M¡ra: Mathematics 1. Secondary DB (muestra)

MUESTRA

N O I T A C U D E Y R SECONDA

S C A I C T I T A M Á E M MATEH José Colera J. Ignacio Gaztelu A. Ramón Colera C.

CLIL

1

1

NATURAL NUMBERS

LEARNING SITUATION

Watch the video

Vicky's skateboard Have you ever planned a project? Think about a real or imaginary 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!

Before starting... review what you know

DO THE

QUIZ!

You want to buy a new skateboard. It costs €165. You only have €56 in your piggy bank. Considering this data, answer the questions.

Income • Allowance → €15 a week • Money for chores → €60 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?

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Expenses • Going out with friends €5-€12 a week • Other expenses (sweets etc.) €4 - 5 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 1-minute explanations videos

What are you going to learn to achieve

Your final

CHALLENGE

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

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

EXPRESSIONS WITH COMBINED OPERATIONS • Order of combined operations

MY MATHS MATHS DICTIONARY MY 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?

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1

Numeral system

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 6

8

9

7

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 GRUPO ANAYA, S.A., 2026 - C/ Valentín Beato, 21 - 28037 Madrid. their level. The Mayan numeral system was partly additive and partly All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or positional. by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the publishers.

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U1

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.

M

H TH

T TH

TH

H

T

U

4

7

8

4

3

0

4

↓ 4 000 000 U

↓ 4 000 U

2 T Th → 20 000

↓ 4U

7 Th →

7 000

4H→

400

7T→

70

3U→

+

3

27 473

The value of 4 changes according to its place value.

THEORY INTO PRACTISE

1 Think about the decimal numeral system. a) How many tens are there in 3 thousands?

4 Translate the following Mayan numbers into the decimal system.

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

T

U

5 Look at the Mayan numbers below, they are in order from lowest to highest. Add four elements to the left of the series. Now, add four to the right.

× 10 × 10

1

0

0

1 Th = 100 T

6 Complete the following in your notebook.

LET’S PRACTISE LET’S PRACTISE

a) 500 T = … H = … Th

2 Write the following numbers using the Egyptian numeral system: 19, 65, 34 120 and 2 523 083.

c) 6 Th = … H = … T

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

3 The following symbols are from an additive system:

1 5 10 10 Write the following numbers using this system: 7, 12, 84 and 1264

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 to 40 001. Check it and find 3 more numbers.

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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

Second ball

Left

We will do an experiment: there is a box with three balls. Their colours are red, yellow and blue. A tree diagram shows all the possible results. Let's see the probabilities of a particular result.

1. Take out two of the balls, one by one.

2. If we repeat the experiment six times, how many times, theoretically, will the red ball be left in the box?

3. We can see that the red ball is left in the box in 2 of the 6 results. So, if we repeat the experiment 6 times we can expect that in 2 of them the ball left in the box will be red.

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LET’S PRACTISE PRACTISE 1LET’S Olivia 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

2 Luca 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.

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

Years

1

2

Flowers

2

2·2=4

…

4 · 2 = 8 8 · 2 = 16

2

…

3

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?

4

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

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

1

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

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?

27

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 trillion 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 (8 100 000 000 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.

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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?

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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 PRACTISE

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 38 3 34 8 85 4 42 5 53 2 23 3 +1 +1+1

8 ≥ 58 ≥ 5 cm 8 ≥ 5

Tens of thousands

Thousands

38 3 34 8 85 4 42 5 53 2 23 3 = ==

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

dm

4 <45<4 5< 5

38 3 34 8 85 4 42 5 53 2 23 3 +1 +1+1um

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

5 ≥55≥5 5≥ 5

... ... ...0 0 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 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

4 Look at the news headline below. Round the number of tourists to the nearest million. Then round the amount they spent to the nearest thousand millions.

In 2024 93 800 000 tourists visited Spain. They spent 126 282 million Euros.

