Mathematical Literacy Level 4 Module 1

Mathematical Literacy NQF 4 Downloaded from hlayiso.com Maths Lit NQF Level 4

Mastering calculations Module 1 Downloaded from hlayiso.com Maths Lit NQF Level 4

Overview 1.1 The calculator 1.2 Calculations and solving problems with positive and negative numbers 1.3 Calculating and solving problems with fractions 1.4 Calculating and solving problems with decimals 1.5 Calculating and solving problems with percentages Downloaded from hlayiso.com Maths Lit NQF Level 4

Overview 1.6 Calculating and solving problems with squares, square roots, cubes and cube roots 1.7 Rounding numbers Downloaded from hlayiso.com Maths Lit NQF Level 4

The Calculator Unit 1.1 Get to know how your calculator works. Identify and find: • the numbers (usually in the middle, with zero at the bottom). • basic maths symbols (usually on the right). • the percentage key, the square root key, the equals key. • The key clears everything on the display. It is sometimes labelled . • The key stands for ‘clear entry’ and it clears the last thing that was entered. Downloaded from hlayiso.com Maths Lit NQF Level 4

The Calculator Unit 1.1 For adding, subtracting, multiplying and division: • Press number key, operation key, number key. • Press the equals sign = for the answer. Important! • Notice that the calculator always uses a decimal point, not a decimal comma. • Markings (or separators) between groups of three digits make the number on the display easier to read. Downloaded from hlayiso.com Maths Lit NQF Level 4

Mixed calculations: the correct order of operations Unit 1.1 A useful rule for the correct order of operations is BODMAS. • Work from left to right in a calculation. • Division and multiplication have the same priority. So when you have division and multiplication in a calculation, work on them from left to right. • Addition and subtraction also have the same priority. Downloaded from hlayiso.com Maths Lit NQF Level 4

Mixed calculations: the correct order of operations Unit 1.1 Important! • Basic or simple calculators do the operations in the order that the keys are pressed. So you must do the operations in the correct order, otherwise the calculator might add numbers before dividing, and so on. • Newer scientific calculators follow the mathematically correct order of operations, so you input the calculation exactly as it is written. Downloaded from hlayiso.com Maths Lit NQF Level 4

Mixed calculations: the correct order of operations Unit 1.1 Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.2 page 3 Unit 1.1 Calculate 25 + (54 ÷ 2) + (1 430 – 1 324) On a basic calculator: 25 + (54 ÷ 2) – (1 430 – 1 394) Calculate both of the expressions in = 25 + 27 – 36 brackets first. = 52 – 36 Do the + and – operations, starting at = 16 the left. Downloaded from hlayiso.com Maths Lit NQF Level 4

Exercise 1.1 Unit 1.1 Complete Exercise 1.1 on page 4 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 4

Addition and multiplication facts Unit 1.1 Commutative Law For adding or multiplying, we can swap numbers around and get the same answer. 40 + 50 = 50 + 40 30 × 20 = 20 × 30 This does not work for division or subtraction: 12 ÷ 3 = 4, but 3 ÷ 12 = ¼ 12 – 3 = 9, but 3 – 12 = –9 Downloaded from hlayiso.com Maths Lit NQF Level 4

Addition and multiplication facts Unit 1.1 Associative Law For adding or multiplying, we can group numbers in different ways without changing the answer. 1. (200 + 41) + 52 = 241 + 52 = 293 and 200 + (41 + 52) = 200 + 93 = 293 2. (13 × 4) × 50 = 52 × 50 = 2 600 and 13 × (4 × 50) = 13 × 200 = 2 600 This does not work for subtraction: (9 – 4) – 3 = 5 – 3 = 2, but 9 – (4 – 3) = 9 – 1 = 8 Downloaded from hlayiso.com Maths Lit NQF Level 4

Addition and multiplication facts Unit 1.1 Distributive Law We get the same answer when we multiply a number by a group of numbers added together, or do each multiplication separately and then add them. 33 × (20 + 40) = 33 × 60 = 1 980 and 33 × 20 + 33 × 40 = 660 + 1 320 = 1 980 This does not work for division: 24 ÷ (4 + 8) = 24 ÷ 12 = 2, but 24 ÷ 4 + 24 ÷ 8 = 6 + 3 = 9 Downloaded from hlayiso.com Maths Lit NQF Level 4

