Global Thinkers: Mathematics 1. Secondary Teacher's Guide (sample)

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THINKERS

© GRUPO ANAYA, S.A., 2024 - C/ Valentín Beato, 21 - 28037 Madrid.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the publishers.

INDEX PRESENTATION

4

THE KEYS OF THE PROJECT MATERIALS FOR THE STUDENTS MATERIALS FOR THE TEACHERS DIGITAL PROJECT

LEARNING EXPERIENCES

34

TRAIN YOURSELF BY SOLVING PROBLEMS

10

NATURAL NUMBERS

24

POWERS AND ROOTS

46

DIVISIBILITY

62

INTEGERS

82

DECIMAL NUMBERS

108

THE METRIC SYSTEM

126

FRACTIONS

144

OPERATING WITH FRACTIONS

158

PROPORTIONALITY AND PERCENTAGES

174

ALGEBRA

192

LINES AND ANGLES

220

GEOMETRIC SHAPES

238

AREAS AND PERIMETERS

264

GRAPHS OF FUNCTIONS

286

STATISTICS

302

1

NATURAL NUMBERS Vicky's skateboard Video. SDG 12: Responsible consumption and production Quiz. Previous knowledge

NUMERAL SYSTEMS

SELF-ASSESSMENT Escape room!

• The decimal numeral system • Value of the digit of a number

EXPRESSIONS WITH COMBINED OPERATIONS • Order of combined operations SOME COUNTING TECHNIQUES

BASIC OPERATIONS WITH NATURAL NUMBERS • Addition and its properties • Multiplication and its properties • Division as sharing and grouping LARGE NUMBERS

• Exact division and integer division • A property of division

ROUNDING NATURAL NUMBERS • Rounding natural numbers 2

NATURAL NUMBERS

t are y ou ha le ar n g W to to

1-minute explanations videos

in

g

TAKE ACTION

12. Responsible consumption and production

Watch the video.

?

LEARNING EXPERIENCE

PROJECTS AND PLANNING

Natural numbers

Learning by doing videos

o

1

NUMERAL SYSTEMS

COUNTING SYSTEMS

BASIC KNOWLEDGE

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

ha

t ca

n y ou

Rounding natural numbers d o to

W

LARGE NUMBERS

TAKE ACTION

?

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!

BASIC OPERATIONS WITH NATURAL NUMBERS

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

DO THE QUIZ!

BEFORE STARTING… REVIEW WHAT YOU KNOW

You want to buy a new skateboard. It costs €165. You only has €96 in your piggy bank.

Considering this data, answer the questions.

Income • Allowance → €15 a week • Money for chores → €20 a week

Expenses

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.

Order of combined operations

MY MATHS DICTIONARY

• Going out with friends €2-€10 a week • Other expenses (sweets etc.) €6 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?

EXPRESSIONS WITH COMBINED OPERATIONS

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 up/down

What is the answer rounded down to the nearest hundred?

24

25

Opening video anayaeducacion.es Watch the video ‘Vicky's skateboard’ about the learning experience.

SDG commitment anayaeducacion.es Watch the video on the sustainable developmen goal 12: Responsible consumption and production.

Gamification anayaeducacion.es Do the quiz ‘Previous knowledge’.

• Structure of the decimal numeral system.

• Reading and writing large numbers.

• Rounding.

• Mental and written arithmetic with the four operations.

• Basic use of the calculator.

• Solving simple expressions with combined operations.

• Solving problems with one and two operations.

LEARNING EXPERIENCE

Before starting the analytical and structured study of the contents, this first double page presents an open and flexible work proposal which will have its definitive resolution at the end of the learning experience. The objective is aimed at the perception of the usefulness of what has been learnt and to reinforce it through practice.

The presentation of this learning experience is divided into three sections: — An initial video, with some reflection on the meaning of the work to be tackled and an example in a context close to the students' world and interests.

The example proposes a problem in a real context: the planning of the purchase of a skateboard, taking into account the missing quantity, in possible weekly savings, which can vary, and from this data, assessing the time needed to achieve the objective.

— The wording of the proposal, in which it is suggested, in principle, to follow the same line as the example. However, it will be at the teacher's own criteria to adapt it to the programme or to the format they consider appropriate.

— A series of activities propose the handling of basic operations with natural numbers, being necessary to combine them on different occasions. From the first activity, students will move in open situations, with data that can vary between certain values that will change depending on decisions to be made about income, expenses, savings, time, etc., as occurs in everyday reality, and which, depending on how they are specified, will give rise to different solutions.

ANSWER KEY

1 (15 + 5) − (10 + 6) = 20 − 16 = €4 2 Considering as expendable those of the ‘others’ section and half of the expenses with the gang: (15 + 5) – (5) = 20 – 5 = 15 3 (165 − 96) : 15 = 69 : 15 → 5 weeks 4 Saving €7 per week, (165 − 96) : 7 = 69 : 7 → 10 weeks MY MATHS DICTIONARY Key language

In context

Add up (to)

40 plus 60 add up to 100.

Take away

38 take away 5 is equal to / equals 33.

Times (by)

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

Divided by

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

Round up/down

What is the answer rounded down to the nearest hundred?

3

NATURAL NUMBERS

1-minute explanations videos

g

TAKE ACTION

12. Responsible consumption and production

Watch the video.

?

LEARNING EXPERIENCE

PROJECTS AND PLANNING

1. Numeral systems

Learning by doing videos

o

t are y ou ha le ar n g W to to

in

1

NUMERAL SYSTEMS

COUNTING SYSTEMS

TEACHING NOTES

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

ha

t ca

n y ou

Rounding natural numbers d o to

W

LARGE NUMBERS

TAKE ACTION

?

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!

BASIC OPERATIONS WITH NATURAL NUMBERS

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

DO THE QUIZ!

BEFORE STARTING… REVIEW WHAT YOU KNOW

You want to buy a new skateboard. It costs €165. You only has €96 in your piggy bank.

Considering this data, answer the questions.

Income • Allowance → €15 a week • Money for chores → €20 a week 1. What is the minimum amount you can save a week? Look at your expenses.

Are they all necessary? What is the maximum you could save by cutting your expenses?

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

EXPRESSIONS WITH COMBINED OPERATIONS

Order of combined operations

MY MATHS DICTIONARY Key language

In context

Add up (to)

40 plus 60 add up to 100.

Take away

38 take away 5 is equal to / equals 33.

Times (by)

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

Divided by

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

Round up/down

What is the answer rounded down to the nearest hundred?

24

25

Learning by doing anayaeducacion.es Watch the video ‘The decimal numeral system’ to see a practical example of this.

Learning by doing anayaeducacion.es Watch the video 'The value of the digit of a number' to see a practical example of this.

Using different numeral systems that were developed in different periods and cultures will help students understand the continuous effort done to construct tools we use today without even thinking about the difficulties in the process, and that form part of our cultural heritage, which is continuously changing and is passed from one generation to the next.

We can also highlight the fact that each culture used a numeral system that met their needs. We cannot think of any situation in which a primitive human being, who was a hunter and gatherer, would have to handle number with more than seven figures. But we simply have to open any journal or scientific treatise to see that these are the same numbers that are essential for today’s society. By this, we mean that numeral systems were perfected as needs to number and calculate developed (trade, building, statistics, etc.). At the same time, each development has enabled new scientific fields to open up and has brought with it the emergence of new numerical requirements.

In order to fully appreciate the advantages of our decimal numeral system, we must compare it to other systems, especially additive systems. Show the difficulty of representing large numbers and decimal numbers with these systems, and also the difficulties of operating with them.

We recommend emphasising that the use of positional systems was a huge leap forward in the use of more concise symbols and their ability to express amounts. The importance of the late appearance of zero will also be noted. It was an abstract symbol used to fill a place where there was nothing, but at the same time, it was key for the development of these systems. Zero allowed different values to be assigned to the figures.

ANSWER KEY

1 a) 3 thousands are 300 tens. b) 1 ten of thousands are 100 hundreds. c) 5 one million units are 50 000 hundreds.

