Mathematics for Academic Studies 3 Teacher's Guide sample unit

DEMO

3

SECONDARY EDUCATION

Mathematics for Academic Studies J. Colera Jiménez, M.ª J. Oliveira González, I. Gaztelu Albero, R. Colera Cañas

T e a c h e r’s G u i d e

Building

Blocks

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: Beatriz Fuentes, Sara Gascón Martín and Joaquín Montón EDITORIAL COORDINATION: Paloma Rodríguez Esteban Design, technical drawings and maps: Miguel Ángel Díaz-Rullo, Patricia G. Serrano, Juan Carlos Quignon Illustrations: Alberto Hoyos Layout: DiScript and Isabel Román Graphic edition: Olga Sayans Translation: Montero Language Services and Robin Munby

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.

Commitment to Sustainable Development Goals 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.

© GRUPO ANAYA, S.A., 2021 - C/ Juan Ignacio Luca de Tena, 15 - 28027 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.

CONTENTS Building blocks and the project keys ..............................

4

Our project ........................................................................................

6

Units

1. FRACTIONS AND DECIMALS ................................................ 8 2. POWERS AND ROOTS ............................................................ 20 3. ARITHMETIC PROBLEMS ..................................................... 30 4. PROGRESSIONS ........................................................................ 42 5. ALGEBRAIC LANGUAGE ........................................................ 54 6. EQUATIONS ................................................................................. 68 7. SYSTEMS OF EQUATIONS .................................................... 82 8. CHARACTERISTICS OF FUNCTIONS ............................... 94 9. LINEAR AND QUADRATIC FUNCTIONS .......................... 108 10. METRIC PROBLEMS IN PLANE GEOMETRY ............... 124 11. GEOMETRIC SHAPES .............................................................. 138 12. GEOMETRIC TRANSFORMATIONS ................................... 150 13. STATISTICAL TABLES AND GRAPHS .............................. 166 14. STATISTICAL PARAMETERS ............................................... 178 15. CHANCE AND PROBABILITY .............................................. 190

g n i d l i u b

blocks

g n i n r a e l p ee k o t A project that is rooted in skills-based learning and the development of student commitment within the realities of available time.

Building Blocks is a new competence-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.

Project keys SDG

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

romotes entrepreneurial thinking in its three P 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...

Our project THE STUDENT’S BOOK The books present content and activities that are adjusted to the curricular development of our community and that develop mathematical reasoning based mainly on solving problems taken from everyday situations. Using a skills-based methodology, it allows us to express, in a creative and innovative way, our commitment to the Sustainable Development Goals, demonstrating how Mathematics contribute to the achievement of these objectives.

THE DIGITAL PROJECT FOR TECHERS THE web www.anayaeducacion.es The teacher’s website is a tool that facilitates and enriches teaching work.

With it, you can adapt the contents to the needs of the students through additional resources or reinforce the most relevant educational aspects: • Course plan, the teacher’s guide and project information.

•D iversity and inclusion, to meet the diversity of students’ motivations, interests and learning styles through: - Key concepts and additional worksheets.

- Extension worksheets and competence development worksheets.

- Tasks, workshops and other resources.

• Assessment, a core value of the project, is supported through: - Predesigned assessment tests and records.

- Specific documentation.

- Assessment tools.

• Resource bank, with a wide variety of digital resources such as: - GeoGebra activities - Video tutorials - Self-assessments - Glossaries - Key concepts - Learn by playing - Problem solving practice Language Bank and more.

•L anguage Bank: created to further develop linguistic skills, this new educational tool can be used both by the teacher and the language assistant.

1

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Building

Blocks

Building

Blocks

Teacher’s Guide

1

g

Buildin

Mathematics

THE TEACHER’S GUIDE

1

SECONDARY EDUCATION

Blocks

learning to keep

www.anayaeducacion.es

J. C olera Jiménez, I. Gaztelu Albero, R. C olera Cañas

SECONDARY EDUCATION

T e a c h e r’s G u i d e

I S B N 978-84-698-7408-0

9

788469 874080

3100508

There is a teacher’s guide for each student book with the solutions to the activities, methodological guidelines, suggestions for applying the project keys, etc.

Mathematics

Building

Blocks

THE DIGITAL BOOK All Building Blocks books have a digital version.

You can download the entire book or download it by units, with all its resources or in its lighter version (without the largest digital resources), which allows offline and online use. The resources of each unit are grouped by type and allow direct access to the resource bank and the Teacher’s guide.

3

AND FOR THE S TUDENTS? Resource bank

at www.a nayaedu cacion.es • Resourc con: es for the project k • Subject eys. key conc epts.

• Resourc es by unit .

Digital b ook A digital version o f the textb be down ook that loaded in can full or by its resourc u nits; with es or a lig all hter vers the heavie ion (with st digital o u t resources) .

1 Fractions and decimals Unit presentation In this unit we aim to consolidate much of what students already know about numbers, their uses and operating with them. We also aim to look at certain aspects in more detail and explain how they work in practice.

By this level, students usually have a reasonably good understanding of what fractions are, what they mean and how they are used. This is not the case, however, for operating with fractions, which students will still have difficulty with. We start by reviewing the concept of a fraction, before moving on to introduce the concept of rational numbers.

We remind students of the concept of fractions as operators. Students do not usually have difficulty calculating a fraction of an amount, but we recommend going over the inverse process in more detail: finding a total amount based on a given part.

We also review equivalent fractions and their properties, ensuring that students know how to reduce fractions to a common denominator and can do this with ease.

Converting fractions into decimals and vice versa, particularly converting recurring decimals into fractions, is a fundamental skill for this level.

It is important to focus on and help students develop their mental calculation skills using both decimal numbers and fractions, since this will help them to develop their confidence and fluency with mental arithmetic.

The majority of students will already have used a calculator, but at this stage they should be starting to acquire a deeper understanding of how calculators are used and an appreciation for their enormous potential when working with fractions.

Basic knowledge • Working with fractions: operations and uses.

• Transforming fractions into decimals.

• Distinguishing between types of decimal numbers.

• Expressing a terminating decimal as a fraction.

• Solving arithmetic problems using fractions.

• Understanding the calculator and its basic uses.

Complementary knowledge • Using the number line to represent fractional numbers.

• Transforming fractions into a recurring decimal number and vice versa.

• Recognising irrational numbers.

Task preparation • Reviewing students’ understanding of the order of operations and the use of brackets.

• Reminding students of some basic concepts and procedures relating to divisibility (exact division, multiple, divisor, etc.)

• Reviewing some basic mental calculation methods.

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KC: key competences, CLC: competence in linguistic communication, CMST: competence in mathematics, 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.

CONTENTS AND COMPETENCES Unit contents

Key competences

Opening page • Sexagesimal fractions in Mesopotamia • Finally, decimal fractions

CLC CMST LL SIE CAE

1. Fractions • Simplifying fractions • Equivalent fractions • Comparing fractions

CLC CMST LL

2. Operations with fractions • Adding and subtracting fractions • Multiplying and dividing fractions • Combined operations with fractions • Fraction of an amount

CLC CMST SCC SIE

3. Decimal numbers • Types of decimal numbers • Transforming fractions into decimals • Transforming decimals into fractions

CLC CMST LL SIE

4. Fractions and decimals with the calculator • Configuration • Fractions • Operating with fractions • Decimals

CMST DC LL SIE

Final pages • Exercises and problems solved • Exercises and problems • Maths Workshop • Self-assessment

CLC CMST DC LL SCC SIE CAE

PROJECT KEYS

1

SDG commitment • Quality education, target 4.a

fractions and decimals

Finally, decimal fractions

In ancient Mesopotamia, the Babylonians used the sexagesimal system to write numbers. Sexagesimal fractions have denominators that are powers of base 60.

Integers have been written as decimals in the Western world since the 13th century. However, sexagesimal fractions were still used to write parts of the unit! Even today, we still use them to measure time: a minute is onesixtieth of an hour, and a second is one-sixtieth of a minute. Therefore, 3 h 36 min 45 s written in decimal time would be 3 + 36 + 452 = 3.6125 h.

60 60 Using decimals to write parts of the unit did not become popular until the end of the 16th century. The French François Viète and the Flemish Simon Stevin were the main driving forces behind the change.

To express 2 they wrote 24 .

5 60

1 In the text, you can find the expression ‘to be a driving force’. Discuss

Reading and listening

Sexagesimal fractions in Mesopotamia

Linguistic plan • Skills: Listening and Reading (receptive skills).

Speaking and Writing (productive skills)

its meaning with your classmates and write a sentence using it in which its meaning is clear.

To express 1 they wrote 45 = 452 .

80 3 600 60

2 True or false? Make the false ones right.

How the Babylonians wrote numbers

a) Sexagesimal fractions were used by the ancient Egyptians. They are fractions with denominators that are powers of base 62.

To understand how the Babylonians wrote numbers on clay tablets, look at some examples in this table. It shows the place values of sexagesimal units: 602

60

1

1/60

b) The sexagesimal system only uses 6 symbols to write the numbers from 10 to 60.

1/602

c) Integers started to be used in the Western world at the beginning of the 13th century.

→ 1 · 602 + 16 · 60 = 4 560 → 24 = 2 = 0.4 60 5 → 1 + 24 = 1.4 60 → 1 + 24 + 452 = 1.4125 60 60

d) Nowadays, we keep using sexagesimal fractions to measure time. e) The mathematicians Viète and Steving had a big influence on the popularisation of decimal numbers at the beginning of the 16th century.

Notice that this system only uses two symbols: = 10 and = 1 to write the numbers from 1 to 59. Depending on the position of those numbers, their values are multiplied by 1, 60, 602… or by 1/60, 1/602… This is what we call a positional system.

