Examination Preparation for Grade 12. Mathematical Literacy Foundational Knowledge for Paper 1 & 2. Learner Booklet.

Transcription

1 Examination Preparation for Grade 12 Mathematical Literacy Foundational Knowledge for Paper 1 & 2 Learner Booklet for the learner

2

3 Grade 12 Mathematical Literacy Foundational Knowledge CONTENTS Learning Outcome 1: Number and Operations in Context (Code A1 A37) 4 1. Number Operations 2. Decimal fractions, common fractions and percentage 3. Rate, ratio and proportion 4. Working with finances 5. Foreign exchange rates 6. Examination questions Learning Outcome 2: Functions and Algebra (Code B1 B25) Reading information from a table 2. Reading and drawing graphs 3. Working with formulae 4. Finding the break-even point Learning Outcome 3: Space, shape and measurement (Code C1 C26) Length and distance 2. Perimeter 3. Area 4. Surface area 5. Volume 6. Conversions 7. Maps, grids and compass direction Learning Outcome 4: Data handling (Code D1 D36) Representing data 2. Reading data from tables and graphs 3. Measures of central tendency and spread 4. Probability Winning Teams

4 Learning Outcome 1: Number and Operations in Context 1. Number operations When you multiply a positive number by a positive number, you get a positive answer. For example, 3 5 = 15 When you multiply a positive number by a negative number, you get a negative answer. For example, 3 5 = 15 When you multiply a negative number by a positive number, you get a negative answer. For example, 3 5 = 15 When you multiply a negative number by a negative number, you get a positive answer. For example, 3 5 = = + + = + = = + When you complete a calculation, remember the order of the operations: Work out brackets first, then multiply or divide and only add or subtract after this. Example: Simplify the following: [(3)3 4] [(3)3 3] = [27 3] simplify the inside brackets = simplify the outer brackets = multiply and work out the square root = divide = 21 add 2. Decimal fractions, common fractions and percentage 1. Percent means out of 100. For example, is 63%. The symbol used for percent is %. 2. To change a percentage to a decimal fraction or a common fraction, divide by 100: 9% = To change a fraction to a decimal, divide the numerator by the denominator. You can use a calculator: = = 0,09 3. We can convert between decimal numbers, fractions and percentages: 0,09 = = 9%, 63,3% = = 0,633 = 0,733 = 73,3% 68% = = 4. To find one quantity as a percentage of another, they both need the same units. 70c as a percentage of R8,00: 100 = 8,75% or, 100 = 8,75%, 5. To find the percentage of a quantity, convert to a decimal or common fraction and multiply: 33% of R240 = R240 = R79,20 or 33% of R240 = 0,33 R240 = R79,20 6. Percentage decrease on a price means the normal price less the decrease. Example: A pair of jeans is on sale with a mark down of 20%. If the original price of the jeans was R199,00, what is the sale price? Sale price is normal price less 20%, so sale price is 80% of normal price. 80% of R199 = 0,8 R199 = R159,20 7. Percentage increase on a price means the normal price plus increase. Example 1: The price of petrol increases by 12%. The original price was R7,70 per litre. What is the new petrol price? Method 1: New price = old price + 12% of old price New price = R7, % R7,70 = R7,70 + 0,12 R7,70 = R7,70 + R0,92 = R8,62 Method 2: New price = old price + 12% of old price = 112% of old price = 1,12 old price = 1,12 R7,70 = R8,62 4 of 35

5 Example 2: Jane is offered a salary increase of 7%. If her new salary is R10 600, what was her salary before the increase? New salary = original salary + 7% of original salary So 107% of original salary = R % of original salary is = R99,07 So the original salary is R99, = R More examples of working with percentage: 1. I pay 14% VAT on my telephone bill. If the total before VAT is R378,50, what is my telephone bill after VAT? I will have to pay R378,50 114% = R378,50 1,14 = R431, A clothes shop advertises their sale of 33% off all marked prices. If a jersey is marked R205, what price will I pay on the sale? I will pay 100% 33% = 67% R205 = R137, Percentage increase or decrease So far the examples we have looked at have asked you to find values after a percentage increase or decrease. We have not looked at finding what actual percentage increase or decrease takes place. We can find this using: Percentage increase = 100% Percentage decrease = 100% Example: 1. The number of learners in the school increased from 500 to 700. By what percentage have they increased? Percentage increase = 100% = 100% = 100% = 100% = 40% 2. The number of elephants in the area have decreased from 800 to 600. By what percentage have they decreased? Percentage decrease = 3. Rate, ratio and proportion 100% = 100% = 100% = 100% = 25% A ratio is used to compare quantities with each other. For example, if I share 21 sweets between Thoko and Thandi in the ratio 2 : 5, then for every 2 sweets I give Thoko, I must give Thandi 5 sweets. Thoko will get 6 sweets and Thandi will get 15 sweets. The quantities compared need to have the same units (the ratio of litres to litres, or rands to rands etc). You may have to convert the units to make them the same (the ratio of 5 cm to 20 mm = 50 mm : 20 mm = 5 : 2) We can work with ratios by multiplying or dividing each part of the ratio by the same number. The ratio 1 : 4 is the same as the ratio 5 : 20 and they are called equivalent ratios. A proportion is an equation stating that two ratios are equal (or in proportion ). For example, = Rate compares quantities with different measuring units. We usually reduce the rate to a quantity per one unit. For example, a car s speed might be 60 (km/h) kilometres per hour for every hour of driving, a distance of 60 km is covered. Scale is a ratio which shows the relationship between the actual distance on the ground and the distance on a map. We deal with this in Learning Outcome of 35

