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Using English in Maths

USING ENGLISH IN MATHSIES MEDINA AZAHARA

Fichas de trabajo basadas en la metodología AICLE para 2º de la ESO

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Using Maths in EnglishIntroducción

Desde que soy profesora de matemáticas bilingües, mi trabajo en cuanto a la elaboración de material didáctico, tan necesario en mi caso por la carencia de un libro de texto específico, se caracteriza por haber sido realizado en soledad, con la única ayuda de los recursos que ofrece la red; probablemente creados también en solitario, y que rara vez se adaptan a lo que tengo en mente.Así, quizá envidiando un poco la práctica del “poolingresources”, tan habitual en la educación anglosajona, uno de los objetivos más añorados ha sido el de trabajar junto al resto de mis compañeros del equipo educativo, con el fin de aunar diferentes experiencias, habilidades y conocimientos para un propósito común. Creo que esta sinergia es palpable, especialmente, en las fichas de carácter transversal.Este documento contiene diez fichas de trabajo para desarrollar la competencia lingüística relativa a las matemáticas bilingües de 2º de la ESO. Se trata de actividades en las que se pretende, de forma lúdica y atractiva, trabajar las expresiones y el vocabulario necesario en cada bloque de contenidos matemáticos, aplicando para ello la metodología AICLE.De estas diez fichas, tres de ellas abarcan contenidos transversales, enriqueciéndose así nuestra materia con la historia (Worksheet 2), la música (Worksheet 4) y las ciencias naturales (Worksheet 8). También hemos querido que en otras, sea necesario buscar y tratar información en internet, así como aprender a utilizar el programa informático “Geogebra” (Worksheet 6).En general, las fichas se caracterizan por poder ser llevadas al aula de forma fácil y práctica, pues están elaboradas con el propósito de que ocupen una sesión (tal vez 2), no se han añadido elementos que encarezcan su impresión, y ocupan entre tres y cinco páginas. Las actividades son variadas y atractivas, para ello comprenden sopas de letras, estudio y cumplimentación de tablas, lecturas comprensivas, preguntas, mini dictados, razonamientos, elaboración de resúmenes, crucigramas, mini debates, interpretación y elaboración de gráficos y funciones, pequeñas investigaciones históricas y matemáticas, ejercicios intelectuales y manipulativos para deducir, para descubrir y elaborar conclusiones… y para aprender más allá de las matemáticas y del idioma, pues muchas de ellas tienen un marcado carácter cultural.Finalmente, pueden ser de utilidad para trabajarlas en la sesión con el lector/a.Este trabajo ha supuesto para la mayoría del grupo, una primera incursión en las nuevas metodologías que la enseñanza bilingüe requiere, y que ojalá sirva de base, inspiración y compañía para todos aquellos que deseen avanzar en este camino.La coordinadora del grupo, Elena Valenzuela Campaña

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Participantes

Los participantes de este grupo, a los que desde aquí agradezco su voluntariosa colaboración y dedicación, han sido, por orden alfabético:

Galera Pomeda, María Dolores; profesora de inglés y coordinadora del proyecto bilingüe.

Jiménez Álvarez, Rafael; profesor de Geografía e Historia bilingües. Jiménez Galindo, Narciso Miguel; profesor de Matemáticas bilingües. López-Cózar Aguilar, Mª Isidora; profesora de matemáticas bilingües y

directora del centro. Nicolás Ródenas, Manuel; profesor de Ciencias Naturales bilingües. Rubio Ritoré, Alberto Manuel; profesor de Música bilingüe. Simón Torres, Francisco José; profesor de informática. Valenzuela Campaña, Elena; profesora de matemáticas bilingües y

coordinadora de este grupo de trabajo.

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Contenido

Worksheet 1: Integer numbers. Powers and roots..................................................................2Worksheet 2: Mathematics in Al Andalus...............................................................................2Worksheet 3: Fractions, decimal numbers and proportionality..............................................2Worksheet 4 Divine Proportion and Music..............................................................................2Worksheet 5: Algebraic expressions. Equations.....................................................................2Worksheet 6: Functions..........................................................................................................2Worksheet 7:Measures, Pythagoras and Tales.......................................................................2Worlsheet 8: Planets and scientific notation. Microorganisms and cells size. Absolute and relative errors.........................................................................................................................2Worksheet 9: Measuring perimeters, areas and volumes.......................................................2Worksheet 10: Statistic..........................................................................................................2

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Worksheet 1: Integer numbers. Powers and roots

Level 2nd ESO

Name:


1. Join up the different cells with arrows:

They are greater than or equal to the number

The unit is a factor of itself.

Characteristics of multiples

The number of factors of a number is limited

A number is factor of itself

A number is multiple of itself

The number of multiples of a number is unlimitedThey are smaller than or equal to the number

Characteristics of factors

If a number is a multiple of another, the second one is divider of the first one.If a number is a factor of another, the second one is multiple of the first one.

2. Say if it is true or false:

Situation True / FalseIf a number is multiple of 10, it’s also multiple of 5If a number is multiple of 5, it’s also multiple of 20There is no number multiple of 3 and 11A number divisible by 4 ends in 4.45 is divisible by 9A number is divisible by 2 always ends in 2,4,6 or 8

a) Give reasons for sentence number 1:

b) Give reasons for sentence number 2:

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3. Mark with circles all the prime numbers

Enunciado simple:a. Qqqqb. 11111c. 11111

Enunciado con cuadro de texto:

What is a prime number?

4. Listen carefully to your teacher and fill in the gaps (write the numbers how they are read)

a) The highest common factor of twenty and _______________ is ten.b) The ___________ common ___________of five and seven is

________________.

c) The ___________ common ____________of seventeen and thirty-four is

___________________.

d) Seven is the highest common factor of forty-nine and _____________________.

5. Fill in the table and draw the numbers in the line below:

Absolute value of negative three |−3| 3The opposite number of four Op (4) -4

|0|

Op(|−1|)The absolute value of the opposite number of one

Op[Op(-2)]

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6. Fill in the following table:

General rule Formula Example

Associative property (2 ∙ 3) ∙ 7 = 2 ∙ (3 ∙ 7)

Commutative property A ∙ B = B ∙ A

Distributive property 4 ∙ [9 + (-3) ] = 4 ∙ 9 + 4 ∙ (-3)

Neutral element A ∙ 1 = A

7. Make up problems in which you have to use the following equations to solve them. Then, find the solution.

a. A problem about ages, meters,etc using the following operation:18 :2+6

b. A problem about your daily life with this operation :(4−3 ) ∙5

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8. Fill in the gaps:

a. Three to the power of four written in the short form is ____ and the result is _________________.

b. If the base is negative and the exponent is even, the result is ____________.

c. If the base is negative and the exponent is _______, the result is negative.

d. The short form of 3 ∙3 ∙3 ∙3 ∙3 ∙3 is _________, and it is read_______________________

e. Negative one to the power of 158 is ____________.

