Operación mundo: Mathematics 1. Secondary DF (muestra)

muestra

DUAL FOCUS

1ATHEMATICS N IO T A C U D E Y R A D SECON

M

A N D A L U S IA

O peración

mundo

INDEX 1. Natural numbers ������������������������������������������7

Summary • 1 Numeral systems • 2 Some counting techniquess • 3 Large numbers • 4 Rounding natural numbers • 5 Basic operations with natural numbers • 6 Calculations with combined operations • The final challenge

2. Powers and roots ��������������������������������������17

Summary • 1 Powers • 2 Power with a base of 10� Applications • 3 • Calculations with powers • 4 The square root • The final challenge

3. Divisibility �����������������������������������������������������25

Summary • 1 The relation of divisibility • 2 Multiples and divisors of a number • 3 Prime and composite numbers • 4 Factorising numbers • 5 Lowest common multiple • 6 Greatest common divisor • The final challenge

4. Integers ����������������������������������������������������������33

Summary • 1 Positive and negative numbers • 2 The set of integers • 3 Addition and subtraction with integers • 4 Addition and subtraction with bracket • 5 Multiplication and division with integers • 6 Calculations with combined operations • 7 Power and roots of integers • The final challenge

5. Decimal numbers ��������������������������������������41

Summary • 1 Structure of decimal numbers • 2 Addition, subtraction and multiplication of decimal numbers • 3 Division of decimal numbers • 4 Square roots and decimal numbers • The final challenge

6. The metric decimal system �����������������49

Summary • 1 Magnitudes and measurements • 2 The metric decimal system 3 • Units of measurements in basic magnitudes • 4 Conversion of units • 5 Complex and simple amounts • 6 Measurement of area • The final challenge

7. Fractions �������������������������������������������������������� 57

Summary • 1 What are fractions? • 2 The relationship between fractions and decimals • 3 Equivalent fractions • 4 Some problems with fractions • The final challenge

8. Operations with fractions �������������������65

Summary • 1 Reducing to a common denominator • 2 Adding and subtracting fractions • 3 Multiplying and dividing fractions • 4 Combined operations • 5 Some problems with fractions • The final challenge

9. Proportionality and percentages ����������73

Summary • 1 Proportionality between magnitudes • 2 Direct proportionality problems • 3 Inverse proportionality problems • 4 Percentages • The final challenge

10. Algebra ��������������������������������������������������������81

Summary • 1 Letters instead of numbers • 2 Algebraic expressions • 3 Equations • 4 Solving equations • 5 Solving first-degree equations with one unknown • 6 Some problems with equations • The final challenge

11. Lines and angles����������������������������������������91

Summary • 1 Basic elements of geometry • 2 Two important lines • 3 Angles • 4 Angle measures • 5 Operations with angle measures • 6 Angle relationships • 7 Angles in polygons • 8 Angles in a circumference • The final challenge

12. Geometric shapes ����������������������������������101

Summary • 1 Polygons and other plane shapes • 2 Symmetries in plane shapes • 3 Triangles • 4 Quadrilaterals • 5 Regular polygons and circumferences • 6 Cordovan triangle and related shapes • 7 Pythagorean theorem • 8 Applications of the Pythagorean theorem • 9 Geometric solids • The final challenge

13. Areas and perimeters ���������������������������111

Summary • 1 Measurements in quadrilaterals • 2 Measurements in triangles • 3 Measurements in polygons • 4 Measurements in a circle • 5 The Pythagorean theorem for calculating areas • The final challenge

14. Graphs of functions ����������������������������� 117 Summary • 1 Cartesian coordinates system • 2 Points that provide information • 3 Points that are related • 4 Interpreting graphs • 5 Linear functions� Equation and representation • The final challenge

15. Statistics ��������������������������������������������������� 125

Summary • 1 Statistical analysis process • 2 Frequency and frequency tables • 3 Statistical graphs • 4 Statistical parameters • 5 Position parameters • The final challenge

1

TAKE ACTION!

NATURAL NUMBERS

What activity (a sport, an artistic activity, learning a language���) would you like to do?

Take on the ‘Final Challenge’ and prepare a budget to be able to attend classes and buy the material�

Natural numbers

large numbers

numeral systems

rounding numbers

Egyptian

million

Mayan

billon

decimal

basic operations

order of operations

rounding

10

00

00

Trillon

000

combined operations

brackets

000 multiplication and division addition and subtraction

addition

subtraction

multiplication

commutative property

commutative property

associative property

3 2

5

3

associative property

division

distributive property

exact division

integer division

2 5 Listening Listen and repeat� The vocabulary is at anayaeducacion.es.

Discover Numeral systems were very simple at first� People cut notches (marks) in sticks or drew hands and fingers� As civilisation progressed, symbols and rules were introduced� Do you know any numeral systems?

FOCUS ON ENGLISH

Reading Read the text ‘Numeral systems’ on anayaeducacion�es and answer the questions there�

7

1

Numeral systems

As society evolved, people had to work with larger numbers and needed a more practical system. That is how different numeral systems emerged.