6 4. Working with finances Use the Winning Teams financial literacy cards to make sure that you understand the following vocabulary: Budget: A plan for what income you expect to receive and what expenses you expect to have over a period of time. Interest: The price for using someone else s money or the price for investing your money with someone else. For example, you can borrow money from the bank and pay it back over time. You have to pay back the whole amount borrowed and a percentage interest (interest rate) on top of that. Or if you invest money in an account at the bank, the bank will pay you a percentage interest (interest rate) on your money. VAT (Value Added Tax): Profit: Loss: Mark up: Discount: Break even: VAT is money charged by the government on top of the price of any goods or services made in South Africa. The money you make on selling something after you have subtracted your expenses. The money you lose on selling something after you have subtracted your expenses. The amount you add onto the cost price of goods before you sell them. The amount by which you reduce the selling price of goods. The point at which expenses and income are equal, so there is no profit or loss. Example (March 2009 Paper 2, Question 2): Annabel started working for a construction company on 3 July She earned a net income of R per annum without any bonus. She set aside R8 400 per month for her monthly expenses, and each month set aside 90% of the balance towards a deposit for a car. Table 1: Annabel s monthly expenditure before buying a car Items Monthly expenditure Rent and electricity R2 850 Groceries R1 500 Student loan repayment R900 Public transport to work R700 Clothing R350 Household insurance R420 Entertainment R350 Life insurance R300 Other R1 030 TOTAL R Calculate Annabel s net monthly salary. (2) How much did Annabel save towards the deposit for a car each month? (3) 2.2 Annabel was advised to invest some of her monthly savings in a special monthly savings account that pays more interest. She thus invested R3 000 of her monthly savings each month in this account. This special savings account paid an interest rate of 10,8% per annum, compounded monthly. Use the formula F = () to calculate the total amount she will have for her deposit if she saves monthly for 11 months, where: F = total amount received x = monthly amount invested i = monthly interest rate n = number of months for which the money was invested (5) 2.3 On 1 July 2008, Annabel receives a 10% salary increase. Calculate Annabel s new monthly net salary. (4) 2.4 On 1 July 2008 Annabel buys a car. She finds that she has to budget R3 900 per month for the car to cover the cost of petrol, repayments, insurance and maintenance. However, she no longer has to pay for public transport. Determine her new total monthly expenditure. (3) 6 of 35

7 Mathematical Literacy Grade 12 Foundational Knowledgee Solutions: Net monthly salary = = R per month Amount remaining each month = R R8 400 = R % of R3 600 = 0,9 R3 600 = R F =? x = R i =,% = 0,9% = 0,009 n = 11 months F = F = F = F = () (,), (,), (,), = R , New monthly salary = 112% of R = 1, 12 R = R New monthly expenditure = R R3 900 R700 = R Foreign exchangee rates The exchange rate gives us a way to convert cost in one currency to the equivalent cost in another currency. Countries often compare their currency with the US dollar (USD) to see how well their economy is doing. The rand dollar exchange rate is the number of rands we need to buy something that costs one dollar. A weak rand means that it will cost you more rand to buy 1 USD. A strong rand means that it will cost you less rand to buy 1 USD. On 25 November 2009, these were the currency exchange rates for the rand in a few selected countries: One unit of foreign currency equals x Rand 1 US dollar (USD) = 7,46 ZA Rands 1 Chinese yuan renminbi (CNY)= 1, 09 ZA Rands One rand equals x currency 1 Rand = 0, Pound sterling (GBP) 1 Rand = 6,23588 India Rupees (INR) Examples: Rands to USD: If I have R69 900, this is R ,46 = 9 369,97 USD. Divide the amount in rands by the exchange rate. USD to Rands: If I have USD, this is ,46 = R Multiply the amount in dollars by the exchange rate. Rands to INR: If I have R69 900, this is R ,09 = ,44 INR. INR to Rands: If I have rupees, this is ,09 = R of 35

8 6. Examination questions Fractions, decimals and percentages; rate, ratio and proportion; finances and exchange rates are integrated in the questions from examination papers below. Example (March 2009 Paper 1, Question 1.1): Do the following calculations. Show all calculations in full Write 47% as a common fraction Write as a decimal fraction Simplify: 1,2 m + (23,5 m 5) 4,7 m Simplify: (3) Calculate 14% VAT on R24 650, Convert R1 500 into Euros (). Use the conversion R1 = 0, Increase R1 250,00 by 24% Calculate the number of 30 g portions of jam that can be obtained from a 450 g tin Determine the cost of 6 bus tickets using the formula: Cost of bus tickets = number of bus tickets R12,15. Solutions: % = ,2 m + (23,5 m 5) 4,7 m = 1,2 m + 117,5 m 4,7 m = 114 m (3) = = = R24 650,00 14% = R24 650,00 0,14 = R3 451, R1 = 0,11. R ,11 = R % = R ,24 = R g 30 g = 15 portions Cost of bus tickets = 6 R12,15 = R72,90 Example (November 2008 Paper 1, Question 1): Write 20% as a common fraction in its simplest form Write as a percentage Simplify 120 : Decrease 500 kg by 12%. 1.2 Calculate the following: R450 R32, % of 450 km 1.3 Marie wants to bake coconut tarts for the school fete using her grandmother s old recipe. They have to be baked at 350F for 20 minutes Convert ½ lb to grams (1 lb = 450 g) Convert 9 oz to grams (1 oz = 30 g) One cup of sugar is equal to 250 ml. How many ml of sugar are needed for this recipe? Convert 350F to C using the formula: Temp in C = (temp in F 32) Round your answer off to the nearest 10 degrees How many eggs does Marie need to bake 72 coconut tarts? Coconut tarts (makes 3 dozen) Ingredients: 1 lb self-raising flour 9 oz margarine ¾ cup sugar 4 eggs 10 oz coconut ½ lb apricot jam 1 teaspoon vanilla essence 8 of 35

