exercising the abilities of reasoning, calculating and … · 2019-03-26 · exercising the...

EXERCISING THE ABILITIES OF REASONING,

CALCULATING AND INTERPRETING

60 PISA QUESTIONS EXPLAINED AND RESOLVED IN DETAILS

CHRISTIANE MAZUR DOI

2019

EXERCISING THE ABILITIES OF REASONING, CALCULATING AND INTERPRETING – CHRISTIANE MAZUR DOI

2

Christiane Mazur Doi is PhD in Metallurgical and Materials Engineering, Master in Science -

Nuclear Technology and Chemical Engineer. She is graduated in Mathematics, with Improvement

in Statistics Topics. Chris has more than 20 years of experience in Higher Education, teaching

subjects such as Calculus, Statistics, Physics, Chemistry and Reading. She elabores instructional

materials for undergraduate courses. She is also author of several books.


3

PRESENTATION

Hello, dear readers!

In this book, aiming to exercise the abilities of reasoning, calculating and interpreting, we

analyze, in a didactic and detailed way, PISA (Program for International Student Assessment)

questions. This exam is developed by the OECD (Organization for Economic Cooperation and

Development) to students aged 15 years old, when basic schooling ends in most countries.

PISA, which takes place every three years, aims to evaluate the literacy of students in

three areas: Reading, Mathematics and Science. The term literacy is used because this exam is

not only based on curricular contents, but also on skills and competences related to the analysis

and interpretation of contextualized utterances, to the use of logical reasoning for problem

solving and to the efficient communication of ideas.

With the intention of improving such skills, 60 questions were selected in this text,

classified according to the demands of the following blocks:

• performing basic calculation (addition, subtraction, multiplication and division);

• reading and interpreting tables and graphs;

• proposition, interpretation and application of mathematical formulas;

• characterization and establishment of mathematical relationships in geometric pictures;

• application of rules to solve a problem;

• using probability topics and central tendency measures;

• presentation of critical analysis of information.

Thus, in this publication, written in simple and direct language, Mathematics, Science and

Reading topics are explained with examples focused on daily situations and on subjects that are

part of our lives. As a result, international examination questions, such as those of PISA, are no

longer difficult, complicated or abstract.

Each of the blocks presents a theoretical introduction approached through situations

extracted from the everyday or by means of practical applications of definitions: the concepts are

developed based on cases close to our reality. The issues are resolved in details, as if there is an

informal conversation between us.

In addition, at the beginning of each exercise, we define the topic to be addressed, and at

the end of each exercise, we emphasize what is needed to resolve it.


4

Note. The headings of the Pisa questions studied in this work, freely available on the internet,

have been taken and adapted from the electronic addresses listed below, accessed on May 2018.

<https://nces.ed.gov/surveys/pisa/pdf/items2_reading.pdf>

<https://nces.ed.gov/surveys/pisa/pdf/items_financial.pdf>

<https://nces.ed.gov/surveys/pisa/pdf/items_math.pdf>

<https://nces.ed.gov/surveys/pisa/pdf/items2_math2012.pdf>

<https://nces.ed.gov/surveys/pisa/pdf/items_reading.pdf>

<https://nces.ed.gov/surveys/pisa/pdf/items_science.pdf>

<https://www.oecd.org/pisa/38709418.pdf>

<https://nces.ed.gov/surveys/pisa/pdf/items2_solving.pdf>

https://nces.ed.gov/surveys/pisa/pdf/items_math.pdf

https://nces.ed.gov/surveys/pisa/pdf/items2_math2012.pdf

https://nces.ed.gov/surveys/pisa/pdf/items_reading.pdf

https://nces.ed.gov/surveys/pisa/pdf/items_science.pdf

https://eur02.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.oecd.org%2Fpisa%2F38709418.pdf&data=02%7C01%7C%7Caeb71d0db65541e8047808d5b1c40abd%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C636610377905497692&sdata=%2FG4gY8FOK2xuA%2BTjlDfZl4QPUNGS%2BwnVPX2AIjNMFXI%3D&reserved=0

https://eur02.safelinks.protection.outlook.com/?url=https%3A%2F%2Fnces.ed.gov%2Fsurveys%2Fpisa%2Fpdf%2Fitems2_solving.pdf&data=02%7C01%7C%7Caeb71d0db65541e8047808d5b1c40abd%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C636610377905497692&sdata=Ib76eCMGbOjY58ywuGT4YHSB3UAfLLuPzZFTKKHv2gE%3D&reserved=0


5

INDEX

CHAPTER 1. Basic calculation: addition, subtraction, multiplication and division.

QUESTION 1.1. Height of 14 steps of a staircase.

QUESTION 1.2. Saving weekly receipts for holiday travel.

QUESTION 1.3. Recipe for salad dressing.

QUESTION 1.4. Manufacture of shelf assemblies.

QUESTION 1.5. Choice of pizza toppings.

QUESTION 1.6. Currency conversion rate between currencies of Singapore and South Africa.

QUESTION 1.7. Options for parts for skateboarding.

QUESTION 1.8. Consequences of having loan for a long period of time.

QUESTION 1.9. Insertion of kite sail to NewWave to reduce diesel consumption.

CHAPTER 2. Reading and interpreting tables and graphs.

QUESTION 2.1. Variation in prices of one Rich Rock share.

QUESTION 2.2. Monthly sales of CDs from 4U2Rock, The Kicking Kangaroos, One's Darling and

The Metalfolkies.

QUESTION 2.3. Average heights of women and young men in the Netherlands in 1998.

QUESTION 2.4. Reporter's statement and correct chart analysis of robberies in 1998 and 1999.

QUESTION 2.5. Tree diagram on the structure of the workforce in a country in 1995.

QUESTION 2.6. Increased carbon dioxide emissions and increased temperature of the Earth's

atmosphere.

QUESTION 2.7. Exports from Zedland, a country that uses zeds as a currency, from 1996 to

2000.

QUESTION 2.8. Comparison between outward and return carriage routes in which an accident

involving a cat occurs.

QUESTION 2.9. Scores of two groups of students in science test.

QUESTION 2.10. Variations in levels of Lake Chad and rock art in the Sahara.

QUESTION 2.11. Times of decomposition of some types of garbage.

CHAPTER 3. Proposition, interpretation and application of mathematical formulas.

QUESTION 3.1. Drug administration by intravenous drip.

QUESTION 3.2. Relation between the number of steps per minute and the length of the step of a

man walking.

QUESTION 3.3. Punctuation system for car classification.

QUESTION 3.4. Cultivation of apples in a field surrounded by conifers.


6

CHAPTER 4. Identification of characteristics and mathematical relationships in

geometric pictures.

QUESTION 4.1. Identification of a geometric picture from its description.

QUESTION 4.2. Farm house with pyramidal shaped roof.

QUESTION 4.3. Wooden fence to a garden bed.

QUESTION 4.4. Approximate calculation of Antarctic area.

QUESTION 4.5. Proposal to attach kite sails to the ship and use the force of the wind to help

reduce the diesel consumption.

QUESTION 4.6. Revolving door with three wings which rotate within a circular space.

QUESTION 4.7. Estimate of apartment area by its floor plant.

QUESTION 4.8. Comparison of prices of small pizza and large pizza.

CHAPTER 5. Application of rule to solve a problem.

QUESTION 5.1. Numbers in faces of six dice.

QUESTION 5.2. Construction of die following the rule of "sum 7 on opposite faces".

QUESTION 5.3. Instructions for making an 8-page booklet.

QUESTION 5.4. Numbers of faces of a tower that can be seen from certain positions.

QUESTION 5.5. Construction of stages patterns with squares.

QUESTION 5.6. Computer graphics building tool.

CHAPTER 6. Using probability and measures of central tendency.

QUESTION 6.1. Picking one candy from a bag with 30 candies.

QUESTION 6.2. Probability of occurrence of an earthquake.

QUESTION 6.3. Rainfall forecast.

QUESTION 6.4. Mei Ling average on five science tests.

QUESTION 6.5. Average height of 25 girls in a classroom.

QUESTION 6.6. Surveys to support the president for the next election.

CHAPTER 7. Critical analysis of information.

QUESTION 7.1. Recognition of priority tasks according to budget.

QUESTION 7.2. Identification of bank deposit related to payment of monthly salary.

QUESTION 7.3. Assessment of factors that influence the price of a motorcycle insurance.

QUESTION 7.4. Immunization against influenza in the ACOL company.

QUESTION 7.5. Use of mobile phones and possible damage to health.

QUESTION 7.6. Advantages and disadvantages of telecommuting.

QUESTION 7.7. Semmelweis and death of women when giving birth by puerperal fever.

QUESTION 7.8. Depletion of the ozone layer.


7

QUESTION 7.9. Cultures of genetically modified (GM) corn.

QUESTION 7.10. Grand Canyon National Park, USA.

QUESTION 7.11. Action of acid rain on marble.

QUESTION 7.12. Benefits of regular and moderate exercise.

QUESTION 7.13. Level of sun protection of a photoprotective product.

QUESTION 7.14. History and indication of vaccination.

QUESTION 7.15. Intelligent clothes for children with speech disabilities.

QUESTION 7.16. Footwear specially designed for sportsmen.

CHAPTER 8. Bibliographic indications and consulted links.


8

CHAPTER 1. Basic calculations: addition, subtraction, multiplication and division.

When we remember the beginning of our learning in Mathematics, we soon think the

basic arithmetic operations: addition, subtraction, multiplication and division.

With arithmetic, a Greek word that means "the art of numbers", we can solve several problems,

such as the one presented below.

Example. "Green Leaf" stationery sells the items in the following table, whose unit prices are

given in a fictitious currency called zed.

Item Price per unit

Notebook

30 zeds

Pencil machine

16 zeds

Eraser

5 zeds

Imagine Josie taking a 100 zed bill to "Green Leaf" stationery and buying a unit of each

item from the table. What will be the change received by Josie?

To calculate Josie's spending on a unit of each item on the table, we use addition, and we

need to add the prices of the notebook (30 zeds), the pencil machine (16 zeds) and the eraser (5

zeds), as it follows.

'Josie s spending Notebook price Pencil machine price Eraser price

'Josie s spending 30 16 5


9

'Josie s spending 51 zeds

To calculate Josie's change, we use subtraction, and we need to "remove" the 51 zeds

effectively spent from the 100 zeds used to pay the account as it follows.

