Global Thinkers: Mathematics 1. Secondary (sample)

e h t a M

ARY D N SECO ATION C EDU

. ra J Cole é . s A Jo telu Gaz . C o i c ra Igna Cole n ó Ram

s c i t a m a i s u l a d n A

INCLUDED

DIGITAL PROJECT

1

GLO BAL

sample

THINKERS

What are we going to learn? PAGE

Train yourself by solving problems

10

1

Natural numbers

24

2

Powers and roots

3

Divisibility

66

4

Integers

88

LEARNING SITUATION · SDG

• Understand the statement very clearly! • Make a drawing Vicky’s skateboard

BLOCK 01 Natural numbers and integers

Responsible consumption and production

Saving plans 48

BASIC KNOWLEDGE

Decent work and economic growth

How do we divide it? Partnerships for the goals

Numbers and more numbers! Partnerships for the goals

• Numeral systems • Counting techniques • Powers • Powers of base 10. Uses • The relation of divisibility • Multiples and divisors of a number • Positive and negative numbers • The set of integers • Addition and subtraction with integers

MORE THAN MATHS

BLOCK 02 Decimal numbers and fractions

5

Decimal numbers

116

6

The metric system

136

7

Fractions

156

8

Operating with fractions

172

9

Proportionality and percentages

190

10

Let’s make a product report! Affordable and clean energy

What units of measurement are there in other countries? Partnerships for the goals

Algebra

210

Fractions in real life? Quality education

Fractions and class surveys Reduced inequalities

Proportionality and percentages Good health and well-being

How many balls and sticks do we need? Industry, innovation

• The structure of decimal numbers

• Magnitudes and measurements • The metric system • What are fractions? • The relationship between fractions and decimals • Reducing to a common denominator • Adding and subtracting fractions • Proportionality between magnitudes • Direct proportionality problems

• Letters instead of numbers • Algebraic expressions

and infrastructure

MORE THAN MATHS

Where are we?

11 BLOCK 03 Geometry

Lines and angles

240

Industry, innovation and infrastructure

Tessellations

12

Geometric shapes

13

Areas and perimeters

260

Industry, innovation and infrastructure

288

How big is our mural? Quality education

• Basic elements of geometry • Two important lines • Angles • Polygons and other plane shapes • Symmetries in plane shapes • Triangles • Quadrilaterals • Measuring quadrilaterals • Measuring triangles

MORE THAN MATHS

BLOCK 04 Processing information

14

Graphs of functions

15

Statistics

312

Doctor! What do the graphs tell us? Good health and well-being

330

Do you have a minute for some questions? Quality education

• Cartesian coordinates • Points that provide information • Statistical analysis process • Frequency and frequency tables

MORE THAN MATHS + GLOSSARY

PORTFOLIO • Make a good plan • Expressing data in a diagram

• Proceed systematically • Calculate

• Large numbers • Rounding natural numbers

• Basic operations with natural numbers • Expressions with combined operations

• Operating with powers • Square roots

Choose a project and plan its financing.

Plan a savings method using the learning experience’s resources.

• Prime and composite numbers • Decomposing a number into its prime factors

• Lowest common multiple • Greatest common divisor

Choose a set of elements and report on the possibilities to divide it into packages.

• Addition and subtraction with brackets • Multiplication and division with integers

• Combined operations • Powers and roots of integers

Devise a situation in which positive and negative numbers are used.

• Addition, subtraction and multiplication with decimals

• Dividing decimals • Square roots and decimal numbers

Report on a product.

• Units of measurement for fundamental magnitudes • Conversion of units

• Complex and simple amounts • Measuring surface areas

Gather information on units of measurement. Think about, develop and explain a situation in which the usefulness of decimal numbers and fractions is demonstrated.

• Equivalent fractions • Problems with fractions • Multiplying and dividing fractions • Combined operations

• Problems with fractions

Present and solve questions involving fractions.

• Inverse proportionality problems • Percentages

• Percentage increases and decreases

Create a context in which three magnitudes linked by proportionality relations are involved.

• Equations • First methods for solving equations

• Solving first-degree equations with one unknown • Solving problems through equations

Investigate the elements that make up towers and generalise the results using algebraic tools.

• Angle measures • Operating with angle measures • Angular relationships

• Angles in polygons • Angles in a circumference

Locate yourself on a map with the help of a compass.

• Regular polygons and circumferences • Cordovan triangle and related shapes • Pythagorean theorem • Applications of the Pythagorean theorem

• Geometric shapes • Polyhedra • Solids of revolution

How to fill the plane with regular polygons.

• Measuring polygons • Measuring circles

• The Pythagorean theorem for calculating areas

Solve shaded area problems.

• Points that are related • Interpreting graphs

• Linear functions. Equation and representation

Analyse a medical chart and try to draw consequences.

• Statistical graphs • Statistical parameters

• Position parameters

Survey: How much do you like Elvis Presley?

1

NATURAL NUMBERS WATCH THE VIDEO

12. Responsible consumption and production

t ha

ca n y ou

d o

to

W

LEARNING EXPERIENCE

Vicky’s skateboard.

you ever planned a project? Think about a real or imaginary TAKE ACTION Have ? one. For example, a trip to another country. How would you pay for it? At the end of the learning experience there is a guide to financial planning. But let’s start thinking first!

DO THE QUIZ!

BEFORE STARTING… REVIEW WHAT YOU KNOW

You want to buy a new skateboard. It costs €165. You only have €96 in your piggy bank. Considering this data, answer the questions. Income • Allowance → €15 a week • Money for chores → €20 a week 1. What is the minimum amount you can save a week? Look at your expenses. Are they all necessary? What is the maximum you could save by cutting your expenses? 24

Expenses • Going out with friends €2-€10 a week • Other expenses (sweets etc.) €6 a week 2. You want to buy a new skateboard. It costs €210. You save all your money. In how many weeks can you buy the skateboard? You don’t want to cut all your expenses. Think of a plan to buy the skateboard in 10 weeks.

Learning by doing videos

o

t are y ou hao learn g W t to

in

1-minute explanations videos

g

TAKE ACTION ?

NUMERAL SYSTEMS

COUNTING TECHNIQUES

• The decimal numeral system • Place value of a number ROUNDING NATURAL NUMBERS

LARGE NUMBERS

• Rounding natural numbers BASIC OPERATIONS WITH NATURAL NUMBERS

EXPRESSIONS WITH COMBINED OPERATIONS

• Addition and its properties • Multiplications and its properties • Division • Exact division and integer division • A property of division

• Order of combined operations

MY MATHS DICTIONARY Key language

In context

Add up (to)

40 plus 60 add up to 100.

Take away

38 take away 5 is equal to / equals 33.

Times (by)

What is 8 times (by) / multiplied by 9?

Divided by

11 divided by 4 is 2 remaining 3 / and 3 remaining / and 3 left over.

Round

What is the answer rounded to the nearest hundred?

25

1

Numeral systems

Natural numbers (1, 2, 3,…) started when humans began to count. Then society evolved and people had to work with large numbers. A more practical system was necessary. That is how different numeral systems emerged in different cultures. A numeral system is the set of symbols and rules we use to represent numbers. This Palaeolithic man has written the number 47. What is the value of each symbol?

The Egyptian numeral system

1

10

100

1 000

10 000

100 000

1 000 000

stick

hobble

rope

flower

finger

frog

person

The Ancient Egyptians simply added the necessary symbols to get the number they wanted. This is an additive system. For example, look at the picture on the right. It is the number 1 333 331.

The Mayan numeral system • The Mayans only had three symbols for numbers: (0)

(1)

(5)

• Look at the diagram below. It shows how the Mayans wrote the numbers 0-20 using the additive system. These numbers are the first level. They wrote larger numbers with the same symbols. They just added new levels. In each new level, the value of the symbols was multiplied by 20. 0 1

2

3

4

5

7

8

9

6

10 11 12 13 14 15 16 17 18 19

Second level (× 20) → First level (× 1) → 20

21

27

36

40

100

137

• The Mayan numeral system also has a characteristic of the positional numeral system: the value of the symbols changes depending on their level. The Mayan numeral system was partly additive and partly positional. 26

The decimal numeral system Today, we use the decimal numeral system. It has ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. It follows these rules:

Remember 27 473

• We write the symbols in different levels, called place values. • Ten units of one level form one single unit of the following level. • The value of a figure depends on its place value. The decimal system is positional.

2 T Th →

20 000

7 Th →

7 000

4H→

400

M

H TH

T TH

TH

H

T

U

7T→

70

4

7

8

4

3

0

4

3U→

↓ 4 000 000 U

↓ 4 000 U

↓ 4U

+

3

27 473

The value of 4 changes according to its place value. Theory into practice

4 Translate the following Mayan numbers into the

1 Think about the decimal numeral system.

decimal system.

a) How many tens are there in 3 thousands? b) How many hundreds are there in one ten of thousands? c) How many hundreds are there in 5 one million units? Help H Th T Th

Th

H × 10

1

0

T

5 Look at the Mayan numbers below. Add four elements

U

to the left of the series. Now, add four to the right.

× 10

0

1 Th = 100 T

Let’s practise! 2 Write the following numbers using the Egyptian

numeral system: 19, 65, 34 120 and 2 523 083.

6 Complete the following in your notebook.

a) 500 T = … H = … Th b) 3 000 H = … Th = … T Th c) 6 Th = … H = … T d) 8 H Th = … T Th = … T 7 Read the sentences. Write true or false.

3 The following symbols are from an additive system:

1 5 10 100 Write the following numbers using this system: 7, 12, 84 and 126.

a) You move a number to a different place value. The value of the number changes. b) You add a 0 to the right of a number. The value of the number is now ten times greater. c) You add a 0 to the left of a number. The value of the number is now ten times lower. d) Half a thousand equals 5 tens. e) One thousand thousands equals one million. 8 A number contains five figures that add up to 5. You

change the place value of the units to thousands. The number increases by 999. What is the number? 27

2

Counting techniques

We use natural numbers for counting. When the quantities are large, we don’t count one by one, we use different counting techniques. Different techniques Data collection Example

Sometimes we need to collect data. For example, to count the number of people or the number of points in a game. The tally marks below show a class vote for their student representative. nerea

6

celia

13

aiTor

9

We can see that there are 28 students in total. We can also see that Celia got the most votes. Data table Example

A data table can show us the frequency distribution of a particular value. Look at the following example: • There are 52 students in Year 1 and 52 in Year 2. • In total, there are 49 girls, 54 boys. One person prefers not to respond. Year 1. Group A

Year 1. Group B

Year 2. Group A

Year 2. Group B

oTHer

oTHer

oTHer

oTHer

14

13

0

12

13

0

11

16

1

12

12

0

Tree diagram Example First ball

28

Second ball

Left

We will make an experiment: a box with three balls. Their 1. Take out two of the balls, there one byis one. colours are red, yellow and blue. A tree diagram shows all the possible results. Let's see the probabilities of times, a particular result.times, 2. If we repeat experiment six how many theoretically, will the red ball be left in the box? 1. Take out two of the balls, one by one. 3. We see that the red ball six is left in the box in 2 times, of the 6 results. 2. If wecan repeat the experiment times, how many theoretically, willthe theexperiment red ball be6left in the So, if we repeat times we box? can expect that in 2 of theWe them theball boxis will red.box in 2 of the 6 results. 3. canthe seeball thatleft theinred left be in the So, if we repeat the experiment 6 times we can expect that in 2 of the them the ball left in the box will be red.

