Shape aand motion - AgriSetaMeasure, estimate, and calculate physical quantities in practical situations 1. Reading scales on the measuring instruments correctly. 2. Estimating quantities

LLeeaarrnneerr GGuuiiddee PPrriimmaarryy AAggrriiccuullttuurree

SShhaappee aanndd mmoottiioonn

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

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

NQF Level: 3 US No: 9013

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

Describe, apply, analyse and calculate shape and motion in 2-and 3-dimensional space in different contexts.

Primary Agriculture NQF Level 3 Unit Standard No: 9013 22

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: Describe, apply, analyse and calculate shape and motion in 2-and 3-dimensional space in different contexts.

US No: 9013 NQF Level: 3 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 49048 3 120

National Certificate in Plant Production 49052 3 120

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.

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.

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.




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!!




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.




WWhhaatt aarree wwee ggooiinngg ttoo lleeaarrnn??

What will I be able to do? .....................................................…………………….… 6

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

What do I need to know? .................................................…..……………………… 7

Introduction ……………………………………………………………………………….…… 8

Session 1 Measure, estimate and calculate.....…………………………….….. 9

Session 2 Explore, describe, represent, interpret and justify………..……. 28

Am I ready for my test? ........................................................... 46

Checklist for Practical assessment .......................................... 49

Paperwork to be done .............................................................. 50

Terms and conditions………………………………………………….. 51

Acknowledgements .................................................................. 51

SAQA Unit Standard




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 in life or the workplace.

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

The purpose of this module is to equip the learner with the necessary skills, knowledge and attributes that will enable him/her to describe, apply, analyse and calculate shape and motion in 2-and 3-dimensional space in different contexts.

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

Measure, estimate, and calculate physical quantities in practical situations. This includes:

Using basic instruments such as rulers, measuring tapes, measuring cylinders or jugs, thermometers, spring or kitchen balances, watches and clocks.

Estimating quantities such as length/distance, area, mass, time, speed and temperature.

Estimating the area and volume of simple irregular shapes and objects. The quantities should range from the low or small to the high or large. Using mass, volume temperature, distance, and speed values in practical

situations relevant to the learner or the workplace. The ability to do calculations involving the effects on area and volume when

altering linear dimensions. Calculating heights and distances using Pythagoras` theorem. Calculating surface areas and volumes of right prisms (i.e., end faces are

polygons and the remaining faces are rectangles) and cylinders from measurements in practical situations relevant to the life of the learner or in the workplace.

Using symbols and units in accordance with SI conventions and as appropriate to the situation.

Explore, describe and represent, interpret and justify geometrical relationships and conjectures. This includes:

Taking applications from different contexts such as packaging, arts, building construction, dressmaking.

Using tessellations and symmetry in artifacts and in architecture. Using rough sketches to interpret, represent and describe situations. Using and interpreting scale drawings of plans (e.g., plans of houses or

factories; technical diagrams of simple mechanical household or work related devices such as jacks,




Nets of prisms and cylinders. Using the Cartesian co-ordinate system in determining location and

describing relationships in at least two dimensions such as a road map.

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 2.

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

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IInnttrroodduuccttiioonn WWoorrkksshhoopp ssttrruuccttuurree

My Current Financial

Module 1: Banking

Knowledge components

Module 1: Measure, estimate, and

calculate physical quantities in practical

situations

1. Reading scales on the measuring instruments correctly. 2. Estimating quantities to a tolerance justified in the context

of the need. 3. Choosing the appropriate instrument to measure a

particular quantity. 4. Measuring quantities correctly to within the least step of

the instrument. 5. Carrying out calculations correctly. 6. Using symbols and units in accordance with SI

conventions and as appropriate to the situation.

1. Properties of geometric shapes 2. Length, area, volume, mass, time, temperature, speed 3. The Cartesian system 4. Scale drawing.

Module 2: Geometrical

relationships and conjectures.

1. Analysing shapes and reflecting on the properties of the shapes accurately, clearly and completely.

2. Including quantitative information appropriate to the situation and need.

3. Planning investigations of geometrical properties. 4. Representing problems comprehensively and in

mathematical terms. 5. Achieving results through efficient and correct analysis

and manipulation of representations. 6. Presenting problem-solving methods clearly, logically and

in mathematical terms. 7. Solutions are correct and are interpreted and validated in

terms of the context of the problem.

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

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SSeessssiioonn 11

MMeeaassuurree,, eessttiimmaattee aanndd ccaallccuullaattee..

After completing this session, you should be able to: SO 1: Measure, estimate, and calculate physical quantities in practical situations.

