒏 𝟒 + 𝟐 + 𝟐 + 𝟐 + ⋯+ 𝟐 Pattern =4+𝟐(𝟎) = 𝟒 + 𝟐(𝟏) =4+2(2) =4+2(3) ⋮ =4+2(𝒏 − 𝟏) Step 2: Describe the pattern The number of people seated at 𝑛 𝑡𝑎𝑏𝑙𝑒𝑠 𝑖𝑠 𝑇𝑁 = 4 + 2(𝑁 − 1) Step 3: Calculate the 12th term, in other words, find 𝑻𝒏 𝒊𝒇 𝒏=𝟏𝟐 Compile by Ricardo Boer Page 63 Mathematics and Mathematical Science 𝑻𝟏𝟐=𝟒+𝟐(𝟏𝟐−𝟏) =𝟒+𝟐(𝟏𝟏) =𝟒+𝟐𝟐 =𝟐𝟔 Therefore 26 people can be seated at 12 tables Step 4: Calculate the number of tables needed to seat 20 people in other words find 𝒏 𝒊𝒇 𝑻𝒏=𝟐𝟎 𝑻 𝒏=𝟒+𝟐(𝒏−𝟏) 𝟐𝟎=𝟒+𝟐(𝒏−𝟏) 𝟐𝟎−𝟒=𝟐(𝒏−𝟏) 𝟏𝟔 =𝒏−𝟏 𝟐 =𝟖+𝟏=𝒏 𝒏=𝟗 Therefore 9 tables are needed to seat 20 people. It is important to note the difference between 𝑛 𝑎𝑛𝑑 𝑇𝑛. N can be compared to a place holder indicating the position of the term in the sequence while 𝑇𝑛 is the value of the place held by 𝑛. From the example above, the first table holds 4 people. So for 𝑛 = 1, the value of 𝑇1 =4 and so on: Compile by Ricardo Boer Page 64 Mathematics and Mathematical Science Level: 4. Duration: 60 min Date: __________________________________________________________________________________________________ Activity 1 Us 7448 So 1, 2, 3, 4, 5 All the Ac of the Specific Outcomes  Write down the next three terms in each of the following sequences: a) 5; 15; 25; ………. (2) b) -8;-5; 2; ………… (2) c) 30;27 ;24;…….. (2)  The general term is given for each sequence below. Calculate the missing terms a) 0;3;…….;15;24 𝑻𝒏= 𝒏𝟐 - 1 (2) b) 3;2;1;0;……..;-2 𝑻𝒏= −𝒏+𝟒 (2) c) -11;…..;-7;…..;-3 𝑻𝒏= −𝟏𝟑+ 𝟐𝒏 (2)  Find the general formula for the following sequences and then find 𝑇10 , 𝑇50 And 𝑇100 a) 2; 5; 8; 11;14 ;…….. (3) b) 0; 4 ; 8 ;12 ;16 ; ……… (3) c) 2;-1;-4;-7;-10 ;……. (2) Total: 20/_________ Compile by Ricardo Boer Page 65 Mathematics and Mathematical Science ACTIVITY SHEET __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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__________________________________________________________________________________________________ __________________________________________________________________________________________________ Compile by Ricardo Boer Page 67 Mathematics and Mathematical Science Patterns and conjecture In mathematics, a conjecture is a mathematical statement which appears to be true, but has not been formally proven. A conjecture can be thought of as a mathematicians way of saying “ I believe that it is true, but I have to proof it yet”. A conjecture is a good guess or an idea about a pattern. For example, make a conjecture about the next number in the pattern 2; 6; 11; 17;…. The terms increase by 4, then 5, and then 6. Conjecture: the next term will increase by 7, s3o it will be 17+7=24. Example: Adding even and odd numbers. Question  Investigate the type of number you get if you find the sum of the odd numbers and an even number.  Express your answer in words as a conjecture.  Use algebra to prove this conjecture. Solution: Step 1: First try some examples 23 + 12 = 35 148 + 31 = 179 11 + 200 = 211 Step 2: Make a conjecture The sum of any odd number and any even number is always odd. Step 3: Express algebraically 𝑆𝑢𝑚 = 2𝑥 + (2𝑦 − 1) = 2𝑥 + 2𝑦 − 1 = (2𝑥 + 2𝑦) − 1 = 2(𝑥 + 𝑦) − 1 From this we can see that 2(𝑥 + 𝑦)𝑖𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟. So then 2( 𝑥 + 𝑦) − 1 is an odd number. Therefore our conjecture is true. Compile by Ricardo Boer Page 68 Mathematics and Mathematical Science SECTION 4 This module covers the following Unit Standards, Specific Outcomes and Assessment Criterions: SAQA US ID: 7452(6 credits) = 1𝑐𝑟𝑒𝑑𝑖𝑡 × 10ℎ𝑜𝑢𝑟𝑠 = 60 𝑛𝑜𝑡𝑖𝑎𝑛𝑎𝑙 ℎ𝑜𝑢𝑟𝑠 (Describe, represent and interpret mathematical models in different contexts) So 1: Describe and represent relationships in a variety of contexts using tables AC: 1, 2, 3, 4 So 2: Describe and represent relationships in a variety of contexts using simple algebraic expressions Ac: 1-3 So 3: Describe and represent relationships in a variety of contexts using graphs Ac 1-4 So 4: Describe and represent relationships in a variety of contexts geometrically Ac 1-3 So 5: Analyse and explain the behaviour of graphs in terms of increasing and decreasing trends Ac 1-4 So 6: Analyse and explain the behaviour of general algebraic, equations and formulae Ac 1-3 Compile by Ricardo Boer Page 69 Mathematics and Mathematical Science .“When you say “It’s hard”, it actually means “I’m not strong enough to fight for it”. Stop saying it’s hard. Think positive!” – Unknown “Life is like photography. You need the negatives to develop.” – Unknown Compile by Ricardo Boer Page 70 Mathematics and Mathematical Science Interval notation It is important to note that the notation can only be used to represent an interval of real numbers. Examples: (3;11) Round brackets indicate that the number is not included. This interval includes all real numbers greater than but not equal to 3 and less than but not equal to 11. (−∞; −2) Round brackets are always use for positive and negative infinity. This interval includes all real numbers less than , but not equal to -2. [1;9) A square bracket indicates that the number is included. This interval includes all real numbers greater than or equal to 1 and less than but not equal to 9. Function notation This is a very useful way to express a function. Another way of writing 𝑦 = 2𝑥 +1 is 𝑓(𝑥) = 2𝑥 +1. We say “𝑓 𝑜𝑓 𝑥 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 2𝑥 + 1". 𝐴𝑛𝑦 𝑙𝑒𝑡𝑡𝑒𝑟 𝑐𝑎𝑛 𝑏𝑒 𝑢𝑠𝑒𝑑, 𝑓𝑜𝑟 𝑒𝑥𝑎𝑚𝑝𝑙𝑒, 𝑔(𝑥), ℎ(𝑥), 𝑝(𝑥), 𝑒𝑡𝑐. Determine the output value. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑓𝑜𝑟 𝑥 = −3 can be written as:𝑓𝑖𝑛𝑑 𝑓(−3)". Replace 𝑥 𝑤𝑖𝑡ℎ − 3: 𝑓(−3) = 2(−3) + 1 = −5 ∴ 𝑓(−3) = −5 This means that when 𝑥 = −3, 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 − 5. Determine the input value: 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑥 𝑡ℎ𝑎𝑡 𝑤𝑖𝑙𝑙 𝑔𝑖𝑣𝑒 𝑎 𝑦 − 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 27" 𝑐𝑎𝑛 𝑏𝑒 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑎𝑠 ∶ 𝑓𝑖𝑛𝑑 𝑥 𝑖𝑓 𝑓(𝑥) = 27" We write the following equation to solve for 𝑥: 2𝑥 + 1 = 27 Compile by Ricardo Boer Page 71 Mathematics and Mathematical Science ∴ 𝑥 = 13 𝑡ℎ𝑖𝑠 𝑚𝑒𝑎𝑛𝑠 𝑡ℎ𝑎𝑡 𝑤ℎ𝑒𝑛 𝑥 = 13 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 27. Representations of functions: Functions can be expressed in many different ways for different purposes. 