Let’s practise! 1 Martha is looking out of her window. She is counting

the number of cars passing by colour. Copy and complete the missing data.

3 Each year this tree grows two branches. Each branch

produces a flower.

year 3

year 2 White

12

Black

3

year 1

Grey

year 0

Red Blue

10

Yellow

3

Green Other

15

ToTal

82

• How many flowers will the tree have after 6 years? And after ten years? Years

1

2

3

4

…

Flowers

2

2·2=4

4·2=8

8 · 2 = 16

…

4 This box has four balls. Three are blue and one is red.

To do the experiment: Take out two balls, one by one.

2 Luis is going on holiday. This is what is in his suitcase:

a pair of trainers, a pair of shoes, two pairs of trousers, two shirts and two sweatshirts.

1

a) How many different combinations of clothes can he wear?

2

a) What is the possibility that the two balls left in the box have different colours? b) What is the possibility that the two balls have the same colour?

1

b) What happens if, instead of two trousers, he takes three pairs? And if we also take into account the shoes he was wearing when he left the house?

3

2

3

3

2

2

2 3

1

3

3

1

3

1 3

1

2

2

1

1

2

3

1

2

1

1

2

3

3

2

c) Now you take three balls, not two. In how many of the outcomes will three of the balls be blue? 29

3

Large numbers

HUndreds

Tens

UniTs

3

8

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

billions

Millions

1

Trillion

…

THoUsands

There are numbers with more than nine figures. We can use the decimal number system to write a number as large as we want. The table below shows the place value of some numbers with more than 9 figures:

The Universe formed thirteen billion eight hundred million years ago.

A young person’s brain contains around one hundred billion neurons.

The volume of Earth is approximately one trillion cubic kilometres.

Remember

A billion can be expressed with the prefix giga: 1 000 000 000 bytes = 1 gigabyte

• One million ↔ A 1 followed by 6 zeros. • One billion ↔ A 1 followed by 9 zeros. • One trillion ↔ A 1 followed by 12 zeros. The English words billions, trillions, quadrillions are false friends. They have a different meaning in Spanish and in English.

A trillon can be expressed with the prefix tera: 1 000 000 000 000 bytes = 1 terabyte

Let’s practise! 1 Write the following numbers in words.

a) Population of the Earth (8 000 000 000 people). b) The numbers of seconds in a century (3 153 600 000). c) The number of kilometres in a light year (9 460 800 000 000). 2 Write the following numbers in figures.

a) Twenty-eight million three hundred and fifty thousand. b) One hundred and forty-three million. c) Two thousand seven hundred million. d) Sixteen gigas. e) One and a half trillion. 30

3 Copy and complete in your notebook.

a) One thousand thousand = one... b) One thousand million = one... c) One million million = one... 4 The human body has between ten and seventy

million million cells. Write both figures in trillions.

5 What is the word for a 1 followed by 16 zeros? 6 Scientists estimate that there are 1 330 quadrillion

cubic metres of water in our seas and oceans. What do you think a quadrillion is?

4

Rounding natural numbers

There are 1 046 992 electric vehicles in Spain today.

There are approximately 1 000 000 electric vehicles in Spain today.

=

When a number has a lot of figures, it is difficult to remember. It is also difficult to do calculations with it. This is why we sometimes change it for an approximate value ending in zeros. Rounding is the most frequent and easy method of approximation. To round a number to a specific place value: • We change all the digits to the right of that place value to zeros. • If the first digit we are changing is a five or more we round the next figure up. This means we add a unit to that digit.

Theory into practice

4 Look at the news headline below. Round the number

1 Look at the diagram below. Round the number

384 523 to the nearest hundreds of thousands; to the nearest tens of thousands; and to the nearest thousands. Hundred of thousands

Tens of thousands

3 83 348 854 425 532 23 3 +1 +1+1

8 ≥ 85 ≥8 5≥ 5 cm

3 83 348 854 425 532 23 3 = = =

... ...0...00 00 00 00 00 0

dm4 < 45 <45< 5

... ...0... 00 00 00 00 0

Thousands 3 83 348 854 425 532 23 3 +1 +1+1 um

5 ≥ 55 ≥5 5≥ 5

of tourists to the nearest million. Then round the amount they spent to the nearest thousand millions.

In 2022 71 600 000 tourists visited Spain. They spent 87 061 million Euros.

... ......0 00 00 00 0

Help If we round the number 52 722: – to the nearest tens of thousands → 50 000 – to the nearest thousands → 53 000

5 Round the following numbers to the nearest millions.

a) 24 356 000

b) 36 905 000

6 The sign below shows the price of a house in Euros. FOR SALE €138 290

Let’s practise! 2 Round the following numbers to the nearest thousands.

a) 24 963

b) 7 280

c) 40 274

d) 99 834

3 Round these numbers to the nearest hundreds and to

the nearest tens of thousands. a) 530 298 b) 828 502 c) 359 481

d) 29 935 236

c) 274 825 048 €138 000 €138 300 €140 000

a) Which of the three approximations is closest to the real value? b) Which approximation would you use in an informal conversation? 7 The town hall has a budget of € 149 637 to renovate

the sports centre. Which approximate number would you use to tell a friend about this? 31

5

Basic operations with natural numbers

Addition and its properties Addition means to find the total value of a set of numbers. Look at the picture on the right. To find the total number of people at the football stadium, we add the numbers together: 11 576 + 9 006 = 20 582. These are two properties of addition:

SEATING CAPACITY: 25 342 seats Seats filled: East stands: 11 576 West stands: 9 006

Properties Commutative property

You can change the order of the summands. The sum does not change. a+b=b+a 34 + 16 = 16 + 34 50

Associative property

The way you group the values does not affect the result. (a + b) + c = a + (b + c) (18 + 3) + 17 = 18 + (3 + 17)

50

21 + 17

18 + 20

38

38

Subtraction • Subtraction means to ‘take’ one number from another number. You calculate what is left (the difference). • For example, we want to count the empty seats at the football stadium above. We subtract the number of occupied seats from the total number of seats. 25 342 ← Minuend (M ) – 20 582 ← Subtrahend (S ) 4 760 ← Difference (D )

25 342 – 20 582 = 4 760 ↓ 25 342 = 20 582 + 4 760

M =S + D Relationship between addition and subtraction: M – S = D → * ⎯⎯⎯⎯→ S=M –D 20 582 = 25 342 – 4 760 Let’s practise!

3 Transform.

a) This addition into a subtraction: 48 + 12 = 60

1 Calculate.

a) 254 + 78 + 136 c) 1 526 – 831 + 63

b) 340 + 255 – 429 d) 1 350 – 1 107 – 58

2 Estimate and then check the answer.

Carmen bought a bag for €167, a coat for €235 and a scarf for €32. How much did she spend in total? a) She spent around €350. b) She spent around €450. c) She spent around €550. 32

b) This subtraction into an addition: 22 – 2 – 6 = 14 4 If Alberto was 15 years older, he would still be 18 years

younger than his uncle Tomás. His uncle Tomás is 51 years old. How old is Alberto?

5 If I buy a washing machine, I will have €246 left in

the bank. If I want to buy a TV too I will need another €204. Is it possible to know the price of either item? And, if so, what is the price?

Multiplication If a ticket to the football game from the previous page costs €35, the total price for all 20 582 tickets is: 35 + 35 + 35 + … + 35 = 35 · 20 582 = €720 370 20 582 times

A multiplication is a repeated addition of the same value. It has got three properties: Properties Commutative property

Associative property

The product does not change if we change the order of the factors.

Distributive property

The way we group the factors does not affect the product.

Multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.

a·b=b·a

(a · b) · c = a · (b · c)

a · (b + c) = a · b + a · c

4·5=5·4

(3 · 5) · 2 = 3 · (5 · 2)

35 · (7 + 3) = 35 · 7 + 35 · 3

20 = 20

15 · 2

3 · 10

35 · 10

245 + 105

30

30

350

350

6 Complete the following multiplications in your

notebook.

× 2

5

+ 9 0 1 2 6 0

+

9 8 × 2 8 7 4 2 9 9 3 4

7 Remember that to multiply by 10, 100, 1 000… you

just have to add one, two, three… zeros to the end of the number. a) 19 · 10 b) 12 · 100 c) 15 · 1 000 d) 140 · 10 e) 230 · 100 f ) 460 · 1 000

8 Write a mathematical expression:

To multiply a number by 8, we can multiply it by 10, then subtract double the original number. Which property describes this?

9 Use mental arithmetic to multiply by 9 and 11. Use the

examples to help you.

• 23 · 9 = 23 · 10 – 23 = 230 – 23 = 207 • 23 · 11 = 23 · 10 + 23 = 230 + 23 = 253 a) 12 · 9 b) 25 · 9 c) 33 · 9 d) 12 · 11 e) 25 · 11 f ) 33 · 11 10 A wheel turns at 1 500 revolutions a minute. How

many times does it turn in fifteen minutes? How many times does it turn in an hour? How many times does it turn in an hour and a half?

11 A farmer has 200 peach trees. Each tree produces

about 7 boxes of peaches. Each box contains 5 kilos of peaches. The farmer sells them at €2 a kilo. How much money do they make? 33

Basic operations with natural numbers

5

Division Two possible interpretations of what division means: • Division means distributing a value into equal parts. For example, to know each person’s share, in a group of people. We use 5 625 cubic metres of water to water a park for 15 days. How many cubic metres do we use each day? 5625 112 075 00

15 375

⎯→

5 625 : 15 = 375 m3 each day Dividend Divisor Quotient

• Division means splitting a whole thing into equal portions of a specific size. Then we know how many portions there are. We use 375 cubic metres of water to water the park each day. There are 5 625 cubic metres of water in a tank. For how many days can we water the park? 5 6 2 5 375 1 8 7 5 15 000

⎯→

5 625 : 375 = 15 days

Types of division There are two types of division depending on the value of the remainder: exact division and integer division.

Exact division

We put 35 kg of oranges into 5 kg boxes. We fill 7 boxes and there are no remaining oranges. Remainder

35 0

5 7

Integer division

We put 38 kg of apples into 5 kg boxes. We fill 7 boxes and there are 3 kg remaining. Remainder

38 3

5 7

There are two types of division depending on the value of the remainder: • Exact division (the remainder is zero). d D ⎯→ The dividend is equal to the divisor 0 q multiplied by the quotient. D=d·q • Integer division (the remainder is not zero). d D ⎯→ The dividend is equal to the divisor r q multiplied by the quotient, plus the remainder. D=d·q+r 34

A property of division To water 3 plants, we use 24 litres of water. What happens if we have double the amount of both? 24 litres

48 litres

24 3 0 8

48 6 0 8

If we double the amount of water and the number of plants, the amount of water each plant receives does not change. In a division, if we multiply the dividend and divisor by the same number, the quotient does not change.

×7

8 4

32 0

×7

224 56 00 4

Let’s practise!

16 Find the missing value in each division.

Dividend 39

12 Find the quotient and remainder of each division.

a) 96 : 13 d) 7 029 : 26

b) 713 : 31 e) 49 896 : 162

c) 5 309 : 7 f ) 80 391 : 629

:3

a) 60 : 12 d) 75 : 15 g) 180 : 30

32

b) 180 : 12 e) 90 : 15 h) 240 : 30

8 :4

c) 300 : 12 f ) 180 : 15 i) 390 : 30

14 Copy and complete the diagrams below in your

notebook. (36 : 12) : 3 :

18 A football club pays €1 470 for football shirts for

35 players. a) How much did each shirt cost? b) There are boxes in the equipment room for the shirts. The players put six shirts in each box. How many boxes are there?

