NUMBER. including: mental calculations; pen-and-paper methods; and the use of calculators.

NUMBER Numbers are used all around us. We need to make calculations when we shop, when we plan a party, when we read a timetable, and even when we play sport. We all need to be able to estimate and calculate answers effectively. In this chapter, we will revise our number skills using a variety of strategies including: mental calculations; pen-and-paper methods; and the use of calculators. 890 67890 890 67890 67890 890 06789 7890 890 678 6789067890 67890 67890 67890 67890678906789067890678 6 789067890 67890678 906789067890678906789067890 6789067890678906789067890 678906789067890 67890 678 67890 67890 06789 678906789067890678906 789067890 67890 8 90 678906789067890678906789067890678 678906789067890678906789067890 6789 678 90 6789067890678 906789067890678906 789067890 67890 9067890 067890 6789067890 678 067890 6789067890677 7890 678906789 06789067 890678906789067890678906789067890678906789067890 67890678906 78906789067 890 6789067890 67890 67890 7890 6789067890 67890678906789067 06789067890676788 8906789067890678906789067890678906789067890678906789067890678906789067890 8 678906789067890678906789067890678906789067890678906789067890678906789067 890678906789067890678906789067890678906789067890678906789067890 6789067890 67890678906789067890678906789067890678906789067890678906789067890678906 7 78906789067890678906789067890678906789067890678906789067890678906789067890 6789067890678906789067890678906789067890678906789067890 67890678906789066 678906789067890678906789067890678906789067890678906789067890678906789067890 6 6789067890678906789067890678906789067890678906789067890678906789067890 67890678906789067890678906789067890678906789067890 67890678906789067890678900 06789067890678906789067890678906789067890678906789067890678906789067890 0 67890678906789067890678906789067890678906789067890678906789067890678906789 06789067890678906789067890678906789067890 678906789067890678906789067890 67890678906789067890678906789067890678906789067890678906789067890678906789 906789067890678906789067890678906789067890678906789067890678906789067890 67890678906789067890678906789067890 678906789067890678906789067890678906788 8906789067890678906789067890678906789067890678906789067890678906789067890 6789067890678906789067890678906789067890678906789067890678906789067890678 8906789067890678906789067890 678906789067890678906789067890678906789067890 67890678906789067890678906789067890678906789067890678906789067890678906 7 78906789067890678906789067890678906789067890678906789067890678906789067890 67890678906789067890 6789067890678906789067890678906789067890678906789066 678906789067890678906789067890678906789067890678906789067890678906789067890 6 6789067890678906789067890678906789067890678906789067890678906789067890 678906789067890 6789067890678906789067890678906789067890678906789067890678900

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Worksheet -0 Brainstarters Skillsheet -0 Factors and divisibility Start up Find the answers to these without using a calculator: a 6 9 b 7 c + 0 d 7 + e 8 f 9 9 g 7 h 6 i 6 8 j 6 9 k 6 6 l 9 Rewrite these integers in ascending order: -6, 0, 9, 7, -, -,, - Find the highest common factor of: a and 8 b 0 and c 6 and 8 Find the lowest common multiple of: a 6 and 0 b and c and Convert each of these fractions to a decimal. a b c 6 Rewrite these numbers in ascending order:.80,.08,.8,.0,.08,.08 8-0 Mental calculation shortcuts In the Mental skills sections of New Century Maths 7, you were provided with a variety of strategies for mental calculation to simplify numerical expressions. Some of them are shown in the following table. Skill Examples Multiplying by a multiple of 0 80 = 8 0 = 0 0 = 00 Dividing by a multiple of 0 000 00 = 0 Changing the order when adding or multiplying + 7 + 8 + + = ( + ) + (8 + ) + 7 = 60 + 0 + 7 = 7 7 = 7 ( ) = 7 0 = 0 Adding and subtracting 8 or 9 + 9 = + 0 = 7 = 7 67 8 = 67 0 + = 7 + = 9 Doubling and halving numbers 7 = double 0 + double 7 = 80 + = 9 = half of 0 + half of = 70 + = 7 8 = half of 0 + half of 8 = 60 + 9 = 69 Multiplying and dividing by or 8 Multiplying and dividing by,, 0,, 0 7 8 Double 7 =, double = 68, double 68 = 6. 7 8 = 6 60 Half of 60 = 80, half of 80 = 0. 60 = 0 8 = 9 = 9 0 = 90 00 = 00 00 = = Multiplying by 9,, 99, 0 7 = 7 0 + 7 = 70 + 7 = 87 9 = 0 = 0 = NEW CENTURY MATHS 8 STAGE ISBN: 97807069