Addition and multiplication facts Unit 1.1 Tip We can use the associative law to make it easier to add or multiply mentally. For example: 19 + 36 + 4 = 19 + (36 + 4) = 19 + 40 = 59 2 × 16 × 5 = (2 × 5) × 16 = 10 × 16 = 160 The distributive law gives us a useful trick for doing mental multiplication quickly. For example: 4 × 57 = 4 × (50 + 7) Break up the number 57 = (4 × 50) + (4 × 7) Distribute 4 over the parts of 57. = 200 + 28 = 228 Downloaded from hlayiso.com Maths Lit NQF Level 4

Exercise 1.2 Unit 1.1 Complete Exercise 1.2 on page 5 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 4

Solving problems Unit 1.1 1 Read the problem carefully. 2 List the facts in the problem. 3 Write down what you need to find. 4 Decide which operations you must do and write the number sentence. Apply your knowledge of the operations to do the 5 calculation. Downloaded from hlayiso.com Maths Lit NQF Level 4

Solving problems Unit 1.1 Tips on deciding which operator to use Use addition with words like ‘find the sum’ and ‘what is the total’. Use addition or multiplication with words like ‘if each’ ... ‘find all’. Look carefully at the facts in the problem to decide. Downloaded from hlayiso.com Maths Lit NQF Level 4

Solving problems Unit 1.1 Tips on deciding which operator to use • Use subtraction with words like ‘what is the difference’, ‘how much more’ or ‘how much less’. • Use division with words like ‘how much is each’, ‘if each is ... how many’, ‘find the cost per ...’. • Sometimes you need to rephrase a problem to help you see what operation to use. State the problem in simple words. It can also help to do a sketch of the problem. Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.3 page 5 Unit 1.1 Downloaded from hlayiso.com Maths Lit NQF Level 4

Exercise 1.3 Unit 1.1 Complete Exercise 1.3 on page 6 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 4

Calculating and solving problems with positive and negative numbers Unit 1.2 • Negative numbers have a minus sign in front of them. They show numbers smaller than zero. • Examples: temperatures below 0°C, financial calculations, and to show changes in position down. • As the negative number appears to get 'bigger', its value gets smaller. • So –5 is smaller than –1. Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.4 page 8 Unit 1.2 • Health laws say that shops must keep fridges at 5°C and freezers must be at –18°C. What is the difference between these two temperatures? • The temperature of a freezer in a shop increases from –18°C to –11°C during a power failure. By how much must the temperature decrease before food can be stored in it again? • Adrian has only R230 in his cheque account, but he can withdraw R150 and R750. How much is his overdraft? Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.4 page 8 Unit 1.2 1. Draw a rough number line. 5 °C – (–18 °C) = 5 + 18 = 23 °C Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.4 page 8 Unit 1.2 2. Draw a rough number line. – 18°C – (–7°C) = –18°C + 7°C = –11°C Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.4 page 8 Unit 1.2 3. Draw a rough number line. R230 – R150 – R750 = –R670 Downloaded from hlayiso.com Maths Lit NQF Level 4

Exercise 1.4 Unit 1.2 Complete Exercise 1.4 on page 9 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 4

Types of fractions Unit 1.3 Fraction • A fraction represents part of a whole that has been divided into equal pieces. • Example: One tenth of a group of people means that the group is divided into ten equal parts, and we are referring to the people in one of those parts. Downloaded from hlayiso.com Maths Lit NQF Level 4

Using a calculator to work with fractions Unit 1.3 Downloaded from hlayiso.com Maths Lit NQF Level 4

Converting a mixed number to an improper Unit 1.3 fraction On a basic calculator: 1. Multiply the whole number by the denominator of the fraction. 2. Add this answer to the original numerator to get the new numerator. 3. Write the new numerator over the fraction line and the original denominator under it. On a scientific calculator: Enter the ‘mixed number’ and press the or key. Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.5 page 11 Unit 1.3 Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.5 page 11 Unit 1.3 Downloaded from hlayiso.com Maths Lit NQF Level 4

Converting an improper fraction to a Unit 1.3 mixed number On a basic calculator: 1. Divide the numerator by the denominator until you have a remainder. 2. Your answer becomes the whole number. 3. The remainder forms the numerator of the fractional part. On a scientific calculator: Enter the ‘improper fraction’ and press the or key. Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.5 page 11 Unit 1.3 Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.5 page 11 Unit 1.3 Downloaded from hlayiso.com Maths Lit NQF Level 4

Equivalent fractions Unit 1.3 Definition Equivalent fractions are two or more fractions that have the same value. Downloaded from hlayiso.com Maths Lit NQF Level 4

Equivalent fractions Unit 1.3 Definition Equivalent fractions are two or more fractions that have the same value. Downloaded from hlayiso.com Maths Lit NQF Level 4