9. Join up the different cells with arrows:

78 :73 (11 /3)5The quotient of two powers of the same base is another power with the same base and its exponent is equal to the numerator power exponent minus the denominator power exponent.

25 ∙35 75The product of several powers of the same base is another power with the same base and its exponent is the sum of the exponents of the multiplying powers

(73)2 25 ∙35 ∙55The power of a product equals to the product of the powers of the factors.

(2 ∙3 ∙5)5 75The power of a quotient equals to the power of the quotient set by the numerator base divided by the denominator base.

(11:3)5 76The quotient of two powers with the same exponent is a power with the same exponent and base equals the quotient of the bases.

(11:3)5 (2 ∙3)5The product of two powers with the same exponent equals a power with the same exponent and its base is the product of the bases

72 ∙73 (11 /3)5The power of a power is a power with the same base and exponent equals the product of these two exponents.

10.Mark with circles the numbers with a perfect root square and find out the rest if they don’t have one:

17, 47, 49, 121, 140, 93, 2500, 1024

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W1: Solutions:1

characteristics of multiples

They are greater than or equal to the number

characteristics of the dividers

The unit is divisor of itself.


The number of dividers of a number is limited


A number is divisor of itself


A number is multiple of itself


The number of multiples of a number is unlimited


They are smaller than or equal to the number


if a number is a multiple of another , the second one is divider of the first one.


if a number is a divider of another , the second one is multiple of the first one.

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Situation True / FalseIf a number is multiple of 10, it’s also multiple of 5 TrueIf a number is multiple of 5, it’s also multiple of 10 FalseThere is no number multiple of 3 and 11 FalseA number divisible by 4 ends in 4. FalseIf a number is multiple of 10, it’s also multiple of 5 True

45 is divisible by 9 TrueA number is divisible by 2 always ends in 2,4,6 or 8 False

3The first 25 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 y 97

4a) The highest common factor of twenty and thirty is ten.b) The least common multiple of five and seven is thirty-fivec) The highest common factor of seventeen and thirty-four is seventeend) Seven is the highest common factor of forty-nine and fourteen

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5Absolute value of negative three |−3| 3The opposite number of four Op (4) -4The absolute value of zero |0| 0

The opposite of the absolute value of negative one Op(|−1|) -1

The absolute value of the opposite number of one |op(−1)| 1

The opposite number of the opposite number of negative two

Op[Op(-2)]-2

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General rule Formula Example

Associative property

(A ∙ B) ∙ C = A ∙ (B ∙ C) (2 ∙ 3) ∙ 7 = 2 ∙ (3 ∙ 7)

Commutative property

A ∙ B = B ∙ A 2 ∙ 3 = 3 ∙ 2

Distributive property

A ∙ (B + C ) = A ∙ B + A ∙ C 4 ∙ (9 + (-3) ) = 4 ∙ 9 + 4 ∙ (-3)

Neutral element A ∙ 1 = A 3 ∙1 = 3

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a) If you have 18€, I have half plus 6 euros. How much money do I have?b) In a party David gives me 4 pieces of cake and I give 3 of them to other friends. This

happens 5 times with different cakes. How many pieces of cake do I take at the end?9

Three to the power of four written in the short form is _ 34 __ and the result is 81.If the base is negative and the exponent is even, the result is _positive__.If the base is negative and the exponent is _odd___, the result is negative.The short form of 3 ∙3 ∙3 ∙3 ∙3 ∙3is 36.Negative one to the power of 158 is ____1______.

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78 :73 75The quotient of two powers of the same base is another power with the same base and its exponent is equal to the numerator power exponent minus the denominator power exponent.

25 ∙35 (2 ∙3)5The product of two powers with the same exponent equals a power with the same exponent and its base is the product of the bases

(11:3)5 (11 /3)5 The quotient of two powers with the same exponent is a power with the same exponent and

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base equals the quotient of the bases.

(2 ∙3 ∙5)5 25 ∙35 ∙55The power of a product equals to the product of the powers of the factors.

(73)2 76The power of a power is a power with the same base and exponent equals the product of these two exponents.

(11:3)5 (11 /3)5The power of a quotient equals to the power of the quotient set by the numerator base divided by the denominator base.

72 ∙73 75The product of several powers of the same base is another power with the same base and its exponent is the sum of the exponents of the multiplying powers

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Number

Integer root

Rest

17 4 147 6 1149 7 0121 11 0140 11 1993 9 22500 50 01024 32 0

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Worksheet 2: Mathematics in Al Andalus

Level 2nd ESO

Name:


Instructions: A computer connected to internet is required.

Introduction:

Muslims remained in Spain for more than eight centuries, since they vanquished Don Rodrigo, the last Visigoth king in the battle of Guadalete, and occupied most part of the Iberian Peninsula in only five years.

They went through several important periods, such as the independent Emirate of Córdoba, and the Caliphate of Córdoba-the ones with the greatest splendor regarding politics and culture. But then the Caliphate was divided into the Taifas, the Almoravides and the Almohads invaded the Peninsula and it all ended in the Nazari Kingdom of Granada.

They brought great advances in maths from the East, where they were coming from. For example, they introduced the numeral system which we use, as well as its specific vocabulary. To sum up, their contribution to maths was of paramount significance for this and other sciences.

You can check its importance doing the following activities.

1. Ask the meaning of the word you do not understand. Summarize the text in three or four sentences.

2. Mark the angles that each of the following figures has and write down the cipher. What do you conclude from that?

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3. Surf the net to find the meaning and origin of the word algorithm. You will find the name of an important Arab mathematician of the IXth century, best known as the father of Algebra.

4. This mathematician is famous for solving the quadratic equation, when he didn’t know the formulae we use nowadays. Find out and explain how he could do it.

5. Surf the net for information about the following Muslim mathematicians and say what their contribution to the science is. Mark the mathematician from Al Andalus.

Birthplace ContributionsQalasadi

Maslama

Yusuf- al-Mutaman (King

of Zaragoza)

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Al-Nayirizi

Thabit ibn Qurra

Ahmad ibn Yusuf

6. Write a short summary of the importance of the Muslim contributions to current mathematics (4 or 5 lines)

7. Although most part of mathematical terms come from Greek or Latin, Muslim mathematicians introduced some of them, like: algebra, algorithm, zero, cipher and guarism.Find them in the word game.

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8. Arab mathematicians invented a clever method to multiply big numbers. Find it and use it to multiply 356 x 421.

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W2 Solutions

3. Algorithm: A logical arithmetical or computational procedure that if correctly applied ensures the solution of a problem.The mathematician is al-Khwarizmi

4.

5.

Birthplace ContributionsQalasadi Baza (Granada) He made attempts at creating an algebraic

notation.

Maslama Madrid He took part in the translation of Ptolemy's Planispherium, improved existing translations of the Almagest, introduced and improved the astronomical tables of al-Khwarizmi, aided historians by working out tables to convert Persian dates to Hijri years, and introduced the techniques of surveying and triangulation.

