NCV 4 Mathematical Literacy Hands-On Support Slide Show - Module 1 Part 2

3. NUMBERS IN MEASUREMENTS At the end of this outcome, you will: be able to read measurements be able to estimate measurements use formula to calculate measurements be able to convert measurements between different units

Units of measurement Measuring unit Medium Distance Metre (m) Area Square metre (m 2 ) Volume Cubic metre (m 3 ) Volume Litre

Activity 17 Convert the quantities in the first column to the units in the second column: 450m km 24,051kg g 32.8kg m 31km m 8,257l ml 60kl L 749,5g kg 141g mg 3824ml l 92m mm 2575mm m 79,5l ml 0,45 32 800 8257 0,7495 3,824 2,575 24051 31000 60000 141000 92000 79500

Activity 17 Complete the following sentences: A millionth of a metre is called a A thousand metres is called a A hundredth of a metre is called a Write the value of one teragram, in other words, one teragram = g In computer terminology one megabyte implies bytes One picometre equals metre What is 10 -9 called? One ns is equal to of a second One MW is equal to Watts millimetre kilometre centimetre 1 000 000 000 000 1 048 576 0.000 000 000 001 1 nano litre 0.000 000 001 1 000 000

Activity 17 3. Select the correct measuring unit and instrument to complete the table: Kitchen scale, tape measure, measuring jug, pipette, burette, trundling wheel, vernier calliper, water displacement method, municipal water meter, bathroom scale

Trundling wheel kilometres millilitre Pipette millimetre Vernier caliper kilogram kilogram Bathroom scale grams Kitchen scale tons Nano metres Microscope Kilolitres Municipal water meter Millimetre Burette Years Calendar Minutes Clock Dimension to be measured Unit of measurement Measuring instrument Distance run on a marathon route Fluid measured to exact millimetre amount by sucking out of a container Internal diameter of a pipe Weight of blue marlin caught from a deep-sea fishing boat Mass of a person before and after going on a diet Mass of ingredients used to bake a cake Mass of a ship’s container Size of a virus through a microscope Amount of water used by a municipality during 24 hours Fluid measure in a laboratory by opening a small tap in a glass tube to add drop by drop to a mixture Time slept during a lifetime of 60 years Time necessary to boil water for sterilisation purposes

3.2 Measuring instruments Length Tape measure Ruler Vernier caliper Screw micrometer

3.2 Measuring instruments Volume Household water usage is measured in kilolitres Petrol poured into a tank is measured in litres Injection needles are measured in millimetres

3.2 Measuring instruments Weight Weight is the common name for the gravitational force of an object It is measured in Newton like all other forces. An astronaut with a mass of 70 kg has a weight of 70 X 9,8 ≈ 690 N on earth. That is the strength of the earth’s gravitational pull on the astronaut’s 70 kg of mass. But on the moon his weight will only equal 110 N because the moon has less force to attract the astronaut with. On a massive planet like Jupiter, the astronaut would weigh more than five times as much as on earth, i.e. 3600 N. Wherever the astronaut stands, he will still have the same amount of matter, which is measured as 70 kg. A kilogram bag of sugar has a mass of 1 kilogram, and a weight on earth of 9,8 N. Most scales give a reading in mass units such as kilograms.

3.2 Measuring instruments Mass This is a measure of the amount of matter in a substance. Mass is measured in kilograms (kg). The mass of a truck or a ship is given in metric ton. Your mass is measured in kilograms on a bathroom scale, that of a frog in grams on the kitchen scale and that of a spider in milligrams on a chemical balance.

3.2 Measuring instruments Temperature Temperature is measured with a thermometer in degrees Celsius or centigrade (ºC) or in degrees Fahrenheit (Imperial system) or in Kelvin. Freezing point of water is 0ºC or 32ºF or 273 K. Below zero measurement of temperature is normally given in negative units of degrees Celsius. Boiling point of water at sea level is 100ºC or 212ºF.

3.2 Measuring instruments Gravitation The force of attraction between masses is known as gravitational force. At a given location on earth and in the absence of air resistance, all dropped objects will fall with the same constant acceleration. We call this the acceleration due to gravity of the earth at that particular place on earth, and we give it the symbol “g”. Its magnitude is approximately: g = 9,80 m / s². It is not exactly the same everywhere. It is called the gravitational constant.