Josie's change Value of the bill Josie's spendings

Josie's change 100 51

Josie's change 49 zeds

Alan also went to the stationery "Green Leaf" and there he bought 2 notebooks and 3

erasers. How much did Alan spend?

To calculate Alan's expenditure on the purchase of 2 notebooks and 3 erasers, we use

multiplication (to make the product of the purchased quantity of a given item by the price of the

item) and addition (to add/ all products from the purchased quantity of the item by the price of

the item), as it follows.

Alan's spending (Nº notebooks ) (Notebook price) (Nº eraser) (Eraser price)

Alan's spending 2 30 3 5 60 15

Alan's spending 75 zeds

What was Alan's average spending per purchased item?

To calculate Alan's average spending per purchased item, we use division, and we need to

divide Alan's total spend (75 zeds) by the total number of items purchased (5, which is the result

of the sum of 2 notebooks and 3 erasers), as it follows.

''

Alan s spenging

TAlan s average s

otal numberof buypending

ed items


10

7'

5

5Alan s average spending

' 15 /Alan s average spe zedsnding item

That means that if all 5 items purchased by Alan had the same price, each would cost 15

zeds.

The situation studied is useful in solving questions 1.1 to 1.9, extracted from Pisa and

translated with some adaptations.

QUESTION 1.1.

Theme. Height of 14 steps of a staircase.

Question 1.1 (Pisa). The following picture shows a staircase with 14 steps and a total height of

252cm.

What is the height of each of the 14 steps of the staircase?

Resolution of question 1.1.

We will assume that all stair steps have the same height.

Let's call “h” the height of each of the 14 steps of the full height staircase equal to 252cm, as

shown in the following picture.


11

From the picture, we see that:

252 hhhhhhhhhhhhhh

14. 252h

1814

252 hh

We conclude that each step is 18cm.

Comments on the resolution of question. 1.1.

To resolve question 1.1, the student should:

make the hypothesis that all steps have the same height, which is quite plausible because, in

general, this is how stairs are made, similar to the staircase of the picture showed;

realize that the sum of 14 equal parts to “h” is the same as the multiplier “h” by 14.

QUESTION 1.2.

Theme. Saving weekly receipts for holiday travel.

Question 1.2 (Pisa). Natasha works at a restaurant for 3 nights a week and 4 hours a night.

She makes 10 zeds per hour. She also gets 80 zeds a week on tips. Natasha saves exactly half of

her weekly earnings because she wants to save 600 zeds for her vacation trip.

How many weeks does Natasha have to work to save the 600 zeds?


12


Disconsidering tips, Natasha makes 40 zeds per night, which is the result of multiplying 4 hours

of work each night by 10 zeds per each hour worked:

' 4 10 4( ) 0Natasha s earning per night disconsidz

ering teds

hours zedsh

ipr

sou

Disconsidering tips, Natasha makes 120 zeds per week, which is the result of multiplying 3 nights

of work each week by 40 zeds per each night worked:

3 40 12' ( ) 0Natasha s earning per week disconsiderzeds

nights zedsn

ing tipsight

In addition to these 120 zeds, Natasha makes 80 zeds on tips per week. So her total earning per

week is 200 zeds, which is the result of adding 120 zeds and 80 zeds:

' ( ) 120 80 200Natasha s earning per week counti zeds zedsng tip edss z

Natasha saves half of what she earns per week to make her vacation trip. So she saves 100 zeds

per week, which is the result of dividing 200 zeds by 2:

200100

2Weekly savings zeds zeds

For Natasha to save 600 zeds, she needs to work 6 weeks, as she saves 100 zeds per week:

600 100 6zeds

zeds weekweek

We conclude that the answer is 6 weeks.

Comments on the resolution of question 1.2.



13

recognize and relate the value received per hour, the number of hours worked per night and

the number of nights worked per week;

check that the received tip must be added to the weekly earnings;

realize the existence of multiple integer values, such as 6 weeks of work to save 600 zeds, as

Natasha makes 100 zeds per week.

QUESTION 1.3.

Theme. Recipe for salad dressing.

Question 1.3 (Pisa). Imagine you are making your own dressing for a salad. Find below the

recipe for preparing 100 milliliters (mL) of that dressing.

Salad oil 60mL

Vinegar 30mL

Soy sauce 10mL

How many milliliters (mL) of salad oil do you need to make 150mL of the dressing?


The recipe on the table refers to the preparation of 100mL of dressing. But you want to make

150mL of dressing. Therefore, the volumes of ingredients should be affected by the multiplication

per factor 1.5, as 150mL is the multiplication of 1.5 per 100mL.

Applying this 1.5 multiplication factor to all ingredients, we have the following table.

Salad oil 60mL 1.5x60 = 90mL

Vinegar 30mL 1.5x30 = 45mL

Soy sauce 10mL 1.5x10 = 15mL

Volume of dressing 60+30+10 = 100mL 90+45+15 = 150mL = 1,5x100

From the table, we learn that we need 90mL of salad oil to make 150mL of dressing.

We conclude that the answer is 90mL of salad oil.



differentiate the volume of a component (salad oil) from the total volume of the dressing;


14

realize the existence of multiples of integer values, as in the case of the volume of 150mL is

the multiplication of factor 1,5 by the volume of 100mL.

QUESTION 1.4.

Theme. Manufacture of shelf assemblies.

Question 1.4 (Pisa). To make a set of shelves like the one shown in the picture below, a joiner

needs the following items: 4 long wood panels, 6 short wood panels, 12 small nails, 2 large nails

and 14 screws.

The joiner has in stock 26 long wood panels, 35 short wood panels, 200 small nails, 20 large nails

and 510 screws.

With this material, how many sets of shelves can the joiner do?


We can summarize the data provided by the question in the following table.

Item Quantity of item needed to

make one set of shelves

Quantity of the item that

the joiner has in stock

Long wood panel 4 26

Short wood panel 6 35

Small nail 12 200

Large nail 2 20

Screw 14 510


15

If we want to calculate the number of shelf assemblies that we can make with the quantities of

items that the joiner has in stock, we just divide those quantities by the amount of items needed

to make one set of shelves. We should consider the whole values of these divisions "rounded

down" as there is no sense in making, for example, "16.7 shelves."

Let's look at the following table in which we use the following symbols.

• Qc: quantity of the item needed to make one set of shelves.

• Qm: quantity of the item that the joiner has in stock.

• Nc: possible number of sets of shelves you can make with the quantity of the item that the

joiner has in stock.

Item Qc Qm Division: Qc

Qm Nc

Long wood panel 4 26 26

6.54 6

Short wood panel 6 35 35

5.86 5

Small nail 12 200 200

16.712

16

Large nail 2 20 102

20 10

Screw 14 510 510

36.414

36

See, for example, there are sufficient quantity of long wooden panels, short wooden panels and

large nails to make the 16 sets that the amount of small nails would allow.

Therefore, based on the calculations made, we observe that the limiter for the construction of the

shelf assembly is the short wooden panel, because, with the quantity of this item available in

stock, the joiner makes only 5 sets of shelves.

If the joiner makes 5 sets of shelves, there will be leftovers of materials, but they are not enough

to make the 6th set of shelves.

We conclude that the answer is 5 sets of shelves.




16

understand that the entire part of the division of the quantity of the item that the joiner has

in stock by the amount of item needed to make a set of shelves provides the number of sets

of shelves that can be made with that item;

recognize that there is an item that limits the number of shelf assemblies that can be made

with the available stock of parts.

QUESTION 1.5.

Theme. Choice of pizza toppings.

Question 1.5 (Pisa). In a pizzeria, the basic pizza comes with two toppings: cheese and

tomato. You can also order your own pizza with extra toppings. There are four options of extra

toppings: olive, ham, mushroom and salami.

Imagine that Ross wants to order a pizza with two different extra toppings. How many different

combinations of pizza can Ross choose?


With the four options of extra toppings (olive, ham, mushroom and salami), Ross can make the 6

different pizza combinations with two extra toppings listed in the table below.

Combination Top 1 Top 2

Combination 1 Olive Ham

Combination 2 Olive Mushroom

Combination 3 Olive Salami

Combination 4 Ham Mushroom

Combination 5 Ham Salami

Combination 6 Mushroom Salami

We learn from the table that there are 6 possibilities of combinations. When the amount of items

to be combined is too large, it may be exhaustive to enumerate, one by one, all results of

possible combinations.

So if we only want to calculate the number of possible combinations, without specifying them, we

can use multiplication.


17

For the first topping, we have 4 options. For the second one, we have 3 options, because we

have already used one of the 4 initial options on the first topping. If we multiply 4 by 3, we will

have 12 combinations. At this value of 12, for example "olive topping 1 and ham topping 2" is

counted as a different situation from "ham topping 1 and olive topping 2". But in reality, this

order of topping does not change the type of pizza. That is, the combination given by "olive

topping 1 and ham topping 2" is the same combination given by "ham topping 1 and olive

topping 2". So, to know the actual number of topping combinations, we must divide 12 by 2,

which results in 6.

We conclude that the answer is 6 combinations of toppings.



identify the possible combinations of two extra pizzas toppings;

use multiplication to calculate possible combinations.

QUESTION 1.6.

Theme. Currency exchange rate between currencies of Singapore and South Africa.

Question 1.6 (Pisa). Mei Ling, from Singapore, is preparing to go to South Africa for 3 months

as an exchange student. So she needs to exchange some Singapore Dollars (SGD) for South

Africa currency (ZAR).

Based on this situation, answer the questions in the following items.

a) Mei Ling verifies that the exchange ratio between Singapore dollars (SGD) and South Africa

currency (ZAR) is: 1 SGD = 4.2 ZAR.

The student exchanges 3,000 SGD for ZAR. How much does Mei Ling get in South African

currency?

b) On her return to Singapore, after the 3 months of exchange, Mei Ling brings 3,900 ZAR. She

exchanges this amount for Singapore dollars according to this exchange rate: 1 SGD = 4.0 ZAR.

How much money, in Singapore currency, does Mei Ling receive in this exchange?

c) During the three months of exchange, the exchange rate ranged from 4.2 ZAR per SGD to 4.0

ZAR per SGD. Has this rate change been favorable to Mei Ling when she exchanged her South


18

African currencies again for Singapore dollars? Give an explanation that justifies your answer.


Item a)

When Mei Ling went to South Africa, 1 Singapore dollar (1 SGD) was equivalent to 4.2 South

African currency (4.2 ZAR): 1 SGD = 4.2 ZAR.