Enterprising culture • Personal dimension: Self-knowledge ICT • Worksheets on Language Bank • Resources for each unit (videos, solved exercises, GeoGebra activities, self-assessment, etc.)

• Portfolio: printable version of the unit portfolio Assessment • Exercises and problems • Preparation of the portfolio • Preparation of the portfolio

1 Write 3 , 5 and 5 in the Egyptian style, as a sum of unit fractions.

4 6

8

2 The number 3;8,29,44 is in sexagesimal form. Write it as a sum of

From sexagesimal fractions to decimal numbers

Developing thinking Technique: • RQAS Cooperative learning Technique: • Shared interpretation

Solve

fractions with denominators that are powers with a base of 60 (3 + + 8/60 + …) and convert it into decimal form. Do you recognise this number?

To transform a number written in sexagesimal notation to a decimal, we just have to use what we already know. Observe: N = 1;24,45 (sexagesimal form)

3 What numbers do you see in this

N = 1 + 24 + 452 = 1 + 2 + 1 = 5 80 60 60

tablet? The colours of the columns correspond to the same units as the table on the previous page.

GE BANK LANGUA LANGUAGE BANK BANK GE BANK GE BANK LANGUA LANGUAGE NK LANGUA NK BA NK BA BANK 11 BA GE GE GE LANGUA LANGUAGE NK LANGUA LANGUA BANK NGUAGE BA UAGE BANK NGUAGE LA LANG

N = 1 + 2 : 5 + 1 : 80 = 1.4125 (decimal form)

10

LA

Fractions and decimals In this unit In the Listening and Reading section the students read and listen to an introductory text about the different uses of fractional and decimal numbers in ancient civilisations that will help them to reflect on the role of custom and tradition in hindering or impeding progress. An example of this is the use of decimal numbers, which are so vital to modern society, but which were not popularised until the late 16th century.

Then, students will answer questions related to the text. Some questions are literal with the answers stated in the text. Others are inferential so the answers are indirectly stated.

Questions to detect preconceptions The exercises on page 11 give students the opportunity to work with the types of fractions used by the ancient Egyptians and Babylonians. They will see how impractical they are for performing operations. This will help them to appreciate the modern procedures for operating with fractions that they study in this unit.

Answer key for ‘Reading and Listening’ 1

Open answer. a) False b) False d) True e) False

c) True

Answer key for ‘Solve’ Open answer.

For example: 3=1+1 ; 4 2 4 8 + 29 + 2 3 + 60 60 2 It is π.

1

5=1+1 5=1+1 ; 6 2 3 8 2 8 44 = 3.1415925 60 3

3 1st

row: 4 395 row: 5.5 rd 3 row: 1.005 2nd

9

Unit

1

Integers are useful for counting things, but they are not as good for expressing measurements. To measure things, we often have to use parts of the unit, like one half, three quarters, seven thousandths... We write these measurements as fractions: 7 1 4 2 3 1000 A fraction is a quotient of two integers. This quotient can be an integer d 6 = 3, –12 = – 4n or a fractional number d 17 = 8 + 1 , –13 = –2 – 3 n.

2 3 2 2 5 5 If the numerator is a multiple of the denominator, the fraction is an integer. If not, it is a fractional number.

anayaeducacion.es • Activities to review operations with integers.

• Activities to practise operations with integers.

Measuring with fractional numbers

The set of all the integers and all the fractional numbers is called the set of rational numbers and is represented by the letter Q. Rational numbers are numbers that can be written as a fraction.

To measure means to relate two magnitudes of the same type.

When we say that the volume of the Moon is 1/50 the volume of the Earth, we are taking the volume of the Earth as a unit. And if we say that the visible part of an iceberg is 1/9 of its total, we are taking the entire volume of the iceberg as a unit.

Rational numbers can be placed on a number line. Between any two numbers on the number line there are infinite other rational numbers.

5 –— 2 –5

–4

–3

10 = 1 + — 3 — 7 7

1 –— 2 –2

–1

0

1

2

23 = 4 + — 3 — 5 5 3

4

5

6

It comes from the word quotient in English, because rational numbers are the quotient of two integers.

Think and practise

1 Draw a number line like this one in your notebook and

put these numbers in their approximate place: 17 , – 11 , 20 , 2 , 16 , – 21 , – 7 3 4 5 3 7 5 2 –5 –4 –3 –2 –1

0

1

2

3

4

4 Match each of the following fraction with its

6

corresponding irreducible fraction:

2 What fractions do these points on the number line

represent?

A –3

12

B –2

–1

C 0

D 1

E 2

3

anayaeducacion.es Activities to review simplifying fractions.

Every rational number can be written as many different equivalent fractions.

In fact, each rational number has an infinite number of equivalent fractions: 3/5 = 6/10 = 9/15 = … We therefore need a way to recognise when two fractions represent the same rational number.

18 and 21 are equivalent because 18 = 18 : 6 = 3 and 21 = 21 : 7 = 3 .

30 35 35 35 : 7 5 30 30 : 6 5

ã Comparing fractions Two fractions with the same denominator are very easy to compare just by looking at their numerators. To compare two fractions with different denominators, we ‘reduce them to a common denominator’. In other words, we find two fractions with the same denominator that are equivalent to the ones we have.

Problems solved 1 Indicate

whether these fractions are equivalent: a) 9 and 12 39 52 b) 15 and 38 35 57 8

a) We find their irreducible fractions and check whether they are equivalent: 9 = 9 : 3 = 3 ; 12 = 12 : 4 = 3 → They are equivalent.

39 39 : 3 13 52 52 : 4 13 b) We do cross-multiplication and check whether they are equivalent: 15 · 57 = 855; 38 · 35 = 1 330 → They are not equivalent.

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1. Fractions

Two fractions are equivalent when they can be simplified to the same irreducible fraction, which is the common expression of that rational number.

We take the LCM as the denominator. LCM (12, 8, 16) = 48.

48 : 12 = 4 → 7 = 7 · 4 = 28 12 12 · 4 48 48 : 8 = 6 → 5 = 5 · 6 = 30 8 8 · 6 48 48 : 16 = 3 → 9 = 9 · 3 = 27 16 16 · 3 48

Clearly: 27 < 28 < 30 48 48 48 Therefore: 9 < 7 <5 16 12 8

Think and practise 5 True or false?

3 Simplify these fractions:

2 , 2 , 5 , 10 , –20 , 30 , –30 , 40 4 6 10 15 30 40 –45 –60 5

Cross-multiplication is a method that we use to check if two fractions are equivalent: a = c if a · d = c · b b d For example, 18 and 21 are 30 35 equivalent because: 18 · 35 = 630 = 21 · 30

12

If the numerator and denominator of a fraction can be divided by the same number (other than 1 and –1), when we divide them by that number, we say that we have simplified or reduced the fraction. For example: 25 = 5 ; 8 = 4 = –2 ; 3000 = 30 = 2 15 3 –12 – 6 3 4500 45 3 When a fraction with a positive denominator cannot be reduced further, we say that it is irreducible.

Why do we use Q to name the set of rational numbers?

Cross-multiplication

2 Compare 7 , 5 and 9 .

ã Simplifying fractions Explain the name…

1

ã Equivalent fractions

FRACTIONS

a) 2 > – 7 because the first one is positive and the 5 4 second one is negative. b) 7 > 2 because the first one is greater than 1 and 3 5 the second one is less than 1.

a) 6 18

b) 15 20

III) 3 4

III) –2 5

c) 8 > 7 because the first one is greater than 2 and 3 4 the second one is less than 2.

c) –15 40

d) 14 –35

III) 1 3

IV) –3 8

d) – 8 > – 7 because the first one is greater than –2 3 4 and the second one is less than –2.

6 Use simplification and cross-multiplication to figure

out whether or not these fractions are equivalent: b) 36 and 78 a) 12 and 21 20 35 102 221

7 Find equivalent fractions of 60 …

a) … with numerator of 20.

126

b) … with denominator of 42.

8 Order these fractions from smallest to largest:

7 12

–6 4

4 6

– 3 15

5 9

–1 2

3 4

13 18

13

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

Suggested methodology We review the concept of a fraction as a quotient of two integers, the result of which could be an integer or a fractional number, and its usefulness for expressing a measurement involving a divided unit.

The concepts of positive and negative fractions and integers expressed as fractions lead us on to the set of rational numbers, Q, and the expansion of the numerical field.

It will be useful for students to see the approximate representation of fractional numbers on a number line. We suggest that they also consider the possibility of finding new fractional numbers between any two given numbers, no matter how close they are.

We discuss comparing equivalent fractions, and the procedure of transforming them into irreducible fractions to demonstrate their equivalence. This will help students develop the habit of always giving their answer as an irreducible fraction, even when this is not asked for explicitly.

We explain the concept of reducing to a common denominator in order to compare nonequivalent fractions, a method which has many other important applications.

Students should be encouraged to use mental calculation here whenever possible (when dealing with equivalence, simplification, comparison, etc.).

Reinforcement and extension We recommend: • From the DIVERSITY AND INCLUSION section in the anayaeducacion.es resource bank: Reinforcement: Practice Exercise 1, worksheet A. Practice Exercise 1, worksheet B. Answer key for ‘Think and practise’ 1

–7 –11 –21 — — — 5 2 4

–5

–4

–3

2 — 3

–2

–4.2 –3.5 –2.75 2 3

0

1 0.67

2 2.29

20 — 5

3

4 4

17 — 3

5

6 5.67

A = – 19 ; B = – 5 ; C = 2 ; D = 3 ; E = 16 = 8 6 3 3 2 7 4 2 = 1 ; 2 = 1 ; 5 = 1 ; 10 = 2 ; – 20 = – 2 ; 30 = 3 ; –30 = 2 ; 40 = – 2 30 4 2 6 3 10 2 15 3 3 40 4 –45 3 –60 3

4 a) 5

–1

16 — 7

III

a) T

b) I

c) IV

b) T

c) T

d) II d) F

12 = 3 and 21 = 3 ; 12 · 35 = 420 = 20 · 21. They are equivalent.

