Mathematics4 Teacher’s Guide
José Colera J. M.ª José Oliveira G. IgnacioGazteluA.
Ramón Colera C. Ana Aicardo B.
Andalusia
LEARNING EXPERIENCES
REAL NUMBERS 1
Solar sails and real numbers
Videos. SDG 9: Industry, innovation and infrastructure
Quiz. Previous knowledge
IRRATIONAL NUMBERS
REAL NUMBERS: THE REAL NUMBER LINE
SELF-ASSESSMENT
Escape room!
LOGARITHMS
• Properties of logarithms
SECTIONS OF THE REAL NUMBER LINE: INTERVALS AND HALF-LINES
• Radicals in exponential form
• Simplification and reduction to a common index
NUMBERS IN SCIENTIFIC NOTATION. ERROR CONTROL
• Operations with numbers in scientific notation ROOTS AND RADICALS
• Product and quotient with the same index. Taking out factors from roots
• Product and quotient with a different index
• Power and root of a radical
• Addition and subtraction of radicals
• Rationalising the denominator
APPROXIMATE NUMBERS. ERRORS
Opening video
anayaeducacion.es Watch the video 'Solar sails and real numbers' about the learning experience.
SDG commitment
anayaeducacion.es Watch the videos on the sustainable development goal 9: Industry, innovation and infrastructure.
Gamification
anayaeducacion.es Do the quiz 'Previous knowledge'.
Real numbers
BASIC KNOWLEDGE
• Recognising rational and irrational numbers.
• Representing any real number on the real number line approximately.
• Working confidently with intervals and half-lines.
• Interpreting radicals. Mental arithmetic.
• Using the exponential form of radicals.
• Using a calculator to perform operations with powers and roots.
• Knowing the properties of radicals.
• Rationalising denominators in simple examples.
• Using approximate decimal numbers. Rounding off and truncating. The relationship between the error (absolute or relative) in a measurement and the number of significant figures.
• Expressing and interpreting numbers in scientific notation.
• Finding a logarithm with or without the help of a calculator.
LEARNING EXPERIENCE
One of the best ways to learn about something is to try and explain. In order to write an article about something as amazing and out of the ordinary as solar sails, students will need to look up information, find data, understand and interpret them and then present them in a way that can easily, and clearly, convey that information to their intended audience.
The main goal is to get the students to work with large and small numbers, operate with them in the process of visualising the enormous distances that are supposed to be covered by these sails. Then the students will compare how long the solar sails will take to travel to those distances in comparison to planes, spacecraft and probes. The students will also be operating with very small numbers when looking at how thick the solar sails are. Errors in calculation and rounding etc. will of course occur and students will reflect and comment on that.
The students will then be encouraged to investigate the topic further on their own.
ANSWER KEY
1 Alfa Centauri is 4.13 · 1013 km from the Sun.
2 It would take 1.33 · 106 = 1 330 000 years to reach Alfa Centauri.
MY MATHS DICTIONARY
Key language
Φ (phi) Non-recurring
Margin of error
∈ (is an element of/belongs to/is a member of )
In context
This construction contains a segment that measures the same as Φ (phi) π is an irrational number; it is infinite and non-recurring
The margin of error is less than 0.0001.
x ∈ R: x is an element of the set of real numbers.
MY MATHS DICTIONARY
Pronunciation:
• The pronunciation for the Golden ratio (Φ) is Phi (/fai/).
• The pronunciation for the word catenary is /kә-'tē-n
-rē/.
1. Irrational numbers
TEACHING NOTES
In this section we present irrational numbers as those that cannot be expressed as a fraction.
Here we demonstrate that 2 is irrational by using the method of reduction to absurdity; the teacher should determine if it is appropriate or not to explain this method. However, it should be emphasised that a decimal number with infinite non-periodic numbers is irrational, even in the case that its decimal expression has certain regularities, as in the case of 7.1010010001… or 7.01234567891011…
We must also insist on the recognition of irrational numbers when we approach roots. For example, knowing at a glance that 32 5 is rational and 8 4 is irrational.
The golden ratio, as the first known non-rational number, the number π and the number e, which students use frequently, are especially motivating to use as small research projects for students. There are numerous references and resources on the Internet about them.