Skill Examples Commonly used fractions and 0. = = 6 decimals 0.6 6 = 6 = ( 6) = = Multiplying by factorising = 9 = 9 0 = 80 Exercise -0 TLF L 89 Use the mental calculation shortcuts shown in the table on page and above to evaluate each of the following expressions. a 0. 0 b 8 + 9 c 68 d 8 60 e 6 + 7 + 6 + 9 + f 6 + 7 + 6 + 9 + g 6 h 0. i 6 j 600 k 68 l m 70 n 6 + 8 o 0 p 0. 8 q 88 + + 7 + 7 + 0 r 88 + + 7 + 7 + 0 s 00 0 t Use mental calculation shortcuts to evaluate these: a 7 000 b c 00 0 d e 7 8 f 9 8 g 6 h 7 i 6 9 j 7 k 0. l 80 m 8 n 00 0 o 6 0 p 0.7 0 q 8 r 900 s 0 t + 9 u 8 v 7 99 w 8 + x 6 y 86-0 The four operations The four basic operations of arithmetic are: + addition subtraction multiplication division Wishball challenge: Tens TLF The multiplier: Go figure TLF L 90 L 006 The divider: With or without remainders Example Simplify 0 8. Method : Long division 8 8 0 6 0 ISBN: 97807069 8 into 0 goes 8 into goes 8 0 8 = 8 CHAPTER WORKING WITH NUMBERS

Method : Preferred multiples 8 0 80 0 times 80 0 times 90 times times 0 8 times 0 8 = 8 Exercise -0 Find the answers to the following: a 8 + 0 b 7 + c 899 89 d 6 80 e 8 7 f 60 6 g 6 h 8 + 6 + 0 i 7900 896 j 60 88 k 7 l 0 0 Ex Find the answers to the following: a b 9 c 06 8 d 780 e 6 f g 67 h 76 i 6 6 NEW CENTURY MATHS 8 STAGE ISBN: 97807069

-0 Integers Integers are the positive and negative whole numbers and zero. We have previously learned the rules for operating with integers using the number line. Negative numbers can be entered into a calculator using the sign change key ( ) or +/. Worksheet -0 Integer review Adding a negative number is the same as subtracting its opposite. Subtracting a negative number is the same as adding its opposite.! Example Find the answer to (-). (-) = + Subtracting a negative number is the same as adding its opposite. = 6 On a calculator: ( ) = The answer is 6. Skillsheet -0 Integers Skillsheet -0 Integers using diagrams positive positive = positive positive negative = negative negative positive = negative negative negative = positive + (The above is also true for dividing with integers.) When multiplying or dividing two numbers which have the same sign, the answer is positive. When multiplying or dividing two numbers which have different signs, the answer is negative. + +! + Example Find the answer to: a - b -6 (-) a - = - On a calculator: b -6 (-) = On a calculator: ( ) = ( ) 6 ( ) = The answer is -. The answer is. ISBN: 97807069 CHAPTER WORKING WITH NUMBERS 7

Exercise -0 TLF L 8 Ex Integer cruncher: Addition and subtraction Worksheet -0 Ex Estimation game Find the answers to the following: a - + (-8) b 6 (-) c - (-) d - + e 6 f -7 + 8 g - + h 6 i -8 + 0 j -7 + + 8 k 8 + 6 l - + 8 Work out answers to each of the following: a - b (-6) c (-) (-8) d 6 (-) e - (-) f - g (-9) h -0 7 i - (-) j 6 (-) k - (-) l -7 (-) m 8 (-) (-) n (-) (-) 7 o (-) Find the answers to the following: a 7 b 8 + c - + d (-) + e -8 (-) f 6 8 g + 0 h -8 (-) i 6 (-) (-) Which of the following is the answer to -0 + 8 9? Select A, B, C or D. A -7 B C -9 D - -0 Rounding and estimating There are many situations in which it is impractical or impossible to give an exact answer. If the length of a wall is measured or calculated to be.8 metres, we may approximate it to.8 m or.8 m.! To round a decimal: cut the number at the required decimal place look at the digit immediately to the right of the specified place if this digit is 0,,, or, leave the number in the specified place unchanged if the digit is, 6, 7, 8 or 9, add to the number in the specified place. Example Round.6 correct to one decimal place.. 6 Cut The next digit is 6, so add to the in the tenths place, to give. So.6 is. (correct to one decimal place). Example Round.89 correct to two decimal places..8 9 Cut The next digit is, so the number does not change. So.89 is.8 (correct to two decimal places). 8 NEW CENTURY MATHS 8 STAGE ISBN: 97807069

