Work with numbers. By the end of this topic, you should be able to: ü Identify the number of significant figures in a number. Chobe N.P.

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1 TOPIC 1 Work with numbers By the end of this topic, you should be able to: ü Identify the number of significant figures in a number ü Round numbers to a given number of decimal places or a given number of significant figures ü Approximate computations that involve sums, differences, products and quotients to a specified place value, decimal place or number of significant figures ü Solve mathematical puzzles ü Complete a number sequence that involves a mixture of whole numbers, directed numbers, fractions, decimals and percentages. Oral activity Discuss approximation and estimation 1 Describe approximation and estimation. Discuss with your partner what you remember about approximation and estimation from Form 1. 3 Name some examples in which we use approximation and estimation. 4 Look at the road map of Botswana below. Discuss how you use the scale to estimate distances on the map. PAIR Sepupa Chobe N.P. Nxainxai Tsau Sehithwa Kalkfontein Ghanzi Mamuno Maun Nxai Pan N.P. Nata Gweta Mosetse Orapa Serowe Serule Francistown Selebi- Phikwe Kang Hukuntsi Tshane Tshabong Mahalapye Stockpoort Jwaneng Molepolole Mochudi GABORONE N Kanye Ramotswa Lobatse 180 km Map of Botswana showing major roads

2 Approximation An approximation of a number or measurement is a value that is close to, but not equal to, the actual value. We use approximations to help us estimate the answers of computations. For example, we can use rounding to find the approximate values of numbers. Key concept Approximate answers of computations. The following are three methods we can use to round numbers. 1 Round to the nearest 10, 100, and so on. Round to a given number of decimal places. 3 Round to a given number of significant figures. 1 Round to the nearest 10, 100 or To round to the nearest 10, look at the last digit in the number. The last digit in a whole number is the units digit. If the last digit is less than 5, round down to the lower 10. If the last digit is 5 or more, round up to the higher 10. For example, 36 rounded to the nearest 10 is 330 and 15 rounded to the nearest 10 is 150. To round to the nearest 100, look at the last two digits in the number. The second last digit in a whole number is the tens digit. If the last two digits are less then 50, round down to the lower 100. If the last two digits are 50 or more, round up to the higher 100. For example, 445 rounded to the nearest 100 is 400, 563 rounded to the nearest 100 is 600 and 1 rounded to the nearest 100 is 00. To round to the nearest 1 000, look at the last three digits in the number. The third last digit in a whole number is the hundreds digit. If the last three digits are less than 500, round down to the lower If the last three digits are 500 or more, round up to the higher For example, 780 rounded to the nearest is and 430 rounded to the nearest is 000. Example 1 The highest point of Lentswe La Baratani at Otse is m above sea level. 1 Round this height correct to the nearest 10 m. Round this height correct to the nearest 100 m. 3 Round this height correct to the nearest m. Note Lentswe La Baratani is 15 km from Lobatse m m m SO ;

3 Round to a given number of decimal places We usually round numbers to one or two decimal places. To round a number to one decimal place, look at the second digit after the decimal point. If the second digit is less than 5, round down to the lower tenth. If the second digit is 5 or more, round up to the higher tenth. For example, 3.56 rounded to one decimal place is 3.6. To round a number to two decimal places, look at the third digit after the decimal point. If the third digit is less than 5, round down to the lower hundredth. If the third digit is 5 or more, round up to the higher hundredth. For example, rounded to two decimal places is Example A calculator shows that the answer to 5 3 = Round the answer correct to one decimal place. Round the answer correct to two decimal places We write the answer of 5 3 as 1.66 The means that the answer is a never-ending decimal. The decimal 1.67 is an approximation of the answer of 5 3 so we use the symbol, which means approximately equal to. Therefore, Example 3 Use your calculator to find Round the answer correct to one decimal place. Round the answer correct to two decimal places Activity 1 Find approximate values 1 Find an approximate value for the answer in each expression a) to f) below. Discuss how approximations can help you check your answers. 3 Use a calculator to find each answer correct to two decimal places. a) 8 3 b) 1 7 c) 0 9 d) e) f) PAIR 4 SO