5 5 20 35 b) 36 = 6 and 78 = 6 ; 36 · 221 = 7 956 = 102 · 78. They are equivalent.

102 221 17 17

6 a)

7 8

10

60 = 10 The solution for both parts, a) and b), is the fraction 20 .

42 21 126 3 < 5 < 7 < 4 < 13 < 3 1 –6 <– <– 12 9 15 4 2 6 18 4

Unit

2

Mental arithmetic 1 Find:

To add (or subtract) fractions with the same denominator, we add (or subtract) their numerators and keep the same denominator.

Mental arithmetic a) 2 + 5 – 4 3 3 3 c) 1 + 1 2 4 e) 17 – 3 5

To add (or subtract) fractions with different denominators, we first transform them into equivalent fractions with the same denominator, then we add (or subtract) their numerators.

b) 1 – 2 3 d) 7 – 1 5 f ) 17 – 5 3

7 – 5 + 2 = 42 – 25 + 120 = 42 – 25 + 120 = 137 10 12 60 60 60 60 60

For example:

ã Multiplying and dividing fractions The product of two fractions is a fraction in which the numerator is the product of the numerators and the denominator is the product of the denominators: a · c = a ·c b d b·d For example: 8 · 7 = 8 · 7 = 56 = 28 3 10 3 · 10 30 15 The quotient of two fractions is the product of the first fraction and the inverse of the second fraction: a : c = a · d = a·d b d b c b ·c 6 :3= 6 · 1 = 6 = 2 For example: 9 : 5 = 9 · 7 = 63 ; 11 11 3 33 11 4 7 4 5 20

anayaeducacion.es • Activities to review adding and subtracting fractions.

• Activities to practise adding and subtracting fractions.

Mental arithmetic a) 3 · 7 9 c) 1 · 12 2 13 e) 6 : 3 5 5 g) 6 : 1 5 2

b) 4 · 15 5 8 d) 1 · 2 · 3 2 3 5 f) 6 :6 5 h) 1 : 1 3 6

To do combined operations with fractions, we first solve the expressions in brackets and then the rest of the operations. Remember to solve products and quotients before additions and subtractions. For example: 2 – 1 c1 – 1 m + 2 (–2) = 2 – 1 · 3 + 2 (–2) = 2 – 1 – 4 = 3 3 3 4 3 4 4 3 = 24 – 3 – 16 = 5 12 12 12 12

1 a) 7 + 11

b) 6 – 11 4 e) 4 : 6 5

9 12 d) 6 : 4 5

2 a) d 3 + 7 – 7 n : 25

4

6

8

4 a)

a) 1 of the total is 350.

2 b) 2 of the total is 400.

3 c) 7 of the total is 350.

10

F ocus on English pay off: to give someone all the money you owe them.

All the portions (fractions) of a whole add up to 1.

For example: We divide a cake by giving Ana 1/3, Mark 1/4 and Oliver the rest. How much does Oliver get?

1 – c 1 + 1 m =1 – 7 = 5 12 12 3 4

1 – d 3 – 1n 4 3 +1 4

b)

3 – 1 ·d 3 – 2 n 4 5 15 6 + 4 ·d 1 – 3 n 25 2 4

d 2 – 5 n·d 3 – 5 n 4 6 b) 3 9 d 7 – 5 n · 4 +1 12 6 3

3 a) 2

c) 3 · 4 5 f) 4 : 1 5 6

b) d 13 – 7 n · d 9 + –13 n 15 25 22 33

12

2 Find the total amount:

1 What fraction completes the unit?

anayaeducacion.es Activities to practise combined operations with fractions.

Calculate and simplify the results:

a) 1 of €520 000.

2 b) 3 of 1 000 000 people.

5 c) 7 of 500 buildings.

10

Observe

ã Combined operations with fractions

Think and practise

1

ã Fraction of an amount

OPERATIONS WITH FRACTIONS ã Adding and subtracting fractions

(–3) · d 3 – 1 n 5 3 (–2) · d 4 – 6 n 3 5

a) 1 , 1 and ? b) 2 , 1 and ?

2 4 ?

3 6 ? c) 1 , 1 and ? d) 1 , 1 , 1 and ?

4 6 ?

2 4 8 ?

To find 3 of an amount, like for example of €1 200, we divide it by 5 (to get one 5 fifth) and then, we multiply it by 3. In other words, we multiply the amount by

2. Operations with fractions

3 → 3 · €1 200 = €720 5 5 To find a fraction, a , of an amount, C, we multiply a · C. b b Examples

• A postman delivers 3/28 of a total of 4 004 letters. How many does he deliver?

3 of 4 004 = 3 · 4 004 = 3 · 4 004 = 3 · 143 = 429 letters 28 28 28 • Bertha owns 7/20 of a company. This year, she received €37 800 in profits. What are the company’s total profits?

If she receives €37 800 for owning 7 , then 1 is 37 800 = €5 400.

20 20 7 Therefore, the total d 20 n is 20 · 5 400 = €108 000.

20 We can get the same result by multiplying Bertha’s share of the profits (€37 800) by the inverse of her share of the company, 20 .

7 7 of the total = 37 800 → total = 37 800 · 20 = €108 000 7 20 To find a fraction, a , of another fraction, c , of an amount, C, we multiply b d a · c ·C . b d Example

Three siblings receive an inheritance of €104 000. Albert gets 3/8, Bertha gets 5/12 and Clare gets the rest. Clare uses 2/5 of her portion to pay off * her debts. How much does she have left?

1 – 3 – 5 = 24 – 9 – 10 = 5 is Clare’s portion.

8 12 24 24 Since she spends 2 of her portion, she has 3 left: 5 5 She has 3 · 5 · 104 000 = 1 · 104 000 = €13 000 left.

5 24 8

Think and practise 5 A cyclist has completed 5/9 of today’s 216 km stage.

How many kilometres has he gone?

6 Yesterday, Karen decided to take €3 900 out of the

bank, which is 3/11 of her savings. What are her total savings?

7 A pool with 5 250 litres of water will be used 4/15 by

Barbara, 2/5 by Eric and the rest by Roberta. Roberta uses 3/10 of her portion to water tomatoes and the rest to water her fruit trees. How much water does Roberta use to water her fruit trees?

14

15

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

Focus on English This Focus on English section helps students understand the meaning of the phrasal verb ‘pay off ’.

Suggested methodology Students often reach this school year without an adequate understanding of operations with fractions, especially more complex operations requiring the application of the order of operations and the use of brackets.

For adding and subtracting, we remind students of the importance of using the lowest common denominator.

For multiplying and dividing, we suggest that students write out the multiplications to be performed with the numerator and the denominator, and that they try to simplify the common factors before calculating the products.

We introduce the calculation of a fraction of an amount, which involves dividing the total quantity by the denominator and multiplying it by the numerator. We then introduce the inverse process (calculating a total amount based on a part corresponding to a given fraction). Finally, this leads us to the concept of a fraction of another fraction, which is equivalent to the product of both fractions acting on a total amount.

These complementary concepts, together with the basic idea that the all the fractions of a whole add up to 1, will allow students to solve the exercises of varying difficulty.

Answer key for ‘Think and practise’ 13 61 1 a) b) 4 36 15 12 c) d) 2 5 2 24 e)

f ) 15 5 2

a)

1 2

2 225 3 3 a) 7 b) 3 865 4 a) 1788 b) – 1 72 b)

5

He has gone 120 km.

6 Her 7

Unit

3

To write a fraction in decimal form, we divide the numerator by the denominator.

The quotient can be:

Decimal numbers are often used for measurements because they can express any intermediate value between two integers.

On calculators, like in English, we use a full stop, not a comma, as the decimal point.

1 437.54 → {∫∫‘¢«|…∞¢}

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

anayaeducacion.es Decimal numbers on the number line.

3.81 3.82 3.83 3.84 3.85 3.86 3.87 3.88 3.89

anayaeducacion.es Help with reasoning: transforming fractions into terminating and recurring decimals.

6

• A terminating decimal, if the only prime factors of the denominator of the simplified fraction are 2 and 5 (or either of them).

For example: 3 = 0.375; 123 = 3.075; 42 = 1.68 8 40 25 Look at why this is true:

The red point can be written as a decimal as close to the point as we want (3.857…).

• Terminating decimals have a limited number of decimal places.

For example: 5.4; 0.97; 8; –0.0725

In a number, the group of decimal places that is repeated again and again is called period. We draw an arc over repeating figures to indicate the period:

The quotients and remainders repeat from here on.

!

16.147

• Decimals that are not terminating or recurring have an infinite number of figures that do not repeat regularly. Unlike terminating decimals and recurring decimals, these numbers are not rational. They are called irrational numbers.

1 What type of decimal are these?

!

2.8

3.5222…

# 1. 54

2 Order these numbers from smallest to largest:

3 = 1.7320508… π – 2 = 1.1415926…

!

2.5

2.5

!

2.35

2.505005… !

3 Write three numbers between 2.5 and 2.5 .

• A recurring decimal, if the denominator of the simplified fraction has a prime factor other than 2 or 5. ! # # For example: 11 = 3.6 ; 86 = 7.81 ; 87 = 29 = 1.318 3 11 66 22 If the quotient is not a terminating decimal, why can we be sure that it is recurring? Let’s look at an example: 3 : 7. You can see the division on the left.

When we divide by 7, the remainder can only be 1, 2, 3, 4, 5 or 6, so it will have to repeat at some point. After that, the entire sequence will repeat.

All irreducible fractions can be written in decimal form: • Terminating decimal, if the only prime factors of the denominator are 2

and 5.