ANSWER KEY
1 a) If 3 is rational, b
It is a contradiction. 3b2 contains the factor 3 an odd number of times, which contradicts that a2 contains it an even number of times.
b) and c) They are the result of operating rational numbers with an irrational number and therefore, are irrational numbers.
2
3 We know that 5 is irrational (for the same reason as 2 ). Observe that: If Φ = 2 51 + , then: 2 Φ = 5 + 1 → 5 = 2 Φ – 1
If Φ were rational, then 5 = 2 Φ – 1 would be rational. It is a contradiction.
From the diagram we obtain the proportion:
From which we obtain the equation: x 2 – x – 1 = 0
Whose only positive solution is Φ.
Developing thinking
It is recommended to use the ‘Chain of associated elements’ technique to do activity 2. The instructions on how to use this technique can be found at anayaeducacion.es
2. Real numbers: the real number line
TEACHING NOTES
The set of real numbers is defined for the first time as the union of two sets that are already known to students: the rational and irrational numbers. It is advisable to emphasise the property of density of both sets.
When explaining the representation of n , with the n integer, it is worth insisting on two aspects:
— The need to have n 1 – represented.
— n is the hypotenuse of the right-angled triangle whose legs are 1 and n 1 –.
To illustrate this process, the teacher could finish the explanation by speaking about the Pythagorean spiral: what it is and how it is obtained by means of the consecutive representations of n .
In general, decimal numbers (terminating, recurring or irrational) are situated on the real number line approximately. However, we think it is appropriate to teach students in which cases the representation of decimal numbers on the real number line can be done exactly:
— If the decimal number is terminating or recurring, it is expressed as a fraction and represented.
— If the decimal number is a quadratic radical, it is represented as seen in the previous point.
KEY
2 The number indicated with an arrow is 1.732. The representation of 2.716 is the following:
the Pythagorean theorem:
3. Sections of the real number line: intervals and half-lines
TEACHING NOTES
The concepts of intervals (of the different types there are) and half-lines are introduced for the first time. It is important for students to start mastering the use of this nomenclature, as they will use it later on when giving the solutions of inequalities and the domain of definition of many functions.
A good way to introduce this section is to ask students to think about the values that x can take so that x exists. This is so that students deduce that any value of x ≥ 0 is valid and that there is a need to look for a notation that includes all the solutions.
It is also important to discuss the empty set. We include an example so that students can think about why we need to be able to represent sets for which there are no values. The example we use here is the set of values of x for which x2 < 0.
It is advisable to do numerous exercises in which you move from interval and half-line symbols to their meanings and representations, and vice versa. You can find exercises to practise this at anayaeducacion.es.
Problems solved 1 and 2 on page 39 aim to demonstrate the different ways of expressing and representing a set of numbers. In problem solved 3 on the same page, we give an example of a situation that requires the use of intervals and half-lines.
ANSWER KEY
1-minute explanations
anayaeducacion.es Watch the video 'Radicals in exponential form' to review these contents.
1-minute explanations
anayaeducacion.es Watch the video 'Simplification and reduction to a common index' to review these contents.
1-minute explanations
anayaeducacion.es Watch the video 'Product and quotient with the same index. Taking out factors from roots' to review these contents.
1-minute explanations
anayaeducacion.es Watch the video 'Product and quotient with a different index' to review these contents.
1-minute explanations
anayaeducacion.es Watch the video 'Product and root of a radical' to review these contents.
4. Roots and radicals (I)
TEACHING NOTES
On this page there is a brief review of the concept of the nth root associated with the nth power, as well as some root properties that students already know from previous years.
Particular attention should be paid to getting students to handle exponential notation of radicals in both directions (from root to power, and vice versa). The exercises at the bottom of page 40 are proposed for this purpose.
Operations with radicals are presented on page 41. We highlight each property’s practical aspect: what it is used for. The titles of each of the sections along with the solved examples demonstrate its usefulness in practice.
The formal expression of the properties and their demonstrations can be found in the margins of this page and the next.
The root of a product and of a quotient are properties that students have already managed in the case of square roots; therefore, more emphasis should be placed on the rest.