In this session we explore the following concepts:

Properties of geometric shapes Volume of 3-dimensional figures Measuring basic quantities

11..11 PPrrooppeerrttiieess ooff ggeeoommeettrriicc sshhaappeess SSuurrffaaccee aarreeaa ooff 22--dd 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

Any flat figure can be broken down into combinations of the shapes shown above.

Calculate the surface area of the shape shown on the right The area must be divided up into shapes that we can calculate the area of: Area 1 = area of triangle = ½ x base x height = ½ x 12m x 5m = 30m2 Area 2 = area of rectangle = l x b = 12m x 5m = 60m2 Total area = area 1 + area 2 = 30m2 + 60m2 = 90m2




Please complete Activity 1: Calculate the surface area of each of the following shapes: 1.

2.

3. Calculate the shaded area

SSuurrffaaccee aarreeaa ooff 33--ddiimmeennssiioonnaall ggeeoommeettrriicc sshhaappeess

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

Calculate the surface area of the rectangular prism shown on the right. To begin, we must draw a net of the shape. Total Area = 2xArea 1 + 2xArea 2 + 2x Area 3 = (2x8cmx12cm) + (2x12cmx5cm) + (2x5cmx8cm) = 192cm2 + 120cm2 + 80cm2 = 392cm2




11..22 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.

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




A farmer needs to calculate the volume of cement needed to cast a triangle-shaped concrete slab. The dimensions and shape are given in the drawing on the left.

First we need to identify the base of the prism. The base is a triangle. Volume = area of base x height = area of triangle x height of slab = ½ x 600cm x 600cm x 10cm = 180 000cm3

= 1,8m3 (see later notes on conversions of volumes)

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




Please complete Activity 2:

1. A swimming pool is 7m long, 5m wide, and 3m deep. It has the same depth

everywhere. What is the inside surface of the swimming pool and what is the volume

(in m3)?

Drawing of pool

Net diagram

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?

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 cubic meters of water will the dam hold when it is full?

11..33 MMeeaassuurriinngg bbaassiicc qquuaannttiittiieess A measurement is an act of assigning a specific value to a physical variable. That physical variable becomes the measured variable. A measurement system is a tool used for this quantification of the physical variable. As such, it is used to extend the abilities of the human senses that, while they can detect and recognize different degrees of roughness, length, sound, colour, and smell, are limited and are not very adept at assigning specific values to sensed variables.




MMeeaassuurreemmeenntt ooff lleennggtthh

In South Africa we use the metric system for measuring length. The S I unit of length is the metre. The following table gives you the relationship between the length measurements.

10 millimetres (mm) = 1 centimetre (cm) 10 centimetres (cm) = 1 decimeter (dm) 10 decimetres (dm) = 1 metre (m) 10 metres (m) = 1 decametre (dam) 10 decametres (dam) = 1 hectometre (hm) 10 hectometres (hm) = 1 kilometre (km) 1000 metres (m) = 1 kilometre (km)

Please complete Activity 3 a, b & c:

a. You have to estimate the length and width and height of the classroom in

centimetres as well as metres. Thereafter you have to use a measuring

instrument such as a ruler or a measuring tape to obtain a more accurate

answer. Write down the method you followed to obtain an estimate as well as

the more accurate answer and your answer. b. Why is there a difference in your estimated values when compared to the values you

obtained from the measuring instrument? c. Calculate the floor area of your classroom in centimetres2

Area =

MMeeaassuurreemmeenntt ooff aarreeaa

The general formula for calculating area is:

Area = length x width

• Assume you are working in cm, then cm x cm = cm2

• Assume you are working in m, then m x m = m2

• Assume you are working in cm and you want to convert your area answer into square metre, then you first have to convert cm into metre before performing your calculation, or you can divide your answer by 10000.




Here is an example:

You have a tile measuring 40 cm x 40 cm. What is the area of the tile in cm2 and m2? Area = length x width = 40 cm x 40 cm = 1600cm2 Area = length x width = 0.40 m x 0.40 m = 0.16m2

The following table gives you the relationships between different area measurements.

100 mm2 = 1 cm2 10 000 cm2 = 1 metre2 100 metres2 = 1 are 100 acres = 1 hectare 10 000 metres2 = 1 hectare

100 hectares = 1 kilometre2 1 000 000 metres2 = 1 kilometre2

Please complete Activity 3 d:

d. Calculate the floor area of your classroom in metres2.

Area =

MMeeaassuurriinngg vvoolluummee

Volume is the amount of space an object occupies.

The volume of an object can be calculated geometrically using mathematical equations or by measuring liquid displacement.