1. Words: The relationship between two variables is such that one is always 5 less than the other” 2. Mapping diagram. Input Function −3 {0 5 Output −8 −5 0 𝑥−5 → 3. Table Input variables(𝑥) Output variables((𝑦) -3 -8 0 -5 5 0 4. Set of order number pairs-3;-8),(0;-5),(5;0) 5. Algebraic formula:𝑓(𝑥) = 𝑥 − 5 6. Graph : p.g. 112 Drawing (insert) Domain and range; The domain of a function is the set of all the independent 𝑥 − 𝑣𝑎𝑙𝑢𝑒𝑠 𝑓𝑜𝑟 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑜𝑛𝑒 𝑑𝑒𝑝𝑒𝑛𝑡 𝑦 − 𝑣𝑎𝑙𝑢𝑒 𝑎𝑐𝑐𝑜𝑟𝑑𝑖𝑛𝑔 𝑡𝑜 𝑡ℎ𝑎𝑡 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛. 𝑇ℎ𝑒 𝑟𝑎𝑛𝑔𝑒 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑦 − 𝑣𝑎𝑙𝑢𝑒𝑠 𝑤ℎ𝑖𝑐ℎ 𝑐𝑎𝑛 𝑏𝑒 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑 𝑢𝑠𝑖𝑛𝑔 𝑎𝑛 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑥 − 𝑣𝑎𝑙𝑢𝑒. Compile by Ricardo Boer Page 72 Mathematics and Mathematical Science Level: 4. Duration: 60 min Date: __________________________________________________________________________________________________ Activity 1 Us 7452 So 2 Ac 1-4 Write the following in set notation a) −∞; 7) b) [13;4) c) (35;∞) 3 d) (4;21) 1 1 e) (-2 ;2) (4) (4) (4) (4) (4) Total: 20/________ Compile by Ricardo Boer Page 73 Mathematics and Mathematical Science ACTIVITY SHEET __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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__________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ Compile by Ricardo Boer Page 75 Mathematics and Mathematical Science Linear functions 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚 𝑦 = 𝑥 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚 𝑦 = 𝑚𝑥 + 𝑐 𝑎𝑟𝑒 𝑐𝑎𝑙𝑙𝑒𝑑 𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑙𝑖𝑛𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠. 𝐼𝑛 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛, 𝑦 = 𝑚𝑥 + 𝑐, 𝑚 𝑎𝑛𝑑 𝑐 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠 𝑎𝑛𝑑 ℎ𝑎𝑣𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑒𝑓𝑓𝑒𝑐𝑡𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛. Example: Plotting a straight line graph Question 𝑦 = 𝑓(𝑥) = 𝑥 𝐶𝑜𝑚𝑝𝑙𝑒𝑡𝑒 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑡𝑎𝑏𝑙𝑒 𝑓𝑜𝑟 𝑓(𝑥) = 𝑥 𝑎𝑛𝑑 𝑝𝑙𝑜𝑡 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡𝑠 𝑜𝑛 𝑎 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑥𝑒𝑠. 𝑥 𝑓(𝑥) -2 -2 -1 0 1 3 1) 2) 3) 4) Join the points with a straight line. Determine the domain and range. About which line is 𝑓 𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐𝑎𝑙? Using the graph, determine the value of 𝑥 𝑓𝑜𝑟 𝑤ℎ𝑖𝑐ℎ 𝑓(𝑥) = 4. 𝐶𝑜𝑛𝑓𝑖𝑟𝑚 𝑦𝑜𝑢𝑟 𝑎𝑛𝑠𝑤𝑒𝑟 𝑔𝑟𝑎𝑝ℎ𝑖𝑐𝑎𝑙𝑙𝑦. 5) Where does the graph cut the axes? SOLUTION Step 1: Substitute values into the equation 𝑥 𝑓(𝑥) -2 -2 -1 -1 0 0 1 1 2 2 Step 2: Plot the points and join with a straight line curve From the table, we get the following points and the graph: (-2;-2),(-1;-1),(0;0),(1;1),(2;2) Compile by Ricardo Boer Page 76 Mathematics and Mathematical Science From the table, we get the following points and the graph: (−2; −2),(−1;−1),(0;0),(1;1),(2;2) (2) Step 3: Determine the domain and range Compile by Ricardo Boer Page 77 Mathematics and Mathematical Science 𝐷𝑜𝑚𝑎𝑖𝑛: 𝑥 ∈ 𝑅 𝑅𝑎𝑛𝑔𝑒: 𝑓(𝑥) ∈ 𝑅 Step 4: Determine the value of 𝒙 𝒇𝒐𝒓 𝒘𝒉𝒊𝒄𝒉 𝒇(𝒙) = 𝟒 From the graph we see that when 𝑓(𝑥) = 4, 𝑥 = 4. 𝑇ℎ𝑖𝑠 𝑔𝑖𝑣𝑒𝑠 𝑡ℎ𝑒 point (4;4). Step 5: Determine the intercept. 𝑇ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑓 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 𝑡ℎ𝑒 𝑎𝑥𝑒𝑠 𝑎𝑡 𝑡ℎ𝑒 𝑜𝑟𝑖𝑔𝑖𝑛 (0; 0). Functions of the form 𝒚 = 𝒎𝒙 + 𝒄 Investigation: The effects of 𝑚 𝑎𝑛𝑑 𝑐 𝑜𝑛 𝑎 𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑙𝑖𝑛𝑒 𝑔𝑟𝑎𝑝ℎ. On the same set of axes, plot the following graphs:      𝑦 =𝑥−2 𝑦 =𝑥−1 𝑦=𝑥 𝑦 =𝑥+1 𝑦 =𝑥+2 Use your results to deduce the effect of different values of c on the graph. On the same set of axes, plot the following graphs:;     𝑦 = −2𝑥 𝑦 = −𝑥 𝑦=𝑥 𝑦 = 2𝑥 Use your results to deduce the effect of different values of m on the graph. The effect of m We notice that the value of 𝑚 affects the slope of the graph. As 𝑚 increases, the gradient of the graph increases. 𝑖𝑓 𝑚 > 0 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 𝑡𝑜 𝑟𝑖𝑔ℎ𝑡(𝑠𝑙𝑜𝑝𝑒𝑠 𝑢𝑝𝑤𝑎𝑟𝑑𝑠). Compile by Ricardo Boer Page 78 Mathematics and Mathematical Science 𝑖𝑓 𝑚 < 0 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑓𝑟𝑜𝑚 𝑟𝑖𝑔ℎ𝑡 𝑡𝑜 𝑙𝑒𝑓𝑡( 𝑠𝑙𝑜𝑝𝑒𝑠 𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑𝑠). For the reason,𝑚 is referred to as the gradient of a straight-line graph. The effect of 𝒄 We also notice that the value of 𝑐 𝑎𝑓𝑓𝑒𝑐𝑡𝑠 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ 𝑐𝑢𝑡𝑠 𝑡ℎ𝑒 𝑦 − 𝑎𝑥𝑖𝑠. 𝐹𝑜𝑟 𝑡ℎ𝑖𝑠 𝑟𝑒𝑎𝑠𝑜𝑛, 𝑐 𝑖𝑠 𝑘𝑛𝑜𝑤𝑛 𝑎𝑠 𝑡ℎ𝑒 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡. 𝑖𝑓 𝑐 > 0 𝑡ℎ𝑒 𝑔𝑟𝑎 > 𝑝ℎ𝑡 𝑠ℎ𝑖𝑓𝑡𝑠 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑢𝑝𝑤𝑎𝑟𝑑𝑠. 𝑖𝑓 𝑐 < 0, 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ 𝑠ℎ𝑖𝑓𝑡𝑠 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑𝑠. m<0 m=0 m>0 c>0 c=0 c<0 SECTION 5 Compile by Ricardo Boer Page 79 Mathematics and Mathematical Science This module covers the following Unit Standards, Specific Outcomes and Assessment Criterions: SAQA US ID: 7450(2 credits) = 1𝑐𝑟𝑒𝑑𝑖𝑡 × 10ℎ𝑜𝑢𝑟𝑠 = 20 𝑛𝑜𝑡𝑖𝑎𝑛𝑎𝑙 ℎ𝑜𝑢𝑟𝑠 So 1: Apply relationships between common quantities in various contexts AC: 1, 2, 3 So 2: Use measuring instruments to measure and calculate quantities in various contexts Ac: 1-3 So 3: Solve measuring problems in various contexts Ac 1-3 Compile by Ricardo Boer Page 80 Mathematics and Mathematical Science “Don’t worry about failures, worry about the chances you miss when you don’t even try.” – Jack Canfield “The pain you feel today is the strength you feel tomorrow. For every challenge encountered there is opportunity for growth.” – Unknown “Build your own dreams, or someone else will hire you to build theirs.” – Farrah Gray EUCLIDEAN GEOMETRY Compile by Ricardo Boer Page 81 Mathematics and Mathematical Science Geometry (from the Greek “geo”=earth and “metria” =measure) arose as the field of