mem30012a apply mathematical techniques sample · tel: 02-9942 3200 fax: 02-9942 3257 ... mem30012a...

MEM05 Metal and EngineeringTraining Package

Learner guideVersion 2

Training and Education SupportIndustry Skills Unit

Meadowbank

Product Code: 5806

MEM30012AApply mathematical techniques

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Developed by Training & Education Support Industry Skills Unit, Meadowbank© TAFE NSW 2014

MEM30012A Apply mathematical techniques

AcknowledgmentsThe TAFE NSW Training and Education Support Industry Skills Unit, Meadowbank would like to acknowledge the support and assistance of the following people in the production of this learner resource guide:

Writer:Warren BlackadderTAFE NSW

Reviewers:Jim MilesChris ZvirblisTAFE NSW

Project Manager:Stephen DaviesEducation Programs ManagerTAFE NSW

EnquiriesEnquiries about this and other publications can be made to:

Training and Education Support Industry Skills Unit, Meadowbank Meadowbank TAFE Level 3, Building J See Street MEADOWBANK NSW 2114

Tel: 02-9942 3200 Fax: 02-9942 3257

© The State of New South Wales, Department of Education and Training, TAFE NSW, Training and Education Support Industry Skills Unit,

Meadowbank, 2014.

Copyright of this material is reserved to TAFE NSW Training and Education Support Industry Skills Unit, Meadowbank. Reproduction or transmittal in whole or in part, other than for the purposes of private study or research, and subject to the provisions of the Copyright Act, is prohibited without the written authority of, TAFE NSW Training and Education Support Industry Skills Unit, Meadowbank.

ISBN 978-1-74236-514-5

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Table of ContentsIntroduction ................................................................................... 9

1. General introduction ............................................................................. 9

2. Using this learner guide ......................................................................... 9

3. Prior knowledge and experience ........................................................... 11

4. Unit of competency overview ............................................................... 11

5. Assessment ....................................................................................... 14

Solving mathematical problems ................................................................ 15

Topic 1: The four operations ........................................................ 17Whole Numbers ..................................................................................... 17

The four operations ................................................................................ 17

Review questions ................................................................................... 25

Topic 2: Converting fractions and decimals .................................. 27Introduction to fractions .......................................................................... 27

Equivalent fractions ................................................................................ 29

Simplifying fractions ............................................................................... 30

Mixed to improper fractions ..................................................................... 31

Adding and subtracting simple fractions .................................................... 32

Adding and subtracting mixed fractions ..................................................... 33

Multiplying simple fractions ...................................................................... 34

Multiplying mixed fractions ...................................................................... 35

Dividing simple fractions ......................................................................... 36

Dividing mixed fractions .......................................................................... 37

Review questions .................................................................................. 38

Topic 3: Decimals ......................................................................... 41What is a decimal? ................................................................................. 41

Change decimals to fractions ................................................................... 42

Changing fractions to decimals ................................................................. 43

Adding and subtracting decimals .............................................................. 45

Multiplying decimals ............................................................................... 47

Dividing decimals ................................................................................... 48

Rounding off .......................................................................................... 49

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Rounding rules ....................................................................................... 50

Scientific notation .................................................................................. 51


Topic 4: Percentages .................................................................... 55Introduction .......................................................................................... 55

Converting a fraction to a percentage........................................................ 56

Converting a percentage to a fraction........................................................ 56

Converting a decimal to a percentage ....................................................... 57

Converting a percentage to a decimal ....................................................... 57

Common fractions decimals and percentages ............................................. 58


Topic 5: Angles and triangles ....................................................... 61Types of angles ...................................................................................... 61

Theorems .............................................................................................. 64

Triangles ............................................................................................... 67

Converting degrees and radians ............................................................... 68

Review questions .................................................................................. 70

Topic 6: Perimeters, areas and unit conversion ............................ 73Metric units ........................................................................................... 74

Perimeter .............................................................................................. 78

Area ..................................................................................................... 79

Composite areas .................................................................................... 83


Topic 7: Algebra ........................................................................... 89Introduction .......................................................................................... 89

Index notation ....................................................................................... 91

Substitution into algebraic expressions ...................................................... 91

Adding and subtracting algebraic expressions ............................................. 93

Adding and subtracting terms .................................................................. 94

Multiplication of algebraic expressions ....................................................... 95

Dividing algebraic expressions ................................................................. 97

Solving algebraic equations ....................................................................100

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Simultaneous equations .........................................................................102

Quadratic equations ...............................................................................103