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https://en.wikipedia.org/wiki/Mathematical_notation

https://en.wikipedia.org/wiki/Mathematical_notation


[1]

Al-Majriti was one of the earliest Alchemists to record the usage and experimentation of mercuric oxide.According to Şā'id ibn Ahmad Andalusī he was the best mathematician and astronomer of his time (in Al-Andalus).[2]:64 He also introduced new surveying methods by working closely with his colleague Ibn al-Saffar. He also wrote a book on taxation and the economy of Al-Andalus.[1]

He edited and made changes to the parts of the Encyclopedia of the Brethren of Sincerity when the encyclopaedia arrived in Al-Andalus[3][4]

Al-Majriti also predicted a futuristic process of scientific interchange and the advent of networks for scientific communication. He built a school of Astronomy and Mathematics and marked the beginning of organized scientific research in Al-Andalus. Among his students were Ibn al-Saffar, Abu al-Salt and Al-Tartushi.[2]:64

Yusuf- al-Mutaman

Zaragoza Yusuf ibn Ahmad al-Mu'taman ibn Hūd wrote a mathematical treatise called Kitab al-Istikmāl (Arabic,كتاباإلستكمال, Perfection or Comprehensive Treatise) in mathematics, where it was edited by Maimonides (ca. 1135 – 1204).Ceva's theorem which is often attributed to the Italian mathematician Giovanni Ceva (d. 1734) was proved much earlier by Al-Mu'taman ibn Hūd. It remains unknown whether Giovanni had discovered this geometry on his own or if he had found a translation of al-Mu'taman's treatise.[1]

Al-Nayirizi Nayriz, FarsProvince, Iran. He compiled astronomical tables, writing a

book on atmospheric phenomena.

Nayrizi wrote commentaries on Ptolemy and Euclid. Nairizi used the so-called umbra (versa), the equivalent to the tangent, as a genuine trigonometric line (but he was anticipated in this by al-Marwazi).

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https://en.wikipedia.org/wiki/Al-Marwazi

https://en.wikipedia.org/wiki/Euclid

https://en.wikipedia.org/wiki/Ptolemy

https://en.wikipedia.org/wiki/Iran

https://en.wikipedia.org/wiki/Iran

https://en.wikipedia.org/wiki/Fars_Province

https://en.wikipedia.org/wiki/Nayriz

https://en.wikipedia.org/wiki/Yusuf_al-Mu'taman_ibn_Hud#cite_note-1

https://en.wikipedia.org/wiki/Giovanni_Ceva

https://en.wikipedia.org/wiki/Ceva's_theorem

https://en.wikipedia.org/wiki/Maimonides

https://en.wikipedia.org/wiki/Mathematics

https://en.wikipedia.org/wiki/Arabic


Thabit ibn Qurra

Harran, Turkey Early reformer of the Ptolemaic systemA founder of staticsLength of the siderealyear

Ahmad ibn Yusuf

Baghdad He worked on a book on ratio and proportion. and was a commentary of Euclid's Elements. This book influenced early European mathematicians such as Fibonacci. Further, in On similar arcs, he commented on Ptolemy's Karpos (or Centiloquium); He also wrote a book on the astrolabe. He invented methods to solve tax problems that were later presented in Fibonacci's Liber Abaci.

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https://en.wikipedia.org/wiki/Astrolabe

https://en.wikipedia.org/wiki/Centiloquium

https://en.wikipedia.org/wiki/Ptolemy

https://en.wikipedia.org/wiki/Leonardo_of_Pisa

https://en.wikipedia.org/wiki/Euclid

https://en.wikipedia.org/wiki/Proportionality_(mathematics)

https://en.wikipedia.org/wiki/Ratio


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Worksheet 3: Fractions, decimal numbers and proportionalityLevel 2nd ESO

Name:


1. Say if it’s true or false. Give reasons:

Situation True / FalseTwo equivalent fractions always have the same denominator

An equivalent fraction of 35 is

2745

78 is greater than 6

7

This is a graphic of 3/8:

13+ 3

5=4

8

23· (−3)

2: (−3)

2=2

3

2. Complete the following table:

Thirteen sixths plus a third136

+ 13

Fifteen sixths

12−3

Four twentieths times five

(−4 ) : 35

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3. Solve the following Criss-cross. If the answer is formed by two words, then write them together (no holes), and always use letters, never numbersnor symbols.

Across

8. To get the power of a fraction you raise the numerator and the ___________ to the exponent.

9. 0'5555 is _______ than 0'555...

13. Fraction that is equivalent to a decimal number

Down

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1. Decimal numbers that have a group of non-repeating decimals between the comma and the repeating decimals.

2. 0'4444.... is a ______ decimal number

3. 0'33333333 is a _______ period number

4. It’s a number with limited decimals.

5. To get the squared root of a fraction you calculate the square ___________ of the numerator and the denominator.

6. 3 to the power of negative 2 is ______ than 5 to the power of negative 2

7. If you divide 1:3 you get a decimal number with period ______

10. Three to the power of -2 is a power with _______ exponent

11. Decimal numbers that have a group of decimals repeating just after the comma.

12. Are the numbers with unlimited decimals.

14. 3/4 is the ________ fraction of 4/3

4. Listen carefully to your teacher and fill in the directly proportional chart

a) Here’s the relation between the time you spend painting a Wall and the meters you get painted

Time (minutes)

45 60 77

Metres 1’5 2’25

The constant of direct proportionality is ______, and it means that ________

b) Explain a situation involving two directly proportional magnitudes:

5. Fill in the inversely proportional chart

a) Here’s the relation between the fast you go in a car and the time you spend on a trip to Málaga

Speed (km/h) 100 120Time (hours) 3’25 2 5

The constant of inverse proportionality is _______, and it means that _______

b) Explain a situation involving two inversely proportional magnitudes:

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6. Solve the following problem:A grandfather distributes 450€ among his three nephews of 8, 12 and 16 years old, directly proportional to their ages. How much money does each one get?

7. Solve the following problem:a) The government has been planting trees in a forest the last two years so its surface

has been increased in 20%. And this summer there was a fire that burnt the 30% of the forest surface. Find out the percentage of the forest that remains.

b) If the forest had a surface of 600 km2 at the beginning, what is its actual surface?