Equivalent fractions Unit 1.3 Definition Equivalent fractions are two or more fractions that have the same value. To write fractions in the simplest form: • Write improper fractions as mixed numbers. • Can the common fraction be written in a simpler form. Is there a number which can divide into the numerator and the denominator? Downloaded from hlayiso.com Maths Lit NQF Level 4

Exercise 1.5 Unit 1.2 Complete Exercise 1.5 on page 13 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 4

Calculating with fractions Unit 1.3 Downloaded from hlayiso.com Maths Lit NQF Level 4

Calculating with fractions Unit 1.3 Converting fractions to decimals on a basic calculator • Using a scientific calculator Enter the fraction and then press or to change one form of a fraction to another. If you have a mixed number, press the . key twice. Downloaded from hlayiso.com Maths Lit NQF Level 4

Converting fractions to percentages Unit 1.3 1. Convert the fraction to a decimal first. 2. Then multiply the decimal number by 100 to get the percentage. Remember that one whole is 100%, so a whole number becomes a multiple of 100%. For example, 3 becomes 300%. Downloaded from hlayiso.com Maths Lit NQF Level 4

Converting fractions to percentages Unit 1.3 Downloaded from hlayiso.com Maths Lit NQF Level 4

Adding and subtracting fractions Unit 1.3 Downloaded from hlayiso.com Maths Lit NQF Level 4

Multiplying fractions Unit 1.3 • To find a fraction of a number, multiply. Multiplying fractions means to find a fraction of a fraction. • Multiply the numerators together and the denominators together. • Simplify the resulting fraction. • Use the memory keys for more complex calculations. Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.11 page 15 Unit 1.3 • On a basic calculator: 2 5 65 26 • On a scientific calculator: 2 5 5 26 Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.11 page 15 Unit 1.3 Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.11 page 15 Unit 1.3 Downloaded from hlayiso.com Maths Lit NQF Level 4

Dividing fractions Unit 1.3 • Invert (turn upside down) the fraction you are dividing by. • Multiply the numerators together and the denominators together. • Simplify the resulting fraction. Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.12 page 15 Unit 1.3 Downloaded from hlayiso.com Maths Lit NQF Level 4

Exercise 1.6 Unit 1.2 Complete Exercise 1.6 on page 16 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 4

Calculating and solving problems with Unit 1.4 decimals Decimal numbers have three parts: 12,302 Downloaded from hlayiso.com Maths Lit NQF Level 4

Calculating and solving problems with Unit 1.4 decimals Decimal numbers have three parts: whole number 12,302 Downloaded from hlayiso.com Maths Lit NQF Level 4

Calculating and solving problems with Unit 1.4 decimals Decimal numbers have three parts: whole number 12,302 decimal comma Downloaded from hlayiso.com Maths Lit NQF Level 4

Calculating and solving problems with Unit 1.4 decimals • Decimal numbers have three parts: fraction whole number 12,302 decimal comma Downloaded from hlayiso.com Maths Lit NQF Level 4

Calculating and solving problems with Unit 1.4 decimals • Decimal numbers have three parts: fraction whole number 12,302 decimal comma • Examples: R32,95 12,302 kg Downloaded from hlayiso.com Maths Lit NQF Level 4

Calculating and solving problems with Unit 1.4 decimals To convert a decimal to a common fraction, use a denominator of 10, 100, 1 000, etc and simplify: Examples: 4 • 4 1 = 0,444 ... = 0, 4 0,04 = = 9 100 25 Recurring decimals: 245 • • = 0,245245... = 0, 24 5 999 Downloaded from hlayiso.com Maths Lit NQF Level 4

Adding and subtracting decimals Unit 1.4 1. Work from left to right 2. Add or subtract numbers of the same place value. 3. Line up decimal commas of the numbers under each other. Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.15 page 18 Unit 1.4 1. 18,231 + 4,97 + 6,34 = 29,541 1 8 , 2 3 1 4 , 9 7 + 6 , 3 4 2 9 , 5 4 1 Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.15 page 18 Unit 1.4 2. 6 283,64 – 1 013,464 = 5 270,176 6 2 8 3 , 6 4 - 1 0 1 3 , 4 6 4 5 2 7 0 , 1 7 6 Downloaded from hlayiso.com Maths Lit NQF Level 4

