Learner Guide - AgriSeta · justify geometrical relationships and conjectures to solve problems in...

LLeeaarrnneerr GGuuiiddee PPrriimmaarryy AAggrriiccuullttuurree

PPhhyyssiiccaall qquuaannttiittiieess && ggeeoommeettrriiccaall

rreellaattiioonnsshhiippss iinn 22 aanndd 33ddiimmeennssiioonnaall ssppaaccee

My name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Company: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Commodity: . . . . . . . . . . . . . . . . . . . . Date: . . . . . . . . . . . . . . .

NQF Level: 4 US No: 12417

The availability of this product is due to the financial support of the National Department of Agriculture and the AgriSETA. Terms and conditions apply.

Measure, estimate & calculate physical quantities & explore, critique & prove geometrical relationships in 2 and 3 dimensional space in the life and workplace of adult

with increasing responsibilities.

Primary Agriculture NQF Level 4 Unit Standard No: 12417

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Version: 01 Version Date: July 2006

BBeeffoorree wwee ssttaarrtt…… Dear Learner - This Learner Guide contains all the information to acquire all the knowledge and skills leading to the unit standard:

Title: Measure, estimate & calculate physical quantities & explore, critique & prove geometrical relationships in 2 and 3 dimensional space in the life and workplace of adult with increasing responsibilities.

US No: 12417 NQF Level: 4 Credits: 4

The full unit standard will be handed to you by your facilitator. Please read the unit standard at your own time. Whilst reading the unit standard, make a note of your questions and aspects that you do not understand, and discuss it with your facilitator.

This unit standard is one of the building blocks in the qualifications listed below. Please mark the qualification you are currently doing:

Title ID Number NQF Level Credits Mark

National Certificate in Animal Production 48979 4 120

National Certificate in Plant Production 49009 4 120

This Learner Guide contains all the information, and more, as well as the activities that you will be expected to do during the course of your study. Please keep the activities that you have completed and include it in your Portfolio of Evidence. Your PoE will be required during your final assessment.

WWhhaatt iiss aasssseessssmmeenntt aallll aabboouutt?? You will be assessed during the course of your study. This is called formative assessment. You will also be assessed on completion of this unit standard. This is called summative assessment. Before your assessment, your assessor will discuss the unit standard with you.

Assessment takes place at different intervals of the learning process and includes various activities. Some activities will be done before the commencement of the program whilst others will be done during programme delivery and other after completion of the program.

Are you enrolled in a: Y N

Learnership?

Skills Program?

Short Course?

Please mark the learning program you are enrolled in:

Your facilitator should explain the above concepts to you.




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The assessment experience should be user friendly, transparent and fair. Should you feel that you have been treated unfairly, you have the right to appeal. Please ask your facilitator about the appeals process and make your own notes.

Your activities must be handed in from time to time on request of the facilitator for the following purposes:

The activities that follow are designed to help you gain the skills, knowledge and attitudes that you need in order to become competent in this learning module.

It is important that you complete all the activities, as directed in the learner guide and at the time indicated by the facilitator.

It is important that you ask questions and participate as much as possible in order to play an active roll in reaching competence.

When you have completed all the activities hand this in to the assessor who will mark it and guide you in areas where additional learning might be required.

You should not move on to the next step in the assessment process until this step is completed, marked and you have received feedback from the assessor.

Sources of information to complete these activities should be identified by your facilitator.

Please note that all completed activities, tasks and other items on which you were assessed must be kept in good order as it becomes part of your Portfolio of Evidence for final assessment.

EEnnjjooyy tthhiiss lleeaarrnniinngg eexxppeerriieennccee!!




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HHooww ttoo uussee tthhiiss gguuiiddee …… Throughout this guide, you will come across certain re-occurring “boxes”. These boxes each represent a certain aspect of the learning process, containing information, which would help you with the identification and understanding of these aspects. The following is a list of these boxes and what they represent:

MMyy NNootteess …… You can use this box to jot down questions you might have, words that you do not understand,

instructions given by the facilitator or explanations given by the facilitator or any other remarks that

will help you to understand the work better.

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What does it mean? Each learning field is characterized by unique terms and definitions – it is important to know and use these terms and definitions correctly. These terms and definitions are highlighted throughout the guide in this manner.

You will be requested to complete activities, which could be group activities, or individual activities. Please remember to complete the activities, as the facilitator will assess it and these will become part of your portfolio of evidence. Activities, whether group or individual activities, will be described in this box.

Examples of certain concepts or principles to help you contextualise them easier, will be shownin this box.

The following box indicates a summary of concepts that we have covered, and offers you an opportunity to ask questions to your facilitator if you are still feeling unsure of the concepts listed.




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WWhhaatt aarree wwee ggooiinngg ttoo lleeaarrnn?? What will I be able to do?.....................................................……………………… 6

Learning outcomes…………………………………………………………………………… 6

What do I need to know?.................................................…..……………………… 6

Session 1 Collecting, organising and analysing data………………….…….. 7

Session 2 Measuring instruments.....................……………………………….. 18

Checklist for Practical assessment .......................................... 36

Paperwork to be done .............................................................. 37

Terms and conditions……………………………………………….….. 38

Acknowledgements .................................................................. 38

SAQA Unit Standard




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WWhhaatt wwiillll II bbee aabbllee ttoo ddoo?? When you have achieved this unit standard, you will be able to:

Measure, estimate, and calculate physical quantities in practical situations relevant to the adult with increasing responsibilities in life or the workplace.

