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Transcript
Maths basics Introduction to mathematical language 3 The basic operations of mathematics 3 Another operation: Squaring 5 Words that indicate a number 7 Some other common words 9 Order of operations 10 A look at the number line 12 Negative numbers 13 Suggested answers to activities 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 16 1 2 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 Introduction to mathematical language Once the early mathematicians had developed their Number System, they were able to efficiently calculate. The next step was to develop mathematical language to describe the type of calculation being performed. Today, a combination of both words and symbols are used to describe a calculation and, as long as any given word or symbol means the same thing each time it is used, there can be no confusion. Symbols are a shorthand way of writing a mathematical instruction. For example, most people know the four basic operations in mathematics are addition, subtraction, multiplication and division. Every hand-held calculator shows these basic operations using the symbols +, –, and . The basic operations of mathematics The symbols +, –, , are the building blocks for arithmetic. Let’s look at their meanings. – + add subtract multiply divide sum of take away times share increase decrease product quotient total difference between groups of plus lots of Addition ‘Add 20 and 12’ means ‘increase 20 by 12’ and is written 20 + 12 = 32. 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 3 Subtraction Subtraction reverses the addition process. So if we start with the answer (32) and subtract 12, we arrive back at our original starting point (20). Thus, ‘subtract 12 from 32’ means ‘decrease 32 by 12’ and is written 32 – 12 = 20. Multiplication Mutiplication is just a shortcut to performing many additions. For example, Daniel needs 8 lengths of timber, each measuring 40 cm, for his wood project. In order to calculate how much timber he needs to buy, he worked out the following. 40 + 40 + 40 + 40 + 40 + 40 + 40 +40 = 320. So Daniel should buy 320 cm of timber. Obviously, Daniel did his calculation the hard way. It would have been much more efficient for him to perform a multiplication. 40 cm 8 = 320 cm Division As you might expect, division and multiplication are opposite operations as well. In other words, division reverses the multiplication process. So if we start with the answer (320) and divide by 8, we again arrive back at our original starting point (40). Thus ‘divide 320 by 8’ means the same as ‘share 320 between 8’ or ‘divide 8 into 320’. It is written as follows. 4 Subtraction is the opposite of addition. Multiplication is continuous addition. Division is the opposite of multiplication. 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 Activity 2A Translate each of these word sentences into a number sentence, then use your calculator to find the answer. The first one has been done for you. Word sentence Number sentence Answer (a) add 16, 85 and 9 16 + 85 + 9 = 110 (b) subtract 28 from 72 (c) multiply 12 by 7 (d) find the sum of 36 and 66 (e) find the quotient of 45 and 9 (f) total 101, 1001 and 10 001 (g) share 87 among 3 (h) increase 265 by 58 (i) find the product of 15 and 5 (j) find the difference between 92 and 29 (k) what are 12 groups of 8? (l) by how much does 36 exceed 9? Check your answers against those at the end of the topic. Another operation: Squaring We have seen that the four basic operations in mathematics are +, –, and . A fifth operation, which is really just an extension of ‘multiply’, is so important that we should mention it here. This operation is the squaring operation, which means ‘multiply a number by itself’. So 3 squared = 3 3 = 9 7 squared = 7 7 = 49 The symbol for the squaring operation makes use of index notation, which will be studied in more detail in Topic 4. 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 5 Because squaring a number means two identical numbers are multiplied together, it is written as follows. The opposite operation to squaring We have seen that: The opposite of + is – The opposite of – is + The opposite of is The opposite of is And the opposite of ‘square’ is ‘square root’. The symbol that we use for ‘square root’ is . Squaring a number means multiplying it by itself. The opposite of square is square root. To find the square root, think ‘what number do I multiply by itself to get this?’ Activity 2B Without the use of a calculator find: 1 62 2 22 3 92 4 202 6 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 5 1002 6 7 8 9 10 Check your answers against those at the end of the topic. Words that indicate a number Our everyday language includes many words which indicate a certain operation or number. For example: 1 double and twice mean ‘multiply by 2’ triple and treble mean ‘multiply by 3’ a score is 20 a dozen is 12. It is said that most humans live for ‘three score and ten’. What does this mean? _____________________________________________________________________ _____________________________________________________________________ 2 What does ‘century’ mean? Think of two examples where it is used. _____________________________________________________________________ _____________________________________________________________________ 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 7 3 What is a baker’s dozen? _____________________________________________________________________ _____________________________________________________________________ 1 3 20 + 10 = 60 + 10 = 70 years. 2 A century means one hundred (100). We use ‘century’ when referring to 100 years and in cricket ‘a century’ is 100 runs. 3 A baker’s dozen refers to 12 + 1 = 13. For example, you may want to buy a dozen bread rolls, but the baker gives you an extra one for the price of 12. Our everyday language also includes many words that use prefixes to indicate a certain number. For example, the prefix ‘bi-’ means two (so a bicycle is a two-wheeled machine) and ‘deca-’ means ten (so a decade is a period of 10 years) and so on. Note: A ‘prefix’ is something that comes before a word. Use your dictionary to find the difference between ‘biannual’ and ‘biennial. Activity 2C Rule a line between each prefix and its number meaning. You may find a dictionary helpful. Check your answers against those at the end of the topic. 8 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 Some other common words Some words are used so commonly in mathematics that we should not progress too much further without mentioning them. Average To find the average of a group of numbers, add them all up and divide by how many numbers there are in the group. Example: The average of 4, 5, 7 and 8 is: Evaluate Evaluate means (just as it sounds) ‘find the value of”. Simplify Simplify means that your answer should end up looking simpler than the question. Some hidden meanings Every language has its hidden messages and meanings which may not be obvious at first sight. In the English language, for example, there are certain rules about where to use full stops, capital letters, and apostrophes, and there are customs about how to say different words. We gradually learn the rules and customs by speaking, reading and writing. The language of mathematics is no different. Certain rules must always be followed by mathematicians all around the world so that there can be no confusion about what is meant. Here is an example. Evaluate 2 + 3 5. This calculation could be approached in two ways. Add 2 + 3 first (to get 5), then multiply by 5, giving a final answer of 25. Mutiply 3 5 first (to get 15), then add 2, giving a final answer of 17. As it happens, only one of these is right, much to the disappointment of mathematics students over the years. Let’s look at the question of order. 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 9 Order of operations Calculations with several operations to be performed have to be handled in the correct order. That order is shown in the flowchart below. 1 Do operations in grouping symbols. Grouping symbols can be ( ) or [ ] or { }. 2 Do multiplications and divisions from left to right. 3 Do additions and subtractions from left to right. Example 1 20 (2 + 3) = 20 5 = 100 (first, do the operation in grouping symbols) (next do the multiplication) Example 2 30 2 – 3 4 = 15 – 12 = 3 (do the multiplication and division from left to right) (do the subtraction) Example 3 80 10 4 2 = 842 = 22 = 1 (all the operations are divisions so work from left to right) Example 4 Note: Questions like the last one use the fraction bar as a grouping symbol. Sometimes, more than one set of grouping symbols may occur in the same calculation. In this case, the inner grouping symbols are worked first. 