The general formula for calculating volume is:

Volume = length x width x height

• Assume you are working in cm, then cm x cm x cm = cm3

• Assume you are working in m, then m x m x m = m3

• Assume you are working in cm and you want to convert your volume answer into cubic metre, then you first have to convert cm into metre before performing your calculation, or you can divide your answer by 1,000,000.




You have a cube measuring 40 cm x 40 cm x 20 cm. What is the volume of the cube in cm3 and m3? Volume = length x width x height = 40 cm x 40 cm x 20 cm = 32 000cm3 Volume = length x width x height = 0.40 m x 0.40 m x 0.2 m = 0.032m3

The following table gives you the relationships between different volume measurements.

1000 mm3 = 1 cm3 1 ml

1000 cm3 = 1 decimetre3 1 litre

1000 dm3 = 1 metre3 1 000 litre

1 million cm3 = 1 metre3 1 000 000 litre

Please complete Activity 3 e, f & g:

e. Calculate the volume of your classroom in cm3

Volume =

f. Calculate the volume of your classroom in m3

Volume =

g. Calculate the volume of your classroom in litres

Volume =

MMeeaassuurriinngg mmaassss

The S I unit of mass is the kilogram. Mass is the amount of matter an object has. We often use a triple-balance beam to measure mass.

Here is a picture of a triple beam balance. You probably have used one in school.

Using a triple-balance beam

Carrying the balance

• Be sure all riders are back to the zero point.

• Place one hand under the balance and the other hand on the support (arm) to carry the balance.

Using the balance

• Zero the balance before you determine the mass of any substance. • Slide all of the riders back to the zero point. • Check to see that the pointer swings freely along the scale.




• Use the adjustment screw to obtain an equal swing of the beams, if necessary. You do not have to wait for the pointer to stop at the zero point. The beam should swing an equal distance above and below the zero point.

• Once you have placed the object to be massed on the pan, move the riders along the beams beginning with the largest mass first. If the beams are notched, make sure all riders are in a notch before you take a reading. Remember; the pointer does not have to stop swinging, but the swing should be an equal distance above and below the zero point on the scale.

Because a triple beam balance compares a known mass to an unknown mass it is unaffected by gravity. Unlike a spring scale, which really measures weight, the triple beam balance gives a true measure of mass.

Problem 1:

A block is put onto a triple beam balance. What is the mass of the object (to the closest tenth of a gram? Look on the picture to the right and get a closer look at reading the scale after the scale has been balanced. Remember to find the mass using a triple beam balance you must add the mass values of each of the three beams. The answer is 79.3g

Please complete Activity 4: a. You have to determine the volume of a glue stick. You fill a measuring cylinder with

500 ml of water. Thereafter you add the glue stick into the water. The level of the

water rises to 550 ml. What is the volume of the glue stick in ml and in cm3?

70g0g 9.3g




b. What is the mass of the pineapple on the kitchen scale? c. What is the range that the kitchen scale can measure?

d. A block is put on a triple-balance beam.

What is the mass of this object?

e. What is the range that this triple

balance beam can measure?

MMeeaassuurriinngg ttiimmee

Nowadays we can easily find out the time by looking at a watch or clock, but it has not always been so easy.

How can I tell if my watch is telling the right time?

You could compare your watch with a friend's. If they both say the same, you could check with another friend just to make sure. If the different watches show different times, you could check them by listening to the Greenwich time signal on the radio. At certain times of the day, usually on the hour, six pips are broadcast in quick succession. The last pip sounds slightly different from the rest. If your minute and second hand are both on the twelve when the last pip starts, then your watch is telling the right time.

But how do they know when to broadcast the six pips?

The pips are controlled by the time service at the SABC. Nowadays the time of transmission is worked out by comparing the time shown on more than 200 extremely accurate atomic clocks in different parts of the world.

Why do we want to measure time?

In the security field, you need to report at what time a particular incident took place. It could also be required of you to say how long a particular incident lasted. This information is crucial in a court of law.




Please complete Activity 5 a - c:

a. What is the time on your watch right now? How does this compare with the

times of two other learners in class? b. How can you ensure that the time on your watch is accurate?

c. Go outside of the classroom. Measure the time it takes two learners walking

at different speeds to walk around the classroom. How does your answers

compare to that of the other learners?

MMeeaassuurriinngg tteemmppeerraattuurree

Three temperature scales are in common use in science and industry. Two of those scales are SI metric:

The degree Celsius (°C) scale was devised by dividing the range of temperature between the freezing and boiling temperatures of pure water at standard atmospheric conditions (sea level pressure) into 100 equal parts. Temperatures on this scale were at one time known as degrees centigrade, however it is no longer correct to use that terminology.