Multiplying and dividing decimals Unit 1.4 • If you use a calculator, the process is exactly the same as for whole numbers. • Estimate your answer quickly so that if you make an error, such as putting the decimal point in incorrectly, you can see this immediately. Without a calculator, follow these steps: • Multiply as with whole numbers (ignoring the decimal places for now). • Count the total of all the decimal places to the right of the decimal comma in the question. • Put that number of decimal places in the answer. Start at the right-hand side and count that number of places to the left. Insert the decimal comma. Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.16 page 19 Unit 1.4 2. A plumber charges R225,75 per hour and works for 3,5 hours. What is his fee? • R225,75 × 3,5 = R790,125 • Round off to the nearest cent: R790,13 To multiply by a power of 10, count the number of zeroes and move the decimal comma the same number of places to the right. Examples: 526,679 × 100 = 52 667,9 341,563 × 100 = 34 156,3 Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.16 page 19 Unit 1.4 2. A plumber charges R225,75 per hour and works for 3,5 hours. What is his fee? To divide by a power of 10, count the number of zeroes and move the decimal comma the same number of places to the left. Examples: 526,679 ÷ 100 = 5,26679 341,563 ÷ 1 000 = 0,341563 Downloaded from hlayiso.com Maths Lit NQF Level 4

Exercise 1.7 Unit 1.4 Complete Exercise 1.7 on page 19 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 4

Calculating and solving problems with Unit 1.5 percentages Percentages • A percentage is a fraction with a denominator of 100. • The % symbol means ‘of a hundred’. • Percentages make it easier for us to compare numbers. • Percentages are fractions where the denominator is 100. Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.17 page 21 Unit 1.5 Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.17 page 21 Unit 1.5 Note: There are two calculator methods you can use here: 31 100 2,86 OR 31 2,86 Downloaded from hlayiso.com Maths Lit NQF Level 4

Exercise 1.8 Unit 1.5 Complete Exercise 1.8 on page 21 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 4

Calculate a percentage of a value Unit 1.5 Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.20 page 23 Unit 1.5 1. The petrol price in South Africa is R15,20 per litre. There is a 12% increase. What is the new price? • Increase 15,20 by 12%. R15,20 × 12% = R1,824 • R15,20 + R1,824 = R17,024 ≈ R17,02 Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.20 page 23 Unit 1.5 2. Refiloe makes and sells dresses. She spends R129,00 on fabric, thread and buttons for each dress. Then she adds 35% to cover her time and labour. What is the selling price of each dress? • Selling price is R129,00 + 35% = 129,00 + 45,15 = R174,15 Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.21 page 23 Unit 1.5 Downloaded from hlayiso.com Maths Lit NQF Level 4

Express a part of a whole as a percentage Unit 1.5 Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.21 page 23 Unit 1.5 1. A pair of jeans originally costing R210 is now sold at a 30% discount. What is the new price? Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.23 page 24 Unit 1.5 1. A printer costs R114 250 including VAT. You know that VAT is charged at 14% of the original price. Work out the VAT-exclusive price. Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.23 page 24 Unit 1.5 2. A supermarket is trying to sell 10 kg bags of rice before their ‘sell by date’. The supermarket advertises them for R56,25 after a 25% discount. What was the original price? R56,25 is 100% - 25% discount Cross multiply to solve. Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.24 page 24 Unit 1.5 A competitor claims that it supplies 29% of a certain product in the market. You know the competitor sells 65 000 units per year. Calculate the total number of sales for this product. Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.24 page 24 Unit 1.5 A competitor claims that it supplies 29% of a certain product in the market. You know the competitor sells 65 000 units per year. Calculate the total number of sales for this product. Downloaded from hlayiso.com Maths Lit NQF Level 4

Exercise 1.9 Unit 1.5 Complete Exercise 1.9 on page 25 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 4

Calculating and solving problems with squares, square roots, cubes and cube roots Unit 1.6 Calculating squares and cubes • To square a number, you multiply it by itself 32 = 3 × 3 = 9 • To cube a number you multiply it by itself twice 33 = 3 × 3 × 3 = 27 Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.25 page 26 Unit 1.6 3. Use your calculator. Round off your answers to two decimal places if necessary. Calculate the following cubes: a) (17,1) 3 b) (49,9)3 a) (17,1)3 = 5 000, 211 ≈ 5 000,21 b) (49,9)3 = 124 251,499 ≈ 124 251,50 Downloaded from hlayiso.com Maths Lit NQF Level 4

Exercise 1.10 Unit 1.6 Complete Exercise 1.10 on page 27 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 4