9 Solutions: % = = = = 68% : 150 = 12 : 15 = 4 : kg 12% of 500 kg = 500 ( 500) = = 440 kg OR 100% - 12% = 88% and 88% of 500 kg = 0, kg = 440 kg R450 R32,40 10 = R450 R324 = R = 25 6 = % 0f 450 km = 0, = 153 km ½ lb = ½ 450 g = 225 g oz = 9 30 g = 270 g ¾ cup = ¾ 250 ml = 187,50 ml Temp in C = (temp in F 32) Temp in C = (350F 32) Temp in C = 318F = 176,67C 180C to the nearest 10C eggs makes 3 dozen (36) tarts, so 8 eggs make 72 tarts. Example (March 2009 Paper 1, Question 3.3): Jabulani starts a small business making and selling money boxes. It costs him R25,50 to make ONE money box, including labour and overheads. He intends selling each money box for R30, Calculate the percentage profit Jabulani will make on each money box How many money boxes will Jabulani need to sell in order to make a profit of at least R400,00? Solutions: He makes a profit of R30,00 R25,50 = R4,50 on each money box., 100% = 17,65%,, = 88,89 He will need to sell 89 boxes to make a profit of R400,00., 9 of 35

10 Learning Outcome 2: Functions and Algebra When we want to find out about how one changing value affects another changing value, we can do this using a table of values, a formula or a graph. 1. Reading information from a table: In the example below, use the columns and the rows of the table to read information about times and destinations. Example (March 2009 Paper 1, Question 4): A train timetable is shown for travelling from Durban to Bloemfontein and then to Kimberley. Table 2: The train journey from Durban to Kimberley via Bloemfontein Frequency: From Durban on Wednesdays Town Arrival Departure Time in minutes stopped at station Durban 18:30 Pietermaritzburg 20:53 21:10 17 Ladysmith 00:33 27 Harrismith 03:23 03:53 30 Bethlehem 05:20 05:40 20 Kroonstad 07:49 08:19 30 Hennenman 08:57 08:59 2 Virginia 09:17 09:19 2 Theunissen 09:50 09:52 2 Brandfort 10:25 10:27 2 Bloemfontein 11:15 11:45 30 Kimberley 14: On which day of the week does this train from Durban arrive in Kimberley? (2) 4.2 How long did the total journey take? Write your answer in hours. (3) 4.3 At what time did the train leave Ladysmith? (2) 4.4 Calculate the total time taken for stops between Durban and Kimberley. Give the answer in hours and minutes. (3) 4.5 Not counting stops, the actual travel time for the train journey is 17,6 hours and the distance between Durban and Kimberley is 842 km. Calculate the average speed at which the train is travelling. Use the formula: Speed = (3) 4.6 James travels from Durban to Brandfort on the same train. He needs to board a bus in Brandfort that is leaving the bus station at 11:00. It takes 5 minutes to walk from the train station to the bus station. Determine whether or not James will be in time to board the bus. Show ALL the necessary calculations. (3) Solutions: 4.1 The train leaves Durban on Wednesday evening and travels through the night to the next afternoon, so it arrives in Kimberley on a Thursday. 4.2 From 18:30 to 14:50. From 18:30 to midnight is 5½ hours. From midnight to 14:50 is 14 hours 50 minutes. So the total time taken is: 5 hours 30 minutes + 14 hours 50 minutes = 19 hours 80 minutes = 20 hours 20 minutes = 20 = 20,33 hours. 4.3 It arrived at 00:33 and waited 27 minutes, so it left at 01: Total time taken for stops = = 162 minutes = 2 hours 42 minutes 4.5 speed = speed = = 47,84 km/h., 4.6 The train arrives in Brandfort at 10:25. James only needs 5 minutes to walk to the bus station and there is 35 minutes before the bus leaves at 11:00. So he has plenty of time to board the bus. 10 of 35

11 2. Reading a graph: In the example below, Jane completes a 20 kilometre walk. We want to see how the distance she covers changes over time. The graph gives us this information. We can read information about time and distance from the graph. Example (March 2009 Paper 1, Question 1.2): Jane participated in a sponsored 20 km walk to raise funds for Aids orphans. The organiser encouraged the walkers to have a fifteen-minute rest during the walk. The graph showing the distance covered by Jane against the time take by her, is given below. Time taken for 20 km walk 20 Distance covered in km A C 07:00 07:30 08:00 8:30 09:00 09:30 10:00 10:30 Time of day D At what time did the walk start? (1) How many kilometres did Jane walk during the first hour? (2) How far had Jane walked by 10:00? (2) How long did Jane take to walk the first 9 km? (2) After how many hours of walking did Jane rest? (2) Give an estimate of the time at which Jane finished the walk. (2) B Solutions: Read this from the graph: At 07:00, Jane had not started walking yet She had walked 6 km. Read this at A on the graph She had walked 16,5 km. Read this at B on the graph She took 1½ hours to walk the first 9 km. Read this at C on the graph Jane rested after two hours. Read this from the graph at D She finished the walk just before 10:30, probably at about 10: of 35

12 3. Working with formulae A formula shows a relationship between two sets of values. If you are given a formula, you can use it to determine and calculate particular values. You can substitute numbers into a formula to find a particular value. You can also substitute numbers into a formula to be able to complete a table of values or plot points on a graph. Example 1: If y = 2x + 3 and x = 5, determine the value of y. y = 2x + 3 y = 2(5) + 3 = 13 Example 2: If y = 2x + 3, complete the table of values below and use this to draw the graph. x y = 2x + 3 2( 2) + 3 = 1 2( 1) + 3 = 1 2(0) + 3 = 3 2(1) + 3 = 5 2(2) + 3 = 7 We can plot the points from the table of values onto a graph. The points are ( 2; 1), ( 1; 1), (0; 3), (1; 5) and (2; 7). These points show the pattern for the rest of the points represented by this formula, and so we can join them to form a straight line graph. y y = 2x x 1 Drawing a line graph: - Use a scale for each axis that works for your data. - Label the axes. - Plot the coordinates. - Join the plotted points if this works for your data. Example 3: Tim rides his motorbike for 4 hours and travels 280 km in that time. Work out his average speed. Use the formula: speed = Speed = = 70 kilometres per hour (km/h) 12 of 35

