Pupil Book 1.1
Kevin Evans, Keith Gordon, Trevor Senior, Brian Speed, Chris Pearce
37716_P001_003.indd 1 7537716 PUPIL BOOK title 1.1.indd 1
19/02/14 8:22 PM 05/02/2014 13:52
Contents 1
Introduction
4
Using numbers
6
1.1 1.2 1.3 1.4 1.5 1.6
The calendar The 12-hour and 24-hour clocks Managing money Positive and negative numbers Adding negative numbers Subtracting negative numbers Ready to progress? Review questions Problem solving Where in the UK?
2
Sequences
2.1 2.2 2.3 2.4 2.5
Function machines Sequences and rules Finding terms in patterns The square numbers The triangular numbers Ready to progress? Review questions Mathematical reasoning Valencia Planetarium
3
Perimeter and area
3.1 3.2 3.3
Length and perimeter Area Perimeter and area of rectangles Ready to progress? Review questions Problem solving Design a bedroom
4
Decimal numbers
7 9 13 16 18 21 24 24 26 28 29 34 37 39 41 44 44 46 48 49 53 56 60 60 62 64
4.1
Multiplying and dividing by 10, 100 and 1000 4.2 Ordering decimals 4.3 Estimates 4.4 Adding and subtracting decimals 4.5 Multiplying and dividing decimals Ready to progress? Review questions Financial skills Shopping for leisure
2
65 68 71 75 77 80 80 82
5
Working with numbers
84
Square numbers Rounding Order of operations Long and short multiplication Long and short division Calculations with measurements Ready to progress? Review questions Problem solving What is your carbon footprint?
85 88 91 94 96 98 102 102
5.1 5.2 5.3 5.4 5.5 5.6
6
Statistics
6.1 6.2 6.3 6.4 6.5 6.6
Mode, median and range Reading data from tables and charts Using a tally chart Using data Grouped frequency Data collection Ready to progress? Review questions Challenge Trains in Europe
7
Algebra
7.1 7.2 7.3 7.4
Expressions and substitution Simplifying expressions Using formulae Writing formulae Ready to progress? Review questions Problem solving Winter sports
8 8.1 8.2 8.3 8.4
Fractions
Equivalent fractions Comparing fractions Adding and subtracting fractions Mixed numbers and improper fractions 8.5 Calculations with mixed numbers Ready to progress? Review questions Challenge Fractional dissections
104 106 107 110 114 116 118 121 124 124 126 128 129 133 137 141 144 144 146 148 149 152 155 160 163 166 166 168
Contents
37716_P001_003.indd 2
19/02/14 8:22 PM
9
Angles
170
9.1 9.2 9.3 9.4 9.5
Using the compass to give directions 171 Measuring angles 176 Drawing angles 182 Calculating angles 185 Properties of triangles and quadrilaterals 190 Ready to progress? 196 Review questions 196 Investigation Snooker tables 198
10 Coordinates and graphs 10.1 10.2 10.3 10.4
Coordinates From mappings to graphs Naming graphs Graphs from the real world Ready to progress? Review questions Challenge Global warming
11 Percentages 11.1 11.2 11.3 11.4 11.5
Fractions and percentages Fractions of a quantity Percentages of a quantity Percentages with a calculator Percentage increases and decreases Ready to progress? Review questions Financial skills Income tax
12 Probability 12.1 12.2 12.3
Probability words Probability scales Experimental probability Ready to progress? Review questions Financial skills School Christmas Fayre
13 Symmetry 13.1 13.2 13.3
Line symmetry Rotational symmetry Reflections
200 201 203 206 209 212 212 214 216 217 220 222 225 227 230 230 232 234 235 241 245 248 248 250 252 253 256 259
13.4
Tessellations Ready to progress? Review questions Activity Landmark spotting
264 266 266 268
14 Equations
270
14.1 14.2 14.3 14.4
Finding unknown numbers Solving equations Solving more complex equations Setting up and solving equations Ready to progress? Review questions Challenge Number puzzles
15 Interpreting data
288 290
15.1 15.2
Pie charts Comparing data by median and range 15.3 Statistical surveys Ready to progress? Review questions Challenge Dancing competition
16 3D shapes
291 296 299 302 302 304 306
16.1 16.2 16.3
3D shapes and nets Using nets to construct 3D shapes 3D investigations Ready to progress? Review questions Problem solving Delivering packages
17 Ratio
307 310 313 314 314 316 318
17.1 17.2 17.3 17.4
Introduction to ratios Simplifying ratios Ratios and sharing Ratios and fractions Ready to progress? Review questions Problem solving Smoothie bar
319 323 329 332 336 336
Glossary
340
Index
346
338
Contents
37716_P001_003.indd 3
271 274 279 283 286 286
3 19/02/14 8:22 PM
How to use this book Learning objectives See what you are going to cover and what you should already know at the start of each chapter. About this chapter Find out the history of the maths you are going to learn and how it is used in real-life contexts. Key words The main terms used are listed at the start of each topic and highlighted in the text the first time they come up, helping you to master the terminology you need to express yourself fluently about maths. Definitions are provided in the glossary at the back of the book. Worked examples Understand the topic before you start the exercises, by reading the examples in blue boxes. These take you through how to answer a question step by step. Skills focus Practise your problem-solving, mathematical reasoning and financial skills.