knowledge dealing with spatial relationships. Geometry can be split into Euclidean geometry and analytical geometry. Analytical geometry deals with space and shape using algebra and a coordinate system. Euclidean geometry deals with space and shape using a system of logical deductions. ANGLES An angle is formed when two straight lines meet at a point, also known as a vertex. ̂ Angles can also be labelled Angles are labelled with a caret on a letter, for example, 𝐵. according to the line segments that make up the angle, for example, C𝐵̂A or A𝐵̂C. The ∠ symbol is a short method of writing angle in geometry and is often used for phrases such as “sum of ∠𝑠 𝑖𝑛 𝑎 △ "angles are measured in degrees which is denoted by ∘ , a small circle raised above the text, similar to the exponent. PROPERTIES AND NATATION In the diagram below two straight lines intersect at a point, forming the four angles𝑎̂,𝑏̂,𝑐̂ and 𝑑̂. CB ‸ A or AB ‸ C The ∠ symbol is a short method of writing angle in geometry and is often used for phrases such as “sum of ∠s in a △”. Angles are measured in degrees which is denoted by ∘ , a small circle raised above the text, similar to an exponent. Figure 1 Properties and notation In the diagram below two straight lines intersect at a point, forming the four angles a ‸ b ‸, c ‸ and d ‸. Compile by Ricardo Boer Page 82 Mathematics and Mathematical Science Figure 2 The following table summarises the different types of angles, with examples from the figure above. Term Property Examples 0 ∘ <angle<90 ∘ a‸ ;c‸ Acute angle Angle =90 ∘ Right angle 90 ∘ <angle<180 ∘ b‸ ;d‸ Angle =180 ∘ a ‸ +b ‸ ; b ‸ +c ‸ 180 ∘ <angle<360 ∘ a ‸ +b ‸ +c ‸ Obtuse angle Straight angle Reflex angle Compile by Ricardo Boer Page 83 Mathematics and Mathematical Science a ‸ and d ‸ ; c ‸ and d ‸ Adjacent angles Angles that share a vertex and a common side. a ‸ =c ‸ ; b ‸ =d ‸ Vertically Angles opposite each opposite angles other when two lines intersect. They share a vertex and are equal. Supplementary Two angles that add up a ‸ +b ‸ =180 ∘ ; to 180 ∘ . b ‸ +c ‸ =180 ∘ angles Two angles that add up Complementary to 90 ∘ . angles a ‸ +b ‸ +c ‸ +d ‸ =360 ∘ Revolution The sum of all angles around a point. Compile by Ricardo Boer Page 84 Mathematics and Mathematical Science Table 1 Note that adjacent angles on a straight line are supplementary. Parallel lines and transversal lines Two lines intersect if they cross each other at a point. For example, at a traffic intersection two or more streets intersect; the middle of the intersection is the common point between the streets. Parallel lines are always the same distance apart and they are denoted by arrow symbols as shown below. Figure 3 In writing we use two vertical lines to indicate that two lines are parallel: AB∥CD and MN∥OP (1) Compile by Ricardo Boer Page 85 Mathematics and Mathematical Science A transversal line intersects two or more parallel lines. In the diagram below, AB∥CD and EF is a transversal line. Figure 4 The properties of the angles formed by these intersecting lines are summarised in the following table: Name of angle Definition Examples Notes a ‸ , b ‸ , c ‸ and Interior angles Angles that lie in d ‸ are interior Interior means between the parallel angles. inside. lines. e ‸ , f ‸ , g ‸ and Exterior angles Angles that lie h ‸ are exterior Exterior means outside the parallel angles. outside. lines. Compile by Ricardo Boer Page 86 Mathematics and Mathematical Science a ‸ and e ‸ , Corresponding Angles on the same b ‸ and f ‸ , c ‸ and angles side of the lines and g ‸ , d ‸ and h ‸ are the same side of the pairs of transversal. If the corresponding angles. lines are parallel, the a ‸ =e ‸ , b ‸ =f ‸ , corresponding angles c ‸ =g ‸ , d ‸ =h ‸ . will be equal. a ‸ and d ‸ , Co-interior Angles that lie in b ‸ and c ‸ are pairs angles between the lines of co-interior angles. and on the same side a ‸ +d ‸ =180 ∘ , of the transversal. If b ‸ +c ‸ =180 ∘ . the lines are parallel, the angles are supplementary. a ‸ and c ‸ , b ‸ and Alternate Equal interior angles d ‸ are pairs of interior angles that lie inside the line alternate interior and on opposite angles. a ‸ =c ‸ , sides of the b ‸ =d ‸ . transversal. If the lines are parallel, the interior angles will be equal. Table 2 If two lines are intersected by a transversal such that  corresponding angles are equal; or Compile by Ricardo Boer Page 87 Mathematics and Mathematical Science  alternate interior angles are equal; or  co-interior angles are supplementary Then the two lines are parallel. Example 1: Finding angles Question Find all the unknown angles. Is EF∥CG ? Explain your answer. AB ∴x ‸ y ‸ +160 ∘ ∴y ‸ p ‸ ∴p ‸ r ‸ +p ‸ ∴r ‸ ∥CD =60 ∘ =180 ∘ =20 ∘ =y ‸ =20 ∘ =180 ∘ =160 ∘ (given) (alt.int.∠'s) (co-int.∠'s) (vert.opp.∠'s) (sumof∠'sstr.line) s ‸ +x ‸ s ‸ +60 ∘ ∴s ‸ =90 ∘ =90 ∘ =30 ∘ (given) EF∥CG EF ∥CG p ‸ s ‸ p ‸ =20 ∘ s ‸ =30 ∘ EF CG Compile by Ricardo Boer Page 88 Mathematics and Mathematical Science Level: 4. Duration: 60 min Date: __________________________________________________________________________________________________ Activity 1 Us: 7450 So: 2 Ac 1-3 Problem 1: Use adjacent, corresponding, co-interior and alternate angles to fill in all the angles labelled with letters in the diagram: (6) Problem 2: Find all the unknown angles in the figure: Compile by Ricardo Boer Page 89 Mathematics and Mathematical Science (11) Problem 3: Find the value of x in the figure: Compile by Ricardo Boer Page 90 Mathematics and Mathematical Science Total: 20/_______ ANSWER SHEET __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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__________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ The theorem of Pythagoras If △ABC is right-angled with B ‸ =90 ∘ , then b 2 =a 2 +c 2 . Converse: If b 2 =a 2 +c 2 , then △ABC is right-angled with B ‸ =90 ∘ . Example 1: Triangles Question Determine if the two triangles are congruent. Use the result to find x, y ‸ and z. Show me this worked solution △CDE≡△CBA △CDE D ‸ +C ‸ +E ‸ 90 ∘ +35 ∘ +E ‸ ∴E ‸ =180 ∘ =180 ∘ =55 ∘ (sumo