13 Example (March 2009 Paper 1, Question 6): Lourens and his five friends are planning a five-day holiday. As none of them own a car, they have decided to hire a Toyota Avanza from Oom Piet s car hire for 5 days. They will have to pay a rental cost plus an additional cost for petrol used. As part of their rental costs, Oom Piet s car hire offers 200 km free per day. Total rental cost for 5 days = R (number of kilometres over km) R3,00 Lourens worked out the total rental cost for 5 days for various distances travelled. He then drew a graph to illustrate his calculations. Total rental cost B Cost in rand A Kilometres travelled 6.1 Use the graph to: Write down the total rental cost over the five days if Lourens and his friends travel: (a) 950 km (1) (b) km (2) Determine the maximum number of kilometres they could travel in five days if they set aside R3 500 to pay for car hire. (2) 6.2 A Toyota Avanza covers 10 km on 1 litre of petrol. Suppose the friends travelled km and petrol costs R10,40 a litre, calculate the petrol bill for the journey, using the following formula: Petrol bill = cost of 1 l of petrol (3) 6.3 On a particular day Lourens and his friends covered a distance of 360 km. Table 3 illustrates the time taken to travel the 360 km at different speeds. Table 3: Time taken to travel 360 km for different speeds Speed in km B per hours Time taken in hours A , Determine the values of A and B. Use the formulae: time = or speed = (4) 13 of 35

14 6.3.2 Use the grid below to draw a line graph representing the information in Table 3. Time taken to travel 360 km Time in hours Speed in kilometres per hour 6.4 The bill for the car hire came to R4 236,00. The six friends decided to share the bill equally. How much did each one have to pay? (2) Solutions: (a) If they travel 950 km, this is below the km free, so it is R You can read this from the graph at A. (b) If they travel km, they pay R (300 km R3,00) = R R900 = R2 900 You can read this from the graph at B R3 500 R2 000 = R1 500 R1 500 R3,00 = 500 km They can travel km km = km. 6.2 Petrol bill = Petrol bill = For A, time = For B, speed = cost of 1 l of petrol R10,40 per litre = R1 456,00 = / = = 12 hours = 60 km per hour 14 of 35

15 Time taken to travel 360 km Time in hours R friends = R706 each Speed in kilometres per hour 15 of 35

16 Example (March 2009 Paper 2, Question 1): The Hospitality Studies department of Ses fikile High School bakes brown bread in order to raise funds for the shortfall incurred in their day-to-day expenses. The school charges the Hospitaliy Studies department a fixed weekly cost of R400,00 for water and electricity. The cost of producing one loaf of brown bread, including labour and ingredients, is R3,50. The brown bread is sold at R6,00 a loaf. 1.1 If one loaf of brown bread requires 450 g of flour, determine the maximum number of loaves of brown bread that can be baked from a 12,5 kg bag of flour. (4) 1.2 The table below shows the weekly cost of making the bread. Table 4: Weekly cost of making brown bread Number of loaves B 300 Total cost (in rand) A The formula used to calculate the total cost per week is: Total cost per week = fixed weekly cost + (number of loaves of bread cost per loaf) Use the formula to determine the values of A and B in Table 4. (4) 1.3 The table shows the weekly income from selling the bread. Table 5: Weekly income received from selling bread Number of loaves D Total income (in rand) C Determine the values of C and D in Table 5. (4) 1.4 Use the values from Table 4 and Table 5 to draw TWO straight line graphs on the same grid (see next page), showing the total COST per week of making bread and the INCOME per week from selling bread. Label the graphs COSTS and INCOME. (8) 1.5 Use the tables or the graphs to answer the following questions: How many loaves of bread must they sell to break even and describe what is happening at the break-even point? (3) What income would they receive if 230 loaves were sold? (2) Estimate the number of loaves baked if the total cost is R840. (2) Determine, by calculation, whether Ses fikile High School will make a profit or a loss if they bake 300 loaves of bread during the week, but only sell 250 of these loaves of bread. (3) Solutions: ,5 kg = g g 450 g = 27,77 So you can make 27 loaves of bread from 12,5 kg of flour. 1.2 Total cost per week = fixed weekly cost + (number of loaves of bread cost per loaf) At A, Total cost per week = R400 + (120 loaves R3,50) = R400 + R420 = R820 At B, R1 240 = R400 + (number of loaves R3,50) R1 240 R400 = (number of loaves R3,50) R840 = (number of loaves R3,50) R840 R3,50 = 240 loaves 1.3 A loaf of bread is sold at R6,00 per loaf. At C, 120 loaves R6,00 = R720 At D, R960 = number of loaves R6,00 R960 R6,00 = 160 loaves 16 of 35

17 Income and costs Amount in rands Number of loaves of bread 17 of 35

18 Income and costs B 1300 Income Costs A Amount in rands Number of loaves of bread loaves must be sold. At this point, both the cost and the income are the same and are equal to R loaves will sell for R1 380 (at B) Read this from the graph at A. It is about 125 loaves Cost of making 300 loaves = R3,50 = R1 450 Income from selling 250 loaves = 250 R6,00 = R So they will make a profit of R of 35