5 Round the following numbers to the nearest millions. a) 24 356 000

b) 36 905 000

c) 274 825 048

6 The sign below shows the price of a house in Euros. €138 000 FOR SALE €138 300 €138 290 €140 000 a) Which of the three approximations is closest to the real value? b) Which approximation would you use in an informal conversation?

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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:

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

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

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

1 Calculate.

3 Transform.

a) 254 + 78 + 136

b) 340 + 255 – 429

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

c) 1 526 – 831 + 63

d) 1 350 – 1 107 – 58

b) This subtraction into an addition: 22 – 2 – 6 = 14

2 Estimate and then check the answer.

Lucía's grandparents bought her three video games for €167, a console for €235 and headphones for €32. How much did they spend? a) They spent around €350. b) They spent around €450. c) They spent around €550.

30

4 If Dani was 15 years older, he would still be 18 years younger than his Dad. His dad is 51 years old. How old is Dani?

5 If I buy a mountain bike, I will have €246 left in the bank. If I want to buy a electric scooter too I will need another €204. Is it possible to know the price of either item? And, if so, what is the price?

U1

Multiplication If a ticket to the football game from the previous page costs €65, the total price for all 20 582 tickets is: 65 + 65 + 65 + … + 65 = 65 · 20 582 = €1 337 830 20 582 times

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

Properties Commutative property

Associative property

Distributive property

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

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

LET’S PRACTISE

6 Complete the following multiplications in your notebook. × 2

9

5

+ 9 0

×

+

1 2 6 0

8

2 8 7 4

2 9 934

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.

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

a) 19 · 10

e) 25 · 11

b) 12 · 100

f) 33 · 11

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?

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?

31

5 Basic operations with natural numbers

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 5625 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 5 0 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 5 3 7

There are two types of division depending on the value of the remainder: • Exact division (the remainder is zero).

D 0

d q

⟶ The dividend is equal to the divisor multiplied by the quotient.

D=d·q • Integer division (the remainder is not zero).

D r

32

d

q

→ The dividend is equal to the divisor multiplied by the quotient, plus the remainder.

D=d·q+r

U1 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 0

48 0

3 8

6 8

If we double the amount of water and the number of plants, the amount of water each plant receives does not change. ×7

In a division, if we multiply the dividend and divisor by the same number, the quotient does not change.

32 0

8 4

×7

224 56 00 4

LET’S PRACTISE

12 Find the quotient and remainder of each division. a) 96 : 13

b) 713 : 31

c) 5 309 : 7

d) 7 029 : 26

e) 49 896 : 162

f) 80 391 : 629

13 Follow the example. Use mental arithmetic to divide these numbers. : 12

• 96 :3

32

8 :4

a) 60 : 12

b) 180 : 12

c) 300 : 12

d) 75 : 15

e) 90 : 15

f) 180 : 15

g) 180 : 30

h) 240 : 30

i) 390 : 30

14 Copy and complete the diagrams below in your notebook.

(36 : 12) : 3

36 : (12 : 3)

:

:

16 Find the missing value in each division.

Dividend

53

1 000

39

15

12

Divisor 38

17 Solve the following problems without making notes. 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?

18 A school library pays €1 470 for 35 boxes of textbooks. a) How much did each box cost? b) The boxes are put on to shelves. Six boxes can fit on one shelf. How many shelves are there?

19 "Your final challenge" Raúl has got €65. He is saving to buy a skateboard that costs €105.

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?

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?

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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 PRACTISE

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

4 · 10 – 8 · 3 + 2

– (

+2

+2

18

26

+2

4·

4 · 10 – (8 · 3 + 2) 4 · 10 – 8 · (3 + 2) 4 · (10 – 8) · (3 + 2)

·3+2

–

34

4 · (10 – 8) · 3 + 2

– 8 ·

4 ·

–

–

4

14

0

+ 2)

· · 40

U1

2 Copy and complete in your notebook. Check your answers with a calculator. Make sure you enter the operations in the correct order.