It is beneficial for students to think about the process followed in the demonstration of the first five properties:
1. Converting the root into an exponent.
2. The application of the properties of powers.
3. Converting the obtained powers into a root.
ANSWER KEY FOR ‘MENTAL ARITHMETIC’
k = 16/81
ANSWER KEY 1 a) x 1/5
a 2/5
x 1/6
x 10/3
a 7/2
24/5
1-minute explanations
anayaeducacion.es Watch the video 'Addition and subtraction of radicals' to review these contents.
1-minute explanations
anayaeducacion.es Watch the video 'Rationalising the denominator' to review these contents.
4. Roots and radicals (II)
TEACHING NOTES
Demonstrating the reason for rationalising a denominator can be very informative for students.
Before the invention of calculators, in order to find the value of 5 1 it was necessary to divide 1 : 2.2360679… by hand.
The quotient of this division, as well as being long and complicated, varies when adding decimal places to the divisor. So, whenever a number was added to the divisor to fine tune the calculation, it had to be calculated from the start again.
This can be avoided by rationalising it.
Once the denominator has been rationalised, it is much easier to perform the calculations and, in addition, as many decimal digits as wanted can be added without changing the quotient obtained so far.
ANSWER KEY
Cooperative learning
If convenient, activity 1 can be done in groups using the resource ‘Cooperative learning: 1 – 2 – 4’.
MY MATHS DICTIONARY
Vocabulary:
To help students verbalise their answers in activity 1, they should use ‘error is’, or ‘error of’.
Examples:
• The absolute error is less than 0.002.
• It has an absolute error of less than 0.002.
5. Approximate numbers. Errors
TEACHING NOTES
Working with decimal numbers forces us to choose an approximation of them. For this reason, an error is made by lack or by excess.
More than the exact application of what the errors and the margin of errors are, it is better to pay attention to the relationship of the absolute error with the order of the last significant figure used, and of the relative error with the quantity of significant figures. By way of reflection, it could be said that the level of relative error is ε/ (estimation – ε), ε being the margin of absolute error. In practice, it is not necessary for students to fine tune to such an extent, it is sufficient that they take ε/estimation as a margin of relative error.
The concepts of significant figures, approximation, errors and margin of error are difficult for students to understand when explained from a theoretical point of view. Students understand them much better with practical examples.
This is what we try to do with the problems solved on these pages, in which the solutions are given in a reasoned way so that they can analyse how convenient each solution is.
ANSWER KEY
2 It is impossible to obtain such precision in measurements.
Most suitable expression: 12 million.
Absolute error < 500 000
Relative error < 4.2 %
3 Absolute error < 0.00005
Relative error < 0.002 %
1-minute explanations
anayaeducacion.es Watch the video 'Operations with numbers in scientific notation' to review these contents.
6. Numbers in scientific notation. Error control
TEACHING NOTES
Students have already seen scientific notation in previous years. This year, we must verify that they can identify, express and operate with numbers in scientific notation, both manually and with a calculator. The problem solved on this page is of good help to check this.
It is necessary to think about the idea of approximation when working with scientific notation as well as about what the error made is. It is easy to recognise the number of significant figures used in numbers expressed in scientific notation. This fact is shown in the ‘Attention’ box on the left-hand side of page 47. Then, the margins of absolute and relative values are worked on in the problem solved.
ANSWER KEY FOR ‘TEST YOURSELF’
KEY
case b) A.e. < 0.05 Ten billones of centimetres.
R.e. < . . 15 005 < 0.033
case c) A.e. < 0.0005 Ten thousandths billones of centimetres.
R.e. < 1 490 0 0005 < 0.000336
1-minute explanations
anayaeducacion.es Watch the video 'Addition and subtraction of radicals' to review these contents.
7. Logarithms (I)
TEACHING NOTES
The concept of logarithm is presented in an intuitive way from powers and their properties. Logarithms are defined as exponents to which one must raise a number to obtain another. It is necessary to remember the properties of powers in order to apply this definition and to justify the properties that students will study next.
We will see logarithms again in learning experience 8, where the exponential functions and inverse logarithmic functions will be studied.
Decimal and Napierian logarithms are introduced as interesting and special cases. Student already learnt about the number e earlier on in this learning experience. On the next page, they will be told about the importance of decimal logarithms in more detail.