19 Learning Outcome 3: Space, shape and measurement 1. Perimeter The perimeter of a flat shape is the length around its outer edge. Perimeter is measured in units of length e.g. mm, cm, m and km. Any formulae for perimeter, area, surface area or volume that are needed to calculate answers will be provided in the examination paper. perimeter of a rectangle = 2l + 2b perimeter of a square = 4l length (l) breadth (b) length (l) length (l) circumference of a circle = 2πr where π 3,14 perimeter of a triangle = a + b + c and r is the radius. r b a c 2. Area Area is measured in square units: mm 2 ; cm 2 ; m 2 ; km 2 This square has an area of 1 cm by 1 cm. It is one square cm or 1 cm 2 in area. area of a rectangle = length x breadth area of a square = length x length length (l) breadth (b) length (l) length (l) area of a rhombus = base height area of a parallelogram = base height h h b area of a trapezium = ½h (a + b) b area of a triangle = ½b x h b h h h a area of a circle = πr 2 where π 3,14 and r is the radius. r b b 19 of 35

20 3. Surface area Surface area is the total amount of area on the surface, the exposed part of a 3-D object. If you were asked to paint the object, the paint would cover the surface area. To calculate the surface area, find the area of each face of the object and add the areas together. face edge vertex Rectangular prism: A right rectangular prism is an object with equal parallel bases; all faces are rectangles; length (l) breadth (b) side faces which are perpendicular to the bases. breadth (b) You can unfold the prism to find its net. This is one possible net, depending on how you unfold the prism. Surface area = 2(length breadth) + 2(length height) + 2(breadth x height) = 2lb + 2lh + 2bh height (h) length (l) l b Triangular prism: Surface area = 2(½ b h) + 2(H S) + b H H S h H h S b H H b H r Cylinder: The length of the rectangular part of the net fits around the circumference of the circular part, so Length = 2πr Surface area = 2πr h + 2(πr) 2 2πr h r h = 2πrh + 2πr 2 where r is the radius, h is the height and π 3, of 35

21 4. Volume: The amount of space a solid shape takes up is called its volume or its capacity. The standard unit of measurement for volume is a unit cube, with each edge one unit long. Volume is measured in cubic units e.g. mm 3 ; cm 3 ; m 3 ; km 3 Example: Volume of a unit cm cube = 1 cm 1 cm 1 cm = 1 cm 3 1 cm height (h) Volume of a rectangular prism = area of base height = length breadth height b Volume of triangular prism = area of base height = (½ b h) H Volume of a cylinder = area of circle base height = r 2 h where π 3,14 and r is the radius. 5. Conversions We use the following metric units of length and distance. We usually use kilometres (km) for very long distances, metres (m) and centimetres (cm) for shorter lengths and millimetres (mm) for very small measurements. thousands hundreds tens ones tenths hundredths thousandths kilometre (km) length (l) breadth (b) hectometre decametre metre (m) decimetre 1. To convert to a smaller unit of length, multiply by 10, 100, etc. Examples: From kilometres to metres: 1,34 km = 1, m = m From metres to centimetres: 7,4 m = 7,4 100 cm = 740 cm From centimetres to millimetres: 33 cm = mm = 330 mm. 2. To convert to a bigger unit of length, divide by 10, 100, etc. Examples: From metres to kilometres: m = km = 1,34 km From centimetres to metres: 740 cm = m = 7,4 m From millimetres to centimetres: 330 mm = cm = 33 cm. 3. To convert units of area, multiply or divide by 10 2, 100 2, (1 000) 2 etc. Examples: From km 2 to m 2 : 1,34 km 2 = 1,34 ( )m = m 2 From m 2 to cm 2 : 7,4 m 2 = 7,4 ( )cm = cm 2 From cm 2 to m 2 : 7,4 cm 2 = 7,4 ( )m 2 = 0,00074 m 2 4. To convert units of volume, multiply or divide by 10 3, 100 3, (1 000) 3 etc. Examples: From m 3 to cm 3 : 7,4 m 3 = 7,4 (100 cm) 3 = cm 3 5. From cm 3 to m 3 : 7,4 cm 3 = 7,4 (100 m) 3 = 0, m 3 S 1 cm 1 cm centimetre (cm) The same conversions apply to units of capacity: kilolitres (kl), litres (l), milliliters (ml) The same conversions also apply to units of mass: kilograms (kg), grams (g) and milligrams (mg). h millimetre (mm) Sometimes we need to convert between different measurement systems. For example, measurements in recipes are sometimes written in pounds (lb) and ounces (oz). You will be given the conversion factor needed. H r h 21 of 35

22 Volume can also be written in litres. It is useful to know cm 3 = 1 litre. 1 m 3 = 1 m 1 m 1 m = 100 cm 100 cm 100 cm = cm 3 = litres = 1 kilolitre. Example: If a cylindrical bucket has a radius of 10 cm and a height of 20 cm, how many litres will it contain? 10 cm Volume = r 2 h = 3,14 (10) 2 20 = cm 3 = 6,28 litres. 20 cm Examples: 1. A recipe uses 2,2 lbs of flour. Convert this to grams if 1 lb = 450 g. 2,2 lbs = 2,2 450 g = 990 g From pounds to grams: multiply 2. How many lbs is 2 kg of flour? 2 kg = g g 450 g = 4,44 lb From grams to pounds: divide 3. Grandpa says that he is 5 foot 7 inches tall. How tall is this in metres if 1 foot = 12 inches and 1 inch = 2,54 cm? 5 foot 7 inches = (5 12) + 7 = 67 inches 67 2,54 cm = 170,18 cm From inches to cm: multiply 170, = 1,7018 m 1,7 m 4. How many inches long is a 30 cm ruler? 30 cm 2,54 cm = 11,8 inches. From cm to inches: divide Example (March 2009 Paper 2, Question 3): As a result of load shedding, Wayne, a chicken farmer, goes back to using a generator to provide dependable power for his chicken shed and his farmhouse. He buys a second-hand diesel tank with a radius of 1 m and a length of 2 m to store the fuel for the generator. 3.1 He decides to paint both the outside surface area of the tank and the stand on which it rests. The surface area of the stand is 1 m 2. It takes 1 litre of paint to paint 3 m 2 of the surface area Calculate the surface area (SA) of the tank in m 2. Use the formula: SA = 2πrh + 2πr 2 where r is the radius, h is the height and π 3,14. (3) Calculate the quantity of paint (in litres) needed to paint both the outside of the tank and the stand. Round off your answer to the nearest litre. (5) If a one litre tin of paint costs R23,63 and a 5 litre tin of paint costs R113,15, calculate the most economical way to purchase the amount of paint needed in Question (3) Calculate the capacity (volume) of the diesel tank in litres where 1 m 3 = litres. Use the formula: V = r 2 h where π 3,14, h is the height and r is the radius. (4) Farmer Wayne fills the diesel tank to 80% of its capacity. The generator used 72 litres of diesel in 36 hours. Calculate the amount of diesel in litres remaining in the tank after 7 days of the generator running continuously. (8) Solutions: SA = 2πrh + 2πr 2 r = 1 m and h = 2 m SA = 2(3,14)(1)(2) + 2(3,14)(1) 2 SA = 12,56 m 2 + 6,28 m 2 = 18,84 m 2 22 of 35