Explore analyse and critique, describe and represent, interpret and justify geometrical relationships and conjectures to solve problems in two and three-dimensional geometrical situations.

LLeeaarrnniinngg OOuuttccoommeess At the end of this learning module, you must is able to demonstrate a basic knowledge and understanding of:

Properties of geometric shapes Surface area and volume Mathematical argument and evaluation based on logical deduction Spatial interrelationships.

WWhhaatt ddoo II nneeeedd ttoo kknnooww?? It is expected of the learner attempting this unit standard to demonstrate competence against the unit standard:

The credit value is based on the assumption that people starting to learn towards this unit standard are competent in Mathematical Literacy and Communications at NQF level 3.




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SSeessssiioonn 11

CCoolllleeccttiinngg,, oorrggaanniissiinngg aanndd aannaallyyssiinngg ddaattaa After completing this session, you should be able to: SO 1: Measure, estimate, and calculate physical quantities.

In this session we explore the following concepts:

Properties of geometric shapes Spatial interrelationships

11..11 PPrrooppeerrttiieess ooff ggeeoommeettrriicc sshhaappeess SSuurrffaaccee aarreeaa ooff 22--ddiimmeennssiioonnaall ffiigguurreess

Rectangle

Area = length x breadth

Square

Area = side x side

Triangle

Area = ½ x base x height

Circle

Area = π x radius x radius = π x (radius)2

Trapezium

Area = ½ (side a + side b) x height

Parallelogram

Area = base x height

Most complicated shapes can be broken down into the six basic shapes shown above.




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Calculate the surface area of the shape below. Step 1: Construct lines to divide the shape into any combination of the six basic shapes. Step 2: Calculate the missing lengths Step 3: Calculate the areas of shapes A, B and C Area A (triangle) = ½ x base x height = ½ x 7cm x 5 cm = 32,5cm2 Area B (trapezium) = ½ (7cm + 6cm) x 5cm = ½ x 13cm x 5cm = 32,5cm2 Area C (rectangle) = length x breadth = 10cm x 2cm = 20cm2 Step 4: Add up Areas A, B and C to obtain total area Total surface area = Area A + area B + area C = 32,5cm2 + 32,5cm2 + 20cm2 = 85cm2

MMyy NNootteess ……

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SSuurrffaaccee aarreeaa ooff 33--ddiimmeennssiioonnaall ffiigguurreess

To calculate the surface area of a three dimensional figure, we need to break the shape up into known shapes by creating a net diagram. If you had to cut out the net and fold it, it would give you a 3-d shape.

3-D shape Net diagram

Rectangular prism

Cylinder

Triangular prism

Cone




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Sphere

Area of sphere = π x d2

Calculate the surface area of the triangular prism below: The length of the longest side of the triangle needs to be calculated by Pythagoras: Hypotenuse2 = 62 + 82 = 36 + 64 = 100 Hypotenuse = 10cm We prepare a net diagram: Total surface area = 2xarea 1 + 2xarea 2 + area 3 = 2x2cm x 6cm + 2 x ½ x 8cm x 6cm + 2cm x 10cm = 24cm2 + 48cm2 + 20cm2 = 92cm2

VVoolluummee ooff 33--ddiimmeennssiioonnaall ffiigguurreess

Many three dimensional shapes are right prisms. If a right prism is cut into slices that run parallel to the base, then all the slices have the same cross section. A right prism can also be described as a three dimensional shape in which the sides are at right angles to the base. All the sides must then have the same height.

For example, the triangular prism is a right prism. The shaded area is the base, and the sides all form 900 with the base. The sides are all the same height.




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The triangular prism is lying with its base pointing towards you.

The same prism is now lying on its base. The dark grey line indicates a slice that can be cut. The slice has the same shape as the base. The sides of the prism are all the same height. The sides are all perpendicular to the base.

The rectangular prism, the cylinder, the triangular prism and a prism that has any other shape as its base are all right prisms. The cone and the sphere are NOT right prisms.

3-D shape Volume Rectangular prism

Volume = area of base x height = area of rectangle x height = length x breadth x height

Cylinder

Volume = area of base x height = area of circle x height = π r2 x height

Triangular prism

Volume = area of base x height = Area of triangle x height of prism = ½ x base of triangle x height of triangle x height of prism

Cone The volume of a cone is one third of the volume of a cylinder. Volume = (Area of Circle x height)/3

or Volume = (π x r2 x height)/3




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Sphere

Volume = π x (diameter)3 6

A farmer needs to calculate the volume of a walk-through dip tank that he plans to build. The dimensions and shape are given in the drawing below.

First we need to identify the base of the prism. The base is a trapezium. Volume = area of base x height = area of trapezium x breadth of tank = ½ x (7m + 5m) x 1,2m x 1,5m = 10,8m3

A farmer has a round water reservoir on his farm. He wants to work out how much water it holds. The reservoir has a diameter of 15m and is 2m deep.

We need to identify the base. The base surface is either the top surface of the reservoir, or the bottom, i.e. it is a circle. Now we need to calculate the radius. The radius is half of the diameter, i.e. 7,5m Volume = area of base x height = area of circle x depth of tank = π r2 x depth of tank = π x 7,5m x 7,5m x 2m = 353,43m3




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11..22 SSppaattiiaall iinntteerrrreellaattiioonnsshhiippss OOrrtthhooggrraapphhiicc ddrraawwiinngg

Orthographic drawing is the basis of all engineering drawing, and it is also the basis for the study of Descriptive Geometry. A well-trained engineer or technician must be able to pick up a drawing and understand it. This understanding, of necessity, involves the basic principles of orthographic drawing.