Help

≤ → Add screen value to memory. µ → Subtract screen value from memory. Ñ → Recover memory value.

40 – 12 : 4 + 2 · 3

40 –

+ +

6 Calculate. Write down the steps you followed.

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

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

: 4 +

A sound technician worked at a music festival this month for 7 hours a day for 12 days. On another 5 days he worked for 6 hours, and another 5 night shifts, of 3 hours. How many hours did he work in total during the month?

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

+

normal rate

night pay

12 · 7 + 5 · 6 + 5 · 3 = 84 + 30 + 15 = 129 40 - 12 =/ 4 ≤ 2 * 3 ≤Ñ= {∫∫∫∫∫∫‘«}

12 days

5 days

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

LET’S PRACTISE

3 Follow the examples to solve the following operations.

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

b) 15 – 10 : 5

c) 4 · 6 – 13

d) (15 – 3) : 4

e) (8 + 2) · 3

f) 18 : (10 – 4))

4 Calculate the following using mental arithmetic.

Compare your results. a) 2 + 3 · 4

(2 + 3) · 4

b) 6 – 2 · 3

(6 – 2) · 3

c) 18 – 10 : 2

(18 – 10) : 2

d) 24 : 6 + 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

b) 3 · (10 – 7) + 4

c) 4 + (7 – 5) · 3

d) 18 – 4 · (5 – 2)

e) 8 – (9 + 6) : 3

f) 22 : (7 + 4) + 3

8 Write the following statements as mathematical expressions and solve them. a) For a birthday party, we buy 4 boxes of pizza, 6 boxes of pastries, and 3 boxes of cakes. The pizza boxes weigh 3 kg, and the others weigh 2 kg each. How many kilograms of food did we buy in total? b) Marco buys some snacks for him and his friends.

He buys 5 packs of chocolate milkshakes, 3 packs of strawberry milkshakes, and 2 packs of vanilla milkshakes. Each pack contains 6 bottles. How many bottles are there in total? 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) In a bakery, they make 60 muffins and put them on trays. Each tray holds 12 muffins. They also make 24 gluten-free muffins, they put them on trays that hold 6 muffins. How many trays have they used in total?

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MY SU AL UAL VIS MY VI SU A RY MMAR SUMM 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 an additive system. This means they add the necessary symbols to get the number they want.

30 000

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

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

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

36

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.

U1

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

3 8 4 5 2 3

3 8 4 5 2 3 +1

Thousands

Ten thousands

=

8≥5

4<5

3 8 0 0 0 0

4 0 0 0 0 0

5≥5

+1

3 8 5 0 0 0

Basic operations with natural numbers Commutative property

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

1482

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

576 + 1333

1909

1909

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

576 + 906 = 906 + 576 1482

Remember Relationships between addition and subtraction: M=S+D M–S=D →* S=M –D

Associative property

Multiplication means repeatedly adding the same value.

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

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) 880 × 3 = 16 × 165 2640 = 2640

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

Exact division The dividend is equal to the divisor multiplied by the quotient (the remainder is 0)

D=d.q+r

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 37

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

2

3894 FBG

b) 35

d) How many cars have the same letters as the blue plate after it?

Write the following in Roman numerals. a) 87 b) 425

Addition and substraction

c) 2 600

10

d) 54 528 4

Write the number ‘fifty-seven’ using at least three different numeral systems.

5

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

11

Rounding Copy and complete the table in your notebook.

12

Rounding Number

4389 GFB

c) How many plates are there between the red and the green ones?

c) 2 130

6

E

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

a) 48

7

E

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

Write the following numbers using the Egyptian additive system.