2 19 = 65 = 34 120 = 2 523 083 = 3 7= 12 = 4 6

84 = 126 = 11

120

126

5 To the left:

To the right:

6 a) 500 T = 50 H = 5 Th c) 6 Th = 60 H = 600 T

b) 3 000 H = 300 Th = 30 T Th d) 8 H Th = 80 T Th = 80 000 T

7 a) True

b) True

c) False

8 40 001, so 41 000 – 40 001 = 999

4

d) False

e) True

2

Counting systems

Let’s practise!

1 Martha is looking out of her window. She is counting

We use natural numbers for simple calculations. For more difficult calculations we have counting systems.

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.

Different systems Data collection

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.

Black

13

Grey

9

12

year 0

Blue

10

Yellow

3

Green

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

oTHer

0

Year 2. Group A

oTHer

12

13

0

Year 2. Group B

oTHer

11

16

1

Other

15

ToTal

82

12

1

2

3

4

…

Flowers

2

2·2=4

4·2=8

8 · 2 = 16

…

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

Read the questions. Explain your answers.

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

a) What is the probability that the outcome (result) is one blue and one red ball? a) How many different combinations of clothes can he wear?

Second event

b) What is the probability that the outcome is two blue balls or two red balls?

Outcome

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?

28

Years

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

0

Example First event

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

And after ten years?

oTHer

12

Tree diagram

A tree diagram can show us the probability of a particular result. Here we can see that the probable outcome of a red ball is 2 out of 6.

year 1

3

Red

A data table can show us the frequency distribution of a particular value.

Look at the following example:

13

White

6

celia aiTor

Data table

TEACHING NOTES

year 2

nerea

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

14

2. Some counting techniques

year 3

Example

c) Now you take three balls, not two. What are the probablities this time?

29

This section shows an aspect of natural numbers, different from calculus or problem solving, which will be dealt with in the following sections.

The aim here is to show strategies that facilitate the task of counting sets or subsets of a total, providing information about their composition and structure. Each of these techniques is introduced with an example and reinforced with activities in the ‘Let’s practice!’ section.

Data counting is used to collect data coming in successively, and offers the possibility of sorting them into pre-decided sections.

The double-entry table is used to classify and analyse the elements of a set by simultaneously using two different criteria. It is sometimes combined with the counting technique mentioned above. It is often used as a resource for sorting and for displaying results in statistical work.

The tree diagram is used as an organising resource in the study and analysis of all possible cases of a situation. It is frequently used in probability calculations.

ANSWER KEY

1

White

12

Black

3

Grey

26

Red

8

Blue

10

Yellow

3

Green

5

Others

15

total

82

2 a) 2 · 2 · 4 = 16. 16 ways b) 2 · 3 · 4 = 24. 24 ways c) 3 · 3 · 4 = 36. 36 ways 3 64 flowers in the sixth year and 1024 flowers in the tenth year.

4 a) Theoretically, out of 12 possible cases, in 6 of them, two balls of different colours remain in the urn. b) n the other 6 cases there are two balls of the same colour. That is, half of the time in each case. c) Out of 24 possible cases, only in 6 cases would the red ball remain in the urn and, therefore, the three blue balls would be left out. In other words, this will theoretically happen one out of every four times the experience is repeated.

5

3

4

Large numbers

1

0

0

0

Tens

0

UniTs

Millions

8

HUndreds

3

THoUsands

Trillion

THoUsands

1

of Millions

…

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

3. Large numbers

Rounding natural numbers

There are numbers with more than nine figures. We can use the decimal number system to write a number with all the figures we want. The table below shows the place value of some numbers with more than 9 figures: There are 1 046 992 electric vehicles in Spain today.

There are approximately 1 500 000 electric vehicles in Spain today.

=

TEACHING NOTES

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 numbers to the right of that place value to zeros.

The Universe formed thirteen thousand eight hundred million years ago.

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

The volume of Earth is approximately one trillion cubic kilometres.

• If the first number we are changing is a five or more we round the next figure up. This means we add another digit to that number.

Theory into practice • One million ↔ A 1 followed by 6 zeros.

4 Look at the news headline below. Round the number

1 Look at the diagram below. Round the number

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

• One trillion ↔ A 1 followed by 12 zeros.

• One quintillion ↔ A 1 followed by 18 zeros.

The English words billions, trillions, quadrillions are false friends. They have a different meaning in Spanish and in English.

Hundred of thousands

Tens of thousands

3 83 348 854 425 532 23 3 8 ≥ 85 ≥8 5≥ 5 cm

+1 +1+1

Thousands

3 83 348 854 425 532 23 3 = = =

dm4 < 45 <45< 5 ... ...0... 00 00 00 00 0

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

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

5 ≥ 55 ≥5 5≥ 5

3 Copy and complete in your notebook.

the following numbers in figures. a) The formation of Earth. b) The numbers of seconds in a century. c) The number of kilometres in a light year.

5 Round the following numbers to the nearest millions.

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

a) One thousand thousand = one... b) One thousand million = one... c) One million million = one...

a) 24 356 000

million million cells. Write both figures in trillions.

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.

5 What is the word for a 1 followed by 30 zeros?

6 Scientists estimate that there are 1.2 million

quadrillion litres of water in our seas and oceans.

What do you think a quadrillion is?

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

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

4 The human body has between ten and seventy

2 Write the following numbers in figures.

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

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

Let’s practise!

1 Read the information in the boxes above. Then, write

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

d) 29 935 236

a) Which of the three approximations is closest to the real value? b) Which approximation would you use in an informal conversation?

7 The town hall has a budget of € 149 637 to renovate

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

30

31

MY MATHS DICTIONARY

Use the plural forms -s + of + noun for large, non-specific numbers.

Examples: • Millions of people, billions of cats, trillions of stars, etc.

• There are millions of people in Spain. vs.

There are 47.42 million people in Spain.

Large number (with six, nine, twelve and more figures) often appear in scientific, sociological, economic data and many more. This is why it is important to prepare and interpret messages related to resources that students already use.

Students must read and write numbers with many figures and work with the corresponding place values (millions, thousands of millions, trillions, etc.) and their equivalents with ease.

In the next learning experience, they will learn to use the abbreviated notation for these place values with the help of powers of base ten.

We also recommend reinforcing the difference between the Spanish term billón and the term ‘billion’ which tends to appear in English texts and media and often leads to mistakes in translations. Counterintuitively, ‘billion’ is the same as a thousand millions (mil millones). In an attempt to differentiate it from billón, and to have an equivalent term in translations, the new term milliard (mil millones) has been coined, although it is not often used.

ANSWER KEY

1 a) Seven thousands of millions b) Three thousand, one hundred fifty-three million six hundred thousands c) Nine trillions, four hundred and sixty thousand eight hundred millions 2 a) 28 350 000 d) 16 000 000 000

b) 143 000 000 e) 1 500 000 000 000

3 a) Million

b) Thousand millions

c) 2 700 000 000 c) Trillion

4 Between 10 and 70 trillion cells 5 Ten thousand trillions 6 A 1 followed by 24 zeros → a trillion trillions

3

4

Large numbers

3

8

0

0

0

0

0

0

Tens

0

UniTs

HUndreds

Millions

THoUsands

Trillion

THoUsands

1 1

of Millions

…

4. Rounding natural numbers

Rounding natural numbers

There are numbers with more than nine figures. We can use the decimal number system to write a number with all the figures we want. The table below shows the place value of some numbers with more than 9 figures: There are 1 046 992 electric vehicles in Spain today.

There are approximately 1 500 000 electric vehicles in Spain today.

=

TEACHING NOTES

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

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 numbers to the right of that place value to zeros.

The Universe formed thirteen thousand eight hundred million years ago.

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

The volume of Earth is approximately one trillion cubic kilometres.

• If the first number we are changing is a five or more we round the next figure up. This means we add another digit to that number.

Theory into practice • One million ↔ A 1 followed by 6 zeros.

4 Look at the news headline below. Round the number

1 Look at the diagram below. Round the number

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

• One trillion ↔ A 1 followed by 12 zeros.

• One quintillion ↔ A 1 followed by 18 zeros.

The English words billions, trillions, quadrillions are false friends. They have a different meaning in Spanish and in English.