36 : (12 : 3) : 19

What did you notice? 15 Solve the following divisions and compare your results.

Then, answer the question. a) (50 : 10) : 5 50 : (10 : 5) b) (36 : 6) : 2 36 : (6 : 2) Does the associative property apply to division?

Divisor 38

a) We divide 150 grams of salami between three sandwiches. How many grams are there in each sandwich? b) How many minutes are there in 180 seconds? c) A car travels 240 kilometres in three hours. How many kilometres did it travel each hour? d) We put 250 kg of apples in 10 kg boxes. How many boxes are there?

these numbers.

: 12

1 000 12

17 Solve the following problems without making notes.

13 Follow the example. Use mental arithmetic to divide

• 96

53 15

Take Action. Raúl has got €65. He is saving to buy a skateboard that costs €105. a) He saves €5 a week. In how many weeks can he buy the skateboard? What if he saves €6? b) He wants to buy the skateboard in 4 weeks. How much must he save, a week? c) He decides to save €13 euros a week. Can he buy the skateboard in 3 weeks? 35

6

Expressions with combined operations

Order of operations to solve an expression When solving expressions with combined operations, you must remember the rules of mathematical notation. These rules help us to make sure that each expression has a unique meaning and solution. The order of combined operations is always: Parentheses or brackets Multiplications and divisions, from left to right Additions and subtractions

48 : 3 + 5 – 2 · 3 = = 16 + 5 – 6 = 21 – 6 = 15

48 : (3 + 5) – 2 · 3 = = 48 : 8 – 6 = 6 – 6 = 0

48 : 3 + (5 – 2) · 3 = = 16 + 3 · 3 = 16 + 9 = 25

48 : 3 + 5 – 2 · 3

48 : (3 + 5) – 2 · 3

48 : 3 + (5 – 2) · 3

16 + 5 – 6

48 : 8 – 6

16 + 3 · 3

21 – 6

6–6

16 + 9

15

0

25

How to use a calculator It may seem strange, but different calculators can give you a different answer. Here you can get 20 or 14. 2+3 *4 =

{∫“≠}

{∫‘¢}

The calculator performs each operation in the order you enter it.

The calculator performs the multiplication first. It follows the correct order of operations.

(2 + 3) · 4 = 5 · 4 = 20

2 + 3 · 4 = 2 + 12 = 14

As you can see, not all calculators work in the same way. Find out which method your calculator uses. Remember this when you use it. Theory into practice 1 Complete each box in your notebook. Check that you have the right answers.

4 · 10 – 8 · 3 + 2

4 · 10 – (8 · 3 + 2)

4 · 10 – 8 · (3 + 2)

·3+2

– (

– 8 ·

+2

+2

–

–

18

26

14

0

–

36

4 · (10 – 8) · 3 + 2

+2

4·

+ 2)

4 · (10 – 8) · (3 + 2) 4 · 4

· · 40

2 Copy and complete in your notebook. Check your

answers with a calculator. Make sure you enter the operations in the correct order.

6 Calculate. Write down the steps you followed. Check

your answers on the right. If they do not coincide, do them again.

Help

≤ → Add screen value to memory. µ → Subtract screen value from memory. Ñ → Recover memory value. 40 – 12 : 4 + 2 · 3 40 –

+ +

a) 6 · 4 – 2 · (12 – 7)

→ 14

b) 3 · 8 – 8 : 4 – 4 · 5

→ 2

c) 21 : (3 + 4) + 6

→ 9

d) 26 – 5 · (2 + 3) + 6

→ 7

e) (14 + 12) : 2 – 4 · 3

→ 1

f ) 2 · (6 + 4) – 3 · (5 – 2)

→ 11

g) 30 – 6 · (13 – 4 · 2)

→ 0

h) 3 · [13 – 3 · (5 – 2)]

→ 12

7 Problem solved

: 4 +

This month, a man worked 7 hours a day for 12 days. He got the standard pay on those days. On another 5 days he worked for 9 hours. For 6 of those hours he got the standard pay. For the other 3 hours he got night pay. How many hours did he work in total this month?

+

We can solve this by writing it in two different ways:

40 ≤ 12 / 4 µ 2 * 3 ≤Ñ= {∫∫∫∫∫∫¢«} (40 – 12) : 4 + 2 · 3

normal rate

12 · 7 + 5 · 6 + 5 · 3 = 84 + 30 + 15 = 129

40 - 12 =/ 4 ≤ 2 * 3 ≤Ñ= {∫∫∫∫∫∫‘«}

12 days

Let’s practise!

a) 8 + 5 · 2 d) (15 – 3) : 4

b) 15 – 10 : 5 e) (8 + 2) · 3

8

c) 4 · 6 – 13 f ) 18 : (10 – 4))

4 Calculate the following using mental arithmetic.

Compare your results. a) 2 + 3 · 4 b) 6 – 2 · 3 c) 18 – 10 : 2 d) 24 : 6 + 2

(2 + 3) · 4 (6 – 2) · 3 (18 – 10) : 2 24 : (6 + 2)

5 Follow the example to calculate:

• 4 · (7 – 5) – 3 = 4 · 2 – 3 = 8 – 3 = 5 a) 2 · (7 – 3) – 5 c) 4 + (7 – 5) · 3 e) 8 – (9 + 6) : 3

b) 3 · (10 – 7) + 4 d) 18 – 4 · (5 – 2) f ) 22 : (7 + 4) + 3

5 days

12 · 7 + 5 · (6 + 3) = 84 + 5 · 9 = 84 + 45 = 129 Answer: In total, he worked 129 hours.

3 Follow the examples to solve the following operations.

• 12 – 2 · 4 = 12 – 8 = 4 • (17 – 5) : 3 = 12 : 3 = 4

night pay

Write the following statements as mathematical expressions and solve them. a) There are 8 boxes of bananas, 20 boxes of oranges and 6 boxes of apples in a fruit shop. Each box of bananas weighs 15 kg. Each box of oranges and each box of apples weighs 8 kg. How many kilograms of fruit are there in the shop? b) A supermarket orders 20 boxes of orange juice, 15 boxes of apple juice and 10 boxes of carrot juice. Each box contains 6 1-litre bottles. How many bottles did the supermarket order? c) There are 15 tables, 55 chairs and 12 bar stools in a cafe. How many legs are there in total? (each bar stool has 3 legs) d) We put 1 500 eggs into boxes of 10 eggs. We put another 1 500 eggs into boxes of 6 eggs. Then we put 300 brown eggs into boxes of 6 eggs. How many boxes are there? 37

My visual summary Numeral systems

A numeral system is a set of symbols and rules we use to represent numbers. Some examples of numeral systems are: Mayan

Egyptian

It is partly an additive system and partly a positional system. Each symbol has a different value depending on its level.

It is an additive system. This means they add the necessary symbols to get the number they want. 30 000 3 000 1 333 331

There are three key symbols: (5)

(1)

(0)

For numbers greater than 20:

1 000 000

300 000

300 30

2nd level (× 20)

3 × 20

1st level (× 1)

0 ×1

+

60

1

Decimal M

It is a positional system. In 4 the example, the figure 4 has a ↓ different value depending on what position it is in. 4 000 000 U

H TH

T TH

TH

H

T

U

7

8

4

3

0

4

↓

↓

4 000 U

4U

Large numbers

We can use the decimal number system to write a number as large as we want.

38

1 000 000

1 000 000 000

1 000 000 000 000

One million A 1 followed by 6 zeros.

One billion A 1 followed by 9 zeros.

One trillion A 1 followed by 12 zeros.

Rounding natural numbers

To round a number to a specific place value: - We change all the digits to the right of that place value to zeros. - If the first digit we are changing is a 5 or more, we round the next figure up. This means we add a unit to the digit. Hundred thousands 3 8 4 5 2 3 +1

=

8≥5

Ten thousands

Thousands

3 8 4 5 2 3

3 8 4 5 2 3

4<5

+1

3 8 0 0 0 0

4 0 0 0 0 0

5≥5

3 8 5 0 0 0

Basic operations with natural numbers

Addition means to find the total value of a set of numbers.

(576 + 906) + 427 = 576 + (906 + 427) 576 + 1333

1909

Subtraction means to ‘take’ one number from another number or to find the difference.

576 + 906 = 906 + 576 1482

Associative property 1482 + 427

Commutative property

1482

Remember Relationships between addition and subtraction:

M =S +D M–S=D →* S=M –D

1909

Associative property

8 342

← Minuend (M )

– 1 909

← Subtrahend (S )

6 433

← Difference (D )

Dividing means distributing a value into equal parts.

(16 × 55) × 3 = 16 × (55 × 3) Multiplication means . value same the g addin repeatedly 880 × 3 = 16 × 165

Distributive property

2640 = 2640

35 × 7 + 35 × 3 = 35 × (7 + 3)

Commutative property

245 + 105

=

35 × 10

16 × 55 = 55 × 16

350

=

350

880 = 880

Integer division The dividend is equal to the divisor multiplied by the quotient plus the remainder D=d.q+r

Exact division The dividend is equal to the divisor multiplied by the quotient (the remainder is 0) D=d.q

Calculations with combined operations

The order of combined operations must always be: First

Then

Finally

Brackets

Multiplication and division from left to right

Addition and subtraction from left to right 39

Exercises and problems DO YOU KNOW THE BASICS?

What can numbers tell us?

Numeral systems

8

1

Translate the following Egyptian numbers into the decimal numeral system. A

Here are some hotel room numbers: 401; 235; 724; 231. a) Which one is at the end of the corridor?

B

b) Which one is on the top floor? c) Which ones are on the same floor? D

C

9

Look at these car number plates: E

3948 FBG

E

3894 FBG

E

4389 GFB

a) Which plate is the oldest? And the newest? 2

3

4 5

b) Which plate number comes directly after the red one? Which plate number comes directly before?

Write the following numbers using the Egyptian additive system. a) 48 b) 35 c) 2 130 Write the following in Roman numerals. a) 87 b) 425 c) 2 600 d) 54 528 Write the number ‘fifty-seven’ using at least three different numeral systems.

c) How many plates are there between the red and the green ones? d) How many cars have the same letters as the blue plate after it? Addition and substraction 10

11

How many figures are there in a trillion? And in a in quintillion? How many zeros are there in each number?

Rounding 6

Copy and complete the table in your notebook.