23 3.1.2 Area to be painted = surface area of stand + surface area of tank = 1 m ,84 m 2 = 19,84 m 2 1 litre paints 3 m 2 19,84 m 2 3 m 2 = 6,613 litres of paint 7 litres of paint If you buy 7 one litre tins, it will cost 7 R23,63 = R165,41. If you buy a 5 litre tin and 2 one litre tins, it will cost R113, R23,63 = R160,41. This is more economical than buying 7 one litre tins V = r 2 h V = (3,14)(1 m) 2 (2 m) = 6,28 m 3 = 6, litres = litres In one hour the generator uses = 2 litres. 7 days = 7 24 hours = 168 hours So in 7 days, the generator will use litres = 336 litres. Amount of diesel in the tank to start: 80% of litres = 0, = litres. So after 7 days, the amount of diesel remaining in the tank is = litres. 6. Maps, grids and compass direction A map represents an area of land, a city or even a street on paper. The map uses a scale factor to reduce the real size of the area to a representation on paper. A scale on a map shows how the lengths on the ground have been reduced to be represented on the map. For example, if a map has a scale of 1 : , this means that every unit of measurement on the map represents units on the ground. So 1 cm on the map represents cm (which is metres or 5 km) on the ground. Example (November 2008 Paper 2, Question 5): Naledi High School decides to tile the floor of their school hall using black and grey tiles. The hall is L-shaped. A rectangular stage is located against one wall of the hall as shown in the sketch. The width of the stage is 5 m and the length is 10 m. The Floor plan of school hall stage is not going to be tiled. The size of a square tile is 50 cm by 50 cm. The school needs 18 m to calculate how many tiles to buy. N 5.1 A scale drawing is to be made of the hall. Determine the scale (in simplified form) to be used if the length of the north wall of the hall is 60 mm. (2) 5.2 Calculate the area of the floor (excluding the stage) to be tiled. (8) Area of a rectangle = length breadth 5.3 The tiler requires that 5% more tiles must be purchased to allow for cutting and breakages. Calculate how many tiles must be bought. (7) 5.4 The ratio of black tiles to grey tiles needed is 4 : 1. Tiles come in boxes of 12. Calculate the number of full (complete) boxes of black tiles that must be bought. (4) 9m 10 m 6 m 18 m 23 of 35

24 Solutions: mm represents 18 m. Convert to the same units of measurement: 18 m = cm = cm = mm = mm. So the scale is 60 : This can be simplified to 1 : Total area of floor (including stage area) Total area of floor (excluding stage area) = (18 m 18 m) (9 m 6 m) = 270 m 2 (10 m 5 m) = 324 m 2 54 m 2 = 270 m 2 50 m 2 = 270 m 2 = 220 m The tiles must cover 220 m 2. One tile is 50 cm 50 cm = 0,5 m 0,5 m = 0,25 m 2 = 880 tiles, 5% more tiles is 0, = 44 tiles. So = 924 tiles must be bought tiles divided in a ratio of 4 : 1 Number of black tiles needed is of = 739,2 tiles 740 black tiles will be needed. Number of boxes needed: = 61,67 So 62 boxes of black tiles will be needed. When we give directions or describe a route, it helps to use direction words such as left and right. It also helps to use the compass directions e.g The church is North-east of the school. North (N) Learn these compass directions! North-West (NW) North-East (NE) West (W) East (E) South-West (SW) South-East (SE) South (S) It is useful to place a grid of lines that cross each other at right angles over the map. This allows us to find a specific point on a map and label it with its grid reference. We name a point using its horizontal coordinate first and then its vertical coordinate. 24 of 35

25 Mathematical Literacy Grade 12 Foundational Knowledgee Example (March 2009 Paper 2, Question 5): 5.5 Gerrie van Niekerk is a primary school learner who lives in Krugersdorp. He lives on the corner of Wishart Street and 5 th Street. Use the map of part of Krugersdorp, Gauteng to answer the following questions Give a grid reference for the Jays Shopping Centre wheree Gerrie and his mother do their weekly grocery shopping. (1) Gerrie s grandmother lives with them and goes to the hospital for her medication once a month. What is the relative position of Krugersdorp Central Hospital with respect to Gerrie s home? (1) Gerrie s father drives from Jays Shopping Centre to the petrol station to buy petrol for his car. Describe his route if the exit from Jays Shopping Centre is in 4 th Street. (3) Gerrie walks from home to Paardekraal Primary School by: Crossing 5 th Street and walking in an easterly direction along Wishart Street; Turning right and walking in a southerly direction along 4 th Street; Turning left and walking in an easterly direction along Onderste Street; Turning right, and walking in a southerly direction along 3 rd Streett The school s entrance is on the corner of 3 rd Street and Pretoria Street. (a) Measure the total walking distance on the map between Gerrie s house and the Paardekraal Primary School in centimetres. (b) Use the scale 1 : to calculate the actual distance Gerrie walks to school. Give your answer in kilometres. (3) (4) 25 of 35