Hundred of thousands

Tens of thousands

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

8 ≥ 85 ≥8 5≥ 5 cm

3 83 348 854 425 532 23 3 = = =

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

dm4 < 45 <45< 5 ... ...0... 00 00 00 00 0

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

5 ≥ 55 ≥5 5≥ 5

the following numbers in figures. a) The formation of Earth. b) The numbers of seconds in a century. c) The number of kilometres in a light year.

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.

3 Copy and complete in your notebook.

a) One thousand thousand = one... b) One thousand million = one... c) One million million = one...

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

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 30 zeros?

6 Scientists estimate that there are 1.2 million

quadrillion litres of water in our seas and oceans.

What do you think a quadrillion is?

30

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

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

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

Let’s practise!

1 Read the information in the boxes above. Then, write

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

d) 29 935 236

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?

7 The town hall has a budget of € 149 637 to renovate

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

31

Learning by doing anayaeducacion.es Watch the video ‘Rounding natural numbers’ to see a practical example of this.

6

In addition to learning the meaning of the term rounding and become proficient at rounding amounts, students must get used to doing these operations to correctly express, remember or record data related to the information and answers to calculations they use daily.

When we are told on television that, for example, ‘the winners of 14 will collect 119 274 Euros’, we remember, and if necessary communicate, the information: ‘those of 14 will collect 120 000 Euros’. It is a different situation when one of the winners collects the prize.

Here the exact amount will be needed.

In order for the learning to be incorporated into the students’ competences, we can give them the activity of preparing a list of situations (like the one in the example) where rounding is suitable and effective (prices, budgets, statistical population data, economy, etc.).

MY MATHS DICTIONARY

ANSWER KEY

The activity proposes rounding the same number to different place values.

The provided example, as well as a helpful diagram, aims to make the task easier and comprehensively strengthen the method. In addition, comparing the different cases will help create new connections and expound on the structure of numbers.

General useful expressions: • Rounded to • Round up • Round down

1 Hundreds of thousands: 400 000 Tens of thousands: 380 000 Thousands: 385 000

• Rounded to the nearest integer

2 a) 25 000

b) 7 000

c) 40 000

3 a) 530 298 → 500 000 and 530 000 c) 359 481 → 400 000 and 360 000

d) 100 000

b) 828 502 → 800 000 and 830 000 d) 29 935 236 → 29 900 000 and 29 940 000

4 There were 83 000 000 tourists, approximately.

They spent approximately 90 000 000 million Euros.

5 a) 24 000 000

b) 37 000 000

6 a) 138 300

b) 140 000

c) 275 000 000

7 €150 000 5

Basic operations with natural numbers

5. Basic operations with natural numbers (I)

Multiplication If a ticket to the football game from the previous page costs €35, the total price for all 20 582 tickets is:

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

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

Properties

35 + 35 + 35 + … + 35 = 35 · 20 582 = €720 370 20 582 times

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

Commutative property

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

Commutative property

Associative 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.

16 × 55

20 = 20

Subtraction

Distributive property

The product of a calculation does not change if we remove the parentheses.

(a · b) · c = a · (b · c)

a·b=b·a 4·5=5·4

245 + 105

35 · 10

88 × 10

350

350

• Subtraction means to ‘take’ one number from another number. You calculate what is left (the difference).

880

• 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.

6 Complete the following multiplications in your

Remember

notebook.

25 342 ← Minuend (M ) – 20 582 ← Subtrahend (S )

× 2

4 760 ← Difference (D )

+ 9 0 1 2 6 0

M =S +D Relationship between addition and subtraction: M – S = D → * S=M –D Let’s practise!

3 Transform.

1 Calculate.

a) 254 + 78 + 136 c) 1 526 – 831 + 63

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

2 Calculate and check the answer:

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

5

a) This addition into a subtraction: 48 + 12 = 60 b) This subtraction into an addition: 22 – 2 – 6 = 14 4 When Alberto is 15 years older, he will still be 18 years

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

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

bank. If I want to buy a TV too I will need another €204. Is it possible to say how much these items cost?

How much do these items cost?

+

9 8 × 2 8 7 4 2 9 9 3 4

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

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

8 Write a mathematical expression:

To multiply a number by 8, we can multiply it by 10, then subtract double the original number.

Which property describes this?

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

examples to help you.

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

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

11 A farmer has 200 peach trees. He puts the peaches into

boxes. Each box contains 5 kilos of peaches. Each tree uses seven boxes. He sells all the peaches at €2 a kilo.

What is his profit?

32

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Learning by doing anayaeducacion.es Watch the video ‘Addition and its properties’ to see a practical example of this.

Learning by doing anayaeducacion.es Watch the video ‘Multiplication and its properties’ to see a practical example of this.

MY MATHS DICTIONARY

For answers use is equal to or equals not is equals to.

Examples: • Five plus five is equal to ten.

• Five plus five equals ten.

TEACHING NOTES

35 · 7 + 35 · 3 = 35 · (7 + 3)

8 × 2 × 5 × 11

Here we find a space to consolidate learning that started in previous years, which will serve as preparation to tackle operations with integers and fractions later on, where similar techniques will be used to those practised here.

We review the algorithms, as well as the properties and relations to addition and subtraction for two reasons: — Its automatic and spontaneous implementation to improve calculation.

— Its theoretical formalisation (expression with letters) so that students go beyond the specific example and apply them to all numbers.

Understanding properties and implementing them at a practical level must be done at these ages through experimentation and practise, rather than through analytical reasoning.

Therefore, the theoretical explanation will be given after comprehension, and it will be the last step of the learning process.

To support this practical implementation, it is a good idea to point out to students the advantages of applying the properties to make calculating products easier. This is especially important when developing mental arithmetic strategies, as shown in these examples: • The product of 35 × 12 can be made into a simpler one, 42 × 10, by combining the associative and commutative properties.

35 × 12 = (7 × 5) × (2 × 6) = 7 × (5 × 2) × 6 (1) = 7 × 10 × 6 = 7 × 6 × 10 = 42 × 10 (2) (1) Associative property (2) Commutative property • The product 125 × 23 is easier to calculate with the distributive property: 125 × 23 = 125 × (20 + 3) = 125 × 20 + 125 × 3 = 2 500 + 375 = 2 875 To expand upon this content, we propose taking out the common factor. This applies the distributive property in the opposite way to which it is usually found: a · b + a · c = a · (b + c) 7

ANSWER KEY

1 a) 468

b) 166

c) 758

d) 185

2 The correct answer is b) 167 + 235 + 32 = €434.

3 a) 48 + 23 = 60 → 60 – 48 = 12 b) 22 – 2 – 6 = 14 → 14 + 2 + 6 = 22 4 51 – 18 – 15 = 18 years 5 The price of the TV is 204 + 246 = €450.

6

4 5 ×2 8 3 6 0 + 9 0 1 2 6 0 9 5 8 ×7 3 2 8 7 4 + 6 7 0 6 6 9 9 3 4

7 a) 190 d) 1 400

b) 1 200 e) 23 000

c) 15 000 f ) 460 000

8 x · 8 = x · (10 – 2) = x · 10 – x · 2 We have applied the distributive property to this equality.

9 a) 25 · 9 = 25 · 10 – 25 = 250 – 25 = 225 b) 25 · 9 = 25 · 10 – 25 = 250 – 25 = 225 c) 33 · 9 = 33 · 10 – 33 = 330 – 33 = 297 d) 12 · 11 = 12 · 10 + 12 = 120 + 12 = 132 e) 25 · 11 = 25 · 10 + 25 = 250 + 25 = 275 f ) 33 · 11 = 33 · 10 + 33 = 330 + 33 = 363 10 In 15 minutes: 1 500 × 15 = 22 500 turns In an hour: 22 500 × 4 = 90 000 turns In one and a half hours: 22 500 × 6 = 135 000 turns 11 200 × 7 × 5 × 2 = €14 000

8

Basic operations with natural numbers

5

5. Basic operations with natural numbers (II)

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

Division

24 litres

48 litres

24 3 0 8

48 6 0 8

Look at two ways to solve common divisions: • 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?