12

Rounding Number

To the nearest hundreds of thousands

c) 831 – 392 – 76

d) 1 648 – 725 – 263

Mental arithmetic. a) 5 + 7 – 3 – 4

b) 18 – 4 – 5 – 6

c) 10 – 6 + 3 – 7

d) 8 + 5 – 4 – 3 – 5

e) 12 + 13 + 8 – 23

f ) 40 – 18 – 12 – 6

Calculate.

c) 128 – (86 – 45 – 12) d) 237 – (152 + 48 – 14) e) 348 – (148 – 86 + 29)

399 675 000

40

b) 651 + 283 – 459

b) 52 – (36 – 27)

19 270 000

You see an advert for a house that costs €293 528. You want to tell a friend, but you can’t remember the exact price. Which of the following sentences would you say? Why? a) It is almost three hundred thousand euros. b) It is just over two hundred thousand euros. c) It is two hundred and ninety thousand euros.

a) 6 070 + 893 + 527

a) 47 – (35 – 28)

To the nearest million

2 830 554

7

Calculate.

f ) 235 – (340 – 152 – 84) Multiplication and division 13

Multiply. a) 16 · 10

b) 128 · 10

c) 60 · 10

d) 17 · 100

e) 85 · 100

f ) 120 · 100

g) 22 · 1 000

h) 134 · 1 000

i) 140 · 1 000

14

Calculate the quotient and remainder. a) 2 647 : 8 b) 1 345 : 29 c) 9 045 : 45 d) 7 482 : 174 e) 7 971 : 2 657 f ) 27 178 : 254

15

TRAINING AND PRACTICE 21

True or false? a) One million is equal to one thousand hundreds. b) One hundred million is equal to one thousand hundreds of thousands. c) One thousand times a million is equal to one giga. d) One hundred gigas are equal to one trillion. e) One trillion is equal to one million millions.

22

Target 11.c. The population of Cairo was 19 487 245 in June 2018. What is the approximate population of Cairo? If the population of Cairo keeps growing, what will the population be by 2030? What measures can Cairo take to become a sustainable city by 2030?

23

The table below contains data on vegetable consumption in Spain in 2016:

Copy and complete in your notebook. 8

5 6

3 6

14

8 2 9

5 7 6

16

Mental arithmetic. a) 3 · (10 : 5) b) (4 · 6) : 8 d) (30 : 5) · 3 e) 10 : (40 : 8)

17

Mental arithmetic. a) One bucket holds 5 litres of water. How many buckets hold 100 litres of water? b) One kilo of almonds costs €12. How much do 5 kilos cost? c) There are 24 cans of soft drinks in a box. How many cans are there in 10 boxes? d) It costs €360 to get four new tyres for your car. How much does each tyre cost?

c) 20 : (2 · 5) f )(40 : 8) : 5

Combined operations 18

19

20

Weight (tonnes)

Value (thousands of )

Fresh fruit

4 369 449

6 195 054

Vegetables and potatoes

3 626 510

5 214 031

Total

7 995 959

11 409 085

Copy the table in your notebook. Round the figures to the nearest million tonnes and to the nearest hundred million Euros.

Calculate. a) 8 + 7 – 3 · 4 c) 15 – 2 · 3 – 5 e) 22 – 6 · 3 + 5 g) 36 – 8 · 4 – 1

b) 8 : 4 + 7 – 3 d) 10 – 12 : 6 – 4 f ) 8 + 10 : 5 – 10 h) 11 – 2 – 9 : 3

24

Star A is five light years away. Star B is five trillion kilometres away. Which star is the furthest?

Calculate. a) 2 · (4 + 6) c) 8 : (7 – 5) e) (5 + 6) · 4 g) (19 – 7) : 2

25

b) 2 · 4 + 6 d) 5 · 7 – 5 f) 5 + 6 : 3 h) 18 – 7 · 2

Copy, calculate and complete in your notebook. a) 48 + … = 163 b) … + 256 = 359 c) 628 – … = 199 d) … – 284 = 196

26

Calculate. a) 5 – [7 – (2 + 3)] b) 3 + [8 – (4 + 3)] c) 2 + [6 + (13 – 7)] d) 7 – [12 – (2 + 5)] e) 20 – [15 – (11 – 9)] f ) 15 – [17 – (8 + 4)] Check your answers: a) 3; b) 4; c) 14; d) 2; e) 7; f ) 10

27

Copy and complete in your notebook. a) 123 · … = 5 904 b) … · 86 = 1 548 c) … : 57 = 26 d) 1 862 : … = 133

Calculate. a) 30 – 4 · (5 + 2) b) 5 + 3 · (8 – 6) c) 5 · (11 – 3) + 7 d) 3 · (2 + 5) – 13 e) 2 · (7 + 5) – 3 · (9 – 4) f ) 4 · (7 – 5) + 3 · (9 – 7) g) 3 · 5 – 3 · (10 – 4 · 2) h) 2 · 3 + 5 · (13 – 4 · 3) Check your answers: a) 2; b) 11; c) 47; d) 8; e) 9; f ) 14; g) 9; h) 11

41

Exercises and problems 28

Calculate the following using mental arithmetic. Remember that dividing by 5 is the same as dividing by 10, and then multiplying by 2. :5

• 90 : 10

a) 60 : 5 d) 140 : 5 g) 210 : 5 29

9

b) 80 : 5 e) 170 : 5 h) 340 : 5

c) 120 : 5 f ) 200 : 5 i) 420 : 5

Copy and complete in your notebook. a) 6 · (8 + 2) = 6 · 8 + 6 · 2 = … · … = … b) 5 · 9 – 5 · 6 = 5 · ( ... – ...) = 5 · … = … c) (10 – 8) · 4 = 10 · 4 – … · 4 = 40 – … = … d) 7 · 12 – 2 · 12 = (… – …) · 12 = … · … = … Which property did you use?

31

True or false? a) We get the same results by multiplying a number by three and by doubling the number and adding it to the original number. b) Three times fifteen is the same as fifteen times three. c) Multiplying by ten is the same as multiplying by five twice. d) Multiplying by ten is the same as multiplying by five, then by two. e) The commutative property only applies to even numbers.

42

B

·2

Investigate: In division, if we multiply the dividend and divisor by the same number, the quotient remains the same. What happens to the remainder?

33

Use the numbers 9, 3 and 1 in an operation to produce the values shown on the scales: A

18

30

32

34

Calculate. a) 4 · 7 – 13 – 2 · 6 c) 5 · 4 + 12 – 6 · 4 e) 5 · 6 – 4 · 7 + 2 · 5 g) 8 · 8 – 4 · 6 – 5 · 8

b) 15 : 3 + 7 + 4 : 2 d) 12 : 4 – 1 – 6 : 3 f) 9 : 3 + 8 : 4 – 7 : 7 h) 18 : 2 – 12 : 3 – 6 : 2

Calculate. a) 7 · 4 + 2 · (10 – 6) c) 5 · (11 – 7) – 6 e) 3 · (8 – 5) – 12 · (6 – 4) g) 9 : (6 – 3) + 8 · (6 – 5) i) [(24 : (9 – 2 – 3)] : 2

b) 8 : 2 + 4 : (9 – 5) d) 12 : (11 – 7) – 2 f) 15 : (2 + 3) + 5 · (2 + 2) h) 6 · (9 – 5) – 18 : (9 – 7) j) 24 : [(9 – 2 – 3) : 2]

INTERPRET, DESCRIBE AND EXPRESS 35

Match each statement to two of the mathematical expressions below: I. There are 50 people on a bus. 16 people get off the bus at the first stop and 4 people get on. II. There are 50 students in a music class. Today, 4 are absent without an excuse. Another 16 are absent because they are at a concert. III. Ernest buys a T-shirt for €16 and a hat for €4. He pays with a €50 note. IV. Last night, 50 clients stayed at a hotel. This morning, 16 more people arrived and 4 people left. a) 50 – 16 – 4 b) 50 – 16 + 4 c) 50 – (16 + 4) d) 50 – (16 – 4) e) 50 + (16 – 4) f ) 50 + 16 – 4

36

Which operation or operations solve the following problem? This morning, a supermarket sold 24 kg of apples for €2/kg, 12 melons for €4 each and 13 pineapples for €2 each. How much profit did it make from selling all the fruit? a) 24 · 12 + 4 · 13 + 2 b) 24 · 2 + 12 · 4 + 13 · 2 c) (24 + 13) · 2 + 12 · 4 d) (24 + 13 + 2) · (2 + 4)

37

Read the problem and look at the resolution. Can you explain what each operation shows? There are horses, cows and chickens at a farm. In total, there are 714 legs, 168 horns and 137 beaks. How many horses are there at the farm? Answer: 1st 168 : 2 = 84 2nd 84 · 4 = 336 3rd 137 · 2 = 274 4th 336 + 274 = 610 5th 714 – 610 = 104 6th 104 : 4 = 26

SOLVE SIMPLE PROBLEMS 38

49

The local tourist industry employs 12 845 people. Three in every five of them are women. How many women work in the tourist industry?

Builders took 14 months to build house A. They began 4 months after starting to work on house B. House B took them 15 months to build. They finished building house A in June. When did they finish building house B?

50

39

Last week, María sent 40 text messages. She sent five to her brother, Pepe. She sent her parents three more messages than she sent to Pepe. She sent the rest to her best friend. How many messages did she send to her best friend?

A farmer has 140 peach trees in her field. She gets 35 kg of peaches from each tree. She sells them in boxes of 10 kg and she gets €12 for each box. What will be her profit?

40

On a farm, there are double the number of cows as horses. In total, there are 36 heads. How many cows are there? How many horses?

41

A van transports 15 boxes of orange juice and 12 boxes of apple juice. Each box contains 24 bottles. How many bottles are there in total?

42

Rosa is two years older than her brother Julián. She is two years younger than her brother Alberto. Their mother is 42 years old. The ages of the three children also add up to 42. How old is each child?

43

A train carrying goods travels at 55 km/h. On the track next to it a passenger train crosses it in the opposite direction at 105 km/h. What is the distance between the two trains after half an hour?

44

A car and a motorbike leave a cafe at the same time in the same direction. The car travels at 90 km/h and the motorbike travels at 100 km/h. What is the distance between them after an hour and a half?

45

A jam factory makes 250 kg of jam. They put the jam into 200 g jars. During the process, 17 of the jars are damaged. How many jars can they sell?

46

A gardener has 50 trays for planting seeds. He plants 100 seeds in each tray. However, 20 seeds in each tray are damaged. How many plants will the gardener obtain?

47

Clara gets €28 to deliver 7 boxes of publicity. How much does she get if she delivers an extra box?

48

There are 7 boxes of cans on a supermarket shelf. There are also 4 cans not in the boxes. Each box contains 6 cans. An worker puts 12 more boxes of cans on the shelf. How many cans are there in total?

Problem solved

Take Action. Making the meaning of each step, each operation and each result clear.

– Kilos (140 trees at 35 kg a tree): 140 · 35 = 4 900 kg – Boxes (4 900 kilos in 10 kg boxes): 4 900 : 10 = 490 boxes – Profit (490 boxes at €12 a box): 490 · 12 = €5 880 Answer: Her profit is €5 880. 51

The Smith family live in California. Jonathan earns $1 940 a month. He earns $720 dollars more than his son John. He earns $880 more than his daughter Cathy. He earns $280 less than his wife, Catherine. How much money does the Smith family earn each month in total?

52

Take Action. A hockey club has €4 000 in the bank. They buy two goal posts, 18 sticks and 18 pairs of skates. This is what they pay: €60

€129

€140

How much money do they have in the bank now? 43

Exercises and problems 53

54

55

A supermarket pays €2 000 for 150 bags of potatoes. Each bag contains 30 kg of potatoes. A shop assistant sees that 300 kg of the potatoes are damaged. He puts the rest into 5 kg bags, and sells each bag for €4. What is the supermarket’s profit? Every day, a baker bakes five trays of muffins. Each tray contains three dozen muffins. The bakery is closed on Mondays. How many muffins does she bake in a week? A tanker has to transport 3 000 litres of oil to a school 40 km away. A pump at the refinery fills and empties the tank. The pump moves 150 litres per minute. The lorry travels at an average speed of 80 km/h and stops for a quarter of an hour on the way. How long does it take to empty the oil and return to the refinery?