PS
FS
Take it further Stretch your thinking by working through the Investigation, Problem solving, Challenge and Activity sections. By tackling these you are working at a higher level.
4
How to use this book
37716_P004_005.indd 4
19/02/14 8:31 PM
Progress indicators Track your progress with indicators that show the difficulty level of each question.
Ready to progress? Check whether you have achieved the expected level of progress in each chapter. The statements show you what you need to know and how you can improve. Review questions The review questions bring together what you’ve learnt in this and earlier chapters, helping you to develop your mathematical fluency.
Activity pages Put maths into context with these colourful pages showing real-world situations involving maths. You are practising your problem-solving, reasoning and financial skills.
Interactive book, digital resources and videos A digital version of this Pupil Book is available, with interactive classroom and homework activities, assessments, worked examples and tools that have been specially developed to help you improve your maths skills. Also included are engaging video clips that explain essential concepts, and exciting real-life videos and images that bring to life the awe and wonder of maths. Find out more at www.collins.co.uk/connect
How to use this book
37716_P004_005.indd 5
5 19/02/14 8:31 PM
1
Using numbers
This chapter is going to show you: • how to use number skills in real life • how to use number in everyday money problems • how to use a number line to understand negative whole numbers • how to use a number line to calculate with negative whole numbers • how to add negative numbers • how to subtract negative numbers.
You should already know: • how to write and read whole numbers • how to add and subtract positive numbers • multiplication tables up to 12 × 12 • how to use a calculator to do simple calculations.
About this chapter If you travel east across the world you go forward in time! If you travel west you go backwards. Because of the Earth’s rotation the day starts at different times across the world. So when it is 8 am in London it is 11 am (+ 3 hours) in Moscow and 3 am (- 5 hours) in New York. If you fly to one of them you can work out your arrival time by adding the time taken by the flight to the positive or negative time difference at your destination. Understanding negative numbers is also important if you’re travelling to a destination with sub‑zero temperatures so you can pack the right clothes. There are 24 time zones in the world. The World Clock in Berlin, shown here, tells you what the time is in all of them at any one moment! 6
37716_P006_027.indd 6
19/02/14 3:33 AM
1.1 The calendar Learning objective
Key word
• To read and use calendars
calendar
The calendar that everyone uses is based on the movement of the Earth around the Sun. One orbit takes about 365 41 days or 1 year. Each year is divided into 12 months or approximately 52 weeks. Each week is 7 days long. Each day is 24 hours long and each hour is 60 minutes long. Each minute is 60 seconds. Because it is not possible to work to a year that includes a fraction of a day, most years have 365 days. Then, every four years, the extra quarters are added together to form an extra day. This happens in a leap year, that has 366 days. The extra day is added into February. Because 365 cannot be divided exactly by 7, particular dates fall on different days each year. Most months have 31 days. Four months - April, June, September and November - have 30 days, and February has 28 days in most years or 29 days every leap year.
Example 1 The calendar shows January and February for 2014.
RY JANUA
FEBRUARY
S S M T W T F T F S T W 4 S M 3 1 2 1 2 3 4 0 11 5 6 7 8 9 1 7 8 8 5 6 9 10 11 1 6 17 1 2 13 14 1 4 15 1 1 5 3 1 5 16 17 2 12 4 2 3 1 8 2 1 9 2 2 2 0 2 1 1 2 2 2 19 20 23 24 25 30 31 26 27 28 28 29 26 27
a How do you know that 2014 is not a leap year? b How many Thursdays are there in January 2014? c Bjorn goes on holiday for two weeks. He leaves on 28 January. On what date does he return? d International Women’s Day is on 8 March 2014. What day of the week is this? a 2014 is not a leap year because February has 28 days. In a leap year, February has 29 days. b There are five Thursdays, starting on 2 January and ending on 30 January. c He leaves on Tuesday 28 January and returns on Tuesday 11 February. d 1 March is a Saturday so 8 March is also a Saturday.
1.1 The calendar
37716_P006_027.indd 7
7
19/02/14 3:33 AM
Exercise 1A 1
How many days are there in: a
2
52 weeks?
b
4 days
a week
c
July?
b
2 hours
5 hours
c
a day?
5 minutes
b
24 minutes
c
one hour?
How many minutes are there in: a
6
c
How many seconds are there in: a
5
12 weeks
How many minutes are there in: a
4
b
How many hours are there in: a
3
3 weeks
120 seconds
b
300 seconds?
How many hours and minutes are there in: a
200 minutes
b
350 minutes?
7
2016 is a leap year. When is the next leap year after that?
8
How many days are there in June, July and August altogether?
9
Judy was born on 5 August 1996. Punch was born on 6 September 1996. How many days older than Punch is Judy?
10
The calendar shows June, July and August of 2014. Saturday 12 July is circled.
JUNE
JULY
F S S M T W T W T T 7 F S M 6 S 5 1 4 2 3 3 4 1 4 2 5 6 7 8 1 12 13 9 10 1 10 11 1 1 2 9 1 0 1 2 2 3 14 15 8 19 16 17 1 17 18 8 2 6 8 1 19 20 21 2 15 26 27 2 23 24 24 25 3 2 5 2 2 2 2 6 7 2 28 29 3 0 31 0 3 29
a
ST
AUGU
F S W T T S M 1 2 9 1 3 7 8 5 6 4 5 16 3 3 14 1 1 2 1 3 10 11 1 22 2 9 20 2 1 8 1 3 9 0 17 7 28 2 2 6 2 24 25
What day of the week is 8 July?
b What date is the third Saturday in August? c
A family went on holiday on 4 August and returned on 18 August. How many nights were they away?
d My last doctor’s appointment was on 29 May. I was told to come back in two weeks. What date is my next appointment? e
8
The first day of the school holidays is Thursday 24 July. The first day back at school is Wednesday 3 September. How many days do the school holidays last?