15 375

5625 112 075 00

⎯→

5 625 : 15 = 375 m3 each day

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

×7

We use 375 cubic metres of water to water the park each day. There are 5 625 cubic metres of water in a tank. For how many days can we water the park?

5 6 2 5 375 ⎯→

000

8 4

32 0

×7

224 56 00 4

Let’s practise!

1 8 7 5 15

16 Find the missing value in each division.

Dividend 39

12 Find the quotient and remainder of each division.

5 625 : 375 = 15 days

a) 96 : 13 d) 7 029 : 26

Types of division

b) 713 : 31 e) 49 896 : 162

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

Exact division

Remainder

35 0

5 7

Integer division

We put 38 kg of apples into 5 kg boxes.

We fill 7 boxes and there are 3 kg remaining.

Remainder

38 3

5 7

There are two types of division depending on the value of the remainder: • Exact division (the remainder is zero). d D ⎯→ The dividend is equal to the divisor 0 q multiplied by the quotient.

D=d·q • Integer division (the remainder is not zero). d D ⎯→ The dividend is equal to the divisor r q multiplied by the quotient, plus the remainder.

D=d·q+r

: 12 :3

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

32

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

8 :4

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

14 Copy and complete the diagrams below in your

notebook.

(36 : 12) : 3 :

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

: 19

What did you notice?

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

Divisor 38

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

36 : (12 : 3)

15 Solve the following divisions and compare your results.

1 000 12

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

these numbers.

• 96

53 15

17 Solve the following problems without making notes.

13 Follow the example. Use mental arithmetic to divide

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

We put 35 kg of oranges into 5 kg boxes.

We fill 7 boxes and there are no remaining oranges.

TEACHING NOTES

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

Dividend Divisor Quotient

• Division means splitting a whole thing into equal portions of a specific size. Then we know how many portions there are.

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

34

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Learning by doing anayaeducacion.es Watch the video ‘Division as sharing and grouping’ to see a practical example of this.

Learning by doing anayaeducacion.es Watch the video ‘Exact and integer division’ to see a practical example of this.

Learning by doing anayaeducacion.es Watch the video ‘A property of division’ to see a practical example of this.

ICT anayaeducacion.es In ‘My web resources’ there are interactive activities to practise mental arithmetic.

Students should already know how to use the division algorithm, but we will use this section to detect any possible gaps in their learning that would prevent them from acquiring subsequent content.

The division concepts will be reviewed using activities in suitable contexts (problem solving): — Division as sharing: this means finding out how many elements correspond to each part when a whole is divided into a specific number of equal parts.

— Division as splitting: this means finding out how many parts of a specific size can be made from the elements of a whole. This concept requires special attention as it is harder to grasp.

Relations between the terms of exact and integer division will be consolidated through checking and applying them to specific situations (for example, division test).

The section ends with an important property of division: what happens if we multiply the dividend and the divisor by the same number? Students can assimilate it through contextualised and simple examples. It will also be important to answer the question: what happens with the remainder? Applying this property will be fundamental to justify the division algorithms with decimal divisors, and will link to other content, such as the equivalence and simplification of fractions.

ANSWER KEY

12 a) c = 7; r = 5 d) c = 270; r = 9

b) c = 23; r = 0 e) c = 308; r = 0

c) c = 758; r = 3 f ) c = 127; r = 508

13 a) 60 : 3 = 20 : 4 = 5 c) 300 : 3 = 100 : 4 = 25 e) 90 : 3 = 30 : 5 = 6 g) 180 : 10 = 18 : 3 = 6 i) 390 : 10 = 39 : 3 = 13

b) 180 : 3 = 60 : 4 = 15 d) 75 : 3 = 25 : 5 = 5 f ) 180 : 3 = 60 : 5 = 12 h) 240 : 10 = 24 : 3 = 8

14 (36 : 12) : 3

36 : (12 : 3)

3 : 3

36 : 4

1

9

We can see that the division does not follow the associative property.

15 a) (50 : 10) : 5 = 5 : 5 = 1 50 : (10 : 5) = 50 : 2 = 25 b) (36 : 6) : 2 = 6 : 2 = 3 36 : (6 : 2) = 36 : 3 = 12 Division does not follow the commutative property.

16 53 · 15 + 39 = 834 (1 000 – 12) : 38 = 988 : 38 = 26 17 a) 50 g

b) 3 minutes

c) 80 km/h

d) 25 boxes

18 a) Each strip cost €42. b) He uses 6 shelves.

19 a) If he saves €5 a week it will take 8 weeks. If he saves €6 it will take 7 weeks. b) €10 c) No, he will be €1 short.

9

6

Expressions with combined operations

2 Copy and complete in your notebook. Check your

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

Order of operations to solve an expression

40 – 12 : 4 + 2 · 3

Multiplications and divisions

40 –

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

a) 8 + 5 · 2 d) (15 – 3) : 4

4·

·3+2

4 · 10 – (8 · 3 + 2)

4 · 10 – 8 · (3 + 2)

– (

– 8 ·

+ 2)

b) 15 – 10 : 5 e) (8 + 2) · 3

→ 0 → 12

+2

+2

–

–

18

26

14

0

4 · (10 – 8) · (3 + 2) 4 · 4

· · 40

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 c) 4 + (7 – 5) · 3 e) 8 – (9 + 6) : 3

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

night pay

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

5 days

Write the following statements as mathematical expressions and solve them. a) There are 8 boxes of bananas, 20 boxes of oranges and 6 boxes of apples in a fruit shop. Each box of bananas weighs 15 kg. Each box of oranges and each box of apples weighs 8 kg. How many kilograms of fruit are there in the shop? b) A supermarket orders 20 boxes of orange juice, 15 boxes of apple juice and 10 boxes of carrot juice.

Each box contains 6 1-litre bottles. How many bottles did the supermarket order? c) There are 15 tables, 55 chairs and 12 bar stools in a cafe. How many legs are there in total? (each bar stool has 3 legs) d) We put 1 500 eggs into boxes of 10 eggs. We put another 1 500 eggs into boxes of 6 eggs. Then we put 300 brown eggs into boxes of 6 eggs. How many boxes are there?

36

37

1-minute explanations anayaeducacion.es Watch the video ‘Order of combined operations’ to review these contents.

Cooperative learning Technique: Pencils in the middle Make groups of four students to do exercise 8. Each member of the team is given a section and asked to put their pencil in the middle of the table. Taking turns, the students read their sections and propose an answer. The rest of the group gives their opinion and they discuss how to solve it.

When they are sure of their answer, everyone in the group takes their pencil and answers the question without talking.

MY MATHS DICTIONARY

For the order of operations in English we use the following acronyms to help students remember how to do this: PEDMAS: Parentheses, exponents, division and multiplication, addition and subtraction BODMAS: Brackets, order, division and multiplication, addition and subtraction

6. Expressions with combined operation TEACHING NOTES

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

8

c) 4 · 6 – 13 f ) 18 : (10 – 4))

4 Calculate the following using mental arithmetic.

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

4 · (10 – 8) · 3 + 2

g) 30 – 6 · (13 – 4 · 2) h) 3 · [13 – 3 · (5 – 2)]

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

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

2 + 3 · 4 = 2 + 12 = 14

Theory into practice

+2

→ 11

normal rate

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.

–

→ 7 → 1

f ) 2 · (6 + 4) – 3 · (5 – 2)

+

3 Follow the examples to solve the following operations.

{∫‘¢} The calculator performs the multiplication first. It follows the correct order of operations.

→ 9

d) 26 – 5 · (2 + 3) + 6 e) (14 + 12) : 2 – 4 · 3

: 4 +

40 - 12 =/ 4 ≤ 2 * 3 ≤Ñ → {∫∫∫∫∫∫‘«} Let’s practise!

2+3 *4 =

{∫“≠}

→ 14 → 2

c) 21 : (3 + 4) + 6

This month, a man worked 7 hours a day for 12 days. He got the standard pay on those days.

On another 5 days he worked for 9 hours. For 6 of those hours he got the standard pay. For the other 3 hours he got night pay. How many hours did he work in total this month?

(40 – 12) : 4 + 2 · 3

It may seem strange, but different calculators can give you a different answer.