4 3 Round to a given number of significant figures What are significant figures? Significant figures are digits in a number that show how accurate the number is. We count significant figures in a number from the first non-zero digit on the left. For example, 3.8 has three significant figures and has two significant figures. We use significant figures to measure quantities such as length, mass and capacity. An answer has more significant figures when you use a more accurate measuring instrument. For example, a scale can measure accurately to one-tenth of a kilogram and gives the mass of a student as 56.3 kg. More accurate scales can measure to onethousandth of a kilogram. A scale like this gives the mass of the student as kg. Rules to find the number of significant figures The number 56.3 has three 1 Count the number of significant figures in a number from the significant figures. The number first non-zero digit on the left. For example, has two has five significant significant figures. figures. Trailing zeros at the end of whole numbers are not significant figures. For example, the four zeros in are trailing zeros, so has only one significant figure. 3 The zeros within a number are significant figures. For example, the zero in 5.08 is within the number, so 5.08 has three significant figures. 4 Trailing zeros after the decimal point are significant figures. For example, 5.30 has one trailing zero after the decimal point, so 5.30 has three significant figures. Example 4 Find the number of significant figures in each number Activity Find the number of significant figures 1 Read the rules for finding the number of significant figures. Find the number of significant figures in each number. a) 630 b) c) 8.0 d) e) f) g) h) i).00 PAIR SO

5 How to round to a given number of significant figures To round to a given number of significant figures, look at the next significant figure in the number. If the next significant figure is less than 5, round down. If the next significant figure is 5 or more, round up. Example 5 1 Round to three significant figures. Round to four significant figures Activity 3 Round to a given number of significant figures Round each number to three significant figures PAIR Emerging issue Information in surveys tells us about national health problems such as HIV/AIDS. This information encourages and helps us to support people in need. Exercise 1 Practise approximating 1 Find the number of significant figures in each number. a) 1.45 b) c) d) 3.80 e) Round each number correct to two significant figures. a) b) 5 68 c) 73.0 d) e) f) Round as follows. a) The nearest 10 b) The nearest 100 c) The nearest d) The nearest e) The nearest A survey showed that by 001, children had been orphaned by HIV/AIDS. Round as follows. a) The nearest 10 b) The nearest 100 c) The nearest d) The nearest Round each decimal correct to two decimal places. a) b) c) d) e) f) Round each decimal correct to the number of decimal places given in brackets. a) 5.84 (one) b) (two) c) (three) d) (four) 7 Round each number correct to the number of significant figures given in brackets. a) 3 18 (one) b) (two) c) (three) d) (four) SINGLE 6 SO ;

6 Estimation In mathematics, we first estimate the answer to a computation to get an idea of the magnitude, or size, of the answer. Knowing the magnitude of an answer helps us make fast decisions. To estimate the answer to a computation, first round the numbers to simplify the computation. For example, a worker earns P53.85/h and he works for 9.3 h. So, his estimated income will be P50/h 9 h = P450. Estimation is an approximation of a quantity. When we estimate, we guess the answer instead of actually counting, measuring or calculating the correct answer. You can estimate the number of cattle in a herd, the distance to a nearby town or the quantity of fuel you need to drive from Gaborone to Lobatse. In the context of this chapter, estimation is a rough calculation to find an approximate answer by rounding the numbers in a calculation. Always estimate the answer to a computation before you calculate the answer. Then you can use your estimate to check your answer. When you use a calculator, estimation prevents you making mistakes if you accidently press the wrong keys. Example 6 1 Estimate the value of Use a calculator to find the answer correct to two decimal places. 3 Compare the answer to your estimate = The estimate and the answer are different because we rounded the numbers. However, 150 and are of a similar magnitude. Key concept Estimation helps us to check the answers to computations. Note Of the same magnitude means that the numbers are of almost the same size. Example 7 1 Estimate the value of Use your calculator to find the answer. Give the answer correct to one significant figure = = 400 (correct to one significant figure) SO ; ;