1 Using numbers
37716_P006_027.indd 8
19/02/14 3:33 AM
Problem solving: The day you were born A Suppose the date you are looking at is 12 August 2005. Work through this example. Step 1: Take the last two digits of the year, multiply by 1.25 and ignore any decimals.
05 × 1.25 = 6.25
Ignoring the decimal gives 6.
Step 2: Add the day.
6 + 12 = 18
Step 3: Add 0 for May, 1 for August, 2 for February, March or November, 3 for June, 4 for September or December, 5 for April or July, 6 for January or October.
18 + 1 = 19
Step 4: If the date is in the 1900s, add 1.
The date is not in the 1900s.
Step 5: If the year is a leap year and the month is January or February, subtract 1.
2005 is not a leap year.
Step 6: Divide by 7 and get the remainder.
19 ÷ 7 = 2 remainder 5
Step 7: The remainder tells you the day. 0 = Sunday, 1 = Monday, 2 = Tuesday, 3 = Wednesday, 4 = Thursday, 5 = Friday, 6 = Saturday
So 12 August 2005 was a Friday.
B Check by doing an internet search for ‘Calendar 2005’. C Now work out the day for your birthday or a friend’s birthday.
1.2 The 12‑hour and 24‑hour clocks Learning objectives
Key words
• To read and use 12‑hour and 24‑hour clocks
12‑hour clock
24‑hour clock
• To convert between the 12‑hour and 24‑hour systems
analogue
digital
There are two types of clock. An analogue clock uses the 12‑hour clock system. A digital clock can use either the 12‑hour or 24‑hour clock system. The time is usually represented by four digits, with a colon to separate the hours and minutes. You can work out afternoon times as 24‑hour clock times by adding 12 hours to the 12‑hour clock time.
11
12
1 2
10
3
9 4
8 7
6
5
1.2 The 12‑hour and 24‑hour clocks
37716_P006_027.indd 9
9
19/02/14 3:33 AM
Example 2 It is half past two in the afternoon. How many ways can you show this time, using 12‑hour and 24‑hour clocks?
11
12
1 2
10
3
9
Here are some of the ways.
4
8 7
5
6
Example 3 The four clocks below show the times that Jamil gets up in the morning (clock A), has lunch (clock B), finishes work (clock C) and goes to bed (clock D). 11
12
1
11 2
10
7
6
2 3
9
4
8
1
10 3
9
12
4
8 7
5
Clock A
Clock B
Clock C
6
5
Clock D
a Fill in the table to show the time on each clock as 12‑hour times and 24‑hour times.
Clock
A
12‑hour
24‑hour
B
C
6:30 am 12:45
D 11:00 pm
17:15
b How long is the time between when Jamil gets up and when he goes to bed? c If Jamil starts work at 08:30 and takes an hour for lunch, how many hours is he at work? a 6:30 am is 06:30, 12:45 is 12:45 pm, 17:15 is 5:15 pm and 11:00 pm is 23:00.
Clock
A
B
C
D
12‑hour
6:30 am
12:45 pm
5:15 pm
11:00 pm
24‑hour
06:30
12:45
17:15
23:00
b Jamil gets up at 06:30 and goes to bed at 23:00. You can use a time line to work out the difference. 6
10
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23
1 Using numbers
37716_P006_027.indd 10
19/02/14 3:33 AM
From 06:30 to 07:00 is 30 minutes. From 07:00 to 23:00 is 16 hours. The answer is 16 hours and 30 minutes. c 08:30 to 09:00 is 30 minutes, 09:00 to 17:00 is 8 hours and 17:00 to 17:15 is 15 minutes. 6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23
Total time is 8 hours plus 30 minutes plus 15 minutes less 1 hour for lunch. This is 7 hours and 45 minutes.
Minutes written as fractions or decimals of an hour The table shows some conversions between hours and minutes. Minutes Hours
6
10
15
20
30
40
45
Fraction
1 10
1 6
1 4
1 3
1 2
2 3
3 4
Decimal
0.1
0.167
0.25
0.333
0.5
0.667
0.75
Example 4 Write each time as a fraction and a decimal of an hour. a 4 hours 30 minutes
b 2 hours 40 minutes
a Using the table above, 4 hours 30 minutes is b 2 hours 40 minutes is
2 32
4 12
hours = 4.5 hours
hours = 2.667 hours
6 c 36 minutes is 6 × 6 minutes so 36 minutes is 10 or
1 hour 36 minutes is
1 35
c 1 hour 36 minutes
3 5
hour = 0.6 hour
hours = 1.6 hours
Note: You can use the ° ’ ” button on your calculator to input hours, minutes and seconds. For example, pressing 4 ° ’ ” 30 ° ’ ” and then the button will input 4 hours 30 minutes. Now press ° ’ ” again and it will convert your time to a decimal.