Here you can get 20 or 14.

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

a) 6 · 4 – 2 · (12 – 7) b) 3 · 8 – 8 : 4 – 4 · 5

7 Problem solved

40 ≤ 12 / 4 µ 2 * 3 ≤Ñ→ {∫∫∫∫∫∫¢«}

How to use a calculator

4 · 10 – 8 · 3 + 2

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

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

Parentheses or brackets

(2 + 3) · 4 = 5 · 4 = 20

6 Calculate. Write down the steps you followed. Check

Help

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:

Like any other language, mathematical language requires sequenced learning, which is checked through practice and that takes time.

Interpreting and producing arithmetic expressions with combined operations and brackets is not obvious for students. On the other hand, experience tells us that we must pay special attention to this aspect to ensure that learning errors that would affect subsequent learning do not occur.

In order to analyse different expressions and compare their differences, we recommend using diagrams that highlight their structure, as shown in the examples.

After calculating the value of an expression it is important for students to get used to expressing all the steps through successive horizontal equalities. Here we must look out for writing errors (these lead to equalities with partial answers by writing, for example 4 · (8 – 6) · 3 = 4 · 2 = 8 · 3 = 24).

Analysing the behaviour of different calculators when doing combined operations is also interesting. By showing students two calculators (one which follows the correct order of operation, and another simpler one that performs each operation as it is entered), they will be surprised to find that the same sequence of keys gives a different result in each: Calculator that respects order of operation: 4 + 6 × 3 → 22 Calculator that performs each operation in the order it is entered: 4 + 6 × 3 → 30 We reach the conclusion that to ensure we use a calculator correctly, we must have indepth knowledge about it and take into account how it works.

ANSWER KEY

Exercise 1 gives different expressions with the same numbers and operations, but with different answers. The guided solution aims to help students internalise the role of brackets and the order of operations. Once the activity is completed, it can be complemented by horizontally writing out each process using successive equalitie 1 4 · 10 – 8 · 3 + 2

4 · (10 – 8) · 3 + 2

4 · 10 – (8 · 3 + 2)

40 – 24 + 2

4· 2 ·3+2

40 – ( 24 + 2)

16 + 2

24 + 2

40 – 26

18

26

14

4 · 10 – 8 · (3 + 2) 4 · (10 – 8) · (3 + 2) 40

– 8· 5

40 – 40

4 · 10

0

40

2 40 – 12 : 4 + 2 · 3 40 – 3

10

·

4 · 2

+

6

5

(40 – 12) : 4 + 2 · 3 28 : 4 +

37 + 6

7 + 6

43

13

6

3 a) 8 + 5 · 2 = 8 + 10 = 18 b) 15 – 10 : 5 = 15 – 2 = 13 c) 4 · 6 – 13 = 24 – 13 = 11 d) (15 – 3) : 4 = 12 : 4 = 3 e) (8 + 2) · 3 = 10 · 3 = 30 f ) 18 : (10 – 4) = 18 : 6 = 2 4 a) 14 and 20 b) 0 and 12 c) 13 and 4 d) 6 and 3 Comparing the answers shows that the brackets change the value of the expression.

Exercises 5 and 6 aim to bring together the abstract nature of arithmetic expression and the daily life of students by giving them meaning in recognisable contexts that they can relate to.

5 a) 2 · (7 – 3) – 5 = 2 · 4 – 5 = 8 – 5 = 3 b) 3 · (10 – 7) + 4 = 3 · 3 + 4 = 9 + 4 = 13 c) 4 + (7 – 5) · 3 = 4 + 2 · 3 = 4 + 6 = 10 d) 18 – 4 · (5 – 2) = 18 – 4 · 3 = 18 – 12 = 6 e) 8 – (9 + 6) : 3 = 8 – 15 : 3 = 8 – 5 = 3 f ) 22 : (7 + 4) + 3 = 22 : 11 + 3 = 2 + 3 = 5 6 a) 6 · 4 – 2 · (12 – 7) = 24 – 2 · 5 = 24 – 10 = 14 b) 3 · 8 – 8 : 4 – 4 · 5 = 24 – 2 – 20 = 22 – 20 = 2 c) 21 : (3 + 4) + 6 = 21 : 7 + 6 = 3 + 6 = 9 d) 26 – 5 · (2 + 3) + 6 = 26 – 5 · 5 + 6 = 26 – 25 + 6 = 1 + 6 = 7 e) (14 + 12) : 2 – 4 · 3 = 26 : 2 – 12 = 13 – 12 = 1 f ) 2 · (6 + 4) – 3 · (5 – 2) = 2 · 10 – 3 · 3 = 20 – 9 = 11 7 Problem solved.

8 a) 8 · 15 + (20 + 6) · 8 = 120 + 26 · 8 = 120 + 208 = 328 kilos b) (20 + 15 + 10) · 6 = 45 · 6 = 270 boxes c) (15 + 55) · 4 + 12 · 3 = 70 · 4 + 36 = 280 + 36 = 316 legs d) 1 500 : 10 + 1 500 : 6 + 300 : 6 = 150 + 250 + 50 = 450 boxes

11

My visual summary

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

Hundred thousands

Mayan

Egyptian

For numbers greater than 20:

300 000

300 30

1st level (× 20)

3 × 20

2nd level (× 1)

0 ×1

+1

3 8 0 0 0 0

1482

1482 + 427

576 + 1333

1909

1909

1482

Remember Relationships between addition and subtraction:

M =S +D M–S=D →* S=M –D Associative property

M

H TH

T TH

7

8

TH

H

T

4

3

0

3 8 5 0 0 0

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

576 + 906 = 906 + 576

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

+

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

4 000 000 U

TEACHING NOTES

5≥5

Commutative property

Associative property

60

1

4<5

4 0 0 0 0 0

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

(5)

(1)

3 8 4 5 2 3

3 8 4 5 2 3

=

8≥5

Basic operations with natural numbers

There are three key symbols: (0)

Thousands

Ten thousands

3 8 4 5 2 3 +1

It is partly an additive system and partly a positional system.

Each symbol has a different value depending on its level.

It is an additive system. This means they add the necessary symbols to get the number they want.

30 000 3 000 1 333 331

1 000 000

My visual summary

Rounding natural numbers

To round a number to a specific place value: - We change all the numbers to the right of that place value to zeros.

- If the number we are changing is a 5 or more, we round the next figure up. This means we add another digit to the number.

Numeral systems

Multiplication means repeatedly adding the same value.

U

4

← Minuend (M ) ← Subtrahend (S )

6 433

← Difference (D )

Dividing means distributing a value into equal parts.

(16 × 55) × 3 = 16 × (55 × 3) 880 × 3 = 16 × 165

↓

↓

Distributive property

4 000 U

4U

35 × 7 + 35 × 3 = 35 × (7 + 3)

Commutative property

245 + 105 = 35 × 10

16 × 55 = 55 × 16

=

880 = 880

350

8 342 – 1 909

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

2640 = 2640

350

Large numbers

We can use the decimal number system to write a number with all the figures we want.

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

Calculations with combined operations

1 000 000

1 000 000 000 000

1 000 000 000 000 000 000

One million A 1 followed by 6 zeros.

One trillion A 1 followed by 12 zeros.

One quintillion A 1 followed by 18 zeros.

The order of combined operations must always be: First

Then

Finally

Brackets

Multiplication and division

Addition and subtraction

38

39

ICT anayaeducacion.es Review the contents of this learning experience through the ‘My interactive visual summary’.

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

14

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

B

D

C

9

3948 FBG

E

3894 FBG

3

Write the following in Roman numerals. a) 87 b) 425 c) 2 600 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?

6

12

Rounding To the nearest hundreds of thousands

b) 651 + 283 – 459

c) 831 – 392 – 76

d) 1 648 – 725 – 263

17

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

How much does each tyre cost?

Mental arithmetic. a) 5 + 7 – 3 – 4

b) 18 – 4 – 5 – 6

c) 10 – 6 + 3 – 7

d) 8 + 5 – 4 – 3 – 5

18

f ) 40 – 18 – 12 – 6

Calculate.