Example 6 a Estimate the answer to 6.0.6.99 b Use your calculator to find the exact answer, then round it to two decimal places. a 6.0.6.99 6 = 7 = 8. Estimated answer = 8. b On a calculator: 6.0.6.99 = 9.8. Rounded answer = 9. (correct to two decimal places). Note: Most scientific calculators have a FIX mode that rounds the number on its display to a given number of decimal places. You may like to investigate the FIX mode. Exercise -0 Round each of these to two decimal places: a.6 b 9.78 c 0.078 d 6.0909 e.088 f.796 Round each of these, correct to one decimal place. a.8 b.076 c 0. d 7. e.080 f.99 Round each of these, correct to three decimal places. a 9.70 b.67 c 0.088 Which of the following is the best estimate for. 7.8? Select A, B, C or D. A 6 8 B 6 7 C 8 D 7 For each of these questions, make an estimate of the answer and then use your calculator to evaluate the answer to the number of decimal places shown in brackets. a.9. + 8.66 [] b (9.7.). [] c 0.60 98. [] d 7.09 0.8 [] e 9.9.7 [] f.6.08. [] g 9.6..8 [] h 6. 0.6 + [] i 0 8. [] Ex Ex Ex 6 Working mathematically Questioning, communicating and reflecting Rounding up or down? Sometimes, whether to round an answer up or down depends upon the situation. In groups of two to four, discuss each of the following situations and decide whether it is more appropriate to round up or to round down. You must give a reason for your choice. Seven friends have dinner at a restaurant and the total bill is $6. They decide to share the bill evenly: $6 7 = $.8 7 How much should each friend pay, to the nearest dollar: $ or $? You have a budget of $ for buying drinks for a party. One can of drink costs $0.9 at the supermarket: $ $0.9 = 7.87 How many whole cans of drink can you buy: 7 or 8? ISBN: 97807069 CHAPTER WORKING WITH NUMBERS 9

You need to find the average number of people living in each home in your street. You survey 8 homes and count a total of 66 people: 66 8 =.8 What is the average whole number of people living in each house: or? You need to paint the walls of a house, with a total surface area of m. One tin of paint covers 6 m : 6 =.8 7 How many whole tins of paint do you need for the job: or 6? Jodie has $098 in her bank account and the bank pays her.7% in interest:.7% $098 = $98.88 How much interest will the bank pay Jodie: $98.8 or $98.8? Worksheet -0 Order of operations puzzle! -0 Order of operations The order of operations First: Grouping symbols (innermost brackets first) Second: or (working left to right) Third: + or (working left to right) Skillsheet -0 Order of operations Example 7 Find answers for each of the following: a + 9 b [ ( 8)] a + 9 = 0 + 7 = 7 On a calculator: + 9 = b [ ( 8)] = [ ] = = On a calculator: ( ( 8 ) ) = Example 8 8 Evaluate + 6 8 8 + 6 = - = 6 8 Divide all of 8 + 6 by all of 8. On a calculator: ( 8 + 6 ) ( 8 ) = The answer is 6. 0 NEW CENTURY MATHS 8 STAGE ISBN: 97807069

Exercise -0 Which of the following is the answer for -0 + 8 9? Select A, B, C or D. A -6 B -8 C 6 D -6 Calculate: a 8 + b 7 c 6 d 6 e 6 + f 7 + g 8 h ( 9) 6 i (0 ) j 0 + 6 k - 6 l 6 ( + ) m (8 ) (7 + ) n (7 0) 0 o 7 (- + 6) 7 p - [ + ] q -6 [ ( )] + r [6 ( )] [ ( + ) + 7] Simplify each of the following. Give your answers to one decimal place where necessary. a + 9 b + c - 8 8 6 00 + 0 d -66 e - f - + + - + 8 6 + 0 [ g - 8 ( ) ] h 7 ( ) i 96 ( 6 0) 0 [( 7 ) ] 8 + Ex 7 Ex 8 Using technology Review of spreadsheets Think of a spreadsheet as a giant calculator. A spreadsheet is made up of many cells. A, B, C, are referred to as columns, and,,, are referred to as rows. The cell shown on the right is read as A. We read cells in the same way we read a street directory. The basic operations used to do calculations are shown in the table below. Mathematical operation Symbol used in spreadsheet Addition (+) + Subtraction ( ) - Multiplication ( ) Division ( ) / Skillsheet -0 Spreadsheets ISBN: 97807069 CHAPTER WORKING WITH NUMBERS

We can make three types of entries into a spreadsheet. They are: Value: a number entered into a cell Label: text entered into a cell Formula: an equation entered into a cell to do a calculation (such as find an average, or sum). A formula always starts with an equals (=) sign. Enter the values into the cells, as shown below. Enter the following formulas into the spreadsheet you began in Question. a In cell B, enter =A+A+A [This represents 0 + 6 +.] b In cell B, enter =A/A8 [This represents 6 (-6).] c In cell B, enter =A*A7 [This represents 0 7.] d In cell B, enter =A6-A8 [This represents - (-6).] e In cell B, enter =sum(a:a8) [This formula adds all cell values from A to A8.] f In cell B6, enter =average(a:a8) [This formula will find the average of all cell values from A to A8.] Convert the following statements into formulas to enter into the given cells in column C of the spreadsheet you began in Question : a Cell C: The sum of the values in cells A, A and A8 b Cell C: Multiply the value in cell A by the value in cell A c Cell C: Divide the value in cell A by the value in cell A8 d Cell C: Subtract the value in cell A from the sum of the values in cells A and A7 e Cell C: Multiply the value in cell A by the value in cell A then divide the result by the value in cell A8 f Cell C6: The sum of the value in cells A6 and A multiplied by the sum of the values in cells A and A g Cell C7: Multiply the values in all cells, A to A8 (Can you predict the answer?) h Cell C8: The square root of the value in cell A [Hint: =sqrt( )] i Cell C9: The average of the values in cells A to A8 j Cell C0: Square the value in cell C k Cell C: Square the value in cell A, then subtract the value in cell A7 l Cell C: Divide the value in cell A by the value in cell A, then add the value in cell A divided by the value in cell A m Cell C: Multiply the value in cell A6 by the value in cell A7 and add, then subtract the value in cell A multiplied by NEW CENTURY MATHS 8 STAGE ISBN: 97807069