Numbers in the Mayan system ❶

The Egyptian numeral system

• Numbers under 20 are written like

The Egyptian numeral system is an additive system. This means that to write a number, they added the necessary symbols until they had written the desired amount.

this:

0 1

2

3

4

5

7

8

9

6

10 11 12 13 14 15 16 17 18 19

1

10

100

1 000

10 000

100 000

1 000 000

stick

hobble

rope

flower

finger

frog

person

The Mayan numeral system

20 20

The Mayan numeral system was partly additive, partly positional. It only had three symbols with different values depending on their level. (Look at figure ❶).

(0) (0)

(1) (1)

• Numbers over 20 are written like this:

Ò 20 → Ò1 → 20 20 21 20 21 27 27 36 36

21 100 137 21 27 36 40 21 100 27 137 36 40 100 137

(5) (5)

The decimal numeral system

Place values 4 304 ❷

We currently use the decimal numeral system.

th

h

t

u

• It has ten symbols or figures (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9). The symbols are

4

3

0

4

written in different place values: ones, tens, hundreds, etc.

↓

• Ten units of one level form a single unit of the following level.

• The value of a figure depends on its place value (meaning that it is a positional

system). (Look at figure ❷).

1

True or false? a) The Egyptian numeral system is additive because the position of the symbols changes their value. b) The decimal numeral system is positional because the value of a figure depends on its place value. c) The Mayan numeral system is positional.

2 Match each number with its numeral system:

a) Mayan 1)

8

40 40

↓

4U 4 000 U The value of 4, in the number 4 304, changes according to its place value.

3 Writing. Copy and complete.

A numeral system is (...) if adding symbols adds their represented amount. A system is positional if the value of each symbol depends on its (...). One example of a numeral system that is both additive and positional is (...). One example of a numeral system that is only positional is the (...) numeral system. One example of a system that is not (...) but is additive is the (...) numeral system.

b) Egyptian c) Decimal 0 1 2 3 4 0 1 2 3 4 4 Speaking. Work in pairs to explain the numeral system 2) 1 923 3) we use today.

5 6 7 8 9 5 6 7 8 9 10 11 12 13 14 10 11 12 13 14

15 16 17 18 19

100 100

Unit 1

2

Some counting techniques

Let’s review some counting techniques:

Data counting On the right you have the results of the vote for the election of a delegate in a 1st ESO class.

Note that: • There are 28 girls and boys in the class.

• Celia obtained 4 more votes than Aitor.

nerea

6

celia

13

aitor

9

Tables and operations The table shows data on the distribution of two groups of 1st ESO and two groups of 2nd ESO in a secondary school.

137 137

1.A 14

13

1.B 0

12

13

2.A 0

11

16

2.B 1

12

12

0

Note that: • There are 52 people in the first year and 52 in the second year.

• In total we counted 49 girls, 54 boys and one person who did not wish to be counted in either of these two groups.

Remenber

Remember the meaning of these symbols: Female → Male → Transgender →

Tree branches There are three different coloured balls (red, blue and yellow). We propose an experiment: take out two balls one after the other.

• Suppose we carry out the experiment six times and ask: How many times, theoretically, should the red ball remain in the urn?

What if we carry out the experiment 30 times?

We rely on a diagram like the one on the right (tree diagram) to explore all the possibilities.

We see that we should expect the red ball to be in the urn two of the six times. If we carry out the experiment 30 = 6 Ò 5 times, the red ball should remain in the urn 2 Ò 5 times.

1

We have an urn with three blue balls and one red ball, and perform the following experiment: we draw two balls one after the other.

1st ball

2nd ball remaining

a) What is the chance that there are two balls of different colours left in the urn?

1

2

3

b) What about the two remaining balls being the same colour?

9

3

Large numbers

The decimal numeral system allows us to write numbers that contain as many figures as we want.

Observe

• One million ↔ A 1 followed by 6 zeros.

↔ A 1 followed by 12 zeros.

• One trillion ↔ A 1 followed by 18 zeros.

0 0 0

0 0 0

0 0 0

units

0 0 0

tens

8 0 0

hundred

3 0 0

thousands

1

1 0

1 0 0

millions

billons

...

thousand millions

• One billion

0 0 0

0 0 0

0 0 0

0 0 0

The English words billions, trillions, quadrillions… are false friends. They do not mean the same in Spanish as in English.

For example: 1 billion ↔ A 1 followed by 9 zeros.

1 trillion ↔ A 1 followed by 12 zeros.

The universe was formed thirteen thousand eight-hundred million years ago.

A young person's brain contains around one hundred thousand millions of neurons.

The volume of the Earth is approximately one billon cubic kilometres.

1

Read the first few lines of this page. Then write how to express the following: a) The number of seconds in a century. b) The number of seconds in a century. c) The number of kilometres in a light year.

2 Write in figures.

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

10

3 Copy in your notebook and complete.

a) A thousand thousands make... b) A thousand millions make a... c) One million thousands make a... d) One million millions is a...

4 The human body has between ten and seventy million

times a million cells. Express these numbers in billions.

5 How would you read the number expressed by a 1

followed by 16 zeros?

6 Scientists estimate that the Earth’s seas and oceans

contain three quadrillion kilograms of water. What do you think a quadrillion is?