f∠'sof△) △CDE △CBA DE ‸ C CD ‸ E DE ∴△CDE =BA ‸ C=55 ∘ =CB ‸ A=90 ∘ =BA=3 ≡△CBA (p roved) (given) (given) (AAS) Compile by Ricardo Boer Page 93 Mathematics and Mathematical Science △CDE CE 2 5 2 x 2 ∴x =DE 2 +CD 2 =3 2 +x 2 =16 =4 (Pythagoras) △CBA B ‸ +A ‸ +y ‸ 90 ∘ +55 ∘ +y ‸ ∴y ‸ =180 ∘ =180 ∘ =35 ∘ (sumof∠'sof△) △CDE ∴CE ∴z ≡△CBA =CA =5 (proved) Level: 4. Duration: 60 min Date: __________________________________________________________________________________________________ Activity: 2 Us: 7450 So: 2 Ac 1-3 Calculate the unknown variables in each of the following figures. (a)=3;(b)=2;(c)=3;(d)=2;(e)=2;(f)=3;(g)=3 Compile by Ricardo Boer Page 94 Mathematics and Mathematical Science Total:20/__________ ANSWER SHEET __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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__________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ We learn something from everyone who passes through our lives.. Some lessons are painful, some are painless .but, all are priceless.” – Unknown “Being happy doesn’t mean that everything is perfect. It means that you’ve decided to look beyond the imperfections.” – Unknown “Nobody ever wrote down a plan to be broke, fat, lazy, or stupid. Those things are what happen when you don’t have a plan.” – Larry Winget Compile by Ricardo Boer Page 97 Mathematics and Mathematical Science Area of a polygon Definition 1: Area Area is the two dimensional space inside the boundary of a flat object. It is measured in square units. Name Shape Formula Square Area=s2 Rectangle Area= b × h Triangle Area=2b×h 1 Table 1 Compile by Ricardo Boer Page 98 Mathematics and Mathematical Science Name Shape Formula 1 Trapezium Area=2(a+b)×h Parallelogr Area=b ×h am Area=πr2(1) Circle (Circumference=2πr )(2) Table 2 Area and perimeter Area of a circle Example 1: Finding the area of a polygon Question Find the area of the following parallelogram: Compile by Ricardo Boer Page 99 Mathematics and Mathematical Science Level: 4. Duration: 60 min Date: __________________________________________________________________________________________________ Activity 1 Us: 7450 So: 2 Ac 1-3 So 3 Ac1-3 Problem 1: Find the areas of each of the polygons below: Compile by Ricardo Boer Page 100 Mathematics and Mathematical Science Total: 30/_____________ ANSWER SHEET __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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__________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __ Right prisms and cylinders Definition 1: Right prism A right prism is a geometric solid that has a polygon as its base and vertical sides perpendicular to the base. The base and top surface are the same shape and size. It is called a “right” prism because the angles between the base and sides are right angles. A triangular prism has a triangle as its base, a rectangular prism has a rectangle as its base, and a cube is a rectangular prism with all its sides of equal length. A cylinder is another type of right prism which has a circle as its base. Examples of right prisms are given below: a rectangular prism, a cube, a triangular prism and a cylinder. Compile by Ricardo Boer Page 103 Mathematics and Mathematical Science Figure 1 Different viewpoints Surface area of prisms and cylinders Compile by Ricardo Boer Page 104 Mathematics and Mathematical Science Definition 2: Surface area Surface area is the total area of the exposed or outer surfaces of a prism. This is easier to understand if we imagine the prism to be a cardboard box that we can unfold. A solid that is unfolded like this is called a net. When a prism is unfolded into a net, we can clearly see each of its faces. In order to calculate the surface area of the prism, we can then simply calculate the area of each face, and add them all together. For example, when a triangular prism is unfolded into a net, we can see that it has two faces that are triangles and three faces that are rectangles. To calculate the surface area of the prism, we find the area of each triangle and each rectangle, and add them together. In the case of a cylinder the top and bottom faces are circles and the curved surface flattens into a rectangle with a length that is equal to the circumference of the circular base. To calculate the surface area we therefore find the area of the two circles and the rectangle and add them together. Below are examples of right prisms that have been unfolded into nets: Compile by Ricardo Boer Page 105 Mathematics and Mathematical Science Figure 2: Rectangular prism A rectangular prism unfolded into a net is made up of six rectangles. Compile by Ricardo Boer Page 106 Mathematics and Mathematical Science Figure 3: Cube A cube unfolded into a net is made up of six identical squares. Compile by Ricardo Boer Page 107 Mathematics and Mathematical Science Figure 4: Triangular prism A triangular prism unfolded into a net is made up of two triangles and three rectangles. The sum of the lengths of the rectangles is equal to the perimeter of the triangles. Compile by Ricardo Boer Page 108 Mathematics and Mathematical Science Figure 5: Cylinder A cylinder unfolded into a net is made up of two identical circles and a rectangle with a length equal to the circumference of the circles. Compile by Ricardo Boer Page 109 Mathematics and Mathematical Science Example 1: Finding the surface area of a rectangular prism Question Find the surface area of the following rectangular prism: large rectangle 2×smallrectangle =perimeterofsmallrectangle×length =(2+5+2+5)×10 =1 4×10 =140cm 2 =2(5×2) =2(10) =20cm 2 Large rectangle+2×smallrectangle=140+20=160 2 Example 2: Finding the surface area of a triangular prism Question Find the surface area of the following triangular prism: x 2 x 2 ∴x ∴perimeteroftriangle =3 2 +(8 2 ) 2 =3 2 +4 2 =25 =5cm =5+5+8 =18cm Compile by Ricardo Boer Page 110 Mathematics and Mathematical Science ∴ area of large rectangle =perimeter of triangle × length =18×12 =216 𝑐𝑚2 1 1 Area of triangle =2 b × h =2 ×8×3 =12𝑐𝑚2 surface area =area large rectangle+(2×area of triangle) =216+2(12) =240𝑐𝑚2 