Just for the record The abacus The abacus is often called the first computer. It was invented by the Chinese in the th century and it is still used today to add, subtract, multiply, divide and to solve mathematical problems involving fractions and square roots. The word abacus comes from the Greek word abax meaning calculating board. The abacus is composed of three sections: the upper beads, the lower beads and the horizontal An abacus uses place value to represent numbers. centre bar called the beam. Only the beads which have been moved to touch the two sides of the beam represent numbers. Each vertical row of beads represents a power of 0 (that is 0 000, 000, 00, 0, ). The beads below the beam represent one unit of that row. The beads above the beam represent five units of that row. Study the examples shown: 0 000 000 00 0 Abacus showing (One 0 unit bead and one unit bead) 0 000 000 00 0 Abacus showing 7 (One 00 unit bead, one 0 unit bead, one unit bead and two unit beads) Represent, 6 and 66 on an abacus. -06 Decimals Converting a decimal to a fraction To convert a decimal to a fraction: count the number of decimal places use that number of zeros in the number in the denominator of the fraction simplify the fraction if required. ISBN: 97807069 CHAPTER WORKING WITH NUMBERS

Example 9 Convert each of these decimals to a fraction: a 0.7 b 0.08 a 0.7 = 7 0.7 has two decimal places, so the fraction has a denominator of 00. 00 b 0.08 = - 8 0.08 has three decimal places, so the fraction has a denominator of 000. 000 = 9 Simplifying the fraction. 00 Worksheet -0 Decimal review Addition and subtraction Make sure you keep place-value columns correct by placing the decimal points underneath each other. Example 0 Skillsheet -06 Decimals Evaluate: a. +. +. b..6 a. +. +... +. 9.6 b..6.0.6.78 Multiplication and division Example Evaluate: a.. b 6. c. 0.0 a Multiply without decimal points first. Then make sure you have the same number of decimal places in the answer as there were at the start of the question... The question has three decimal places. Multiplying without decimal points: 9 0 690 So.. = 6.90 Writing the answer with three decimal places. NEW CENTURY MATHS 8 STAGE ISBN: 97807069

b. 6. 6. =. c When dividing by a decimal fraction, make the decimal fraction a whole number by moving the decimal point the appropriate number of places to the right. In this case: 0.0 Move the decimal point in the other number the same number of places:. 0. Now divide 0 by :. 0.0 = 0 = 0. Exercise -06 Write each of these as a fraction in its simplest form: a 0. b 0.07 c 0.0 d 0.009 e 0. f 0.8 g 0. h 0.06 i 0.00 j. k 0.0 l.0 The decimal. can be written as which of the following mixed numerals? Select A, B, C or D. A B - C D - 0 00 000 Work out these calculations: a. + 0.8 b. +.6 c 8. 6.9 d.9 0.9 e.6 0.6 f + 0.6 +. g 0.0.06 h.6 9.88 i. + 7.0 + 6. j. +.6 +.00 k.8 0.0 l.6.8 Find the answers to the following: a. b.6 c.8 d 8. 0. e. 0.6 f 0.06 0. g 6.. h 0.. i 8. j 0.87 k 0.. l.7. Complete each of these number grids by finding the missing numbers. (Round decimals to two places, when required.) a + 9.6..07 b top row minus left-hand column c 0.6 8.6. 8 6 7.6.8. d.0. 70.07 0.6.9 e top row divided by left-hand column 0. 0.8 0. 0.07 TLF L 87 Wishball challenge: Ultimate Ex 9 Ex 0 Ex ISBN: 97807069 CHAPTER WORKING WITH NUMBERS