7 Note Estimation skills help us manage and understand situations that involve numbers and measuring. Exercise Estimate and calculate answers 1 Estimate the value of each expression, then calculate the value correct to two decimal places. Use your estimates to check your answers. a) b) c) d) Estimate the value of each expression, then calculate the value correct to two decimal places. Use your estimates to check your answers. a) b) c) d) Use your calculator to calculate the value of each expression. Give the answers correct to: i) Two decimal places ii) Two significant figures. a) b) c) d) SINGLE Multi-step computations Multi-step computations involve brackets. When you use penand-paper methods or a calculator, you have to do multi-step computations in separate steps. When you use pen-and-paper methods, work with an extra digit in the intermediate steps than what the answer requires. For example, if the answer must be correct to two decimal places, work to three decimal places in the intermediate steps. In multi-step computations, the order of operations is important. We use the BODMAS rule for multi-step computations. First calculate the expressions in brackets, followed by of, followed by the four basic operations in the order of the BODMAS list. B: Brackets O: Of D: Division M: Multiplication A: Addition S: Subtraction 8 SO ; ;

8 Example 8 1 Calculate ( ).51. Give the answer correct to two decimal places. Calculate ( ). Give the answer correct to two decimal places. Use these calculator key sequences to find the answers on your calculator ( ).51 = M RM ( ) = 5.39 If your calculator has bracket keys, then you can use this key sequence ( ) When you do not have a calculator, use pen-and-paper methods to do the computations. Your teacher will show you how to do these examples using pen-and-paper methods. Write down the examples and use them when you need to revise your work. Exercise 3 Calculate multi-step computations Calculate each computation correct to two decimal places ( ) ( ) 5 ( ) ( ) ( ) SINGLE Case study Programme summary of the Youth Health Organisation The Youth Health Organisation, YOHO, is a Botswana youth programme that is part of the Theatre and Arts Programme. YOHO helps to educate young people about sexual health. YOHO aims to help reduce the number of HIV/AIDS infections, teenage pregnancies and sexually transmitted infections, or STIs, among 14 to 9 year olds. YOHO uses programmes such as media, theatre or edutainment, and peer education. It educates and promotes the artistic community, writes edutainment scripts, and performs plays, poetry, and modern Emerging issue Sexual health is important to win the fight against HIV and AIDS. SO ;

9 and traditional dances. YOHO works with local music artists to inspire youth to fight HIV/AIDS and live positively. The Peer Education Programme, PEP, uses peer-support methods to educate youth through talk shows, street parties, debates, condom education, interpersonal communication, motivational talks and life skills training. Their main focus is HIV/AIDS, STI education and prevention, and sexuality. PEP works with other groups and nongovernmental organisations, or NGOs, to train and educate youth about sexual health and HIV/AIDS. [Source: Adapted from on 3 June 009.] One of the aims of Vision 016 is to stop the spread of HIV so that there are no new infections in that year. 1 Do you feel a need to be involved in the YOHO programme? Discuss this with a parther. Name two programmes that YOHO offers to youth. 3 Estimate then calculate the cost to get to a debate at a rural school if the organiser travels km at a rate of P.84/km. Round the answer correct to two decimal places. 4 Estimate then calculate the cost of a radio programme that lasts 3 min at a rate of P11.17/min. Round the answer correct to two decimal places. Key concept Identify features of number sequences. Number sequences A number sequence is an ordered set of numbers that follows a certain rule. For example, ; 4; 6; 8; 10; ; 30 is a number sequence where the rule is that each member of the sequence is more than the previous member. The first member of the sequence is and the last member of the sequence is 30. There are two types of sequences: finite sequences and infinite sequences. A finite sequence has a last member. For example, 100 is the last member of the sequence 10; 0; 30; ; 100. An infinite sequence has a never-ending number of members or we can say it extends to infinity. An example of such a sequence is 1; 3; 5; 7; You can always find the next member by adding to the previous member. 10 SO ;

