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## Preparation for Commercial / ATP Pilots

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**Your instructor for this Ground School…Brand WesselsCell:**073-591 3907Email: brandwessels@yahoo.com**Agenda:**• Personal Introductions – name, background, qualifications • What is our expectations from this course • Course Rules – be on time, be prepared and participate constructively • Be Professional • Course Schedule • Have FUN!**Mathematics**What do we need to know? We need to have basic understanding of: • Basic Algebra - cross-multiplication, cross-addition and -subtraction, averaging, powers and roots, bracketing, percentages, inverse calculation and vectors. • Basic Trigonometry – Triangles, Ratio’s, Pythagoras. • Basic Interpolation. • How to operate the Navigation Computer and Scientific Calculator.**The Myth – “this is difficult…”**• If you passed Mathematics up to Grade 10 standard grade, you have covered everything you will need. • There is NOTHING in the CAA theory syllabus, that is as difficult as passing a current Grade 12 Higher Grade Mathematics' exam – go try one….**That Being Said….**• You have to know the basics WELL. • You have to know your calculators WELL. • You have to stay CURRENT. • You have to show enough RESPECT for the basics required. • The CPL/ATPL exam is quite a lot of work – about the same volume as the first 3 month’s of Engineering studies at University. If you are not on top of the required mathematics, you will waste time.**The ABC of this course….**• APPLY your • BACKSIDE to the • CHAIR….**Trig Example: You are taking off from a runway, with a hill**300’ high, 6000’ from the threshold. What angle of climb must you maintain to clear the hill? tan c = b/a And y=300’ and x=6000’ Tan x = 0.05 Divide by tan same as inverse (or cot, or tanˉ¹) Thus c = 2,86º Let's Go... Push this button just before you choose a 2nd function button Inverse Button Force of 3 Force of 2 10 to the force … Square Root Brackets Degree, minutes, seconds – also hours, minutes, seconds % Button**A fraction is an ordered pair of whole numbers, the 1st one**is usually written on top of the other, such as ½ or ¾ . numerator denominator The denominator tells us how many pieces the whole is divided into, thus this number cannot be 0. The numerator tells us how many such pieces are being considered.**Variable – A variable is a letter or symbol that**represents a number (unknown quantity). • 8 + n = 12**A variable can use any letter of the alphabet.**• n + 5 • x – 7 • w - 25**An Equation is like a balance scale. Everything must be**equal on both sides. = 10 5 + 5**When an amount is unknown on one side of the equation it is**an open equation. = 7 n + 2**When you find a number for n you change the open equation to**a true equation. You solve the equation. = 7 5 n + 2**Simple Algebra**• Remember Rules: • The sum of two positive numbers is always positive. • The sum of two negative numbers is always negative. • Multiplication/Division of two positive numbers is always positive. • Multiplication/Division of two negative numbers is always positive. • Multiplication/Division of a positive and a negative number is always negative.**Addition and Subtraction**26+(-38)-(-55)+(-61)-(23) = -41 On the calculator – type it all without the brackets….**Powers and Roots**2 x 2, same as 2², same as 2 to the power of 2, same as 4. The root of 4, same as √4, same as 2. On the calculator: 2x², enter = 4 √4, enter = 2**Inverse operation**the opposite operation used to undo the first. • 4 + 3 = 7 7 – 4 = 3 • 6 x 6 = 36 36 / 6 = 6 • Use “xˉ¹” on you calculator.