Unit 1

4

Rounding natural numbers

When a number contains a lot of figures, it is difficult to remember. It also makes calculations more difficult. For this reason, we often replace large numbers with rounded values. These are more manageable numbers that are very close in value to the original number, but end in zeros.

Rounding is the most commonly used method.

Let’s look at an example of how to round.

Rounding 52 722 to the nearest thousand.

… 000

All the numbers to the right of that place value must be replaced with zeros.

If the first figure being replaced is greater than or equal to five, we add a one to the previous figure.

+1

52 722 7≥5

53 000

It is approximately 53 000 of them.

Round these numbers to the nearest thousand: a) 24 963 b) 7 280 c) 40 274 d) 99 834 2 Approximate to the nearest hundreds and tens of thousands. a) 530 298 b) 828 502 c) 359 481 d) 29 935 236 e) 47 142 900 f) 730 927 106 3 Reading. Read this news item and round the number of tourists there are to the nearest millions and the amount they spent to the nearest thousands of millions.

1

, In 2018tourists 0 0 0 0 0 82 6 ited Spain. vis y spent The million 89 678 uros e

4 Round to the nearest million.

a) 24 356 000 b) 36 905 000 c) 274 825 048 5 Here are some approximations of the price of a flat

for sale:

FOR SALE

€138 000

138 290

€138 300 €140 000

a) Which is closer to the real price? b) Which one do you think is more appropriate for everyday information, if you cannot remember the exact amount?

6 A local council has budgeted €149 637 to refurbish

a sports area.

How would you say this in an informal conversation?

11

5

Basic operations with natural numbers

Addition and its properties

Properties of addition ❶

Remember that addition means to find the total value of a set of numbers.

For instance, to find the total number of people at the football stadium, we must add the numbers together.

Addition follows two properties: • Commutative property: The sum does not change if the order of the addends changes.

• Commutative property

a+b=b+a • Associative property: The sum is not affected by how the addends are grouped.

34 + 16 = 16 + 34

50

50

• Associative property

(18 + 3) + 17 = 18 + (3 + 17)

21 + 17 38

18 + 20

38

(a + b) + c = a + (b + c) (Look at figure ❶).

SEATING CAPACITY: 25 342 seats

Subtraction and how it is related to addition Remember that subtraction means to ‘take away’ one number from another number, or to find the difference.

For example, if we want to count the empty seats at the football stadium in the picture on the right, we must subtract the number of full seats from the total number of seats: 25 342 – 20 582 = 4 760 Note that 25 342 = 20 582 + 4 760 and that 20 582 = 25 342 – 4 760.

– 20 582 ← Subtrahend (S ) 4 760 ← Difference (D )

M=S+D S=M–D

Calculate. a) 254 + 78 + 136 c) 340 + 255 – 429

3 Transform.

b) 1 526 – 831 + 63 d) 1 350 – 1 107 – 58

2 Reading. Read the problem and calculate the answer.

Check it afterwards.

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

12

Remember

25 342 ← Minuend (M )

Relationships between addition and subtraction: M – S = D

1

Seats filled East stands: 11 576 West stands: 9 006

a) this addition into a subtraction: 48 + 12 = 60. b) this subtraction into an addition: 22 – 2 – 6 = 14.

4 If Alberto were 15 years older, he would still be 18

years younger than his uncle Thomas, who is 51 years old. How old is Alberto?

5 If I only bought a washing machine, I would still

have €246. However, if I also bought a TV, I would be €204 short. Could you say how much is any of these items?

Unit 1

Multiplication and its properties Remember that multiplication is basically a repeated addition of the same value.

For example, if a ticket to the football game costs €35, the total amount for all 20 582 tickets sold would be: 35 + 35 + 35 + … + 35 = 35 · 20 582 = €720 370 20 582 times

Multiplication follows these properties:

Remember

• Commutative property: The product does not change if the order of the

Properties of multiplication • The associative property allows us to regroup terms and the commutative property allows us to change their order.

factors changes.

a·b=b·a • Associative property: The result of a multiplication is not affected by how

the factors are grouped. Look at the first image on the right.

16 ×

8 × 2 × 5 × 11

(a · b) · c = a · (b · c)

88 × 10

• Distributive property: The product of a calculation does not change if we

remove the parentheses. Look at the second image on the right. a · (b + c) = a · b + a · c

55

a · (b – c) = a · b – a · c

880 • Distributive property

of multiplication

35 · 7 + 35 · 3 = 35 · (7 + 3)

245 + 105 350

35 · 10

350

On Thursday a group of friends bought 7 tickets to the game. On Friday, they bought another 3. How much did all the tickets cost?

We can calculate the cost of the tickets in two different ways: • Price of 7 tickets + Price of 3 tickets • Price of (7 + 3) tickets

35 · 7 + 35 · 3 = 35 · 10

6 Copy and complete in your notebook.

× 2

5

+ 9 0 1 2 6 0

+

9 8 × 2 8 7 4 6 9 9 3 4

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

we add one, two, three... zeros to the end of the figure.

a) 19 · 10 b) 140 · 10

b) 12 · 100 e) 230 · 100

c) 15 · 1 000 f ) 460 · 1 000

Copy into your notebook and write a mathematical equality:

8 Writing.

Multiplying a number by eight is the same as first multiplying it by ten, then subtracting double the original number.