Review questions .................................................................................105

Topic 8: Ratios and rates ............................................................ 109What is a ratio?.....................................................................................109

Equivalent ratios ...................................................................................111

Simplifying ratios ..................................................................................111

Problem solving using ratios ...................................................................112

Rates ...................................................................................................114

Speed, distance and time .......................................................................114


Topic 9: Scale drawings .............................................................. 119Scale ...................................................................................................119


Topic 10: Volumes and surface areas ......................................... 125Use of volume formulae in industry ..........................................................125

Use of surface area formulae in industry ...................................................126

Square or rectangular prism ...................................................................126

Cylinder ...............................................................................................128

Triangular prism ....................................................................................129

Pentagonal prism ..................................................................................130

Cone ...................................................................................................131

Pyramid ...............................................................................................132

Torus ...................................................................................................133

Sphere ................................................................................................134


Topic 11: Pythagoras theorem .................................................... 137Composite shapes .................................................................................138


Topic 12: Trigonometry .............................................................. 143Angles .................................................................................................144

Trigonometric functions ..........................................................................144

Inverse trig ratios..................................................................................148

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Sine and Cosine rules ............................................................................149

Estimating ............................................................................................151


Topic 13: Coordinates and statistics ........................................... 155Cartesian coordinates ............................................................................155

Polar coordinates ..................................................................................157

Converting polar and Cartesian coordinates ..............................................159

Length and mid-point of a line ................................................................161

Statistics ..............................................................................................162

Review questions ..................................................................................165

Topic 14: Industrial applications ................................................ 167Material weights ....................................................................................167

Areas ..................................................................................................171

Component volumes ..............................................................................178

Flow rates and velocity ..........................................................................181

Member lengths ....................................................................................183


Practice competency test ........................................................... 191

Answers to review questions ..................................................... 195

Resource Evaluation Form .......................................................... 205

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of 208 © TAFE NSW (TES, Industry Skills Unit Meadowbank) 2014


Unit elements and performance criteria

ElementsElements are the essential outcomes of the unit of competency.

Performance criteriaTogether, performance criteria specify the requirements for competent performance.

1. Use concepts of arithmetic in the solution of engineering problems.

1.1 Units of physical quantities are converted to facilitate engineering calculations.

1.2 Calculations are performed to solve problems involving rational and irrational numbers.

1.3 Scientific notation is used to represent numbers.1.4 Calculations are checked for reasonableness using

estimating and approximating techniques.

2. Solve engineering problems involving algebraic expressions with one independent variable.

2.1 Algebraic expressions are manipulated using mathematical operations in their correct order.

3. Use two-dimensional geometry to solve practical problems.

3.1 Angles expressed in degrees are correctly converted to radians and vice versa.

3.2 The perimeter, area, length and angles of a range of two-dimensional figures are correctly calculated.

3.3 The volume and surface area of complex figures are correctly calculated.

3.4 Points identified in terms of Cartesian coordinates can be converted to polar coordinates and vice versa.

4. Use trigonometry to solve practical problems.

4.1 Basic trigonometry functions are used to calculate the lengths of the sides of right-angled triangles.

4.2 Inverse trigonometry functions are used to determine angles in a right-angled triangle given the lengths of two sides.

4.3 The sine rule is used to determine the lengths of the sides of acute and obtuse angled triangles given one side and two angles.

4.4 The cosine rule is used to determine the lengths of the sides of acute and obtuse angled triangles given two sides and one angle.

5. Graph linear functions.

5.1 Linear functions are solved graphically and equations of straight lines are determined from the slope and one point, or two points.

5.2 Two linear functions are solved simultaneously both algebraically and geometrically.

5.3 The length and mid point of a line segment are determined.

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© TAFE NSW (TES, Industry Skills Unit Meadowbank) 2014 Page 13 of 208


6. Solve quadratic equations.

6.1 Quadratic equations are solved.6.2 Simultaneous linear and quadratic equations are

solved.

7. Perform basic statistical calculations.

7.1 Mean, median and mode are calculated from given data.

7.2 Standard deviation is calculated and interpreted employing graphical representation.

Required skills and knowledge

Essential knowledge:

Look for evidence that confirms knowledge of:• Transposing and evaluating formulae• Polynomials• Straight line coordinate geometry• Introduction to indices• Introduction to trigonometry• Circular functions• Trigonometry of oblique triangles• Trigonometric identities• Introduction to functions and their graphs.