The kelvin (K) temperature scale is an extension of the degree Celsius scale down to absolute zero, a hypothetical temperature characterized by a complete absence of heat energy. Temperatures on this scale are called kelvins, NOT degrees kelvin, kelvin is not capitalized, and the symbol (capital K) stands alone with no degree symbol. kelvin is 273.15 less than °C.

The degree Fahrenheit (°F) non-metric temperature scale was devised and evolved over time so that the freezing and boiling temperatures of water are whole numbers, but not round numbers as in the Celsius temperature scale.

Some baseline temperatures in the three temperature scales:

temperature kelvins degrees Celsius degrees Fahrenheit

symbol K °C °F

boiling point of water 373.15 100. 212.

melting point of ice 273.15 0. 32.

absolute zero 0. -273.15 -459.67




Mechanical thermometers: Mechanical devices used to measure temperature are classified in various ways. In this section, we will discuss only the expansion thermometer types. Expansion thermometers operate on the principle that the expansion of solids, liquids, and gases has a known relationship to temperature change. Liquid-in-glass thermometers are the oldest, simplest, and most widely used devices for measuring temperature. A liquid-in-glass thermometer has a bulb and a very fine-bore capillary tube. The tube contains alcohol or some other liquid that uniformly expands or contracts as the temperature rises or falls. The selection of liquid is based on the temperature range for which the thermometer is to be used.

Almost all liquid-in-glass thermometers are sealed so atmospheric pressure does not affect the reading. The space above the liquid in this type of thermometer may be a vacuum, or this space maybe filled with an inert gas, such as nitrogen, argon, or carbon dioxide. The capillary bore may be round or elliptical. In either case, it is very small; therefore, a relatively small expansion or contraction of the liquid causes a relatively large change in the position of the liquid in the capillary tube. Although the capillary bore has a very small diameter, the walls of the capillary tube are quite thick. Most liquid-in-glass thermometers have an expansion chamber at the top of the bore to provide a margin of safety for the instrument if it should accidentally overheat.

The South African Weather Service use the following terms when sharing weather forecasts on radio and TV.

Temperatures

Summer (October to March)

Winter (April to September)

Very Hot >35 ºC >35 ºC

Hot 30 - 34 ºC 27 - 34 ºC

Warm 26 - 29 ºC 23 - 26 ºC

Cool 20 - 25 ºC

Mild 17 - 22 ºC

Cold 15 - 19 ºC 12 - 16 ºC

Very Cold <15 ºC <12 ºC




Please complete Activity 6: a. What is the temperature indicated on the thermometer? b. Describe the difference between oC and Kelvin. c. At what temperature does water boil at sea level? d. How would you rate the following summer day temperatures? Mark the appropriate

box with an x. Temperature Very cold Cold Cool Warm Hot

8oC

28oC

30oC

22oC

16oC

34oC

MMeeaassuurriinngg ssppeeeedd

Speed is calculated once you have the distance travelled as well as the time it took to complete the distance. The equation for speed is

v = ∆s/∆t

The SI unit for speed is m/s or ms-1.

Calculate the speed of a motor vehicle after it took 2 minutes to travel 2.4 km. v = ∆s/∆t = 2400m / 120s = 20ms-1

An odometer in a motor vehicle to is used to measure the distance of the vehicle.

Mechanical odometers are turned by a flexible cable made from a tightly wound spring. The cable usually spins inside a protective metal tube with a rubber housing. On a bicycle, a little wheel rolling against the bike wheel turns the cable, and the




gear ratio on the odometer has to be calibrated to the size of this small wheel. On a car, a gear engages the output shaft of the transmission, turning the cable.

The cable snakes its way up to the instrument panel, where it is connected to the input shaft of the odometer.

The gearing

This odometer uses a series of three worm gears to achieve its 1690:1 gear reduction. The input shaft drives the first worm, which drives a gear. Each full revolution of the worm only turns the gear one tooth. That gear turns another worm, which turns another gear, which turns the last worm and finally the last gear, which is hooked up to the tenth-of-a-kilometer indicator.

Please complete Activity 7: 1. Calculate the speed of a motor vehicle after it took 3 minutes to travel 1,8 km. 2. What instrument will you use to measure the following quantities?

Quantity Measuring instrument

Time

Speed

Length

Volume

Area

Mass

Temperature




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

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

a

b

c

θ




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. 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.

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

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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

1. It is 11:00 on a sunny summer day. You are standing next to a block of flats. Calculate the height of the block of flats (y) based on the information in the diagram. Calculate the hypotenuse of the shaded area by using Pythagoras’ theorem.