Example 5 Look at this picture.
a What time is the train to Epping due? b Give this time according to both the 24‑hour and the 12‑hour clock. a The time on the board is 20:20. The train is due in 12 minutes, at 20:32. b The train is due at 20:32 or 8:32 pm.
1.2 The 12‑hour and 24‑hour clocks
37716_P006_027.indd 11
11
19/02/14 3:33 AM
Exercise 1B 1
2
Write these 12‑hour clock times as 24‑hour clock times. a 5 am b 5 pm c 11:20 am
d 12 noon
e
h 12 midnight
f
16:20
f
g
22:30
10:45 am
h 19:25
11:07
How many hours and minutes are there between each pair of times? a 09:30 to 13:30 b 08:15 to 10:45 c 18:20 to 22:30 d 06:22 to 09:33
e
f
06:45 to 15:30
11:12 to 23:28
4
A TV programme starts at 17:50 and finishes at 18:25. How long is the programme on for?
5
A TV programme starts at 09:55 and is 25 minutes long. At what time does it finish?
6
Copy the diagrams and fill in the missing values. a b c
7
PS
g
11:05 pm
Write these 24‑hour clock times as 12‑hour clock times, using am and pm. a 18:00 b 07:00 c 12:15 d 00:25 e
3
5:15 pm
8
Time 09:23
+ 20 minutes
Time …
Time 10:18 10:18 Time
+… … minutes minutes +
Time 11:08 11:08 Time
Time …
+ 22 minutes
Time 12:54
Write each time as a fraction and as a decimal. a 2 hours 15 minutes b 3 hours 45 minutes
c
1 hour 20 minutes
d 3 hours 30 minutes
f
2 hours 18 minutes
e
1 hour 40 minutes
This timetable shows the times of flights from Heathrow to Newcastle.
a
Depart Heathrow
07:20
10:05
13:20
13:45
16:30
21:00
Arrive Newcastle
08:25
11:15
14:30
14:55
17:40
22:05
The 13:20 departure is delayed by 15 minutes. What is the new departure time?
b The 22:05 arrival is 10 minutes early. What is the new arrival time? c
Alan arrives at Heathrow at 09:20 to catch the next flight to Newcastle. What time does he arrive in Newcastle?
d Sarah arrives at Heathrow at 13:15. She is told she is too late for the 13:20, as check‑in has closed. How long does she have to wait for the next flight that she can check in for, to depart? 9
12
Jonas walks to the station each morning to catch the 07:18 train. It takes him 15 minutes to walk to the station. On the way he stops to buy a paper, which takes him 2 minutes. He likes to arrive at the station at least 5 minutes before the train is due. What is the latest time he should leave home?
1 Using numbers
37716_P006_027.indd 12
19/02/14 3:33 AM
10
Mobile phone calls are charged at 25p per minute for the first 3 minutes, then 10p per minute for each minute over 3 minutes. a
How much will the following mobile phone calls cost? i
ii A call lasting 8 minutes
A call lasting 2 minutes
b After finishing a call, Jonas had a message from the phone company saying: ‘Your call cost 95p.’ How many minutes did the call take?
Problem solving: Times A A school operates a two-week timetable. Year 7 have three 55-minute mathematics lessons one week and two 1-hour mathematics lessons in the second week. How much time do they spend in mathematics lessons over the two weeks? B Jo left home at 8:05 am. It took her 8 minutes to walk to Sara’s house. Sara took 5 minutes to get ready, then they both walked to school, arriving at 8:35 am. How long did it take them to walk from Sara’s house to school? C After his birthday, Adam wrote ‘thank you’ letters to five aunts. He took one hour to write the letters. Each letter took the same amount of time to write. How long did it take him to write each letter? D Jonas takes 4 minutes to read each page of a comic. The comic has 16 pages. How long will it take him to read it? E
Sonia was 12 minutes late for a meeting. The meeting lasted 1 hour and 45 minutes and finished at 11:50 am. What time did Sonia arrive at the meeting?
F
The first three lessons at an academy last 1 hour and 45 minutes altogether. Each lesson is the same length. How long is each lesson?
1.3 Managing money Learning objective • To work out everyday money problems
EN CI M SP E
EN IM SP EC
SP E
CI M
EN
EN CI M
SP E
SP EC IM EN
SP EC IM EN SP EC IM EN
SP
EC I
M
EN
In England you will use these coins and notes.
The English £1 notes were replaced by £1 coins in 1984, but £1 notes are still used in the Channel Islands. 1.3 Managing money
37716_P006_027.indd 13
13
03/03/14 6:00 PM
Example 6 How can you make up 12p, using exactly four coins? The only combination, using four coins, is: • 1p + 1p + 5p + 5p.
Example 7 Show that there are six different ways to make 7p, using coins. You can only use 1p, 2p and 5p coins. The combinations are: • 1p + 1p + 1p + 1p + 1p + 1p + 1p • 1p + 1p + 1p + 1p + 1p + 2p • 1p + 1p + 1p + 2p + 2p • 1p + 2p + 2p + 2p • 1p + 1p + 5p • 2p + 5p.
.. ... .. ...............
. ..............T...............T.........T........T ... ........T...TE. E....TE. E.T...E.T...E...TE. .E...ET E ..E
...... ... .... . .... ................... . . . . . ................. ...... . ... ...... ... ..... .. .. ... ................ .. ....... ............... .. N
.. ....