Essential skills:

Look for evidence that confirms skills in:• Using and applying mathematical formulae:• Logical thinking• Problem solving• Calculating• Applying statistics• Using computer numerical methods• Drawing graphs.

Employability skillsBy successfully completing this unit of competency you will also be demonstrating a range of Employability Skills that are addressed under the Employability Skills Framework identified as appropriate for the MEM05, Metal and Engineering Training Package. These skills apply generally to work in metal and allied manufacturing industries and are specifically customised to concentrate on work at different levels and sectors of industry.

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Solving mathematical problemsAn equation is a combination of math expressions.

Word problems are a series of expressions that can be converted into an equation.

There are three steps to solving math word problems:

1. Layout your work in a logical manner using separate lines for each operation and keeping the numbers aligned in neat columns.

2. Translate the wording into a numeric equation that combines smaller "expressions".

3. Solve the equation!

Suggestions:

Read the problem fully - Get a feel for the whole problem.

List information and the variables you identify. Attach units of measure to the variables (gallons, miles, inches, litres, kilograms, metres etc.).

Define what answer you need; as well as its units of measure.

Work in an organized manner; working clearly will help you think clearly.

Draw and label all graphs and pictures clearly.

Note or explain each step of your process; it will help you track variables and remember their meanings.

Look for the "key" words; certain words indicate certain mathematical operations:

Key words for addition ( + )

Increased by; more than; combined together; total of; sum; added to

What is the sum of 8 and y?

Express the total weight of the mast (x) and antenna (y).

Key words for Subtraction ( - )

Less than, fewer than, reduced by, decreased by, difference of

What is four less than y?

What is the difference in the mass of the beams x and the columns y?

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Key words for multiplication ( * or x )

Times, multiplied by

What is y multiplied by 13?

A car drives at 105 kilometres per hour. How far will it go in x hours?

Key words for division ( ÷ or / )

Per, a; out of; ratio of, quotient of; percent (divide by 100).

What is the quotient of x and 3?

y machines produce a total of 3457. Express their average production.

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Topic 1: The four operations

Required skills

On completion of the topic you will be able to:

• Determine the result of a mathematical expression using addition.• Determine the result of a mathematical expression using subtraction.• Determine the result of a mathematical expression by multiplication.• Determine the result of a mathematical expression by division.• Calculate the result of a compound mathematical expression consisting of

addition, subtraction, multiplication and division.

Required knowledge• The difference between a whole number, decimal percentage and fraction.• Multiplication and division is carried out before addition and subtraction.• Laying out calculations on paper in a logical manner.

Topic aim

The aim of this topic is to reintroduce and revise the basics of mathematic operations.

Whole NumbersEssentially the definition of a whole number is based around what it does not contain. A whole number can not be a fraction of a number, a percentage, or have a decimal. If you have a number like 16.25 or 16¼, it has a whole number portion (16). But in itself, this number is not a whole number because it contains a decimal (.25) or a fraction (¼).

Whole numbers are simply the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 .......and so on, they continue and contain no fractions (½) or decimals (0.50).

The four operationsWith the increased usage of calculators, mobile phones and computers into the workplace and educational institutions to compute a simple set of numbers or formula, it is still very important to have a thorough understanding of the basic mathematical skills.

The four operations in mathematics are:

AddingSubtractingMultiplyingDividing

Everyone must know the 4 basic operations of addition (+), subtraction (-), multiplication (×) and division (÷), and how they are applied to whole numbers, fractions, and decimals.

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Adding whole numbers

Addition is the process of uniting a series of two or more numbers into one sum or total, and is represented by the symbol +.

Each number being added is called an addend while the total, which is the answer to the addition problem, is called the sum.

3 addend 8 addend+ 2 addend + 3 addend

5 sum 11 sum

Adding numbers with 1 digit is pretty straightforward. In general, you can do it mentally or use your fingers. Adding zero to a number never changes the number. Look at the following problems and select the incorrect answer.

A B C D E F G H I5 6 4 3 2 6 7 1 8

+ 3 + 4 + 7 + 9 + 8 + 5 + 6 + 9 + 38 10 12 12 10 11 13 10 11

If you selected C you are correct as 4+7=11; NOT 12.

Adding numbers with 2 or more digits is a bit more challenging. So when we have to add 45 and 27 together the result can be determined using the Expanded Notation method or placing the numbers into columns and carrying numbers over.

Expanded Notation

To calculate the sum the numbers are broken down into basic elements:

45 = 40 (four tens) + 5 (ones)

27 = 20 (two tens) + 7 (ones)

The ones are totalled first 5 + 7 = 12 or 10 (ten - to be carried over) + 2 (ones).