ANSWER KEY
1 a) 3 b) –2 c) 7 d) – 4 e) 0 f) – 4 g) –1/2 h) –1 i) 2/5
2 a) a = 100 b) b = 6 c) c = 5 d) d = 9
3 log 7 = 0.84509804 log 70 = 1.84509804
They have the same decimal part because log 70 = log 7 + 1.
7. Logarithms (II)
TEACHING NOTES
The study of logarithms can be presented along with its history, so that students see mathematics as a living science that evolves over time.
We explain two different ways of finding a logarithm with any base on a calculator:
— Using the button and then applying the base change property that appears on the previous double page.
— Using the button included in many newer calculators, which allows us to find the logarithm of a number with any base.
It is important for students to learn both methods, as this will help them to understand the base change property.
The text in the left-hand margin on decimal logarithms draws students’ attention to the fact that the decimal logarithms of two numbers which are identical except for the position of the decimal place will only differ by an integer.
ANSWER KEY
4 a) 9.531381461
b) 4.191806549 c) –0.390964976
ICT
anayaeducacion.es Review the contents of this learning experience through the ‘My interactive visual summary’.
My visual summary
TEACHING NOTES
The My visual summary section aims to summarise all the contents the students have studied throughout the learning experience.
Visual summaries are a useful teaching tool that facilitates the learning process in many ways:
• They provide students with non-linear visual ways to understand, produce and represent knowledge.
• They help students develop higher-order thinking skills including analytical skills.
• They facilitate organisation, recall and processing of information.
• They help students externalise their knowledge and show their understanding.
• They make explicit structural form of knowledge and relationships between concepts, and therefore enhance students’ comprehension.
• They attend to different learning styles.
• They are effective organisational tools students can apply to organise their knowledge. Visual summaries can be used at the beginning and at the end of each learning experience. They are an excellent way to organise and make prior knowledge evident while setting a good bases for your class to begin.
The My interactive visual summary digital resource allows to display images and texts or images only. This is a great way to begin reviewing content and to motivate learning. It also allows cooperative learning, interaction, and communication. At the end of the learning experience, you can display images and go through concepts and relationships with students. In this way you will summarise the content of the learning experience in a whole class activity.
Exercises and problems solved
TEACHING NOTES
On the ‘Exercises and problems solved’ section we give students strategies, suggestions, clues and ways of thinking, which will be useful to them when solving the activities proposed in the final pages of the learning experience.
The ultimate goal is for students to be able to replicate similar procedures whenever they are faced with a problematic situation.
In particular, in this learning experience they focus on the properties of and operations with powers and radicals and calculations with logarithms, which are presented in the form of very simple logarithmic equations.
Problems solved 4 is of special interest because it proposes solving a real situation in the context of calculating compound interest, which was studied in the previous year. Several of the concepts worked on in this learning experience intervene in the solution: scientific notation, errors and calculations with powers and roots.
Exercises and problems
DO YOU KNOW THE BASICS?
Number sets
1 a) Integer: –17, – 16 , 7 56
b) Natural: 7 56
c) Irrational: 18
2 a) Rational: 08 7 ! ; – 4 ; – 3 7
Irrational: 2 3 ; 2 1 ; 2π; e 3 b) –3 7 ; – 4 ; 2 1 ; 2 3 ; . 08 7 ! ; e 3 ≈ 0.90; 2π
c) They are all real.
3 Open answer. For example: Integer: –3; Rational not integer: – 7 2 ; Irrational: – 2 3 e
4 N: 152 Z: 152; –4
Q: –4; 6 13 ; 2.7; 152 Á: They are all real.