3

3948 FBG

To the nearest hundreds of thousands

Calculate. a) 6 070 + 893 + 527

b) 651 + 283 – 459

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. a) 47 – (35 – 28)

To the nearest million

b) 52 – (36 – 27)

2 830 554

c) 128 – (86 – 45 – 12)

19 270 000

d) 237 – (152 + 48 – 14)

399 675 000

e) 348 – (148 – 86 + 29)

You see a new mobile phone you like, it costs €293.50. You want to tell a friend, but you can't remember the exact price. Which of the following sentences would you say? Why?

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

Multiply.

a) It is almost three hundred euros.

a) 16 · 10

b) 128 · 10

c) 60 · 10

b) b) It is just over two hundred euros.

d) 17 · 100

e) 85 · 100

f ) 120 · 100

c) c) It is about two hundred and ninety euros.

g) 22 · 1 000

h) 134 · 1 000

i) 140 · 1 000

38

U1

14

TRAINING AND PRACTICE

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

21

a) One million is equal to one thousand hundreds.

Copy and complete in your notebook.

5

8 6

8 2

3

b) One hundred million is equal to one thousand hundreds of thousands.

14 9

c) One thousand times a million is equal to one giga.

5

d) One hundred gigas are equal to one trillion.

6

e) One trillion is equal to one million millions.

7 6 16

17

22

Mental arithmetic. a) 3 · (10 : 5) b) (4 · 6) : 8

c) 20 : (2 · 5)

d) (30 : 5) · 3

f )(40 : 8) : 5

e) 10 : (40 : 8)

23

b) One kilo of almonds costs €12. How much do 5 kilos cost?

20

The table below contains data on vegetable consumption in Spain in 2025: Weight (tonnes)

Value (thousands of €)

c) There are 24 cans of soft drinks in a box. How many cans are there in 10 boxes?

Fresh fruit

3 605 334

7 288 242

d) It costs €360 to get four new tyres for your car.

How much does each tyre cost?

Vegetables and potatoes

335 756

5 843 489

Total

6 956 090

13 131 731

Combined operations

19

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?

Mental arithmetic. a) One bucket holds 5 litres of water. How many buckets hold 100 litres of water?

18

True or false?

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

b) 8 : 4 + 7 – 3

c) 15 – 2 · 3 – 5

d) 10 – 12 : 6 – 4

e) 22 – 6 · 3 + 5

f ) 8 + 10 : 5 – 10

g) 36 – 8 · 4 – 1

h) 11 – 2 – 9 : 3

Calculate. a) 2 · (4 + 6)

b) 2 · 4 + 6

c) 8 : (7 – 5)

d) 5 · 7 – 5

e) (5 + 6) · 4

f) 5 + 6 : 3

g) (19 – 7) : 2

h) 18 – 7 · 2

24

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

25

Copy, calculate and complete in your notebook.

26

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

a) 48 + … = 163

b) … + 256 = 359

c) 628 – … = 199

d) … – 284 = 196

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 39

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

29

34

A

18

a) 60 : 5

b) 80 : 5

c) 120 : 5

d) 140 : 5

e) 170 : 5

f ) 200 : 5

g) 210 : 5

h) 340 : 5

i) 420 : 5

Copy and complete in your notebook. a) 6 · (8 + 2) = 6 · 8 + 6 · 2 = … · … = …

INTERPRET, DESCRIBE AND EXPRESS

b) 5 · 9 – 5 · 6 = 5 · ( ... – ...) = 5 · … = …

35

Which property did you use?

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.

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

III. Ernest buys a T-shirt for €16 and a hat for €4.

He pays with a €50 note.

True or false?

IV. Last night, 50 clients stayed at a hotel. This morning, 16 more people arrived and 4 people left.

a) We get the same results by multiplying a number by three and by doubling the number and adding it to the original number.

a) 50 – 16 – 4

b) 50 – 16 + 4

b) Three times fifteen is the same as fifteen times three.

c) 50 – (16 + 4)

d) 50 – (16 – 4)

c) Multiplying by ten is the same as multiplying by five twice.

e) 50 + (16 – 4)

f) 50 + 16 – 4

d) Multiplying by ten is the same as multiplying by five, then by two.