19

b) 52 – (36 – 27) c) 128 – (86 – 45 – 12) d) 237 – (152 + 48 – 14) e) 348 – (148 – 86 + 29)

You see an advert for a house that costs €293 528.

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

f ) 235 – (340 – 152 – 84)

20

Multiplication and division 13

Multiply. a) 16 · 10

22

Target 11.c. The population of Cairo was 19 487 245 in June 2018. What is the approximate population of Cairo?

If the population of Cairo keeps growing, what will the population be by 2030? What measures can Cairo take to become a sustainable city by 2030?

23

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

c) 20 : (2 · 5) f )(40 : 8) : 5

Weight (tonnes)

Combined operations

a) 47 – (35 – 28)

To the nearest million

399 675 000

7

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

Calculate. a) 6 070 + 893 + 527

e) 12 + 13 + 8 – 23

Copy and complete the table in your notebook.

Number 2 830 554 19 270 000

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

5

16

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

Rounding

9

4389 GFB

Addition and substraction

11

21

14

8 2

7 E

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

10

3 6

Look at these car number plates: E

5 6

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

Write the following numbers using the Egyptian additive system. a) 48 b) 35 c) 2 130

TRAINING AND PRACTICE

Copy and complete in your notebook.

8

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

2

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

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

b) 128 · 10

c) 60 · 10

d) 17 · 100

e) 85 · 100

f ) 120 · 100

g) 22 · 1 000

h) 134 · 1 000

i) 140 · 1 000

4 369 449 3 626 510

5 214 031

Total

7 995 959

11 409 085

6 195 054

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

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

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

24

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

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

25

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

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

26

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

27

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

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

40

Exercises and problems DO YOU KNOW THE BASICS? Numeral systems

1 a) 57

Value (thousands of )

Fresh fruit Vegetables and potatoes

The My visual summary section aims to summarise all the contents the students have studied throughout the learning experience.

Visual summaries are a useful teaching tool that facilitates the learning process in many ways: • They provide students with non-linear visual ways to understand, produce and represent knowledge.

• They help students develop higher-order thinking skills including analytical skills.

• They facilitate organisation, recall and processing of information.

• They help students externalise their knowledge and show their understanding.

• They make explicit structural form of knowledge and relationships between concepts, and therefore enhance students’ comprehension.

• They attend to different learning styles.

• They are effective organisational tools students can apply to organise their knowledge.

Visual summaries can be used at the beginning and at the end of each learning experience.

They are an excellent way to organise and make prior knowledge evident while setting a good bases for your class to begin.

The My interactive visual summary digital resource allows to display images and texts or images only. This is a great way to begin reviewing content and to motivate learning. It also allows cooperative learning, interaction, and communication.

At the end of the learning experience, you can display images and go through concepts and relationships with students. In this way you will summarise the content of the learning experience in a whole class activity.

41

b) 234

c) 2 540

d) 3 430 000

2 a)

b)

SDG commitment Watch the video for target 11.c before doing exercise 22.

Start a class discussion on actions we can do to make cities more sustainable.

c)

3 a) 87 = LXXXVII b) 425 = CDXXV c) 2 600 = MMDC d) 54 528 = LIVDXXVIII 4 Decimal: 57 Roman: LVII Egyptian: 5 One trillion → 1 000 000 000 000 → 13 figures, 12 zeros One quintillion → 1 000 000 000 000 000 000 → 19 figures, 18 zeros

12

Rounding

6

Rounding Number

To the nearest hundreds of thousands

To the nearest million

2 830 554

2 800 000

3 000 000

19 270 000

19 300 000

19 000 000

399 675 000

399 700 000

400 000 000

7 The third option is the closest. But it does not say it is a rounded figure.

The first option is not as accurate as the third, but says that it is a rounded figure.

What can numbers tell us?

8 a) 235

b) 724

c) 235 and 231

9 a) The oldest is 3948 FBG. The newest is 4389 GFB. b) The one after the red one is 3949 FBG. The one before the red one is 3947 FBG. c) There are 45 number plates. d) The last number plate with the same letters would be 9999 GFB. 9 999 – 4 389 = 5 610 5 610 cars have number plates with the same letters.

Addition and subtraction

10 a) 7 490 c) 363

b) 475 d) 660

11 a) 5 d) 1

b) 3 e) 10

c) 0 f) 4

12 a) 40 d) 51

b) 43 e) 257

c) 99 f ) 131

Multiplication and division

13 a) 160 d) 1 700

b) 1 280 e) 8 500

c) 600 f ) 12 000

13

g) 22 000 14 a) c = 330; r = 7 d) c = 43; r = 0 15 8 1 6 0 6 6 1 6

h) 134 000

i) 140 000

b) c = 46; r = 11 e) c = 3; r = 0

c) c = 201; r = 0 f ) c = 107; r = 0

2 5 3 2

8 2 9 5 6 1 2 9 0 3 5 0 7 6 0 6

16 a) 6 d) 18

b) 3 e) 2

14

5 9 2 5

c) 2 f) 1

17 a) 100 : 5 = 20 barrels b) 12 · 5 = 60 Euros c) 10 · 24 = 240 bottles d) 360 : 4 = 90 Euros Combinend operations

18 a) 3 e) 9

b) 6 f) 0

c) 4 g) 3

d) 4 h) 6

19 a) 20 e) 44

b) 14 f) 7

c) 4 g) 6

d) 30 h) 4

20 a) 30 − 4 · 7 = 30 − 28 = 2 b) 5 + 3 · 2 = 5 + 6 = 11 c) 5 · 8 + 7 = 40 + 7 = 47 d) 3 · 7 − 13 = 21 − 13 = 8 e) 2 · 12 − 3 · 5 = 24 − 15 = 9 f ) 4 · 2 + 3 · 2 = 8 + 6 = 14 g) 15 − 3 · (10 − 8) = 15 − 3 · 2 = 15 − 6 = 9 h) 6 + 5 · (13 − 12) = 6 + 5 · 1 = 6 + 5 = 11 TRAINING AND PRACTICE

21 a) False d) False

b) True e) True

c) True

22 Approximately 20 million inhabitants.

Open answer.

23 weight

value

(tonnes)

(thousands of €)

fresh fruit

4 369 449

6 195 054

vegetables

3 626 510

5 214 031

7 995 959

11 409 085

and potatoes total

24 1 light year → 9 and a half trillion kilometres 9 500 000 000 000 Star A → 5 light years ≈ 45 trillion kilometres

14

Star B → 5 trillion kilometres Star A is further away than star B.

Exercises and problems 28

Calculate the following using mental arithmetic.

Remember that dividing by 5 is the same as dividing by 10, and then multiplying by 2. :5

• 90 : 10

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

9

A

SOLVE SIMPLE PROBLEMS

30

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

31

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

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

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

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

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

36

37

Match each statement to two of the mathematical expressions below: I. There are 50 people on a bus. 16 people get off the bus at the first stop and 4 people get on.

II. There are 50 students in a music class. Today, 4 are absent without an excuse. Another 16 are absent because they are at a concert.

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

He pays with a €50 note.

IV. Last night, 50 clients stayed at a hotel. This morning, 16 more people arrived and 4 people left. a) 50 – 16 – 4 b) 50 – 16 + 4 c) 50 – (16 + 4) d) 50 – (16 – 4) e) 50 + (16 – 4) f ) 50 + 16 – 4 Which equation or equations solve the following problem?

This morning, a supermarket sold 24 kg of apples for €2/kg, 12 melons for €4 each and 13 pineapples for €2 each. How much profit did it make from selling all the fruit? a) 24 · 12 + 4 · 13 + 2 b) 24 · 2 + 12 · 4 + 13 · 2 c) (24 + 13) · 2 + 12 · 4 d) (24 + 13 + 2) · (2 + 4) Read the problem and look at the solution. Can you explain what each operation shows?

There are horses, cows and chickens at a farm. In total, there are 714 legs, 168 horns and 137 beaks. How many horses are there at the farm?

Answer: 1st 168 : 2 = 84 2nd 84 · 4 = 336 3rd 137 · 2 = 274 4th 336 + 274 = 610 5th 714 – 610 = 104 6th 104 : 4 = 26

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

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

50

39

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

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

What is her profit?