10 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 Example 5 3 + [10 – (4 + 3)] = 3 + [10 – 7] = 3+3 = 6 (inner grouping symbols worked first) (other grouping symbols then worked) Activity 2D Questions 1–19 Simplify each of the following: 1 15 – 4 – 5 – 3 2 120 6 2 2 3 20 – (29 – 17) 4 (230 + 170) 10 5 5 6 6 (9 + 6) 3 7 100 – 40 2 8 32 + 42 9 10 4 + 10 5 10 10 – 6 3 + 3 11 10 – 4 2 2 12 10 – (4+ 2) 2 13 (10 – 4) + 2 2 14 3 42 15 16 (8 + 3) (8 – 4) 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 11 17 (5 + 2) (8 8) 18 4 [12 – (6 + 1)] 19 [20 – (4 + 2)] 2 Questions 20–24 Put in grouping symbols where necessary to make each of the following statements true: 20 15 – 3 5 = 60 21 8 + 4 – 3 2 = 10 22 8 + 4 – 3 2 = 6 23 8 + 4 – 3 2 = 18 24 24 4 3 2 = 1 Challenge Using four 4’s and any mathematical operations we can get an answer of 1. For example: 4 4 4 4 = 1 Now can you use four 4’s and any mathematical operations to get an answer of 7? Check your answers against those at the end of the topic. A look at the number line If we were to try to map the types of numbers we have met so far, we would end up with a number line like this. Notice that: the numbers are spaced at equal distances. the arrow on the end of the number line is used to show that it continues forever. The number line above shows ‘the set of counting numbers’, in other words, the numbers we would use if we were counting. 12 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 There are many more numbers than these. In fact, as you will soon see, there are an infinite number of numbers! Let’s explore some of them, now. Negative numbers Consider these questions. 1 Adam’s cheque account had a balance of only $230 when he wrote a cheque for $250 to pay for two new tyres. What is the balance of his account now? _____________________________________________________________________ _____________________________________________________________________ 2 At 5 am one winter’s day at Cowra the temperature was 2ºC. By 6 am, the temperature had fallen 3º. What was the temperature then? _____________________________________________________________________ _____________________________________________________________________ 1 $20 overdrawn ($ – 20) 2 1ºC below zero ( – 1ºC). These situations show the need for negative numbers, or numbers less than zero. On the number line, negative numbers are shown to the left of zero. When we travel around the number line we move to the right for positive numbers and to the left for negatives. This is as easy as moving east and west across a map or up and down on a thermometer. 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 13 Activity 2E 1 (a) At 4 am the temperature was –2ºC. Mark this on the thermometer below. (b) By 6 am the temperature had fallen another degree. What’s the temperature now? (c) Over the next two hours, the temperature rose 2 degrees. What was the temperature at 8 am? (d) By midday, the temperature was 9ºC. By how many degrees did the temperature rise between 8 am and midday? 2 Look at this diagram of a department store. 14 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 If ground level is floor 0, then the third floor would be + 3 (you would go up three floors, or add three floors to your height above ground level). Parking level P2, two floors below ground level, would be – 2. (a) Mary parks her car on parking level P1. She then takes the lift up three floors. What will she be shopping for on this level? (b) Quan works in this department store. He parks on parking level P2, then takes the lift up two floors. What floor does Quan work on? (c) Rhett enters the store on the ground level. He takes the lift up four floors then down two floors, which departments does he visit? (d) On Monday Anelise met her friend for lunch in the Skyline Café. After lunch she took the lift down seven floors to go home. Where did she park her car? 3 Sven has an overdraft account with Starbank. His bank statement for the month of October is shown below. Complete the balance column. Check your answers against those at the end of the topic. 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 15 Suggested answers to activities Activity 2A (b) 72 – 28 = 44 (c) 12 × 7 = 84 (d) 36 + 66 = 102 (e) 45 9 = 5 (f) 101 + 1001 + 10 001 = 11 103 (g) 87 3 = 29 (h) 265 + 58 = 323 (i) 15 × 5 = 75 (j) 92 – 29 = 63 (k) 12 × 8 = 96 (l) 36 – 9 = 27 Activity 2B 1 36 2 4 3 81 4 400 5 10 000 6 2 7 10 8 4 9 9 10 12 16 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 Activity 2C 1 = uni 7 = hept or sept 2 = bi 8 = oct 3 = tri 9 = nona 4 = quad 10 = deca 5 = pent 12 = dodec 6 = hex or sex Note: Did you notice that there is no prefix to indicate 11? Activity 2D 1 3 13 7 2 5 14 48 3 8 15 6 4 800 16 44 5 13 17 7 6 30 18 20 7 20 19 7 8 25 20 (15 – 3) × 5 = 60 9 90 21 8 + (4 – 3) × 2 = 10 10 11 22 (8 + 4) – 3 × 2 = 6 11 6 23 (8 + 4 –3) × 2 = 18 12 7 24 24 (4 × 3) 2 = 1 Challenge 4+4–44=7 Note: It is, in fact, possible to get all the numbers from 0 to 10 using four 4’s in this manner. Try it! 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398 17 Activity 2E 1 (a) (b) –3C (c) –1C 2 (d) 10 (a) Ladies’ fashions (b) Ground floor – shoes (c) Manchester then Ladies Fashions (d) P2 3 18 4930CD: 2 Recognising Everyday Mathematics 2005/008/014/05/2006 P0025398