Intervals an hailf-lines 5 A
Powers, roots and radicals
9 a) 3 125 b) 2 401 c) 3 1 d) 16
10 a) x 2/5 b) 21/2 c) 102 d) 201/2 e) (–3)3/5 f ) a 1/4 g) x –6/5 h) a 1/3
11 a) 5 b) ()39 –2 3 3 = c) 3 4 3 d) a 3 4 e) aa 3 6 = f ) a 3 5 –
12 a) 2–2 = 1 4 b) / 13 2 ≈ 1.26 c) / 32 2 ≈ 2.83
13 a) a / 910 b) x / 16 c) a / –34
14 a) 3 b) a 2 3 c) a 3 d) ab 2 4 e) a 2 3 f ) a 2b 3
Approximate numbers. Scientific notation
15 a) A.e. = 41/3 – 13.67 = . 00 03 # ; R.e. = 13 67 0 003 = 2.19 · 10–4
b) A.e. = 8.5 – 72 = 0.015; R.e. = . . 85 0 015 = 0.001765 = 1.765 · 10–3
c) A.e. = 1 123 7 – 0.65 = 0.0024; R.e. = 065 0 0024 = 0.00677
16 a) A.e. < 500; R.e. < 0.014 < 0.02
b) A.e. < 500; R.e. < 8.9 · 10–5 < 9 · 10–5
c) A.e. < 5 000; R.e. < 0.0056 < 0.006
d) A.e. < 0.5; R.e. < 6.3 · 10–6 < 6 · 10–6
e) A.e. < 500 000; R.e. < 0.056 < 0.06
f) A.e. < 500; R.e. < 0.0037 < 0.004
17 a) 3 ⋅ 1012 b) 1.5 ⋅ 10–5 c)
18 a) 14 1015 = 1.4 1016; A.e. < 0.05 1016 = 5 1014 b) 12.5 ⋅ 10–3 = 1.25 ⋅ 10–2; A.e. < 0.005 ⋅ 10–2 = 5 ⋅ 10–5 c) 0.24 1012 = 2.4 1012; A.e.. < 0.05 1012 = 5 1010 d) 36 10–14 = 3.6 10–13; A.e. < 0.05 10–13 = 5 10–15
Logarithms
a)
6
21 a) 4.548 b) 5.732 c) –1.748 d) 1.694 e) 1.495 f) –0.402
TRAINING AND PRACTICE
22 1 –104 3.5 1 – √ 2 7 23 ( π 4 1 4 11 9 √ 6 √
a) –7 b) 23 – 4 15 c)
41 a) x ∈ [7, +∞) b) x ∈ (– ∞, 5] d) x ∈ (– ∞, 0]
42 a) (–1, 1) b) (–3, 3) c) (–2, 8) d) (–3.5; 0,5) e) (–3.3; –2,7) f) (–0.9; 5.1) g) (–5.5; 5.1)
43 a) E(0, 3) b) E(3, 1) c) E(3, 3) d) E(1.5; 2.5) e) E(–0.5; 2.5) f) E(3.75; 3.75) g) E(–3.6; 1.4) h) E(2.95; 1.75)
44 a) M = x 2 (x – 3) b) M = () y x 31 + c) M = x xy 23 2 +
45 a) A total = 24 + 30 6 b) D = 53 ≈ 7.3 c) A.e. = 0.01989 < 0.02; R.e. = 0.0027 < 0.003
46 P = 2 5 51 + 5 3
THINK A LITTLE MORE
47 Take Action. a) 1 · 1018 km b) 1 580 millennia c) 7.85 · 1035 electrons
48 Height = 8 3 2
49 V = 4 3 cm3
50 (1) (2)
(1):
(2):
We can find the relationship between a and b from the equilateral triangle from side b: b2 = b 2 2cm + a2 → b = a 3 4 2 = 3 4 a
And returning to a (2): Areahexagon = 3ab = 3 ∙ 3 4 a2
Now by dividing the two areas, we can look for their relationship and we find the relationship is 4/3.
51 a) x ∈ (–3, 3) b) x ∈ [– 4, 6] c) x ∈ (–7, 1)
When the inequalities are the opposite of the above, they will be represented through joining two intervals. a) x ∈ (– ∞, –3] ∪ [3, +∞) b) x ∈ (– ∞, – 4) ∪ (6, +∞) c) x ∈ (–
52 a) The results are: 8 13 and 13 21
b) They are indeed numbers in the sequence.
c) The quotients are close to the golden ratio: 2 15 + ≈ 1.62
MY MATHS DICTIONARY
Vocabulary:
To help students verbalise their answers you can provide them with useful phrases.
Examples:
• Due to...
• The reason for that is...
• The correct answer should be...