10 Example 9 Give the first member, last member and the rule for each number sequence. 1 1; 13; 14; 15; ; 40 1; 4; 7; 10; 3 5; 10; 15; 0; ; ; 8; 4; ; 8 5 ; 4; 8; 16; 6 1; ; 4; 7; 11; ; 79 1 First member = 1; last member = 40; rule: add 1 First member = 1; last member: continues to infinity; rule: add 3 3 First member = 5; last member = 100; rule: add 5 4 First member = 3; last member = 8; rule: subtract 4 5 First member = ; last member: continues to infinity; rule: multiply by 6 First member = 1; last member = 79; rule: add 1; ; 3; Note The rule of a number sequence describes the relationship between the numbers in the sequence. Example 10 Find the rule for each number sequence and use the rule to find the missing numbers. 1 1; 4; 7; 10; ; ; ; ; 5 1; 4; 16; 64; ; ; ; ; ; 5; 0; 5; ; ; ; ; 30 1 Rule: add 3; missing numbers: 13; 16; 19; Rule: multiply by 4; missing numbers: 56; 1 04; 4 096; Rule: subtract 5; missing numbers: 10; 15; 0; 5 Activity 4 Describe number sequences 1 A number sequence is an ordered set of numbers that follows a certain rule. a) Explain what an ordered set of numbers means. b) Explain what follows a certain rule means. Write any number sequence that has six members. a) Write the rule for the number sequence. b) Write the first member of the number sequence. c) Write the last member of the number sequence. 3 You are given the number sequence 1; 7; 14; ; ; ; ; a) Find the missing numbers. b) Find the total of the numbers in the number sequence. PAIR SO

11 Exercise 4 Work with number sequences 1 Describe each number sequence. a) 1; 4; 7; 10; 13; 16; 19; b) 1; ; 4; 8; 16; 3; 64; 18 c) 18; 6; ; 3 ; 9 ; 7 ; 81 Find i) the rule and ii) the missing numbers in each number sequence. a) 1 ; 1; ; 1 ; ; ; ; b) 1%; 3%; 5%; ; ; ; c).3; 3; 3.7; 4.4; ; ; ; 3 Find the missing numbers in each number sequence. a) ; 5; 8; 11; ; ; ; b) 1 ; 11 ; 1 ; ; ; ; c) 3.5; 5.5; 7.5; ; ; ; 4 Find the sum of the numbers in each number sequence. a) 1; 3; 5; 7; 9; 11; 13; 15; 17; 19; 1; 3 b) ; 6; 18; 54; 16; 486; 1 458; 4 374; 13 1 c) 3; 15; 7; 1; 9; 17; 5; 33; 41; 49; 57 5 Look at the calendar of September 01 and answer the questions. a) Write and describe the number sequence in MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY SATURDAY the second row of the calendar. 1 b) Write and describe the number sequence in the first column of the calendar c) Start at Monday, 4 September 01 and move repeatedly one position to the right and one position down. Write the numbers as you move. Do these numbers form a sequence? If so, describe the sequence d) Find two similar sequences on the calendar and write them down. e) Start at Friday, 1 September 01 and move repeatedly one position to the left and one position down. Write the numbers as you move and describe the number sequence. f) Add the numbers in the first two columns of the calendar. Look out for short methods such as = 10 8 = 18. g) Explain why the total of the second column is 4 more than the total of the first column. h) Use the fact in g) to write the totals of the other columns of the calendar. SEPTEMBER SUNDAY 01 4 SINGLE 1 SO.1.8..

