Our publications contain carefully selected content, illustrations and language to comply with
non-discrimination
on the grounds of gender, culture or opinion.
Grupo Anaya considers social and environmental responsibility to be one of its fundamental values.
For this reason, we are committed to:
· continually improving our contents and materials related to the environment.
· reducing our carbon emissions.
· using natural resources responsibly.
· making sure that our activity has no negative consequences for endangered forests.
These commitments, among others, mean that 100% of the paper used in our books has the PEFC
label.
Commitment to Sustainable Development Goals
Building Blocks is an educational project of Anaya Educación for Secondary Education with the participation
of: K. Chambers, Danny Latimer, R. Oakes, J. Roe, Hannah Peat, Karen Piper, Deborah Spencer, Denise Suárez,
Begoña Fuente Larrazabal, Sara Gascón Martín, C. Ordóñez, Ana Villarroya, J. Colera Jiménez, Ignacio Gaztelu
Albero and Ramón Colera Cañas.
The following people have worked on this book:
Editorial team:
Federica Cocco, Beatriz Fuentes and Sara Gascón Martín
INCLUSION AND ANTI-DISCRIMINATION ADVISOR:
Víctor Díez
Design, technical drawings and maps:
Miguel Ángel Castillejos, Patricia G. Serrano, Juan Carlos Quignon
Illustrations:
Alberto Hoyos
Layout:
DiScript and Isabel Román
Graphic edition:
Olga Sayans
Translation:
Montero Language Services
Academic and Professional Orientation:
created in conjunction with Fundación Bertelsmann.
Coordinator: Juan José Juárez Calvo. Expert collaborators: Sara Lozano Santiago, Belén Pérez
Castro and Pilar Vázquez Hernández.
M
atheMatics
1 t
eacher
´
s
G
uide
P
roMocional
- Bl001062 - 9223491
© Text: José Colera Jiménez, Ignacio Gaztelu Albero, Ramón Colera Cañas, 2020.
© This edition: GRUPO ANAYA, S.A., 2020 - Juan Ignacio Luca de Tena, 15 - 28027 Madrid - Printed in Spain.
Building Blocks and the project keys
... 4
INDEX
Unit
1.
NATURAL NUMBERS
... 6
A project that is rooted in skills-based
learning and the development of student
commitment within the realities
of available time.
to keep learning
building
blocks
Building Blocks is a new skills-based approach, with
the utmost curricular rigour and a coherent and
coordinated content sequence in the areas throughout
the entire educational stage. It promotes linguistic
communication skills, which are essential for assessing
the knowledge that allows us to understand the world
around us and develop social awareness.
Building Blocks offers the possibility of incorporating
active methodologies, using cooperative learning and
thinking strategies, promoting personal and social
skills for emotion management and the development
of entrepreneurship in a flexible way. It attends to
academic and professional orientation, whilst
embracing equality and inclusion, all within the
framework of the Sustainable Development Goals that
we must keep focused on over the coming years.
SDG Commitment
Establishes the Sustainable Development Goals
as a framework for learning that prepares students towards committed
citizenship.
Developing thinking
Proposes strategies to stimulate reflection,
“learning how to think” and the development of critical and creative
thinking habits.
Cooperative learning
Offers techniques to develop the skills that
allowus to work together and efficiently in a diverse society.
Emotional education
Offers emotional management tools to face the
challenges of this complex educational stage.
Enterprising culture
Promotes entrepreneurial thinking in its three
dimensions: personal, social and productive.
ICT
Integrates the use of ICT in the learning process itself in a responsible,
intelligent and ethical way.
Academic and professional orientation
Helps students to get to
know themselves, to understand the environment and to make decisions
that allow them to confidently enter the labour market.
Assessment
Incorporates strategies that allow students to participate in
the assessment of their learning, analysing “what they have learned” and
“how they have learned it”.
Linguistic Plan
Develops communication skills both written and spoken
and the tools needed to describe, present, instruct, comment, defend or
refute ideas...
Project keys
Unit presentation
Natural numbers do not seem to obey any human intellectual
‘construction’. They have always naturally appeared in all cultures to
count, order, measure, and so on.
The unit starts by comparing some of the most well-known numeral
systems. This enables us to show, in addition to the historical development
of how numbers are represented, that the concept of natural numbers is
the same for all of them regardless of how it is expressed.
After reviewing the structure of the decimal numeral system, and
discussing its advantages compared to other numeral systems, we work on
reading and writing numbers with nine or more figures. The students are
also reminded how to round and its advantages.
We then review basic operations with natural numbers and some of their
properties, paying special attention to division as this is where more errors
and gaps are detected both conceptually and operationally with regards to
the algorithm.
When reviewing the operations, we practice calculations and prioritise
problem solving. This ensures students review and improve concept
construction.
Lastly, we build on solving expressions with brackets and combined
operations.
There are three types of contents in this unit:
•
Theoretical aspects:
– Numeral systems. The decimal numeral system.
– Properties of operations and their advantages when calculating.
•
Calculation:
– Algorithms for operations.