Answer Height of building tan θ = a/b tan 30o = a / 20 a = tan 30o x 20

= 0.577 x 20 = 11.54m

Hypotenuse According to Pythagoras c2 = a2 + b2

c2 = (11.54)2 + (20)2

= 133.17 + 400 = 533.17 c = √533.17 = 23.09m

Please complete Activity 8:

a) 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

Shaded

b =

30o

a? c?




b) 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 =

c) It is 11:00 on a sunny summer day. You are standing next to a block of flats. Calculate the height of the block of flats (y) based on the information in the diagram. Calculate the hypotenuse of the shaded area by using Pythagoras’ theorem.

Concept (SO 1) I understand this concept

Questions 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.

Calculations are carried out correctly.

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

Shaded

b = 40

30o

a? c?




SSeessssiioonn 22

EExxpplloorree,, ddeessccrriibbee,, rreepprreesseenntt,, iinntteerrpprreett aanndd jjuussttiiffyy..

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

In this session we explore the following concepts:

Tessellations The four types of symmetry in the plane Nets of prisms and cylinders Cartesian co-ordinate system

22..11 TTeesssseellllaattiioonnss A dictionary* will tell you that the word "tessellate" means to form or arrange small squares in a checkered or mosaic pattern. The word "tessellate" is derived from the Ionic version of the Greek word "tesseres," which in English means "four." The first tilings were made from square tiles.

A regular polygon has 3 or 4 or 5 or more sides and angles, all equal. A regular tessellation means a tessellation made up of congruent regular polygons. [Remember: Regular means that the sides of the polygon are all the same length. Congruent means that the polygons that you put together are all the same size and shape.]

Only three regular polygons tessellate in the Euclidean plane: triangles, squares or hexagons. We can't show the entire plane, but imagine that these are pieces taken from planes that have been tiled.

Here are examples of

A tessellation of triangles

A tessellation of squares

A tessellation of hexagons




When you look at these three samples you can easily notice that the squares are lined up with each other while the triangles and hexagons are not. Also, if you look at 6 triangles at a time, they form a hexagon, so the tiling of triangles and the tiling of hexagons are similar and they cannot be formed by directly lining shapes up under each other – a slide (or a glide!) is involved.

You can work out the interior measure of the angles for each of these polygons:

Shape

triangle square pentagon hexagon more than six sides

Angle measure in degrees

6090

108120

more than 120 degrees

Since the regular polygons in a tessellation must fill the plane at each vertex, the interior angle must be an exact divisor of 360 degrees. This works for the triangle, square, and hexagon, and you can show working tessellations for these figures. For all the others, the interior angles are not exact divisors of 360 degrees, and therefore those figures cannot tile the plane.

MMaakkee yyoouurr oowwnn tteesssseellllaattiioonn

Your tessellation must be based on a shape that will tessellate. Look at the foxes. Can you see the shape they are based on?

This is the fundamental, or primary, cell. Parallelograms will all tessellate. You can see that every part needed to make up the full foxes picture is contained in the parallelogram. All you need to do is to repeat them and you get the full tessellation. By joining similar points in a tessellation, it should usually be possible to work out the fundamental cell. In 2 motif tessellations it will contain both of them!

So what shapes will tessellate?

There are only 3 regular geometric shapes that will. 'Regular', means that all sides and internal angles are equal. These are the equilateral triangle, square and hexagon.

The 'F's show that the shapes can be flipped and rotated without affecting the ability to tessellate. This can make whole pictures more varied and interesting. Not all the possibilities are shown here.

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

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Other well known geometric shapes that will tessellate are the rectangle, parallelogram, and diamond (or rhomboid). Note that the parallelogram and diamond can be squashed or thick, it makes no difference to their ability to tessellate. The parallelogram could have alternate rows tipped the other way.

Hexagons do not have to have equal sides. Those with opposite sides equal will all tessellate as shown below left.

So far 3 other 6 sided figures have been found that will tessellate. One is shown below right.




Regular pentagons will not tessellate, as shown below, but there are 14 known 5 sided shapes that will. Here is one quite pretty one...

What about 4 sided figures? We know squares, rectangles, parallelograms and rhomboids are OK but are there any others?

Well, kites are well known to tessellate. In fact, any 4 sided shape will tessellate if put back to back with a copy of itself - it forms a hexagon with opposite sides equal and, as we have seen, these will always tessellate.

How about triangles other than equilateral ones? Once again, if used in pairs, they will form parallelograms or kite shapes and we have seen that these do tessellate! (In the case of kites, you will need 4 to make 2 parallelograms).

Enough of geometric shapes. What about irregular shapes like animals and birds – much more interesting – how do we find shapes like these and make picture tessellations? Well, you must start with a basic shape that will tessellate – that's why geometric shapes are important.