The tens are totalled next, 40 + 20 + 10 (carried over) = 70 or 70 (tens) + 2 (ones) = 72.

Columns

The same result can be achieved by arranging the numbers into columns where the ones, tens, hundreds are lined vertically below each other.

45+27

72

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The ones (right hand) column is totalled first, then the tens (next left), then the hundreds (3rd column from the right) and so on. Where the ones column totals more then 10, the number 0 to 9 is placed in the answer line and the ten number carried over to the ten column.

1

4 5 5 + 7 = 1 2

+ 2 7

7 2

Subtracting whole numbers

Subtract means to take away from. Subtracting whole numbers is the opposite of adding whole numbers and is represented by the symbol - (minus).

Instead of adding 2 numbers to get a sum, one number is removed from the other to get the difference. The following are simple subtractions:

8 9 6 10 15- 4 - 3 - 1 - 2 - 5

4 6 5 8 10

In the first example, if 8 – 3 = 5 is related to money, if you had $8 and bought an item for $3 you would have $5 remaining.

Subtractions with one digit are fairly simple however subtractions with 2 or more digits can become more complicated especially when a digit being subtracted is larger than the other digit; e.g. 82 – 7.

As in addition the result can be determined using the expanded notation or column methods.

Expanded Notation:

As with additions, the numbers are broken down into basic elements:

82 = 8 tens + 2 ones or 7 tens + 12 ones- 7 = - 7 ones - 7 ones

7 tens + 5 ones =75

In the above example, the 8-10’s is reduced to 7-10’s and the 2-1’s is increased to 12-1’s. The ones can then be subtracted (12 – 7 = 5) and added to the 7-10’s to give 75.

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Columns:

Since 7 can not be subtracted from 2, 10 is borrowed from the 10’s column and then paid back.

128 2

-1 77 5

In the above example, 10 is borrowed from the 8 to turn the 2 into 12 therefore 12–7=5; the borrowed 10 is then deducted from the tens column, 8 – 1 = 7. The final answer is 75.

Multiplying whole numbers

Numbers being multiplied are called Factors, while the result or answer to the multiplied figures is called the Product. A solid knowledge of the multiplication tables is recommended to successfully multiply.

The multiplication symbol is × (times).

3 factor 8 factor× 2 factor × 3 factor

6 product 24 product

The easiest multiplication we can perform is the one with one digit because all we need is a good recall of a multiplication table.

8 9 6 10 15× 4 × 3 × 1 × 0 × 532 27 6 0 25

Multiplying a two-digit number by a one-digit number is a little bit more fun but can easily be calculated using the Expanded Notation method.

Let’s look at 53 x 7 using the Expanded Notation and Columns methods.

Expanded Notation:

53 = 5 tens and 3 ones× 7 = × 7 ones

35 tens and 21 ones35 + 2 tens and 1 one

which gives us an answer of 371

The 21 ones break down to 2 tens and 1 one with the 2 tens being added to the 35 tens to give 37 tens and 1 one or 371.

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Columns:

As with addition and subtraction, multiplication can be done by laying the calculation out in column form.

Method 1:

53 factor× 7 factor21 Ones - 3 × 7

350 Tens - 5 × 70371 product

Method 2:

2 carry over53 factor 7×3=21; place 1 on the product line and carry the 2

to the 10's column.× 7 factor371 product 7×5=35, 35+2=37; place 1 on the product line next

to the 2

Keep the columns neat and carry over numbers clearly and neatly.

Dividing whole numbers

Dividing whole numbers is the opposite of multiplying whole numbers. It is the process by which we try to find out how many times a number (divisor) is contained in another number (dividend). The resulting answer is called the quotient. The symbol used to designate divide is ÷.

6 quotient5 )30 dividend

divisor

Long Division

At times when the number or dividend is too long to simply divide, the Long Division method should be used to calculate the answer or quotient. The division is laid out in the same manner but there are several lines underneath, breaking up the long number into shorter sections.

Example 1-1

Divide 1645 by 7 or 1645 ÷ 7

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Step 1:

7 )1645

Lay the division out as shown.