– Expressions with brackets and combined operations.
– Mental arithmetic.
•
Using a calculator and knowledge of basic techniques.
Basic 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.
Task preparation
•
Find information about different numeral systems (ancient civilisations,
binary system used for computer languages, etc.).
•
Review how to work with the four operations (detecting gaps).
•
Show the different types of calculators.
•
Look back at general strategies and methods for solving problems and
describing the solving processes.
1
NATURAL NUMBERS
Contents and competencies
Unit contents
competencies
Core
Opening page
• Different ways of writing the
same number
• Different multiplication
methods
• Combined operations
CLC
CMST
1. Numeral systems
• The Egyptian numeral system
• The Mayan numeral system
• The decimal numeral system
CLC
CMST
DC
LL
SCC
CAE
2. Large numbers
CLC
CMST
LL
3. Rounding natural numbers
CLC
CMST
SCC
4. Basic operations with natural
numbers
• Addition and its properties
• Subtraction and its relation to
addition
• Multiplication and its properties
• Division
• Exact division and integer
division
• A property of division
CMST
LL
5. Expressions with combined
operations
• Order of operations to solve an
expression
• How to use a calculator
CLC
CMST
DC
LL
SCC
Final pages
• Exercises and problems
• Maths Workshop
• Self-assessment
CLC
CMST
LL
SCC
SIE
CAE
KC: key competences, CLC: Competence in linguistic
communication, CMST: mathematic competence and basic
competences in science and technology, DC: digital competence,
LL: learning to learn, SCC: social and civic competence, SIE:
sense of initiative and entrepreneurship and CAE: cultural
awareness and expression.
7
Natural numbers
Starting the unit
In the Listening and Reading section the students will start reading and
listening an introductory text about the different ways of expressing
natural numbers. It also proposes that students think about the usefulness
of numeral systems, how they are different and the role they have played
in different cultures and ages.
It also presents a multiplication algorithm that is different to the ones that
students already know about but which is similarly based on the decimal
numeral system. Here we can take the opportunity to compare it with the
algorithm used by the Egyptians (see page 26 of the student book) and
show the relationship between the way numbers are represented and the
advantages or disadvantages when operating with them.
Questions to detect preconceptions
•
Create a signs system to be able to code any number lower than 50 (or
100…).
•
Read and write number that have a maximum of eight figures.
•
Calculate using basic operations.
•
Compare very simple expressions varying the position of the brackets.
•
Invent problems for a given operation.
Answers
1
Open answer.
2
1 346 + 834 = 2 180. 2 180 = MMCLXXX
3
The number is 3 059.
MMMLX
CM
DM
UM
C
D
U
4
The first and third.
5
Open answer.
6
a) 208 × 34
0
0
0
6
0
7
8
2
0 3
2 4
2
6 10
1 0
2
0
8
3
4
0 7 0 7 2
0 0
b) 453 × 26
0
0
2
8
1
7
4
8
0 1
0 6
8
11 7
4
5
3
2
6
1 1 7 7 8
3 0
7
a) 15 · 3 b) 20 + 15 · 3 c) (20 + 15) · 3
LANGUAGE BANK
LANGUAGE BANK
LANGUAGE BANK
LANGUAGE BANK
LANGUAGE BANK
LANGUAGE BANK
LANGUAGE BANK
LANGUAGE BANK
LANGUAGE BANK
LANGUAGE BANK
LANGUAGE BANK
LANGUAGE BANK
9NATURAL NUMBERS
Numeral systems throughout historyWe use numeral systems to express quantities since prehistoric times. All the major civilisations used them, from the Egyptians, Greeks, Romans, Arabs and Mayans to the people of Babylon, China and India. Each civilisation developed its own numeral system. These systems were then passed on from village to village and changed over time. At first, numbers were only used to count natural products like sheep, fruit and coins. Early numeral systems were very simple. For example, people drew hands and fingers, or cut marks on sticks. Romans had a particular way of writing numbers: they used letters instead of the numbers we use today! Each letter had a different value. Imagine how the Romans would use letters in calculations. For example:
MCCCXLVI + DCCCXXXIV Does it seem complicated? Multiplications would be even more difficult! Luckily for us, things became simpler with the Hindu-Arabic numeral system, which we call positional notation. It introduced decimal numbers and established the bases of the system we use today. The new systems were more complex, but they made counting more practical. Numeral systems help us to write numbers, remember specific amounts and express them to others. Therefore, we must be able to understand and solve calculations when we see them in their written form. 1 In the title of the text you can find the concept of numeral systems. Try
to define it using your own words. 2 You are already familiar with Roman numerals. However, here is a quick
reminder of the value of each letter: Now, translate the Roman addition in the text into the decimal numeral system, do the operation and translate it back into the Roman numeral system.
1 5 10 50 100 500 1 000
Different ways of writing the same number
These are three different ways of writing the same number:
3 Which number is it? Write the number that follows in all three ways. 4 Do any of them use the decimal numeral system? 5 Can you think of any other ways to write this number?
Different multiplication methods
Look at the example of an ancient Indian multiplication below. It shows how they calculated 346 × 57.