• Recurring decimal, if the denominator has prime factors other than 2

— In pure recurring decimals, all the figures after the decimal point are # repeated. For example: 7.81818181... = 7. 81

For example: 2 = 1.4142135…; π = 3.14159265…

16

7 0.428571

and 5.

Therefore, they are both rational numbers. However, decimals with infinite decimal places that do not repeat are irrational numbers.

— In mixed recurring decimals, at least one of the figures after the decimal ! point is not repeated. For example: 18.35222222... = 18.352

Think and practise

3.52

3.0 20 60 40 50 10 3

• Recurring decimals have an infinite number of decimal places that repeat periodically.

Remember

2.7

Example

it repeats

ã Types of decimal numbers Let’s look at the different types of decimal numbers:

# 5. 68

123 = 123 = 123 · 5 = 123 · 25 = 3 075 = 3.075 40 2 3 · 5 2 3 · 5 3 1000 10 3 If the only factors are 2 and 5, we can always write the denominator as a power of base 10.

3.9

Writing numbers in decimal form gives us a very easy and effective way to assess them, compare them and operate with them.

3. Decimal numbers (I)

For example: 72 = 8; –240 = –16 15 9

2

3.8

Roberta uses 1 225 litres to water her fruit trees.

• An integer, when the numerator is a multiple of the denominator.

Decimal numbers can be represented on the number line, and we can use them to get as close as we want to any number on the line:

Remember

1

ã Transforming fractions into decimals

DECIMAL NUMBERS

total savings are €14 300.

Think and practise 4 True or false?

! a) 1 = 0.333… = 0.3 3 3 = 3 · 0.333… = 0.999… = 0.!

9 3 !

Because 3 = 1, then 0.9 = 1.

3

! # b) 5.4 = 5.44 #

# c) 3.72 = 3.7272727… = 3.727 ! ! d) 0.3 + 0.6 = 1

5 Without doing the division and looking only at the

denominator of the simplified fraction, say whether these fractions transform into terminating decimals or into recurring decimals: b) 42 c) 101 d) 1001 a) 44 150 150 1024 500

6 Write a value of k that makes the fraction 84 :

a) An integer.

k

b) A terminating decimal. c) A recurring decimal.

17

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

Suggested methodology We start by reminding students how to represent decimal numbers on the number line and how to get as close as possible to any point we want on the line using a decimal number.

We do this by using smaller and smaller intervals, which, when expanded and divided into ten equal parts, give a new decimal figure.

We also remind students of the different types of decimal numbers and their notation.

The calculator is a powerful instrument for investigating the transformation of fractions into decimals. Using the division keys, applying a constant factor and converting fractions into decimals, students will come to identify certain patterns in the solutions they obtain.

It is important they take the way calculators round their answers into account to avoid confusion regarding recurring fractions.

11

Here are some examples: • Ask students to divide the first ten natural numbers by 3, so that they learn the period for any fraction of the type a/3, given the relationship of a to the multiples of 3.

• Obtain the quotient of 1/9 and use it to describe the decimal expression of a/9 no matter what the value of a.

Working in analogue mode, students will learn which fractions give rise to terminating and recurring decimals, and that all of these are rational numbers.

The activities at the end of each page are useful for consolidating the concepts and procedures studied.

Answer key for ‘Think and practise’ 3.52 8 Terminating decimal. !

2. 8 8 Pure recurring decimal. # 1. 54 8 Pure recurring decimal.

3 = 1.7320508… 8 Decimal that is not terminating or recurring.

2.7 8 Terminating decimal.

3.5222… 8 Mixed recurring decimal. π – 2 = 1.1415926… 8 Decimal that is not terminating or recurring. ! !

2 2.35 < 2.5 < 2.505005… < 2. 5

1

3

Open answer.

4 a) T 5

a) Recurring

b) T

c) T

d) T

b) Terminating

c) Terminating

d) Terminating

6 Open

answer. For example: a) k = –12 b) k = 5

3 DECIMAL NUMBERS

Unit

#

• To transform N = 2.563 into a fraction:

We have just seen that if we divide the numerator of a fraction by its denominator, the result is a terminating or recurring (pure or mixed) decimal number. Now, let’s look at the inverse case: How do we transform a decimal into a fraction? ➜ from terminating decimals to fractions

Writing a terminating decimal as a fraction is very easy because the denominator is a power of base 10.

25.636363… Now, we multiply by 100 to get another number with the same decimal part.

100N =

10N = 54.444…

3 The decimal portion disappears when we subtract: 10N = 5.444…

10N – N = 54 – 5 → 9N = 49 → N = 49 9 & • Period with more than one figure: N = 6.207 = 6.207207207…

1000N = 6 207.207207… The decimal portion disappears when we 3 1000N = 6.207207… subtract:

anayaeducacion.es Help with reasoning: transforming mixed recurring decimals into fractions.

7.324324… We get a pure recurring decimal.

100 000N = 7 324.324324… Another pure recurring decimal, with the same decimal portion.

100 000N – 100N = 7 324 – 7 → 99 900N = 7 317 → N = 7 317 99 900 Check both cases with a calculator.

To write a mixed recurring decimal, N, as a fraction:

anayaeducacion.es GeoGebra. Examples of how to write decimal numbers as fractions.

• We multiply N twice by powers of base 10 to get two pure recurring

decimals with the same period.

• When we subtract them, the result is an integer.

• We isolate N to get the fraction.

1 000N – N = 6 207 – 6 → 999N = 6 201 → N = 6 201 999 You can check these two examples by doing the divisions on a calculator. anayaeducacion.es Help with reasoning: transforming from pure recurring decimals into fractions.

2.5636363… We multiply by 10 to get a pure recurring decimal.

1 000N – 10N = 2 563 – 25 → 990N = 2 538 → N = 2 538 990 & • Another example: N = 0.07324 = 0.07324324324…

Let’s look at two examples of the process: !

• Period with one figure: N = 5.4 = 5.4444…

When we multiply N by 1 000, the result is another number with the same decimal portion.

N= 10N =

1 000N = 2 563.636363… When we subtract this number from the previous number, the decimal part disappears.

In other words, the result is an integer.

For example: 2.5 = 25 = 5 ; 3.41 = 341 ; 0.004 = 4 = 1 100 1000 250 10 2 ➜ from pure recurring decimals to fractions

When we multiply N by 10, the result is another number with the same decimal portion.

3. Decimal numbers (II)

1

➜ from mixed recurring decimals to fractions

ã Transforming decimals into fractions

➜ summary

To write a recurring decimal (pure or mixed) as a fraction, we use the given number to find two pure recurring decimals with the same period. When we subtract them, the result is an integer.

To write a pure recurring decimal, N, as a fraction: • We multiply N by a power of base 10 to find another number with the

same decimal portion.

We already know that decimal numbers with infinite non-repeating decimal places are irrational numbers, so they cannot be written as fractions.

• When we subtract them, the result is an integer.

• We isolate N to get our answer.

Think and practise Think and practise

&

7 Write as fractions:

a) 6.2 ! d) 3.5 # g) 0.23 ! j) 5.9

18

&

&

8 Notice that 0.208 + 0.791 = 0.999 = 1.

b) 0.63 ! e) 0.1 & h) 41.041 & k) 7.009

c) 1.0004 ! f ) 2.7 & i) 40.028 # l) 0.99

Check by writing each addend as a fraction and then adding the fractions together.

9 Transform these decimals into fractions and then

calculate: ! & # a) 3.5 + 1.76 – 2.103

! ! b) 1.3 : 2.16

10 Complete the process to write these numbers as

fractions:

N = 6.21777… ! a) 6.217 * 100N = 621.77777…

1000N = 6 217.7777…

N = 0.0316262… # b) 0.03162 * 1000N = 31.626262…

100 000N = 3162.626262…

11 Write these decimals as fractions:

! a) 6.25

! b) 0.001

# c) 5.018

12 Which of these numbers are rational? Write them as

a fraction: a) 3.51

b) 5.202002000…

d) 0.3212121…

e) π = 3.141592…

# c) 5. 03 & f ) 7.4331 # #

13 Transform into fractions and check that 5.48 = 5.484 .

19

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

12

c) k = 9

Suggested methodology Students already know that rational numbers are numbers that can be represented by a fraction, and that these fractions correspond to an integer or a terminating or recurring decimal. In these pages we address the inverse problem, trying to find a fraction that corresponds to a terminating or recurring decimal number.

In the case of terminating decimals, we have to find an equivalent fraction whose denominator is a power of base 10 and then simplify it. This is a very simple process. The same cannot be said for recurring decimals, so here we explain the process in more detail to ensure students understand it. If students have the opportunity to apply the procedure to a sufficient number of examples, they will eventually learn to apply it automatically, while also understanding the concepts behind it.

It may help students to do some preliminary activities before studying the standard procedure, such as: — Dividing the digits 1 to 9 by 9 and observing the period. This way we see that: ! !

0.8 = 8 and therefore, 5.8 = 5 + 8 = 53 9 9 9

— Dividing the numbers 10 to 100 by 99 and observing the two-figure period. This demonstrates that: # # 5. 17 = 5 + 0. 17 = 5 + 17 = 512 99 99 — For mixed recurring decimals, we can use a similar method: # 21.# 37 8 21.# 2.137 = 37 = 21 + 37 = 2 116 10 99 99 # 2 116/99 2 116 2.137 = = 10 990 We recommend emphasising that this process is only applicable to terminating or recurring decimals, and reminding students that decimals with an infinite number of figures that are not recurring cannot be transformed into a fraction.