Example 3: Finding the surface area of a cylindrical prism Question Find the surface area of the following cylinder (correct to 1 decimal place): area of large rectangle area of circle =circumference of circle × length =2πr × l =2π(10)×30 =1884,96cm 2 =πr 2 =π10 2 =314,16𝑐𝑚2 surface area =area large rectangle +(2 ×area of circle) =1884,96+2(314,16) = 2513,28 𝑐𝑚2 Compile by Ricardo Boer Page 111 Mathematics and Mathematical Science Level: 4. Duration: 60 min Date: __________________________________________________________________________________________________ Activity 2 Us: 7450 So: 2 Ac 1-3 So 3 Ac 1-3 Problem 1: Calculate the surface area of the following prisms: (12) Problem 2: If a litre of paint covers an area of 2 m2, how much paint does a painter need to cover: Compile by Ricardo Boer Page 112 Mathematics and Mathematical Science 1. a rectangular swimming pool with dimensions 4 m ×3 m ×2,5 m (the inside walls and floor only); (4) 2. The inside walls and floor of a circular reservoir with diameter 4 m and height 2,5 m. (4) Total: 20/__________ Compile by Ricardo Boer Page 113 Mathematics and Mathematical Science ANSWER SHEET __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ Compile by Ricardo Boer Page 114 Mathematics and Mathematical Science Volume of prisms and cylinders Definition 3: Volume Volume is the three dimensional space occupied by an object, or the contents of an object. It is measured in cubic units. The volume of a right prism is simply calculated by multiplying the area of the base of a solid by the height of the solid. Calculating volume Volume =area of base ×height =area Rectangular of rectangle ×height =l ×b ×h prism (8) Volume =area of base ×height =area 1 Triangular of triangle ×height =( b ×h)×H 2 prism (9) Volume =area of base ×height =area Cylinder of circle ×height =πr 2 ×h (10) Table 1 Compile by Ricardo Boer Page 115 Mathematics and Mathematical Science Example 4: Finding the volume of a cube Question Find the volume of the following cube: Area of square =s 2 =3 2 = 9𝑐𝑚2 Volume =area of base ×height =9×3 =27𝑐𝑚3 Example 5: Finding the volume of a triangular prism Question Find the volume of the triangular prism: 1 1 Area of triangle =2 b× h = (2 ×8)×10 =40𝑐𝑚2 Volume 1 =area of base ×height =2 𝑏 ×h ×H =40×20 =800𝑐𝑚3 800cm 3 Example 6: Finding the volume of a cylindrical prism Question Find the volume of the following cylinder (correct to 1 decimal place): Area of circle = πr² =π(4)² =16πcm² Volume =area of base ×height =πr² ×h =16π×15 =754,0cm³ Compile by Ricardo Boer Page 116 Mathematics and Mathematical Science Level: 4. Duration: 60 min Date: __________________________________________________________________________________________________ Activity 3 Us: 7450 So: 3 Ac 1-3 Problem 1: Calculate the volumes of the following prisms (correct to 1 decimal place): Total: 15/___________ Compile by Ricardo Boer Page 117 Mathematics and Mathematical Science ANSWER SHEET __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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__________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ Compile by Ricardo Boer Page 119 Mathematics and Mathematical Science Right pyramids, right cones and spheres Definition 1: Pyramid A pyramid is a geometric solid that has a polygon as its base and sides that converge at a point called the apex. In other words the sides are not perpendicular to the base. The triangular pyramid and square pyramid take their names from the shape of their base. We call a pyramid a “right pyramid” if the line between the apex and the centre of the base is perpendicular to the base. Cones are similar to pyramids except that their bases are circles instead of polygons. Spheres are solids that are perfectly round and look the same from any direction. Examples of a square pyramid, a triangular pyramid, a cone and a sphere: Compile by Ricardo Boer Page 120 Mathematics and Mathematical Science Level: 4. Duration: 60 min Date: __________________________________________________________________________________________________ Activity 4 Us: 7450 So: 3 Ac 1-3. Find the total surface area of the following objects (correct to 1 decimal place if necessary): Total: 20/_____________ Compile by Ricardo Boer Page 121 Mathematics and Mathematical Science ANSWER SHEET __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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__________________________________________________________________________________________________ __________________________________________________________________________________________________ Compile by Ricardo Boer Page 123 Mathematics and Mathematical Science Chapter summary  Area is the two dimensional space inside the boundary of a flat object. It is measured in square units.  Area formulae: o square: s 2 o rectangle: b ×h 1 triangle: 2 b ×h o 1 o trapezium: 2 (a+b)×h o parallelogram: b ×h o circle: πr²  Surface area is the total area of the exposed or outer surfaces of a prism.  A net is the unfolded “plan” of a solid.  Volume is the three dimensional space occupied by an object, or the contents of an object. It is measured in cubic units. o o  Volume of a rectangular prism: 1l ×b ×h 1 Volume of a triangular prism: (2 b ×h)×H o Volume of a square prism or cube: s³ o Volume of a cylinder: πr² ×h A pyramid is a geometric solid that has a polygon as its base and sides that converge at a point called the apex. The sides are not perpendicular to the base.  