Complete a table to show all the other ways.
N
One way is 5 × 10p and 0 × 2p.
N
b There are six ways to make 50p, using just 10p and 2p coins.
.. .. ... .. .. .........E.....C..E......E......E.......E.........E...E......... ...C.NE.C. .CN.E...CN .CE.N.CN..NC.CN.E.CN C
a Explain why it is not possible to make an odd number of pence with these coins.
........ PE ..N..... ..... P.. E. . . ...... . P.. E.. ..N.N. .. .. .. .. P ..E.... ...N . . P . . E... .N.... P. E ..10 . . . . . .... . N P ..N. ..E . . . . . . . 10 ..E ...P .. . . N...P 10 ..E ...N ...... .... P .. ... E 10 . N P E .. ...10
10 10 10 10 10 10
N
10 p coins
O O O O O O O O O O O
P P P P P
E E
N N N N N E N E N E N E N E N E N
E E E
2 2 2 2 2 2 2 2 2 2 2 P
P
P
P
P
P
E E E EC E E E E E E C C C C C C EC C C C
Suppose you have been collecting 10p and 2p coins.
T WT WT WT WT WT WT WT WT WT WT W
Example 8
2 p coins
a You cannot make an odd number of pence, as both 10 and 2 are even numbers, so any combination will also be even. b
14
10p coins
2p coins
5
0
4
5
3
10
2
15
1
20
0
25
1 Using numbers
37716_P006_027.indd 14
28/02/14 12:52 AM
Exercise 1C FS
1
Write down all the ways of making a total of 4 pence, using just 1p and 2p coins.
FS
2
Write down all the ways of making a total of 12 pence, using just 2p and 5p coins.
FS
3
Mark buys a magazine for 25p and receives 75p change from a £1 coin. He receives exactly four coins. What coins are they?
PS FS
4
Zara has two 20p coins, one 2p coin and a 1p coin. Ali has a 50p coin, a 10p coin and a 5p coin.
FS
Ali owes Zara 24p. Explain how he can pay her. 5
.. ..
I
V
F
NC
E
PE
50
TY P
ENCE
IF
... .... T.E...
..
PE
........
Y
NC
E
Zara
10
....
Y
T W E NT
20 20
1
........ PE .... E
N
Y
T W E NT
E
......... ...C E
N
.N
N
ON
E
2
..... ...... P
.......... ..N C E
C
E
P E
....F.....
N
TW
E
.......
P
.. ...
O
Ali
FS
5
What is the smallest number of coins that you need to make a total of 68 pence?
FS
6
Don and Donna have the same amount of money as each other. Don has two coins that are the same, and Donna has five coins that are the same. What could their coins be?
FS
7
A purse contains equal numbers of 1p, 5p and 10p coins. In total, the purse contains £1.44. How many of each type of coin are there?
PS
8
A purse contains three times as many 20p coins as 1p coins. In total, the purse contains £1.83. How many of each type of coin are there?
FS
9
Art has three coins that are the same. Bart has five coins that are the same. Explain why it is not possible for them to have the same amount of money.
PS
10
Grapes cost 54p for a kilogram. I buy one kilogram and pay with a £1 coin. I receive four coins as my change. What coins do I get?
11
I buy two cans of cola for 37p each and pay with a £1 coin. I receive three coins as my change. What coins do I get?
12
I buy three pencils for 21p each and pay with a £1 coin. I receive four coins as my change. What coins do I get?
Problem solving: USA money The coins used in the USA are worth 1 cent, 5 cents, 10 cents and 25 cents. These are known as pennies, nickels, dimes and quarters. A Mario has $1.32 in pennies, nickels and dimes. He has a total of 25 coins. There are twice as many nickels as pennies, so how many of each kind of coin does Mario have? B Joss has 35 coins, which consist of nickels, dimes and quarters. The coins are worth $5.55. She has five more dimes than nickels. How many nickels, dimes and quarters does she have? 1.3 Managing money
37716_P006_027.indd 15
15
28/02/14 12:52 AM
1.4 Positive and negative numbers Learning objectives
Key words
• To use a number line to order positive and negative whole numbers
negative number positive number
• To solve problems involving negative temperatures
temperature
Look at the two pictures. What can you see? What is the difference between the temperatures? Temperature +20 °C
Temperature -5 °C
Every number has a sign. Numbers greater than 0 are called positive numbers. Although you do not always write it, every positive number has a positive (+) sign in front of it. Numbers less than 0 are called negative numbers and must always have the negative (-) sign in front of them. You can show the positions of positive and negative numbers on a number line. The value of the numbers increases as you move from left to right. For example, 8 is greater than 2, 2 is greater than -5 and -5 is greater than -10. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1
0
1
2
3
4
5
6
7
8
9
10
You can use the number line to compare the sizes of positive and negative numbers. You can also use it to solve problems involving addition and subtraction.
Example 9 Which number is greater, -7 or -3? –10 –9 –8 –7 –6 –5 –4 –3 –2 –1
0
Because -3 is further to the right on the line, it is the larger number. Notice that -3 is closer to zero than -7 is.