**Parentheses and Brackets)**Use brackets when you want to do certain calculation before the rest: b² = 60² - (35000÷6080)² b² = 3566,8. Now press √ and b = 59,72 C = 2 x ((9÷3) + (4+3)²) C = 2 x (3 + 49) C = 104**Order of Algebraic Operation:**“PEMDAS” Solve in the following sequence: • P for solving Parentheses(or brackets) • E for solving Exponents next • MD for Multiplication and Division next • AS for Adding and Subtracting next.**Example:**• y = ((4³ + √((3+27) – (25÷5))) ÷ 3) + 273 • P is y = ((4³ + √(30 – 5)) ÷ 3) +273 And y = ((4³ + √25) ÷ 3) +273 • E is y = ((64 + 5) ÷ 3) +273 • MD is y = (69 ÷ 3) + 273 • AS is y = 23 + 273 = 296 Prove it by typing the whole equation into your calculator at once….**Procedure**• Isolate the variable by performing the inverse operation on that variable. • The inverse of subtraction is adding. The inverse of adding is subtracting. • Perform the same operation on the side of the equal sign that does not have a variable.**We want to get the y by itself. Perform the inverse**operation. The inverse of adding is subtracting. Example y + 13 = 25 - 13 - 13 Do the same operation on the other side of the equal sign. y = 12**Check the answer in the original equation.**y + 13 = 25 12 + 13 = 25 25 = 25**Example 2**To get k by itself, we perform the inverse operation. The opposite of “minus 12” is “plus 12.” k – 12 = 4 + 12 + 12 k = 16**Check**k – 12 = 4 16 – 12 = 4 4 = 4**Procedure**• Isolate the variable by performing the inverse operation on the number that is attached to the variable. • The inverse of multiplication is division. The inverse of division is multiplication. • Use the “Golden Rule.” Perform the same operation on the other side of the equal sign.**Example**m ÷ 3 = 10 The inverse of division is multiplication. x 3 x 3 Repeat the operation on the other side. 30 m =**Check. Use the original equation.**m ÷ 3 = 10 30 ÷ 3 = 10 10 = 10**Example 2**The inverse of multiplying is dividing. 7b = 105 ÷ 7 ÷ 7 1 5 105 7 b = 15 7 3 5 35 0**Check**7b = 105 7(15) = 105 105 = 105**Cross Multiplication**Moving the variable around in a function, until the unknown variable is isolated. Example: In a² = b² + c², if we have to solve for c we have to isolate it on one side of the equal sign. Important: What you do on one side of the equation has to be done on the other side. Thus: a² = b² + c² - b² leaves c² isolated, but then we have to subtract b² on the left side of the equation as well: a² - b² = c²**And solving for B in the following function:**x b b x Something divided by itself = 1 SIN B =**Remember “of” means multiply in mathematics.**• “Per” means division in mathematics.**Solve the Problems**3a = 21 To solve a, divide both sides by 3: a = 7 b + 17 = 59 To solve a, subtract 17 from both sides: b = 42 c – 22 = 100 To solve c, add 22 to both sides d = 50 5 To solve for d, multiply both sides by 5 d = 250**Vocabulary**exponent – the number of times a number is multiplied by itself. base – the number that is being multiplied. exponent 83 base This is read “8 to the 3rd power” or “8 cubed.” It means 8 x 8 x 8.**Evaluating Exponents**25 = 2 x 2 x 2 x 2 x 2 = 32 = 6 x 6 x 6 = 216 63 1.34 = 1.3 x 1.3 x 1.3 x 1.3 = 2.8561**Exponents with a base of 10**• Any multiple of ten can be expressed as an exponent with a base of ten. • The base is 10. The number of zeroes gives us the exponent. • Example: 100 = 102 • 10,000 = 104 1,000,000 = 106**Writing in Expanded Form Using Powers of 10**• First, write the problem in expanded form. • Then, change each term to a multiplication of the value and its place. • Change the place values to exponents with powers of 10.**Example**7, 946 7, 000 + 900 + 40 + 6 (7 x 1,000) + (9 x 100) + (4 x 10) + (6 x 1) 7 x 103 + 9 x 102 + 4 x 101 + 6**Percentages**• Simply a fraction of 100 (meaning “cent) • Examples: • 1/3 = 33.33% (1÷3x100) • ¾ = 75% (3÷4x100) • 1½ = 150% (3÷2x100) • 15% of 3267 = 490 • 230 expressed as a % of 430 = 230÷430x100 = 53,5% On the calculator – use “shift, %” to do it faster.