Surface area formulae: o o square pyramid: b(b+2h) 1 triangular pyramid: 2b(h b +3h s ) o right cone: πr(r +h s ) o sphere: 4πr² Compile by Ricardo Boer Page 124 Mathematics and Mathematical Science SECTION 6 This module covers the following Unit Standards, Specific Outcomes and Assessment Criterions: SAQA US ID: 7449(2 credits) = 1𝑐𝑟𝑒𝑑𝑖𝑡 × 10ℎ𝑜𝑢𝑟𝑠 = 20 𝑛𝑜𝑡𝑖𝑎𝑛𝑎𝑙 ℎ𝑜𝑢𝑟𝑠 (Critically analyse how mathematics is used in social, political and economic relations) So1: Critically analyse the use of mathematical language and relationships in the workplace Ac 2 Compile by Ricardo Boer Page 125 Mathematics and Mathematical Science Three things you cannot recover in life: the WORD after it’s said, the MOMENT after it’s missed and the TIME after it’s gone. Be Careful!” – Unknown “Though no one can go back and make a brand new start, anyone can start from now and make a brand new ending.” – Carl Bard Compile by Ricardo Boer Page 126 Mathematics and Mathematical Science Level: 4. Duration: 60 min Date: __________________________________________________________________________________________________ Activity 1 Us: 7449 So: 2 Ac 2 1: Calculate the mean, median and mode of the following data sets: 1. 2;5;8;8;11;13;22;23;27 2. 15;17;24;24;26;28;31;43 3. 4;11;3;15;11;13;25;17;2;11 4. 24;35;28;41;32;49;31 (12) 2: The ages of 15 runners of the Comrades Marathon were recorded: 31;42;28;38;45;51;33;29;42;26;34;56;33;46;41 Calculate the mean, median and modal age. (4) 3: In the first of a series of jars, there is 1 sweet. In the second jar, there are 3 sweets. The mean number of sweets in the first two jars is 2. 1. If the mean number of sweets in the first three jars is 3, how many sweets are there in the third jar? (4) Compile by Ricardo Boer Page 127 Mathematics and Mathematical Science 2. If the mean number of sweets in the first four jars is 4, how many sweets are there in the fourth jar? (3) Problem 4: Find a set of five ages for which the mean age is 5, the modal age is 2 and the median age is 3 years. (3) Problem 5: Four friends each have some marbles. They work out that the mean number of marbles they have is 10. One friend leaves with 4 marbles. How many marbles do the remaining friends have together? (4) Total: 30/__________ Compile by Ricardo Boer Page 128 Mathematics and Mathematical Science ANSWER SHEET _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ 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Mathematics and Mathematical Science When the past calls, let it go to voicemail, believe me, it has nothing new to say.” – Unknown “Rule 1 of life. Do what makes YOU happy.” – Unknown “Walk away from anything or anyone who takes away from your joy. Life is too short to put up with fools.” – Unknown ..“Love what you have. Need what you want. Accept what you receive. Give what you can. Always remember, what goes around, comes around…” – Unknown .. “Just remember there is someone out there that is more than happy with less than what you have.” – Unknown “The biggest failure you can have in life is making the mistake of never trying at all.” – Unknown “Life has two rules: #1 never quit #2 always remember rule # 1.” – Unknown “No one is going to hand me success. I must go out & get it myself. That’s why I’m here to dominate to conquer.Both the world, and myself.” – Unknown Compile by Ricardo Boer Page 131 Mathematics and Mathematical Science INSTRUCTIONS AND INFORMATION 1. Answer ALL the questions in your ANSWER BOOK. 2. Calculators may be used, but you must show ALL calculations. 3. Read the questions carefully before you write your answers. 4. Write legibly and present your work clearly. 5. Number the answers correctly and clearly. 6. Write the answers in blue or black ink. Compile by Ricardo Boer Page 132 Mathematics and Mathematical Science QUESTION 1 1.1 1.2 Is the number pattern: 950; 925; 900; … increasing or decreasing? Explain your answer. (2) Continue the following number patterns by adding two more terms to each pattern. 1.2.1 21; 28; 35; ... ; ...; (2) 1.2.2 8; 24; 72; ... ; ...; (2) 1.2.3 1; 4; 9; … ; ...; (2) 1.3 Describe the pattern in QUESTION 1.2.3 in your own words. 1.4 Triangles are flipped to form the patterns below. Study the patterns of triangles and their sides to answer the questions that follow: 2 triangles 1 triangle 1.4.1 1.4.2 (2) 3 triangles Continue the pattern by drawing the patterns for 4 triangles and 5 triangles in the ANSWER BOOK. (4) The table below will show the number of sides for different combinations of triangles in QUESTION 1.4 and 1.4.1. Copy and complete the table in your answer book. Number of triangles (t) Number of sides (s) 1 3 2 3 4 5 (4) 1.4.3 Describe the pattern in your own words. (2) 1.4.4 How many sides will 8 triangles, arranged in this manner, have? (2) Compile by Ricardo Boer Page 133 Mathematics and Mathematical Science 1.4.5 Determine a general rule for the Number of sides (s) in a certain Number of triangles (t) in the form s = … 1.4.6 Use your rule, or any other method, to determine how many triangles can be made if there are 81 sides. (2) (3) [27] QUESTION 2 2.1 Write down the answers to complete the following flow chart for the given input values: 2.1.1 8 ½ 93 2.2 + 17 (4) 2.1.2 A restaurant charges a minimum amount just to enter but once you are inside all items on the menu cost R10. The graph below shows the total cost in rand (C), after having n items to eat. Restaurant prices 120 110 100 Cost in Rand 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 No. of items 2.2.1 Copy the following table in your answer book. Use the graph above to complete the table: No. of items (n) Cost in rand (C) 2.2.2 0 20 1 2 5 (4) 7 (a) The graph does not start at the point (0;0). Why not? Compile by Ricardo Boer Page 134 (1) Mathematics and Mathematical Science (b) How much does it cost to enter the restaurant? (1) 2.2.3 Is the given graph increasing or decreasing? Motivate your answer. (2) 2.2.4 If your bill comes to R100, how many items did you eat? (1) 2.2.5 Show by substitution that the formula C = 10n + 20 can be used to calculate the cost for eating a certain number of items. Select your own number for substitution. (3) 2.2.6 How much will the total cost be to eat 12 items? (3) [19] QUESTION 3 3.1 Choose the correct word from those given in brackets: 3.1.1 3.1.2 3.2 The theorem of Pythagoras is applicable to (scalene/right-angled) triangles. (1) The longest side of a right-angled triangle is called the (hypotenuse/right-angled side). (1) Write down the theorem of Pythagoras for the triangle below. Use AB, AC and CB in your answer. A B (2) C Compile by Ricardo Boer Page 135 Mathematics and Mathematical Science 3.3 The diagram below shows a perpendicular cliff JK next to a river with a boat sailing on the river. 