Number skills 2. Objectives. Before you start this chapter

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1 This chapter explores different mental and written methods of calculation. The Pontcysyllte Aqueduct in North Wales was built more than 00 years ago, long before calculators were invented. All of the engineering calculations had to be done using pencil and paper! Before you start this chapter Put your calculator away! Number skills Objectives This chapter will show you how to develop a range of strategies for mental calculation recall all positive integer complements to 100 recall all multiplication facts to 10 10, and use them to derive quickly the corresponding division facts multiply or divide any number by powers of 10 use a variety of checking procedures including working the problem backwards and considering whether a result is of the right order of magnitude understand and use negative integers both as positions and translations on a number line use standard column procedures for addition, subtraction and multiplication of integers add, subtract, multiply and divide integers, including negative numbers estimate answers to problems involving integers and decimals D C 1 What is the value of the 7 in each of these numbers? a 7 b 750 c.67 d Work out a 6 b 8 6 c 1 ( ) Arrange these numbers in order, from smallest to largest. a 1, 8, 10, 7 b 6,.5, 9, c, 6.5, 1, HLP Chapter

2 11.1 Adding and subtracting whole numbers Why learn this? Many careers require basic maths skills. or example, trainee helicopter pilots must pass a mental maths test before they can move on to further training. Objectives Add and subtract mentally Recall positive integer complements Keywords integer complement, partition Use standard column procedures to add and subtract whole numbers Skills check 1 What is the value of the in each of these numbers? HLP Section.1 a b 50 c 1 d 85 Copy and complete. a b c Work out a 11 6 b 19 7 c 18 1 d 1 1 Adding and subtracting mentally Integer complements are useful when adding and subtracting mentally. 6 and are complements in and 0 are complements in 100. You can use partitioning to add mentally. The -digit number 5 can be partitioned into 0 5 or You can find a difference mentally by counting on from the smaller number to the larger number. Try it with 7 and 11. xample 1 Use a mental method to work out a 6 b 78 6 a b Add the tens together and then add the units together. Start at 6 and count up to 78 in easy steps xercise 11A 1 Use a mental method to add these numbers. a 7 b 66 c 5 8 d 5 Use a mental method to work out these subtractions. a 19 b 7 5 c 67 9 d Adding and subtracting whole numbers 185

3 Copy and complete these additions. a 10 b c d e f Look at the numbers in the box a Select two numbers with a total of 100. b Select two numbers with a difference of 50. Written methods for adding and subtracting whole numbers To add or subtract whole numbers, line up the units, tens, hundreds and so on, then add or subtract. xample Work out a b Always write down the carried digit, otherwise you may forget it. Remember, it is the top number minus the bottom number. a b Set out the calculation in columns. Always add the units column first Write down the units and carry the 1 ten to the tens column. Set out the calculation in columns. Always subtract the units column first. is smaller than 6 so you need to borrow 10 from the tens column: The 80 in the tens column becomes 70. xercise 11B 1 Work out these additions. a b 7 c 1 7 d Work out these subtractions. a 6 b c 5 8 d Number skills

4 U N C T I O N A L U N C T I O N A L U N C T I O N A L What is 199 plus 7? A shop sold 87 packs of toilet roll on Monday, 6 packs on Tuesday and 1 packs on Thursday. How many packs of toilet roll will the shop need to order to replace the stock sold? U N C T I O N A L 5 6 A postman had 6 letters to deliver. He has delivered 189 so far. How many letters does he have left to deliver? Copy and complete. a b 1 c 5 d Copy and complete. a 8 5 b 5 9 c 5 6 d AO Keywords for adding and subtracting Look for keywords to help you decide which operations to use to solve a problem. The table reminds you of some helpful keywords to look for. Operation add difference plus minus sum subtract total take away xample Nathan has 87 football stickers. He gives 9 away and buys a pack of 8. How many football stickers does Nathan now have? Nathan now has 5 football stickers. Identify the keywords: gives away implies, buys implies. Would give the same answer? 11.1 Adding and subtracting whole numbers 187