Mei Ling exchanged 3,000 SGD per X ZAR, according to the rate already given. To calculate the X

value of ZAR currency that was purchased with 3,000 SGD, we can make as below.

1 4.2

3,000

SGD ZAR

SGD X ZAR

To find the result of X, we make the following equality, coming from the “cross-multiplication” of

the values present in the previous scheme:

1 4.2 3.000xX x

12,600X ZAR

We conclude that Mei Ling got the value of 12,600 ZAR on the way to South Africa.

Item b)

Upon Mei Ling's return to Singapore, Singapore dollar (1 SGD) was equivalent to 4.0 South

African currency (4.0 ZAR): 1 SGD = 4.0 ZAR.

On her return, Mei Ling brought 3,900 ZAR and exchanged them for Y SGD.

To calculate the Y value of SGD that was exchanged with 3,900 ZAR, we can make as below.

1 4.0

3,900

SGD ZAR

Y SGD ZAR

To find the result Y, we make the following equality, coming from the “cross-multiplication” of the

values present in the previous scheme:

4 1 3,900xY x


19

3,900

4Y

975Y SGD

We conclude that Mei Ling obtained the value of 975 SGD upon returning to Singapore.

Item c)

Mei Ling was favored by the exchange rate that ranged from 4.2 ZAR per SGD to 4.0 ZAR per

SGD.

This reduction on the exchange rate made Mei Ling, in her return, get more Singapore dollars

with the currencies she brought from South Africa than she would have obtained with the first

quote.

On this way, she received 4.2 ZAR for 1 SGD. In her return, she paid only 4.0 ZAR for 1 SGD.

Therefore, on the return of Mei Ling to Singapore, each SGD was 0.2 ZAR cheaper than the

amount practiced when she went to South Africa.



recognize and operationalize relationships that change over time regarding the equivalences

of SGD and ZAR currencies;

evaluate the impact of the changes over time between the SGD and ZAR currency equivalents

in monetary exchanges.

QUESTION 1.7.

Theme. Options for parts for skateboarding.

Question 1.7 (Pisa). Eric, a big fan of skateboards, goes to a store called "SKATERS" to check

out some prices. In this shop, you can buy a complete skateboard or you can buy a base, a set of

4 wheels, a set of 2 trucks and a set of hardware to assemble your own skateboard. The prices of

these items are shown in the following table.


20

Product/part Price (zeds) Picture

Complete skateboard 82 or 84

Base 40, 60 or 65

Wheel set 14 or 36

Truck set 16

Hardware set 10 or 20

Based on this situation, answer the questions in the following items.

a) Eric wants to set up his own skateboard. In that case, what are the minimum price and the

maximum price that he will pay for the skateboard?

b) "SKATERS" shop offers 3 different bases, 2 different wheel sets and 2 different hardware sets.

There is only one option for the base. In that case, how many different types of skateboards can

Eric build?

c) Eric has 120 zeds to spend and he wants to buy the most expensive skateboard possible with

this value. In that case, how much Eric will spend on each of the four parts that make up the

skateboard? Put your answer in the table below.

Part Value (zeds)

Base

Wheel set

Truck set

Hardware set


21


Item a)

In the table below, there are the minimum and maximum values of each product (part) to mount

the skateboard. Our intention is to sort out the interesting data to resolve the problem.

Part Minimum value (zeds) Maximum value (zeds)

Base 40 65

Wheel set 14 36

Truck set 16 16

Hardware set 10 20

If we add the minimum values of each product (40, 14, 16 and 10), we have the minimum price

of 80 zeds:

40 14 16 10 80 zedsMinimum price

If we add the maximum values of each product (65, 36, 16 and 20), we have the maximum price

of 137 zeds:

65 36 16 20 137Maximum pri zedsce

Item b)

Let's call the 3 different bases of B1, B2 and B3; the 2 different sets of wheels of R1 and R2; and

the 2 different sets of hardware F1 and F2.

Based on this, we can have the skateboard mounting combinations shown in the following table.

Combination Base Wheels Hardware

Combination 1 B1 R1 F1









22

Combinação 9 B3 R1 F1




From the table, we see that there are 12 different types of skateboards that can be assembled.

Alternatively, we might think so: there are 3 different bases, 2 different wheel sets and 2

different sets of hardware. The product of these quantities (3.2.2 = 12) provides the possible

combinations of skateboards.

We conclude that the answer is 12 different types of skateboards.

Item c)

In the table below, we have the prices for each part of the skateboard for the 12 possible options

of being assembled. In the last column, we have the total price of each of the options.

Option Base

(zeds)

Wheels

(zeds)

Trucks

(zeds)

Hardwares

(zeds)

Total value

(zeds)

Option 1 40 14 16 10 80

Option 2 40 14 16 20 90

Option 3 40 36 16 10 102

Option 4 40 36 16 20 112

Option 5 60 14 16 10 100

Option 6 60 14 16 20 110

Option 7 60 36 16 10 122

Option 8 60 36 16 20 132

Option 9 65 14 16 10 104

Option 10 65 14 16 20 114

Option 11 65 36 16 10 127

Option 12 65 36 16 20 137

If Eric has 120 zeds to spend and he wants to buy the most expensive skateboard possible, then,

seeing the table, we detect that the option that comes closest to this value, without exceeding it,

is option 10, with a total price of 114 zeds. This option is shown in the following table.


23

Part Value (zeds)

Base 65

Wheel set 14

Truck set 16

Hardware set 20



add parts prices to check the total assembly price of a skateboard;

check the possible combinations of parts for assembling a skateboard;

calculate the minimum price and the maximum price when you are assembling a skateboard;

evaluate the impacts of the prices of the parts on the final price of a skateboard;

sort out the interesting data to solve a specific problem.

QUESTION 1.8.

Theme. Consequences of having loan for a long period of time.

Question 1.8 (Pisa). Mrs. Jones has a loan of 8,000 zeds with First Zed Finance. The annual

interest rate on the loan is 15%. Her repayments are 150 zeds each month. After one year, Mrs.

Jones still owes 7,400 zeds. Another finance company called Zed Best will give Mrs. Jones a loan

of 10,000 zeds with an annual interest rate of 13%. Her repayments would also be 150 zeds each

month.

What is one possible negative financial consequence for Mrs. Jones if she agrees to the Zed Best loan?


First, let's analyze the situation of Mrs. Jones with First Zed Finance after a year of having a loan

of 8,000.

In the 8,000 zed loan that Mrs. Jones borrowed from First Zed with an annual interest rate of

15%, the absolute interest in one year was 1,200 zeds, as the result of 15 percent of 8,000 is

1,200:

1515% 8.000 8.000 0,15 8.000 1.200

100of zeds zeds


24

So in the end of the first year, the debt amount went from 8,000 zeds to 9,200 zeds, which is the

sum of the initial 8,000 zeds with the absolute interest of 1,200 zeds:

zedszedszeds 200.9200.1000.8

As Mrs. Jones made 12 monthly payments of 150 zeds to First Zed in a year, she paid 1,800 zeds

to this financial institution, which is the multiplication of 12 by 150:

zedszeds 800.115012

If we remove the 1,800 zeds Mrs. Jones paid to First Zed from the due amount of 9,200 zeds, we

see that in the end of a year she still owes 7,400 zeds to the financial intitution:


Now, let's analyse the situation of Mrs. Jones with Zed Best Finance if she takes on a new loan. If

Mrs. Jones borrows 10,000 zeds from Best Zed, with an interest rate of 13 percent a year, the

absolute value of interest in a year would be 1,300 zeds, as the result of 13 percent of 10,000 is

1,300:

1313% 10.000 10.000 0,13 10.000 1.300

100of zeds zeds

So in the end of the first year, the debt amount will go from 10,000 zeds to 11,300 zeds, which is

the sum of the initial 10,000 zeds with the absolute interest of 1,300 zeds:


As Mrs. Jones will make 12 monthly payments of 150 zeds to Zed Best in one year, she will have

paid 1,800 zeds to this financial institution, which is the result of multiplying 12 by 150:

zedszeds 800.115012

If we remove the 1,800 zeds that Mrs. Jones will have paid to Best Zed from the due amount of

11,300 zeds, we see that at the end of a year she still owes 9,500 zeds to the financial:


25


We conclude that if Mrs. Jones accepts the new loan, offered by Zed Best Finance, she will have

more debts to pay, she will take more time to repay the full debt and she will pay higher absolute

interest. That is, there are already negative consequences for the interest paid only with the first

loan, which will be aggravated by having a new loan.



make calculations involving percentages;

distinguish interest rate from absolute value of interest;

evaluate critically the impacts of having loan for a long period of time and/or high interest

rates.

QUESTÃO 1.9.

Theme. Insertion of kite sail to NewWave to reduce diesel consumption.

Question 1.9 (Pisa). Due to the high cost of diesel as fuel of 0.42 zeds per liter, the owners of

NewWave, whose characteristics are listed in the following chart, are thinking of equipping their

ship with a kite sail. It is estimated that this kite sail has the potential to reduce diesel

consumption by about 20%.

The cost to equip NewWave with the kite sail is 2,500,000 zeds. After how many years will the

values of diesel fuel savings cover the cost of kite sailing? Make calculations to justify your

answer.


26


Without the sail, NewWave consumes, per year, 3,500,000 liters of diesel, which costs 0.42

zeds/liter.

To calculate the X value of zeds spent per year on diesel to supply NewWave without sail, we can

make the scheme below.

0,42 1

3.500.000

zed liter

X zed liters

To find the result of X, we make the following equality, coming from the “cross-multiplication” of

the values in the scheme:

1 0.42 3,500,000xX x

1,470,000X zeds

The insertion of the sail into the ship can reduce the diesel consumption by about 20%.

Therefore, this installation will annually reduce diesel consumption by 700,000 liters, because 20

percent of 3,500,000 liters result in 700,000 liters:

2020% 3.500.000 3,500,000 700,000

100of liters x liters

To calculate the Y value of zeds saved per year in diesel to power NewWave with sail, we can

make the scheme below.

0.42 1

700,000

zed liter

Y zed liters

To find the result of Y, we make the following equality, coming from the “cross-multiplication" of

the values in the scheme:

1 0.42 700,000xY x


27

294,000Y zeds

The cost to equip NewWave with the sail is 2,500,000 zeds. If we divide this amount by the

amount saved annually with the diesel consumption, which is 294,000 zeds, we have the amount

of years necessary to cover the cost of the investment, not considering inflation effects.