40

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

41

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

42

Rosa is two years older than her brother Julián.

She is two years younger than her brother Alberto.

Their mother is 42 years old. The ages of the three children also add up to 42. How old is each child?

43

A train carrying goods travels at 55 km/h.

On the track next to it there is a passenger train travelling at 105 km/h. What is the distance between the two trains after half an hour?

44

A car and a motorbike leave a cafe at the same time in the same direction. The car travels at 90 km/h and the motorbike travels at 100 km/h.

What is the distance between them after an hour and a half?

45

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

46

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

INTERPRET, DESCRIBE AND EXPRESS 35

49

38

B

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

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

Which property did you use?

33

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

·2

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

29

32

34

18

47

Clara gets €28 to deliver 7 boxes of publicity.

How much does she get if she delivers an extra box?

48

There are 7 boxes of cans on a supermarket shelf.

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

Problem solved

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

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

51

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

52

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

€129

€140

How much money do they have in the bank now?

42

43

Enterprising culture Productivity (productive dimension): My project.

In exercise 30, students are asked to do some research for which they need to prepare a plan of action in order to be successful.

25 a) 48 + 115 = 163 c) 628 − 429 = 199

b) 103 + 256 = 359 d) 480 − 284 = 196

26 a) 5 − [7 − 5] = 5 − 2 = 3 c) 2 + [6 + 6] = 2 + 12 = 14 e) 20 − [15 − 2] = 20 − 13 = 7

b) 3 + [8 − 7] = 3 + 1 = 4 d) 7 − [12 − 7] = 7 − 5 = 2 f ) 15 − [17 − 12] = 15 − 5 = 10

27 a) 123 · 48 = 5 904 c) 1 482 : 57 = 26

b) 18 · 86 = 1 548 d) 1 862 : 14 = 133

28 a) 12 d) 28 g) 42

b) 16 e) 34 h) 68

c) 24 f ) 40 i) 84

29 a) 6 · (8 + 2) = 6 · 8 + 6 · 2 = 48 · 12 = 60 b) 5 · 9 − 5 · 6 = 5 · (9 – 6) = 5 · 3 = 15 c) (10 − 8) · 4 = 10 · 4 − 8 · 4 = 40 – 32 = 8 d) 7 · 12 − 2 · 12 = (7 − 2) · 12 = 5 · 12 = 60 We used the distributive property.

30 D = d · q + r k · D = k · (d · q + r) = k · d · q + k · r The distributive property tells us that the remainder is also multiplied by the same number.

31 a) True

b) True

c) False

d) True

e) False

32 a) 3 e) 12

b) 14 f) 4

c) 8 g) 0

d) 0 h) 2

33 a) 36 e) −15 i) 3

b) 5 f ) 23 j) 12

c) 14 g) 11

d) 1 h) 15

34 a) 9 + (3 − 1) = 11

b) 9 − (3 + 1) = 5

III → a) and c)

IV → e) and f )

INTERPRET, DESCRIBE AND EXPRESS

35 I → b) and d)

II → a) and c)

36 b) and c) 37 1st The number of cows is equal to half the number of horns: Cows → 168 : 2 = 84 nd 2 Cow legs → 84 · 4 = 336 3rd The number of chicken legs is double the number of beaks: Chicken legs → 137 · 2 = 274 th 4 Cow legs + chicken legs → 336 + 274 = 610 5th The number of horse legs is equal to the total of legs minus cow and chicken legs: Horse legs → 714 – 610 = 104 th 6 The number of horses is obtained by dividing the previous number by 4: Horses → 104 : 4 = 26

15

SOLVE SIMPLE PROBLEMS

38 House B was finished in March.

39 She sent 27 messages.

40 There are 12 horses and 24 cows.

41 There are a total of 648 cans.

42 Julián is 12 years old, Rosa is 14 years old and Alberto is 16 years old.

43 The distance between them after half an hour is 80 km.

44 The distance between them after an hour and a half is 15 km.

45 It produced 1 233 valid jars.

46 He expects to obtain 4 000 plants.

47 If she delivers an extra pile, she will be paid 32 Euros.

48 Now there are 118 cans.

49 There are 7 707 women.

50 Problem solved.

51 The family earns 6 440 dollars monthly.

52 There are €142 left.

53 The retailer makes a profit of €1 360.

Exercises and problems 53

A supermarket pays €2 000 for 150 bags of potatoes. Each bag contains 30 kg of potatoes. A shop assistant sees that 300 kg of the potatoes are damaged. He puts the rest into 5 kg bags, and sells each bag for €4. What is the supermarket’s profit?

54

Every day, a baker bakes five trays of muffins.

Each tray contains three dozen muffins. The bakery is closed on Mondays. How many muffins does she bake in a week?

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

62

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

63

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

64

56

a) How many thousands of visitors were there in 2022? b) How many visitors were there on average, a day?

There are 54 tourists on a group visit to Madrid.

They arrive at the airport and take taxis to their hotel. Each taxi can transport four people each. How many taxis do they need?

58

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?

59

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

Why?

60

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

44

1 0

111 110 101 100

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

65

c) Approximately how many visitors were there in 2021?

57

rd 33.ª 1 0 1 0

nd 22.ª

1

The Real Alcázar of Sevilla got an average of 144 404 visitors a month in 2022. This is more than double the number of visitors in 2021.

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

74

The graph below shows the colour of 30 690 cars produced in three months:

69

Valentina has a farm with ducks and geese.

Today, she sold 21 of her animals for €350.

She sold double the number of ducks as geese. One goose costs three times the price of a duck.

How much does one duck cost? How much does one goose cost?

75

Aarón is a travel agent. This year, he sells these holidays: — 56 % of people go to the beach.

— 47 % go to a village.

— 23 % go to both destinations.

What percentage of people did not go to the beach or to a village?

70

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

76

71

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

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

77

72

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

We know this information about the Year 1 students at a Secondary school: — 44 eat in the cafeteria, 58 take the school bus and 47 participate in extracurricular activities.

— 24 students eat in the cafeteria and participate in extracurricular activities.

— 23 eat in the cafeteria and take the school bus; 25 take the school bus and participate in extracurricular activities.

— 11 students do all three things and 17 students do not do any.

How many students are there in total?

Grey

Antonio, Beatriz, Cora and David go to the cinema. They have four seats next to each other. In how many different ways can they sit?

First, let’s solve an easier problem: In how many different ways can they sit if Antonio sits in the 1st seat?

1st 2nd

66

67

Draw a grid to help you. For example:

A B

B

C

C

3rd

C

4th

D

D

D

D

B

C

D

D

B

C

B

C

B

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.

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

White

Green

Blue

Red

Others

54 She bakes 1 080 muffins a week.

How many red cars are there?

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

1st 1.ª

THINK A LITTLE MORE

68

20 15

73

15

30

20 20

30

Draw a Venn diagram like this one to help you.

20

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?

YEAR 1 CAFETERIA

SCHOOL BUS

EXTRA ACT.

78

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

45

55 The lorry will take 1 hour 55 minutes.

THINK A LITTLE MORE

56 a) 1 733 millions b) Between 4 800 and 4 850 visitors c) Around 860 000, or between 850 000 and 860 000 57 They need 14 taxis.

58 The van still has more than half of its load.

59 It travels 1 950 m in 78 seconds. Yes, the driver exceeded the speed limit.

60 It will take her 4 week to buy the skateboard.

61 It makes 174 cars every day.

62 180 students study a second language.

60 students study German.

63 You can choose 30 possible menus.

64 There are 8 four-figure numbers that only contain 0 and 1.

65 They can sit in 24 different ways.

66 There are 10 ways of making €1: 0

10 cts.

16

20 cts.

5

50 cts.

0

1

2

3

4

5

6

7

8

9

10

0

2

4

1

3

0

2

−

1

−

0

2

1

0

1

0

1

0

−

0

−

0

67 Marta has spent €66; Julián, €36, and Rosa, €106.

68 They need to order 32 boxes to fill the order.

69 Each duck costs €10, and each goose costs €30.

70 There are 15 cars and 5 motorbikes.

71 She won’t make it. Since she moves forward 80 metres per minute, it will take her 125 minutes (2 h 5 min) to cover the ten kilometres.