61

A car manufacturer makes 15 660 cars between January, February and March. On average, how many cars does it make each day?

62

There are 450 students at a school. Two in every five students study a second language. One in three of these students studies German. How many students study a second language? How many study German?

63

There are five starters, three main courses and two desserts on a menu. How many different combinations of meal are there?

64

It is possible to form four different numbers of three figures only using zeros and ones. 1stª 1.

56

The Real Alcázar of Sevilla got an average of 144 404 visitors a month in 2022. This is more than double the number of visitors in 2021. a) How many thousands of visitors were there in 2022? b) How many visitors were there on average, a day?

58

65

59

A car travels 2 km in 78 seconds. The speed limit is 90 km/h. Does the driver exceed the speed limit? Why?

60

Take Action. Marta has saved €162. She wants to buy a skateboard that costs €199. If she saves €10 each week, in how many weeks can she buy the skateboard?

44

Antonio, Beatriz, Cora and David go to the cinema. They have four seats next to each other. In how many different ways can they sit? First, let’s solve an easier problem: In how many different ways can they sit if Antonio sits in the 1st seat?

There are 54 tourists on a group visit to Madrid. They arrive at the airport and take taxis to their hotel. Each taxi can transport four people each. How many taxis do they need? A van transports 27 boxes of drinks. Each box contains 24 bottles. There is a traffic accident and 311 bottles break. Does the van have a minimum of half the number of bottles after the accident?

0

111 110 101 100

How many four figure numbers are there that only contain zeros and ones? And five figure numbers?

c) Approximately how many visitors were there in 2021? 57

1

1

THINK A LITTLE MORE

33.rdª 1 0 1 0

22.ndª

1st

A

2nd

B

B

C

C

D

D

3rd

C

D

B

D

B

C

4th

D

C

D

B

C

B

66

You have a large pile of 50, 20 and 10 cent coins. In how many different ways can you add them together to make 1 Euro? Explain your answer.

67

Marta, Julián and Rosa go shopping. Marta spends €30 more than Julián and €40 less than Rosa. In total, they spend €208. How much does each friend spend?

68

A company that organises events wants to buy 150 dozen roses. A local florist has 40 boxes of 25 roses in stock at the moment. How many more boxes of 25 roses does the florist need?

74

The graph below shows the colour of 30 690 cars produced in three months. How many red cars are there?

Grey

69

Valentina has a farm with ducks and geese. Today, she sold 21 of her animals for €350. She sold double the number of ducks as geese. One goose costs three times the price of a duck. How much does one duck cost? How much does one goose cost?

70

In a garage there are three times as many cars as motorbikes. How many vehicles are there of each type if there are 70 wheels?

71

A woman walks 100 steps a minute. She moves 80 cm with each step. She is going to walk 10 km and she wants to arrive in two hours. Is that possible? Why?

72

A rectangular field measures 150 m × 300 m. A farmer wants to plant trees in it. She plants them in rows, parallel to the fences around the field. There is a distance of 5 metres between each row. The distance between the first row and the fence is also 5 metres. How many trees can she plant?

15

73

15

30

20 20

Can you add up the numbers 1 to 100?

Red

Others

76

A number contains four figures that add up to 4. If you change the place value of the ones and hundreds, the number increases by 99. What is the number? There is more than one possible answer.

77

We know this information about the Year 1 students at a Secondary school: — 44 eat in the cafeteria, 58 take the school bus and 47 participate in extracurricular activities. — 24 students eat in the cafeteria and participate in extracurricular activities. — 23 eat in the cafeteria and take the school bus; 25 take the school bus and participate in extracurricular activities. — 11 students do all three things and 17 students do not do any. How many students are there in total? Draw a Venn diagram like this one to help you.

20

1+ 2 + 3 + 4 + 5 + 6 + 7 8 · 7 = 56 + 7 + 6 + 5 + 4 + 3 + 2 + 14 56 : 2 = 28 8+8+8+8+8+8+8

Blue

Aarón is a travel agent. This year, he sold these holidays. These are the results of a statistic made in a town about where people went on holidays: — 56 % of people went to the beach. — 47 % went to a village. — 23 % went to both destinations. What percentage of people did not go to the beach or to a village?

30

Martina added the first seven natural numbers together as follows:

Green

75

Draw a grid to help you. For example: 20

White

YEAR 1 CAFETERIA

SCHOOL BUS

EXTRA ACT.

78

Four friends weigh themselves in pairs. They do this in all possible combinations and write down the results in a random order: 83 kg - 87 kg - 91 kg - 80 kg - 84 kg - 88 kg The heaviest child weighs 46 kg. How much does each child weigh separately? 45

Self-assessment

GO TO THE ESCAPE ROOM AND TEST YOURSELF!

anayaeducacion.es Answer key.

1 Complete the following table in your notebook:

a) 154 · = 462 b) : 27 = 98 c) 30 275 : = 35 d) 1 508 = · 125 + 8

Numeral systems Egyptian

0 1

2

3

4

5

7

8

9

6

Mayan

10 11 12 13 14

Decimal

15 16 17 18 19

10 Copy and complete in your notebook.

11 You can put two and a half dozen eggs in a box. 528

2 Which of the above systems are additive? Which

systems are positional? What is the difference?

How many boxes do you need for 350 eggs?

12 There are 60 seats in a cafe. There are three times

more chairs than stools. How many chairs are there? How many stools?

3 Look at these amounts:

• The surface area of Brazil is eight million five hundred and fourteen thousand eight hundred and seventy-seven square kilometres. • The population of the world in April of 2018 was 7 601 767 200 people. Express the first in figures and the second in words. 4 a) Round the amounts from activity 3 to the nearest

tens of thousands. b) Round them to the place value you think is best. Write the place value you chose.

5 Calculate.

a) 12 + 3 · 5 – 2 b) 7 · 3 – 4 · 2 + 2 6 Write an expression with the numbers 2, 3 and 5 with

a result of 18. You have to use all three numbers and you can repeat any of them.

7 a) Transform this sum into a subtraction: 18 + 42 = 60

b) Transform this subtraction into a sum: 55 – 45 = 10 8 Do the following combined operations.

a) 19 – 5 · (10 – 7) + 4 · 7 b) 10 · [7 · 5 – (4 + 6 · 3)] 9 Copy and fill in the blanks in your notebook.

= 180 a) 18 · · 100 = 27 000 b) = 40 c) 4 000 : : 10 = 38 d) 46

13 A lorry travelling at 60 km/h passes a car travelling

at 90 km/h in the opposite direction. What is the distance between them after ten minutes?

14 A beekeeper has 187 hives. Each hive produces

approximately 9 kilos of honey every six months. a) She puts the honey in half-kilo jars. How many jars of honey does she produce each year? b) She puts the jars in boxes. There are six jars in each box. She sells each box for €18. What is her annual profit? c) Round the profit to the place value you think is best.

15 Which of the expressions below solves the following

problem? A farmer has two fields of apple trees. One field has 180 trees and the other has 170 trees. Each tree produces an average of 35 kg of apples. She puts the apples into 10 kg boxes. How many boxes does she need? a) (180 + 70 + 35) · 10 b) (35 · 180 + 35 · 70) · 10 c) (35 · 180 + 35 · 70) : 10

I TAKE ACTIO

N

Vicky's skateboard

LEARNING EXPERIENCE

€165

Now you are going to plan a budget to buy something. It can be real or imaginary. For example, a trip to a another country. You can use the example below to help you. Vicky loves to skateboard and she wants to buy a new one that costs €165. She only has €96 in the piggy bank. She analyses her finances to plan the purchase. Income • Weekly allowance ➝ €15. • Vicky’s grandma gives her €20 a month. • Between €0 and €20 every two weeks for helping her neighbours (watering plants, walkings dogs, etc.).

Expenses • Between €5 and €12 a week to go out with her friends. • Between €16 and €20 a month for other expenses (sweets, comics, etc.).

Now analyse Vicky’s finances: 1 Copy and complete the table to calculate her minimum and maximum savings. Income

Expenses

Allowance

Grandma

Neighbours Going out

Other

MINIMUM SAVING/weekly

15

20 : 4 = 5

0

12

20 : 4 = 5 (15 + 5 + 0) – (12 + 5) = 3

MAXIMUM SAVING/weekly

15

20 : 4 =5

20 : 2 = 10

5

16 : 4 = 4 (… + … + …) – (… + …) = …

2 If Vicky saves the minimum, in how many weeks can she buy the skateboard?

She has to save ➝ Cost minus what she has in the piggy bank ➝ 165 . ... = ... In 10 weeks she saves ➝ 10 . 3 = €30 ➝ is that enough? ... In 20 weeks she saves ➝ 20 . ... = €... ➝ is that enough? ... In ................................................................................. If Vicky saves the maximum, how much more money does she need after two weeks? ... 3 Imagine you want to achieve your goal in two months. You don’t want to cut

all your expenses. What income and expenses plan would you adopt?

Maths

in context

Apply your language skills www.anayaeducac ion.es

Now, it's your turn... • Follow the same process. Calculate your finances (savings and income). How long will it take you to buy the item of your choice?

• Alternatively, imagine she wants to buy a helmet and a set of knee and elbow pads. Investigate the costs of these items. Then follow the same process as above. 47

2

POWERS AND ROOTS WATCH THE VIDEO

8. Decent work and economic growth

t ha

ca n y ou

d o

to

W

LEARNING EXPERIENCE

Saving plans.

unit, you are going to try out a saving plan. You’ll use the tools TAKE ACTION Inandthisresources of the mathematical language in this learning experience. ? At the end of the topic there is a guide to help you. But first, think of some of your own ideas.

DO THE QUIZ!

BEFORE STARTING… REVIEW WHAT YOU KNOW

Let’s look at this saving plan from a student! On day one, I put a €1 coin in my piggy bank. On day two, I put €2 in my piggy bank. On day three, €4. On day four, €8. As you see, each day I save double the amount as the day before. 1. How much will go in the piggy bank on day five? How much money will be in there in total? 2. How many days will it be before the student has to put in €100? And €200? 3. Copy and complete.

48

Expression

22

23

24

…

Meaning

2×2

2×2×2

2×2×2×2

…

Value

4

8

…

32

Learning by doing videos

o

t are y ou hao learn g W t to

in

1-minute explanations videos

g

TAKE ACTION ?

POWERS

• Perfect square and cube POWERS OF BASE 10. USES

OPERATING WITH POWERS

• Polynomial factorisation of a number • Product and quotient of powers with the same exponent • Product and quotient of powers with the same base

SQUARE ROOTS

• Powers of another power

• Exact root • Integer root

MY MATHS DICTIONARY Key language

In context

bn to the power of

1 000 000 000 is ten to the power of nine (109).

x 2 squared

72, Seven squared is forty nine.

x 3 cubed

33, Three cubed is twenty seven.

square root of

9, The square root of nine is three.

≠ is not equal to

a ≠ 0, a is not equal to zero.

≈ is approximately

π ≈ 3.1415, Pi is approximately three point one four one five.

49

1

Powers Powers What are powers?

• Powers are a shortered form of writing a product of equal factors. For example: a · a · a · a · a = a5

25

Exponent Base

Squared and cubed powers

• Raising a number to the power of 2 is the same as squaring it: 7 · 7 = 72 → 7 squared is 49. • Raising a number to the power of 3 is the same as cubing it: 7 · 7 · 7 = 73 → 7 cubed is 343.