5 U N C T I O N A L xercise 11C 1 Work out a b c 19 7 d A shop had 16 tins of dog food in stock one morning. During the day they sold 7 tins and received a delivery of 55 tins. How many tins of dog food were there at the end of the day? AO On Saturday morning leri had 16 in her bank account. She then paid 5 into the account. On Sunday morning she withdrew 80. How much was then in leri s account? A car s milometer showed a total mileage of On Monday the car was taken for a test run by a customer. The customer did 5 miles on the test run. On Tuesday the same car was taken for another test run. The car was returned with 870 on its milometer. What distance was covered in the second test run? U N C T I O N A L 11. L Multiplying whole numbers Keywords product, grid Why learn this? Objectives method, standard Race engineers multiply the amount of fuel needed Multiply whole numbers by 10, 100 and 1000 column method per lap by the number Remember and use multiplication facts up to of laps to work out how Multiply mentally much fuel is required for the whole race. Multiply whole numbers using written methods Skills check HLP Section.1 1 What is the value of the 8 in each of these numbers? a 80 b 18 c 857 d 8 96 Double these numbers. a 6 b 1 c 15 d Copy and complete these multiplications. a b 1 c d 8 Multiplying by 10, 100 and 1000 To multiply whole numbers by 10, move the digits one place to the left by 100, move the digits two places to the left by 1000, move the digits three places to the left. 188 Number skills

6 xample Work out a 6 10 b c TT TH H T U A place value diagram can help you multiply by 10, 100 and (digits move one place to the left) (digits move two places to the left) (digits move three places to the left) Remember to add 0 as a place holder. xercise 11D 1 Work out a 10 b c d e Copy and complete these multiplications. a 6 60 b c What must 75 be multiplied by to get ? a Work out 10. b What is the value of the digit in the answer to part a? 5 What is the value of the digit in the answer to 80 10? Multiplication facts and partitioning You can use the multiplication facts up to to work out other multiplications. You can use partitioning to multiply mentally. The -digit number 16 can be partitioned into So 16 5 is the same as (10 5) (6 5). xample 5 Use a mental method to work out Partition 5 into 0 and 5, then multiply each by Add the products to get the total. 11. Multiplying whole numbers 189

7 xercise 11 1 Copy and complete these multiplication grids. a b 5 7 c Use partitioning to work out these multiplications. a 1 6 b 15 8 c 7 d 6 8 e 5 f 9 Make jottings when you do mental calculations. Written methods for multiplication You can use the grid method or the standard column method for written multiplication. Different words can be used for multiplication. Keywords to look out for include multiply, times and product. xample 6 Work out 5 9. rid method Split 5 into hundreds (00), tens (0) and units (5). This is This is 0 9. This is Add the separate answers. So Standard method Always multiply from right to left Write down 5 and carry over ( tens) Write down 1 and carry over ( hundreds). 190 Number skills

8 xercise 11 1 Work out a 51 7 b 87 9 c 6 6 d 09 6 What is 189 times? Look at ernando s maths homework. a b c Mark his answers. If an answer is incorrect, explain what ernando has done wrong. A football season ticket costs 56. How much would five season tickets cost? xample 7 Work out. rid method This is Work out. This is Standard method Add the separate answers. 7 1 ( ) 8 0 ( 0) Then write down 0 and work out. xercise 11 1 Work out a 1 8 b 5 6 c 5 6 d 7 8 Work out a 6 b 51 c 75 d Multiplying whole numbers 191

9 U N C T I O N A L U N C T I O N A L AO AO ind the product of 6 and 18. To work out the area of a rectangular plot you multiply the length 19 m by the width. What is the area of this plot? m Jamie travels to Crewe 17 times a year to visit his family. He travels by train and each return ticket costs 58. He has saved 950 towards travel for the next year. a What is the total cost of Jamie s travel for 1 year? b Has Jamie saved enough money? If not, how much more does he need to save? Javier travels to Liverpool times a year to support his football team. He travels by train and each return ticket costs 16. He has saved 860 towards travel for the season. Has Javier saved enough money? If not, how much more does he need to save? The units will be m. U N C T I O N A L U N C T I O N A L L 11. Dividing whole numbers Why learn this? Dividing the total cost of a meal between a group of friends ensures everyone pays the same amount. Objectives Divide whole numbers by 10, 100 and 1000 Derive division facts from multiplication facts Use repeated subtraction for division of whole numbers Keywords divisor, remainder Skills check 1 What is the value of the 9 in each of these numbers? a 9 b c 0.9 d Copy and complete these multiplications. HLP Sections.1,. a 6 b 6 0 c 9 7 d 7 56 HLP Section 11. Work out a 10 b 5 c d 9 Dividing by 10, 100 and 1000 To divide whole numbers by 10, move the digits one place to the right by 100, move the digits two places to the right by 1000, move the digits three places to the right. 19 Number skills