Which property describes this?

13

5

Basic operations with natural numbers

Division

Integer division and exact division ❷

• Division means sharing a value between several people/things, in equal parts,

• Integer division (the remainder is not

to work out how much is received by each.

• Division means splitting a whole thing into portions of a specific size, to work out how many portions there are.

zero).

D d r q The dividend is equal to the divisor multiplied by the quotient plus the remainder:

Division: exact and integer

There are two types of division depending on the value of the remainder: exact division and integer division. (Look at figure ❷).

D=d·q+r • Exact division (the remainder is

Properties of division

zero).

Look what happens if we multiply the dividend and the divisor by the same number:

D d 0 q

The dividend is equal to the divisor multiplied by the quotient:

3 plants need 24 litres of water. What would happen if we had twice the number of plants and twice the amount of water?

D=d·q

Because we divided twice the amount of water by twice the number of plants, the amount of water each plant receives stays the same.

24 litres

48 litres Observe

24

3

0

8

Ò2 Ò2

48

6

0

8

9 Reading. True or false?

11

a) The quotient must be greater than the divisor. b) The remainder is always lower than the divisor. c) In exact division, multiplying the dividend by two doubles the quotient. d) Multiplying the dividend and the divisor by 3 will triple the quotient. e) Division follows the commutative property.

10 Find the missing factor in each division:

dividend 39 14

53 15

In a division, if the dividend and divisor are multiplied by the same number, the quotient remains the same.

1 000 12

divisor 38

Copy the diagrams below into your notebook.

Complete the calculations.

(36 : 12) : 3

36 : (12 : 3)

:

:

Do you notice anything?

12 A farmer collects 1 274 eggs. He puts 30 eggs in each

tray, and 10 trays in each box.

How many eggs are left over in a partially empty tray?

How many trays are left over in a partially empty box?

Unit 1

Calculations with combined operations

6

Order of operations to solve an expression When solving expressions with combined operations, you must remember the rules of mathematical notation. These rules ensure that each expression has a single meaning and solution.

Look at the order followed in the calculations below: although each calculation contains the same numbers and operations, the answers are not the same.

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

This happens because a different hierarchical priority is being respect in each expression.

So the order of combined operations must always be: • First, brackets.

• Then, multiplication and division.

• Lastly, addition and subtraction.

1

Complete each box in your notebook. Check that you have the right answers.

4 · 10 – 8 · 3 + 2 –

+2 +2 18

Write the following statements as a mathematical expression and calculate. a) A van is carrying 8 boxes of bananas, 20 boxes of oranges and 6 boxes of apples. Each box of bananas weighs 15 kilos. Each box of oranges and each box of apples weighs 8 kilos. How many kilograms of fruit is the van carrying?

2 Reading.

b) A supermarket orders 20 crates of full-fat milk, 15 crates of skimmed milk and 10 crates of semiskimmed milk. Each crate holds six one-litre bottles.

How many bottles did the supermarket order?

Using the calculator

Type this sequence into a calculator: 2 + 3 . 4 = It may seem strange, but different calculators give different answers.

You could have got 20 or 14.

{∫“≠} → The calculator performs each operation in the order it was entered.

(2 + 3) · 4 = 5 · 4 = 20 {∫‘¢} → The calculator performs

the multiplication first. In other words, it respects the order of operations.

2 + 3 · 4 = 2 + 12 = 14 As you can see, not all calculators have the same internal logic. Find out which method your calculator uses. Remember this when you are using it.

c) There are 15 tables, 55 chairs and 12 stools in a cafe.

How many legs are there in total? (note: tables and chairs have 4 legs and stools have 3 legs). d) A farmer packs 1 500 eggs into boxes that hold 10 eggs. He packs another 1 500 eggs into boxes that hold 6 eggs. Finally, he packs 300 free-range eggs into boxes that hold 6 eggs. How many boxes did he fill?

3 Look at the example and calculate.

4 · (7 – 5) – 3 = 4 · 2 – 3 = 8 – 3 = 5 a) 2 · (7 – 3) – 5

b) 5 · 2 + 4 · (7 – 5)

c) 3 · (10 – 7) + 4

d) 18 : 2 – 2 · (8 – 6)

e) 4 + (7 – 5) · 3

f ) 30 – 4 · (5 + 2)

g) 18 – 4 · (5 – 2)

h) 5 + 3 · (8 – 6)

i) 8 – (9 + 6) : 3

j) 5 · (11 – 3) + 7

k) 22 : (7 + 4) + 3

l) 3 · (2 + 5) – 13 15

TAKE ACTION!

The final challenge

DRAW UP A BUDGET

SCAN THIS CODE TO CONSULT THE

Alone or in a team, play the role of Diego, do the activities and, at the end, draw up a budget to be able to attend classes in the activity of your choice�

GLOSSARY FOR THIS UNIT�

This year Diego wants to sign up for tennis lessons and has checked the prices of the two tennis schools in his town on the Internet.