• They used a table with the individual numbers along the sides.
• First, each number on the top was multiplied by each number along the side. The answer was written in the corresponding box. For example, the yellow box shows:
4 × 7 = 28
• Then, the numbers in each vertical column were added together. There could only be one digit in each column. 6 Use this method for the following multiplications:
a) 208 × 34 b) 453 × 26
Combined operations
7 To play sports at your local sports centre, you have to pay a €20 registration fee and €15 per month. Match the following mathematical expressions to the descriptions below:
(20 + 15) · 3 20 + 15 · 3 15 · 3 a) The amount you pay in the second term. b) The amount you pay in the first term. c) The amount you pay in the first month for three children. 1 12 5 2 12 2 1 2 0 4 3 0 2 9 6 1 3 46 57 1 9 7 2 2 2 8
1
8Reading and listening
Learning to think
Technique: Think and share with a partner.
It is beneficial for students to think about what they believe the answer is and to
listen to their classmates’ ideas to compare opinions.
Once they have thought about it, we ask them to share their answers with the
classmate sitting next to them, explaining how they reached their answer.
KEY ELEMENTS
SDG commitment
• Target 11.c.
• Target 13.3.
Linguistic Plan
• Skills: oral expression (argumentative text) and
reading comprehension (expository text)
Learning to think
• Think and share with a partner
Cooperative learning
Technique:
• Pencils in the middle
Entrepreneurial culture
• Productivity (productivity dimension): my
project
• Initiative (productivity dimension): I support
changes
ICT
• Interactive activities
• Mental arithmetic
• Answers for the self-assessment
• Sheets on Language Bank
Assessment
• Exercises and problems
• Preparation of the portfolio
Numeral systems
Methodological suggestions
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…). 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.
Answers for ‘Consolidating ideas’
The activity included in this section aims to review the structure of the DNS and detect
gaps in their understanding, providing support to overcome them.
1
a) 3 thousands are 300 tens.
b) 1 ten of thousands are 100 hundreds.
c) 5 one million units are 50 000 hundreds.
Answers for 'Let's practice!'
1
19 =
65 =
34 120 =
2 523 083 =
2
7 =
84 =
12 =
126 =
3
6 11 120 126
4
To the left:
To the right:
5
a) 500 T = 50 H = 5 Th
b) 3 000 H = 300 Th = 30 T Th
c) 6 Th = 60 H = 600 T
d) 8 H Th = 80 T Th = 80 000 T
6
a) True b) True c) False d) False e) True
7
40 001, so 41 000 – 40 001 = 999
11 1 Unit Natural numbers (1, 2, 3…) were introduced because people needed to count.
They changed over time to adapt to new cultures and times. Prehistoric civilisations already had counting systems. For example, they used their fingers, cut notches into sticks, put beads on strings… As society evolved, people had to work with larger numbers and needed a more practical system. This is how different numeral systems emerged in different cultures.
A numeral system is the set of symbols and rules used to represent numbers.
ãThe Egyptian numeral system Ancient Egyptians used the following symbols:
1 10 100 1 000 10 000100 0001 000 000 stick hobble rope flowerfinger frog person The way they wrote numbers was simple. They added the necessary symbols until they had the number they wanted. Numeral systems, like the Egyptian, that add symbols together until they get the desired value are called additive systems.
ãThe Mayan numeral system The Mayan people lived in modern-day Guatemala and the south of Mexico. Before Columbus arrived in the Americas, the Mayans only used three symbols to write numbers:
Look at the diagram on the left. It shows how numbers under 20 were written using an additive system. These numbers are refered to as the first level. Larger numbers were written using the same symbols, but adding new levels. Each time they added a new level, the value of the symbols was multiplied by 20. Second level (× 20) → First level (× 1) →
The Mayan numeral system also has a characteristic of the positional numeral
system because the value of the symbols changes depending on their level. The Mayan numeral system was partly additive and partly positional.
(0) (1) (5) (0) (1) (5) (0) (1) (5) (0) (1) (5) (0) (1) (5) 2021273640100 137 20 2127 3640100137
ãThe decimal numeral system Nowadays, we use the decimal numeral system. It has ten symbols or figures (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) and follows these rules:
• We write the symbols in different levels called place values: units, tens, hundreds…
• 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. For example:
The value of 4 changes according to its place value.
1
NUMERAL SYSTEMS
This Palaeolithic man has written the number 47. What is the value of each symbol?
This is the number 1 333 331.
Remember
We can break down a number into units and its different place values:
27 473 2 T Th → 20 000 7 Th → 7 000 4 H →400 7 T → 70 3 U → + 3 27 473 10 M H Th T Th Th H T U 4 7 8 4 3 0 4 ↓ 4 000 000 U 4 000 U↓ 4 U↓ Let’s practise!
1 Write the following numbers using the Egyptian numeral system: 19, 65, 34 120 and 2 523 083. 2 The following symbols are used in an additive system:
1 5 10 100 Write the following numbers using this system: 7, 12, 84 and 126. 3 Translate the following Mayan numbers into the
decimal system: 4 Look at the Mayan numbers below. Add four elements
to the left of the series. Now, add four to the right.