10 xample 8 Work out a b c H T U. t h A place value diagram can help you divide by 10, 100 and (digits move one place to the right) (digits move two places to the right) (digits move three places to the right) Use zero as a place holder to show there are no units. xercise 11H 1 Work out a b c d 5 10 e Copy and complete these divisions. a 80 8 b c d What must 80 be divided by to get 8? a Work out b What is the value of the digit in the answer to part a? What is the value of the digit 5 in the answer to 5 10? Repeated subtraction You can use repeated subtraction for division. The number you divide by is called the divisor. Keep subtracting multiples of the divisor until you cannot subtract any more. Then see how many lots of the divisor you subtracted altogether. The multiples of a number are the numbers in its times table. Different words can be used for division. Keywords to look out for include divide and share. 11. Dividing whole numbers 19

11 U N C T I O N A L U N C T I O N A L xample 9 A group of four friends hire a car for 0. The cost is shared equally. How much does each person have to pay? is the divisor Subtract multiples of the divisor. Working with multiples of 10 keeps things easy ach person pays 85. The aim is to get to zero. xercise 11I 1 Work out a 6 b 18 7 c d 51 9 Share 96 between 8 people. Nathan has 19p credit on his phone. It costs 6p to send a text message. How many text messages can he send? ight friends go to a restaurant. The total bill is 1. They share the bill equally. How much does each person have to pay? U N C T I O N A L U N C T I O N A L Rounding up and down Sometimes you will not be able to get to zero by subtracting multiples of the divisor. The number you have left is called the remainder. When solving a problem you may have to decide whether to round up or down. or example, when working out the number of buses needed, the exact answer might be.6. But you cannot have part of a bus, so you have to decide whether you need buses or buses. xample 10 armer Brawn has to put 7 eggs in boxes. An egg box holds 1 eggs. a How many egg boxes can armer Brawn fill? b How many egg boxes does armer Brawn need to pack all the eggs? 19 Number skills

12 U N C T I O N A L U N C T I O N A L U N C T I O N A L is bigger than 10 so you cannot get to zero. 10 is called the remainder remainder 10 a armer Brawn can fi ll egg boxes. There are 10 eggs left over. b armer Brawn would need egg boxes to pack all the eggs. xercise 11J 1 Work out these divisions. They all have remainders. a 9 5 b c 7 6 d 51 7 Work out these divisions. Some of them have remainders. a b 19 c 78 d Cakes are packed in boxes of 1. There are 60 cakes to be packed. a How many full boxes will there be? b How many cakes will be left over? U N C T I O N A L A group of 8 people travel by minibus. ach minibus holds 1 people. a How many minibuses are needed? b How many spare seats will there be? AO U N C T I O N A L 5 Nathan is laying laminate flooring in his office. The area of the office is 600 m. ach length of laminate covers an area of m. ach pack of laminate has 1 lengths. a What is the total area covered by one pack of laminate? b How many packs of laminate will Nathan need? AO U N C T I O N A L 6 wenno is laying laminate flooring in her office. The area of the office is 50 m. ach strip of laminate covers an area of m. ach laminate pack contains 16 strips. How many packs of laminate flooring will wenno need? AO 11. Dividing whole numbers 195

13 11. stimation L Why learn this? You can use estimation to check you have enough materials to complete a DIY job. Objectives Check a result by working the problem backwards Keywords BIDMAS, inverse, estimate, approximate D C Make estimates and approximations of calculations Skills check 1 Work out a 5 6 b (11 9) c 5 1 d 8 ( ) Round each number to one significant figure. a b 5500 c 19 d.8 HLP Section.6 HLP Section. Working backwards Calculations must be done in the right order: Brackets Indices (powers) Division and Multiplication Addition and Subtraction. If there are several multiplications and divisions, or additions and subtractions, do them one at a time from left to right. You can check the answer to a calculation by Operation Inverse working it backwards (using the inverse operation). Inverse means opposite. xample 11 Use inverse operations to check these calculations. a b If is correct, then a so is correct. If is correct, then b (16 7) 16 0 (16 10) so is incorrect. You can write means is not equal to. 196 Number skills