School A No. of people

1

MWF

School B T Th

No. of people

3 days per week

2 days per week

1 day per week

1

60 €/month 40 €/month

1

63 €/month

42 €/month

25 €/month

2

48 €/month 33 €/month

2

51 €/month

35 €/month

18 €/month

3

36 €/month 24 €/month

3

39 €/month

26 €/month

14 €/month

A Diego’s grandparents have offered to pay for his tennis lessons, but he should not spend more than €400 per year. a) If he wants to attend classes for the whole school year, i.e. from September to June, what are your options?

b) You want to buy a racket that costs €35 and include the cost in your annual budget, what are your options? c) Diego wants to take the most classes and share them with the fewest people. Which option do you think he will choose? Why?

€9 2 With the money left over from his budget, Diego

€

0

€6

has decided to buy some sports accessories. If he wants to buy as many items as possible and spend all the money, which of these will he choose?

2

€2

€

Final product Choose an activity such as a sport, an artistic activity or learning a language, look for information on the Internet about schools that offer classes in that activity and draw up a budget in which you

16

4

consider how much you would spend on the classes, according to your needs and preferences, and the cost of the supplies necessary to attend.

3

2

TAKE ACTION!

POWERS AND ROOTS

What is your favourite team sport?

Take on the ‘Final Challenge’ and organise a sports tournament in your school�

Powers and roots

powers

base exponent

factorisation of polynomials

4

operations with powers

square root

power of a product

8

power of a quotient

square number cubic number

product of powers quotient of powers

(a··b) =a ·b (a x

x

x

root

radicand

(a:b)x=ax:bx m+nn am·an=am+

m-nn am:an=am-

perfect squares

exact root

power of another power

integer root

power with zero as the exponent

estimating

50=80=200=1 Listening Listen and repeat� The vocabulary is at anayaeducacion.es

Discover

FOCUS ON ENGLISH

The Pythagoreans were the first to use the words square and cube in relation to numbers� But these words and their derivatives have other uses and meanings� Try to name some of them�

Reading Read the text ‘The first mathematicians’ on anayaeducacion.es and answer the questions�

17

1

Powers

Powers are a shortened form of writing a product of the same factors. For example: a · a · a · a · a = a5 The number to be multiplied is called the base. The number written in superscript is called the exponent. The exponent tells us how many times to multiply the base by itself.

ab

Exponent

Remember

3 · 3 · 3 · 3 = 34 We say: three to the power of four, or three raised to the fourth power.

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

Base

Two very important powers: ‘squared’ and ‘cubed’ • Squaring a number means raising the number to the power of 2.

Squares and cubes in geometry ❶

For example: 72 = 7 · 7 = 49 → 7 squared is 49.

• Cubing a number means raising the number to the power of 3.

For example: 73 = 7 · 7 · 7 = 343 → cubed is 7 es 343.

Look at figure ❶.

5

5 squared is: 52 = 5 · 5 = 25 (25 squares)

5

5

Using the calculator

With the exception of a few simple examples, powers usually produce very large numbers. For example: 96 = 9 · 9 · 9 · 9 · 9 · 9 = 81 · 9 · 9 · 9 · 9 = 729 · 9 · 9 · 9 = … = 531 441

5

Calculators can help us to solve these tedious problems quickly.

• On a simple calculator, you need the following buttons: * and =.

96 ⎯→ 9

*

*

=

↓ 92

=

=

↓ 93

↓ 94

=

↓ 95

• On a scientific calculator, you need this button: ‰.

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

= ⎯→ {∫∞«‘¢¢‘}

↓ 96

96 ⎯→ 9 ‰ 6 = ⎯→ {∫∞«‘¢¢‘}

1

Listening. Complete the activity ‘Calculating powers’ at anayaeducacion.es.

2 Complete the table in your notebook.

Power

Base

Exponent

5

3

26 a4

a) Cubing a number is the same as multiplying it by itself three times. b) Raising a number to the fourth power is the same as multiplying it by four. c) 10 squared is 20. d) 10 cubed is 1 000.

m

18

3 Reading. True or false?

5

e) Two to the power of five is the same as five squared.

Unit 2

2

Power with a base of 10.

Applications

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 with a base of ten are the same as the digit in question, followed by the number of zeros in the exponent.

1 Speaking. In pairs, take turns

reading these numbers and writing them out in full. b) 15 · 109 a) 4 · 105 c) 86 · 1014 d) 12 · 104

2 Write the following numbers as

powers of base ten:

Abbreviating large numbers

a) One thousand

We can use powers with a base of 10 to abbreviate large numbers that end in zero. To do this, we simply count the zeros, then replace them with a power of base 10 raised to the same number as the number of zeros that the number has.

For example: 400 000 = 4 · 100 000 = 4 · 105 This helps us to write and understand large numbers more easily.

b) One million c) One thousand millions d) One billon 3 Find the value of x.

a) 2 936 428 ≈ 29 · 10x b) 3 601 294 835 ≈ 36 ·10x

One light year is 9 460 800 000 000 km.

c) 19 570 000 000 000 ≈ ≈ 20 · 10x

• We round to two significant figures → 9 500 000 000 000 • We factorise the number → 95 · 100 000 000 000 • We write the second factor as a power with a base of 10 10 → 95 · 1011

Therefore, one light y ear is 95 · 1011 km.

This makes it easier to read, write and remember what one light year is.