3,4 m J L 8m K Calculate the distance (JL) from the top of the cliff to the bottom of the boat if the cliff is 3,4 m high and the boat is 8 m away from the cliff. Round off your answer to one decimal place. (6) [10] QUESTION 4 4.1 4.2 Themba is 3 years older than Basil. The sum of Basil’s and Themba’s ages is 27 years. 4.1.1 If Basil is x years old, how old is Themba in terms of x? (1) 4.1.2 Use the information above to determine an equation in terms of x. (1) 4.1.3 Determine the ages of Basil and Themba. (4) Tn = –3n is a formula to calculate the terms of a row of terms. 4.2.1 4.2.2 4.3 Substitute n = 1, 2 and 3 into the formula to determine the first 3 terms of the row. (3) Do the terms increase or decrease? (1) Solve 2x – 6  3x – 4 and represent the solution on a number line. (5) [15] QUESTION 5 5.1 The formula for the area of a rectangle is given by A = l  b , where A = area, l = length and b = breadth. Calculate the breadth if A = 312,5 m2 and l = 25 m. Compile by Ricardo Boer Page 136 (4) Mathematics and Mathematical Science 5.2 Simplify the following expressions: 5.2.1 5.2.2 5.3 5.4 (3) a(a 2  2b 2  c) 3m 2 (2) m3 Factorise the following expressions completely: 5.3.1 2( x  1)  3x( x  1) (2) 5.3.2 18 x 3  2 x (4) The area of a rectangle is 6 m2. Determine the possible lengths and breadths of the rectangle. Let the possible lengths and breadths be only natural numbers. (2) [17] QUESTION 6 6.1 The shapes inside a beehive can be drawn geometrically as shown in the C below. diagram E A Source: Discovering Geometry- Serra (2003) Compile by Ricardo Boer B D F Page 137 Mathematics and Mathematical Science 6.1.1 6.1.2 6.2 Select and write down the name of the shape drawn on the right (triangle, pentagon, rectangle, hexagon). (1) Select and write down the correct answer: The following are lines of symmetry: (AB only) OR (AB and EF) OR (AB, CD and EF). (1) Study the diagram of a part of a pattern on a necklace and then answer the questions that follow. A B C D K 6.3 J I 6.2.1 Name THREE different shapes that can be found in the diagram. (3) 6.2.2 Describe the transformation that transformed the figure on the left of line BJ to that on the right. (2) 6.2.3 Are ∆AKD and ∆CID congruent or not? Motivate your answer. (2) 6.2.4 What effect does the different shades of black and white have on the picture? (1) Why are houses usually built on rectangular and not on round plots? TOTAL: Compile by Ricardo Boer Page 138 (2) [12] 100 Mathematics and Mathematical Science __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ __________________________________________________________________________________________________ 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Write in black or blue ink. All the drawings must be done in pencils Need to get 60% to be found competent in this Exam No use of calculators Question 1 Factorise the following      𝒂𝟐 − 𝒂 𝒎𝟐− 𝟑𝟔 𝟗𝒃𝟐 − 𝟖𝟏 𝒂𝟐 − 𝟏𝟎 + 𝟐𝟓 𝟏𝟔𝒃𝟐 + 𝟓𝟔𝒃 + 𝟒𝟗 (2) (2) (2) (3) (3) Question 2 Simplify the following:    (𝒂 − 𝟐)𝟐 -𝒂(𝒂 + 𝟒) (𝟓𝒂 − 𝟒𝒃)(𝟐𝟓𝒂𝟐 + 𝟐𝟎𝒂𝒃 + 𝟏𝟔𝒃𝟐 (𝟐𝒎 − 𝟑)(𝟒𝒎𝟐 + 𝟗)(𝟐𝒎 + 𝟑)  𝟐 𝒙 𝒙 +𝟐 - 𝟐𝒙 𝟑 (3) (3) (3) (3) Question 3 Solve the following equations:       𝟐𝒚 − 𝟑 = 𝟕 −𝟑𝒚 = 𝟎 𝟏𝟔𝒚 + 𝟒 = −10 𝟐𝟑𝒙 − 𝟏𝟐 = 𝟔 + 𝟑𝒙 𝟏𝟐 − 𝟔𝒙 + 𝟑𝟒𝒙 = 𝟐𝒙 − 𝟐𝟒 − 𝟔𝟒 𝟒 𝒑 𝟏𝟔 =𝟐𝟒 Compile by Ricardo Boer (2) (2) (2) (3) (4) (3) Page 143 Mathematics and Mathematical Science Question 4 Complete the following: Find the 6th term in each of the following sequence:    4;13;22;31 (2) 5;2;-1;-4 (2) Find the general term of the following sequences:    3;7;11;15 (3) -2;1;4;7 (3) The seating of sports stadium is arranged so that the first row has 15seats; the second row has 19 seats, the third row 23 seats and so on. Calculate how many seats will be in the twenty-fifth row. (3) Question 5 Instructions: Please write in black or blue ink. Show all the calculations of the sums. Problem 1: Calculate the volumes of the following prisms (correct to 1 decimal place): Compile by Ricardo Boer Page 144 Mathematics and Mathematical Science (15) Question 8 There are 240 children at a party. The ratio of the number of boys to the ,number of girls at the party is 3 : 1. How many boys are there at the party? (2) Total Marks 100 Compile by Ricardo Boer Page 145 Mathematics and Mathematical Science ANSWER SHEET: _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ 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The question paper consists of 6 questions. Answer all the questions Please write in black or blue ink All drawings must be in pencils Show all calculations You may use a calculator Please number correctly No use off tipex Draw a line thru if you made a mistake with a question QUESTION 1 Write each decimal as a simple fraction a) 0,12 b) 0,006 c) 1,59 (6) ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ __________ Use your calculator and write the following numbers to 3 decimal places a) √2 b) √5 c) √6 (6) ________________________________________________________________________________ ________________________________________________________________________________ Compile by Ricardo Boer Page 149 Mathematics and Mathematical Science ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ __________ Find the 6th term in each of the following sequences; a) 4; 13; 22;31;…… b) 5;2;-1;-4;…….. c) 7,9;9,7;12;14,3;………. (6) ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________ Find the general term of the following sequences; a) 3;7;11;15 b) -2;1;4;7 (6) ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ____________ QUESTION 2 Solve x in the following; a) 𝑥 5 = 32 b) 2𝑥 3 = 128 c) 𝑥½ = 16 (6) ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ Compile by Ricardo Boer Page 150 Mathematics and Mathematical Science ________________________________________________________________________________ ________________ Work out the following sums; a) (𝑥 + 3)(𝑥 2 + 𝑥 + 2) b) (𝑥 + 𝑦)³ = (4) (4) ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ______________________ QUESTION 3 Factorise the following; a) b) c) d) 6𝑥 2 − 