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Across

3. A percentage is equivalent to a ratio with denominator equals _____

6. The value of X in the proportion 4/5 = 16/X is _______.

Down

1. You have a ______ among numbers A,B,C,D if the ratio between A and B equals the ratio between C and D.

2. The _________ of a total is the amount that is in every 100 units.

4. The time and the distance are are _______ proportional magnitudes.

5. The ratio between two numbers A and B is the quotient A/B.

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W3 SOLUTIONS :

1FalseTrueTrueTrueFalseTrue

2A half less three; negative five over two(4/20)·5; OneNegative four divides by three fifths; negative twenty over three

3.-

denominator To get the power of a fraction you raise the numerator and the ___________ to the exponent.

root To get the squared root of a fraction you calculate the square ___________ of the numerator and the denominator.

negative three to the power of -2 is a power with _______ exponent

inverse 3/4 is the ________ fraction of 4/3

greater 3 to the power of negative 2 is ______ than 5 to the power of negative 2

excactdecimal It's a number with limited decimals.

perioddecimalnumbers Are the numbers with unlimited decimals.

pureperioddecimals Decimal numbers that have a group of decimals repeating just after the comma.

mixedperioddecimals Decimal numbers that have a group of non-repeating decimals between the comma and the repeating decimals.

generatingfraction Fraction that is equivalent to a decimal number

three If you divide 1:3 you get a decimal number with period ______

exact 0'33333333 is a _______ period number

periodic 0'4444.... is a ______ decimal number

smaller 0'5555 is _______ than 0'555...

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4

Time (minutes) 45 60 77 90Meters 1’125 1’5 1’925 2’25

The constant of direct proporcionality is ____40___., that means that you take 40 minutes to paint 1 m of wall

5

Speed (km/h) 61’54 100 120 40Time (hours) 3’25 2 1’67 5

The constant of inverse proporcionality is ___200__. It means a distance of 200 kilometres.

7A grandfather distribute 450€ among his three nephews of 8, 12 and 16 years old, directly proportional to their ages. How much money get each one?450/36 = 12’5€ per year, so: 8x12’5=100 ; 12x12’5=150 ; 16x12’5=200 ;

81’2 x 0’7 = 0’84. There’s only 84% leftActual surface 600 x 0’84 = 504 km2

9.-

ratio The ratio between two numbers A and B is the quotient A/B.

proportion You have a ______ among numbers A,B,C,D if the ratio between A and B equals the ratio between C and D.

twenty The value of X in the proportion 4/5 = 16/X is _______.

directly The time and the distance are are _______ proportional magnitudes.

percentage The _________ of a total is the amount that is in every 100 units.

onehundred A percentage is equivalent to a ratio with denominator equals _____

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Worksheet 4 Divine Proportion and Music

Level 2nd ESO

Name:


Instructions: Computer and internet are required.

1. Look for the legend of Pythagoras’ Hammers and talk in pairs about it.

2. Read the following short text from Aristotle and ask for the meaning of the words that you don´t know.

a. Choose the correct Pythagorean definition of harmony among the following ones.

A state of agreement, cooperation and peacefulness.

The proportion among the parts and the whole.

An orderly arrangement of things.

A simultaneous combination of notes.

b. Explain the use of the musical instrument invented by Pythagoras, named monochord, which, in fact, was hardly suitable to play music.

c. If we regard the tone as string, the following graphic represents the relationship

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“Since the Pythagoreans saw that the properties and ratios of the musical scales are based on numbers, and since it seemed clear that all other things have their whole nature modeled upon numbers, they thought that numbers and harmony are the ultimate essence of the whole physical universe.”

ToneOctave

Fourth Fifth


between the tone and the octave, the fourth and the fifth. If the tone represents the one, write the fractions (in its simplest terms) that represent the Octave, the fourth and the fifth.

3. Let´s meet the Golden number:

The golden number is represented by the letter φ (phi), and it stablishes the golden ratio, that is considered the perfect proportion between two elements.

φ=1+√52

1.6180339… φ is an irrational number

Now, let’s investigate on our own:

a) What is a Golden Rectangle? Draw it.

b) Look up examples of Golden Rectangles in the real world.

c) Put your Identity Card on the paper and draw it. Indicate its measures and prove that it is a Golden Rectangle.

d) φ in Music

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The fifth is the golden point of the scale, since it comes from multiplying the scale’s twelve semitones by the number φ. If then we look for the golden proportion of the fifth, we will obtain another interval which, together with the fundamental sound and the fifth, forms the perfect chord. Which interval is it?

e) φ in Architecture

Some architects have also applied the Golden Proportion to their works. Find them in that design of the CapillaPazzi by Bruneslleschi, and answer the following questions:

f) How many Golden Rectangles can you observe?

g) Prove that all of them are Golden Rectangles.

1st Golden Rectangle

2+33

=1.6 3rd Golden Rectangle

2nd Golden Rectangle

4th Golden Rectangle

h) Look at the numerical series of Fibonacci, and find out the following number:

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1 – 1 - 2 – 3 – 5 – 8 – 13 – 21 – 34 -

i) Do the following divisions:

3421

=¿ 2113

=¿

What do you conclude from that?

j) Look for more examples of the Golden Number:

Arts

Nature

Human body

4. The basis of tonal harmony.

The seven sounds of the diatonic scale and the twelve sounds of the chromatic one are the result of a concatenation of fifths. That’s why this “circle of fifths” is at the root of our melodies. How can we say that this circle also rules the relations among the different keys of our tonal system?

5. Put the words in this Polycleitus’ sentence in order:

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numbers/ of / question / a / is / perfection.

6. Physics of the Sound agrees with Pythagoras.

Look at the following graphic of the sounds so called “harmonics”. Take the three first and explain how they hold the Pythagorean theory.

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W4 Solutions

Question 2

a) The proportion among the parts and the whole.

b) The monochord was intended to show the numerical proportions of the musical intervals through the mathematical divisions of the string.

c) The interval of octave is the half of the string; the fifth is three quarters of the string; the fourth two thirds of the strings.

Question 3

a)b) ID card, windows, Unites Nations Building, i-podc) 8,4 x 5,2 cm 8,4/5,2 = 1,6153 = golden numberd) It is the third. e) The director rectangle of this spiral has the module “fi”. Being the reason of

development of this rectangle equal to “fi”, the modules of the so obtained waxing rectangles are the elements of the Fibonacci sequence. In this drawing we can see how the whole is the best element to establish the proportion between the two other ones, such as Pythagoras thought.

f) 4 rectanglesg)

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h) 55i) 1,6j) The golden number appears when dividing any term in Fibbonacci series by the

previous term.

Arts and architecture : Pyramid, parthenon, Dalí, violins, Miguel Ángel, Durero, Da Vinci, Wolfgang Amadeus Mozart sonates, The fifth symphony of Ludwig van Beethoven…

Nature: crystal structures, spiral of the galaxies, the nautilus shell…..

Human body: Distances in our body (leg- thigh, arm-forearm)

Question 4

The most natural modulation of each key in our tonal system is always toward the “dominant”, that is to say, towards the scale (or key) built on the fifth note of the main one.

Question 5

“Perfection is a question of numbers”.

Question 6

In this drawing we can see how the proportions 1/2, 2/3 and 3/4 of the three first harmonic sounds coincide with the divisions made by Pythagoras on the string in order to find the intervals of “octave”, “fifth” and “fourth”. In such a way, acoustic stands for mathematics.