Using technology Order of operations Enter the values shown into your own spreadsheet. Use spreadsheet formulas with appropriate mathematical notation (* for multiplication, / for division, + for addition and - for subtraction) to complete the calculations below in the given cells. Try to predict the answer before you enter each formula. The first one has been done for you. C: 6 + 7 In cell C enter: =A+A-A6. C: (7 + 8.) 0.006 Convert the answer in cell C to a mixed numeral by clicking on Format Cells, Fraction (as shown below). C: 7 + 8. 0.006 Convert the answer in cell C to a mixed numeral (as shown above). C: 0 - C: 0.006 7 + 6 8. 6 C6: 0.006 (7 + 6) 8. 7 C7: 0 Convert the answer in cell C7 to a decimal by clicking on Format + 0.006 Cells, Number and Decimal places (as shown below). 8 C8: 0 + 0.006 9 C9: 7 + 0 0 C0: 0 ( + 6 + 8.) - C: 0-0.006 C: (0 + 8.) [ (-)] C: 0 + 8.6 (-) C: 0 7 6 8. Convert the answer in cell C to a fraction, by clicking on Format Cells, Fraction, and Up to three digits. C: + (-) Convert the answer in cell C to a fraction. Choose Up 7 + 8. + to digits. 6 NEW CENTURY MATHS 8 STAGE ISBN: 97807069

Mental skills A Maths without calculators Time before and time after Examine these examples. a What is the time hours and minutes after 6:0pm? 6:0pm + hours = 0:0pm Count: 6:0, 7:0, 8:0, 9:0, 0:0 0:0pm + minutes = 0:pm b What is the time 7 hours and 0 minutes after :am? :am + 7 hours = 6: pm Count: :, :, :, :, :, :, :, 6: 6:pm + 0 minutes = 6:pm + minutes + minutes = 7:00pm + minutes = 7:pm or minutes 7 hours minutes = 7 hours 0 minutes :am :00noon 7:00pm 7:pm Now find the time of day: a hours 0 minutes after 9:0am b hours 0 minutes after 7:0pm c hours minutes after 6:pm d hours 0 minutes after :am e hours minutes after 0 hours f 7 hours minutes after 70 hours g 8 hours 0 minutes after :0am h hours minutes after 0:0pm i 6 hours minutes after 0 hours j hours minutes after 00 hours k 9 hours 0 minutes after :0pm l hours 0 minutes after 8:am Examine these examples. a What is the time hours and minutes before :0am? :0am hours = 8:0am Count back: :0, 0:0, 9:0, 8:0 8:0am minutes = 8:0am b What is the time hours and 0 minutes before 7:0pm? 7:0pm hours = :0pm Count back: 7:0, 6:0, :0 :0pm 0 minutes = :0pm 0 minutes 0 minutes = :00pm 0 minutes = :0 pm or 0 minutes hours 0 minutes = hours 0 minutes :0pm :00pm 7:00pm 7:0pm ISBN: 97807069 CHAPTER WORKING WITH NUMBERS 7

Now find the time of day: a hour minutes before 7:0pm b hours 0 minutes before :0am c hours 0 minutes before :0pm d hours minutes before 8:am e hours 0 minutes before hours f hours minutes before 070 hours g hours minutes before :am h 9 hours 0 minutes before 9:pm i hours 0 minutes before 00 hours j hours minutes before 06 hours k hours minutes before :0pm l hours 0 minutes before :00 noon Worksheet - Crossnumber challenges Skillsheet -07 Indices -07 Powers Remember that powers are used as a shorthand way of writing repeated multiplication. We write as.! On a calculator: the square of a number can be found using the x key. the cube of a number can be found using the x key. x y x ^ any power of a number can be found using the, or key. Example Use your calculator to find: a b c 6 a = 96 On a calculator, enter: x =. b = Enter: x =. c 6 = 96 Enter: 6 x y =. Exercise -07 Ex Evaluate each of the following: a b c 6 d e 7 f g 8 h i 0 j 9 k l 6 Find the missing power each time. a = 8 b = 7 c 0 = 00 d = 096 e = f = 8 NEW CENTURY MATHS 8 STAGE ISBN: 97807069

Which of the following is the value of (.)? Select A, B, C or D. A.9 B.69 C.6 D.09 Calculate: a b c d 6 e f g h i 6 8 j + k l + a Find ( ). b Find: i ii c Does ( ) =? Explain your answer. 6 a Find: i ( ) ii iii b Does ( ) =? Explain your answer. 7 Use what you found in Questions and to complete this pattern: ( 8) = 8 Write three examples of your own to show that (ab) = a b. 9 Copy and complete the following: a 8 = (6 ) b = ( ) c 0 = ( 0) = 6 = = = = = d 6 = ( ) e 8 = ( 7) f = ( ) = = = = = = -08 Square roots and cube roots The square root ( ) of a given number is the positive value which, if squared, will give that number. The cube root ( ) of a given number is the value which, if cubed, will give that number. Skillsheet -08 Square roots and cube roots Example Find the square root of 6. 6 = 6 because 6 = 6 6 = 6 On a calculator: 6 = ISBN: 97807069 CHAPTER WORKING WITH NUMBERS 9