5 Complete the following in your notebook: a) 500 T = … H = … Th b) 3 000 H = … Th = … T Th c) 6 Th = … H = … T d) 8 H Th = … T Th = … T 6 True or false?
a) If you move figures to a different place value, the value of the number they represent changes. b) If you add a zero to the right of a number, its value
becomes ten times greater. c) If you add a zero to the left of a number, its value
becomes ten times lower. d) Half a thousand equals 5 tens. e) One thousand thousands equals one million. 7 A number contains five figures that add up to 5. If
you change the place value of the units to thousands, the total increases by 999. What is the number?
Consolidating ideas
1 Think about the decimal numeral system. a) How many tens are there in 3 thousands? b) How many hundreds are there in one ten of thousands? c) How many hundreds are there in 5 one million units? Help H Th T Th ThHTU 1× 100× 10 0 1 Th = 100 T 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
9
Large numbers
Methodological suggestions
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, billones…) and their equivalents with ease.
In the next unit, 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 our 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 millardo (mil millones) has been coined, although it is not often used.
Answers for ‘Let's practice!’
1
a) Seven thousands of millions.
b) Three thousand, one hundred fifty-three million six hundred thousands.
c) Nine billones, four hundred and sixty thousand eight hundred millions.
2
a) 28 350 000
b) 143 000 000
c) 2 700 000 000
d) 16 000 000 000 e) 1 500 000 000 000
3
a) Million
b) Thousand millions c) Thousand millions d) Billón
4
Between 10 and 70 billones of cells.
5
Ten thousand billones.
6
A 1 followed by 24 zeros → one billón of billones.
13 1 Unit Many numbers contain more than nine figures. For example, there are
7 000 000 000 people on the Earth, 3 153 600 000 seconds in a century and 9 460 800 000 000 kilometres in a light year. The decimal numeral system allows us to write numbers that contain as many figures as we want. The table below shows the place value for some numbers with more than 9 figures:
… billón th ous an ds of mil lions
millions thousands hundredstensunits
13800000000 100000000000 1000000000000
The universe was formed thirteen thousand eight hundred million years ago.
A young person’s brain contains around one hundred thousand millions of neurons.
The volume of the Earth is approximately one billón cubic kilometres.
• One million ↔ A 1 followed by 6 zeros.
• One billón ↔ A 1 followed by 12 zeros.
• One trillón ↔ A 1 followed by 18 zeros.
LARGE NUMBERS
An uncommon name for a thousand millions (1 000 000 000) is a milliard. Sometimes, the prefix giga is also used. For example: 1 000 000 000 bytes = 1 gigabyte
Interesting fact
2
12 Let’s practise!
1 Read the first paragraph on this page. Then, write the following numbers in words: a) The number of people on Earth. b) The number 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 thousands. b) One hundred and forty-three millions. c) Two thousand seven hundred millions. d) Sixteen gigas. e) One and a half billón.
3 Copy and complete in your notebook: a) One thousand thousands = one ... b) One thousand millions = one ... c) One million of millions = one ... 4 The human body contains between ten and seventy
million of millions cells. Express both amounts in billones. 5 How do you say the number that is written as a 1
followed by 30 zeros? 6 Scientists estimate that there are three cuatrillones
kilograms of water in our seas and oceans. What do you think a cuatrillón is?
ROUNDING NATURAL NUMBERS
3
Let’s practise!
1 Round the following numbers to the nearest thousands: a) 24 963 b) 7 280 c) 40 274 d) 99 834 2 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 3 Look at the newspaper in the picture. Round the number of tourists to the nearest millions and the amount they spent to the nearest thousands of millions.
4 Round the following numbers to the nearest millions: a) 24 356 000 b) 36 905 000 c) 274 825 048 5 The sign below shows the price of a house in Euros.
138 290€138 290 €
€138 000 €138 300 €140 000 a) Which of the three approximations is closest to the
real value? b) Which approximation would you use in an
informal conversation? 6 The town hall has a budget of €149 637 to refurbish
its sports centre. Which approximate number would you use to tell a friend about this?
In 2018, 82 600 000 tourists visited Spain. They spent 89 678 million Euros.
anayaeducacion.es Practise rounding numbers.
Consolidating ideas
1 Use the diagram below to help you round the number 384 523 to the nearest hundreds of thousands; to the nearest tens of thousands; and to the nearest thousands.
Help If we round the number 52 722: – to the nearest tens of
thousands → 50 000 – to the nearest thousands → 53 000
FOR SALE
hundreds of thousands tens of thousands thousands
Th T Th H Th +1 8 ≥ 5 3 8 4 5 2 3 0 0 0 0 0 ... = 4 < 5 3 8 4 5 2 3 0 0 0 0 +1 5 ≥ 5 3 8 4 5 2 3 0 0 0 ... ... +1 8 ≥ 5 3 8 4 5 2 3 0 0 0 0 0 ... = 4 < 5 3 8 4 5 2 3 0 0 0 0 +1 5 ≥ 5 3 8 4 5 2 3 0 0 0 ... ... +1 8 ≥ 5 3 8 4 5 2 3 0 0 0 0 0 ... = 4 < 5 3 8 4 5 2 3 0 0 0 0 +1 5 ≥ 5 3 8 4 5 2 3 0 0 0 ... ...