Generally speaking, a course in Engineering Drawing consists of drawing various objects in two or more views utilizing the principles of orthographic projection. These views may be projected on the three principal planes – horizontal, frontal and profile – or on auxiliary planes. In turn, the views may or may not be sectioned.

Many students entering a learnership have had limited experience in orthographic drawing in the high school or technical school which may have prepared them for the learnership. It may have only consisted of several weeks of Mechanical Drawing, but this previous contact with the principles involved in orthographic drawing forms a frame of reference which usually proves valuable in solving Engineering Drawing problems.

The question might then be asked, "Well, what is Descriptive Geometry?" Very briefly, Descriptive Geometry is the graphical solution of point, line and plane problems in space. These solutions are accomplished by means of the same principles of orthographic drawing which are involved in making a simple three-view drawing of an object.

Orthographic Projection – the use of parallel lines of sight at 90o to an image plane.

On the following page is a picture of a 3-dimensional object that is represented by top, front and side views.


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CCuuttss aanndd ffiillllss

Of the many types of problems encountered by farmers, one of the most common is that of reading contour maps. Sometimes soil needs to be removed from one place (cut) to be placed in another spot (fill), for example when land is being leveled (see Fig. 3 below).

The following are some of the terms used in locating cuts and fills:

• Profile – a vertical section of the earth's surface containing a given line which may be either straight or curved. The length of the profile must be equal to the true length of the given line. (See Fig. 2)

• Section – a vertical section at right angles to the profile line.

• Cut – earth removed to obtain a required slope or elevation.

• Fill – earth added to existing contour in order to obtain a required slope or elevation.

Fig 1 Principal Planes

Top view

Front view Side view

Fig 2 Profile of contour map




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Fig 3 Three dimensional view of contour map


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Please complete Activity 1: A swimming pool is 7m long, 5m wide, 1m deep at the shallow end, and 3m deep at the deep end. The floor slopes evenly. What is the inside surface of the swimming pool and what is the volume (in m3)? Drawing of pool

Net diagram

Please complete Activity 2: Calculate the volume of a cylinder with a radius of 3 metre and a depth of 5 metre. What is the surface area of the wall of the cylinder?




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Concept (SO 1) I understand

this conceptQuestions that I still would

like to ask

Scales on the measuring instruments are read correctly.

Quantities are estimated to a tolerance justified in the context of the need.

The appropriate instrument is chosen to measure a particular quantity.

Quantities are measured correctly to within the least step of the instrument.

Appropriate formulae are selected and used.

Calculations are carried out correctly and the least steps of instruments used are taken into account when reporting final values.

Symbols and units are used in accordance with SI conventions and as appropriate to the situation.


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SSeessssiioonn 22

MMeeaassuurriinngg iinnssttrruummeennttss

After completing this session, you should be able to: SO 2: Explore, analyse & critique, describe & represent, interpret and justify geometrical relationships.

In this session we explore the following concepts:

General measurement system Calibration Measuring basic quantities Distinction between certain quantities Calculating heights and distances Hydraulic jacks House plan Road maps World maps Time zones Cartesian co-ordinate system

22..11 GGeenneerraall mmeeaassuurreemmeenntt ssyysstteemm It is the nature of human beings to measure things in order to understand them better, or to be able to make comparisons. We measure time in years, hours, minutes or seconds, distances in terms of kilometres, meters, centimetres or millimetres. Every single variable that we measure has its own units and is measured by some kind of tool. You are invited to refer back to the Level 2 module 12444 in which a number of different tools were described.

22..22 CCaalliibbrraattiioonn The most commonly used measuring instrument is the ruler. The ruler is by nature very imprecise. Any other measuring instrument has to be calibrated to make sure that the results obtained are accurate. Just think of your wristwatch! Every now and again you set the time on your watch according to the time given on television or radio. You are in fact calibrating a measuring instrument.




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The Vernier and micrometer also need to be calibrated. When they are used a great deal, their measuring faces wear out. How can the user be sure that the measurement is correct? Let us take the Vernier as an example. If you close the Vernier completely you should get a reading of zero. If you do not get a reading of zero, you adjust a little screw at the back of the instrument until you do get zero. Now you know that your instrument is accurate.

Any scale measuring mass is provided with some kind of standard. The standard is a block, usually of metal, of known mass. You simply place the standard on the scale and see what reading you get. If the reading is different to what the standard is supposed to be, then you adjust the scale until the reading is correct.

Complicated measuring equipment can be calibrated periodically by the manufacturer, or it can be adjusted by the South African Bureau of Standards (SABS). You will then be supplied with a certificate of calibration.

Calibration is so important, that you can win or lose court cases based on the correctness of your measuring apparatus. For example, you are entitled to ask for proof of calibration of the machines that police use for setting speed traps. Any laboratory result will only be declared accurate if the laboratory calibrated its measuring equipment.

SSttaattiicc ccaalliibbrraattiioonn

The most common type of calibration is known as a static calibration. The term "static" refers to a calibration procedure in which the values of the variables involved remain constant during a measurement, that is, they do not change with time. In static calibrations, only the magnitudes of the known input and the measured output are important. An example is a mass scale.