26 Solutions: C South-East Here are some possible answers: Turn left in 4 th Street. Turn left into Buiten Street. After passing Gerrie Visser Street, turn right into the next street. You will see the petrol station ahead of you. OR Turn left into 4 th Street. Turn left into Wishart Street. Turn right into Gerrie Visser Street. Turn left into Buiten Street. At the next street turn right. You will see the petrol station ahead of you. OR Turn in a northerly direction along 4 th Street. Turn in a westerly direction along Buiten Street. After passing Gerrie Visser Street, turn in a northerly direction into the next street you come to. You will see the petrol station ahead of you (a) Total walking distance on the map = (1,8 cm + 2,7 cm + 1,8 cm + 1,5 cm) = 7,8 cm (b) Scale is 1 : So 7,8 cm on the map is 7, cm on the ground. 7, cm = cm = m = 858 m = km = 0,858 km. Learning Outcome 4: Data handling Data handling is the study of statistics, or data. We collect, organise, analyse and interpret data and we can interpret the data in different ways. 1. Representing data 1.1 Frequency tables Example: In a mathematics class, 30 learners completed a test that was out of 20 marks. Here is a list of their results: 14; 10; 11; 19; 15; 11; 13; 11; 9; 11; 12; 17; 10; 14; 13; 17; 7; 14; 17; 13; 13; 9; 12; 16; 6; 9; 11; 11; 13; 20. We can show these results in a frequency table (horizontal or vertical). A frequency table shows how many learners there are in each category. We can tally the number of learners getting each possible mark out of 20 and then write this as a frequency. Mark out of 20 Tally Frequency (number of learners) 6 / 1 7 / /// 3 10 // 2 11 //// / 6 12 // 2 13 //// 5 14 /// 3 15 / 1 16 / 1 17 /// / 1 20 / 1 With this data set, it is more useful to group the data. We can use class intervals of 5 and make this frequency table (including a column for the tallies): Class Frequency interval = = = 6 26 of 35

27 Data can be represented with a graph. This helps us to read all the data easily and to compare parts of the data with each other. 1.2 Bar graphs Leave equal spaces between the bars. Use a scale with equal intervals to mark the vertical axis. The graph needs a heading. Both axes need labels. The height or length of the bar represents the frequency of that category of information. The bars can be drawn vertically or horizontally Test marks Class intervals of test marks Example (March 2009 Paper 1, Question 2.2): Learners were invited to enter a national essay-writing competition. The 70 winners attended a Youth Forum in Johannesburg. A survey was done to find out how many winners came from each province. The results are given in Table 6. Table 6: Number of winners from each province Province Number of winners Eastern Cape 8 Free State 6 Gauteng 10 Kwazulu-Natal 11 Limpopo 8 Mpumalanga 7 Northern Cape 5 North West 6 Western Cape What percentage of the winners came from Mpumalanga? (2) Calculate the ratio of the number of winners from Gauteng to the number of winners from Northern Cape. Give your answer in simplified form. (2) Suppose one of the winners is chosen randomly. What is the probability that the learner is from: (a) The Eastern Cape (b) South Africa (4) Draw a vertical bar graph to represent the data in Table 6 on the grid provided. (5) Number of winners attending the Youth Forum Number of learners EC FS GP KZN LP MP NC NW WC Province 27 of 35

28 Solutions: Mpumulanga: 100% = 10% Gauteng : Northern Cape = 10 : 5 = 2 : (a) P(learners from Eastern Cape) = (b) P(learners from South Africa) = bar graph = = 0,114 or 11,43% = = 1 or 100% 12 Number of winners attending the Youth Forum 10 Number of learners EC FS GP KZN LP MP NC NW WC Province 1.3 Pie Charts A pie chart uses sectors of the circle to represent the categories of data. It is easy to compare the categories of data to each other and to the total data in a pie chart. A pie chart shows a value as a fraction of the total. Here is the pie chart for the test results of the mathematics class (refer to frequency table and bar graph). Each frequency has been converted to a percentage in this pie chart % % % % Class intervals for test results 2. Reading data from tables and graphs When working with data, we can read and interpret the data represented in tables and graphs. We can also compare two sets of data with each other using two tables or two graphs. 28 of 35

29 2.1 Comparing two sets of data represented in tables Example (March 2009 Paper 2, Question 4): One of the aims of the Arrive Alive Campaign is to increase safety on South African roads. The Arrive Alive team decide that one of the ways of alerting the public to the dangers of road travel is to publish the data on fatalities on South African roads. Table 7: Number of fatalities per province YEAR PROVINCE GAU KZN WC EC FS MPU NW LIM NC RSA [Source: ] The Arrive Alive team also decide to compare the estimated million vehicle kilometres (mvk) travelled in a province to the number of fatalities in that province. Table 8: Million vehicle kilometres (mvk) travelled per province YEAR PROVINCE GAU KZN WC EC FS MPU NW LIM NC RSA [Source: ] Note: This means that a total of km were travelled by all the vehicles on Gauteng roads in Use the data in TABLE 7 and TABLE 8 to answer the following questions: 4.1 Which province had a decrease in both the number of fatalities from 2005 to 2006 and the million vehicle kilometres (mvk) travelled? (1) Which TWO provinces had the highest number of fatalities in 2005 and 2006? (2) In which TWO provinces were the highest million vehicle kilometres (mvk) travelled? (2) Describe the possible relationship between the number of fatalities and the number of million vehicle kilometres (mvk) travelled per province in the two provinces indicated in Question (2) What percentage of the total number of fatalities in South Africa in 2006 occurred in Gauteng (GP)? (3) Calculate the number of fatalities per million vehicle kilometres travelled in 2006 in: (a) Gauteng (rounded off to THREE decimal places) (4) (b) The province with the lowest number of fatalities (rounded off to THREE decimal places) (4) Which of these two provinces do you think is the safest in terms of kilometres travelled and fatalities suffered? Give ONE valid reason for your answer. (3) Solutions: 4.1 Limpopo Gauteng and KwaZulu-Natal Gauteng and KwaZulu-Natal The higher the number of million vehicle kilometres (mvk) travelled, the higher the number of fatalities Fatalities in Gauteng = 100% = 22,17% (a) Gauteng: = = 0,077 (b) Northern Cape: = = 0, ,077 < 0,134 so fewer fatalities occur in Gauteng than in Northern Cape per million vehicle km. So Gauteng is safer. 29 of 35