Reinforcement and extension We recommend: • From the DIVERSITY AND INCLUSION section in the anayaeducacion.es resource bank: Extension: Practice Exercises 2 and 3, worksheet A. Answer key for ‘Think and practise’ 7

a) 31/5 e) 1/9 i ) 39 988/999

b) 63/100 f ) 25/9 j ) 54/9

c) 10 004/10 000 g) 23/99 k) 7 002/999

d) 32/9 h) 41 000/999 l ) 99/99 = 1

208 + 791 = 999 = 1 999 999 999 ! # & 9 a) 3. 5 + 1. 76 – 2. 103 = 32 + 175 – 2 101 = 35 386 9 99 999 10 989 ! ! 4 13 8 = b) 1. 3 : 2.16 = : 3 6 13 8

10

a) 5 526/900 = 1 399/225

11

a) 563/90

b) 3 131/99 000 b) 1/900

c) 4 968/990 = 276/55

b) It is not a rational number. d) 318/990 = 53/165 f ) 74 257/9 990 _ # 543 bb 13 5.48 8 100N – N = 543 8 N = # # 99 ` 5.48 = 5.484 # 5.484 8 1 000M – 10M = 5 430 8 M = 5 430 = 543 b 990 99 a 12

Unit

4 Secondary functions On scientific calculators, most keys have two secondary functions (they appear above the key).

The two functions are usually in different colours: • SHIFT → yellow • ALPHA → red For example, this key:

When you press it calculates the cube root. you can write When you press recurring decimals. We will learn more about this on the next page.

From now on, when we speak about a secondary function, we will refer to it with its key. For example, to speak about the cube root function we will write:

This school year is a good time to start working with a scientific calculator, which will be very useful to you during the rest of Secondary and Bachillerato. We will be referring to the CASIO CLASSWIZ calculator in many of our instructions because it is the most widely used by students of this level. But you could use any other calculator with similar characteristics.

It is also important to configure the calculator to OUTPUT in fractions and not in mixed numbers. To do this, go into the configuration menu ( �) and use the ’ arrow to move to the next screen. Once you are there, select 1:Fraction result. Then, select 2:d/c.

3’4”

3 If you press = 4

1 Enter the expressions on the right into a calculator and

check that when you press the = key, the fractions are simplified or you get the corresponding fractions.

1 ’ 6 ”-

7 12

If you make a mistake when entering data, you can always use the � key to go back. In other words, the � key deletes what is to the left of the cursor.

3 4

If you want to go back to INPUT once you have finished entering data, press the “ arrow. You will be able to enter additional addends or correct any errors.

Non-recurring decimals are written in the usual way, remembering to use a full stop (.), not a comma, for the decimal point.

Be careful!

If you enter a terminating decimal in which one or more figures repeat ‘many’ times, the calculator will probably interpret it as a recurring decimal: 5.43434343434343 =�

key and the ”’‘“ arrows.

If you press � with a number in the OUTPUT, it will transform the number from a fraction into a decimal or vice versa.

3.875 → 3 . 875 = Use the

5.43434343434343 5.43

3.875 31 8

�

3.875 3.875

keys to enter a recurring decimal.

# 5.491 → 5 . 4

3 4

91 =

5.491 5437 990

�

5.491 5.491

Calculators have limits on entering recurring decimals and fractions due to the excessive size of the INPUT or OUTPUT. You can explore and find these limits on your own.

Think and practise

3 4

2 Use the calculator to find the fractions that generate

If you enter a fraction that is not simplified, you can press = to simplify it: 6 → 8

2 ’ 3 ”+ 5 *

2 1 11 11 ’ 12 ”= 3 + 5 x 6 – 12

ã Decimals

6’8”

6 8

6

= 8

3 4

a) 3 5 d) 3.25

b) 8 12 e) 0.27

these decimal numbers: # b) 3.002 ! f ) 2.09 a) 2.354 # e) 0.125

# c) 0.0243 # g) 0.1233

# d) 3.701 ! h) 1.1

3 Use a calculator to solve the following. Write the

result in fractional and decimal form.

Think and practise

20

8= 8

output

To do this, press the configuration key �. This select 1:Input/Output and then, 1:I Mat/O Mat (INPUT and OUTPUT in mathematics mode).

To enter fractions, use the

The sum 3 + 2 can be written as: 5 3 2 . This is called a mixed number.

5 They are not used often nowadays, so we are not going to spend more time on them.

6

input

c) 27 15 f ) 0.321

d 4 + 1n : 2 5 5 d 4 – 5 n· 7 9 3 8 –1 3

4. Fractions and decimals with the calculator

For example, to get 2 + 5 · 1 – 11 : 3 6 12

example, to enter 6 you can press: 8

6

It is essential to configure the calculator to receive data (INPUT) and express results (OUTPUT) in the format that we need. We suggest configuring both in mathematics mode.

This way, all fractions, roots and powers will be displayed in the way we are used to seeing them.

ã Fractions

Recall what mixed numbers are

You can try different methods on the calculator to find new, and sometimes easier, ways of doing things. For

� and choose 1:Calculate.

Enter 3 → 4

To do operations with fractions, just enter the chain of operations in INPUT and press the = key.

Some simplifications

ã Configuration We mostly use the calculator to do arithmetic calculations. To do so, enter the

1

ã Operating with fractions

FRACTIONS AND DECIMALS WITH THE CALCULATOR

a ) 351/100 c) 498/99 = 166/33 e) It is not a rational number.

4 Use a calculator to do these operations with fractions

and decimal numbers. Get the results as fractions and decimal numbers (terminating or recurring). a) 5 – 2 4 7

b) d 4 + 2n · –3 9 5

c) d–3 + 1 n : 2 3 5

d) d –2 – 3 n – 2 5 7 & f ) 0.218 : d2 – 5 n 3 ! h) d 2 – 3. 3n · 1 7 8

e) 2 – d 1 + 3n : 1 7 8 3 ! g) –5 – 3.25 2

21

Suggested methodology If they have not done so already, this is the year when students should learn to use a calculator efficiently.

We use the CASIO CLASSWIZ as our calculator of reference, since surveys show it to be the most widely used model in Secondary education. In this double page we will introduce some of the basic functions of this model.

We describe some new functions for working with fractions and decimals, such as: 13

• Entering fractions with the key, which, together with the operations and brackets keys, allows us to view combined operations on the screen, such as the following: c 45 + 1m : 25 5 7 c 49 – 3 m – 8 –1 3 • Entering recurring decimal numbers using the combination of these two keys .

This allows us to view and perform combined operations with decimal numbers and fractions, for example: ! c 27 – 3, 3m $ 18 • When introducing the basic functions of a calculator, we place special emphasis on the INPUT/OUTPUT settings. We recommend that in this unit students configure both input and output in mathematics mode so that they obtain their results in the form of a fraction. In this format, applying the � key to the result expresses it in the form of a decimal. This key is very useful for transforming decimals into fractions and vice versa.

In other words, it calculates the original fraction instantly.

Deciding when it is appropriate to use calculators in class is of course a decision for teachers. Nevertheless, with this in mind, we strongly recommend that students try using calculators to check the solutions to exercises they have previously solved manually (or using mental arithmetic). This will be especially useful in helping familiarise students with their use.

Answer key for ‘Think and practise’ !

27 13 321 1 a) 0.6 b) 0.6 c) 1.8 d) e) f ) 100 4 1000 1177 500

37 1 832 1 486 241 21 62 c) d) e) f) g) 300 495 9 900 10 495 495 >;;;;?

3 – 135 = 0.8766233 154 >;;;;? ! !

27 = 4 a) 0.96428571 b) – 22 = –1.46 c) – 20 = –6. 6 28 15 3 > ;;; ; ? > ;;; ; ? & d) – 99 = –2.8285714 e) – 509 = –9.089285714 f ) 218 = 0. 654 56 333 35 >;;;;? !

259 8 = –5.75 g) – h) – = –0. 380952 45 21

2

, you will find

exercises and

‘Portfolio’ resource bank cacion.es In the anayaedu to create your portfolio on how guidance

d

PROBLEMS solve

Exercises and

1 Operations with fractions

Calculate and simplify: 1+

1+

1 1 1+ 1 2

1+

1+

1 c1 + 1 m 2

=1+

1 1+

1 2 +1 2

=1+

1

1 + c1 : 3 m 2

=1+

1

1+ 2 3

=

= 1 + c1 : 5 m = 1 + 3 = 8 3 5 5 1 Your turn Calculate: 3+ 5 3+ 1 2

2 Operations with fractions and decimals

Calculate and give your answer as a fraction: !

1 + 0.!

3 – 3 · 1.02 + 1.5 5 2 Your turn Calculate: ! ! ! !

(4.28 – 0.12) 0. 3 : 1. 6 + 7 · 0.4 3

We transform all the decimals into fractions and complete the calculations. !

0. 3 = 1 ; 1.5 = 3 3 2 ! ! !

N = 1.02 → 10N = 10.2 → 100N = 102.2

Fractions and decimals

Simplify these fractions and group the ones that are equivalent:

2

Reduce to a common denominator and order from smallest to largest:

24 36

1 – c 2 + 1 m = 15 – 10 – 3 = 2 3 5 15 15 15 15 If 2 of her monthly pocket money is €10, her monthly pocket money is: 15 10 · 15 = 150 = €75 2 2

4 Taps and fractions

Tap A fills a water tank in 2 hours, and tap B fills the same tank in 3 hours. The tank has a drain that takes 6 hours to empty the tank with the taps closed. If we open both taps and the drain, how long will it take to fill the tank?

22

1/2 + 1/3 – 1/6 = 2/3 of the tank So the time they will take is: 1 : 2 = 3 h = 1.5 h = 1 h 30 min.