DDyynnaammiicc ccaalliibbrraattiioonn

In a broad sense, dynamic variables are time dependent in both their magnitude and frequency content. The input-output magnitude relation between a dynamic input signal and a measurement system will depend on the time-dependent content of the input signal. When time-dependent variables are to be measured, a dynamic calibration is performed in addition to the static calibration. A dynamic calibration determines the relationship between an input of known dynamic behaviour and the measurement system output. Usually, such calibrations involve either a sinusoidal signal or a step change as the known input signal.

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The accuracy of a system can be estimated during calibration. If we assume that the input value is known exactly, then the known input value




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can be called the true value. The accuracy of a measurement system refers to its ability to indicate a true value exactly.

By definition, accuracy can be determined only when the true value is known, such as during a calibration.

PPrreecciissiioonn aanndd bbiiaass eerrrroorrss

The repeatability or precision of a measurement system refers to the ability of the measuring instrument to give the same result again and again and again. If a measuring instrument always provides the same wrong value every time, then the instrument is considered to be precise, but not accurate.

The average error in a series of repeated calibration measurements defines the error measure known as bias. Bias error is the difference between the average and true values. Both precision and bias errors affect the measure of a system's accuracy.

The concepts of accuracy, and bias and precision errors in measurements can be illustrated by the throw of darts. Consider the dart board of Figure 6 where the goal will be to throw the darts into the bull's-eye. For this analogy, the bull's-eye can represent the true value and each throw can represent a measurement value. In Figure 6(a), the thrower displays good precision (i.e., low precision error) in that each throw repeatedly hits the same spot on the board, but the thrower is not accurate in that the dart misses the bull's-eye each time. This thrower is precise, but we see that low precision error alone is not a measure of accuracy. The error in each throw can be computed from the distance between the bull's-eye and each dart.

(a) High repeatability gives low precision error but gives no direct indication of accuracy

(b) High accuracy means low precision and bias errors

(c) Bias and precision errors lead to poor accuracy

Figure 6. Throws of a dart: illustration of precision and bias errors and




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The average value of the error yields the bias. This thrower has a bias to the left of the target. If the bias could be reduced, then this thrower's accuracy would improve. In Figure 6(b), the thrower displays high accuracy and high repeatability, hitting the bull's-eye on each throw. Both throw scatter and bias error are near zero. High accuracy means low precision error and low bias errors as shown. In Figure 6(c), the thrower displays neither high precision nor accuracy with the errant throws scattered around the board.

22..33 MMeeaassuurriinngg bbaassiicc qquuaannttiittiieess The Unit Standard requires of you to estimate or measure quantities such as length/distance, area, mass, time, speed, acceleration and temperature.

Please complete Activity 3. A farmer decided to build a dam on this farm. He wanted to have a cylindrical shaped dam. The diameter of the dam was to be 20m and the depth was to be 4m. How many litres of water will the dam hold when it is full?

22..44 DDiissttiinnccttiioonn bbeettwweeeenn cceerrttaaiinn qquuaannttiittiieess MMaassss

The mass of a body refers to “how much matter” is found in the object. Mass causes an object to have weight. Mass is a quantitative measure of the property described in everyday language as inertia. Mass is measured in kilogram. It only has a quantity and no direction. The mass of an object will be the same anywhere in the universe.

WWeeiigghhtt

All things are attracted to the earth. Objects fall because earth exerts a downward force on them. This force is called gravity. The force of gravity is the force of attraction between two bodies because of their masses. Thus the weight of an object is the force exerted on that object by gravity.

Weight = Forcegravity = mass x gravity. Gravity on earth is 9,8m.s-2

The weight of a body is a force, and must be expressed in terms of the unit of force which in Newton.

Some interesting facts:




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The weight of an object will be less on the moon than on earth, as the moon’s gravity is less than the one on earth. This is why moon walkers seem to float across the moon’s surface.

The weight of a given body varies by a few tenths of a percent from point to point on the earth's surface, partly because of local deposits of ore, oil, or other substances whose density differs from the average, and partly because the earth is not a perfect sphere but is flattened somewhat at the poles.

The weight of a given body decreases inversely with the square of its distance from the earth's centre, and at a radial distance of two earth radii, for example, it has decreased to one-quarter of its value at the earth's surface.

Calculate the weight of an object with a mass of 1 kilogram, at a point where g = 9.80ms-2, is

w = mg = 1 kg x 9.80ms-2 = 9.80 N

MMoottiioonn

Mechanics deals with the relations of force, matter and motion. Motion may be defined as a continuous change of position. In most actual motions, different points in a body move along different paths. The complete motion is known if we know how each point in the body moves, so to begin we consider only a moving point, or a very small body called a particle.

SSppeeeedd

Speed is a scalar quantity as the direction is not indicated in the unit. The equation for speed is

v = ∆s/∆t

This can also be written as follows:

Speed = distance

Time

The triangle on the right allows you to do any calculation regarding speed, distance or time. D = distance, T = time and s = speed.

According to the triangle, s=D/t, D = s x t and t = D/s




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Calculate the speed of a car if it travels 300 km in 2 hours. S = D/t = 300km/2hrs = 150km/h Calculate the distance a runner covers if he runs at 8km/h for 3 hours. D = sxt = 8km/h x 3 h = 24km

AAcccceelleerraattiioonn

When the velocity of a moving body changes continuously as the motion proceeds, the body is said to move with accelerated motion. The average acceleration of the body as it moves from P to Q is defined as the ratio of the change in velocity to the elapsed time.