Step 2:

27 )1645 7 will go into 16, twice so 2 is placed above

the line

Step 3:

27 )1645 2×7=14; 14 is placed below the 16

14

Step 4:

27 )1645

14 Subtract 14 from 16 = 2 2

Step 5:

27 )1645

14 24 Carry the 4 down next to the 2 to gave a

figure of 24

Step 6:

237 )1645

14 24 24 ÷ 7 is 3 so 3 is placed in the quotient

next to 2

Step 7:

237 )1645

14 24 21 3 × 7 = 21; 21 is placed below the 24

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Step 8:

237 )1645

14 24 21 3 Subtract 21 from 24 = 3

Step 9:

237 )1645

14 24 21 35 Carry the 5 down next to the 3 to give a

figure of 35

Step 10:

2357 )1645

14 24 21 35 35 ÷ 7 is 5, so 5 is placed in the quotient

next to 23

Step 11:

237 )1645

14 24 21 35 35

0

The final answer is 235.

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Order of operations

Since the early days of ancient Egypt, Greece and Rome, Mathematicians have agreed on a definite order of doing the 4 operations (+, – , ×, ÷ ) otherwise confusion would occur.

Working from left to right, multiplication and division are calculated before additions and subtractions.

• First: Work out any grouping of symbols or brackets.

• Second: Work out any multiplication and division as they occur from left to right.

• Third: Work out any addition and subtraction as they occur from left to right.

From the following example, which is the correct answer; A or B?

A 5 + 3 × 4 = 17B 5 + 3 × 4 = 32

As any multiplication must be done before any addition, then the answer would be 5+12=17, therefore answer A is correct. For B to be correct brackets must be placed around the 5 + 3 with the question appearing as (5 + 3) x 4 = 32.

The agreed Order of Operations is often remembered as B O D M A S or Brackets Of Division Multiplication Addition Subtraction.

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Review questions

The following questions have been included to help you revise what you have learnt in Topic 1: The four operations.

Note: Questions should be answered without the use of calculators.

1. Determine the following answers:

a. 7+4 b. 17+24 c. 34+58 d. 62+74 e. 53+96f. 85+65 g. 105+89 h. 186+178 i. 275+349 j. 597+627k. 1017+527 l. 297+2489 m. 2481+92 n. 1579+6982 o. 8715+2547


a. 6-4 b. 12-8 c. 84-24 d. 98-72 e. 53-7f. 85-36 g. 207-104 h. 179-36 i. 285-147 j. 987-321k. 1489

-268l. 2478

-1587m. 2481

-1478n. 2579

-1587o. 8715

-2547


a. 5×4 b. 3×9 c. 8×7 d. 9×6 e. 12×8f. 7×8 g. 14×6 h. 85×24 i. 96×48 j. 57×57k. 118×47 l. 249×58 m. 758×68 n. 1485×69 o. 1479×247


a. 15 ÷ 3 b. 48 ÷ 8 c. 16 ÷ 4 d. 81 ÷ 9 e. 12 ÷ 6

f. 8 ) 448 g. 6 ) 252 h. 7 ) 868 i. 4 ) 116j. 9 )3132 k. 3 ) 501 l. 10 )1000 m. 12 ) 2904


a. 8 × 6 + 4 b. 4 x 7 – 9 c. 4 + 7 × 8d. 12 × 3 + 5 x 6 e. 36 – 4 × 8 f. 28 × 2 ÷ 8g. 46 + 24 ÷ 4 h. 6 × 8 + 4 × 3 i. 15 × 5 + 25 ÷ 5

6. Name the 4 major operations in used in mathematics.

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NOTES

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Topic 2: Converting fractions and decimals

Required skills

On completion of the topic you will be able to:

• Simplify fractions

• Convert mixed and improper fractions

• Add and subtract simple fractions

• Add and subtract mixed fractions

• Multiply and divide simple fractions

• Multiply and divide mixed fractions.

Required knowledge

• The differences between simple and mixed fractions

• The lowest common denominator

• Methods of basic addition, subtraction, multiplication and division

• Transposition of formulae and expressions

• Laying out calculations on paper in a logical manner.

Introduction to fractionsIt would be nice if all measurements and numbers in everyday engineering projects involved only whole numbers known as integers. However, in engineering this is an impossibility.

Some people may ask “In an age of metric measuring systems, why do we need to know about fractions?”.

Although the majority of drawings and manufactured objects produced in Australia are metric, some countries (e.g. United States of America) still work using the Imperial System and many drawings prepared for American companies, or drawings and components produced by American companies may be in Imperial measurements.

A fraction is part of a entire object which is broken into a number of equal parts, generally 2, 4, 8, 10, 12, 16, 32, 64 etc.

A fraction is shown as 2 whole numbers, one over the other. The number on top is called the Numerator while the number on the bottom is called the Denominator.

5 Numerator8 Denominator

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