14 xercise 11K 1 Use an inverse operation to check each of these calculations. a b 9 17 c d 1 6 Use the inverse operation to find the inputs of each function machine. a 18 b c 8 d Checking answers by approximating stimating the answer to a calculation gives you an approximate answer. So you can use estimation to check that an answer is about right. To estimate: round all the numbers to one significant figure do the calculation using these approximations. xample 1 Use approximation to estimate the answer to each of these calculations. a 8 b c d a b c d The symbol means approximately equal to rounded to 1 s.f. is 50. rounded to 1 s.f. is rounded to 1 s.f. is rounded to 1 s.f. is rounded to 1 s.f. is rounded to 1 s.f. is rounded to 1 s.f. is rounded to 1 s.f. is 5. 6 rounded to 1 s.f. is 50. D xercise 11L 1 Suzanne wants to buy some notepads priced at 1.75 each. She has 10. By rounding to the nearest whole number, estimate how many notepads Suzanne can buy. 11. stimation 197

15 U N C T I O N A L U N C T I O N A L Ryan is doing his a maths homework. Copy and complete stimate: is roughly Ryan s homework. Check: 518. is close to 500 b stimate: 9.8 is roughly Check: c stimate: Check: By rounding to one significant figure, decide which is the best estimate for each calculation. stimate A stimate B stimate C a b c d ind an approximate value for each calculation. a b 6. c 5.. d Use approximation to estimate the answer to each calculation. a 7 b 18 6 c 5 76 d A pack of 500 sheets of printer paper costs stimate the cost of 8 packs. D 7 stimate the answer to each calculation. a b c d C AO C AO stimate the answer to each calculation. a b c A litre of matt emulsion paint will cover an area of about 15.5 m. Rashid needs to paint a sports hall with a total wall area of 65 m. d a Which is the best calculation to estimate how much paint he needs? A B C ive a full reason for your answer. b Use estimation to decide how many litres of paint Rashid will need for two coats. Pepe is on his way home. He has a further 8 miles to travel. His car has 1.8 litres of fuel remaining. Pepe s car travels 5.5 miles per litre of fuel. Use estimation to decide whether Pepe has enough fuel to get home. U N C T I O N A L U N C T I O N A L 198 Number skills

16 11.5 Why learn this? You need negative numbers to record very low temperatures. Negative numbers Objectives Calculate a temperature rise and fall Order negative numbers Add and subtract negative numbers Multiply and divide negative numbers Keywords positive, negative, minus sign HLP Section. Skills check 1 Arrange these numbers in size order, from biggest to smallest. 1, 8, 9.5, 1,.6 Write the colder temperature in each pair. a 8 C, 6 C b 5 C, 0 C c C, 1 C d C, 10 C Calculating temperature rise and fall Numbers greater than zero are positive numbers. Numbers less than zero are negative numbers. Negative numbers are written with a minus sign in front of the number. You can show positive and negative numbers on a number line This thermometer shows a reading of C. This means that the temperature is C below 0 C C xample 1 At 10 am the temperature is C. a At noon the temperature is 9 C. What is the rise in temperature? b By evening the temperature has fallen by 1 C. What is the new temperature? a C The rise in temperature is 6 C. Start at C and count the steps up to 9 C. b C Remember to count 0 C as one of the steps. The new temperature is C. Start at 9 C and count down 1 steps Negative numbers 199

17 Ordering negative numbers To order negative numbers, think of a number line. Remember that the further to the left you go, the lower the numbers are. So 10 is lower than 5. xample 1 Arrange these temperatures in order, coldest first. 0 C, 8 C, C, 6 C Order: 8 C, 6 C, 0 C, C Use a number line to help you order positive and negative numbers. xercise 11M 1 Write the number that each arrow is pointing to. C D A B H Write the colder temperature in each pair. a C, 1 C b 5 C, 0 C c 8 C, C d C, 6 C Calculate the new temperature if the temperature now is a 8 C and it warms up by C b 5 C and it cools down by 6 C c C and it rises by 7 C d 5 C and it falls by 9 C. Use the thermometer to help you C Put the correct sign, < or >, between each pair of temperatures. a C 0 C b C C c C 8 C d 1 C 5 C 5 Write each set of temperatures in order, coldest first. a C, 1 C, 5 C, C b 9 C, C, 6 C, 0 C, 10 C c 5 C, C, 1 C, C, 8 C Always check that you have included all the numbers in your answer. 00 Number skills