34

1st Golden Square

2+33

=1.6 3rd Golden Square

8+58

=1.6

2nd Golden Square

5+35

=1,6 4th Golden Square

13+813

=1.6

Worksheet 5: Algebraic expressions. Equations.

Level 2nd ESO

Name:


1. Listen carefully to your teacher and fill in the gaps:

Why are we learning algebra?

Algebraic language use numbers, ___________ and mathematical __________. A letter represent any _______________. Algebra is a tool to _______________ problems, we use it to find the numerical ____________ of something that we don´t know.Algebra allows us to express the ________________ between the parameters in a ____________. We can put values into those formulas to find any ___________. This could be the area of a ____________, the ___________ of medicine to give a child based on their ___________, the blood-______________ of a person according to their age, or the monthly mortgage _______________.Putting all these ideas together, we can conclude that ____________ is a language for science, _____________, economics, business, engineering and ____________. It is the language to describe how the world __________!


Situation Algebraic expression

As double as many as your ageFive times the sum of two numbers

x4

Ana is seven years older than her brother(3 x)2

He has twice as much money as she hasTake the quantity negative five plus x, and then multiply by eightThe area of a rectangle whose height is half the base.The perimeter of an equilateral triangle whose sides are x

A=π r2

3. Work out the following numerical values:

a. The blood pressure of a girl aged 18, using the formula: BP=x :2+110 b. Volume of

35


c. The volume of the Earth, for R ≈ 6000 Km

V= 43π R3

d. Interests you obtain from a bank account with a capital (C) of €1000, at a rate (R) of 12% for 180 days (T):

I=C·R·T36500

4. Join with arrows the different cells:

(3 x+2)2 (x+3)(x−3)The square of a sum equals the sum of the squares plus twice the product of the addends.

x2−9 9 x2+12x+4The square of a difference equals the sum of the squares less twice the product of the minuend and the subtrahend.

9 x2−12 x+4 (3 x−2)2The difference of two squares equals the product of the sum and the difference.


a. A problem about ages, money or whatever you want, using the following system of equations:x+ y=20y=3 x

b. A problem involving a directly proportional distribution, using the equation :3000

5= x

2= y

3

36



Across

37


2. A polynomial formed by two monomials.5. Ten times the square of three.

7. Fifty Farenheit degrees into Centigrades degrees ℃=5 ·(℉−32)

98. If I am eighteen, five less than twice my age.9. The degree of the polynomial −3 x7−5 x3+4 x

11. The numerical value of x2−13 for x = -1

13. Choose the positive solution: 2x2−16 x=015. Choose the negative solution: 2 x2−3x−5=0(clue is in the draw under).17. If my parent is forty-five, and I have a third of his age, write my age.

Down1. Area of a rectangle whose height is two fifths his base plus two, the base measures ten.3. Literal part of the monomial 3ab4. Coefficient of the monomial -5xy5. Solve the equation: (3+x)/6 + 5 = 76. The degree of the monomial 4 x2 y10. The perimeter of an isosceles triangle whose two equal sides are triple the distinct side, whose length is two.12. Choose the positive solution: x2+9=13014. The value of the unknown: (5x/2) + 7 = 17

38


16. Half the difference of sixty and twelve.

39


W5 SOLUTIONS

1. Why are we learning algebra?

Algebraic language use numbers, letters and mathematical symbols. A letter represent any number.Algebra is a tool to solve problems, we use it to find the numerical value of something that we don´t know.Algebra allows us to express the relationship between the parameters in a formula. We can put values into those formulas to find any result. This could be the area of a triangle, the dose of medicine to give a child based on their weight the blood-pressure of a person according to their age, or the monthlymortgage payment.Putting all these ideas together, we can conclude that algebra is a language for science, technology, economics, business, engineering and medicine. It is the language to describe how the world works!


Situation Algebraic expression

As double as many as your age 2xFive times the sum of two numbers 5(x+y)Fourth of a number x

4Ana is seven years older than her brotherThe square of triple a number (3 x)2

He has twice as much money as she has 2xTake the quantity negative five plus x, and then multiply by eight (-5+x)8The area of a rectangle whose height is half the base. 3xThe perimeter of an equilateral triangle whose sides are x 3xThe area of a circle A=π r2

7. Work out the following numerical values:

a. The blood pressure of a girl aged 18, using the formula: BP=x :2+110

b. Volume of c. The volume of the Earth, for R ≈ 6000 Km

V= 43π R3

d. Interests you obtain from a bank account with a capital (C) of €1000, at a rate (R) of 12% for 180 days (T):

I=C·R·T36500

40

119

9,8x1011 Km3

59,18€


8. Join with arrows the different cells:

(3 x+2)2 (x+3)(x−3)The square of a sum equals the sum of the squares plus twice the product of the addends.

x2−9 9 x2+12x+4The square of a difference equals the sum of the squares less twice the product of the minuend and the subtrahend.

9 x2−12 x+4 (3 x−2)2The difference of two squares equals the product of the sum and the difference.


e. A problem about ages, money or whatever you want, using the following system of equations:x+ y=20y=3 x

Ex: Sonia y Maria want to buy a book that cost €20. If Sonia is paying the triple of Maria, how much many is putting every one?

f. A problem involving a directly proportional distribution, using the equation :3000

5= x

2= y

3Ex: A total amount of €5000 is going to be distributed between the partners of a business according the years that they have been working in it. If they have been in the business for 2 and 3 years. How much money gets everyone?

10.Solve the following Criss-cross. If the answer is formed by two words, then write them together (no holes), and always use letters, never numbers.

Across2. A polynomial formed by two monomials.BINOMIAL5. Ten times the square of three.NINETY

41


7. Fifty Farenheit degrees into Centigrades degrees ℃=5 ·(℉−32)9

TEN8. If I am eighteen, five less than twice my age. THIRTYONE9. The degree of the polynomial −3 x7−5 x3+4 x SEVEN 11. The numerical value of x2−1

3 for x = -1 TWOTHIRDS

13. Choose the positive solution: 2x2−16 x=0 EIGTH15. Choose the negative solution: 2 x2−3 x−5=0(clue is in the draw under). NEGATIVE ONE17. If my parent is forty-five, and I have a third of his age, write my age.FIFTEEN

Down1. Area of a rectangle whose height is two fifths his base plus two, the base measures ten. FIFTY3. Literal part of the monomial 3ab AB4. Coefficient of the monomial -5xy NEGATIVEFIVE5. Solve the equation: (3+x)/6 + 5 = 7 NINE6. The degree of the monomial 4 x2 y THREE10. The perimeter of an isosceles triangle whose two equal sides are triple the distinct side, whose length is two. FOURTEEN12. Choose the positive solution: x2+9=130 ELEVEN14. The value of the unknown: (5x/2) + 7 = 17 FOUR16. Half the difference of sixty and twelve. SIXTEEN

42

Worksheet 6: Functions

Level 2nd ESO

Name:


Instructions: Computer with Geogebra is required.