Remember what you learnt about the powers of base 10 and the positional value of digits in a number. In this example, we show you a transformation called polynomial factorisation of a number.

836 279 =

the following.

• 180 000 = 18 · 104

a) 5 000 b) 1 700 000

Polynomial factorisation of a number

800 000 + 30 000 + 6 000 + 200

4 Follow the example to transform

+

c) 4 000 000 000

70 + 9

8 · 105 + 3 · 104 + 6 · 103 + 2 · 102 + 7 · 101+ 9

5 Abbreviate the data below:

a) There are 334 326 000 000 000 000 000 000 molecules in one litre of water. b) The Alpha Centauri star system is approximately forty billones kilometres from the Sun.

6 Speaking. Discuss as a group how to order the following

amounts from smallest to largest: a) 8 · 109

b) 17 · 107

c) 98 · 106

d) 1010

e) 16 · 108

f ) 9 · 109 19

3

Calculations with powers

The power of a product (Product of powers with the same exponent) The power of a product is equal to the product of the powers of its factors.

(2 · 3)3 = 63 = 6 · 6 · 6 = 216 23 · 33 = (2 · 2 · 2) · (3 · 3 · 3) = 8 · 27 = 216

(a · b)n = an · bn

We multiply the powers of the factors.

(2 · 3)3 = 23 · 33

Or also: 23 · 33 = (2 · 2 · 2) · (3 · 3 · 3) = (2 · 3) · (2 · 3) · (2 · 3) = (2 · 3)3

The power of a quotient (Quotient of powers with the same exponent) The power of a quotient is equal to the quotient of the powers of the dividend and the divisor.

(6 : 3)3 = 23 = 2 · 2 · 2 = 8 63 : 33 = (6 · 6 · 6) : (3 · 3 · 3) = 216 : 27 = 8

(a : b)n = an : bn

We divide the powers of the dividend and the divisor.

(6 : 3)3 = 63 : 33

Or also: 63 : 33 = (6 · 6 · 6) : (3 · 3 · 3) = (6 : 3) · (6 : 3) · (6 : 3) = (6 : 3)3

1

Use the example to help you fill in the blanks.

(4 · 3) 2 = 12 2 = 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…

20 3 : 4 3 = …

2 Think and calculate.

a)53 · 23 d) 203 · 53

b) 42 · 52 e) 84 · 34

c) 252 · 42 f ) 183 · 63

20

b) 243 : 63 e) 1003 : 503

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 = … = … e) (363 : 93) · 253 = (… : …)3 · 253 = …3 · 253 = = (… · 25)3 = …3 = … f ) (542 : 32) : 22 = (… : …)2 : …2 = …2 : …2 = = (… : …)2 = …2 = …

5 Writing. Copy and complete in your notebook.

• The (...) is equal to the product of the powers of the

3 Think and calculate.

a)165 : 85 d) 352 : 52

4 Think and calculate.

c) 214 : 74 f ) 275 : 35

factors.

• The (...) is equal to the quotient of the powers of the dividend and the divisor.

Unit 2

The product of powers with the same base When you multiply two powers of the same number, it produces another power of that number.

54 · 53 = (5 · 5 · 5 · 5) · (5 · 5 · 5) = 57 4 times

54 · 53 = 54 + 3 = 57

3 times

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

am · an = am + n

We add the exponents.

The quotient of powers with the same base Remembering the relationship between multiplication and division, we have to:

54 · 53 = 57 ↔

57 : 53 = 54 57 : 54 = 53

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

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

6 Reduce the following expressions.

am · an = am + n

We subtract the exponents.

a) x 8 : x 3

b) m4 · m2

Reduce to just one power. a) x · x2 · x3 b) m2 · m4 · m4

c) x 5 · x 5

d) k 6 : k 4

c) (k9 : k5) : k3

d) (x5 : x3) : x

e) m6 : (m8 : m4)

f ) (k2 · k5) : k6

11

7 Copy the expressions in your notebook. Replace each

asterisk with the corresponding exponent. b) a5 · a3 = a* a) 64 · 63 = 6* c) m3 · m* = m9 d) 26 : 24 = 2*

8 Calculate.

a) (35 · 33) : 36

b) (62 · 65) : (63 · 64)

9 Reduce to just one power.

a) 52 · 52

b) 32 · 35

c) 105 · 102

d) a5 · a5

e) m7 · m

f ) x2 · x6

10 Express with just one power.

a) 26 : 22 c) 107 : 106 e) m5 : m

b) 38 : 35 d) a10 : a6 f ) x8 : x4

12 Complete in your notebook, then reduce.

a) a3 · a5 = a… + … = a… b) a8 : a5 = a… – … = a…

13 Calculate and reduce to just one power.

a) a12 : (a4 · a4) = a12 : a… = a… b) 59 : (54 · 53) = 5… : 5… = 5… c) (m10 : m8) · (m5 : m4) = m… · m… = m…

14 Writing. Copy and complete in your notebook.

a) To (...) two (...), we keep the same base and subtract the exponents. b) To (...) two (...), we keep the same base and add the exponents.

21

3

Calculations with powers

Powers of other powers When we raise one power to another power, it produces a new power with the same base.

(54)3 = 54 · 54 · 54 = 54 + 4 + 4 = 54 · 3 = 512 To raise a power to another power, we keep the same base and multiply the exponents.