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

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TTeesssseellllaattiioonn ccoonnssttrruuccttiioonn

Line Method This method is capable of producing some unexpected tessellations! As explained in 'basics', you must start with a shape that will tessellate – in this case an equilateral triangle. You can lay down a grid if you want but as each side is equal... 1. First – draw a wavy line.

2. Now if the software package you are using allows layers, it is easy to copy the line, paste it back on a new layer and then rotate it 60°. This can be in either direction - it makes no difference. Move it so the ends meet as shown. If you have not got layers the principle is the same but more fiddly. You now have 2 sides of the triangle.

5. Draw a straight construction line joining the ends of the 2 wavy lines. Mark the centre.

This line will be removed later. 4 If you have layers, use a new one. Draw another wavy line from one end of the

construction line to the middle. 5 Copy and paste this line. Now you need to rotate it 180°. Move it to bridge the gap...




6. This is the arty part – can you see the shape of an animal or flower? Different people will see different things. There is no right answer! If you really can't see anything with yours then try again with another start line. Did you see an animal here? If you use your imagination you will see that I tried to create the head of a calf. Whatever you have, it will now tessellate. Copy and paste another...

7. If you make alternate ones a different colour they will show up better and not run into

each other. You will need to rotate some of them through ± 60° to get them to fit. 8. Continue filling them in until you have filled the page.

9. Here is the complete picture with the calves completely filling all the space with no gaps – a true tessellation! Carry on to see the framework on which it is based and to see a larger picture of the finished tessellation...

10. If we overlay a grid, you can see how each individual picture is based on an equilateral triangle.




TTeesssseellllaattiioonnss iinn aarrcchhiitteeccttuurree

Brick-like tessellations can be seen on the mid-bottom portions as well as at the very top. The regular tessellation can be seen on the mid-upper portion in the picture to the right.

Tessellations abound in the design shown here. You can find the semi regular tessellation 3.6.3.6 on the third, fifth, seventh, and ninth steps in the image on the left.

Please complete Activity 9. a. What is a tessellation?

b. What is the important principle for the interior angle in order for a

tessellation to work?

c. Give examples where you have seen tessellations being used in

architecture in buildings or in the home.

Islamic tower Date of Artwork:

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

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22..22 TThhee ffoouurr ttyyppeess ooff ssyymmmmeettrryy iinn tthhee ppllaannee

A pattern is symmetric if there is at least one symmetry (rotation, translation, reflection, glide reflection) that leaves the pattern unchanged.

RRoottaattiioonn

To rotate an object means to turn it around. Every rotation has a center and an angle.

TTrraannssllaattiioonn

To translate an object means to move it without rotating or reflecting it. Every translation has a direction and a distance.

RReefflleeccttiioonn

To reflect an object means to produce its mirror image. Every reflection has a mirror line. A reflection of an "R" is a backwards "R".




GGlliiddee rreefflleeccttiioonn

A glide reflection combines a reflection with a translation along the direction of the mirror line. Glide reflections are the only type of symmetry that involve more than one step.

Symmetries create patterns that help us organize our world conceptually. Symmetric patterns occur in nature, and are invented by artists, craftspeople, musicians, choreographers, and mathematicians.

In mathematics, the idea of symmetry gives us a precise way to think about this subject. We will talk about plane symmetries, those that take place on a flat plane, but the ideas generalize to spatial symmetries too.

Plane symmetry involves moving all points around the plane so that their positions relative to each other remain the same, although their absolute positions may change. Symmetries preserve distances, angles, sizes, and shapes.

A figure, picture, or pattern is said to be symmetric if there is at least one symmetry that leaves the figure unchanged.

For example,

ATOYOTA The letters in ‘ATOYOTA’ form a symmetric pattern: if you draw a vertical line through the center of the "Y" and then reflect the entire phrase across the line, the left side becomes the right side and vice versa. The picture doesn't change. If you draw the figure of a person walking and copy it to make a line of walkers going infinitely in both directions, you have made a symmetric pattern. You can translate the whole group ahead one person, and the procession will look the same. This pattern has an infinite number of symmetries, since you can translate forward by one person, two people, or three people, or backwards by the same numbers, or even by no people. There is one symmetry of this pattern for each integer (positive, negative, and zero whole numbers).

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

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Please complete Activity 10.

a. Classify all the capital letters in English (in their simplest forms) according to their

symmetries. For example, "A" has a reflection in a vertical line, and "R" has no

symmetry (except rotation by 0 degrees). Letter Classification

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

b. Use the box below to make a symmetric pattern by starting with an asymmetric shape (a letter is fine) and repeating a single translation over and over (also translate it backwards). That is, decide on a direction and distance for your translation (for example, 3 cm to the right).