53 a) a1 = 131/2; a2 = 133/4; a3 = 131/2 + 1/4 + 1/8 = 13(4 + 2 + 1)/8 = 137/8
b) When the ration, r, of a geometric series is between 0 and 1, 0 < r < 1, the sum of its terms is finite and convergent. If we have a geometric progression whose ratio is 1 2 : a n = a n – 1 ∙ r, with r = 1 2 and a0 = 1 2 , its terms are: 1 2 , 1 4 , 1 8 , … Therefore, the exponents of our sequence are the sum of the terms of the new sequence.
c) It is close to 1.
54 It will take approximately 37 years and 2 months to triple the investment. YOU CAN ALSO DO THIS
55 1.25 10128
56 a) x = 56 b) x = –1
57
DO YOU UNDERSTAND IT? THINK
61 The triangle inscribed in the semicircle is a right triangle and the height, h, above the hypotenuse divides it into two similar right triangles. The following can be fulfilled in a similar way:
62 x +
(–1, 7)
3)
Gamification
anayaeducacion.es Go to the Escape Room and test yourself!
ICT
anayaeducacion.es Answer key for the selfassessment.
Self-assessment
ANSWER KEY
1 a) and b) 33 //33 4 6 46 23 2 3 === → Real (irrational). 2π → Real (irrational) , loglog log 05 10 5 2 1 –1 2 2 == = 2 → Does not exist . 347 # → Rational and real 2.0333… → Rational and real 81 = 9 → Natural and real 4 3 → Real (irrational)
5 → Real (irrational)
2 A = 5 ; B = 6 ; C = 7 ; D = 11
3 a) Spain: population 4.7432805 ∙ 107; annual GDP 1.206842 ∙ 106 Estonia: population 1.331796 ∙ 106; annual GDP 3.1445 ∙ 104 Finland: population 5.548241 ∙ 106; annual GDP 2.51367 ∙ 105 France: population 6.7842582 ∙ 107; annual GDP 2.500870 ∙ 106 b) Estonia’s GDP is about 2.6 % of Spain’s, 12.5% of Finland's and 1.3% of France’s. 4 a) {x / –2 ≤ x ≤
I take action: Do the maths and compare the data to make it easier to understand
TEACHING NOTES
In order to prepare to write the scientific article, there are activities for the students to complete which deal with errors, very large and very small numbers that they will need to operate with, dividing, multiplying, taking roots, etc.
Finally, students will be encouraged to investigate other aspects of this topic that opens multiple learning opportunities.
ANSWER KEY
• We say that Alpha Centauri is 4.37 light years from the Sun. Give a value for the absolute error, in kilometres, of this measurement and another value for the relative error.
Absolute error is: 0.005 light years = 4.73 · 1010 km.
The relative error is: 0.005/4.37 = 0.0011 = 0.11 %.
• The distance from the Earth to the Sun is 1.5 · 108 km. Is there any difference between the distance from Alpha Centauri to the Sun or to the Earth considering the absolute error calculated above?
The absolute error is 0.005 light years ≈ 5 - 1010 km.
There is no difference because the distance is less than the error.
• Solar sails have reached accelerations of 1 mm/s2. With this acceleration, how much does the speed (in km/h) increase over one day? Write the solution to one significant figure.
It increases its speed by 300 km/h per day (with a significant figure).
• The fastest probe, powered by engines, reached 22.88 km/s. How many days does the solar sail above need to reach this speed?
It needs about 265 days.
• One of the solar sails was 7.5 microns thick. An average human hair is 0.07 mm thick. Compare the size of the two. (1 micron = 0.001 mm)
The thickness of a hair is almost 10 times larger that of the sail.
• The space-time equation for uniformly accelerated motion, if the initial velocity is 0, is as follows: s = 2 a t 2 . How many years would it take the spacecraft to travel the distance to Alpha Centauri with a constant acceleration of 1 mm/s2? Is it enough to get there in 20 years? The distance is 4.37 light years. It would take 289 years. That acceleration is not enough.
• What constant acceleration would it need to reach Alpha Centauri in 20 years? It would require an acceleration of 208 mm/s2
• What mistake are we making when we say that the speed of light is 300 000 km/s if it is in fact 299 792 458 m/s?
We are making an absolute error of 207.542 km/s.
The relative error is 0.07 %.
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.
© GRUPO ANAYA, S.A., 2025 - C/ Valentín Beato, 21 - 28037 Madrid.