When a number has many figures, it is difficult to remember and makes calculations more difficult. We often replace it with an approximate value ended in zeros. That number is more manageable. For example:
31 853 000 × 500 = 15 926 500 000
There are 31 853 000 €500 notes in circulation
in Spain. Approximately, how many thousand million Euros is this?
Approximately, they are sixteen thousand millions.
Rounding is the most frequent and easy method of approximation. To round a number to a specific place value:
• We replace with zeros all the numbers to the right of that place value.
• If the first number being replaced is bigger than or equal to five, we add one more unit to the following figure. Focus on English
The English words billions, trillions, quadrillions… are false friends. They do not mean the same in Spanish as in English. For example:
1 billion ↔ A 1 followed by 9 zeros. 1 trillion ↔ A 1 followed by 12 zeros.
Focus on English In English, we write the € symbol before the amount (not after it like in Spanish). However, when we say the amount, we say Euro after it. For example:
€500 ↔ Five hundred Euros.
13 1 Unit Many numbers contain more than nine figures. For example, there are
7 000 000 000 people on the Earth, 3 153 600 000 seconds in a century and 9 460 800 000 000 kilometres in a light year. The decimal numeral system allows us to write numbers that contain as many figures as we want. The table below shows the place value for some numbers with more than 9 figures:
… billón th ous an ds of millions
millions thousands hundredstensunits
13800000000 100000000000 1000000000000
The universe was formed thirteen thousand eight hundred million years ago.
A young person’s brain contains around one hundred thousand millions of neurons.
The volume of the Earth is approximately one billón cubic kilometres.
• One million ↔ A 1 followed by 6 zeros.
• One billón ↔ A 1 followed by 12 zeros.
• One trillón ↔ A 1 followed by 18 zeros.
LARGE NUMBERS
An uncommon name for a thousand millions (1 000 000 000) is a milliard. Sometimes, the prefix giga is also used. For example: 1 000 000 000 bytes = 1 gigabyte
Interesting fact
2
12 Let’s practise!
1 Read the first paragraph on this page. Then, write the following numbers in words: a) The number of people on Earth. b) The number 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 thousands. b) One hundred and forty-three millions. c) Two thousand seven hundred millions. d) Sixteen gigas. e) One and a half billón.
3 Copy and complete in your notebook: a) One thousand thousands = one ... b) One thousand millions = one ... c) One million of millions = one ... 4 The human body contains between ten and seventy
million of millions cells. Express both amounts in billones. 5 How do you say the number that is written as a 1
followed by 30 zeros? 6 Scientists estimate that there are three cuatrillones
kilograms of water in our seas and oceans. What do you think a cuatrillón is?
ROUNDING NATURAL NUMBERS
3
Let’s practise!
1 Round the following numbers to the nearest thousands: a) 24 963 b) 7 280 c) 40 274 d) 99 834 2 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 3 Look at the newspaper in the picture. Round the number of tourists to the nearest millions and the amount they spent to the nearest thousands of millions.
4 Round the following numbers to the nearest millions: a) 24 356 000 b) 36 905 000 c) 274 825 048 5 The sign below shows the price of a house in Euros.
138 290€138 290 €
€138 000 €138 300 €140 000 a) Which of the three approximations is closest to the
real value? b) Which approximation would you use in an
informal conversation? 6 The town hall has a budget of €149 637 to refurbish
its sports centre. Which approximate number would you use to tell a friend about this?
In 2018, 82 600 000 tourists visited Spain. They spent 89 678 million Euros.
anayaeducacion.es Practise rounding numbers.
Consolidating ideas
1 Use the diagram below to help you round the number 384 523 to the nearest hundreds of thousands; to the nearest tens of thousands; and to the nearest thousands.
Help If we round the number 52 722: – to the nearest tens of
thousands → 50 000 – to the nearest thousands → 53 000
FOR SALE
hundreds of thousands tens of thousands thousands
Th T Th H Th +1 8 ≥ 5 3 8 4 5 2 3 0 0 0 0 0 ... = 4 < 5 3 8 4 5 2 3 0 0 0 0 +1 5 ≥ 5 3 8 4 5 2 3 0 0 0 ... ... +1 8 ≥ 5 3 8 4 5 2 3 0 0 0 0 0 ... = 4 < 5 3 8 4 5 2 3 0 0 0 0 +1 5 ≥ 5 3 8 4 5 2 3 0 0 0 ... ... +1 8 ≥ 5 3 8 4 5 2 3 0 0 0 0 0 ... = 4 < 5 3 8 4 5 2 3 0 0 0 0 +1 5 ≥ 5 3 8 4 5 2 3 0 0 0 ... ...