Translate your letter 3 cm to the right, then translate the new letter 3 cm to the right, etc.

Also translate the original letter 3 cm. to the left, etc. Did you get any other types of

symmetries (reflections, glide reflections, or rotations) in the process?

c. Make a symmetric pattern by starting with an asymmetric shape and repeating

a single glide reflection over and over (also glide it backwards). That is, pick a

reflection line and a translation in a direction parallel to the reflection line.

Keep applying the same glide reflection to the new shapes that you generate

until you run out of paper. Did you get any other types of symmetries

(reflections, translations, or rotations) in the process?




d. House plan 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 floor space 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?

22..33 NNeettss ooff pprriissmmss aanndd ccyylliinnddeerrss Group work: You will be given the following materials:

• Solid figures of prisms and pyramids

• Nets which cover the prisms and pyramids

• Graph paper

• Scissors

• Masking Tape

• Pencil

• Ruler

Instructions:

1. Start with a blank piece of paper.




2. Place your prism on top of the paper, and trace around the side, which is touching the paper.

3. Without lifting the solid, 'roll' it to an adjacent side, and then trace that side.

4. Continue 'rolling' and tracing until you have traced all of the sides of your solid.

5. After you trace a side, you may wish to put a small piece of masking tape on that side to ensure that you don't trace it twice!

6. You will need to plan ahead to ensure that you have traced all of the sides and that you can roll from one to the next.

7. After you have drawn your net, take scissors and cut the net apart. Try to reconstruct the figure with tape.

Please complete Activity 11. a. Draw the nets of your prism and pyramid.

b. Trace the following net on to a separate piece of paper. Cut it out and stick it

together with masking tape. Draw windows and doors on the net.

• Make a sketch of the house, orienting it so that the front of the house faces

north. Label E, W, and S for the three remaining sides of the house. Fold the net to make the house and confirm the 2-D sketch.

• Determine the area of the net in cm2 by using a cm2 grid/transparency and use the area information to calculate the cost of painting the outside of the house if it costs

R50 to paint one m2 (note; 1 cm2 corresponds to 1 m2 of the actual house).




Can you imagine a net for this shape?

Did it look like this?

22..44 CCaarrtteessiiaann ccoo--oorrddiinnaattee ssyysstteemm The modern Cartesian coordinate system in two dimensions (also called a rectangular coordinate system) is commonly defined by two axes, at right angles to each other, forming a plane (an xy-plane). The horizontal axis is labeled x, and the vertical axis is labeled y. In a three dimensional coordinate system, another axis, normally labeled z, is added, providing a sense of a third dimension of space measurement. The axes are commonly defined as mutually orthogonal to each other (each at a right angle to the other). (Early systems allowed "oblique" axes, that is, axes that did not meet at right angles.) All the points in a Cartesian coordinate system taken together form a so-called Cartesian plane.

The point of intersection, where the axes meet, is called the origin normally labeled O. With the origin labeled O, we can name the x axis Ox and the y axis Oy. The x and y axes define a plane that can be referred to as the xy plane. Given each axis, choose a unit length, and mark off each unit along the axis, forming a grid. 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).

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 12.

a. Study the copy of a map of part of Roodepoort. Answer the following questions on the

next page:




1. What important landmark will you find at the Cartesian coordinates – DT 84? 2. Give the Cartesian coordinates of Bennie Reinecke Pleasure Park. 3. What streets border the Vehicle Testing Ground? 4. Name three community services to be found at DQ 84. b. Plot the following values on the Cartesian plane provided below. Label the points

that you have plotted. (i) (5,2) (ii) (-4,-3) (iii) (-2,4) (iv) (1,-4)

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.

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.

Solutions are correct and are interpreted and validated in terms of the context of the problem.

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

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AAmm II rreeaaddyy ffoorr mmyy tteesstt?? Check your plan carefully to make sure that you prepare in good time. You have to be found competent by a qualified assessor to be declared

competent. Inform the assessor if you have any special needs or requirements before

the agreed date for the test to be completed. You might, for example, require an interpreter to translate the questions to your mother tongue, or you might need to take this test orally.

Use this worksheet to help you prepare for the test. These are examples of possible questions that might appear in the test. All the information you need was taught in the classroom and can be found in the learner guide that you received.