3.2 Measuring instruments Force This is an action capable of accelerating an object. It is the amount of push or pull to speed up the object. It is measured in Newtons. One Newton is the force required to impart an acceleration of 1 m / s² to a mass of 1 kg. Newton is mass times acceleration. Thus 1 N = 1 kg . m / s². When we speak of the weight of something it is a force towards earth. Weight = mass times the gravitational constant. Weight = mass x 9,80 N

3.2 Measuring instruments Speed Speed is a measure of the time taken to travel a certain distance. It is also called a rate of change. If you run 40 km in 4 hours, then it means that in every hour you run 10 km, or that you run 10 km/h. To get to the ten km/h you had to divide the 40 km by the 4 hours. Therefore, speed is calculated by dividing the distance travelled, by the time taken to travel the distance: Speed = distance time

3.2 Measuring instruments Velocity The velocity of a moving object is given by stating both its speed and the direction in which the object is moving. If the velocity is stated without reference to any direction, it must be assumed that the object is travelling in a straight line and that the overall direction of that line does not matter.

Time Distance Zero velocity Object at rest Constant Velocity

3.2 Measuring instruments Acceleration The acceleration of a moving object is a measure of how its velocity/speed changes in relation to time. Acceleration may be positive (for speeding up), or negative (for slowing down). The SI unit of acceleration is metres per second per second, abbreviated to m/s2. Acceleration equals difference in speed divided by difference in time. It is the gradient of the speed function or in other words it is the second derivative of the function: distance against time. A velocity-time graph is a graph that shows the relationship between the velocity of an object in relation to time . The gradient of a line drawn on the graph is a measure of acceleration. If the line of the relationship is curved, the particular acceleration at some moment in time can only be found by drawing a tangent to the curve at that point and measuring its gradient. The area under the curve between two ordinates drawn from the time scale, measures the distance travelled in that time interval.

Conversion of speed units m.h -1 km.h -1 m.s -1 ft.s -1 m.h -1 1,6093 0,44704 1,4667 km.h -1 0,62137 0,27778 0,91134 m.s- 1 2.2369 3,6 3,2808 ft.s -1 0,68182 1,0973 0,3048

Activity 18 Read the following scales and give the answers with the correct abbreviations of measurement a. Volume 0.5ml 5ml 1,05ml 1,5ml 1,9 ml

b. Mass c. length 0,30 kg 0,90kg 1,50kg 1,95kg 0,05kg 0,35kg 0,55kg 0,90kg 0,2 kg 0,7kg 1,4 kg 1,9 kg 0,25kg 0,63kg 1,12 kg 1,75kg

Activity 18 There are many different rates If you travelled 2 500 km in 16 hours, at what speed did you travel? A cyclist completed 42,5 km in 2 hours and 15 minutes. At what speed did he travel? Remember that you should work only in hours to give an answer in km/h. A flight of 6 000 km takes 22 hours. The aircraft stops for refuelling twice for a total of 45 minutes. At what speed did the aircraft fly? A typist types 600 words in 27 minutes. What speed is she typing at? An employee takes two and a half hours at KWV to bottle 7,5 kilolitres of wine at his station in the bottling chain. What is the bottling speed in litres per hour? 156,25 km /h 18.88 km/h 282,35km/h 22.2 words per minute 3000 litres per hour

Activity 18 There are many different rates A seamstress completes 6 dresses in 8 hours. At what speed is she working (dresses per hour) g. Interpret the following graphs of distance against time and speed against time 0,75 dresses per hour Constant speed to point A. Stationary from A to B. Constant speed to point C

Activity 18 There are many different rates Water leaks from a tank at the rate of 5ml per second. If the tank has 1950ml of water, how long will it take before it is empty? A train travels 144km in two hours. Find the speed of the train in km /h m /s Which is the better rate for a tourist R2490 for 3 days, or R2790 for 4 days at a Safari Lodge in Botswana R2 850 for 4 days, or R3 590 for 5 days at a Victoria Falls Hotel 390 seconds = 6 ½ minutes 72 km/h 144 x 1000 ÷ 3600 = 40m/s R830 per day vs R697,50. 4 days is better value per day R712.50 vs 718; 4 days is better value for money per day

Activity 19 At the end of December your mother or friend’s mother receives a water and electricity bill of over R800-00 where she normally pays more or less R350-00. She asks you to investigate her water usage and to advise her on ways to save water. Locate the water meter and take the meter reading. There are three different scales of units on this meter. Find out from your local municipality which measurements of volume the different scales represent. See whether the reading remains constant when no taps are turned on. If the meter ticks on steadily even though no taps are open, find out where the leak in the system is. If the reading remains constant, give a reason for the elevated water usage. Investigate water usage in the sprinkler irrigation lines in the garden. Turn them on separately for 20 minutes and for each twenty minute period, record the meter reading. See if there are any leaks in the system, e.g. has a garden fork pierced a hole in the hose? Calculate how much water will be saved by watering only every second day.

Activity 19 When next the washing machine is used on a full load, measure the water usage with no other taps running. In a similar fashion establish how much water is used in the toilet, the shower and the bath. Draw a table of the results. Discuss possible methods of saving water/ reducing water wastage. Take into account the frequency of watering the garden, the washing of dishes or clothes, water usage during showers and baths. Plan the household’s activities to keep the water bill down to R450-00. Price of water varies on a sliding scale according to the volume used.