Example Find the cube root of. = = because = = On a calculator: Example Between which two consecutive whole numbers does 0 lie? Looking at the square numbers, =, 6 = 6, 7 = 9, we can tell that between 6 and 7. 0 must lie Exercise -08 Copy and complete the following table: Number 6 7 8 9 0 Number squared 6 Number cubed Evaluate: Ex a 78 b 6 c 89 d 089 Ex 8 e f g 97 h 6 Use your answers for Question to help you answer Questions to 7. Between which of the following consecutive whole numbers does lie? Select A, B, C or D. Ex 80 A 0 and B 9 and 0 C 79 and 8 D 8 and 9 Between which of the following consecutive whole numbers does lie? Select A, B, C or D. A and B and 6 C 6 and 7 D 8 and 9 Between which two consecutive whole numbers does lie? 6 Between which of the following consecutive whole numbers does 0 lie? Select A, B, C or D. A and B 0 and C and D and 6 7 Between which two consecutive whole numbers does each of the following lie? a 6 b 0 c 0 d 00 e 76 f 800 0 NEW CENTURY MATHS 8 STAGE ISBN: 97807069

8 Give the answer to each of these to one decimal place: a 7 b 00 c 0 d 6. e 9 f 000 g. h 0 9 a Find 6. b Find: i ii 9 c We know that 6 = 9. Does 6 = 9? Explain your answer. 0 a Find: i ii iii 9 b We know that = 9. Does = 9? Explain your answer. Use what you found in Questions 9 and 0 to complete each of the following: a 6 = 6 b 8 = = = = = c 900 = 00 d = 8 = = = = e 0 = f 76 = = 9 = 6 7 = = Write three examples of your own to show that ab = a b. We can use a spreadsheet to find powers and roots of numbers. Use the links provided to go to an activity that allows you to practise this skill. Spreadsheet -0 Powers and roots -09 Fractions 7 numerator denominator! Fractions can be entered into a calculator using the fraction key: or. Some types of fractions proper: the numerator is smaller than the denominator. For example, -, - 78 00 improper: the numerator is equal to or larger than the denominator. For example, -, 7 mixed numeral: a whole number and a common fraction. For example, 7 8 a b c Skillsheet -09 Fractions Skillsheet -0 Fractions and decimals ISBN: 97807069 CHAPTER WORKING WITH NUMBERS

Example 6 Change these improper fractions into mixed numerals: 7 7 a b - 7 a 7 = 7 b - = 7 = = 6 On a calculator: 7 a b c = On a calculator: 7 a b c = Example 7 Change these mixed numerals into improper fractions: a b a 6 = - + b 0 = + 7 = = - On a calculator: On a calculator: a b c a b c = d/c a b c a b c = d/c Pressing d/c ( SHIFT a b c or ndf a b c ) converts a mixed numeral into an improper fraction. Worksheet -06 Fractions puzzle Example 8 0 Simplify -. To simplify a fraction, keep dividing the numerator and the denominator by a common factor until they are as small as possible. 0-0 = = On a calculator: 0 a b c = NEW CENTURY MATHS 8 STAGE ISBN: 97807069

Example 9 Change these fractions to decimals: a b - 00 000 The number of decimal places is the same as the number of zeros in the denominator. a = 0. 00 Two zeros in the denominator, so two decimal places in the answer. b - = 0.0 000 Three zeros in the denominator, so three decimal places in the answer. Exercise -09 Write each of these improper fractions as a mixed numeral: a b - 9 c d - 0 e - 7 f - 00 g 7 h - Write each of these mixed numerals as an improper fraction: a b c d e 6 f 7 g 0 h 7 Arrange these fractions in order, starting with the smallest.,,, 8, 8, 8 7 8 Simplify the following: a - b - c - 0 6 8 d - e - 7 f - g - h 60 i - 8 00 77 Copy and complete each of the following: a =? b = -? c = 6 d - = e? = f - = 60? 8 8 g - = - 6 h - = - 9 i -? =? 0? 0 6 -?? - 0 Ex 6 Ex 7 Ex 8 TLF L Dynamic fractions ISBN: 97807069 CHAPTER WORKING WITH NUMBERS

TLF L Design a farm Worksheet -08 Fraction review Ex 9 6 Change these fractions to decimals: a - b 7 0 00 c d e 00 00 f 7 a Change the following fractions to decimals: - 000 00-8 000-0 b Place the fractions in part a in ascending order (from smallest to largest). -0 Operations with fractions - 000-6 0 000 Addition and subtraction To add or subtract fractions, the fractions must have common denominators. If necessary, convert the fractions so that they have the same denominators. Example 0 Worksheet -09 Fractagons Evaluate: a b c 7 + a 7 = - 7-7 7 = = - - - On a calculator: a b c 7 a b c = b + = + + + 0 = + - + - = - On a calculator: a b c a b c + a b c a b c =. NEW CENTURY MATHS 8 STAGE ISBN: 97807069

c = + = + = On a calculator: a b c a b c a b c a b c =. Multiplication and division To multiply fractions, multiply the numerators together and multiply the denominators together. Convert any mixed numerals to improper fractions first. To divide by a fraction, multiply by its reciprocal. Convert any mixed numerals to improper fractions first. Example Evaluate: a b c 7 a b 7 6 = - = 7 - = - 0 = - 0 On a calculator: On a calculator: a b c a b c 7 = a b c a b c a b c a b c = c = - 6 = = On a calculator: a b c a b c =. ISBN: 97807069 CHAPTER WORKING WITH NUMBERS