(an)m = an · m

We multiply the exponents.

Powers with zero as the exponent Look what happens when we divide any power (for example, 53) by itself: • Using the quotient rule → 53 : 53 = 53 – 3 = 50 = 1 • Using regular calculation → 53 : 53 = 125 : 125 = 1

We therefore assign 50 a value of 1.

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

a0 = 1(a ≠ 0)

Any number (not zero) raised to the power of 0 is equal to 1.

20 = 1; 80 = 1; 100 = 1; 340 = 1

15 Calculate and observe that results of the following

calculations are not the same. a) (6 + 4)2 ; 62 + 42 b) (5 + 2)3 ; 53 + 23

16 Copy the following problems into your notebook.

Fill in the blanks with either ‘=’ or ‘≠’. b) (4 + 1)3 a) (4 + 1)3 43 + 13 c) (6 – 2)4 e) 102

64 – 24

52 · 22

g) (12 : 3)2

122 : 32

d) 73

(10 – 3)3

f ) 104

52 · 22

h) 122 : 62

17 Reduce to just one power.

22

53

a) (52)3

b) (25)2

c) (103)3

d) (a5)3

e) (m2)6

f ) (x4)4

64

18 Reduce.

a) (x · x2 · x3)5 c) (k9 : k5)3 e) (m8 : m4)2 g) (x2)5 : x7 i) (k2)6 : (k3)4

b) (m2 · m4 · m4)2 d) (x5 : x3)4 f ) (k2 · k5)6 h) m10 : (m3)3 j) (x5 : x3)2

19 Solve

the following expressions with combined operations. b) 24 – 38 : 36 – 22 a) 62 + 22 – 22 + 5 c) 10 + (52)3 : (53)2 d) (105 : 55) – (22 · 22) e) [(8 – 5)2 · (9 – 6)3] : 35 f ) [(7 – 4)3 – (9 – 4)2]4

20 Writing. Copy and complete in your notebook.

a) To (...) one (...), we keep the same base and multiply the exponents. b) The (...) of a number is equal to one.

Unit 2

4

The square root

Calculating the square root simply means doing the opposite to squaring a number. b2 = a ) a = b

Using the calculator

On some calculators, you have to press the buttons in this order to calculate: 105 674 is:

Root Radicand

105 674 $ → {«“∞…≠|∞………}

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

• 42 = 16 → • 152 = 225 →

On others, you must press:

$ 105 674 = → {«“∞…≠|∞………}

16 = 4 → The square root of 16 is 4.

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

Exact and approximate roots The square roots of natural numbers are called perfect squares: 12 22 32 42 52 … 82 … 112 … 202 …

1 4 9 16 25 64 121 400 The square root of a perfect square is called an exact root.

These are some exact roots:

9=3

121 = 11

400 = 20

For most numbers, the root is not an exact quantity of whole units. The closest natural number is called the approximate root. It must be lower than the original number.

Let’s find the square root of 40: 6 2 = 36 < 40 The square root of 40 is a number 4 → 6 < 40 < 7 → 2 between 6 and 7.

7 = 49 > 40 40 ≈ 6 → The approximate root of 40 is 6.

Estimating square roots Estimate 3 900 .

• The squares of 62 and 63 are around 3 900 62 2 = 3 844 < 3 900 63 2 = 3 969 > 3 900 • The square root of 3 900 is a number between 62 and 63.

3900 ≈ 62 → The approximate root of 3 900 is 62.

1

Calculate mentally. b) 9 a) 4 c) 400 d) 900 e) 6 400 f ) 8100

2 Calculate the aproximate roots:

a) 5 c) 32 e) 68

b) 10 d) 39 f ) 92

3 Estimate.

a) 90 c) 1521 e) 700

b) 150 d) 6 816 f ) 10 816

4 Calculate using a calculator.

a) 2 936 b) 10 568 c) 528 471 23

TAKE ACTION!

ORGANIZE A SPORTS TOURNAMENT

SCAN THIS CODE TO CONSULT THE

Alone or in a team, do the role play, complete these activities and, at the end, organise a sports tournament in your school�

FOR THIS UNIT�

GLOSSARY

WEST SIDE

A

SOUTH FUND

The seaside town where Ana lives is going to hold a beach volleyball tournament. The matches will take place in the new stadium, next to the harbour. Ana is excited because she won tickets to the final, where the two best teams will compete for the gold medal.

B

C

D

D

A

E

B

F

C E

F

G

NORTH FUND

The final challenge

H

EAST SIDE

1

Look at the stadium layout. Each lateral section has 100 seats distributed in the same number of rows as columns. The end sections have 25 seats each, also distributed in the same number of rows as columns.

What is the capacity of the stadium?

2 These are the ticket prices for the final:

SIDE

BACK

A, D, E, H

B, C, F, G

A, C, D, F

B, E

€15

€20

€5

€10

If all the tickets sell, how much money would the tournament raise? Calculate this using a single expression.

3 There are 5 rounds in the tournament and the

losing teams drop out in each round. If the two best teams play in the final, how many teams have played in each round? How many teams entered the tournament?