• We call base to the number that we multiply and the exponent tells us how many times we have to multiply the base by itself. • We say: a to the power of b. In the example above: Two to the power of five or Two to the fifth power. • We calculate powers: a) 73 = 7 · 7 · 7 = 343 b) 104 = 10 · 10 · 10 · 10 = 10 000

Numbers and geometry cube 5 squared is: 52 = 5 · 5 = 25 (25 squares)

5

5

square 5

5 cubed is: 53 = 5 · 5 · 5 = 125 (125 cubes)

5

5

Draw a similar diagram to show 32 and 33. Can you think of a way to represent 34? How to calculate powers Powers usually produce very large numbers. Calculators can help us to solve them quickly. 96 = 9 · 9 · 9 · 9 · 9 · 9 = 81 · 9 · 9 · 9 · 9 = 729 · 9 · 9 · 9 = … = 531 441 a) On a simple calculator, you need the following buttons: * and =. 96 ⎯→ 9 * * = = = = = ⎯→ {∫∫∞«‘¢¢‘} ↓ ↓ ↓ ↓ ↓ 92 93 94 95 96

b) On a scientific calculator, you need this button: ‰. 96 ⎯→ 9 ‰ 6 = ⎯→ {∫∫∞«‘¢¢‘}

note: When a number is too big for the screen, the calculator tells us that there is an error. But, scientific calculators can show the result like this: 458 ⎯→ [VCWHJCGCDGEKÀÍÏ] This tells us that we must multiply the decimal number by ten thirteen times. To do this, we move the decimal point 13 places to the right. 50

x2 x3

Theory into practice 1

9 Use a calculator to calculate the following powers.

a) 311 d) 134

Calculate 75. Complete in your notebook. 75 = 7 · 7 · 7 · 7 · 7 = (7 · 7) · (7 · 7) · 7 = 49 · 49 · 7 = = ·7=…

a) 2x = 64 c) 6z = 36 e) 10n = 10 000

b) 5x = 3 125 → x = …

3 Calculate and complete.

2 · (112 – 92) – 62 = 2 · (121 – = – =…

) – 62 = 2 ·

–

=

12 ↓ 1

52 = 5 · 5 = 25 53 = (5 · 5) · 5 = 25 · 5 = … 54 = (5 · 5) · (5 · 5) = … 7x = 2 401 → What is the value of x? With a

Let’s practise!

Power

Base

Exponent

5

3

m

5

26 a4

b) 33 – 32 d) 92 – (72 + 42) f ) (82 – 72)2 – 2 · 102

Cut out two squares from a piece of graph paper, one 10 × 10 and the other 5 × 5. Are there twice as many (double the number of ) squares in the first square as in the second? Explain your answer.

f ) 204

6 Complete this table in your notebook.

202 ↓ 400

15

5 Read these powers aloud and write them as products.

e) 106

… ↓ …

Take Action. A woman arrives at a hostel. She pays one coin for the night. The manager of the hostel doubles the price of a room on the second night. He doubles the price again on the third night and again on the fourth night. a) How much does the woman pay? Write an expression with powers. b) Calculate the value of this expression.

b) 7 · 7 · 7 d) 3 · 3 · 3 · 3 · 3 · 3 d) 152

32 ↓ 9

14

4 Write the following as a power.

c) 93

22 ↓ 4

a) 82 + 8 c) 53 – 52 + 5 e) (26 – 24)5 – 24

7 * * = = = {∫∫∫∫“¢≠‘}

b) 27

c) a3 = 64 f ) a10 = 1 024

13 Calculate and write the steps you followed.

simple calculator:

a) 34

b) a2 = 25 e) a3 = 1 000

12 Find the square of the first twenty natural numbers.

1 Complete in your notebook.

a) 6 · 6 c) 4 · 4 · 4 · 4

b) 3 y = 81 d) 8m = 512 f ) 30t = 810 000

11 Find the value of the base, a, in these examples.

a) a4 = 16 d) a4 = 2 401

Help

2

c) 95 f ) 2052

10 Find the value of these exponents.

2 Find the value of x.

a) x 3 = 125 → x = …

b) 66 e) 353

16 Use powers to express the number of cubes in the

following polycubes.

BB

7 Calculate using mental arithmetic. Then order the

powers from highest to lowest. b) 52 a) 23 d) 203 e) 104 b) 35 e) 152 h) 304

BB

AA

c) 43 f ) 112

8 Calculate the following in your notebook.

a) 28 d) 94 g) 103

AA

CC

DD

CC

DD

c) 123 f ) 852 i) 1003 51

2

Powers of base 10. Uses

As you already know, to multiply by 10 we simply have to add a zero. Therefore: 102 = 10 · 10 = 100

103 = 10 · 10 · 10 = 1 000

105 = 100 000

109 = 1 000 000 000 9 zeros

Powers of base 10 are the same as the unit followed by the number of zeros in the exponent. Abbreviating large numbers We can abbreviate numbers that end in zeros. We write the product of the number multiplied by a power of base 10. This helps us to write and understand large numbers more easily. 400 000 = 4 · 100 000 = 4 · 105 Example

One light year: 9 460 800 000 000 km. Look at the steps. This is how we make this number easier to read, write and remember: • First, we round to two significant figures → 9 500 000 000 000 • Then, we factorise the number → 95 · 100 000 000 000 • Finally, we abbreviate this to a power of base 10 → 95 · 1011 One light year is equivalent to 95 · 1011 km.

In one gram of oxygen there are 37 638 383 060 000 000 000 000 atoms. 37 638 383 060 000 000 000 000 21 digits

Polynomial factorisation of a number You know about the powers of base 10 and the positional value of digits in a number. Now look at the polynomial factorisation of a number. We transform a bigger number into smaller numbers. 800 000 + 30 000 + 6 000 + 200 + 836 279 =

70

+ 9

In one gram of oxygen there are 38 · 1021 atoms.

8 · 105 + 3 · 104 + 6 · 103 + 2 · 102 + 7 · 10 + 9

Let’s practise! 1 Write the following numbers as powers of base 10.

a) One thousand. c) One thousand million. b) 15 · 109

c) 86 · 1014

3 Find the value of x.

b) 3 601 294 835 ≈ 36 · 10x a) 2 936 428 ≈ 29 · 10 x c) 19 570 000 000 000 ≈ 20 · 10 x 52

following numbers. a) 74 238 c) 4 528 926

b) One million. d) One trillion.

2 Write these numbers with all their figures.

a) 4 · 105

4 Use the polynomial factorisation method for the

5

b) 680 290 d) 46 350 000

Abbreviate the following data. a) In one litre of water there are 334 326 000 000 000 000 000 000 molecules. b) The Alpha Centauri star system is approximately forty trillion kilometres from the Sun.

3

Operating with powers

Product of powers with the same exponent • Look at the following examples. See how they both give the same result. (2 · 3)3 = 63 = 6 · 6 · 6 = 216 23 · 33 = (2 · 2 · 2) · (3 · 3 · 3) = 8 · 27 = 216 23 · 33 = (2 · 2 · 2) · (3 · 3 · 3) = (2 · 3) · (2 · 3) · (2 · 3) = (2 · 3)3 The power of a product is equal to the product of the powers of the factors. (a · b)n = an · bn • Do not get confused: the power of an addition (or subtraction) is not the same as the sum of the powers of the summands. (a + b)n ≠ an + bn (a – b)n ≠ an – bn (2 + 3)4 = 54 = 625 24 + 34 = 16 + 81 = 97 (2 + 3)4 ≠ 24 + 34 Quotient of powers with the same exponent • Here are two more expressions that also give the same result. (6 : 3)3 = 23 = 2 · 2 · 2 = 8 63 : 33 = (6 · 6 · 6) : (3 · 3 · 3) = 216 : 27 = 8 Or: 63 : 33 = (6 · 6 · 6) : (3 · 3 · 3) = (6 : 3) · (6 : 3) · (6 : 3) = (6 : 3)3 The power of a quotient is equal to the quotient of the powers of the dividend and the divisor. (a : b)n = an : bn Theory into practise 1 Look at the examples. Use the same method to complete and solve the expressions in your notebook:

• 56 · 26 = (5 · 2)6 = 106 = 1 000 000 • 123 : 43 = (12 : 4)3 = 33 = 27 • 54 · 44 = (5 · 4)4 = 204 = (2 · 10)4 = 24 · 104 =16 · 10 000 = 160 000 • (66 · 56) : 156 = (6 · 5)6 : 156 = 306 : 156 = (30 : 15)6 = 26 = 64 a) 25 · 55 = (… · …)5 = …5 = … b) 184 : 94 = (… : …)4 = …4 = … c) 63 · 53 = (… · …)3 = …3 = (… · 10)3 = …3 · 103 = … · 1 000 = … d) (85 · 65) : 245 = (… · …)5 : 245 = …5 : 245 = (… : 24)5 = …5 = … e) (363 : 93) · 253 = (… : …)3 · 253 = …3 · 253 = (… · 25)3 = …3 = … f ) (542 : 32) : 22 = (… : …)2 : …2 = … 2 : … 2 = (… : …)2 = … 2 = … 53

3

Operating with powers

Product of powers with the same base • The product of power with the same base is another power of that number. 54 · 53 = (5 · 5 · 5 · 5) · (5 · 5 · 5) = 57

54 · 53 = 54 + 3 = 57

• Notice that the exponent of the final product is the sum of the exponents of the factors. To multiply two powers with the same base, the base stays the same and we add the exponents together. am · an = am + n Quotient of powers with the same base • To calculate the quotient of powers with the same base, remember the relationship between multiplication and division. 54 · 53 = 57

57 : 53 = 54 → 57 : 53 = 57 – 3 = 54 57 : 54 = 53 → 57 : 54 = 57 – 4 = 53

• Notice that the exponent of each quotient is the difference between the exponent of the dividend and the exponent of the divisor. To divide two powers with the same base, the base stays the same and we subtract the exponents: am : an = am – n Powers of another powers When we raise one power to another power, we get a new power with the same base. The new exponent is the product of the exponents in the initial expression. (54)3 = 54 · 54 · 54 = 54 + 4 + 4 = 54 · 3 = 512 To raise one power to another power, the base stays the same and we multiply the exponents. (an)m = an · m Powers with zero as the exponent Look what happens when we divide any power by itself: • Applying the quotient property. 53 : 53 = 53 – 3 = 50 = 1 • Using the regular calculations. 53 : 53 = 125 : 125 = 1 The zero power of a number different from zero is equal to one: a0 = 1 (a ≠ 0)

54

Theory into practice

8 Copy the following problems in your notebook. Fill in

the gaps with either = or ≠.