1. Read carefully the definition of function below. Then, look at the pictures and discuss with your partner whether the graphs are functions.

A function is a relation between two variables or magnitudes (x and y) where the value of the magnitude y depends on the value of the magnitude x; and each value

of x is related to an only value of y.

2. In pairs, discuss whether the following relations, are functions. Give reasons and use examples.

a. Every number is assigned to his square root.

b. Every number is assigned to his inverse.

c. Every book is assigned to his author.

d. Every author is related to his books.

3. Anna has done this route on bicycle, where the y-axis represents the distance from home, and the x-axis, the time since she left. Study the graph and

43


answer the questions below.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

25

30

35

40

A bicycle route

Time (hours)

Dist

ance

from

hom

e (K

m)

a) Does Anna always run at the same speed? On which intervals does she maintain the same speed?

b) What is the period of time in which Anna runs at the highest speed? What is this speed?

c) When did Anna decide to go back home?

d) What happened during the interval (1´5, 2)?

e) How far away has Anna gone from home?

4. Now, using the same graph, fill in the gaps using only the correct words from box.

The ____________ of the function is (0, 4.5), and the _____________ is (0, 35). It is a _______________function. It is _______________ on (0 , 1.5), _____________ on (1.5, 2) U (2.5, 4.5), and not increasing or decreasing on (2, 2.5). It has a _______________ of 35 at x = 1.5 and it has no__________________.

44


decreasing maximum discontinuous rangedomain minima

continuous minimum increasing maxima

5. A phone company charges 50 cents per call plus 20 cents per minute. a. State the formula that represents the relationship between the number

of minutes and the total cost.

b. If we talk for 8 minutes, how much does it cost?

c. Plot the graph.

6. A group of friends are buying a videogame that cost €25. State the function that relates the amount of money per friend (y) to the number of friends involved(x). Then, plot the graph.

45


WORKING WITH GEOGEBRA

Now, click on the following link: https://youtu.be/tHmSYbE15Vo , watch and listen to the video and do the following exercises:

7. Start a Geogebra session and repeat the steps as shown in the video.

8. Now start anothernew session. Introduce the following function:

f(x)=(x+2)(x-4)(x+1)2

8.1. Change the line colour into red.

8.2. Name the graph with “Exercise 8”.

8.3. Set the basics dimensions to x(-10,10) and y(-150,5).

8.4. Fix the spacing between values on the axes: 1 for x, 15 for y.

8.5. Export the graph and paste it in a MS Word or Writer document and send it to the teacher via email.

8.6. What will happen if you change (x+1) to (x-1) in the function? (Write the answer in the same document you send to the teacher).

46


W6Solutions

1.a. Is a functionb. Is not a function

2.

a. Every number is assigned to his square root. Notb. Every number is assigned to his inverse. Yesc. Every book is assigned to his author. Yesd. Every author is related to his books. Not

3.

a) Does Anna always run at the same speed? On which intervals does she maintain the same speed? Not, ( 0, 0.5 ) U (0.5, 1) U (1, 1.5) U ( 1.5, 2) U (2, 2.5 ) U (2.5, 3.5) U (3.5 U 4)b) What is the period of time in which Anna runs at the highest speed? What is this speed? (0.5 , 1) 50 Km/hc) When did Anna decide to go back home? After an hour and halfd) What happened between the intervals (1.5 , 2) ? She is at the same place, she does not movee) How far away has Anna gone from home?35 Km

4. Now, using the same graph, fill in the gaps using only the correct words from box.

The domain of the function is (0, 4.5), and the range is (0, 35). It is a continuous function. It is increasing on (0 , 1.5), decreasing on (1.5, 2) U (2.5, 4.5), and not increasing or decreasing on (2, 2.5). It has a maximum of 35 at x = 1.5 and it has no minima.

5. A phone company charges 50 cents per call plus 20 cents per minute. a.State the formula that represents the relationship between the number of minutes and the total cost.y = 20x + 50

b.If we talk for 8 minutes, how much does it cost? 160 + 50 = 210 cents. 2 euros and 10 cents

c.Plot the graph.

47


0 2 4 6 8 10 120

50

100

150

200

250

300

6. A group of friends are buying a videogame that cost €25. State the function that relates the amount of money per friend (y) to the number of friends involved (x). Then, plot the graph.

0 5 10 15 20 25 30 3505

1015202530

48

Worksheet 7: Measures, Pythagoras and Tales.

Level 2nd ESO

Name:


1. Have you ever read “Gulliver´s travels”, by Jonathan Swift? Do you know the story of Gulliver in Lilliput? Talk in pairs about the story.

2. Read the following short text from Gulliver´s travel and ask for the meaning of the words that you don´t know.

a. Lilliputians are 12 times as small as human beings. i. If 1 inch ≈ 2.5 cm, how many centimetres tall, more or less, are the

Lilliputians?

ii. Using the Lilliputians height, calculate the human beings’.

b. Gratefully, Lilliputians provided food, clothes and shelter to Gulliver.

i. If the daily amount of food is proportional to the surface area of the body, how many Lilliputian portions would Gulliver eat?

49

After a fierce storm had wrecked his ship Adventure, Gulliver was washed ashore

on an island called Lilliput. He awoke to a strange feeling all over his body. He

found he couldn’t move. He looked down and saw hundreds of tiny beings, less

than six inches tall, swarming all over his body. The tiny Lilliputians had tied him

to the ground so that he could not harm them. “I won’t harm you,” said Gulliver but

the Lilliputians would not set him free. Gulliver promised to help them fight a

neighbouring country called Blefuscu. Gulliver captured the enemy’s navy with

ease and helped the Lilliputians win the war. He became their hero and they made

him part of the king’s court.


ii. How many Lilliputian handkerchiefs make one for Gulliver?

iii. If a human handkerchiefs measures 24 x 24 cm, what would be the measures for a Lilliputian´s?

iv. How many Lilliputian houses would fill a standard-size house?

3. Let´s read something more from the story:

a. Draw a ground plan of Mildendo, according to the given description.

b. If 1 foot ≈ 30,5 cm, convert the given measures to cm and m. Then, how many metres would be these measures for a standard-size city?

foots cm m Measures for a real city (m)

Length of the wall sides

Width of the great streets

c. Using the Pithagoras´ theorem, calculate the length of the two great streets.

4. Something more about Mildendo.50

Mildendo, the capital of Lilliput is an exact square, each side of the Wall being

five hundred foot long. The two great streets, [which run across from corner to

corner], and are five foot wide, divide it into four quarters. The emperor´s Palace

in just in the centre, where the two great streets meet.


a. Calculate the area of Mildendo in Km2. Then, what would be the area for a real city?

b. Now let’s thing. For a city having the measures of Mildendo, it is possible to have a population of five thousand? Here you have same data to help you. Fill in the table:

City Population Area (Km2)

Population density (inhabitants/Km2)

Córdoba 300 000 1255Málaga 570 000 400Segovia 53 000 164Mildendo

i. Which city is Mildendo comparable to?

ii. Why is the population density of Segovia greater than Córdoba´s? What that means?