4 Each of the teams that participated in the 3rd and

4th round will be awarded 12 T-shirts valued at €12 each. The tournament champion team will receive €6 000 and the runner-up team will receive € 3 000. a) Express the total cost of the T-shirts with an exponential. b) How much money will the prizes cost?

Final product Organise a sports tournament in your school. Calculate the capacity, ticket prices, prize money, the number of participating teams and draw up a report.

24

DUAL FOCUS

1 mathematics sECONDaRY EDUcAtION

Contents 1. Natural numbers ...................................................................................2 2. Powers and roots ................................................................................ 4 3. Divisibility ................................................................................................................6 4. Integers ....................................................................................................................8 5. Decimal numbers ............................................................................................. 10 6. The metric decimal system ...........................................................................12 7. Fractions ................................................................................................................ 14 8. Operations with fractions ............................................................................ 16 9. Proportionality and percentages ............................................................. 18 10. Algebra ............................................................................................................... 20 11. Lines and angles ................................................................................................22 12. Geometric shapes........................................................................................... 24 13. Areas and perimeters ................................................................................... 26 14. Graphs of functions ...................................................................................... 28 15. Statistics ............................................................................................................30

1

Natural numbers

1

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

Egyptian

What does it mean for a numbering system to be positional and additive?

It is partially an additive system and partially 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 until they have the desired amount.

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

Decimal

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

300 30

2nd level (× 20)

3 × 20

1st level (× 1)

0 × 1

+

60

1

M

H TH

T TH

TH

H

T

U

4

7

8

4

3

0

4

↓ 4 000 000 U

↓ 4 000 U

↓ 4U

Large numbers The decimal numeral system allows us to write numbers that contain as many figures as we want.

1 000 000

1 000 000 000 000

1 000 000 000 000 000 000

One million A 1 followed by 6 zeros.

One billon A 1 followed by 12 zeros.

One trillon A 1 followed by 18 zeros.

2 How do you write in text the number 7 126 000 123 437 140 001?

2

Rounding natural numbers To round a number to a specific place value: • The figures to the right of that place value are replaced with zeros.

• If the first figure being replaced is greater than or equal to 5, we add one more unit to the previous figure.

Hundred thousands

3 8 4 5 2 3 +1

8≥5

Ten thousands

=

4 0 0 0 0 0

Thousands

3 8 4 5 2 3

3 8 4 5 2 3 4<5

5≥5

+1

3 8 0 0 0 0

3 8 5 0 0 0

3 Approximate

the number 473 277 to the tens of thousands and explain how you did it.

Basic operations with natural numbers Addition means to find the total value of a set of numbers.

576 + 906 = 906 + 576 1482

Associative property (576 + 906) + 427 = 576 + (906 + 427) 1482 + 427

576 + 1333

1 909

1 909

Multiplication means repeatedly adding the same value.

Distributive property 35 × 7 + 35 × 3 = 35 × (7 + 3) 245 + 105

=

35 × 10

350

=

350

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

Commutative property 1482

8 342 ← Minuend (M )

Remember

– 1 909 ← Subtrahend (S )

Relationships between addition and subtraction:

6 433 ← Difference (D )

M =S +D M–S=D →* S=M –D Dividing means splitting something into several equal portions.

Associative property (16 × 55) × 3 = 16 × (55 × 3) 880 × 3 = 16 × 165 2 640 = 2 640

Commutative property 16 × 55 = 55 × 16 880

=

880

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

4 Indicate which property applies in this equation: 42 · (5 + 6) = 42 · 5 + 42 · 6

D=d.q

Calculations with combined operations The order of combined operations must always be: First

Brackets

Then

Multiplication and division

Finally

Addition and subtraction

5 Calculate the

following operation: 56 + 2 · (45 – 15) : 3 3

2

Powers and roots 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

1

32 = 9

Geometrically represent the number 73.

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 the digit in question, followed by the number of zeros in the exponent.

3 zeros 2 zeros 103 = 10 . 10 . 10 = 1 000 102 =10 . 10 = 100 105 = 100 000

5 zeros

109 = 1 000 000 000

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 +

2 Use the polynomial

factorisation method for the following numbers: a) 86 432 b) 57 092 103

200

+

70

+

9

836 279 = 8 . 105

4

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

Calculations with powers • The power of a product is equal to the product

(a . b)n = an . bn

• The power of a quotient is equal to the quotient

(a : b)n = an : bn

• To multiply two powers with the same base,

am . an = am+n

• To divide two powers with the same base, we

am : an = am-n

• To raise a power to another power, we keep the

(an)m = an · m

• Any number (not zero) raised to the power of

a0 = 1 (a ≠ 0)

of the powers of its factors.

of the powers of the dividend and the divisor.

we keep the same base and add the exponents.

keep the same base and subtract the exponents.

3 Reduce to a

power of one. a) 22 · 32 b) 79 : 73 c) 90 · (43)2

same base and multiply the exponents.

zero is equal to one.

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

b2 = a

a=b

Root

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

4 Calculate the integer

Radicand

root of the following numbers. a) 50 b) 35 c) 42

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

9=3

121 = 11

400 = 20

• The closest natural number that is less than the original number is called the approximate root.

40 ≈ 6

The approximate root of 40 is 6.

5

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