Activity 19 Use the table below to: Do the calculations with both the 1005 as well as the 2008 sliding indices Calculate the % difference between 2005 and 2008 in the price of the total amount of water used

Kilolitres used Cost per kilolitre/2005 0-6 No cost 7-10 R1,41 11-30 R2,32 31-56 R3,95 57-80 R4,46 Above 80 R6,00 Kilolitres used Cost per kilolitre/2008 0-6 No cost 7-10 R1,95 11-30 R3,88 31-56 R6,23 57-80 R8,04 Above 80 R10,83

3.3 Using a formula to calculate a measurement Shape Formula for perimeter Triangle, trapezium and any other odd shape Add the length of the sides. If some lengths are not given, you will be able to calculate the distances from indirect measurements Square 4 times side length = 4l = 4s Rectangle 2 lengths + 2 breadths = 2l + 2b Circle 2  r

3.3 Using a formula to calculate a measurement 2-D shape2 Formula for area Rectangles l x b Parallelogram B x H Trapezium h/2 (P1 + P2) Triangle ½bh Circle  r 2

3.3 Using a formula to calculate a measurement 3-D object Formula for volume Rectangular solid length x breadth x height Cylinder  r 2h

Activity 20 Calculate the perimeter of the rectangles with the following dimensions: length = 12cm; breadth = 8cm length = 45mm; breadth = 2,6 cm length = 16,25m; breadth = 1238cm 40cm 142 mm 5726cm

Activity 20 Now calculate the area of the rectangles with the dimensions in the previous question (Use the formula: area of a rectangle = length times breadth) 96cm 2 1179cm 2 201.175m 2

Activity 20 In a fruit juice processing factory, the different juice types have to be kept at different temperatures for different time periods to ensure complete sterilisation. The constant temperature pipe through which the juice flows, is: 5 metres long for pear juice (juice remains in the pipe for 45 seconds), and 6 metres long for tomato juice (juice remains in the pipe for 72 seconds). Calculate the speed through the pipe system of the two different juices. Pear juice: 0,11 m /s Tomato juice: 0,083 m/s

Activity 20 Measure the diameter of a R5,00 coin to the nearest centimetre Then, measure the diameter of the coin to the nearest millimetre Which measurement is more accurate? When the coin is measured to the nearest centimetre what is the greatest possible error? When the coin is measured to the nearest millimetre, what is the greatest possible error? (The precision of the measurement is always the same as the unit used to measure, and the greatest possible error is always half of this unit) 3cm 26mm Measuring to millimetres 0,50 cm 0,5 mm

Activity 20 If something is correctly measured as 5 m to the nearest metre, what is the smallest it could be? How much shorter than 5 m could it be? Explain why? 4,5m 0,5m

Activity 20 You have to calculate the area of stainless steel sheeting necessary to make a cylinder which will be used as a fermentation tank in the making of wine. The cylinder has to have a capacity of 230 000 litre. The radius of the tank is 1,5 m. Use the different values for  given in the example above and compare the answers for total external surface area. Which value would you use to order the stainless steel sheeting? Formula for total external surface area of a cylinder = 2  r 2 + 2  rh Formula for the volume of a cylinder =  r 2 h Radius of 1,5 metres

Activity 20 You have to order the correct volume of paint for the outside surface of a cylinder. The cylinder has a radius of 1,5 m and a height of 3,25 m. Use a value of 3,1519 for  and calculate how much paint to order. Formula for total external surface area of a cylinder =  r2 + 2  rh. 2  (1,5)(1,5 + 3,25) = 44.77m 2

Activity 21 A manufacturer wants to sell fruit juice in cylindrical one-litre cans, using the least possible amount of metal. For the litre can, he has to find the radius and height which will give the smallest surface area of metal. Work to the nearest millimetre and approximate to the third decimal place for height of the cylinder and to the second decimal place for the total external surface area of the cylinder. Use the formulae: Volume of a cylinder =  r²h Total external surface area of cylinder = 2  r² + 2  rh

Activity 21 Complete the table by calculating the total external surface area of the cans Decide on dimensions for the cylinder which will use the least metal. If you had worked to the nearest centimetre and had approximated all calculations to one decimal place, calculate how the answer you chose would have differed 318,310 79,577 35,368 19,894 12,732 8,842 6,496 4,974 3,930 3,183 2006,28 1025,13 723,21 600,52 577,08 559,53 593,59 652,12 731,12 828,32 6cm Radius (cm) 1 2 3 4 5 6 7 8 9 10 Height of can (cm) Tot ext surf area

NCV 4 Mathematical Literacy Hands-On Support Slide Show - Module 1 Part 2

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NCV 4 Mathematical Literacy Hands-On Support Slide Show - Module 1 Part 2