3 2

66 165

6

Order from smallest to largest: ! ! # a) 3.56; 3.56 ; 3.5 ; 3.56 ! ! # b) –1.32; –1.32 ; –1.32 ; –1.3 ! c) 2. 3; 8 ; 2.34; 32 ; 21 3 15 10

11

Write as fractions: ! a) –1.03 b) 14.3 # # e) 0.012 f ) – 3.15

343 539

5 –5 12 3

12

13

e) – 7 3

Write each of these fractions as a decimal number: 5 233 13 17 9 13 23 25 9 6 200 7 990 22

7

Without doing the division, determine which of these are terminating decimals and which are recurring decimals:

8

Classify these rational numbers as terminating decimals or recurring decimals (try to answer before doing the division): 13 17 4 2 1 81 3 5 50 11 60 250

3 2

4 5

13 9

7 · 11 3 · 52

19 22 · 5

3 · 7 2 · 23 5· 7

b) 1 – 1 3 5 e) 12 : 3 7

c) 2 · 9 3 4 f) 8 · 5 15

Mental arithmetic: a) Half of 2 . b) One third of 12 .

3 7 c) Two-thirds of a number is 22. What is the number? d) Five-fourths of a number is 35. What is the number?

14

Reduce to a fraction: a)

d) – 3 2

! d) 0.32 ! h) 9.09

Mental arithmetic: a) 1 + 1 2 4 d) 2 of 60 3

b) 32 = 12 x 15

Write as a sum of an integer and a fraction like in the example:

! c) – 2.5 ! g) 5.345

Operating with fractions

Find the value of x :

• 8 = 6+2 = 6 + 2 = 2 + 2 3 3 3 3 3 a) 8 b) 15 c) 16 5 7 8

If tap A fills the tank in 2 hours, it will fill 1/2 of the tank in one hour.

If we open all three at the same time, in 1 hour they will fill:

26 39

3 –1 8 6

a) x = 35 18 42

Tap B fills 1/3 of the tank in one hour.

The drain empties 1/6 of the tank in one hour.

225 400

a) x and 26 b) 3 and 51 x 17 6 4 a) The fractions are equivalent if their crossed products coincide: x · 4 = 26 · 6 → 4x = 156 → x = 39 b) We use cross-multiplication again: 3 · 17 = 51 · x → 51 = 51x → x = 1

5

1

Write three numbers between each pair of decimals: ! a) 0.345 and 0.346 b) 2.3 and 2.4 c) – 4.5 and – 4.4

10

Problem solved

Calculate the value of x that makes these fractions equivalent:

4

If she spends 2 , she has 1 left over.

3 3 We calculate 3 of 1 → 3 · 1 = 1 5 3 5 5 3 At the end of the month she has:

26 65

11 – 7 24 4 3

1 + 1 – 3 · 46 + 3 = 1 + 1 – 23 + 3 = 6 + 10 – 46 + 45 = 15 = 1 5 3 2 45 2 5 3 15 2 30 30 30 30 30 2

Your turn We take half the oil from a bottle, and then one fifth of the oil that remains. If there are 3 L in the bottle now, what is its capacity?

9

1

! !

100N – 10N = 102.2 – 10.2 = 92 → 90N = 92 → N = 92 = 46 90 45 Now, we calculate:

3 Calculate the total

Annabelle spends 2/3 of her monthly pocket money in the first half of the month. Of what remains, she spends 3/5 in the second half, and has €10 left to save. How much is her monthly pocket money?

Unit

problems

Practise

1

15

3+ 1 2 7– 3 2

1–2 b) 4 3 5– 7 6 12

7·3 c) 8 5 1–1 5 2

Calculate and simplify by factorising, like in the example: • 15 · 7 = 15 · 7 = 3 · 5 · 7 = 1 21 25 21 · 25 3 · 7 · 5 · 5 5

16

a) 3 · 20 5 21

b) 6 · 5 25 18

c) 12 · 35 7 36

d) 9 · 20 16 27

e) 13 · 84 12 65

f ) 90 · 14 35 36

Transform into fractions and calculate: ! ! b) 0.12 – 0.2 a) 3.5 + 2.3 # ! ! ! d) 3.42 + 7.6 c) 1.6 – 1.02 Check that your answers, in order, are –7/90, 35/6, 122/11 and 29/45.

b)

h)

10 9

Exercises and problems solved Suggested methodology In the ‘Exercises and problems solved’ section we offer strategies, suggestions, hints and approaches that will help students solve the subsequent activities in the final pages of the unit.

Ultimately, we want students to be able to reproduce these procedures, or similar ones, each time they come across a difficult problem.

23

Assessment anayaeducacion.es In ‘My web resources’ there are documents to prepare a portfolio.

14

a)

Answer key for ‘Your turn’ 7 1 31 53 2 30 3 7.5 L

, you will find

exercises and

‘Portfolio’ resource bank cacion.es In the anayaedu to create your portfolio on how guidance

d

PROBLEMS solve

Exercises and

1 Operations with fractions

Calculate and simplify: 1+

1+

1 1 1+ 1 2

1

1+

1 c1 + 1 m 2

=1+

1 1+

1 2 +1 2

=1+

1

1 + c1 : 3 m 2

=1+

1

1+ 2 3

=

= 1 + c1 : 5 m = 1 + 3 = 8 3 5 5 1 Your turn Calculate: 3+ 5 3+ 1 2

2 Operations with fractions and decimals

Calculate and give your answer as a fraction: !

1 + 0.!

3 – 3 · 1.02 + 1.5 5 2 Your turn Calculate: ! ! ! !

(4.28 – 0.12) 0. 3 : 1. 6 + 7 · 0.4 3

We transform all the decimals into fractions and complete the calculations. !

0. 3 = 1 ; 1.5 = 3 2 3 ! ! !

N = 1.02 → 10N = 10.2 → 100N = 102.2

1

2

1 – c 2 + 1 m = 15 – 10 – 3 = 2 3 5 15 15 15 15 If 2 of her monthly pocket money is €10, her monthly pocket money is: 15 10 · 15 = 150 = €75 2 2

4 Taps and fractions

Tap A fills a water tank in 2 hours, and tap B fills the same tank in 3 hours. The tank has a drain that takes 6 hours to empty the tank with the taps closed. If we open both taps and the drain, how long will it take to fill the tank?

22

1/2 + 1/3 – 1/6 = 2/3 of the tank So the time they will take is: 1 : 2 = 3 h = 1.5 h = 1 h 30 min.

3 2

26 39

66 165

Reduce to a common denominator and order from smallest to largest: 3 –1 8 6

5 –5 12 3

11

Write as fractions: ! a) –1.03 b) 14.3 # # e) 0.012 f ) – 3.15

12

13

Write as a sum of an integer and a fraction like in the example:

e) – 7 3

7

Without doing the division, determine which of these are terminating decimals and which are recurring decimals: 3 2

4 5

13 9

7 · 11 3 · 52

19 22 · 5

3 · 7 2 · 23 5· 7

Classify these rational numbers as terminating decimals or recurring decimals (try to answer before doing the division): 13 17 4 2 1 81 3 5 50 11 60 250

Fractions and decimals

c) 2 · 9 3 4 f) 8 · 5 15

Mental arithmetic: a) Half of 2 . b) One third of 12 .

3 7 c) Two-thirds of a number is 22. What is the number? d) Five-fourths of a number is 35. What is the number?

14

Reduce to a fraction: a)

d) – 3 2

Write each of these fractions as a decimal number: 5 233 13 17 9 13 23 25 9 6 200 7 990 22

b) 1 – 1 3 5 e) 12 : 3 7

15

3+ 1 2 7– 3 2

1–2 b) 4 3 5– 7 6 12

Exercises and problems Practise

! d) 0.32 ! h) 9.09

Mental arithmetic: a) 1 + 1 2 4 d) 2 of 60 3

b) 32 = 12 x 15

• 8 = 6+2 = 6 + 2 = 2 + 2 3 3 3 3 3 a) 8 b) 15 c) 16 5 7 8

! c) – 2.5 ! g) 5.345

Operating with fractions

Find the value of x :

6

8

Order from smallest to largest: ! ! # a) 3.56; 3.56 ; 3.5 ; 3.56 ! ! # b) –1.32; –1.32 ; –1.32 ; –1.3 ! c) 2. 3; 8 ; 2.34; 32 ; 21 3 15 10

Problem solved

a) x = 35 18 42 5

1

Write three numbers between each pair of decimals: ! a) 0.345 and 0.346 b) 2.3 and 2.4 c) – 4.5 and – 4.4

10

343 539

a) x and 26 b) 3 and 51 x 17 4 6 a) The fractions are equivalent if their crossed products coincide: x · 4 = 26 · 6 → 4x = 156 → x = 39 b) We use cross-multiplication again: 3 · 17 = 51 · x → 51 = 51x → x = 1

Tap B fills 1/3 of the tank in one hour.

The drain empties 1/6 of the tank in one hour.

225 400

Calculate the value of x that makes these fractions equivalent:

If tap A fills the tank in 2 hours, it will fill 1/2 of the tank in one hour.

If we open all three at the same time, in 1 hour they will fill:

26 65

11 – 7 24 4 3

4

If she spends 2 , she has 1 left over.

3 3 We calculate 3 of 1 → 3 · 1 = 1 5 3 5 5 3 At the end of the month she has:

Simplify these fractions and group the ones that are equivalent: 24 36

1 + 1 – 3 · 46 + 3 = 1 + 1 – 23 + 3 = 6 + 10 – 46 + 45 = 15 = 1 5 3 2 45 2 5 3 15 2 30 30 30 30 30 2

Your turn We take half the oil from a bottle, and then one fifth of the oil that remains. If there are 3 L in the bottle now, what is its capacity?

9

Fractions and decimals

! !

100N – 10N = 102.2 – 10.2 = 92 → 90N = 92 → N = 92 = 46 90 45 Now, we calculate:

3 Calculate the total

Annabelle spends 2/3 of her monthly pocket money in the first half of the month. Of what remains, she spends 3/5 in the second half, and has €10 left to save. How much is her monthly pocket money?