5. Discover this hidden Jonathan Swift´s quote:

51

The town is capable of holding [five thousand] souls. The houses are from three to five stories. The shops and the markets are well provided.


W7 Solutions:

2.a) i) 2.5 · 6 = 15 cma) ii) 15 · 12 = 180 cmb) i) 12·12=144 portionsb) ii) 12· 12 =144 handkeerchiefsb) iii) 12·12·12 = 1728 houses

3.

a)

b)

foots cm m Measures for a real city (m)

Length of the wall sides 500 1500 150 1800

Width of the great streets 5 30 0,3 3,6

c) h2=5002+5002→h=707m

4. a) A = 150 · 150 = 22500 m2 = 0,0225 Km2 ; A = 0,0225 · 144 = 3,24 Km2

b)

City Population Area (Km2)

Population density (inhabitants/Km2)

Córdoba 300 000 1255 239Málaga 570 000 400 1425Segovia 53 000 164 323Mildendo 5000 3,24 1543

It is comparable to Málaga

6. EVERY MAN DESIRES TO LIVE LONG, BUT NO MAN WISHES TO BE OLD

52

Worksheet 8: Planets and scientific notation. Microorganisms and cells size. Absolute and relative errors.

Level 2nd ESO

Name:


PLANETS AND SCIENTIFIC NOTATION

53


Complete this chart using the information provided:

Name of planet

Distance from Sunin standard notation

Distance from Sunin scientific notation

Radius of planetin standard notation

Radius of planet in scientific notation

Mass of planetin standard notation

Mass of planetin scientific notation

ABSOLUTE AND RELATIVE ERRORS.

54


1. What absolute and relative errors are committed if you take the Earth-Sun distance of 150 million km?

2. What absolute and relative errors are committed if you take the Earth diameter of 13000 km?

3. What absolute and relative errors are committed if you take the Earth mass of 6 x 1024 kg?

55


MICROORGANISMS AND CELLS SIZE.

Complete this chart:

Name Size in standard notation Size in m Size in scientific notation

Ebola virus 1 mm

Escherichia coli(Bacterium present in the stomach)

3 micrometres

Acarus(Microscopic spiders) 0.5 mm

AIDS virus 0.1 micrometres

Flu virus 0.000 000 1 m

Red blood cell 0.000 006 m

Sperm 4 micrometres

Meningitis bacterium 0.000 000 5 m

Tuberculosis bacterium 0.000 003 m

56


W8 SOLUTIONS.

Complete this chart using the information provided:

Name of planet

Distance from Sunin standard notation

Distance from Sunin scientific notation

Radius of planetin standard notation

Radius of planet in scientific notation

Mass of planetin standard notation

Mass of planetin scientific notation

Mercury 57 910 000 km 5.791 x 107 km 2 439 km 2.439 x 103 km 320000000000000000000000 kg 3.2 x 1023

Venus 108 210 000 km 1.082 x 108 km 6 052 km 6.052 x 103 km 4880000000000000000000000 kg 4.88 x 1024

Earth 149 600 000 km 1.496 x 108 km 6 378 km 6.378 x 103 km 5979000000000000000000000 kg 5.979 x 1024

Mars 227 920 000 km 2.279 x 108 km 3 397 km 3.397 x 103 km 642000000000000000000000 kg 6.42 x 1023

Jupiter 778 570 000 km 7.785 x 108 km 71 398 km 7.139 x 104 km 1901000000000000000000000000 kg

1.901 x 1027

Saturn 1 433 530 000 km

1.433 x 109 km 60 000 km 6 x 104 km 568000000000000000000000000 kg

5.68 x 1026

Uranus 2 872 460 000 km

2.872 x 109 km 26 200 km 2.62 x 104 km 86800000000000000000000000 kg

8.68 x 1025

Neptune 4 495 060 000 km

4.495 x 109 km 25 225 km 2.522 x 104 km 103000000000000000000000000 kg

1.03 x 1026

57


ABSOLUTE AND RELATIVE ERRORS.

4. What absolute and relative errors are committed if you take the Earth-Sun distance of 150 million km?5. What absolute and relative errors are committed if you take the Earth diameter of 13000 km?6. What absolute and relative errors are committed if you take the Earth mass of 6 x 1024 kg?

4. Absolute error = 150 000 000 - 149 600 000 = 400 000 kmRelative error = (400 000 / 149 600 000) x 100 = 0.267 %

5. Absolute error = 13 000 – 12 756 = 244 kmRelative error = (244 / 12 756) x 100 = 1.912 %

6. Absolute error = 5979000000000000000000000 – 6000000000000000000000000 = 21000000000000000000000 = 2.1 x 1022 kgRelative error = (2.1 x 1022 / 5.979 x 1024) x 100 = 2.1 x 1022 x 100 = 2.1 x 1024 %

58


MICROORGANISMS AND CELLS SIZE.

Complete this chart:

Name Size in standard notation Size in m Size in scientific notation

Ebola virus 1 mm 0.001 1 x 10-3 m

Escherichia coli(Bacterium present in the stomach)

3 micrometres 0.000 003 3 x 10-6 m

Acarus(Microscopic spiders)

0.5 mm 0.000 5 5 x 10-4 m

AIDS virus 0.1 micrometres 0.000 000 1 1 x 10-7

Flu virus 0.1 micrometres 0.000 000 1 1 x 10-7

Red blood cell 6 micrometres 0.000 006 6 x 10-6

Sperm 4 micrometres 0.000 004 4 x 10-6

Meningitis bacterium 0.5 micrometres 0.000 000 5 5 x 10-7

Tuberculosis bacterium 3 micrometres 0.000 003 3 x 10-6

59

Did you know that one of The Seven Wonders of The Ancient World still stands outside Cairo, Egypt? The Great Pyramid, tomb of Khufu, stands 450 feet high, even with some of its upper stones missing. It once rose 481 feet high! The pyramid was constructed around 2680 BC and contains over two million blocks of stone, each weighing about 2.5 tons.

How would mathematics have been used in building such structure?

Some modern day structures are also shaped like pyramids. Look at the pictures of the Washington Monument and the Transamerica Pyramid.

How are these structures similar to the great Pyramid? How are they different? A pyramid is a space figure whose base is a polygon and whose faces are triangle with a common vertex. What kinds of polygons serve as the bases for these pyramids?

Talk in pairs or groups of three.

Worksheet 9: Measuring perimeters, areas and volumesLevel 2nd ESO

Name:


Relating Perimeter and Area

1. Fill in the table according to the instructions below:

a) Draw a rectangle on grid paper. Record its length, width, perimeter, and area in a chart like the one below.

Rectangle Length Width Perimet

er Area

a)

b)

c)

d)

b) Draw another rectangle, with sides increased by a factor of 2. Record your findings in the table.

c) Draw a third rectangle, with sides increased by a factor of 3. Record your findings.