Let's see:

294,0008.5

3,500,000

We concluded that, without including any inflation occurring in the period, after 8 to 9 years of

installation of the sail, the cost of the sail is recovered as consequence of the economy in the

consumption of diesel in NewWave.




understand the economic impact over the diesel consumption with the installation of a sail to

a ship;

assess the period of time required to recover financially an investment.


28

CHAPTER 2. Reading and interpreting tables and graphs.

In many communication vehicles, such as magazines and newspapers, and in many online

disclosures, such as news feeds, we come across information given in tables and graphs. They

can represent a lot of data succinctly.

It is not enough just to look at the contents displayed in tables and graphs: we need to

interpret them. Let's look at an example of this kind of interpretation.

Example. Pisa, an international exam developed and applied by the Organization for Economic

Cooperation and Development (OECD), assesses the literacy of students at the age of 15 in the

areas of Mathematics, Science and Reading. Pisa scores are presented according to the

performance scale of the table below, which shows six levels of proficiency.

Performance scale of Pisa participants.

Level Score

1 From 358 to 420 points





6 More than 669 points

Students with less than 358 points are classified as "below level 1" because they cannot

understand even the simplest headings in the exam. Pisa periodicity is triennial, and the following

graph shows the Mathematics scores of some countries that participated in all editions of Pisa.

Scores of some countries in Mathematics in various editions of Pisa.

Can we say that Brazil has already presented, in Pisa, results in Mathematics

classified as "below level 1", on a scale of 1 to 6?


29

Yes. From the graph, we can see that, in 2000, Brazil's score in Mathematics in Pisa was lower

than 358 points, which, according to the performance table, classifies it as "below level 1 ".

Can we assure that the results in Mathematics of Brazil and Mexico in the various

editions of Pisa have never exceeded level 1, in a scale of 1 to 6, and that these

countries have always maintained growth in scores?

The first assertion can be considered true, because, by analyzing the graph, we see that both

Brazil and Mexico never reached 420 points in Mathematics, minimum score, according to the

given performance table, to rank a country in "level 1".

The second assertion is false because, from the analysis of the graph, we see that from 2012 to

2015 both Brazil and Mexico had decreased in Mathematics scores in Pisa.

Can we say that Poland presented significant growth in Mathematics scores in Pisa

from 2009 to 2012, but nevertheless has never surpassed the performance of China

(Hong Kong)?

Yes. From the graph, we can see that Poland presented significant growth in Mathematics scores

in Pisa from 2009 to 2012, going from almost 500 points to about 520 points. China (Hong Kong)

has always had about 550 points or more.

The situation studied is useful to assist in the resolution of questions 2.1 to 2.11, extracted from

Pisa and translated with some adaptations.

QUESTION 2.1.

Theme. Variation in prices of one Rich Rock share.

Question 2.1 (Pisa). The following graph shows the price variation (in zeds) of one Rich Rock

share over a 12-month period.


30

Based on data from the Rich Rock share graph, highlight the "True" or "False" indication for the

following statements.

Statement “True” or “False”?

The best month to buy stocks was September. “True”/“False”

The share price rose about 50% in a year. “True”/“False”


Analysis of the first statement.

By reading the graph, we see that the lowest value for which one Rich Rock share was sold

occurred in September and it was about 1.70 zeds. So September was the best month to buy

shares from that company.

Threfore, the first statement is "True."

Analysis of the second statement.

By reading the graph, we see that, in June, once Rich Rock share was worth the "starting price"

of 2.00 zeds. After a year, in May, this one Rich Rock share was worth the "final price" of 2.50

zeds. Thus, in this period, this action increased by 25%:

" " " "100%

" "

final price initial pricePercentage increase x

initial price

2.50 2.00100%

2.00Percentage increase x

0.50100% 25%

2.00Percentage increase x

Threfore, the second statement is "False."

Table with responses.

Statement “True” or “False”?

The best month to buy stocks was September. “True”/“False”

The share price rose about 50% in a year. “True”/“False”


31



read price values of a stock given in a graph;

identify the month in which the minimum price of a share occurs;

calculate the percentage increase in the price of a share in the period of one year.

QUESTION 2.2.

Theme. Monthly sales of CDs from 4U2Rock, The Kicking Kangaroos, One's Darling

and The Metalfolkies.

Question 2.2 (Pisa). In January, new CDs of the bands 4U2Rock and The Kicking Kangaroos

were released. In February, new CDs from No One's Darling and The Metalfolkies were released.

The graph below shows the sales of these bands' CDs from January to June.

Based on the graph, answer the following items.

a) How many CDs did The Metalfolkies sell in April?

b) In which month, for the first time, did No One's Darling band sell more CDs than The Kicking

Kangaroos band?


32

c) The manager of The Kicking Kangaroos is worried because the number of CDs this band sold

decreased from February to June. What is the estimated amount of CDs sold by The Kicking

Kangaroos in July if the same rate continues?


Item a)

By reading the graph, as highlighted below, we see that the band The Metalfolkies sold 500 CDs

in April.

Item b)

By reading the chart, as highlighted in the following picture, we see that in April, for the first

time, the band No One's Darling sold more CDs than the band The Kicking Kangaroos.


33

Item c)

By reading the chart, we see that the band Kicking Kangaroos sold about 1,850 CDs in February

and 680 CDs in June, as summarized in the table below.

Month Number of CDs sold by The Kicking Kangaroos

February 1,850

June 680

The monthly average decrease rate on the number of CDs sold by The Kicking Kangaroos can be

calculated as the division of the variation of the number of CDs sold from February to June for

the number of months from February to June, as it follows.

" " " "sales in February salesAverage decrease rat

in June

number of mo hse

nt

1,850 680293

4Average decrease rate


34

So, on average, The Kicking Kangaroos sold about 293 CDs less per month between February

and June. If this is kept from June to July, in July, the band will sell 293 CDs less than it sold in

June.

As in June the Kicking Kangaroos sold about 680 CDs, it is estimated that in July the band will sell

approximately 387 CDs, which is the result of subtraction 680-293.



read, graphically, the numbers of CDs sold by 4 bands from February to June;

calculate the average decrease rate of the number of CDs sold by The Kicking Kangaroos

from February to June;

estimate the number of CDs sold by The Kicking Kangaroos band in July.

QUESTION 2.3.

Theme. Average heights of women and young men in Netherlands in 1998.

Question 2.3 (Pisa). The chart below shows the average heights of women and young men in

1998 in Netherlands.


35

Based on the exposed situation, answer the following items.

a) In Netherlands, from 1980 to 1998, the average height of 20-year-old women increased by

2.3cm, to 170.6cm in 1998. What was the average height of 20-year-old women in 1980?

b) Explain, according to the graph, how the growth rate of girls varies after 12 years of age.

c) According to the graph, on average, during what period of life are women taller than men,

both being at the same age?


Item a)

In the Netherlands, from 1980 to 1998, the average height of 20-year-old women increased by

2.3cm, reaching 170.6cm in 1998. That is, in 1980, this average height was 168.3cm, which is

the result of subtraction 170.6-2.3=168.3.

Item b)

In the picture below, we highlighted the average height of women as a based on age.

We see that, from 12 to 20 years old, although the height of women increases with age, at this

period, the rate of height increased with age is lower than the rate of height increased with age

from 10 to 12 years old.

Item c)

In the following picture, we have highlights of the ages among which the average height of

women exceeds the average height of men: this occurs between 11 and 13 years old.


36



read, on a graph, average values of women and men heights, according to age, in

Netherlands in 1998;

evaluate variations in the rates of increase in the average height of women in certain periods;

check the age range in which the average height of women exceeds the average height of

men.

QUESTION 2.4.

Theme. Reporter's statement and correct chart analysis of robberies in 1998 and

1999.

Question 2.4 (Pisa). A TV reporter showed the following picture and said as follow.

The graph shows that there is a huge increase in the number of robberies from 1998

to 1999.


37

Do you consider the reporter's statement to be a reasonable interpretation of the graph? Give an

explanation that justifies your answer.


The reporter's statement is unreasonable. The justifications for this are in the following analyzes,

regarding characteristics of the graph and aspects in the reporter's statement.

Characteristics of the graph.

There are some important features to be observed in the graph, such as those below.

• The data in the graph refer to absolute values, but no relative values. As we can see below, we

had about 508 robberies in 1998 and 516 robberies in 1999.


38

• The chart scale does not start at zero (there is a "break" indicated by a singularity on the

vertical axis).

• The scale is linear and varies from "5 by 5".

• The percentage increase in robberies was 1.6%, as we calculated below.

1999 1998% 100%

1998

Number of robberies in Number of robberies inIncrease x

Number of robberies in

516 508% 100% 1.6%

508Increase x

Content of the reporter's statement.

There are also some aspects to be considered in the reporter's statement, such as those below.

• Use of the adjective "gigantic".

• Biased opinion.

• Misinterpretation of data.

• Possible induction of public opinion.

Conclusion.

The reporter's statement does not correspond to a reasonable interpretation of the data

presented in the graph.




distinguish absolute value from relative value of robberies;

check a graph scale;

evaluate a reporter's statement.

QUESTION 2.5.

Theme. Tree diagram in the structure of the workforce in a country in 1995.

Question 2.5 (Pisa). The tree diagram below shows the structure of a country’s labor force or

“working-age population”. The total population of the country in 1995 was about 3.4 million.


39

The Labor Force Structure year ended 31 March 1995 (000s)1

Notes.

1. Numbers of people are given in thousands (000s).

2. The working-age population is defined by persons between 15 and 65-years old.

3. People "Not in labor force" are those not really looking for work an/ or are not available to work.

Source. Miller, D. Form 6 Economics. ESA Publications. Box 9453. Newmarker, Auckland, NZ, p.64.


a) What are the two main groups into which the working-age population is divided?

A. Employed and unemployed.

B. Of working age and not of working age.

C. Full-time workers and part-time workers.

D. In the labor force and not in the labor force.

b) How many people of working age were not in the labor force? Write the absolute number of

people, not the percentage.

c) In which part of the tree diagram, if any, would each of the people listed in the table below be

included?

The first one has been done for you.


40

d) Suppose that information about the labor force was presented in a tree diagram like this every

year.

Listed below are four features of the tree diagram. Show whether or not you would expect these

features to change from year to year, by reforcing either “Change” or “No change”. The first one

has been done for you.