Self-assessment

anayaeducacion.es Answer key.

2

3

4

5

7

8

9

6

Mayan

10 11 12 13 14

Decimal

15 16 17 18 19

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

528

2 Which systems are additive? Which systems are

positional? What is the difference?

3 Look at these amounts:

I TAKE ACTION

Vicky's skateboard

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?

• The surface area of Brazil is eight million five hundred and fourteen thousand eight hundred and seventy-seven square kilometres.

• The population of the world in April of 2018 was 7 601 767 200 people.

Express the first in figures and the second in words.

€165

LEARNING EXPERIENCE

0 1

10 Copy and complete in your notebook.

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

Numeral systems Egyptian

Self-assessment

GO TO THE ESCAPE ROOM AND TEST YOURSELF!

1 Complete the following table in your notebook:

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. Viki only has €96 in the bank. She analyses her finances to plan the purchase.

Income • Weekly allowance ➝ €15.

• Viki�s grandma gives her €20 a month.

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

5 Calculate.

1 Copy and complete the table to calculate her minimum and maximum savings.

Income

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

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

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

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

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

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

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

1

Now analyse Vicky’s finances:

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.

ANSWER KEY

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

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

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 this number.

15 Which of the expressions below solves the following

problem?

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

Expenses

Allowance

Grandma

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

Neighbours Going out 20 : 2 = 10

5

16 : 4 = 4 (… + … + …) – (… + …) = …

Other

0 1

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

She has to save ➝ Cost minus what she has in the bank ➝ 165 . ... = ...

In 10 weeks she saves ➝ 10 . 3 = €30 ➝ is that enough? ...

In 20 weeks she saves ➝ 20 . ... = €... ➝ is that enough? ...

In .................................................................................

If Vicky saves the maximum, how much more money does she need after two weeks? ...

3 Imagine you want to achieve your goal in two months. You don’t want to cut

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

Maths

in context

Apply your language skills www.anayaeducacio n.es

egiptian

5

• 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.

mayan 47

Gamification anayaeducacion.es Go to the Escape Room and test yourself!

ICT anayaeducacion.es Answer key for the self-assessment.

decimal

2

3

4

7

8

9

10 11 12 13 14

Now, it's your turn...

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

46

6

numeral systems

3 042

15 16 17 18 19

13

528

The Egyptian numeral system is additive. The Mayan system is partly additive and partly positional. And the decimal numeral system is positional.

2 In an additive system each sign has a fixed value and the value is obtained by adding up the value of all the signs involved. In a positional system the value of each sign depends on the place it occupies.

3 Surface area of Brazil: 8 514 877 km2.

Global population: Seven thousand six hundred and one million seven hundred and sixty-seven thousand two hundred people.

4 a) 8 510 000 and 7 601 780 000 b) Surface area of Brazil → Rounded to the hundreds of thousands: 8 500 000 km2 Global population → Rounded to the hundreds of millions: 7 600 000 000 5 a) 25 b) 15 6 For example: 2 · (5 − 2) · 3 7 a) 60 − 42 = 18 b) 45 + 10 = 55

60 − 18 = 42

8 a) 32

b) 130

9 a) 10

b) 270

c) 100

d) 380

10 a) 3

b) 2 646

c) 865

d) 12

11 We need 12 cartons of 30 eggs: 11 will be full and the remaining carton will contain 20 eggs and 10 empty places.

12 There are 15 stools and 45 chairs.

13 There is a distance of 25 km between them after 10 minutes.

14 a) She produces 6 732 jars of honey per year. b) She makes an annual profit of €20 196. c) The rounded profit is €20 000.

15 The third expression, c), solves the problem: (35 · 180 + 35 · 70) : 10 = 875 She expects to fill 875 boxes of apples.

17

TAKE ACTION Self-assessment

anayaeducacion.es Answer key.

0 1

2

3

4

5

7

8

9

6

Mayan

10 11 12 13 14

Decimal

15 16 17 18 19

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

528

2 Which systems are additive? Which systems are

positional? What is the difference?

3 Look at these amounts:

Vicky's skateboard

10 Copy and complete in your notebook.

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

Numeral systems

I TAKE ACTION

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?

• The surface area of Brazil is eight million five hundred and fourteen thousand eight hundred and seventy-seven square kilometres.

• The population of the world in April of 2018 was 7 601 767 200 people.

Express the first in figures and the second in words.

€165

LEARNING EXPERIENCE

1 Complete the following table in your notebook:

Egyptian

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. Viki only has €96 in the bank. She analyses her finances to plan the purchase.

Income • Weekly allowance ➝ €15.

• Viki�s grandma gives her €20 a month.

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

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.

46

TEACHING NOTES

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

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

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

4 a) Round the amounts from activity 3 to the nearest

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

I take action: Vicky’s skateboard

GO TO THE ESCAPE ROOM AND TEST YOURSELF!

Income

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 this number.

15 Which of the expressions below solves the following

problem?

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

Expenses

Allowance

Grandma

Neighbours Going out

Other

MINIMUM SAVING/weekly

15

20 : 4 = 5

0

12

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

MAXIMUM SAVING/weekly

15

20 : 4 =5

20 : 2 = 10

5

16 : 4 = 4 (… + … + …) – (… + …) = …

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

She has to save ➝ Cost minus what she has in the bank ➝ 165 . ... = ...

In 10 weeks she saves ➝ 10 . 3 = €30 ➝ is that enough? ...

In 20 weeks she saves ➝ 20 . ... = €... ➝ is that enough? ...

In .................................................................................

If Vicky saves the maximum, how much more money does she need after two weeks? ...

3 Imagine you want to achieve your goal in two months. You don’t want to cut

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

Maths

in context

Apply your language skills www.anayaeducacio n.es

Now, it's your turn...

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

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

47

The proposal made on the first page of the learning experience is recalled here: Have you ever planned a project? Think about a real or imagined one (a purchase, a trip or a gift, etc.) and how are going to finance it. At the end of the learning experience you will find a guide to plan its financing. But think first!

Because it is open-ended, following the teacher's indications, it may have been developed by varying the initial data or starting from a different context than the one offered. It will have been specified at the beginning or during the time spent working with the content, or perhaps now at the end.

The aim of this proposal is for students to handle the learning activated or revised throughout the learning experience, to perceive its usefulness and to apply it in a context that has meaning in their environment.

In any case, the page shows a concreteness that can be taken as a directed activity to be carried out or as an example that helps to devise a new one.

ANSWER KEY

1

Income Allow. Grandma

Expenses Tips

Going out

Other

0

12

20 : 4 = 5

(15 + 5 + 0) – (12 + 5) = = €3

5

16 : 4 = 4

(15 + 5 + 10) – (5 + 4) = = €21

Minimum weekly savings

15

20 : 4 = 5

Maximum weekly savings

15

20 : 4 = 5 20 : 2 = 10

2 She has to save ➝ Cost minus what she has in the piggy bank ➝ 165 – 96 = 69 Now try it out: In 10 weeks she saves ➝ 10 – 3 = €30 ➝ Is that enough? No, 30 < 69 In 20 weeks she saves ➝ 20 – 3 = €60 ➝ Is that enough? No, 60 < 69 In 23 weeks she saves ➝ 23 – 3 = €69 ➝ Is that enough? Yes, she gets the €69.

If Viki saves the maximum, how much more money does she still need after two weeks?

She would need 69 – 2 – 21 = 69 – 42 = €27.

In 3 weeks she would save ➝ 3 – 21 = €63 ➝ Is that enough? No, 63 < 69 In 4 weeks she would save ➝ 4 – 21 = €84 ➝ Is that enough? Yes, 84 > 69 3 For example: Getting €4/week in tips, and spending €8/week going out and €4 in other expenses, the weekly saving would be (15 + 5 + 4) – (8 + 4 = 24 – 12 = €12.

15 + 5 + 4) – (8 + 4) = 24 – 12 = €12 You would get the missing €69 in 6 weeks (6 – 12 = €72).

18

NOTES

19