1. I am sure of this and understand it well 2. I am unsure of this and need to ask the Facilitator or Assessor to explain what it means

Questions 1. I am sure 2. I am unsure

1. Calculate the area of the shaded regions a. b.




2. A farmer wants to build a shed on his farm. He wants to draw it according to the diagram below.

a) Calculate how many square meters of roofing he

needs to buy. b) If corrugated iron sheeting costs R70 per m2,

calculate the cost of the roof.

c) Each brick measures 0,22m x 0,11m x 0,076m. Calculate how many bricks are needed for one of the side walls.

d) Calculate the volume of the rectangular part of the shed (i.e. ignore the triangular part under the roof).

e) Calculate the area of the floor in m2 f) Inside the shed he wants to place a drum of radius

1m and height 1,5m. Calculate the volume of the drum in m3.

g) Convert the volume of the drum to litres. h) Inside the shed he also wants to build a loft. To get

up to the loft he also needs to build a staircase. i. What is the distance AD? ii. What is the distance AB? iii. Calculate the size of angle A

3. The farmer has a tractor that travels 10 km in 50 minutes. Calculate the speed of the tractor in km/hr.




4. Study the picture of the tessellation. a) Explain, in words, how the tessellation has been

achieved. b) Name an example of where the use of this particular

pattern could be suitable.

5. Plot the points (1; 5) (2; -3) (-3; -4) (-4; 1) on the Cartesian plane provided below.




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.




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

Learner Information Form

Unit Standard 9013

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:




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.

AAcckknnoowwlleeddggeemmeennttss

PPrroojjeecctt MMaannaaggeemmeenntt::

M H Chalken Consulting

IMPETUS Consulting and Skills Development

DDeevveellooppeerr::

Kwapele Learning and Consulting

AAuutthheennttiiccaattoorr::

Ms C Almeida

TTeecchhnniiccaall EEddiittiinngg::

Ms C Almeida

OOBBEE FFoorrmmaattttiinngg::

Ms P Prinsloo

DDeessiiggnn::

Didacsa Design SA (Pty) Ltd

LLaayyoouutt::

Ms A du Plessis




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:

Describe, apply, analyse and calculate shape and motion in 2-and 3-dimensional space in different contexts

SAQA US ID UNIT STANDARD TITLE

9013 Describe, apply, analyse and calculate shape and motion in 2-and 3-dimensional space in different contexts

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 3 4

REGISTRATION STATUS REGISTRATION START DATE REGISTRATION END DATE

SAQA DECISION NUMBER

Reregistered 2004-12-02 2007-12-02 SAQA 1657/04

PURPOSE OF THE UNIT STANDARD

This unit standard is designed to provide credits towards the mathematical literacy requirements of the NQF at level 3. 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 in life or the workplace • Explore 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 2.




UNIT STANDARD RANGE

The scope of this unit standard includes length, surface area, volume, mass, speed ; ratio and proportion; making and justifying conjectures. Contexts relevant to the adult, the workplace and the local community. 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 in practical situations.

OUTCOME NOTES

Measure, estimate, and calculate physical quantities in practical situations relevant to the adult 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 and temperature. • Estimate the area and volume of simple irregular shapes and objects. • 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 learner or the workplace. • Calculations involving the effects on area and volume when altering linear dimensions. • 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) and cylinders from measurements in practical situations relevant to the life of the learner 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. Calculations are carried out correctly.


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




SPECIFIC OUTCOME 2

Explore, describe and represent, interpret and justify geometrical relationships and conjectures.

OUTCOME NOTES

Explore, 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 use of tessellations and symmetry in artifacts and in architecture. • Use rough sketches to interpret, represent and describe situations. • Use and interpret scale drawings of plans (e.g., plans of houses or factories; technical diagrams of simple mechanical household or work related devices such as jacks, • Nets of prisms and cylinders. • Road maps relevant to the local community. • 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. Conjectures as appropriate to the situation, are based on well-planned investigations of geometrical properties.


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


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


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


7. Solutions are correct and are interpreted and validated in terms of the context of the problem.

UNIT STANDARD ACCREDITATION AND MODERATION OPTIONS

Accreditation Option: 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 • Length, area, volume, mass, time, temperature, speed • The Cartesian system • Scale drawing

Critical Cross-field Outcomes (CCFO):

UNIT STANDARD CCFO IDENTIFYING

• Identify and solve problems using critical and creative thinking: Solve a variety of problems involving space, shape and time using geometrical techniques related to the life or workplace of the learner

UNIT STANDARD CCFO ORGANIZING

• Collect, analyse, organise and critically evaluate information: Gather, organise, and interpret 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 SCIENCE

• Use mathematics: Use mathematics to analyse, describe and represent realistic and abstract situations and to solve problems relevant to the adult, the workplace and the local community.

UNIT STANDARD ASSESSOR CRITERIA

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.

UNIT STANDARD NOTES

N/A

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.

Shape aand motion - AgriSetaMeasure, estimate, and calculate physical quantities in practical situations 1. Reading scales on the measuring instruments correctly. 2. Estimating quantities

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