Rounding numbers Unit 1.6 Definition • Rounding is a way of simplifying numbers, by reducing the digits in a number while trying to keep its value similar. • We round numbers all the time, without realising it. Rounding makes it easier to describe and understand numbers. • Example: Sibu pays about R600 per month for electricity and water. • Rounding off makes numbers easier to compare at a glance. It is easier to compare 2 113 000 with 2 117 000, rather than comparing 2 113 987 with 2 117 999. Downloaded from hlayiso.com Maths Lit NQF Level 4

Rounding numbers Unit 1.6 • You can use rounding to check calculations without a calculator. If you pay for groceries of R313,65 with two R200 notes, you can estimate your change, by rounding up to R314. R400 – R314 = R86. You should get just less than R90 change. • Rounding helps identify mistakes in your calculations. If you make quick estimates, you can see if you have a ‘sensible’ answer. 782 ÷ 4. You know that 800 ÷ 4 = 200, so you expect an answer just under 200. If it is wrong, do the calculation again. Downloaded from hlayiso.com Maths Lit NQF Level 4

Rounding to the nearest 10, 100, 1Unit000 1.6 and more 1. Underline or circle the digit in the place value you are rounding off to and the digit on its right. 2. Write down which two numbers it lies between. 3. Decide to round up or down. You can use a number line to check. 4. 1, 2, 3 or 4 are rounded down and 5, 6, 7, 8 or 9 are rounded up. Downloaded from hlayiso.com Maths Lit NQF Level 4

Rounding to the nearest 10, 100, 1Unit000 1.6 and more Examples: • 6 143 rounded to the nearest 10: 6 143 is between 6 140 and 6 150 3 is less than 5, so round down to 6 140. • 10 485 rounded to the nearest 10 is 10 490. • 292 rounded to the nearest 100 is 300. • 7 635 rounded to the nearest 100 is 7 600. • 3 641 rounded to the nearest 1 000 is 4 000 • 62 212 rounded to the nearest 1 000 is rounded down to 62 000. Downloaded from hlayiso.com Maths Lit NQF Level 4

Exercise 1.11 Unit 1.6 Complete Exercise 1.11 on page 29 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 4

Rounding in everyday life Unit 1.6 • There are some cases where we must decide whether to round up or down, depending on the context. > Example 1.26 page 30 1. How many egg boxes can Sara fill if she has 255 eggs and each egg box can hold 6 eggs? • 255 ÷ 6 = 42 and 3 left over. • She can fill only 42 boxes. Downloaded from hlayiso.com Maths Lit NQF Level 4

Rounding in everyday life Unit 1.6 2. Etienne calculates that he needs 42,357 cubic metres of concrete for the foundations of a house. The concrete is made in batches of 10 cubic metres. How many batches should he make? • 42,357 m3 ÷ 10 = 4 and 0,357 m3 left over. • He needs to make 5 batches to make sure that there is enough concrete. In general, the rule is: • round down when we can get rid of the remainder (we can’t sell a box with 3 eggs in it) • round up when we must include the remainder (we can’t leave some of the foundation without concrete). Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.27 page 30 Unit 1.6 1. Write 2,6470588 to 2 decimal places (dp). 1. 2,6470588 The second decimal digit is 4. So the number lies between 2,64 and 2,65. Look at the next digit. The next digit is 7, so we have to round up. So the answer is 2,65 (2 dp). Downloaded from hlayiso.com Maths Lit NQF Level 4

Example 1.27 page 30 Unit 1.6 3. Round off the number 3,9762645 to 1 dp. 3. 3,9762645 The first decimal digit is 9. The number lies between 3,9 and 4,0. The 7 after the 9 means you have to round up. So the answer is 4,0. Downloaded from hlayiso.com Maths Lit NQF Level 4

Exercise 1.12 Unit 1.6 Complete Exercise 1.12 on page 31 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 4

Exercise 1.13 Unit 1.6 Complete Exercise 1.13 on page 32 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 4

Module assessment Module 1 Complete Module assessment on page 34 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 4

TERMS AND CONDITIONS This PowerPoint Presentation has been developed by Macmillan Education South Africa (Pty) Ltd. All texts, images, videos, animations, audio and vector simulations contained in the slides are property of Macmillan South Africa. Reproducing, reselling and redistributing this material without the written permission of Macmillan South Africa is prohibited. Lecturers are granted permission to: (i) modify the slides by adding and removing content; (ii) print copies of the presentation; and (iii) download and save the slides to a computer or local server. Nothing in this copyright notice constitutes permission to assert or imply that your use of the materials is sponsored or endorsed by Macmillan South Africa. Downloaded from hlayiso.com Maths Lit NQF Level 4