2. How do perimeter and area change when you double the length of the sides of a rectangle? Is there a pattern to how an increasing in length of sides affects perimeter and area?

3. Extension Activity: A gardener has 20m of fence. What is the largest rectangular garden she can enclose? Investigate relationships

60

5ft


between the perimeter and area.

4. Find the area of each figure:

10 in.13ft 4in.

18 in.

Working with area

5. Alex entered “Hearts ´n Flowers” in the Garden Club´s annual competition. To complete the entry form, he needs the area of the flower bed. How would you find the area?

61

Π=3.141592653589793…..

Two mathematicians in New York used the computer to extend the number of decimal places for π from 201 million to 480 million (No, they didn´t find any repeating pattern!).

In 280 B.C. Archimedes was happy to estimate its value as between 223/71 and 22/7


MATHEMATICS AND HISTORY

6. What area can be covered by a paint roller 5 cm in diameter and 20 cm wide as it makes 6 full revolutions?

7. A manufacturer wants to make a can whose label is as large as possible. The height and diameter of the can may not be more than 24 m together. What dimensions should the can be?

8. A cylinder has a diameter of 20 cm and a height of 35 cm. What will change the surface area more, adding a centimeter to the diameter or adding a centimeter to the height? Explain your answer.

VOLUME OF PRISMS AND CYLINDERS.

The Peachtree Plaza Hotel in Atlanta has a cylindrical tower and a rectangular prism base. The volume of each is important for

62


efficient functioning of the air conditioning system.

The volume of a prism or a cylinder is the area of the base multiplied by the height of the figure.

Application:

9. A prism with a square base has a height of 42 cm. The volume is 156,282 cm3. Find the area of the base.

10.A cylindrical silo is 38 feet high. The volume of the silo is 11,938 cubic feet. What is the area of the base? What is the diameter of the base?

11. A cube has a surface area of 150m2. What is its volume?

63


W9 SOLUTIONS:Reading: Answers will vary. Possible answers include using inclined planes, and measuring angles and sizes of blocks.

Possible answers include squares and triangles as bases for pyramids; other examples of pyramids are some New York buildings and Mexican pyramids.

Relating Perimeter and AreaPerimeter increases by the same factor as the sides. Area increases “faster” by the square of the side´s increase.

d)Possible answers:9cm, 1cm, 9cm2; 8cm, 2cm, 16cm2

3) 30ft2 56in2

Working with Area4) 64.26m2

5)2,120 cm2

6)h= 12 cm d=12 cm7) Adding to the diameter. This increases radius and radius is aquared to find surface area8) 3,721 cm2

9) base: 314ft2 diameter: 20ft10) 125m3

64

Worksheet 10: Statistic

Level 2nd ESO

Name:


1. Find the following words, related to statistic. Do you know their meanings? Talk in pairs.

Range Frequency, Sampling, Median, Mode, Mean, Circle graph, Bar graph, Line graph

Reading

Which peanut butter tastes best? Do you buy CDs? What dog food do you feed your dog? These are some of the questions a company doing market research might ask. Businesses want to find out what products people like and don´t like. They ask people to try different brands and pick the one they like the best. Companies then use this information to create new products or improve old ones.

How might market researchers use mathematics? When you buy one product rather than another, you are making a choice. The marker researcher who records your decision is gathering facts or data. Statistics is the science of collecting, organizing and analyzing data.

How could statistic be used to determine the top ten albums? What other information could statistics describe?

65

R U M G B N O D U M S PV H T D A A H W H I S MY I E E Z E R Q P I L ES A M P L I N G A D U DH Y C N E U Q E R F X IR P U F V C S L G A G AX Z A K I I E G E E P NZ U J R V Q P A N D H HM Q C E G N A R I O F YA L X E B Y M Q L M D ME Z X F V F R Y V Y N JX F N E G Q W N L A F I


Exercises

1. Write True or Falsefor each statement. If false, give a set of data for which the statement is false.

a. The mode is always close to the median

b. The median is always in the middle of the line plot

c. The median is sometimes not included in the data

d. If a mode exists, it is always included in the data.

2. Name the type of graph that would best display each set of data. Explain each choice.

1. Daily high temperatures for a month.

2. Mathematics test scores on eachchapter.

3. Categories of expenses and amounts spent

4. Numbers of male and female students in each year

3. Here we have the language´s marks for two classes from the same school obtained in the same exam:

Class 2nd A2,5,7,8,2,5,6,9,3,4,6,7,8,7,6,8,6,8,4,7,6,4,5,6,4,5,6,3,9,8.Class 2nd B8,9,7,8,7,8,9,7,8,9,6,9,8,7,4,4,4,5,6,7,8,9,6,9,9,8,5,7,8,6.

a) Build a frequency distribution table for each one and then draw one graph that compares the two classes.

66


b) Answer the questions: Which one has the best results? How do you know?

Find the median and range of ages for each class

4. Observe the graph

a) How many women earn in Utopia between 50,000 and 54,000 pounds?

b) Are there more women or men who earn 12,000 pounds? How many more?

c) True or False and Why

There are more women who earn more than 85,000 pounds than men.

There are the same amount of women who earn 36,500 pounds and 41,500 pounds.

The amount of men who earn less than 4,000 pounds is less than won.

67


68


W10 SOLUTIONS:

Y + + + + + E + + + + R + + + + C + + + + H D + + A + + + G + I N + + + P + O N + + + N + + R + E + + A + G M + + I + + + C + + U + R E + + + L + + + + L + + + Q G + + + P + + + + + E + + + + E + + M + + + + + + + + + + + N R A + + + + + + + + + + + + I S F M + + + + + + + + + + + L + + + E + + + + + N + + + + H P A R G D + + + B A R G R A P H + + + + I + + + E + + + + + + + + + + + A + + M + + + + + + + + + + + + N + + + + + + + + + + + + + + +

Reading:Answers will vary. Possible answers: compile data, create charts and graphs, estimate costs and profits.Exercises:

1- a) False b) False c) True d) True2- a)line or bar b)bar c)circle d)double bar

3-Class A Range: 7 Median: 6Xi fi Fi hi Hi

2 2 2 2/30 2/30 3 2 4 2/30 4/304 4 8 4/30 8/305 4 12 4/30 12/306 7 19 7/30 19/307 4 23 4/30 23/308 5 28 5/30 28/309 2 30 2/30 1

class B is better because it has got less range and higher median

Class B Range: 5 Median: 8Xi fi Fi hi Hi

4 3 3 3/30 3/305 2 5 2/30 5/306 4 9 4/30 9/307 6 15 6/30 15/308 8 23 8/30 23/309 7 30 7/30 1

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2 3 4 5 6 7012345678

clase Aclase b

5. a) 400 thousandsb)Men 400 thousands and women 390 approximatelyc)True, True and False

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