30 2.2 Comparing two sets of data represented in two pie charts, bar graphs or broken line graphs Example (November 2008 Paper 2, Question 4.1): The graphs below represent the provincial data on average annual incomes (in thousands of rand) and unemployment rates (as a percentage). Key: WC = Western Cape KZN = KwaZulu-Natal MP = Mpumalanga EC = Eastern Cape NW = North West LIM = Limpopo NC = Northern Cape GP = Gauteng FS = Free State RSA = whole of South Africa Use the graphs to answer the questions below: Calculate the difference in value between the average annual income for Gauteng and the Eastern Cape. (4) What relationship, if any, exists between the unemployment rate and the average annual household income for the provinces? (2) Give a valid reason why a person would choose to work in Gauteng rather than in the Eastern Cape. (2) Solutions: Difference between average annual income between Gauteng and Eastern Cape = R R = R The higher the average income, the lower the unemployment rate Gauteng has a higher average annual income and lower unemployment rate than the Eastern Cape. It is likely that a person will earn a better salary in Gauteng and the chances of being employed in Gauteng are better. 30 of 35

31 2.3 Interpreting a compound bar graph Compound and multiple bar graphs are used to compare two or more sets of data. Example (March 2009 Paper 1, Question 2.3): The compound bar graph below shows the percentage of South African children from age seven to thirteen, enrolled in primary schools during 1996 and Percentage enrolment of South African children in primary school Percentage ,1 81,8 87,9 91,3 93,6 94,4 94, ,8 95,6 95,9 96,3 96, ,8 Ages in years [Source: 8 November 2007] Use the graph to answer the following questions: What percentage of the 10-year-olds was enrolled during 1996? (1) Calculate the percentage increase in enrolment of 11-year-olds from 1996 to (3) Which age group had: (a) The largest percentage enrolment during (1) (b) The smallest percentage enrolment during 2007 (1) (c) The greatest increase in percentage enrolment between 1996 and (2) If there were year-old children in South Africa in 1996, calculate the number of 10-year-old children enrolled in primary school. (2) Solutions: ,3% (read this from the table) Percentage increase = 96,3% 93,6% = 2,7% (a) 13-year-olds (b) 7-year-olds (c) 7-year-olds ,3% of =, = learners 31 of 35

32 2.4 Using graphs to represent a point of view Sometimes graph s misrepresent or appear to change the facts This may be done purposely to mislead or persuade the audience to a point of view. Some of the reasons why graphs can be misleading: The scales used are too big or too small for the data Scales that use changing intervals between markings on the axes Scales that do not start at zero Unlabelled axes Leaving out important information Example (March 2009 Paper 2, Question 6.1): Bathwizz is a company that installs and renovates bathrooms. The general manager had to present the company s earnings for the first three quarters of the year to the company directors. He drew the graphs below. GRAPH 1 GRAPH 2 Use the graphs to answer the following questions: What possible trend do you notice with regard to Bathwizz s quarterly income? (2) Calculate the average (mean) monthly income for Bathwizz for the first nine months of the financial year. (4) The general manager wanted to prove to the company directors that Bathwizz s income was increasing and that the company was doing well. Which graph would be the better one to show to the company directors? Give a reason for your answer. (3) Solutions: There is a steady increase in income The three quarters shown are nine months. Mean monthly income (in R s) =,, = = R1,22 hundred thousand = R It will be better to show the company directors Graph 2 because the increase between quarters looks bigger. This is because the scale of the graph starts at 2,5 and not at of 35

33 5. Measures of central tendency and spread: These are different measures of finding the middle of a set of data and how much the set of data is spread out. In a mathematics class, 13 learners completed a test out of 25 marks. Here is a list of their results: 14; 10; 23; 21; 11; 19; 13; 11; 20; 21; 9; 11; 17. There are three ways to find the average of their results. We can use the mean, the median or the mode. 5.1 To find the mean, we add up all the values (test marks in this example) and divide by the number of values (number of learners in this example) ( ) mean = = () = = 15,3846 = 15,38 rounded off to two decimal places 5.2 The median is the middle number in an ordered data set. If you order the results in ascending order, you get: 9; 10; 11; 11; 11; 13; 14; 17; 19; 20; 21; 21; 23. The middle number is the 7 th number out of 13 numbers which is 14. So the median is 14. When there is an even number of values in the data set, the median lies halfway between the middle two values. We can add these two values and divide by 2. Suppose another learner wrote the test and her result was 9. Then we can add this to the ordered data set: 9; 9; 10; 11; 11; 11; 13; 14; 17; 19; 20; 21; 21; 23. Now the middle two numbers are 13 and 14. = =13,5 So the median is 13, The mode is the number or value that appears most frequently in the data set. The mode of the results of the test is 11. The measures of dispersion or spread give us information about how spread out the data is. The range is the difference between the highest value (or maximum) and the lowest value (or minimum) in a data set. It shows how spread out the whole data set is. Range = largest value in the data set smallest value in the data set For example, the range of the test results: 23 9 = of 35