Unit

problems

Practise

1+

7·3 c) 8 5 1–1 5 2

1

Calculate and simplify by factorising, like in the example: • 15 · 7 = 15 · 7 = 3 · 5 · 7 = 1 21 25 21 · 25 3 · 7 · 5 · 5 5

16

a) 3 · 20 5 21

b) 6 · 5 25 18

c) 12 · 35 7 36

d) 9 · 20 16 27

e) 13 · 84 12 65

f ) 90 · 14 35 36

24 = 2 36 3

225 = 9 400 16

26 = 2 65 5

343 = 7 539 11

66 = 2 165 5

26 = 2 39 3

Equivalent fractions: 24 and 26 ; 26 and 66 36 39 65 165

Transform into fractions and calculate: ! ! b) 0.12 – 0.2 a) 3.5 + 2.3 # ! ! ! d) 3.42 + 7.6 c) 1.6 – 1.02 Check that your answers, in order, are –7/90, 35/6, 122/11 and 29/45.

23

Assessment anayaeducacion.es In ‘My web resources’ there are documents to prepare a portfolio.

9 11 – 42 – 4 24 24 24 24 7 5 5 3 1 11 In order from lowest to highest: – < – < – < < < 4 3 12 24 8 6 3 Problem solved.

2

In the order in which they appear:

4

a) x = 15

5

a) 1 +

6

7

3 5

10 24

– 40 24

b) x = 40 b) 1 +

9 = 0.36 25 >;;;;?

5 = 0.714285

7

7 8

c) 2 +

13 = 1. !

4 9 # 233 = 0.235

990

2 7

d) –1 –

1 2

e) –2 –

!

23 = 3. 83 17 = 0.085 200 6 # 13 = 0.590 22

1 3

3 $ 7 2 $ 23 , Terminating decimals 8 3 , 4 , 19 2 5 22 $ 5 5$7 Recurring decimals 8 13 , 7 $ 112 9 3 $5

8

Terminating decimals 8 2 , 1 , 81 5 50 250 Recurring decimals 8 4 , 13 , 17 3 11 60

Open answer. ! ! # 10 a) 3.5 < 3.56 < 3.56 < 3.56 ! ! # b) –1.3 < –1.32 < –1.32 < –1.32 32 2. !

8 21 c) < < 3 < 2.34 < 15 3 10 9

11

23 103 43 b) c) – d) 9 100 3 2 104 4 811 e) f ) – g) h) 165 33 900 a) –

29 90 91 10

Operating with fractions 12

a)

3 3 2 b) c) 15 4 2

d) 40

13

a)

1 4 b) 3 7

d) 28

14

a)

7 7 5 b) – c) – 11 4 3

15

a)

5 7 5 4 1 b) c) d) e) 12 5 3 7 15

16

a)

35 7 29 122 b) – c) d) 6 90 45 11

c) 33

e)

8 4 f ) 3 7

f) 1

15

Unit

Exercises and problems 17

Calculate and give each result as an irreducible fraction: a) 3 – 1 d–1 + 2 n – 7 : d 4 – 1 + 2 n 5 2 3 15 5 3 b) d1 + 1 n – d 3 + 1 n · d 1 – 1 n : 1 3 4 2 3 4 6

22

23

c) d 3 + 1 n – >1 – d 3 – 1 n + 2 – 3 H 5 3 4 2 3 20 d) – 4 · 1 + 3 – d 1 + 1 : 2 n 3 2 3 3 2 4 e) d 5 – 5 + 2 · 1 n : >2 – 1 d1 + 5 nH 2 2 6 3 4 3 f ) 5 : d 2 + 1n – 3 : d 1 – 1 n 4 2 4 g) – 3 >3 – 3 – d 17 – 1n · d 1 – 3nH 3 8 5 20

24

19

If m = 1 and n = –7 , calculate: 3 2 4m + 1 n 2 1 a) b) 1 + m·n – m 1– 1 n m– 1 n True or false?

! a) 4 – (0.75 + 0. 6) + 13 = 1 3 12 ! ! b) d 5 + 0.16nd– 4 n + 65 d0. 1 – 0.2 – 1 n = 17 6 3 8 3 36 ! !

1 : 3 – 1. 3 : 1.1 3 4 80 c) ! 2 ! = – 51 15 · 0.02 + – 1.09 3

Problem solving 20

21

24

3/5 of a theatre’s seats are stalls, 1/4 are in the first balcony and the remaining 90 are in the second balcony. How many seats does the theatre have in total?

Of the 28 students in a class, 4/7 passed all their subjects. Of them, 1/4 got ‘excellent’ as their average score. How many students got ‘excellent’ as their average score? What portion of the class failed at least one subject?

26

Anne used some of her money to buy some comics. All the comics that she bought cost the same.

If she used one fifth of her money to pay for one third of the comics that she bought, what fraction of her money was left after paying for all the comics?

27

We sold 2/3 of a property and then we sold 3/5 of the rest. The remaining 600 m2 will be used to make paths and gardens. What is the total area of the property?

28

One third of the people who attend a conference are from Spain and 3/10 are from France. Of the remaining attendees, 6/11 are from Switzerland and 25 people are from Italy. How many people attended the conference?

29

Michael spends 3/5 of his monthly pocket money in the first 2/3 of the month. If he spends at the same rate during the rest of the month, what fraction of his monthly pocket money will be left at the end of the month?

A tank holds 1 500 L of water. We use 5/12 of the water one day and 500 L of water the next day.

What fraction of the total water is left in the tank?

Julie receives €120 for her birthday. If she spends 2/5 on clothes, 1/4 on books and 3/20 on food, how much did she spend on each? What fraction of the money does she have left?

The nutritional information written on the bottles of a brand of milk states that there are 120 mg of calcium in every 100 mL of milk. That is 3/20 of the recommended daily amount of calcium that a person should take every day. What is the recommended daily amount of calcium?

25

The solutions, in order, are: 11/4, –7/30, –1, –26/3, 17/24, –3/4 and –1/3.

18

One barrel of wine fills 480 bottles that measure 2/5 litre each. How many 3/4 litre bottles can be filled with the same barrel of wine?

30

Two boxes of apples sell for €2.50 per kilo. The first box has 5/12 of the total and sells for €50. How many kilos of apples were there in each box?

Advanced problem solving

Let’s think!

31

39

Estimate to complete these equalities in your notebook with the missing figures: a) 436 = 44 2156 77 c) 343 = 4 534 11

32

b) 84 = 44 = 4 315 45 45 d) 75 = 45 = 4 445 84 27

a) 2 4 2 4 3 = 17 5 3 10 30

34

A tap fills a water tank in 9 hours. If the tap and drain are both open at the same time, it takes 36 hours to fill the tank. How long does the drain take to empty the tank if the tap is closed?

35

Two 600-millilitre bottles have orange juice in them. One bottle is one-third full and the other is two-fifths full. We add water to each bottle to fill them completely. Then, we empty them both into a larger bottle. What fraction of the liquid in the larger bottle is orange juice?

36

x=

I spend 1/10 of the money in my piggy bank.

Then, I deposit 1/15 of what I have left in the bank.

I still need €36 to have the original amount again.

What was the original amount?

38

A group of friends go to a pizzeria and order three kinds of pizza: A, B and C. Each of them eats 1/2 of A, 1/3 of B and 1/4 of C. They order a total of 17 pizzas and there are no whole pizzas left over. a) Did each friend eat more or less than a whole pizza? How many friends are they? b) How many pizzas of each kind did they order?

Were there leftovers? c) Answer the same questions if the friends ordered a total of 20 pizzas.

non-recurring portion of the number without the decimal point as many zeros as there are figures before the period

# 18.2573 = 182 573 – 1825 9900

Check that it works for these recurring numbers: & ! & ! b) 0.7 c) 3.2501 d) 0.02171 a) 11.123 40

Find four fractions between 1 and 1 . How 12 11

many are there?

41

True or false? Explain and give examples.

17 7 b) 24 30 7 5 18 a) b) – 18 8 17

a) –

19

a) T

c) –

1 3

d) –1

11 4

e)

f) –

26 3

g) –

3 4

a) Some decimal numbers are not rational. b) The quotient of two terminating decimals is always a terminating decimal. c) When you add up two pure recurring decimals, you always get a pure recurring decimal. d) All integers can be written as fractions. e) If two positive fractions are less than 1, their product can be greater than 1.

At a party, 2/3 of the guests are boys, 3/5 of the girls have a partner and 6 of the girls are single. How many guests were at the party?

37

as many nines as there are figures in the period

# 3.27 = 327 – 3 99

b) 4 4 10 4 1 = 4 5 3 2 3

Two farmers, a father and his daughter, take 2 hours to plough a field. The father takes 6 hours to do the same work alone. How long would it take the daughter to do it alone?

–

decimal without the decimal point

Fill in the gaps in your notebook with the signs +, –, · or : to make each equality true:

33

1

There is a general rule for writing recurring decimals as fractions. It uses a different method from the one you learnt in this unit:

f ) When you divide two recurring decimals, you always get a recurring decimal.

42

Divide the numbers from 1 to 10 by 11. a) How many different decimals can you get? b) Is that related to the fact that we are dividing by 11?

b) F

c) T

Problem solving

c) Can you predict the result of 23 : 11 and 40 : 11?

43

If we write the fraction 20/13 as a decimal number, what number is in the 50th position? Is the same number in the 100th position?

44

If 0 < a < c < 1, which of these statements is b d true? a) a · c < a < 1 b d b

b) a · c > c b d d

c) a · c > 1 b d 25

Linguistic plan The ‘Expository text’ resource allows students to practise the oral exposition of their solutions.