12 Patterns in nature Nature consists of patterns. You can see examples of these patterns in the spirals of pineapples, pine cones, sunflower seeds, seashells and snail shells. You can also see nature s patterns in the number of petals on flowers or leaves on a twig. Fibonacci numbers Nature s patterns are based on Fibonacci numbers. The Fibonacci sequence is 1; ; 3; 5; 8; 13; 1; 34; The rule is to add the previous two members to get the next member of the sequence. Activity 5 Investigate patterns in nature 1 Find examples of patterns in plants, flowers, fruits and shells. Study the examples and identify the spirals. Count the number of petals on different flowers. Use a table to record these numbers and the names of the flowers. 3 Look at the number of petals on flowers you recorded. Are any of these numbers Fibonacci numbers? Discuss your results. PAIR Mathematical puzzles Mathematical puzzles are puzzles that need mathematics to solve them. You need to work with numbers and relationships between numbers, draw shapes, complete tables, and use logical reasoning to solve mathematical puzzles. Activity 6 Investigate square numbers You need 50 counters. Use your counters to build this sequence. GROUP Key concept Apply your mathematical knowledge and skills to solve mathematical puzzles Draw the sequence in your exercise book. Write the number of counters below each group in the sequence. Describe the pattern in words. Determine how many counters you need to build the fourth group in the sequence. Use your counters to build the fourth and fifth groups in the sequence. Draw these two groups in your exercise book. Write the number of counters in each group below each drawing. Describe the pattern in the number sequence below the drawings. Patterns occur in nature in different ways. SO ;

13 7 8 Determine the sixth and seventh numbers in the sequence. Write these numbers in your exercise book. We call these numbers square numbers. Explain why we call them square numbers. Exercise 5 1 Solve mathematical puzzles Naledi uses counters to build this sequence. Look at her sequence and answer the questions. a) Draw the sequence in your exercise book. Write the number of counters below each group in the sequence. b) Determine how many counters Naledi needs to build the fourth group. c) Draw the fourth and fifth groups and write the number of counters below each drawing. d) Look at the numbers only. Determine how many counters Naledi needs to build the sixth group. e) Draw the sixth group in your exercise book and check your answer in d). The rows, columns and diagonals of each square add up to the number shown next to the square. Copy the squares into your exercise book and fill in the missing numbers. SINGLE a) 1 14 b) The rows, columns and diagonals of this square add up to the same number. Each of the digits 1 to 9 can appear in the square only once. Copy the 8 square into your exercise book then fill in the 5 7 missing numbers. The sum along each side of the triangle is 4 9 the same and the sum of the digits at the vertices is 6. Fill in the digits 1 to 9 in the circles to make these statements true. Calculate the values of A, B and C that make all of these statements true. a) A B = 10 b) A C = 4 c) C = B 14 SO

14 Summary ü We approximate answers of computations by rounding numbers to the nearest power of 10, to a given number of significant figures or to a given number of decimal places. ü We identify significant figures in the following ways. ü We count significant figures from the first non-zero digit on the left. For example, 0.13 has three significant figures. ü All zeros within a number are significant. For example, has five significant figures. ü Trailing zeros are not significant unless they are the last digits after a decimal point. For example 0.70 has two significant figures. ü To round a number to two decimal places or to two significant figures, look at the next digit. If the next digit is 5 or more, round up. For example rounded to two decimal places is If the next digit is less than 5, round down. For example, rounded to two decimal places is ü The rule of a number sequence describes the relationship between the members of the sequence. We use the rule of a number sequence to find missing members of the sequence. Revision Round each number correct to two significant figures. a) b) c) 3.87 d) A piece of rope is cm long. You want to cut the rope into 19 equal pieces. a) Round the length of the rope to two significant figures. b) Estimate the length of each piece of rope. c) Calculate the actual length of each piece of rope correct to two decimal places. Calculate correct to two decimal places ( ) Find the missing numbers in each number sequence. a) 1; 3; 5; 7; ; ; b) 1; 3; 9; 7; ; ; c) 100; 10; 0.1; ; ; Calculate the answers correct to two decimal places. a) b) c) d) e) Calculate the answers correct to two decimal places. a) b) c) d) 1.43( ) 11.7 e) ( ) 1 Round to the nearest: a) 10 b) 100 c) Round each decimal correct to two decimal places. a).8741 b) c)