16
1 Using numbers
37716_P006_027.indd 16
19/02/14 3:33 AM
Example 10 Write these temperatures in order, from lowest to highest. 8 °C, -2 °C, 10 °C, -7 °C, -3 °C, 4 °C First, draw up a number line. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1
0
1
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
Then mark the numbers on the number line. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1
0
1
You can see that the order is: -7 °C, -3 °C, -2 °C, 4 °C, 8 °C, 10 °C
Example 11 Work out the difference between each pair of temperatures. a -4 °C and 6 °C
b -1 °C and -8 °C
Look at each of these pairs on a number line. a –6 –5 –4 –3 –2 –1 –6 –5 –4 –3 –2 –1
0 0
1 1
3 3
4 4
5 5
6 6
7 7
8 8
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1
0 0
1 1
2 2
3 3
4 4
The difference is 10 degrees.
2 2
b
The difference is 7 degrees.
Exercise 1D 1
Write down the highest and lowest temperatures in each group. a
2
b
-2 °C, 6 °C, -10 °C
c
-15 °C, -10 °C, -5 °C
Work out the difference between the temperatures in each pair. a
3
5 °C, -6 °C, 4 °C -4 °C and 8 °C
b
-20 °C and -10 °C
c
5 °C and -8 °C
On Monday the temperature at noon was 8 °C. Over the next few days, these temperature changes were recorded. Monday to Tuesday
down 5 degrees
Tuesday to Wednesday
up 2 degrees
Wednesday to Thursday
down 6 degrees
Thursday to Friday
up 1 degree
What was the temperature on Friday? 1.4 Positive and negative numbers
37716_P006_027.indd 17
17
19/02/14 3:33 AM
4
Put the numbers in each group in order, from smallest to largest. a 2, 7, 0, 9, 5 b -5, 7, 9, 2, 3 c 0, 4, -6, -8, 12 d 1, -1, 8, -9, -15
5
6
e
-3, -9, -12, 1, -5
f
-9, -3, 5, -8, 4
State whether each of these statements is true or false. a
5 °C is lower than 8 °C
b -5 °C is lower than 8 °C
c
5 °C is lower than -8 °C
d -5 °C is lower than -8 °C
e
8 °C is lower than 5 °C
f
g
8 °C is lower than -5 °C
h -8 °C is lower than -5 °C
-8 °C is lower than 5 °C
Put these temperatures in order, from highest to lowest. 16 °C, -5 °C, 10 °C, -10 °C, -6 °C
PS
7
Find the number that is halfway between the numbers on each line. a
8
c
–2
–6
0
–9
–3
b
–8
–2
–9
–3
Challenge: Temperatures on the Moon Temperatures on the Moon vary depending on where light from the Sun reaches it. When parts of the Moon are turned away from the Sun, the temperatures there can be as low as -150 °C. When sunlight reaches the surface of the Moon, temperatures can be as high as 120 °C. When men first landed on the Moon, on 11 July 1969, the temperature was about -20 °C. A What is the difference between the lowest and highest temperatures on the Moon? B Is -20 °C closer to the lowest temperature or the highest temperature?
1.5 Adding negative numbers Learning objectives
Key word
• To carry out additions and subtractions involving negative numbers
brackets
• To use a number line to calculate with negative numbers You can use a number line to add and subtract positive and negative numbers.
18
1 Using numbers
37716_P006_027.indd 18
19/02/14 3:33 AM
Example 12 Use a number line to work out the answers. a 2 + (-5)
b (-7) + 4
c 3 + (-2) + (-5)
a Starting at zero and ‘jumping’ along the number line to 2 and then back 5 gives an answer of -3.
–5 2 –3
0
2
–7
b Similarly, (-7) + 4 = -3. Notice that you can use brackets so that you do not mistake the negative sign for a subtraction sign.
4 –7
–3
0
–5
c Using two steps this time, 3 + (-2) + (-5) = -4.
–2 3
–4
0
3
Look at these patterns. 4+2=6
2+4=6
4+1=5
1+4=5
4+0=4
0+4=4
4 + (-1) = 3
(-1) + 4 = 3
4 + (-2) = 2
(-2) + 4 = 2
Notice that
4 + (-1) = 3 and (-1) + 4 = 3 have the same value as 4 - 1 = 3
and
4 + (-2) = 2 and (-2) + 4 = 2 have the same value as 4 - 2 = 2.
Adding a negative number gives the same result as subtracting a positive number.
Example 13 Work out the answers. a 6 + (-1)
b 20 + (-5)
c (-2) + (-3)
a 6 + -1 = 6 - 1 =5 b 20 + -5 = 20 - 5 = 15 c (-2) + (-3) = (-2) - 3 = -5
1.5 Adding negative numbers
37716_P006_027.indd 19
19
19/02/14 3:34 AM
Exercise 1E 1
Work out the answer to each of these. a
10 - 6
b 7-2
c
6 - 10
d 4+8
e
2-5
f
6-6
g
-2 + 8
h -3 - 2
i
-5 - 1
j
-2 + 9
k
-5 - 5
l
m -1 - 4 + 4 2
o -8 + 3 - 5
b 9 + (-5)
7 + (-1)
c
= 10 -
=
=
=
=
=
=
=
a
2-4
b 6 + (-3)
c
5 + (-4)
d 7 + (-3)
e
(-3) + (-3)
f
5-9
g
(-3) + 9
h 0-8
i
10 + (-4)
j
(-2) + (-6)
k
12 + (-9)
l
n 10 + (-20)
o 0 + (-4) 1
2
3
5 + (-5)
p (-2) + (-9)
4
5
6
7
8
9 10 11 12 13 14 15
Work these out. a
10 + (-5)
b (-5) + (-10)
c
40 - 100
d (-10) + (-10)
e
20 + (-10)
f
(-50) + 50
g
25 + (-30)
h 100 - 200
i
12 + (-7)
j
(-100) + (-90)
k
15 + (-45)
l
(-5) + (-15)
Work these out. 4 + 5 + (-8)
b 10 - 1 + (-1)
d 15 + 3 + (-10)
e
5 + 3 + (-2)
c
20 + (-10) + (-5)
f
8 + (-1) + (-9)
c
-3, -8, 7
Find the total of the numbers in each list. a
7
(-4) + (-5)
Use the number line below to help you work out the answers.