Exercise -0 TLF L 80 Fraction fiddle: Hit the apple Ex 0 Evaluate: a + b + c 8 8 d + e + f 7 g h i + - 0 j k l 6 + 6 Ex a b c 7 d e f g h i j k l 8-6 a 8 b c d 60 e f - 8 7 6 NEW CENTURY MATHS 8 STAGE ISBN: 97807069

Mental skills B Time differences Maths without calculators Examine this example. What is the time difference between :0am and 6:pm? From :0am to :0pm = 6 hours Count: :0, :0, :0, :0, :0, :0, :0 From :0am to 6:00pm = 0 minutes From 6:00pm to 6:pm = minutes hours + 0 minutes + minutes = 6 hours minutes or 0 minutes 6 hours minutes = 6 hours minutes Spreadsheet -0 Employee timesheets :0am :00noon 6:00pm 6:pm Now find the time difference between: a :0am and 7:0pm b 6:0pm and :00 midnight c :pm and 8:0pm d :0am and 0:am e :0pm and :0am f 9:am and :am g 0 hours and 09 hours h 0 hours and 0 hours i 7:am and :0pm j :pm and 0:0pm - Applying numbers Exercise - Six friends at a restaurant split a $ bill evenly. How much did each person pay? The temperature changed from 6 C to -8 C overnight. Which of the following describes the change? Select A, B, C or D. A it decreased by C B it decreased by C C it increased by C D it increased by C A heart beats 80 times in one minute. How many times will it beat in one hour? How many times must 0 be added to 7 to give 97? Select A, B, C or D. A B C 6 D Worksheet -0 Magic squares Worksheet - Crossnumber puzzles Worksheet - Crossnumber challenges TLF TLF L 768 Fish market L 99 School canteen: Two traders: level ISBN: 97807069 CHAPTER WORKING WITH NUMBERS 7

The table on the right shows the temperature of four towns at two different times of day. Which of the following towns had the smallest change in temperature? Select A, B, C or D. A Echo Bay B Cambridge Bay C Dawson Creek D Fort Liard 6 Michael went shopping and bought the following items: an exercise book for $.70, two pens for $.60 each, a drink for $.0 and a packet of chips for $.6. a How much did Michael spend in total? b If Michael paid with a $0 note, how much change did he receive? 7 Jessica s car holds litres of petrol. If the price of petrol is 8. cents per litre, how much will Jessica need to pay to fill the tank? Give your answer to the nearest cents. 8 Traci needs to build a wooden rectangle as shown on the right. How much timber will be left from a. m length of timber? Temperature Town :00am 8:00am Echo Bay -6 C - C Cambridge Bay - C C Dawson Creek C 6 C Fort Liard - C C 0. m 9 Thao s mobile phone plan charges $0 per month plus $0.8 for each phone call. How much will Thao need to pay if she made 9 calls in one month? 0.8 m 0 Andrew, Priscilla, Heidi and Edmond shared a $00 000 Lotto win. How much did each person receive? 8 NEW CENTURY MATHS 8 STAGE ISBN: 97807069

In 9, US athlete Donald Lippincott ran 00 metres in 0.6 seconds while, in 00, Tim Montgomery, also from the USA, ran 00 m in 9.78 seconds. a If he could maintain the same speed, how far (to the nearest metre) could Donald have run in one minute? b How far could Tim have run in one minute? c After one minute, how far ahead of Donald would Tim be? Danielle uses half a sheet of adhesive plastic to cover her books, while Christina uses of the same sheet. What fraction of the original sheet remains? Select A, B, C or D. A B C - D - 9 0 0 Spreadsheets can be used to solve problems. Use the link provided to go to an activity in which a spreadsheet is used to plan a movie night with friends. Power plus Scientific notation is a special way of representing very large or very small numbers. This is how your calculator handles this problem: Screen.6 0 means.6 0 that is.6 0 0 0 0 = 600 Screen.678-0 means.678 0 - that is.678 0 0 0 = 0.00 678 a Can you see a quick way of writing the answer each time? b Write each of the following calculator displays as an ordinary number: i. 0 ii. 0 iii 9. -0 iv 6.667 0 v 9.6-06 vi 8.9-06 vii.00-0 viii.698 07 ix. 08 x.70-0 Write each of these numbers in scientific notation: a 000 b 000 000 c 0.007 d 000 e 0.000 f 0.000 g 000 000 h 0.000 i 0.000 000 0 j (. 0 ) (6 0 - ) a What is the largest number that can be displayed on your calculator? b What is the smallest? Spreadsheet -0 Movie night Worksheet - Crossnumber challenges Worksheet - Scientific notation ISBN: 97807069 CHAPTER WORKING WITH NUMBERS 9