ā = ∆v/∆t

Calculate the average acceleration of a car if it can go from 1 to 100km/h in 10 seconds ā = 100km/h ÷ 10s = 10km/h/s i.e. the car accelerates 10 km/h every second.

Please complete Activity 4. Your city council has decided to build a cone shaped reservoir. The plan that was given to the building contractor indicated that the top diameter of the reservoir must be 50m. The depth of the reservoir is indicated at 30m. What volume of water can be stored in the reservoir? What is the surface area of the wall of the cone?

22..55 CCaallccuullaattiinngg HHeeiigghhttss aanndd DDiissttaanncceess In the diagram below side c is called the hypotenuse. The hypotenuse is always the side opposite the 900 angle.

Pythagoras’ theorem states the following for a right-angled triangle:

c2 = a2 + b2

a

b

c

θ




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In words, Pythagoras’ theorem states: In a right-angled triangle the sum of the squares of the shorter two sides is equal to the square of the hypotenuse.

Calculate the size of the side marked x in each case.

x2 = 42 + 32 = 16 + 9 = 25 x = 5m

x2 + 52 = 132 x2 + 25 = 169 x2 = 169 – 25 = 144 x = 12m

The sides of a right angles triangle can also be seen in relation to the other angles:

Side AC lies opposite angle B.

Side BC lies adjacent (next to) angle B

Notice that there are two sides adjacent to angle B: side AB and side BC. We do, however, already know that side AB is the hypotenuse. So only side BC is called “adjacent”.

Side AC lies adjacent to angle A.

Side BC lies opposite angle A.

Trigonometry laws are as follows:

sin θ = opposite/hypotenuse cos θ = adjacent/hypotenuse tan θ = opposite /adjacent

The trigonometric ratios must be learnt. Learning tip: Remember this silly rhyme. The first letters of each word help you to remember.




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Some Old Hags Cackle And Haggle Till Old Age

In triangle ABC on the left, we can work out the three trig ratios.

Sin 300 = o/h = ½ = 0,5

Cos 300 = a/h = √3/2 = 0,867

Tan 300 = o/h = 1/3 = 0,33

Now use a scientific calculator and work out sin 300. You get 0,5.

(key sequence on calculator: sin 30 =)

Work out cos 300 and you get 0,867

Work out tan 300 and you get 0,33.

These trigonometric ratios can be used to work out sides and angles of right-angled triangles.

Work out all the missing sides and angles in the triangle MNL

a) Work out the hypotenuse by using Pythagoras. ML = 5

b) Work out angle L by using Trig: Sin L = o/h = 4/5 = 0,8 L = 53,130 (key sequence on calculator: 2nd function sin 0,8)

c) Work out angle M by using the fact that angles of a triangle add up to 1800 M = 1800 – 900 – 53,130 = 36,870


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Please complete Activity 5. 1. A ladder rests against a wall 24m high. The foot of the ladder is 7m from the foot

of the wall. Calculate the length of the ladder. Diagram

2. In the triangle below

a) name the hypotenuse ____________

b) name the side opposite θ ___________

c) name the side adjacent to θ

____________

d) sin θ = _________

e) cos θ = _________

f) Tan θ = _________

3. Morgan is standing 5 metres away from the base of a tree. The angle between his feet and the top of the tree is 550. How tall is the tree?

4. It is 11:00 on a sunny day. You are standing next to a block of flats.

a. Calculate the height of the block of flats (y) based on the information in the diagram.

b. Calculate the hypotenuse of the shaded area by using Pythagoras’ theorem.




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22..66 HHyyddrraauulliicc JJaacckkss Hydraulic jacks and many other technological advancements such as automobile brakes and dental chairs work on the basis of Pascal's Principle, named for Blaise Pascal, who lived in the seventeenth century. Basically, the principle states that the pressure in a closed container is the same at all points.

Pressure is described mathematically by a Force divided by Area. Therefore if you have two cylinders connected together, a small one and a large one, and apply a small Force to the small cylinder, this would result in a given pressure. By Pascal's Principle, this pressure would be the same in the larger cylinder, but since the larger cylinder has more area, the force emitted by the second cylinder would be greater.

This is represented by rearranging the pressure formula P = F/A, to F = PA. The pressure stayed the same in the second cylinder, but Area was increased, resulting in a larger Force. The greater the differences in the areas of the cylinders, the greater the potential force output of the big cylinder. A hydraulic jack is simply two cylinders connected as described above.

An enclosed fluid under pressure exerts that pressure throughout its volume and against any surface containing it. That's called 'Pascal's Principle', and allows a hydraulic lift to generate large amounts of FORCE from the application of a small FORCE.

Assume a small piston (one square cm area) applies a weight of 1 N to a confined hydraulic fluid. That provides a pressure of 1 N per square cm throughout the fluid. If another larger piston with an area of 10 square cm is in contact with the fluid, that piston will feel a force of 1 N/ cm2 x 10 cm2 = 10 N.

So we can apply 1 N to the small piston and get 10 N. of force to lift a heavy object with the large piston. Is this 'getting something for nothing'? Unfortunately, no. Just as a lever provides more force near the fulcrum in exchange for more distance further away, the hydraulic lift merely converts

w1 = 1 N

w2 = 10 N

A1 = 1 square cm

A2 = 10 square

D1 = 10 cm

D2 = 1 cm




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work (force x distance) at the smaller piston for the SAME work at the larger one. In the example, when the smaller piston moves a distance of 10 cm it displaces 10 cubic cm of fluid. That 10 cubic cm displaced at the 10 square cm piston moves it only 1 cm, so a small force and larger distance has been exchanged for a large force through a smaller distance.