a 6
e
=9-5
–15 –14 –13 –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
5
d 12 + (-4)
10 + (-7)
=7-1
m 7 + (-8)
4
p -10 + 11 - 4
Copy and complete each calculation. a
3
n -2 + 2 - 2
-10 + 8
b
6, −4, 10
−10, 20, 5
In each magic square, all the rows, columns and diagonals add up to the same total. Copy and complete the squares. 1
a
6
-1
b
5 4 8
3 2
c
1 -4 -3
-5
Alf has £20 in the bank. He writes a cheque for £30.
PS
20
4
How much has he got in the bank now?
1 Using numbers
37716_P006_027.indd 20
19/02/14 3:34 AM
Challenge: Number squares A In this 4 by 4 magic square the numbers in all of the rows, columns and diagonals add up to -2. -8
6
3
-3 7
0
-4
2
Copy and complete the square. B In this 4 by 4 magic square the numbers in all of the rows, columns and diagonals add up to the same number. -3
10
7
5 3
1
0
-2
-5
Copy and complete the square.
1.6 Subtracting negative numbers Learning objective • To carry out subtractions involving negative numbers Look at this pattern. 4-3=1 4-2=2 4-1=3 4-0=4 4 - (-1) = 5 4 - (-2) = 6 4 - (-3) = 7 Notice that
4 - (-1) = 5 has the same value as 4 + 1 = 5
and
4 - (-2) = 6 has the same value as 4 + 2 = 6.
Subtracting a negative number is the same as adding the corresponding positive number.
Example 14 Work out the answers. a 3 - (- 4) b 10 - (-6) c -3 - (-7) a 3 - (-4) = 3 + 4 = 7 b 10 - (-6) = 10 + 6 = 16 c -3 - (-7) = -3 + 7 = 4 1.6 Subtracting negative numbers
37716_P006_027.indd 21
21
19/02/14 3:34 AM
Exercise 1F 1
Copy and complete each calculation. a
2
20 - (-10)
8 - (-2)
b 10 - (-5)
=8+2
= 10 + 5
= 20 +
=
=
=
c
a
4 - (-6)
b 8 - (-1)
e
(-4) - (-2)
f
i
8 - (-4)
j
(-5) - (-5)
d 10 - (-2)
(-1) - (-10)
g
(-5) - (-10)
h (-10) - (-3)
(-9) - (-3)
k
(-7) - (-7)
l
n (-10) - (-20)
o 0 - (-10) 3
4
5
6
7
8
9 10 11 12 13 14 15
b (-10) - (-10)
c
20 - (-10)
d (-15) - (-5)
e
21 - (-16)
f
40 - (-20)
g
-25 - (-20)
h -200 - (-140)
i
18 - (-9)
j
(-140) - (-90)
k
12 - (-22)
l
(-4) - (-12)
Work these out. 9 + 5 - (-7)
b (-4) - 2 - (-2)
c
20 - (-10) + (-10)
8 - 9 - (-3)
f
(-1) - (-3) - (-4)
e
Choose a number from each list and subtract one from the other. Repeat for at least four pairs of numbers. What are the biggest and smallest answers you can find? A
3
-6
-4
32
9
B
-2
9
-9
-4
-2
Copy each statement and work out the missing numbers. 9 + ... = 5
b 10 - . . . = 15
c
. . . + (-6) = 10
d (-4) - . . . = 8
Copy each calculation and then fill in the missing numbers. a
22
2
p (-2) - (-6)
10 - (-8)
a 7
1
(-5) - (-5)
a
d 25 + 4 - (-20)
6
=
Work these out.
a
5
(-6) - (-6)
c
–15 –14 –13 –12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0
4
e
Use the number line below to help work out the answers.
m (-8) - (-2)
3
d -15 - (-5)
5 + +1 = 6
b -3 - 1 = -4
c
4-1=3
5+0=5
-3 - 0 = -3
3-0=3
5 + -1 = 4
-3 - -1 = -2
2 - -1 = 3
5 + -2 = . . .
-3 - -2 = . . .
1 - -2 = . . .
5 + ... = ...
-3 - . . . = . . .
0 -... = ...
5 + ... = ...
-3 - . . . = . . .
... - ... = ...
1 Using numbers
37716_P006_027.indd 22
19/02/14 3:34 AM
8
Work out the answers.