When a number has many figures, it is difficult to remember and makes calculations more difficult. We often replace it with an approximate value ended in zeros. That number is more manageable. For example:
31 853 000 × 500 = 15 926 500 000
There are 31 853 000 €500 notes in circulation in Spain. Approximately,
how many thousand million Euros is this?
Approximately, they are sixteen thousand millions.
Rounding is the most frequent and easy method of approximation. To round a number to a specific place value:
• We replace with zeros all the numbers to the right of that place value.
• If the first number being replaced is bigger than or equal to five, we add one more unit to the following figure. Focus on English
The English words billions, trillions, quadrillions… are false friends. They do not mean the same in Spanish as in English. For example:
1 billion ↔ A 1 followed by 9 zeros. 1 trillion ↔ A 1 followed by 12 zeros.
Focus on English In English, we write the € symbol before the amount (not after it like in Spanish). However, when we say the amount, we say Euro after it. For example:
€500 ↔ Five hundred Euros.
Rounding natural numbers
Methodological suggestions
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’ competencies, 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…).
Answers for ‘Consolidating ideas’
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.
1
Hundreds of thousands: 400 000
Tens of thousands: 380 000
Thousands: 385 000
ICT
anayaeducacion.es
In ‘My web resources’ there are
interactive activities to practise this content.
Focus on English
The Focus on English section highlights words and
expressions that can be difficult when trying to
understand the text. The word or concept is always
followed by a clear definition or explanation.
Focus on English
The Focus on English section highlights words and
expressions that can be difficult when trying to
understand the text. The word or concept is always
followed by a clear definition or explanation.
Answers for 'Let's practice!'
1
a) 25 000
b) 7 000
c) 40 000
d) 100 000
2
a) 530 298 → 500 000 and 530 000
b) 828 502 → 800 000 and 830 000
c) 359 481 → 400 000 and 360 000
d) 29 935 236 → 29 900 000 and 29 940 000
3
There were 83 000 000 millions of tourists, approximately.
They spent 90 thousands of millions, approximately.
4
a) 24 000 000
b) 37 000 000
c) 275 000 000
5
a) 138 300
b) 140 000
6
€
150 000€
Basic operations with natural numbers (I)
Methodological suggestions
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 experiments 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)
Answers for 'Let's practice!'
1
a) 468
b) 166
c) 758
d) 185
2
The correct answer is la 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
9 5 8
×
7 3
2 8 7 4
+ 6 7 0 6
15 1 Unit Although you already know how to operate with natural numbers, this section isa revision of some concepts and properties.
ãAddition and its properties
Addition means to find the total value of a set of numbers. Look at the picture on the left. To find the total number of people at the football stadium, we add the numbers together:
11 576 + 9 006 = 20 582 Addition follows these two properties:
• Commutative property: The sum does not change if the order of the sumands changes.
a + b = b + a
• Associative property: The way in which the values are grouped does not affect the result.
(a + b) + c = a + (b + c)
ãSubtraction and its relation to addition
Subtraction means to “take” one number from another number, to calculate what is left. In other words, to find the difference. For example, if we want to count the empty seats at the football stadium above, we subtract the number of full seats from the total number of seats:
25 342 – 20 582 = 4 760 As you can see 25 342 = 20 582 + 4 760. And 20 582 = 25 342 – 4 760.
Relationship between addition and subtraction: M – S = D → M S D S M D– = + = *
4
BASIC OPERATIONS WITH NATURAL NUMBERS
14 Let’s practise!
1 Calculate: a) 254 + 78 + 136 b) 340 + 255 – 429 c) 1 526 – 831 + 63 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.
3 Transform: a) This addition into a subtraction: 48 + 12 = 60 b) This subtraction into an addition: 22 – 2 – 6 = 14 4 If Alberto were 15 years older, he would still be
18 years younger than his uncle Thomas. His uncle Thomas is 51 years old. How old is Alberto? 5 If I only bought a washing machine, I would still have
€246. However, if I also bought a TV, I would be €204 short. Could you say how much is any of these items? anayaeducacion.es Mental arithmetic: addition and subtraction. 25 342 ← Minuend (M )
– 20 582 ← Subtrahend (S ) 4 760 ← Difference (D )
Remember
SEATING CAPACITY: 25 342 seats
Seats filled East stands: 11 576 West stands: 9 006 34 + 16 = 16 + 34 50 50 (18 + 3) + 17 = 18 + (3 + 17) 21 + 17 18 + 20 38 38 Commutative property Associative property
ãMultiplication and its properties A multiplication is basically a repeated addition of the same value. For example, if a ticket to the football game from the previous page costs €35, the total price for all 20 582 tickets purchased is:
35 + 35 + 35 + … + 35 = 35 · 20 582 = €720 370
20 582 times
Multiplication follows these three properties:
• Commutative property: The product does not change if we change the order of the factors.
a · b = b · a
• Associative property: The product of a multiplication is not affected by the way in which we group the factors.