22..77 HHoouussee ppllaann According to the requirements for this unit you need to be familiar with the reading and analysis of house plans.

Please complete Activity 6.

1. How many bedrooms does this house have? 2. How many doors does the owner bedroom have? 3. How many bathrooms are there and where are they situated? 4. What does each bathroom contain? 5. How many windows does the kitchen have? 6. What is included in bedroom 2? 7. What is the room next to Bedroom 2? 8. What is the floors pace area of the living room? 9. What interesting feature does this house have that we do not always in find in a

South African house?


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22..88 RRooaadd mmaappss You are required to have some familiarity with reading road maps.

Please complete Activity 7. Study the map of the centre of Johannesburg on the next page and answer the questions below: 1. What is the distance from the corner of Pretoria & Claim St. to the T-junction of

Bree & End St.? 2. What are the names of the street that surround the Ellis Park Rugby Stadium? 3. What establishment is on the corner of Twist & Wolmarans St?




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22..99 WWoorrlldd MMaappss You are required to have the skill of reading and interpreting world maps.

Please complete Activity 8. 1. The map of Africa indicates a scale of 1: 40,000,000. What does that mean? 2. Name the neighbouring countries to Zambia. 3. What are the names of the two oceans that surround Africa? 4. What is the former name of Namibia? 5. What is the capital of Nigeria?




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22..1100 TTiimmee ZZoonneess You are required to show an understanding of the different time zones in the world.

Please complete Activity 9. 1. What is the Universal Time Constant? 2. What does GMT stand for? 3. How many hours is South Africa ahead of GMT? 4. How many degrees make up one hour time difference? 5. What is the time difference between South Africa and Perth, Australia? 6. What is the International Date Line?


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22..1111 CCaarrtteessiiaann CCoo--oorrddiinnaattee SSyysstteemm The graphs that you have drawn in previous years are always drawn on a set of axes. These axes have the fancy name of “Cartesian co-ordinate system” and the whole graph is known as a Cartesian plane.

The Cartesian co-ordinate system consists of two axes: the horizontal X axis and the vertical Y axis. The X and Y axes cross at 900 at the point zero. This point is also known as the origin.

One can also draw three-dimensional graphs. In this case there are 3 axes: the X, Y and Z axes. The Z axis provides a third dimension in space. We shall, however, only work with X and Y axes.

To specify a particular point on a two dimensional coordinate system, you indicate the x unit first (abscissa), followed by the y unit (ordinate) in the form (x,y), an ordered pair. In three dimensions, a third z unit is added, (x,y,z).

The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values.

An example of a point P on the system is indicated in the picture below using the coordinate (5,2).




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The arrows on the axes indicate that they extend forever in the same direction (i.e. infinitely). The intersection of the two x-y axes creates four quadrants indicated by the roman numerals I, II, III, and IV. Conventionally, the quadrants are labelled counter-clockwise starting from the northeast quadrant. In Quadrant I the values are (x,y), and II:(-x,y), III:(-x,-y) and IV:(x,-y). (see table below.)

Quadrant x values y values

I > 0 > 0

II < 0 > 0

III < 0 < 0

IV > 0 < 0

Please complete Activity 10. Plot the following values (a) (5,2) (b) (-4,-3) (c) (-2,4) (d) (1,-4)

MMyy NNootteess …… . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3344


Please complete Assignment. You are required to draw up the plan for a house with six equal sides. In other words, the outline of the house must be a hexagon. A hexagon is shown below. Your plan must fit onto an A4 page. Your plan must include the following: a) A scale (e.g. 5cm represents 1m) b) At least 2 bedrooms c) A kitchen d) A passage e) A lounge/dining room f) At least one bathroom g) The lengths of all walls must be indicated on the diagram Additional task: a. Calculate the surface area of a room of your choice. b. All plans and calculations must be attached to the pages in this work book. c. Please attach your plan to this page. d. Show all calculations on this page.




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Concept (SO 2) I understand this concept

Questions that I still would like to ask

Descriptions are based on a systematic analysis of the shapes and reflect the properties of the shapes accurately, clearly and completely.

Descriptions include quantitative information appropriate to the situation and need.

3-dimensional objects are represented by top, front and side views.

Different views are correctly assimilated to describe 3-dimensional objects.

Available and appropriate technology is used in producing and analysing representations.

Relations of distance and positions between objects are analysed from different views.

Conjectures as appropriate to the situation, are based on well-planned investigations of geometrical properties.

Representations of the problems are consistent with and appropriate to the problem context. The problems are represented comprehensively and in mathematical terms.

Results are achieved through efficient and correct analysis and manipulation of representations.

Problem-solving methods are presented clearly, logically and in mathematical terms.

Reflections on the chosen problem solving strategy reveal strengths and weaknesses of the strategy.

Alternative strategies to obtain the solution are identified and compared in terms of appropriateness and effectiveness.




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CChheecckklliisstt ffoorr pprraaccttiiccaall aasssseessssmmeenntt …… Use the checklist below to help you prepare for the part of the practical assessment when you are observed on the attitudes and attributes that you need to have to be found competent for this learning module.

Observations Answer Yes or No

Motivate your Answer (Give examples, reasons, etc.)

Can you identify problems and deficiencies correctly?

Are you able to work well in a team?

Do you work in an organised and systematic way while performing all tasks and tests?

Are you able to collect the correct and appropriate information and / or samples as per the instructions and procedures that you were taught?