2 Reduce to just one power. Complete in your notebook.

a) a3 · a5 = a… + … = a…

b) a8 : a5 = a… – … = a

a) (4 + 1)3

43 + 13

b) (4 + 1)3

c) (a2)4 = a… · … = a…

d) (a2)2 = a… · … = a

c) (6 – 2)4

64 – 24

d) 73

(10 – 3)3

f ) 104

52 · 22

e) 102

3 Calculate and reduce to just one power.

52 · 22

g) (12 : 3)2

a) a12 : (a4 · a4) = a12 : a… = a… b) (53)3 : (54 · 53) = 5… : 5… = 5… c) (m10 : m8) · (m5 : m4) = m… · m… = m…

h) 122 : 62

a) 52 · 52

b) 32 · 35

c) 105 · 102

d) a5 · a5

e) m7 · m

f ) x2 · x6

10 Express with just one power.

• x9 : (x3 · x4) = x9 : x7 = x2 (y5)2 : (y2 · y4) = y10 : y6 = y4

a) 26 : 22

b) 38 : 35

(z3 · z5) : (z4 · z2) = z8 : z6 = z2

c) 107 : 106

d) a10 : a6

e) m5 : m

f ) x8 : x4

Let’s practise! 4 Look at the example. Then complete in your

11 Reduce to just one power.

notebook. 2

64

9 Reduce to just one power.

Help • a3 · a3 = a3 + 3 = a6 (a3)3 = a3 · 3 = a9

122 : 32

53

2

• (4 · 3) = 12 = 144 4 " (4 · 3)2 = 4 2 · 3 2 4 2 · 3 2 = 16 · 9 = 144 a) (3 · 5)2 = … 4… 32 · 52 = …

b) (4 · 2)3 = … 4… 43 · 23 = …

c) (12 : 3)2 = … 4… 12 2 : 3 2 = …

d) (20 : 4)3 = … 4… 203 : 43 = …

b) 42 · 52

c) 252 · 42

d) 203 · 53

e) 165 : 85

f ) 183 : 63

g) 214 : 74

h) 352 : 52

i) 1003 : 503

6 Calculate.

b) (25)2

c) (103)3

d) (a5)3

e) (m2)6

f ) (x4)4

12 Reduce.

5 Calculate in the simplest way possible.

a) 53 · 23

a) (52)3

13

a) x · x2 · x3

b) m2 · m4 · m4

c) (k9 : k5) : k3

d) (x5 : x3) : x2

e) m6 : (m8 : m4)

f ) (k2 · k5) : k6

g) (x2)5 : x7

h) m10 : (m3)3

i) (k2)6 : (k3)4

j) (x5 : x3)2

Solve the following expressions with combined operations. a) 62 + 22 – 22 + 5

a) (25 · 35) : 65

b) (64 · 34) : 94

c) (803 : 83) : 53

d) (482 : 22) : 62

b) 24 – 38 : 36 – 22

e) (82 · 122) : (62 · 82)

f ) (33 · 43) : (203 : 53)

c) 10 + (52)3 : (53)2

7 Calculate and note that the results are not the same.

d) (105 : 55) – (22 · 22)

a) (6 + 4)2

b) (5 + 2)3

e) [(8 – 5)2 · (9 – 6)3] : 35

62 + 42

53 + 23

f ) [(7 – 4)3 – (9 – 4)2]4 55

4

Square roots

Square roots and its parts • To calculate the square root we simply do the opposite to squaring a number. Its parts are: Root

— b2 = a ↔ √a = b

Radicand

• We say the square root of a is equal to b. Some examples are: 42 = 16 → 152 = 225 →

16 = 4 → The square root of 16 is 4. 225 = 15 → The square root of 225 is 15.

Types of square roots Remember

Exact roots • The square of natural numbers are perfect squares. The square root of a perfect square is an exact root. 12 22 ↓ ↓ 1 4

32 42 52 ↓ ↓ ↓ 9 16 25

For example, the following square roots are exact roots: 9 =3

121 = 11

400 = 20

Integer roots • The root of most numbers is not an exact amount of whole units. In the example, the square root of 40 is a number between 6 and 7.

12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 72 = 49 82 = 64 92 = 81

102 = 100 112 = 121 122 = 144 132 = 169 142 = 196 152 = 225 162 = 256 172 = … 182 = …

6 2 = 36 < 40 4 → 6 < 40 < 7 7 2 = 49 > 40 • The closest natural number is the integer root. It must be lower than the original number, not higher. In the example above, the root of 40 is 6.

Theory into practise 1 Complete in your notebook. Use the information in

the boxes to help you.

a) 175 ≈ 13 → Integer root. b) 200 … → … c) 225 … → … d) 250 … → … 56

122 = 144 132 = 169 142 = 196 152 = 225 162 = 256

Help • 122 = 144 → 144 = 12 → Exact root. • 150 ≈ 12 → Integer root. • 132 = 169 → 169 = 13 → Exact root. • 230 ≈ 15 → Integer root.

Estimating square roots Example

Estimate 3900. _ 60 2 = 3 600 < 3 900b b b b h h h b b As you can see, 3 900 is higher than 622 and lower ` b 62 2 = 3 844 < 3 900b 2 b b than 63 . 2 b 63 = 3 969 > 3 900b a 3 900 ↓

622

632 3 969

3 844 — √3 900 ↓

62

63

Therefore: 62 < 3 900 < 63 The square root of 3 900 is a number between 62 and 63. 3900 ≈ 62 → The integer root of 3 900 is 62.

Let’s practise! 2 Look at the example. Then copy and complete:

6 Calculate. Use the results in activity 5 to help you.

• 25 = 5 → The square root of 25 is 5.

a) 289

b) 361

c) 484

a) 49 = 7 → …

d) 576

e) 676

f ) 841

b) 64 = … → …

7 Look at the box below. Then, calculate the roots and

c) 81 = … → …

write exact or integer.

d) 121 = … → …

502 = 2 500 532 = 2 809

3 Mental arithmetic.

512 = 2 601 542 = 2 916

522 = 2 704 552 = 3 025

a) 4

b) 9

c) 36

a) 2 550

b) 2 601

c) 2 725

d) 400

e) 900

f ) 3 600

d) 2 815

e) 2 916

f ) 2 929

g) 6 400

h) 8100

i)

a) 90

b) 150

c) 700

d) 1521

e) 6 816

f ) 10 816

10 000

4 Calculate these integer roots.

a) 5

b) 10

c) 24

d) 32

e) 39

f ) 50

g) 68

h) 92

i)

105

8 Estimate.

9 Solve the following expressions.

a) 121 – 100 + 81

5 Write down the perfect squares from 200 to 900 in

b) `4 · 25 – 5 · 9 j: 5

172

182

…

302

c) 4 3 – 2 5 – 5 2 + 7

289

324

…

900

d) (8 – 6) 6 : 4 4 x

your notebook. 162 152 225

256

57

Square roots

4

Square root-finding algorithm Follow these steps to calculate square roots on paper. For example, calculate 105674 : 1 Separate the digits in the radicand. Put them into sets of two-digit numbers

starting from the right. Then, calculate the root of the first pair on the left. _ 10… i. √10 . 56 . 74 3 3 · 3 → –9

6

← A

A = 10 = 3 and 1 remaining.

← B

B: Double the value of A.

1 2 Move the next pair of numbers (56) down. Now, find c so that, 6 c × c

is the closest possible to 156, but lower.

√10 . 56 . 74 3

√10 . 56 . 74 3

–9 ↓↓

–9

6 c × c c =2

1 56

62 × 2 = 124

156

6 2 × 2 = 124 → –124 032 3 Move the value c = 2 up to the answer box. Move the next pair of

numbers (74) down and repeat the process. √10 . 56 . 74 32

√10 . 56 . 74 32

–9

62 × 2 = 124

–9

62 × 2 = 124

156

64 d × d

156

645 × 5 = 3 225

–124 32

–124 d =5

74

• On some calculators, to calculate 105 674 you must press the buttons in this order: 105674 $ → {«“∞…≠|∞………} • On others, you must press: $ 105674 = → {«“∞…≠|∞………}

3274

64 5 × 5 = 3 225 → –3225 0049 4 Move the value

Remember

c = 5 up to the answer box.

Answer: 105 674 = 325 Check: 3252 + 49 = 105 674 Let’s practise! 10 Copy the information below in your notebook. Use

the algorithm to find the roots. √1 1 5 8 4 – 6 × – 2 5 6 0 0

58

√2 7 3 8 5 – 102 × 2 2 3 8 –

11 Calculate by hand. Then check your results with a

calculator. a) 1 444

b) 2 025

c) 2 945

d) 3 974

e) 20164

f ) 126 782

12 Use a calculator to solve.

a) 2 936

b) 10 568

c) 528 471

My visual summary Powers

• Powers are a shortened form of writing a number that is multiplied by itself many times: Exponent

ab = a . a . ... . a

b times

In words, we say: a to the power of b, or a raised to the bth power

Base

• If we raise a number to the power of 2, this is the same as squaring it. Square numbers

22=4

12=1

32=9

42=16

• If we raise a number to the power of 3, this is the same as cubing it. Cubic numbers

13=1

23=8

33=27

43=64

• A power of 10 is the same as 1, followed by the number of zeros in the exponent. 102 =10 . 10 = 100 105 = 100 000

2 zeros 5 zeros

103 = 10 . 10 . 10 = 1 000 109 = 1 000 000 000

3 zeros 9 zeros

Polynomial factorisation of a number

We do this as follows: - Factorising a number based on the place value of its figures. - Powers with a base of 10. Example: 800 000 + 30 000 + 6 000 +

200

+

70

+

9

836 279 = 8 . 105

+ 3 . 104 + 6 . 103 + 2 . 102 + 7 . 10 + 9 59

Calculations with powers

• The power of a product is equal to the product of the powers of its factors.

(a . b)n = an . bn

• The power of a quotient is equal to the quotient of the powers of the dividend and the divisor.

(a : b)n = an : bn

• To multiply two powers with the same base, we keep the same base and add the exponents.

am . an = am+n

• To divide two powers with the same base, we keep the same base and subtract the exponents.

am : an = am-n

• To raise a power to another power, we keep the same base and multiply the exponents.

(an)m = an · m

• Any number (not zero) raised to the power of zero is equal to one.

a0 = 1 (a ≠ 0)

Square roots

• Calculating a square root simply means doing the opposite to squaring a number.

b2 = a

a=b Root

Radicand

In words, we say: the square root of a is equal to b

• The square root of a perfect square is called an exact root.

9=3

121 = 11

400 = 20

• The natural number whose square is closest and less than the original number is called the integer root. The integer root of 40 is 6. 40 ≈ 6

60

Exercises and problems DO YOU KNOW THE BASICS?

Operating with powers

Calculating powers

11

Calculate. b) (5 – 4 + 2 – 1)3 a) 72 – 62 + 52 – 42 c) (10 – 6)2 – (10 – 8)3 d) 34 – (5 – 3)2 – (23)2 e) (13 – 3)2 · (7 + 3)2 + (15 – 5)2 · 10

12

Read and discuss. Write the expression as a mathematical equality or inequility. a) Power of a product. ↔ The product of the powers of the factors. b) Power of an addition. ↔ The sum of the powers of the summands. c) Product of powers with the same base. ↔ The same base raised to the sum of exponents. d) Power of a power. ↔ The same base raised to the product of the exponents. e) Powers with an exponent of zero. ↔ One.

1

Mental arithmetic. b) 63 c) 35

a) 24 2

3

d) 204

e) 300

Copy and complete in your notebook. b) 2 = 4 900 a) 3 = 8 000 d) 4 = 160 000 c) 4 = 10 000 Calculate in your notebook. b) 95 c) 110 d) 153

a) 55 4 5

e) 164

Write down the perfect squares from 1 000 to 1 500 in your notebook. Copy and complete this table in your notebook. a0

a1

a2

a3

a4

a5

3

13

16 1 000 16 1

14

Write these numbers in full. b) 106 c) 1010 d) 1012 a) 102

e) 1016

7

Write as a power of base 10. a) One hundred. b) One hundred millions. c) One hundred trillions. d) One hundred thousand trillions.