Worksheet - Numbers crossword Chapter review Language of maths cube cube root decimal decimal place denominator estimate evaluate factor fraction improper fraction integer long division mental calculation mixed numeral numerator operation order of operations power proper fraction reciprocal round simplify square square root What are the four arithmetic operations? If you round a decimal to the nearest hundredth, how many decimal places is this? What are the order of operations rules? What type of numeral can an improper fraction be converted to? How do you write the cube root of -6? 6 What is the cube root of -6? Topic overview What parts of this chapter did you remember from last year? Are there any parts of this chapter that you still do not understand? Discuss any problems with your teacher or a friend. Copy and complete this chapter summary that has been started for you. Check your work with that of other students and with your teacher. Operations + Mental calculation Fractions + Mixed numerals 9067 678906789 890678906789 67890678906789 6789067890678 0 9067890678906 WORKING 78906789067890 678906789067890 WITH 678906789067890 6789067890678 9 NUMBERS 6789067890 6789067890 8 6 6789067 7 67890 Numerators Denominators Decimals + Powers and roots x Integers -, -, -, 0,,, 0 NEW CENTURY MATHS 8 STAGE ISBN: 97807069

Chapter revision Evaluate each of these expressions without using a calculator. a 7 0 b 0. 6 c 9 d 9 e f 8 g 0. 00 h i 90 0 j 6 k 0.6 60 l 8 8 m 0 n 0. 00 o 0 p 7 8 a Estimate the answer to + 7 + + 60 +. b Find the exact answer to + 7 + + 60 + without using a calculator. Evaluate each of these expressions without using a calculator. a 8 + 6 + 6 b 09 + + 00 c 78 d 6 88 e 7 f 7 g 8 h 7 6 i 8 7 j 80 k 96 8 l 9 m 0 + 099 + 6 n 6 o 80 78 p 79 Evaluate: a - + 6 b 8 c (-) d 0 e - 8 f - 7 g (-) h - (-) i -6 9 j (-6) (-) Round each of these numbers correct to the number of decimal places shown in the brackets. a 0.7 [] b.0 [] c 98.087 [] d 69.97 [] 6 Evaluate: a b c + 6 8 d ( + 8) e -6 8 + f 6 8 g 80 [( + ) 8] h -6 ( + ) Topic test Chapter Exercise -0 Exercise -0 Exercise -0 Exercise -0 Exercise -0 Exercise -0 ISBN: 97807069 CHAPTER WORKING WITH NUMBERS

Exercise -0 7 Evaluate these expressions, giving your answers rounded to two decimal places. a 8 b - 7 6 8 c - d (8 6) (69 + ) 7 + 8 Exercise -06 Exercise -06 Exercise -07 Exercise -08 8 Convert these decimals to fractions: a 0. b 0. c 0.007 9 Evaluate: a. + 6.8 b. + 0.8 +.6 c 7. 6.9 d 8 0.0 e.6 f. 0. g 0.6 0.8 h 9.6 0. i (.) j.. 0 Calculate: a 7 b 6 c d (-) e f Calculate: a 8 b 00 c 7 d - e. f - g 0 000 h Between which two consecutive whole numbers does lie? Select A, B, C or D. A and b and 6 c 7 and 8 d 8 and 9 Exercise -08 Without using a calculator, estimate to one decimal place. Exercise -08 Exercise -08 Exercise -09 Copy and complete the following: a 0 = ( 0) b 6 = 9 = 0 = = = = Convert each of these improper fractions to a mixed numeral. a - b - 7 c d 78 - Exercise -09 6 Convert each of these mixed numerals to an improper fraction. a b c 6 d NEW CENTURY MATHS 8 STAGE ISBN: 97807069

7 Reduce these fractions to their simplest form. a 6 b - 8 c d 8 - e - 8 f 8-6 - 6 Exercise -09 8 Convert each of these fractions to a decimal: a b - 7 c - d 00 0 000 8 00 Exercise -09 9 Evaluate: 7 a + b + c d 7 7 7 8 + e f g h 8 6 6-7 7 7 8 6 0 a Tamara earns $79.0 for working 8 hours a week. How much does she earn each hour? b A light aircraft can climb 0 metres every minute. If it climbed for. minutes after take-off, what height did it reach? c One Friday, the manager of a store added together all the sales figures of the staff. They were: Mario $0, Sue $97.60, Theo $88.0, Frank $8.0, Samantha $0. What was the total of the sales figures? d A gardener took 00 watermelons to market and sold three-quarters of them for $.0 each. The rest were sold for $.90 each. i How many watermelons were sold for $.0 each? ii Calculate the total amount received by the gardener. e Mark is paid $6.7 per hour. How much does he earn if he works for 6 hours? f A petrol tanker holds 0 000 L of fuel. If of the tank is emptied, how much fuel is left in the tank? Exercise -0 Exercise - ISBN: 97807069 CHAPTER WORKING WITH NUMBERS