Features of tree diagram Answer

The labels in each box (e.g. “In labor force”) Change /No change

The percentages (e.g. “64.2%”) Change /No change

The numbers (e.g. “2656.5”) Change /No change

The footnotes under the tree diagram Change /No change

e) The information about the labor force structure is presented as a tree diagram, but it could

have been presented in a number of other ways, such as a written description, a pie chart, a

graph or a table.

The tree diagram was probably chosen because it is especially useful for showing

A. changes over time.

B. the size of the country’s total population.

C. categories within each group.

D. the size of each group.


41


Item a)

As you can see from the diagram below, the two main groups are “in labor force” people and “not

in labor force” people.

Item b)

As you can see from the diagram below, the absolute number of persons of working-age who are

not classified in the workforce category (“not in labor force” people) is 949,900.

See that the numbers of people are given in thousands (000s). Therefore, the reading of 949.9 in

the diagram box corresponds to 949,900.

Item c)

In the following table, we indicate in which parts of the tree diagram, if any, each person will be

included.


42

Item d)

We assume that information about the labor force is given each year in a tree diagram as

presented.

Below we listed four features used for plotting the tree diagram. We also indicated whether or

not these resources are expected to change from year to year.

Features of tree diagram Answer

The labels in each box (e.g. “In labor force”) Change /No change

The percentages (e.g. “64.2%”) Change /No change

The numbers (e.g. “2656.5”) Change /No change

The footnotes under the tree diagram Change /No change

The names in each box of the diagram and the footnotes refer to general definitions and

classifications of workers, which do not vary from year to year. The percentages and the numbers

are values that depend on the reality of a given moment, which vary from year to year.

Item e)

The tree diagram, which presents information divided into classes, is particularly suitable for

displaying categories within groups.

This is the case of the presentation of the structure of the work force in the picture of the

statement in which:

working-age population is divided into workforce people and no workforce people;

workforce people are divided into employed persons and unemployed persons;

employed people are divided into full-time workers and part-time workers;

people who work part-time are divided into people who are looking for full-time work and

people who are not looking for full-time work;

unemployed people are divided into people who are looking for full-time work and people who

are not looking for full-time work.



distinguish absolute value from percentage value;

understand the structure of representations in a tree diagram;

check factors that change and factors that do not change over time in presentations about

the structure of the workforce;

sort out the interesting data in a tree diagram to solve a specific problem.


43

QUESTION 2.6.

Theme. Emissions of carbon dioxide and increase in the temperature of the Earth's

atmosphere.

Question 2.6 (Pisa). Read the text and answer the following questions.

THE GREENHOUSE EFFECT: FACT OR FICTION?

Living things need energy to survive. The energy that sustains life on the Earth comes from the

Sun, which radiates energy into space because it is so hot. A tiny proportion of this energy

reaches the Earth.

The Earth’s atmosphere acts like a protective blanket over the surface of our planet, preventing

the variations in temperature that would exist in an airless world.

Most of the radiated energy coming from the Sun passes through the Earth’s atmosphere. The

Earth absorbs some of this energy, and some is reflected back from the Earth’s surface. Part of

this reflected energy is absorbed by the atmosphere.

As a result of this, the average temperature above the Earth’s surface is higher than it would be if

there was no atmosphere. The Earth’s atmosphere has the same effect as a greenhouse, hence

the term greenhouse effect.

The greenhouse effect is said to have become more pronounced during the twentieth century.

It is a fact that the average temperature of the Earth’s atmosphere has increased. In newspapers

and periodicals the increased carbon dioxide emission is often stated as the main source of the

temperature rise in the twentieth century.

Imagine a student named André becomes interested in the possible relationship between the

average temperature of the Earth’s atmosphere and the carbon dioxide emission on the Earth.

In a library, he comes across the following two graphs.


44


a) André concludes from these two graphs that it is certain that the increase in the average

temperature of the Earth’s atmosphere is due to the increase in the carbon dioxide emission.

What is it about the graphs that supports André’s conclusion?

b) Another student, Jeanne, disagrees with André’s conclusion. She compares the two graphs

and says that some parts of the graphs do not support his conclusion. Give an example of a part

of the graphs that does not support André’s conclusion. Explain your answer.

c) André persists in his conclusion that the average temperature rise of the Earth’s atmosphere is

caused by the increase in the carbon dioxide emission. But Jeanne thinks that his conclusion is

premature. She says: “Before accepting this conclusion, you must be sure that other factors that

could influence the greenhouse effect are constant”. Name one of the factors that Jeanne means.


Item a)

In general, except for some stretches, as in the period from 1940 to 1980, André notes in the

graphs that both the carbon dioxide emission (first graph) and the average temperature of the

Earth's atmosphere (second graph) increase over the years.

Item b)

By reading the graphs, Jeanne notes, for example, that there are stretches of carbon dioxide

emission (first graph) over time without corresponding increase in the average temperature of


45

the Earth's atmosphere (second graph).

Let's look at some specific situations that may not confirm André's conclusions.

• Approximately from 1900 to 1910, there was an increase in the emission of carbon dioxide and

a decrease in the average temperature of the Earth's atmosphere.

• From about 1980 to 1983, there was a drop in the emission of carbon dioxide and an increase

in the average temperature of the Earth's atmosphere.

• Approximately from 1860 to 1900, there were increases and declines in carbon dioxide

emissions and only an increase in the average temperature of the Earth's atmosphere.

• From about 1950 to 1980, there was an increase in the emission of carbon dioxide and

constancy of the average temperature of the Earth's atmosphere.

• In 1940, the average temperature of the Earth's atmosphere was much higher than in 1920,

and from 1920 to 1940 carbon dioxide emissions were similar.

Item c)

Other factors that could influence the greenhouse effect are, for example, the destruction of the

ozone layer by the CFCs (chlorofluorocarbons) from aerosols and the possible increase in the

intensity of solar radiation.



understand and compare graphical information on carbon dioxide emission variation and

average Earth's atmosphere temperature over the years;

cite factors that can cause the greenhouse effect.

QUESTION 2.7.

Theme. Exports from Zedland, a country that uses zeds as currency, from 1996 to

2000.

Question 2.7 (Pisa). The following graphs show information about exports from Zedland, a

country that uses zeds as its currency.


46


a) What was the total value (in millions of zeds) of exports from Zedland in 1998?

b) What was the value of fruit juice exported from Zedland in 2000?


Item a)

As showed in the following graph, in 1998 Zedland exported 27.1 million zeds (27,100,000 zeds).


47

Item b)

From the graph on the left, we can see that, in 2000, Zedland exported 42.1 million zeds

(42,100,000 zeds).

From the graph on the right, we can see that, in 2000, 9% of the total exported by Zedland

corresponded to exports of fruit juice.

Therefore, to know the value (in millions of zeds) that Zedland received in 2000 for exports of

fruit juice, we need to calculate 9 percent of 42,100,000 zeds:

99% 42,100,000 42,100,000 3,789,000

100of zeds x zeds

We conclude that the answer is 3,789,000 zeds or, approximately, 3.8 million zeds.



understand two different graphical representations (bar graph and pie graph);

apply certain percentage of exports to total exports.

QUESTION 2.8.

Theme. Comparison between outward and return carriage routes in which an accident

involving a cat occurs.

Question 2.8 (Pisa). Kelly left home for a drive. During the ride, a cat ran in the front of the

car. Kelly fired the brakes, but ran over the animal. Shaking, she returned home.

The following graphic is a simplified record of car speed during Kelly's driving route.


48


a) What was the maximum speed of the car during the trip?

b) When was the time Kelly set the brakes to try not to run over the cat?

c) Was the Kelly’s journey back home shorter than the trip she made to the place where the cat

accident occurred? Give an explanation that justifies your answer, using the information in the

graph.


Item a)

The maximum speed of the car during the trip was 60km/h, as read in the graph of the the

picture below.

Item b)

At 9:06, Kelly set the brakes to try not to run over the cat, because at that moment, there is a

sudden drop in speed (from 60 km/h to 12 km/h), as shown in the following picture.


49

Item c)

Kelly's journey back home was shorter than the trip she made to where the cat accident

occurred. We can see it in the following picture, where area 2, which corresponds to the return

path, is smaller than the area 1, which corresponds to the outward path.



identify the maximum speed value in a speed graph per time;

check the braking moment on a speed graph by time;

evaluate the space traveled by a car on a speed graph per time.


50

QUESTION 2.9.

Theme. Scores of two groups of students in Science test.

Question 2.9 (Pisa). The following diagram shows the results of two groups of students, called

group A and group B, in a Science test. The mean score for group A is 62 and the mean score for

group B is 64.5. To pass this test, students must score 50 points or more.

Observing the diagram, the teacher says that in the Science test, group B was better than group

A. The students in group A do not agree with their teacher. They try to convince her that group B

may not necessarily have done better.

Give one mathematical argument, using the graph, that the students in group A could use to

prove they are right.


By reading the graph, we can construct the following table.

Score range Number of students in group A Number of students in group B

0 - 9 1 0

10 - 19 0 0

20 - 29 0 0

30 - 39 0 0

40 - 49 0 2

50 - 59 3 1

60 - 69 4 5

70 - 79 2 3

80 - 89 2 1

We see that, although the average score of group A (62 points) is lower than the average score

of group B (64.5 points), more students from group A passed the test.


51

We observed, in group A, only one (1) student failed (who did between 0 and 9 points) and, in

group B, two (2) students failed (who did between 40 and 49 points), because to pass this test,

students must score 50 points or more.

We can also verify that there are more students in group A among those who obtained the

highest grades than in group B (2 students in group A did between 80 and 89 points and only 1

student in group B did between 80 and 89 points).

Thus, there are mathematical arguments extracted from the graph that can be used by group A

to convince the teacher that group B was not better than group A.



read and interpret data from a bar graph;

understand scores expressed in ranges of values;

compare the performance of two groups of students in a Science test using different criteria,

such as average score, number of failing students and number of students with higher scores;

use mathematical arguments to support the idea that one group of students performed better

on one test than another group.

QUESTION 2.10.

Theme. Variations in the levels of Lake Chad and rock art in the Sahara.

Question 2.10 (Pisa). Picture 1 shows changing levels of Lake Chad, in Saharan North Africa.

Lake Chad disappeared completely in about 20,000 BC, during the last Ice Age. In about

11,000 BC it reappeared. Today, its level is about the same as it was in AD 1,000.