(a · b) · c = a · (b · c)
• Distributive property: The product of a calculation does not change if we remove the parentheses.
a · (b + c) = a · b + a · c a · (b – c) = a · b – a · c The following example will help you to understand the distributive property: On Thursday, a group of friends bought 7 tickets to the game. On Friday, they bought 3 more tickets. How much did all the tickets together cost? We can write the calculation in two different ways:
price of 7 tickets + price of 3 tickets ↔ price of (7 + 3) tickets 35 · 7 + 35 · 3 = 35 · 10 35 · 7 + 35 · 3 = 35 · (7 + 3)
245 + 105 35 · 10 350 350
Let’s practise!
6 Complete the following multiplications in your notebook: 5 × 2 + 9 0 1 2 6 0 9 8 × 2 8 7 4 + 6 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:
Multiplying a number by eight is the same as multiplying it by ten, then subtracting double the original number. Which property describes this?
9 Multiply mentally 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 per 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 an orchard with 200 peach trees. He
estimates that each tree can fill seven boxes with five kilos of peaches in each one. How much profit does he make if he sells all of the peaches at €2 per kilo? anayaeducacion.es Mental arithmetic:
multiplication. Mental arithmetic 16 × 55 8 × 2 × 5 × 11 88 × 10 880 The associative property allows us to regroup expressions. The commutative property allows us to change their order.
ICT
anayaeducacion.es
In ‘My web resources’ there are
11
7
a) 190
b) 1 200
c) 15 000
d) 1 400
e) 23 000
f) 460 000
8
x · 8 = x · (10 – 2) = x · 10 – x · 2
We have applied the distributive property to this equality.
9
a) 12 · 9 = 12 · 10 – 12 = 120 – 12 = 108
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
17 16 1 Unit ãDivisionRemember two of the ways in which we can solve divisions that are common in everyday life:
• We use 5 625 cubic metres of water to water a park for 15 days. How many cubic metres do we use each day?
5 6 2 5 15 1 1 2 375 0 7 5 0 0
⎯→ 5 625 : 15 = 375 m3 each day
Division means sharing a value between several people or things in equal
parts to work out how much each one receives.
• We use 375 cubic metres of water to water the park each day. There are 5 625 cubic metres of water in a tank. How many days does the water in the tank last?
5 6 2 5 375 1 8 7 5 15 0 0 0
⎯→ 5 625 : 375 = 15 days
Division means splitting a whole thing into equal portions of a specific
size to work out how many portions there are.
ãExact division and integer division In the previous example, we used 5 625 cubic metres of water to water the park for exactly 15 days. We had no water left.
5 6 2 5 375 1 8 7 5 15 0 0 0 ⎯→ 5 625 = 375 · 15We call this an exact division. However, if there were 5 700 cubic metres of water in the tank, there would be some extra water left, even after 15 days.
5 7 0 0 375 1 9 5 0 15 0 7 5
⎯→ 5 700 = 375 · 15 + 75 We call this an integer division. There are two types of division depending on the value of the remainder:
• Exact division (the remainder is zero). D d
0 q ⎯→ The dividend is equal to the divisor multiplied by the quotient. D = d · q
• Integer division (the remainder is not zero). D d
r q ⎯→ The dividend is equal to the divisor multiplied by the quotient, plus the remainder. D = d · q + r
4 BASIC OPERATIONS WITH NATURAL NUMBERS
WATER FOR DAILY USE
We pack 35 kg of oranges into 5 kg boxes.
35 5 07 We fill 7 boxes and there are no remaining oranges. We pack 38 kg of apples into 5 kg boxes.
38 5 37 We fill 7 boxes and there are 3 kg remaining. Exact division Integer division anayaeducacion.es Mental arithmetic: division. ãA property of division Look at what happens if we multiply the dividend and the divisor by the same number:
To water 3 plants, we need 24 litres of water. What happens if we have twice the number of plants and twice the amount of water?
24 litres 48 litres 24 3
0 8 48 60 8 If we double the amount of water and the number of plants, the amount of water each plant receives does not change. In a division, if we multiply the dividend and divisor by the same number, the quotient does not change.
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 to divide mentally.
• 96 : 12 8
: 332 : 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) : What have you noticed? 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 division follow the associative property?
16 Find the missing value in each division: dividend 53
39 15 1 000 divisor
12 38 17 True or false?
a) The quotient must be greater than the divisor. b) The remainder is always lower than the divisor. c) In an exact division, multiplying the dividend by
2 doubles the quotient. d) Multiplying the dividend and the divisor by 3
triples the quotient. e) Division follows the commutative property. 18 Solve the following problems without making notes:
a) 150 grams of salami are divided between three sandwiches. How many grams are there in each sandwich? b) How many minutes are there in 180 seconds? c) We have travelled 240 kilometres in three hours.
How many kilometres did we travel each hour? d) We put 250 kg of apples in 10 kg boxes. How
many boxes are there? 19 A farmer collects 1 274 eggs. He puts 30 eggs in each
tray, and 10 trays in each box. How many eggs are left over? How many trays are left over?
Example
32 8 0 4 224 56 00 4 The quotient does not change. × 7 × 7
Basic operations with natural numbers (II)
Methodological suggestions
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 going to be 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.