𝑥 − 15 𝑥 2 − 7𝑥 − 18 2𝑎(3𝑥 −2y)-−5𝑏(3𝑥 − 2𝑦) 6𝑎𝑥 − 4𝑏𝑦 − 8𝑥𝑏 + 3𝑎𝑦 (3) (3) (2) (4) ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ Compile by Ricardo Boer Page 151 Mathematics and Mathematical Science ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ __________________________________ Solve for x in each of the following equations; a) 6𝑥 − 3(𝑥 + 2) = 3 + 6𝑥 b) 5(3 − 𝑥) + 2 = 𝑥 + 7 (3) (3) ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ______________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ _______________________________________________________________________________ Compile by Ricardo Boer Page 152 Mathematics and Mathematical Science QUESTION 4 Philani bought a scooter for R15 000. He paid 15% of the amount in cash and signed a hirepurchase agreement to pay the balance in 24 equal monthly instalments. The interest rate is 10% per annum. .4.1 How much did he pay in cash? ___________________________________________________________ ___________________________________________________________ (1) 4.2 Calculate the total amount that he must still pay. ___________________________________________________________ ___________________________________________________________ ___________________________________________________________ ___________________________________________________________ ___________________________________________________________ (4) 4.3 Calculate the monthly instalment. __________________________________________________________ ___________________________________________________________ (2) A bag contains 6 red, 3 blue, 2 green and 1 white balls. A ball is picked at random. Determine the probability that it is: (a) red (b) Blue or white (c) Not green (d) Not green or red (8) A playing card is selected randomly from a pack of 52 cards. Determine the probability that it is: (a) The 2 of hearts (b) A red card (c) A picture card (d) An ace (e) a number less than 4 (10) Even numbers in the range 2 to 100 are written on cards. What is the probability of Selecting a multiple of 5, if a card is drawn at random? (4) Compile by Ricardo Boer Page 153 Mathematics and Mathematical Science _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ TOTAL: 85/________ Compile by Ricardo Boer Page 154 Mathematics and Mathematical Science Compile by Ricardo Boer Page 155 Mathematics and Mathematical Science Compile by Ricardo Boer Page 156 Mathematics and Mathematical Science Compile by Ricardo Boer Page 157 Mathematics and Mathematical Science Compile by Ricardo Boer Page 158 Mathematics and Mathematical Science Compile by Ricardo Boer Page 159 Mathematics and Mathematical Science Compile by Ricardo Boer Page 160 Mathematics and Mathematical Science Compile by Ricardo Boer Page 161 Mathematics and Mathematical Science Compile by Ricardo Boer Page 162 Mathematics and Mathematical Science Compile by Ricardo Boer Page 163 Mathematics and Mathematical Science Compile by Ricardo Boer Page 164 Mathematics and Mathematical Science Compile by Ricardo Boer Page 165 Mathematics and Mathematical Science Reference: EADIE, A. AND LAMPE, G. Mathematics In-text: (Eadie and Lampe, n.d.) Your Bibliography: Eadie, A. and Lampe, G. (n.d.). Mathematics. 1st ed. EVERYTHING MATHS AND SCIENCE This book was derived from the original Free High School Science Texts written by volunteer academics, educators and industry professionals. Everything Maths and Everything Science are trademarks of Siyavula Education. Siyavula Authors Luke Kannemeyer; Alison Jenkin; Marina van Zyl; Dr Carl Scheffler Siyavula and DBE team Heather Williams; Nkosilathi Vundla; Bridget Nash; Ewald Zietsman; William Buthane Chauke; Leonard Gumani Mudau; Sthe Khanyile; Josephine Mamaroke Phatlane Siyavula and Free High School Science Text contributors Dr Mark Horner; Dr Samuel Halliday; Dr Sarah Blyth; Dr Rory Adams; Dr Spencer Wheaton Siyavula Education Iesrafeel Abbas; Sarah Abel; Taskeen Adam; Ross Adams; Tracey Adams; Dr Rory Adams; Andrea Africa; Wiehan Agenbag; Ismail Akhalwaya; Matthew Amundsen; Ben Anhalt; Prashant Arora; Bianca Bôhmer; Amos Baloyi; Bongani Baloyi; Raymond Barbour; Caro-Joy Barendse; Katherine Barry; Dr Ilsa Basson; Richard Baxter; Tara Beckerling; Tim van Beek; Lisette de Beer; Prof Margot Berger; Jessie Bester; Mariaan Bester; Jennifer de Beyer; Dr Sarah Blyth; Sebastian Bodenstein; Martin Bongers; Dr Thinus Booysen; Ena Bosman; Janita Botha; Pieter Botha; Gareth Boxall; Stephan Brandt; Hannes Breytenbach; Alexander Briell; Wilbur Britz; Graeme Broster; Craig Brown; Michail Brynard; Richard Burge; Christina Buys; Jan Buys; George Calder-Potts; Biddy Cameron; Eleanor Cameron; Mark Carolissen; Shane Carollisson; Richard Case; Sithembile Cele; Alice Chang; William Buthane Chauke; Faith Chaza; Richard Cheng; Fanny Cherblanc; Saymore Chifamba; Lizzy Chivaka; Dr Christine Chung; Dr Mareli Claasens; Brett Cocks; Zelmari Coetzee; Phillipa Colly; Roché Compaan; Willem Conradie; Stefaan Conradie; Deanne Coppejans; Rocco Coppejans; Gary Coppin; Tim Craib; Dr Andrew Craig; Tim Crombie; Dan Crytser; Jock Currie; Dr Anne Dabrowski; Laura Daniels; Gareth Davies; Mia de; Tariq Desai; Sandra Dickson; Sean Dobbs; Buhle Donga; William Donkin; Esmi Dreyer; Matthew Duddy; Christel Durie; Fernando Durrell; Dr Dan Dwyer; Frans van Eeden; Kobus Ehlers; Alexander Ellis; Tom Ellis; Dr Anthony Essien; Charl Esterhuysen; Andrew Fisher; Dr Philip Fourie; Giovanni Franzoni; Sanette Gildenhuys; Olivia Gillett; Ingrid von Glehn; Tamara von Glehn; Nicola Glenday; Lindsay Glesener; Kevin Godby; Dr Vanessa Godfrey; Terence Goldberg; Dr Johan Gonzalez; Saaligha Gool; Hemant Gopal; Dr Stephanie Gould; Umeshree Govender; Dr Ilse le Grange; Heather Gray; Lynn Greeff; Jaco Greyling; Martli Greyvenstein; Carine Grobbelaar; Suzanne Grové; Eric Gulbis; Dr Tom Gutierrez; Brooke Haag; Kate Hadley; Alex Hall; Dr Samuel Halliday; Asheena Hanuman; Dr Melanie Dymond Harper; Ebrahim Harris; Dr Nicholas Harrison; Neil Hart; Nicholas Hatcher; Jason Hayden; Laura Hayward; Thomas Haywood; Dr William P. 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