36

33

40

Calculate. a) 4 · 7 – 13 – 2 · 6

b) 15 : 3 + 7 + 4 : 2

c) 5 · 4 + 12 – 6 · 4

d) 12 : 4 – 1 – 6 : 3

e) 5 · 6 – 4 · 7 + 2 · 5

f) 9 : 3 + 8 : 4 – 7 : 7

g) 8 · 8 – 4 · 6 – 5 · 8

h) 18 : 2 – 12 : 3 – 6 : 2

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?

e) The commutative property only applies to even numbers.

32

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.

d) 7 · 12 – 2 · 12 = (… – …) · 12 = … · … = …

31

B

·2

9

c) (10 – 8) · 4 = 10 · 4 – … · 4 = 40 – … = …

30

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

37

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)

Read the problem and look at the resolution.

Can you explain what each operation shows?

a) 7 · 4 + 2 · (10 – 6)

b) 8 : 2 + 4 : (9 – 5)

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?

c) 5 · (11 – 7) – 6

d) 12 : (11 – 7) – 2

Answer:

e) 3 · (8 – 5) – 12 · (6 – 4)

f) 15 : (2 + 3) + 5 · (2 + 2)

1st 168 : 2 = 84

2nd 84 · 4 = 336

g) 9 : (6 – 3) + 8 · (6 – 5)

h) 6 · (9 – 5) – 18 : (9 – 7)

3rd 137 · 2 = 274

4th 336 + 274 = 610

i) [(24 : (9 – 2 – 3)] : 2

j) 24 : [(9 – 2 – 3) : 2]

5th 714 – 610 = 104

6th 104 : 4 = 26

Calculate.

U1

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?

It took 14 months to make a game Jumping Frogs. Work started on Jumping Frogs 4 months after making the Home Gym app, which took 15 months to make. Jumping Frogs was finished in June. In which month was Home Gym finished?

50

39

Last week, Ana sent 40 text messages. She sent five to her brother, Álvaro. She sent her parents three more messages than she sent to Álvaro. 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 her profit be?

40

Twice as many girls as boys are in a musical.

If there are 36 in total, how many girls are there?

And boys?

41

For the end of year school party we bought 15 boxes of orange juice and 12 boxes of lemon juice.

Each contains 24 bottles. How many bottles are there in total?

42

Eva is two years older than her brother Lucas.

She is two years younger than her sister Manuela.

Their father 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

Érika and Gabriel go for a run, they start at the same time from home. Érika runs at 10 km/h and Gabriel at 12 km/h. How far apart will they be after an hour and a half if they run in the same direction?

45

A company bought 250 kg of wax to make candles weighing 200 g each. During the manufacturing process, 17 candles were damaged.

How many candles can they sell?

46

In a science project, students plant 50 trays with 100 seeds each. On average, 20 seeds in each tray are damaged. How many plants will the students get?

47

Clara gets €168 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

"Your final challenge" 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

"Your final challenge" 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?

41

Exercises and problems 53

To make money for a holiday, a group of students buy 150 boxes containing 30 pens each for €2 000. Three hundred pens are broken and so discarded. The rest are stored in packs of five pens.

They sell them for €4 per pack. How much profit do they make from the sale?

54

In a pottery class, they make five trays of pots every day. Each tray holds three dozen pots. There are no classes on Sunday. How many pots do they produce per week?

55

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 mobile phone shop sells 15 660 mobile phones between January, February and March.

How many mobile phones does it sell on average 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

At a burger restaurant you can choose between five types of meat, three types of bread, and the option of adding cheese. How many different ways can you choose a burger?

64

It is possible to form four different numbers of three figures only using zeros and ones. st ª 11.