Answers for ‘Let’s practise!’
12
a) c = 7; r = 5
b) c = 23; r = 0
c) c = 758; r = 3
d) c = 270; r = 9 e) c = 308; r = 0
f) c = 127; r = 508
13
a) 60 : 3 = 20 : 4 = 5
b) 180 : 3 = 60 : 4 = 15
c) 300 : 3 = 100 : 4 = 25
d) 75 : 3 = 25 : 5 = 5
e) 90 : 3 = 30 : 5 = 6
f) 180 : 3 = 60 : 5 = 12
g) 180 : 10 = 18 : 3 = 6
h) 240 : 10 = 24 : 3 = 8
i) 390 : 10 = 39 : 3 = 13
14
(36 : 12) : 3
3 : 3
1
36 : (12 : 3)
36 : 4
9
We can see that the division does not follow the associative property.
ICT
anayaeducacion.es
In ‘My web resources’ there are
Expressions with combined operations
Methodological suggestions
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
in-depth knowledge about it and take into account how it works.
Answers for ‘Consolidating ideas’
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 equalities.
1
4 · 10 – 8 · 3 + 2
40 – 24 + 2
16 + 2
18
4 · (10 – 8) · 3 + 2
4 · 2 · 3 + 2
24 + 2
26
4 · 10 – (8 · 3 + 2)
40 – ( 24 + 2)
40 – 26
14
4 · 10 – 8 · (3 + 2)
40 – 8 · 5
40 – 40
4 · (10 – 8) · (3 + 2)
4 · 2 · 5
4 · 10
19 1 Unit5
EXPRESSIONS WITH COMBINED OPERATIONS
ãOrder 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. Look at the order followed in the calculations below. Although each calculation contains the same values and operations, the answers are not the same.
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 The order of combined operations is always: 1st Brackets. 2nd Multiplication and division. 3rd Addition and subtraction.
ãHow to use a calculator Type this sequence into a calculator: 2 + 3 * 4 =
It may seem strange, but different calculators give different answers. You could have got 20 or 14.
{∫“≠} → The calculator performs each operation in the order it was entered. (2 + 3) · 4 = 5 · 4 = 20
{∫‘¢} → The calculator performs the multiplication first. It follows the correct order of operations.
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.
18 Consolidating ideas
1 Complete each box in your notebook. Check that you have the right answers. 4 · 10 – 8 · 3 + 2 – + 2 + 2 18 4 · (10 – 8) · 3 + 2 4 · · 3 + 2 + 2 26 4 · 10 – (8 · 3 + 2) – ( + 2) – 14 4 · 10 – 8 · (3 + 2) – 8 · – 0 4 · (10 – 8) · (3 + 2) 4 · · 4 · 40 Write any number that contains two
figures, a b . Now, write the figures in the opposite order, b a . Add both numbers together, and divide your answer by the sum of the two figures, a + b.
( a b + b a ) : (a + b) = …? What is the answer? Why?
Why? • 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 Consolidating ideas
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. Let’s practise! 40 – 12 : 4 + 2 · 3 40 – + + (40 – 12) : 4 + 2 · 3 : 4 + + 40 ≤ 12 / 4 µ 2 * 3 ≤ Ñ → {∫∫∫∫∫∫¢«}40 - 12 =/ 4 ≤ 2 * 3 ≤ Ñ → {∫∫∫∫∫∫‘«}
1 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) 2 Calculate the following in your head. 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) 3 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 4 Calculate. Write down the steps you followed. Check
your answers on the right. If they do not coincide, go over what you have done 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 5 Problem solved
This month, an employee worked for 7 hours on 12 days earning his normal pay rate. He also worked 9 hours on 5 days. During 6 of those hours, he earned his normal pay rate. For the remaining 3 hours, he earned the night-time rate. How many hours did he work in total this month?
We can solve this by writing it in two different ways:
Answer: In total, he worked 129 hours. 6 Write the following statements as mathematical
expressions and solve them. a) A van carries 8 boxes of bananas, 20 boxes of
oranges and 6 boxes of apples. Each box of bananas weighs 15 kg. Each box of oranges and each box of apples weighs 8 kg. How many kilograms of fruit does the van carry? b) A supermarket orders 20 crates of full-fat milk,
15 crates of skimmed milk and 10 crates of semi-skimmed milk. Each crate holds 6 one-litre bottles. How many bottles did the supermarket order? c) There are 15 tables, 55 chairs and 12 stools in a
cafe. How many legs are there in total? (hint: each stool has 3 legs). d) A farmer packs 1 500 eggs into boxes that hold
10 eggs, another 1 500 eggs into boxes that hold 6 eggs, and 300 free-range eggs into boxes that hold 6 eggs. How many boxes does he fill? 12 · 7 + 5 · (6 + 3) = 84 + 5 · 9 = 84 + 45 = 129
12 days 5 days
12 · 7 + 5 · 6 + 5 · 3 = 84 + 30 + 15 = 129normal rate night-time rate