Are you able to communicate your knowledge orally and in writing, in such a way that you show what knowledge you have gained?

Can you base your tasks and answers on scientific knowledge that you have learnt?

Are you able to show and perform the tasks required correctly?

Are you able to link the knowledge, skills and attitudes that you have learnt in this module of learning to specific duties in your job or in the community where you live?

The assessor will complete a checklist that gives details of the points that are checked and assessed by the assessor.

The assessor will write commentary and feedback on that checklist. They will discuss all commentary and feedback with you.

You will be asked to give your own feedback and to sign this document.

It will be placed together with this completed guide in a file as part of you portfolio of evidence.

The assessor will give you feedback on the test and guide you if there are areas in which you still need further development.




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PPaappeerrwwoorrkk ttoo bbee ddoonnee …… Please assist the assessor by filling in this form and then sign as instructed.

Learner Information Form

Unit Standard 12417

Program Date(s)

Assessment Date(s)

Surname

First Name

Learner ID / SETA Registration Number

Job / Role Title

Home Language

Gender: Male: Female:

Race: African: Coloured: Indian/Asian: White:

Employment: Permanent: Non-permanent:

Disabled Yes: No:

Date of Birth

ID Number

Contact Telephone Numbers

Email Address

Postal Address

Signature:




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TTeerrmmss && CCoonnddiittiioonnss This material was developed with public funding and for that reason this material is available at no charge from the AgriSETA website (www.agriseta.co.za).

Users are free to produce and adapt this material to the maximum benefit of the learner.

No user is allowed to sell this material whatsoever.

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All qualifications and unit standards registered on the National Qualifications Framework are public property. Thus the only payment that can be made for them is for service and reproduction. It is illegal to sell this material for profit. If the material is reproduced or quoted, the South African Qualifications Authority (SAQA) should be acknowledged as the source.

SOUTH AFRICAN QUALIFICATIONS AUTHORITY

REGISTERED UNIT STANDARD:

Measure, estimate & calculate physical quantities & explore, critique & prove geometrical relationships in 2 and 3 dimensional space in the life and workplace of

adult with increasing responsibilities

SAQA US ID UNIT STANDARD TITLE

12417 Measure, estimate & calculate physical quantities & explore, critique & prove geometrical relationships in 2 and 3 dimensional space in the life and workplace of adult with increasing responsibilities

SGB NAME NSB PROVIDER NAME

SGB for Math Literacy, Math, Math Sciences L 2 -4

NSB 10-Physical, Mathematical, Computer and Life Sciences

FIELD SUBFIELD

Physical, Mathematical, Computer and Life Sciences Mathematical Sciences

ABET BAND UNIT STANDARD TYPE NQF LEVEL CREDITS

Undefined Regular-Fundamental Level 4 4

REGISTRATION STATUS

REGISTRATION START DATE

REGISTRATION END DATE

SAQA DECISION NUMBER

Reregistered 2003-12-03 2006-12-03 SAQA 1351/03

PURPOSE OF THE UNIT STANDARD

This unit standard is designed to provide credits towards the mathematical literacy requirements of the NQF at level 4. The essential purposes of the mathematical literacy requirements are that, as the learner progresses with confidence through the levels, the learner will grow in: • An insightful use of mathematics in the management of the needs of everyday living to become a self-managing person. • An understanding of mathematical applications that provides insight into the learner`s present and future occupational experiences and so develop into a contributing worker. • The ability to voice a critical sensitivity to the role of mathematics in a democratic society and so become a participating citizen. People credited with this unit standard are able to: • Measure, estimate, and calculate physical quantities in practical situations relevant to the adult with increasing responsibilities in life or the workplace. • Explore analyse and critique, describe and represent, interpret and justify geometrical relationships and conjectures to solve problems in two and three-dimensional geometrical situations.

LEARNING ASSUMED TO BE IN PLACE AND RECOGNITION OF PRIOR LEARNING

The credit value is based on the assumption that people starting to learn towards this unit standard are competent in Mathematical Literacy and Communications at NQF level 3.

UNIT STANDARD RANGE

The scope of this unit standard includes length, surface area, volume, mass, speed; ratio, proportion; making and justifying conjectures. Contexts relevant to the adult, the workplace and the country. More detailed range statements are provided for specific outcomes and assessment criteria as needed.

Specific Outcomes and Assessment Criteria:

SPECIFIC OUTCOME 1

Measure, estimate, and calculate physical quantities.

OUTCOME NOTES

Measure, estimate, and calculate physical quantities in practical situations relevant to the adult with increasing responsibilities in life or the workplace.

OUTCOME RANGE

• Basic instruments to include those readily available such as rulers, measuring tapes, measuring cylinders or jugs, thermometers, spring or kitchen balances, watches and clocks. • In situations which necessitate it such as in the workplace, the use of more accurate instruments such as vernier callipers, micrometer screws, stop watches and chemical balances. • Quantities to estimate or measure to include length/distance, area, mass, time, speed acceleration and temperature. • Distinctions between mass and weight, speed and acceleration. • The quantities should range from the low or small to the high or large. • Mass, volume temperature, distance, and speed values are used in practical situations relevant to the young adult or the workplace. • Calculate heights and distances using Pythagoras` theorem. • Calculate surface areas and volumes of right prisms (i.e., end faces are polygons and the remaining faces are rectangles) cylinders, cones and spheres from measurements in practical situations relevant to the adult or in the workplace.