THINK A LITTLE MORE 56

57

58

In 2024, Disneyland Paris got an average of 1 342 874 visitors per month. This is four times more than the number of visitors in 2020 due to COVID-19. Rounded to the nearest hundred thousand, answer the following questions: a) Approximately how many visitors were there in 2024? b) How many visitors were there on average a day in 2024? c) Using the data in the statement, estimate the number of visitors during 2020.

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

"Your final challenge". 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?

42

1 0 1 0

1

1

0

111 110 101 100

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

65

Álvaro, Bárbara, Carmen and Dani 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 Álvaro sits in the 1st seat?

A group of 54 students go on an end-of-year trip to the Pyrenees to go rafting. Each raft can only hold 4 students and the instructor, how many rafts 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?

33.rdª

ndª 22.

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

Rubén, Ana and Marcos go shopping. Rubén spends €30 more than Ana and €40 less than Marcos. In total, they spend €208. How much does each friend spend?

U1

68

At a magicians' convention, the need 150 dozen decks of cards. They already have 40 boxes of 25 decks each. How many more boxes are needed?

69

Olivia sold 21 3D-printed figures, including characters and creatures, for €350. She sold double the number of characters as creatures. One creature costs three times the price of a character.

How much does a character cost? How much does a creature cost?

70

71

72

74

grey

75

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?

My smart watch tells me I move 80 cm with each step. If I walk at a pace of 100 steps per minute, can I walk 10 km in 2 hours?

Solar panels will be installed on a field, its size is 150 m × 300 m. The panels will be in rows and columns parallel to the fences around the field. There is a distance of 5 m between each row. The distance between the first row and fence is also 5 metres, How many solar panels can be installed?

15

20

15

white

green

blue

red

others

1st year secondary school students were surveyed about their 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?

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.

Draw a grid to help you. For example: 20

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

30

— 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?

20 30

Draw a Venn diagram like this one to help you.

20

year 1 CaFeteria

sChool bus

eXtra aCt.

73

Martina added the first seven natural numbers together as follows: 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 Can you add up the numbers 1 to 100?

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?

43

anayaeducacion.es Answer Key.

Review

GO TO THE ESCAPE ROOM AND TEST YOURSELF!

1 Complete the following table in your notebook:

a) 154 ·

Numeral systems Egyptian

b)

: 27 = 98

3

4

c) 30 275 :

= 35

5 6

8

9

d) 1 508 =

· 125 + 8

7

15 16 17 18 19

Decimal

= 462

0 1 2

10 11 12 13 14

Mayan

10 Copy and complete in your notebook.

528

2 Which of the above systems are additive? Which systems are positional? What is the difference?

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

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 June 2025 was 8 062 923 489 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. a) 18 · b)

= 180 · 100 = 27 000

c) 4 000 : d)

44

= 40

: 10 = 38

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?

Two 1st year secondary school classes sell raffle tickets for a school trip. In class A there are 13 volunteers and class B, only 7. Each volunteer sells an average of 35 raffle tickets.

They can buy one bus ticket with the profit from the sale of 10 tickets. How many bus tickets will they get? a) (13 + 7 + 35) · 10 b) (35 · 13 + 35 · 7) · 10 c) (35 · 13 + 35 · 7) : 10

U1

Your final challenge

Create

€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 her piggy bank. She analyses her finances to plan the purchase.

Income

Expenses

• Weekly allowance ➝ €15.

• Between €5 and €12 a week to go out with her friends.

• Between €0 and €20 every two weeks for helping her neighbours (watering plants, walking dogs, giving private classes, etc.).

• Between €16 and €20 a month for other expenses (sweets, comics, etc.).

• Vicky’s grandma gives her €20 a month.

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 her savings ➝ 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 w w w.anayaeduc acion.es

Now, it's your turn...

1 Follow the same process. Calculate your finances (savings and income). How long will it take you to buy the item of your choice?

2 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.

45

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