9
a
+5 - +2
b -3 - -3
+8 - -5
f
-8 - -6
g
-5 + +5
h +10 - -3
i
k
+4 - +5
l
6 - -9
m -1 - -2 + -3
n - +2 + +2 - +2
c
d -6 + -2
e
+9 - +2
-2 + -2
j
-3 + -9
o -1 + -6
These temperatures were recorded at Glasgow Airport over one week in January. Copy and complete the table. Draw a number line to check your answers. Sun
Mon
Tue
Maximum temperature (°C)
5
0
−2
Minimum temperature (°C)
−4
−5
Temperature (°C)
Difference (degrees)
10
Wed −6
9
9
11
Thu
Fri
Sat
1
2
4
−3
−1 5
A fish is 10 m below the surface of the water. A fish eagle is 25 m above the surface of the water. How many metres must the bird descend to get the fish?
Challenge: Marking a maths test A A maths test consists of 20 questions. A correct answer earns 3 points. If an answer is wrong, the mark is -1 point. a
A computer spreadsheet is useful for this activity.
Work out the score for each pupil. i
Ali gets 10 right and 10 wrong.
iii Charlie gets 8 right and 12 wrong.
ii
Bel gets 15 right and 5 wrong.
iv Dawn gets 7 right and 13 wrong.
b What times table are all your answers in? B Investigate what happens when a correct answer earns 3 points and a wrong answer gets -2.
1.6 Subtracting negative numbers
37716_P006_027.indd 23
23
19/02/14 3:34 AM
Ready to progress? I can show my understanding of how the number line extends to include negative numbers. I can carry out addition and subtraction involving negative numbers. I can use my understanding of negative numbers to solve simple real-life problems.
Review questions 1 Write 8:30 pm, using the 24‑hour clock.
PS
2 I have equal numbers of 10p and 20p coins. Altogether I have £3.60. How many 10p coins do I have? 3 A train departs from Vienna at 06:30 and arrives in Munich at 10:30. a How long does the journey take? b It takes me 30 minutes to walk from home to Vienna station. What time do I need to leave home to arrive in time for the train? c Adult tickets cost €100 and child tickets cost €50. How much do two adult tickets and one child ticket cost altogether? 4 The keel (bottom) of a cruise ship is 10 m below the surface of the water. The deck is 20 m above the water. What is the total height of the ship, from deck to keel? 5 Dan is on the 20th floor of a skyscraper. He goes up 5 floors and then down 10 floors. Which floor is he on now?
FS
6 a Maria puts £100 into a new bank account. In the next two weeks she takes out £25 and then £35. How much is left in her account? b Next, Maria puts in £15 and takes out £50. How much is left in her account now?
24
1 Using numbers
37716_P006_027.indd 24
19/02/14 3:34 AM
PS
7 Anaya and Euan have parked their car on level -5 of the shopping centre’s car park. They take the lift to the shops on level +3. Anaya realises she has left her purse in the car so goes back to the car to get it. Then she returns to the shops and meets Euan on level +1. How many levels has Anaya travelled through altogether?
FS
8 This is part of a bank account statement. Copy the statement and complete the balance column. Paid in (£)
Paid out (£)
Balance (£)
100
100
20
120 100
20
30
-10
50 60 10 80
9 The maximum depth of the Mariana Trench in the Pacific Ocean is 11 km below sea level. The summit of Mount Everest is 9 km above sea level. a How much higher than the base of the trench is the summit of Mount Everest? b If Mount Everest was set in the deepest part of the trench, how far would its summit be below sea level?
PS
10 In a popular BBC TV programme, each team has to buy three items at an antiques fair. The items are sold at auction and the team that makes more money wins and keeps any profit they make. Copy and complete each team’s score sheet. a
Red team Item Silver bowl
Buying price (£)
Selling price (£)
-20
50
Walking stick
40
Teddy bear
65
Total
145
Profit (£) +10
55
b Blue team Item
Buying price (£)
Charm bracelet
c
Dinkie toy car
60
Clock
80
Total
150
Selling price (£)
Profit (£)
50
+40
85 -25
Which team won and by how much?
Review questions
37716_P006_027.indd 25
25
19/02/14 3:34 AM
Problem solving Where in the UK? A Comparing temperatures Look at the map of the UK. It shows temperatures (°C) on one winter day. Use it to answer these questions. 1 What is the temperature in Belfast? 2 What is the temperature in Sheffield? 3 Which is the hottest place shown? 4 Which is the coldest place shown? 5 What is the difference in temperature between Portmeirion and London? 6 How much hotter is Plymouth than Swansea? 7 Which place is at –6 °C? 8 Which place is 5 degrees colder than Norwich? 9 Which place is 6 degrees hotter than Aberdeen?
B A trip to Edinburgh A businesswoman wants to travel from London to Edinburgh for a meeting. She wants to return to London on the same day. The table shows some train times. Use these times to plan her day so that she can spend at least 4 hours in Edinburgh. Work out all the possible trains she could use.
26
London (depart)
Edinburgh (arrive)
Edinburgh (depart)
London (arrive)
08:00
12:20
15:30
19:50
09:00
13:20
16:30
20:52
09:30
14:15
17:00
21:43
10:00
14:22
17:30
22:17
1 Using numbers
37716_P006_027.indd 26
19/02/14 3:34 AM
Aberdeen –8
–6
Edinburgh
Belfast –2
Sheffield –3
Portmeirion –4
3 Norwich
Swansea –1
0 Bristol
London 2
Plymouth 1
Problem solving
37716_P006_027.indd 27
27 19/02/14 3:34 AM