18 6 The temperatures at an observation point are taken at set times during the day. The readings are shown in the table. a What is the rise in temperature between 8 am and 1 noon? b Describe the rise and fall of the temperature between 8 am and 8 pm. Temperatures taken at an observation point Time Temperature ( C) 8 am 1 noon 8 pm 6 8 pm 0 c At am that day, the temperature was 5 degrees lower than the temperature recorded at 8 am. What was the temperature at am? d By 1 midnight, the temperature had dropped by 10 C from what it was at 1 noon. What was the temperature at 1 midnight? Adding and subtracting negative numbers If you add a negative number, the result is lower than the number you started with. Adding a negative number is the same as subtracting a positive number. If you subtract a negative number, the result is higher than the number you started with. Subtracting a negative number is the same as adding a positive number. xample 15 Work out a () b 5 () c 8 () d 1 (6) a () b 5 () c 8 () d 1 (6) 6 Start at and count forwards. is the same as. Start at 5 and count back. is the same as. Start at 8 and count back. is the same as. Start at 1 and count forwards is the same as. The first means count back but the second reverses direction so you count forwards. Think of it as taking ice out of a drink the drink will get warmer Negative numbers 01

19 xercise 11N 1 Work out a 5 b (6) c d (7) e 9 f (6) g (6) h 9 (7) i 7 (9) j 11 (6) A number without a sign is always positive. The temperatures of some cities in Temperatures in six cities in December December are shown in the table. City Temperature ( C) a Which is the warmest city? Berlin b Which is the coldest city? Istanbul c What is the temperature difference London 6 between the warmest and coldest cities? Moscow 11 d What is the temperature difference New York 5 between Moscow and New York? Sydney 5 Copy and complete these calculations. a 7 b 0 c 8 d 6 Look at these numbers AO AO a b c i Choose one of the numbers to make this total as large as possible. ii What is the answer? i Choose one of the numbers to make this total as small as possible. ii What is the answer? i Choose one of the numbers to make this difference as large as possible. 1 ii What is the answer? 5 Look at these numbers. a Which two numbers have a difference of 7? b Which two numbers sum to 1? Multiplying and dividing negative numbers When multiplying or dividing two numbers you need to check the signs. If the signs are the same, the answer is positive. If the signs are different, the answer is negative. 0 Number skills

20 xample 16 Work out a () () b () (5) c (18) () d (5) (5) a () () 6 The signs are different, so the answer is negative. b () (5) 0 The signs are the same, so the answer is positive. When the answer is positive, you don t need to write the sign. c (18) () 6 The signs are different, so the answer is negative. d (5) (5) 5 The signs are the same, so the answer is positive. xercise 11O 1 Work out a () b () c () (5) d () () e () () Work out a 10 () b 1 () c (9) d () 6 e (16) () Work out a (6) 6 b 6 (6) 18 c 18 d 18 Copy and complete these calculations. a 6 18 b c 7 1 d 0 10 What number gives an answer of when multiplied by? Here is a sequence of numbers. 1,,, 8, Work out the next three numbers in this sequence. ind two different pairs of numbers that have a difference of 7 and a product of 1. means Negative numbers AO AO 0

21 U N C T I O N A L U N C T I O N A L Review exercise 1 Work out a b c 16 6 Write a number that is a 00 more than 170 b 10 less than 105 c 0 multiplied by 0 d 5 less than 0 e more than 6 [ marks] [ marks] stimate the value of [ marks] A theatre has 8 rows of 18 seats. How many seats are there altogether? [ marks] 5 The temperatures in five towns are shown in the table. Town Temperature ( C) Caernarfon 1 Llandudno Conwy 0 Holyhead 1 Llanberis a Which town was the coldest? b What was the temperature difference between the coldest and the warmest towns? 6 Put these numbers in order, starting with the smallest. 5, 7, 8,.5 [ marks] 7 Lucas needs 50 to buy a laptop computer. He plans to save 15 each week. How many weeks will it take Lucas to save enough money for the computer? U N C T I O N A L [ marks] 8 There are 155 people going to a concert. They are travelling by bus. ach bus can seat 8 people. How many buses will be needed? U N C T I O N A L [ marks] 9 a What is the value of the in the answer to 567 multiplied by 10? b What is the value of the 6 in the answer to 567 divided by 10? 0 Number skills

22 Work out a (6) b 8 () c 5 (1) d (11) Work out a (5) () b () (9) c (18) (6) d (0) (5) stimate the value of ind an approximate value of Show all your working. [ marks] [ marks] D Chapter summary In this chapter you have learned how to add, subtract and multiply mentally recall positive integer complements multiply and divide whole numbers by 10, 100 and 1000 use multiplication facts up to calculate a temperature rise and fall use standard column procedures to add and subtract whole numbers check a result by working the problem backwards order negative numbers add and subtract negative numbers multiply whole numbers using written methods derive division facts from multiplication facts use repeated subtraction for division of whole numbers multiply and divide negative numbers make estimates and approximations of calculations D C Chapter 11 Summary 05