ASSESSMENT CRITERIA

ASSESSMENT CRITERION 1

1. Scales on the measuring instruments are read correctly.


2. Quantities are estimated to a tolerance justified in the context of the need.


3. The appropriate instrument is chosen to measure a particular quantity.


4. Quantities are measured correctly to within the least step of the instrument.


5. Appropriate formulae are selected and used.


6. Calculations are carried out correctly and the least steps of instruments used are taken into account when reporting final values.


7. Symbols and units are used in accordance with SI conventions and as appropriate to the situation.

SPECIFIC OUTCOME 2

Explore, analyse & critique, describe & represent, interpret & justify geometrical relationships.

OUTCOME NOTES

Explore, analyse and critique, describe and represent, interpret and justify geometrical relationships and conjectures to solve problems in two and three-dimensional geometrical situations.

OUTCOME RANGE

• Applications taken from different contexts such as packaging, arts, building construction, dressmaking. • The operation of simple linkages and mechanisms such as car jacks. • Top, front and side views of objects are represented. • Use rough sketches to interpret, represent and describe situations. • The use of available technology (e.g., isometric paper, drawing instruments, software) to represent objects. • Use and interpret scale drawings of plans (e.g., plans of houses or factories; technical diagrams of simple mechanical household or work related devices). • Road maps relevant to the country. • World maps. • International time zones. • The use of the Cartesian co-ordinate system in determining location and describing relationships in at least two dimensions.

ASSESSMENT CRITERIA


1. Descriptions are based on a systematic analysis of the shapes and reflect the properties of the shapes accurately, clearly and completely.


2. Descriptions include quantitative information appropriate to the situation and need.


3. Three-dimensional objects are represented by top, front and side views.


4. Different views are correctly assimilated to describe 3-dimensional objects.


5. Available and appropriate technology is used in producing and analysing representations.


6. Relations of distance and positions between objects are analysed from different views.


7. Conjectures as appropriate to the situation, are based on well-planned investigations of geometrical properties.


8. Representations of the problems are consistent with and appropriate to the problem context. The problems are represented comprehensively and in mathematical terms.


9. Results are achieved through efficient and correct analysis and manipulation of representations.


10. Problem-solving methods are presented clearly, logically and in mathematical terms.


11. Reflections on the chosen problem solving strategy reveal strengths and weaknesses of the strategy.


12. Alternative strategies to obtain the solution are identified and compared in terms of appropriateness and effectiveness.

UNIT STANDARD ACCREDITATION AND MODERATION OPTIONS

Providers of learning towards this unit standard will need to meet the accreditation requirements of the GENFETQA. Moderation Option: The moderation requirements of the GENFETQA must be met in order to award credit to learners for this unit standard.

UNIT STANDARD ESSENTIAL EMBEDDED KNOWLEDGE

The following essential embedded knowledge will be assessed through assessment of the specific outcomes in terms of the stipulated assessment criteria. Candidates are unlikely to achieve all the specific outcomes, to the standards described in the assessment criteria, without knowledge of the listed embedded knowledge. This means that the possession or lack of the knowledge can be inferred directly from the quality of the candidate`s performance against the standards. • Properties of geometric shapes • Surface area and volume • Mathematical argument and evaluation based on logical deduction • Spatial interrelationships.

Critical Cross-field Outcomes (CCFO):

UNIT STANDARD CCFO IDENTIFYING

Identify and solve problems using critical and creative thinking: Solve a variety of problems relevant to the adult with increasing responsibilities involving space, shape and time using geometrical techniques.

UNIT STANDARD CCFO COLLECTING

Collect, analyse, organise and critically evaluate information:

Gather, organise, evaluate and critique information about objects and processes.

UNIT STANDARD CCFO COMMUNICATING

Communicate effectively: Use everyday language and mathematical language to describe properties, processes and problem solving methods.

UNIT STANDARD CCFO CONTRIBUTING

Use mathematics: Use mathematics to analyse, describe and represent realistic and abstract situations and to solve problems relevant to the adult with increasing responsibilities.

UNIT STANDARD ASSESSOR CRITERIA

Notes to assessors: Assessors should keep the following general principles in mind when designing and conducting assessments against this unit standard: • Focus the assessment activities on gathering evidence in terms of the main outcome expressed in the title to ensure assessment is integrated rather than fragmented. Remember we want to declare the person competent in terms of the title. Where assessment at title level is unmanageable, then focus assessment around each specific outcome, or groups of specific outcomes. • Make sure evidence is gathered across the entire range, wherever it applies. Assessment activities should be as close to the real performance as possible, and where simulations or role-plays are used, there should be supporting evidence to show the candidate is able to perform in the real situation. • Do not focus the assessment activities on each assessment criterion. Rather make sure the assessment activities focus on outcomes and are sufficient to enable evidence to be gathered around all the assessment criteria. • The assessment criteria provide the specifications against which assessment judgements should be made. In most cases, knowledge can be inferred from the quality of the performances, but in other cases, knowledge and understanding will have to be tested through questioning techniques. Where this is required, there will be assessment criteria to specify the standard required. • The task of the assessor is to gather sufficient evidence, of the prescribed type and quality, as specified in this unit standard, that the candidate can achieve the outcomes again and again and again. This means assessors will have to judge how many repeat performances are required before they believe the performance is reproducible. • All assessments should be conducted in line with the following well documented principles of assessment: appropriateness, fairness, manageability, integration into work or learning, validity, direct, authentic, sufficient, systematic, open and consistent.

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