Picture 1. Changing levels of Lake Chad, in Saharan North Africa.


52

Picture 2 shows Saharan rock art (ancient drawings or paintings found on the walls of caves) and

changing patterns of wildlife.

Picture 2. Saharan rock art.

Source. Copyright Bartholomew Ltd. 1988. Extracted from The Times Atlas of Archaeology and reproduced by permission of Harper Collins Publishers.


a) What is the depth of Lake Chad today?

b) In what year does the graph in picture 1 begin?

c) Why did the author choose to start the chart at that time?

d) Picture 2 is based on the assumption that

A. the animals in the rock art were present in the area at the time they were drawn.

B. the artists who drew the animals were highly skilled.

C. the artists who drew the animals were able to travel widely.

D. there was no attempt to domesticate the animals which were depicted in the rock art.

e) For this question you need to draw together information from picture 1 and picture 2.

The disappearance of the rhinoceros, hippopotamus and aurochs from Saharan rock art

happened

A. at the beginning of the most recent Ice Age.

B. in the middle of the period when Lake Chad was at its highest level.


53

C. after the level of Lake Chad had been falling for over a thousand years.

D. at the beginning of an uninterrupted dry period.


Item a)

As highlighted in the fowlling picture, we can see that the depth of Lake Chad in AD 1,000 was

about 2 meters. In the beginning of the question, we can read that the level of Lake Chad today

is almost the same as it was in AD 1,000. Therefore, the depth of Lake Chad is about 2 meters

today.

Item b)

As highlighted in the following picture, we can see that the graph of picture 1 begins at about

11,000 BC.


54

Item c)

The author chose to start picture 1 in 11,000 BC, because that was the time when Lake Chad

reappeared. As stated in the beginning of the question, "Lake Chad completely disappeared in

about 20,000 BC ... and in about 11,000 BC it reappeared."

Item d)

Picture 2, which covers the period from 8,000 BC to AD 1,000, is based on the assumption that

the animals in the rock art represented therein were in the Sahara region at the time they were

drawn (alternative A).

Item e)

By reading picture 2, we can see that the disappearance of rhinoceroses, hippopotami and

aurochs in the rock art of the Sahara occurred in about 2,000 BC. By reading picture 1, we can

see that about 2,000 BC corresponds to the date after the level of Lake Chad was falling for over

a thousand years (alternative C).



read and interpret the introductory text of the question and the depth values of Lake Chad

with the time given in the picture;

comprise a graphical scale that uses periods BC and AD;

associate moments in graphic representations of rock art found in the Sahara and of depth

values of Lake Chad.

QUESTION 2.11.

Theme. Time of decomposition of some types of garbage.

Question 2.11 (Pisa). For the accomplishment of a task on the environment, students collected

information about the time of decomposition of some types of trash that people usually generate

and obtained the information presented in the following table.

Type of trash Decomposition time

Banana peel 1 to 3 years

Orange peel 1 to 3 years

Cardboard box 0,5 year

Bubble gum 20 to 25 years

Newspaper A few days

Polystyrene cup More than 100 years


55

One of the students thinks about presenting this information in the form of a bar graph. Give a

reason why a bar graph is inappropriate for the presentation of such data.


There are several reasons why a bar graph is inappropriate for the presentation of the data in the

table. We expose some of them below.

• Since the decomposition time of the various types of garbage has different magnitudes, ranging

from a few days to more than 100 years, the difference between the lengths of the bars in a bar

graph would be very large. For example, if the length of the bar to represent the decomposition

time of the newspaper was on the order of 1 centimeter, the length of the bar to represent the

decomposition time of the polystyrene cup would be on the order of 300 meters.

• Decomposition time of some types of garbage was given in intervals, and it is not possible to

make a bar equivalent to "indeterminate values" such as "between 1 and 3 years" or "over 100

years".



observe the different orders of magnitude of the decomposition time of the various types of

waste;

check that the decomposition time of some types of garbage was given in intervals;

evaluate the characteristics of the data in the table and its inadequacy for presentation in a

bar graph.


56

CHAPTER 3. Proposition, interpretation and application of mathematical formulas.

A mathematical formula can "say a lot" concisely. Let's look at an example.

The kinetic energy of a moving body is defined as the half of the product (multiplication) of the

mass of that body multiplied by the square of its velocity.

Could we write this definition "more directly" by a formula or an equation?

Yes. For this, we need symbols to represent the quantities involved in the calculation of

kinetic energy. We can use, for example, the following symbols.

Ec: kinetic energy of the body, in Joules (J).

m: mass of the body, in kg (kilograms).

v: speed of the body speed, in m/s (meters per second).

Since the kinetic energy (Ec) of a moving body is defined as the half of the multiplication

of the mass (m) of that body by the square of its velocity (v), we have the following formula:

2.

2C

m vE

For example, consider a billiard ball of mass (m) equal to half a kilo (0.5 kg) that moves at

velocity (v) of 0.6 m/s. What is the kinetic energy of this ball?

Photo. Flickr/ Ruth_W – CC BY-NC-ND 2.0

By the formula, we can calculate the kinetic energy of the billiard ball (Ecb), that is 0.09J:

2 2. 0.5 (0.6)0.09

2 2Cb

m v xE J

Now imagine that a truck of mass m, without load, is in a straight and uniform motion

with velocity v. In this case, the kinetic energy of the truck (Ec1) is:


57

2

. 2

1

vmEC

If this truck receives a load, so that its mass is doubled, without any change in velocity,

what will its kinetic energy (Ec2) be?

As the mass of the truck is doubled, it goes from m to 2m. The speed remains v.

Therefore, the kinetic energy Ec2 is twice the kinetic energy Ec1, since

2

1

.

2C

m vE :

2 2

2

2 . .2.

2 2C

m v m vE

2 12.C CE E

Think back to the mass truck without loading. If it doubles the speed, what will its kinetic

energy Ec3 be?

As the speed of the truck is doubled, it goes from v to 2v. The mass remains m.

Therefore, the kinetic energy Ec3 is the quadruple of the kinetic energy Ec1, since

2

1

.

2C

m vE :

2 2 2 2 2

3

.(2 ) .2 . .4. .4.

2 2 2 2C

m v m v m v m vE

3 14.C CE E

See, for example, that the influence of mass change on kinetic energy is "direct", but the

influence of velocity change on kinetic energy has a "squared" impact.

The situation studied is useful to assist in the resolution of questions 3.1 to 3.4, extracted

from Pisa and translated with some adaptations.

QUESTION 3.1.

Theme. Drug administration by intravenous drip.

Question 3.1 (Pisa). Intravenous drips are used to deliver fluids and drugs to patients, as

shown below.


58

Nurses need to calculate the drip rate, D, in drops per minute, for intravenous drips. To do this,

they use the following formula:

n

vdD

.60

.

In this expression, we have what follows.

d is the drip factor, measured in drops per mL.

v is the intravenous drip volume in mL.

n is the number of the intravenous drips required to run.


a) A nurse wants to double the time an intravenous drip takes to run. Based on this intention,

describe how D changes if n is duplicated (doubled), but d and v do not change.

b) Nurses also need to calculate the volume v of intravenous drip, which runs from the drip rate

D. Imagine that an intravenous drip with a drip rate of 50 drops per minute has to be given to a

patient for 3 hours. For this intravenous drip the drop factor is 25 drops per milliliter (25

drops/mL). What is the volume in mL of the intravenous drip?


First, we need to say that a mathematical equation is a sentence that states something with

symbols.


59

Let's see what the equation given in this exercise means: the drip rate (D, in drops per minute) is

equal the multiplication of the drip factor (d, drops per mL) per the intravenous drip volume (v, in

mL) divided by the multiplication of 60 per the number the intravenous drip required to run (n).

Item a)

If a nurse wants to double the time that an intravenous drip takes to run, she needs to make the

number of hours needed for the drip to end to go from n to 2n. It is reported that the drip factor

d and the intravenous drip volume v do not change.

When the drip time is n, the drip rate D1 is given by:

n

vdD

.60

.1

When the drip time is 2n, the drip rate D2 is given by:

2

. 1 ..

60.2 2 60.

d v d vD

n n

2

.. 60.

60.2 2

d vd v nD

n

We can see, regarding the formulas above, 1

2 multiplied by

.

60.

d v

n is equal a half of

.

60.

d v

n.

Since .

60.

d v

n is equal D1, we can write:

12

.

60.2 2

d vDnD

Therefore, if the time an intravenous drip doubled, with no changes in other parameters, the drip

rate becomes half of what it was.

Item b)

An intravenous drip with D drop rate of 50 drops per minute should be administered to a patient

for the time n of 3 hours, with d drop factor of 25 drops per mL. That is, we have what follows.


60

• D = 50 drops/min

• n = 3 hours

• d = 25 drops/mL

We see that D, n and d are already in the units specified in the statement. So:

60

d vD

n

2550

60 3

v

50 60 3 25 v

9,000 25 v

9,000360

25v v mL

Therefore, the intravenous drip volume is 360mL.



observe how the variables behave in a formula for calculating the rate of intravenous drip;

perform intravenous drip volume calculation.

QUESTION 3.2.

Theme. Relation between the number of steps per minute and the length of the step

of a man walking.

Question 3.2 (Pisa). In the following picture, we have footprints of a man walking. The

measure P is the distance between the "backs" of two consecutive footprints.


61

For men, the formula

140P

n provides an approximation for the relationship between n and

P , being n the number of steps per minute and P the pacelength (in meters).


a) Imagine that the formula is applied to Heiko walking. He walks at 70 steps per minute. What

is Heiko’s pacelength?

b) Imagine that the formula is applied to Bernard. Bernard knows his pacelength is 0.80 meters.

What is Bernard's walking speed in m/s and in km/h?


Item a)

The number n

of steps per minute that Heiko performs while walking is 70 steps per minute.

We apply the formula

140P

n to Heiko's case to calculate his pacelength (in meters). So:

14070

140 PP

n

P14070

700.5

140P P m

Therefore, Heiko’s pacelength is half meter (0.5m).


62

Item b)

We apply the formula

140P

n to Bernard's case. Bernard’s pacelength is 0.80 meters. So:

14080,0

140 n

P

n

140 0.80 112 / minn n steps

So Bernard walks 112 steps per minute.

As each step Bernard measures 0.8 meters, his walking speed is 89.6 meters per minute,

because:

112 112 0,80min min

steps m

112 89,6min min

steps m

We have to calculate this speed in km/h. To make this conversion, we must keep in mind that 1

meter equals 1/1,000 kilometers and 1 minute equals 1/60 hours.