15

8

Write these numbers in full. b) 34 · 109 a) 13 · 107

16

9

10

c) 62 · 1011

In a kilometre, there are 103 = 1 000 metres. In a

metre, there are 102 = 100 centimetres. Express the number of centimetres in one kilometre. Use the same format.

Round the numbers of inhabitants to the nearest hundred thousands. Then use a power of base 10 to abbreviate them. casablanca: 5 899 000 paris: 10 858 000 san francisco: 5 929 000 beijing: 21 009 000

Write the correct exponent for each asterisk. b) a5 · a3 = a* d) 26 : 24 = 2* c) m3 · m* = m9 f ) m8 : m* = m6 e) a9 : a8 = a* g) (42)3 = 4* h) (a2)2 = a* j) (x*)2 = x12 i) (m4)* = m12 a) 64 · 63 = 6*

Powers of base 10. Abbreviating large numbers 6

Reduce the following expressions. b) m4 · m2 c) (k2)4 d) x5 · x5 e) (m3)2 f ) k6 : k4

a) x8 : x3

Look at the example. Then reduce to one power. • (m7 : m4) · (m4 : m3) = m3 · m = m4 a) (a7 : a) · a3 b) (x9 : x4) : x3 c) (m2)5 : (m3)2 d) (a5)3 : (a4)3 Problem solved

Reduce to just one power. Then, calculate. 210 : 44 210 : 44 = 210 : (22)4 = 210 : 28 = 22 = 4 17

Write the correct exponent for each asterisk. Then check the results. a) 27 : 16 = 27 : 2* = 2* = 8 b) 81 : 32 = 3* : 32 = 3* = 9 c) 83 : 25 = (2*)3 : 25 = 2* : 25 = 2* = 16 d) 93 : 33 = (3*)3 : 33 = 3* : 33 = 3* = 27 61

Exercises and problems Square roots 18

Estimate these approximate or exact roots. b) 121 c) 1 785 a) 90

19

Calculate the following with a calculator. b) 1024 c) 1369 a) 655 d) 4 225 e) 12 664 f ) 33 856

20

Identify and copy the perfect squares into your notebook. 1 000 1 225 1 600 1 724 1 601 2 464 3 364 3 540 3 773 3 844 4 000 5 625

TRAINING AND PRACTICE 21

Calculate the following using a calculator. 12 b) 510 c) 453 d) 674 e) 993 a) 4

22

Find the exponents of these powers. x b) 10x = 10 000 a) 2 = 256 c) 7x = 2 401 d) 13x = 2 197

23

24

25

Look at the example. Then transform the following: • 180 000 = 18 · 104 a) 5 000 b) 1 700 000 c) 4 000 000 000

Look at the example. Copy them in your notebook with the correct exponent for each asterisk. Then calculate: • 212 : 45 = 212 : (22)5 = 212 : 210 = 22 = 4 a) 36 : 92 = 36 : (3*)2 = 36 : 3* = 3* = … b) 253 : 54 = (5*)3 : 54 = 5* : 54 = 5* = …

29

Copy these expressions. Write the correct exponent for each asterisk. Then calculate. a) (55 · 53) : 253 = (55 · 53) : (5*)3 = … b) (23 · 42) : 8 = [23 · (2*)2] : 2* = [23 · 2*] : 2* = … c) (34 · 92) : 272 = [34 · (3*)2] : (3*)2 = [34 · 3*] : 3* = …

30

8 · 109

17 · 107

98 · 106

1010

16 · 108

9 · 109

Express the following amounts using powers of base 10. a) Eight thousand five hundred millions. b) Two trillion three hundred thousand millions. c) Four quintillion nine hundred thousand trillion. Look at the example. Then calculate: • 123 : 43 = (12 : 4)3 = 33 = 27 b) 26 · 56 c) 253 · 43 a) 82 · 52 d) 65 : 35 e) 153 : 53 f ) 204 : 54

27

Calculate. a) 184 : (24 · 34) c) (154 : 34) : 52 e) (62 · 65) : (63 · 64)

b) (35 · 33) : 36 d) (45)2 : (47 : 43) f ) (407 : 57) : (25 · 45)

Solve. 2

a) 5 2 + 12 2 – ` 5j 31

4

2

b) ` 2j + ` 3j – 5 0

Copy these expressions in your notebook. Complete with = or ≠. a) 2 · 9

36

b) 3 · 4

12

c) 5 · 16

20

d) 4 · 25

10

e) 9 · 9

18

f) 4 · 4

16

32

Take Action. You have a chessboard with 8 × 8 squares. You put a grain of rice on square 1. Then you put two grains of rice on square 2, etc. You double the number of grains of rice in each square. Express your answers below using powers. a) How many grains of rice do you use to complete the first row? b) How many grains of rice are there on the last square of the second row? Compare with the previous result.

33

Write the powers in your notebook and calculate. a) Montse has a box of cubes. The length of the edge of each cube is 1 cm. She uses the cubes to build three equal 3 cm edge cubes. Number of cubes she uses: 3 = … b) A man plants lettuces in his garden. He plants them in 25 rows. There are 25 lettuces in each row. Number of lettuces: 25 = … c) A van carries six pallets of boxes of milk. There are 36 boxes on each pallet. Each box contains 6 one-litre cartons of milk. Number of litres: 6 = …

Order the following amounts from highest to lowest.

26

62

28

34

Target 12.3. There are nine starters, nine main dishes and three desserts on a restaurant menu. How many combinations of meal are there? Give your answer using powers.

35

How many mothers and fathers are there in Juancho’s family up to his great-great grandparents? Express the number using a power.

39

Look at the cube in the picture. It is made of 5 × 5 × 5 cubic units.

a) You paint the outside of the cube red. How many cubes have red paint on them? b) You now want to make a bigger cube. You cover the red cube in a layer of green cubes. How many green cubes do you need? 40 SOLVE SIMPLE PROBLEMS 36

You have 20cm × 20cm square tiles. What is the largest squar floor space you can cover without cutting any tiles? How many tiles are remaining?

37

Marcos has a bag of 50 dice. They are 1cm × 1cm × 1cm. He wants to build a big cube. What is the biggest cube he can build? How many dice are remaining?

38

Problem solved

Copy and complete the intermediate steps. What information do you need for the solution?

Marta has five sheets of stickers. Each sheet has forty stickers. She uses them to decorate the small cube. Are there sufficient stickers to decorate the big cube? • How many stickers has she got? She’s got 5 · 40 = 200 stickers. • How many stickers did she use on the small cube? She used 6 · 32 = … stickers on the small cube. • How many stickers does she have left? She has 200 – … = … stickers left. • How many stickers does she need for the big cube? She needs … stickers for the big cube.

A square field has a surface area of 900 square metres. You want to put a fence around it. How many metres of fence do you need?

THINK A LITTLE MORE 41

Alberto told Nacho and Sara a rumour. Ten minutes later, Nacho told Raquel and Marta; and Sara told Rosa and Pablo. After another ten minutes, all of the second group tell another two people each. How many people know the rumour one hour after Nacho and Sara?

42

The floor of a square bedroom is covered in 484, 15 cm, tiles. They are all white, except for the tiles 15 cm away from the wall. These tiles are red.

How many red tiles are there? 63

Self-assessment

GO TO THE ESCAPE ROOM AND TEST YOURSELF!

anayaeducacion.es Answer key.

1 Copy and complete. Look at the example:

8 Copy and complete this table in your notebook.

• Eight squared → 82

The properties of powers

a) Six cubed → … b) Seven to the fourth → … c) ... → 102 d) ... → 123 e) ... → 29

The power of a product is equal to the product of the powers of the factors. The power of a quotient is equal to the quotient of the powers of the dividend and the divisor. To multiply two powers with the same base, we add the exponents together.

2 Write the following as a power.

a) 1 · 1 · 1 · 1 · 1 c) 5 · 5 · 5 e) m · m g) x · x · x · x

b) 2 · 2 · 2 · 2 · 2 d) 10 · 10 · 10 f) a · a · a h) z · z · z · z · z · z

3 Calculate.

a) 50 d) 70 g) 100

b) 51 e) 71 h) 103

c) 52 f ) 72 i) 107

4 Copy and complete in your notebook.

a) 2 = 8 c)

b) 5 = 125

2 = 81

d)

4 = 81

5 Write down all of the perfect squares between 100

and 300.

6 Copy and complete the table below in your notebook. Number

(a · b)n = an · bn

Polynomial decomposition

456 002 3 · 106 + 7 · 105 + 2 · 103

7 The Spanish population on January 2021 was

47 394 223 inhabitants.

am : an = am – n

To divide… To raise a power to another power…

9 Copy and complete in your notebook.

a) x 3 · y 3 = (

·

)

b) x 4 : y 4 = (

:

)

10 Reduce to just one power.

a) a3 · a2

b) x5 : x4

c) (a3)4

11 Calculate using the fastest method.

a) 24 · 54

b) 183 : 93

12 Reduce.

a) (x 5 · x 2) : x 4

b) (a5)2 : (a2)3

13 Copy and complete in your notebook.

a) 36 = d

b) 400 = d

c) 10 000 = d

d) d = 3

e) d = 8

f ) d = 30

14 Calculate the integer square root of 2 920 on paper.

Then, check your answer with a calculator.

15 Álvaro draws three squares. Square 1 is 5cm × 5cm,

square 2 is 12cm × 12cm and square 3 is 13cm × 13cm. He colours the first two red and the last one green. Which surface area is greater: the red one or the green one?

16 How many 1cm edge dice are there in 10 boxes like

the one below? m

10 c

a) Round the number to the nearest million. b) Abbreviate the number using a power of base 10. 64

10 cm 10

cm

I TAKE ACTIO LEARNING EXPERIENCE

N

Savings plan

Remember the savings plan from the student at the beginning of the learning experience. – Day 1, a €1 coin in the piggy bank. – Day 2, €2 in the piggy bank. – Day 3, €4. – Day 4, €8, etc. Each day, you save double the amount as the day before. First, let’s look at how much the amount grows each day. Day

1

2

3

4

5

6

7

8

9

10

Amount you add

1

2

4

8

16

32

64

?

?

?

Accumulated savings

1

1+2=3

4+3=7

8 + 7 = 15

31

63

?

?

?

?

Can you identify a relationship between the values in the second and third rows? Can you express this as a power? Day

1

2

3

4

5

…

10

Amount added

20

21

22

23

?

…

?

Accumulated savings

21 – 1

22 – 1

…

…

?

…

?

1 How many days will it be before you have to put in more

that €100? (8 days: 27 = € ...)

2 How much will you have in the piggy bank then, including

what you put in that day? (2... – 1 = € ...)

3 Answer activities 1 and 2 above, but for €1 000. 4 Can you do the same with larger amounts? How many days

Ask your teacher if you can use a calculator. If you can, make sure that you are using your model of calculator correctly. All calculators make maths work easier, but not all of them work in the same way. Below you can see two models for calculating powers.

would it take to save €1 000 000? On day ... you will have to put 2... = € ... and with that, you will have accumulated 2... = € ... . Reflect: What do you think about the increase in the amount of money to put in every day? Do you think the saving plan is reasonable?

1 × 2 × 2 × 2 × 2 × 2 = 32

Maths

Now, it's your turn... Repeat the same process as above, but this time, put in three times as much as the day before, each day.

in context

Apply your language skills www.anayaeducac ion.es

65

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