Enterprising culture This task helps to identify personal skills and abilities relating to understanding formulas and mathematical proofs.

Cooperative learning You can use the ‘Cooperative learning: Shared interpretation’ resource for this exercise. In groups, students listen to the different ideas of each participant, then come up with a shared solution.

20 1/4

is left in the tank.

21

She spent €48 on clothes, €30 on books and €18 on food. She has 1/5 of the money left.

22

256 bottles can be filled.

23

The recommended daily amount of calcium is 800 mg.

24 The

theatre has 600 seats in total.

25

4 students got excellent. 3/7 of the class failed at least one subject.

26

After paying for the comics 2/5 of her money was left.

27

The total area of the property is 4 500 m2.

28

150 people.

29

1/10 of his pocket money will be left.

30 There

were 20 kg in the first box and 28 kg in the second.

Advanced problem solving 84 = 12 = 4 31 a) 336 = 12 b) 315 45 15 2156 77 4 10 2 # 2 + 3 = 17 b) # 5 3 5 3 10 30 33 It would take the daughter 3 hours.

32

a)

34 It

#

c)

343 = 7 539 11

d)

75 = 15 = 5 405 81 27

1 = 4 2 3

takes 12 hours.

35

11/30 of the liquid in the larger bottle is orange juice.

36

45 guests were at the party.

37

The original amount was €900.

38

a) Each friend ate 13/12 of a pizza, more than a whole pizza. They are 15 friends. b) 8 of A, 5 of B and 4 of C. There was 1/2 of A and 1/4 of C left over. c) Each person has eaten 13/12 of a pizza, more than a whole one. They are 18 friends.

They ordered 9 of A, 6 of B and 5 of C. There was 1/2 of C left over.

Let’s think!

39

a) 11123 – 11 = 3 704 999 333

40 Open 41

b)

7 9

c)

2171 – 217 = 977 32 501 – 32 = 10 823 d) 90 000 45 000 9 990 3 330

answer. There are an infinite number, for example,

a) T

b) F

42 a) You

c) T

d) T

e) F

5 , 5 , 5 , 5 .

56 57 58 59

f) F

can get 10 different decimals.

b) Yes.

# # c) 23 = 2. 09; 40 = 3. 63 11 11 43 The 44 a) T 16

number in the 50th position is 3, and the number in the 100th position is 4. b) F

c) F

Unit

Maths workshop LET’S LEARN AND CALCULATE Identification codes and check digits

• A jeweller gets a €140 discount on 16 identical brooches.

According to the catalogue, their price is €87.50 each.

Nowadays, codes are used to uniquely identify products, they are similar to an ID card for people or a number plate for cars. Many of these codes have a digit that can detect any errors made when writing them. These are called check digits. For example:

Bar codes You have certainly seen many bar codes. The bars and white spaces form a code in a binary system, which is a system that only contains ones and zeros. An optical device is used to read the code and identify the item.

There are different types of bar codes, but the most common one has 13 figures grouped into three parts. You can see an example on the right.

Let’s see how the check digit is calculated:

Country code (2 or 3 digits) 84 → Spain

Code indicating the company and the product (9 or 10 digits)

Check code (1 digit)

84 3448504835 6

1. We add up the figures in the odd positions, starting from the left (the check digit, x, is written as an addend to the sum).

2. The sum of the digits in the even positions multiplied by 3 is added to the previous result. We give x a value to make the total result a multiple of 10.

Let’s look at an example of a bar code for a book like the one you are reading now. The 978 at the beginning represents the ISBN (International Standard Book Number).

At what price should he sell each brooch if he wants to make €500 in profit?

• Marta buys three biscuits and Beatrice buys two. Their friend Veronica joins them later for breakfast but she does not bring any biscuits. When they divide up the cost, Veronica has to pay €5.

How will Marta and Beatrice divide up the €5?

anayaeducacion.es Answer key.

7 I can fill twenty 3/5-litre bottles of oil with

1 + 5 · c 5 + 1m – 1 : 2 8 3 3 5 2 Write a fraction that transforms into a terminating

decimal, a fraction that transforms into a pure recurring decimal and a fraction that transforms into a mixed recurring decimal.

The total is: 34 + x + 99 = 133 + x. It has to be a multiple of 10.

The only valid single-figure number for x is 7, making the sum 140.

The number would therefore be: 9 7 8 8 4 6 7 8 5 2 1 2 7 • Calculate the missing control digits in the bar codes on the right.

3 Write three numbers between the pairs of numbers

given:

! ! b) 2.7 and 2.8

a) 3 and 4 20 25

Copy the numbers of three bar codes from any three products in your notebook. Check that they are all correct by calculating the control digit.

4 Without doing the division, say whether these are

terminating decimals or recurring decimals: 89 50

INDEPENDENT PROBLEM SOLVING

113 12

23 32

18 7

5 Calculate the result of this operation. First, transform

the decimals into fractions:

! # d0.18 – 1.89 + 8 n · 1.1 11

Commas matter If we put the comma in the correct place, this statement is true: ‘five times four twenty plus one, twenty-two’

6 Zoe spent 1/3 of her money on books and 2/5 on

music. If she has €36 left, how much did she have originally?

Can you clarify it?

Commitment

26

• A group of friends goes to a coffee shop. They all order coffee and one-fifth of them also order cake. A coffee costs €0.85 and a piece of cake costs €1.10.

They give the waiter €11.

Did they leave a tip? If so, how much of a tip did they leave?

• A landowner hires a servant for an annual salary of eleven gold coins and one horse. After four months, the servant quits and receives the horse and one coin. What was the value of the horse?

SELF-ASSESSMENT 1 Calculate and simplify the result:

978846785212x We add up the figures in the odd positions: 9 + 8 + 4 + 7 + 5 + 1 + x = 34 + x We add up the figures in the even positions: 7 + 8 + 6 + 8 + 2 + 2 = 33 We multiply the previous result by 3: 3 · 33 = 99

•

Maths Workshop

1

PRACTICE MAKES PERFECT!

one-third of the oil in a jug. How many litres of oil were in the jug? How many 3/4-litre bottles can I fill with the rest?

8 One-fifth of the members of a gym are over 60

years old, and two in every three are between 25 and 60 years old.

a) What fraction of the members are 25 years old or younger? b) If there are a total of 525 members, how many are in each age group?

9 I buy a bike and pay in three instalments. In the

first instalment, I pay 3/10 of the total, in the second, I pay 4/5 of the rest, and in the third, I only have to pay €21. How much does the bicycle cost?

10 True or false?

a) All fractions are rational numbers. b) All rational numbers are fractions. c) A fraction is always equivalent to a recurring decimal number. d) A recurring decimal is a rational number.

Watch the video for target 4.a. Think of something you can do to contribute to achieve that goal. Make a commitment to put your idea into practice.

27

Developing thinking The resource ‘Reason-Question-Answer-Synthesize (RQAS)’ allows students to argue for and defend their own ideas and consider opposing ideas, encouraging them to rethink their own positions.

Linguistic plan The ‘Expository text’ resource allows students to practise the oral exposition of their solutions.

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

SDG commitment Watch the video for target 4.a.

Let’s learn and calculate The idea of this activity is to give students a very basic introduction to the word of codes.

While students will be used to seeing bar codes, they are unlikely to have thought about how they work.

The check digit can be calculated by following a few simple steps, so it will be easy for students to learn the procedure and explain it to their friends and family, or to do it for themselves with any barcode they come across.

Another interesting code is the one on our National ID Card, with the letter we are assigned. The letter is assigned based on the number, n, by doing the following division n : 23, which gives a remainder between 0 and 22. Each remainder is associated with a letter according to the following table: 0

1

2

3

4

5

6

7

8

9

10

11

T

R

W

A

B

M

Y

F

S

D

X

B

12

13

14

15

16

17

18

19

20

21

22

N

J

Z

S

Q

V

H

L

C

K

E

Answer key

• Code 9788499351421: Odd positions: 9 + 8 + 4 + 9 + 5 + 4 + x = 39 + x Even positions: 7 + 8 + 9 + 3 + 1 + 2 = 30 3 · 30 = 90 39 + x + 90 = 129 + x To make the result a multiple of x = 1, as we hoped to demonstrate.

Code 8413240400295: Odd positions: 8 + 1 + 2 + 0 + 0 + 2 + x = 13 + x Even positions: 4 + 3 + 4 + 4 + 0 + 9 = 24 3 · 24 = 72 13 + x + 72 = 85 + x To make the result a multiple of 10, x = 5, as we hoped to demonstrate.

• Open answer.

Independent problem solving Five times four comma twenty plus one, twenty-two. In this case, it should read: Five times four point twenty plus one, twenty-two.

5 · 4.20 + 1 = 22 Practice makes perfect!

• He should sell each brooch for €110.

• €4 for Marta and €1 for Beatrice.

• They left a tip of 30 cents.

• The value of one horse was 4 coins.

Self-assessment 11 1 6 58 4 2 Open answer. For example: terminating decimal, ; pure recurring decimal, ; 99 25 23 mixed recurring decimal, .

45 3 a) 3 = 0.15; 4 = 0.16 20 25 Open answer. For example: 0.15 < 0.151 < 0.1519 < 0.1531 < 0.16 ! ! b) Open answer. For example: 2.7 < 2.78 < 2.783 < 2.787 < 2.8 17

4 5

89 23 113 18 . Recurring decimals: . and and 50 32 12 7

– 109 100

6

She had €135 originally.

7

There were 36 L of oil in the jug. With the rest I can fill 32 3.4-litre bottles.

8

9 10

18

Terminating decimals:

a) 2 15 b) Over 60 years old: 105. Between 25 and 60 years old: 350. 25 years old or younger: 70.

The bicycle costs €150. a) T

b) T

c) F

d) T

notes

19