Thus Bernard's walking speed is 5,376 kilometers per hour, because:

1601.00089,6 89,6 89,6

1min 1.000

60

kmm km

hh

89,6 5,376min

m km

h



observe how the variables behave in a formula that provides an approximation for the

relationship between the number of steps per minute and the length of a person's step;

perform unit conversions (from m/s to km/h).


63

QUESTION 3.3.

Theme. Punctuation system for car classification.

Question 3.3 (Pisa). A car shop uses a rating system to evaluate new cars, and provides the

title of "The Car of the Year" for the highest rated auto. Five cars (Ca, M2, Sp, N1 and KK) are

being evaluated, and their scores are shown in the table below.

Carro Safety features

(S)

Fuel efficiency

(F)

External appearence

(E)

Internal fittings

(T)

Ca 3 1 2 3

M2 2 2 2 2

Sp 3 1 3 2

N1 1 3 3 3

KK 3 2 3 2

The values in the table have the following interpretation.

3 points: excellent.

2 points: good.

1 point: acceptable.

To calculate the total score for a car, the store uses the following rule, which is a weighted sum

of the individual score points:

Total Score = (3xS) + F + E + T


a) What is the total score for car "Ca"?

b) The manufacturer of car “Ca” thought the rule for the total score was unfair. He wants a new

rule to calculate the total score so that the car "Ca" is the winner. In this rule, all variables (S, F,

E and T) must be included. Write it down, filling in the gaps with positive coefficients.

Total Score = ………× S + ………× F + ………× E + ………× T.


64


Item a)

By reading the table, we have the parameters below for car "Ca".

Safety features: S = 3

Fuel efficiency: F = 1

External appearance: E = 2

Internal fittings: T = 3

The total score for a car is given by the following formula:

Total Score = (3xS) + F + E + T

Therefore, the total score for car "Ca" is 15 points, because:

Total Score for car “Ca” = (3x3) + 1 + 2 + 3 = 15 points

Item b)

There are several formulas that can make car "Ca" the winner. In them, the safety features (S)

and the external appearence (E) must have greater "weights", because they are the items in

which car "Ca" has maximum score. The fuel efficiency (F) should have lower "weight", as it is

the item in which the car "Ca" has a minimum score. The internal fittings (T) should not be very

high, as car "Ca" does not have a maximum score in this item, but cars "Sp", "N1" and "KK" do.

A rule option with all variables included (S, F, E, and T) and makes car "Ca" winner is as it

follows:

Total Socore = 5×S + 1×F + 2×E + 6×T.

With this formula, the scores of the cars are as it follows.

Total Score - car “Ca” = 5x3 + 1x1 + 2x2 + 6x3 = 38 points

Total Score - car “M2” = 5x2 + 1x2 + 2x2 + 6x2 = 28 points

Total Score - car “Sp” = 5x3 + 1x1 + 2x3 + 6x2 = 34 points

Total Score - car “N1” = 5x1 + 1x3 + 2x3 + 6x3 = 32 points

Total Score - car “KK” = 5x3 + 1x2 + 2x3 + 6x2 = 35 points


65



apply data to a formula that calculates the score of cars according to four variables;

propose a new formula that calculates the score of cars according to the same four variables,

but car "Ca" must be the one with the highest score.

QUESTION 3.4.

Theme. Cultivation of apples in a field surrounded by conifers.

Question 3.4 (Pisa). A farmer plants apple trees in a square pattern. In order to protect the

trees against the wind, he plants conifers all around the orchards. In the picture below, there are

schemes of this situation, where n represents the number of rows of apple trees.

n=1

X X X

X O X

X X X

n=2

X X X X

X O O X

X O O X

X X X X

n=3

X X X X X

X O O O X

X O O O X

X O O O X

X X X X X

n=4

X X X X X X

X O O O O X

X O O O O X

X O O O O X

X O O O O X

X X X X X X

LEGEND

X: conifer

O: apple tree



66

a) Complete the table below.

n Number of apple trees Number of conifers

1 1 8

2 4

3

4

5

b) Establish formulas relating the number of apple trees and the number of conifers to the

number n of rows of apple trees.

c) For what number of rows the number of apple trees is equal the number of conifers?

d) Suppose the farmer wants to make a much larger orchard with many rows of trees. As the

farmer makes the orchard bigger, which will increase more quickly: the number of apple trees or

the number of conifers? Explain how you found your answer.


Item a)

We can see, in the following scheme, situations of n equal to 1, 2, 3, 4 and 5 rows of apple trees.

n=1

(8 conifers e 1 apple tree)

X X X

X O X

X X X

n=2

(12 conifers e 4 apple trees)

X X X X

X O O X

X O O X

X X X X

n=3


X X X X X

X O O O X

X O O O X

X O O O X

X X X X X


67

n=3


X X X X X

X O O O X

X O O O X

X O O O X

X X X X X

n=5


X X X X X X X

X O O O O O X

X O O O O O X

X O O O O O X

X O O O O O X

X O O O O O X

X X X X X X X

Based on these schemes, we can complete the following table.

n Number of apple trees Number of conifers

1 1 8

2 4 12

3 9 16

4 16 20

5 25 24

Item b)

We will do separately the studies on the quantities of apple trees and the quantities of conifers.

Apple trees.

Let's call m the number of apple trees.

We want to determine the formula that relates m and the number n of rows of apple trees. For

this, we can see only the arrangement of the apple trees according to number of rows in the

following picture.

n=1

O

n=2

O O

O O

n=3

O O O

O O O

O O O

n=4

O O O O

O O O O

O O O O

O O O O

n=5

O O O O O

O O O O O

O O O O O

O O O O O

O O O O O

We see that the number m of apple trees is the product of the number of rows by the number of

columns. So we have what follows.


68

Number of lines Number of rows Number of apple trees

1 1 1.1 = 12 = 1

2 2 2.2 = 22 = 4

3 3 3.3 = 32 = 9

4 4 4.4 = 42 = 16

5 5 5.5 = 52 = 25

... ... ...

n n n.n = n2 = m

The formula relating m and the number n of rows of apple trees is: 2nm .

Conifers.

Let's call c the number of apple trees.

We want to determine the formula that relates c and the number n of rows of apple trees.

For this, we see only the arrangement of the conifers according to number of rows in the

following picture.

n=1

X X X

X X

X X X

n=2

X X X X

X X

X X

X X X X

n=3

X X X X X

X X

X X

X X

X X X X X

n=4

X X X X X X

X X

X X

X X

X X

X X X X X X

n=5

X X X X X X X

X X

X X

X X

X X

X X

X X X X X X X

We see that the number c of apple trees is 4 times the number n of lines and, to this result, 4

must be added, which corresponds to the quantity of conifers at the "corner". So we have what

follows.


69

Number of lines Number of conifers at the tips Number of conifers

1 4 4.1 + 4 = 8

2 4 4.2 + 4 = 12

3 4 4.3 + 4 = 16

4 4 4.4 + 4 = 20

5 4 4.5 + 4 = 24

... ... ...

n 4 4.n+ 4 = 4.(n + 1) = c

The formula relating m and the number n of rows of apple trees is:

)1(4 nc .

Item c)

To know the value of number n of lines in which the number m of apple trees and the number c

of conifers are equal, we need to equate the expressions that give m and c as a function of n.

)1(42 nncm

442 nn

0442 nn

21 ( 4) ( 4) 0n n

We have a second degree equation (2 0An Bn C ), with A=1, B=-4 and C=-4. So:

2 4

2

B B ACn

A

2( 4) ( 4) 4(1)( 4) 4 16 4(1)( 4)

2(1) 2n

2( 4) ( 4) 4(1)( 4) 4 32

2(1) 2n


70

2222

244

n

We see that there is no integer value of n for which m = c.

Item d)

In this item, there is the supposition that the farmer wants to make much larger orchard with

many rows of trees. To achieve the best results, it should increase m apple trees more rapidly,

because m varies with n squared.


To solve question 3.4, the student should:

note that the number of apple trees varies with the square of the number of rows and that

the number of conifers varies linearly with the number of rows;

solve a second degree equation.


71

CHAPTER 4. Identification of characteristics and mathematical relationships in

geometric pictures.

We will present characteristics of regular geometric pictures and, also, a brief study on the

similarity of triangles.

SQUARE

Dimension: side L.

Area (S): L side squared, i.e., S=L2.

Perimeter (P): sum of the sides, given by four times side L, i.e., P=4L.

RECTANGLE

Dimensions: base B and height H.

Area (S): product (multiplication) of base B by height H, i.e., S=B.L.

Perimeter (P): sum of the sides, given by the sum of twice base B and twice height H, i.e.,

P=2B+2H.


72

PARALLELOGRAM

Dimensions: base B, height H and side A.

Area (S): product (multiplication) of base B by height H, i.e., S=B.L.

Perimeter (P): sum of the sides, given by the sum of twice base B and twice side A, i.e.,

P=2B+2A.

Observation. The perimeter of a parallelogram with side A and base B is greater than the

perimeter of a rectangle with side A and base B, because the side of a parallelogram is

greater than its height.

CIRCLE

Dimension: radius R.

Area (S): product (multiplication) of the irrational number "pi" by squared radius R, i.e.,

.2RS

Perimeter (P): product (multiplication) of twice the irrational number "pi" by radius R, i.e.,

.2 RP


73

TRIANGLE

Dimensions: base B and height H.

Area (S): half of the product (multiplication) of base B by height H, i.e., .2

.HBS

Perimeter (P): sum of the sides.

RECTANGLE TRIANGLE

Dimensions: hypotenuse A, cathetus B and cathetus C.

Pythagorean theorem:222 CBA .

A

Bsen cos ;

A

Csen cos ;

B

Ctg ;

C

Btg


74

TRAPEZIO

Dimensions: larger base B, smaller base b and height H.

Area (S): half of the product (multiplication) of height H by the sum of larger base B and

smaller base b, i.e., .2

).( HbBS

Perimeter (P): sum of the sides.

CUBE

Dimension: edge A.

Volume (V): edge raised to cube, i.e., V=A3.


75

PARALLELEPIPED

Dimensions: length C, width L and thickness E.

Volume (V): product (multiplication) between length C, width L and thickness E, i.e., V=C.L.E.

BALL

Dimension: radius R.

Volume (V): product (multiplication) of four-thirds of the irrational number "pi" by radius R

raised to the cube, i.e., .3

4 3RV


76

CYLINDER

Dimensions: height H and radius R.

Volume (V): product (multiplication) of the irrational number "pi" by squared radius R and by

height H, i.e., .2HRV

Side area (ALAT): double the product (multiplication) of the irrational number "pi" by radius R

and height H, i.e., .2 RHALAT

Base area (ABASE): product (multiplication) of the irrational number "pi" by radius R raised

squared, i.e., .2RABASE

TRIANGLE SIMILARITY

Two triangles BAC and MAN, as represented in the following picture, are similar if and only if their

three angles are congruent in the same order and their homologous sides are proportional.

The ratio between two homologous sides or between two similar triangles is called the

resemblance ratio, denoted by R in the expression: .BC BA CA

RMN MA NA


77

Let's apply some of these situations in questions 4.1 to 4.7, extracted from Pisa and translated

with some adaptations.

QUESTION 4.1.

Theme. Identification of a geometric picture from a description.

Question 4.1 (Pisa). Among pictures A, B, C, D and E, indicate the one that is according to

following description.

Triangle PQR is a right triangle with right angle at R. The line RQ is smaller than the

line PR. M is the midpoint of the line PQ and N is the midpoint of the line QR. S is a

point inside the triangle. The line MN is greater than the line MS.


78


Let's analyze each part of the description to identify according to what the statement gives.

Part 1. Triangle PQR is a right triangle with right angle at R.

In the picture of alternative E, we don’t see right angle in R. In this picture, in R, the angle is less

than 90°.

Therefore, alternative E is ruled out.

Part 2. The line RQ is smaller than the line PR.

Apparently, in the pictures of the alternatives A and C, the length of segment RQ is greater than

the length of segment PR.

Therefore, alternatives A and C are discarded.

Part 3. M is the midpoint of the line PQ and N is the midpoint of the line QR.

In the pictures of alternatives B and E, M can not be the midpoint of the PQ segment.

In the pictures of alternatives A, B and E, N can not be the midpoint of the QR segment.

Therefore, alternatives A, B and E are discarded.

Part 4. S is a point inside the triangle.

In the pictures of alternatives A, B and E, S is not an internal point to the triangle.

Therefore, alternatives A, B and E are discarded.

Part 5. The line MN is greater than the line MS.

Apparently, in the pictures of all alternatives, the length of segment MN is greater than the length

of segment MS.

Conclusion. The only alternative that provides a picture according to the 5 parts of the description

is alternative D.



understand the 5 parts that constitute the description of a geometric picture;

identify the picture that serves the entire description.


79

QUESTION 4.2.

Theme. Farm house with pyramidal shaped roof.

Question 4.2 (Pisa). The photograph below shows a farm house with pyramidal roof.

The following picture is a model made by a student of mathematics for the pyradimal roof, with

inclusion of measurements (in meters).

The attic floor, indicated by ABCD in the model, has a squared shape. The roof supporting

structure forms the edges of a block, represented by the rectangular prism EFGHKLMN. Point E is

the midpoint of segment AT. Point G is the midpoint of segment CT. Point H is the midpoint of

segment DT. All the edges of the model have length equal to 12m.



80

a) Calculate the floor area of the attic (square ABCD).

b) Calculate the length of segment EF, one of the horizontal edges of the block.


Item a)

The attic floor is the square ABCD, with side L equal to 12m.

The area AQ of the side square L is calculated by AQ=L2.

Therefore, the floor area of the attic is 144m2, because, in this case, AQ=122=144.

Item b)

We see, in the following picture, a highlight of EFT triangle and ATB triangle.

We know that BT and BA measure 12m and F is the midpoint of BT. Therefore, FT measures 6m,

which is half of 12m.

Since the ATB and ETF triangles are similar, we have the following relationship:

EF

FT

BA

BT

We know that BT = 12m, BA = 12m and FT = 6m. So:

EFEFEF

FT

BA

BT 61

6

12

12


81

661 EFEF

We conclude that EF segment length is 6m.



observe the presence of regular geometric pictures in a pyradimal roof;

check similarities of triangles;

calculate areas and lengths.

QUESTION 4.3.

Theme. Wooden fence to a garden bed.

Question 4.3 (Pisa). A carpenter has 32 meters of timber and wants to make a border around

a garden bed. He is considering the following designs for the garden bed.


82

Underline "Yes" or "No" for each design to indicate whether the garden bed can be made with 32

meters of timber.

Garden bed design Using this design, can the garden bed be made with 32

meters of timber?

Design A “Yes”/“No”

Design B “Yes”/“No”

Design C “Yes”/“No”

Design D “Yes”/“No”


Let's look at designs A, B, C and D separately to see if it is possible with each of them to make a

border around a garden bed with the available 32 meters of wood.

Design A (picture A).

The perimeter of the picture A is the same perimeter P of a rectangle with base B equal to 10m

and height H equal to 6m. We can see it by the completions in dotted lines made in the picture A.

Perimeter P is 32 meters, because:

326210222 HBP

As the carpenter has 32 meters of wood and it takes 32 meters of wood to make the plot of

drawing A, this option is possible to be executed.


83

Design B (picture B).

Picture B is a parallelogram of base B equal to 10m, height H equal to 6m and side A greater than

6m. Its perimeter, which is the sum of all its sides, is greater than the perimeter of a rectangle

with base equal 10m and height equal 6m, calculated in the case of designer A. So the perimeter

of this parallelogram is greater than 32m.

As the carpenter has 32 meters of wood and it takes more than 32 meters of wood to make the

designer B, this option is not possible to be executed.

Design C (picture C).

The perimeter of the picture C is the same perimeter P of a rectangle with base B equal to 10m

and of height H equal to 6m. We can see it by the completions in dotted lines made in the picture

C. That is, the perimeter of the picture of the designer C is equal to 32m.

As the carpenter has 32 meters of wood and it takes meters of wood to make design C, this

option is possible to be executed.


84

Design D (picture BD).

Picture D is a rectangle with B equal to 10m and height H equal to 6m. That is, the perimeter P of

the rectangle of drawing D is equal to 32m.

As the carpenter has 32 meters of wood and it takes 32 meters of wood to make design D, this

option is possible to be executed.

Based on the analyzes, we can indicate "Yes" or "No" for each design, answering whether or not

is possible to make a border around a garden bed with 32 meters of wood.

Garden bed design Using this design, can the garden bed be made with 32

meters of timber?

Design A “Yes”/“No”

Design B “Yes”/“No”

Design C “Yes”/“No”

Design D “Yes”/“No”



recognize regular geometric shapes;

calculate and compare perimeter values;

evaluate the possible designs of borders around a garden bed and identify those that can be

constructed with 32m of wood.

QUESTION 4.4.

Theme. Approximate calculation of Antarctic area.

Question 4.4 (Pisa). Analyze the following map of Antarctica.


85

Estimate the area of Antarctica using the map scale in the picture. Show your work and explain

how you made your estimative. You can draw over the map if it helps you with your estimative.


The area can be estimated by approaching the map of Antarctica by a circle of radius R of about

2,000km, following the indication of the scale in the map.

This circle is represented in the following picture.


86

The area S of the circle is the product (multiplication) of the irrational number "pi" by the radius

R raised squared, that is, .2RS

In the case under study, we have an area of approximately 13 million km2, because:

222 000.560.12)000.2(14,3 kmRS



recognize the possibility of approaching the Antarctic area by a circle of radius of about

2,000km;

read a map scale;

calculate the area of a circle.


87

QUESTION 4.5.

Theme. Proposal to attach kite sails to ships and use the force of the wind to help

reduce diesel consumption.

Question 4.5 (Pisa). Ninety-five percent of world trade is moved by sea, by roughly 50,000

tankers, bulk carriers and container ships. Most of these ships uses diesel fuel.

Engineers are planning to develop wind power support for ships. Their proposal is to attach kite

sails to ships and use the wind’s power to help reduce diesel consumption and the fuel’s impact

on the environment.

© by skysails

One advantage of using a kite sail is that it flies at a height of 150m. There, the wind speed is

approximately 25% higher than down on the deck of the ship.


a) At what approximate speed does the wind blow into a kite sail when a wind speed of 24km/h

is measured on the deck of the ship?

b) Approximately, what is the length of the rope for the kite sail, in order to pull the ship at an

angle of 45° and be at a vertical height of 150 m, as shown in the

following diagram?


88


Item a)

We know that the wind speed (v) is approximately 25% greater than the speed measured on the

deck of the ship.

Therefore, if we measure the wind speed of 24km/h on the ship's deck, the wind is equal

30km/h, as 25% of 24km/h is 6km/h, and 24km/h plus 6km/h is 30km/h, as indicated in the

calculations below.

2525% 24 / 24 / 6 /

100of km h km h km h

hkmhkmhkmv /30/6/24

Note that 6km/h is how much the wind speed exceeds the 24km/h measured on the deck of the

ship.

So to know the wind speed, we need to do the sum of 24km/h and 6km/h.

Item b)

The following rectangle triangle represents the situation shown in item b.


89

For this configuration, we want to determine the value L of the hypotenuse of the right triangle,

which corresponds to the length of the rope of the kite sail to pull the ship at a 45° angle, at a

vertical height of 150m. So:

15045 45

opposite cathetussin sin

hypotenuse L

The sine of 45° is about 0.71. So:

1500,71

L

0.71 150L

150211

0.71L L m

We conclude that for the situation of item b, the length of the rope of the kite sail to pull the ship

is approximately 211 meters.



understand the meaning of a text that distinguishes wind speed from speed measured on the

deck of the ship;


understand relations in a triangle rectangle;

know the approximate value of the sine of 45 degrees.


90

QUESTION 4.6.

Theme. Revolving door with three wings which rotates within a circular space.

Question 4.6 (Pisa). A revolving door includes three wings which rotates within a circular

space.

The inside diameter of this space is 2 meters (200 centimeters). The three door wings divide the

space into three equal sectors. The plan below shows the door wings in three different positions

viewed from the top.


a) What is the value, in degrees, of the angle formed by two door wings?

b) The two door openings (the dotted arcs in the diagram) are the same size. If these openings

are too wide the revolving wings cannot provide a sealed space and air could then flow freely

between the entrance and the exit, causing unwanted heat loss or gain. This is shown in the

following diagram.


91

What is the maximum arc length in centimeters (cm) that each door opening can have, so that

air never flows freely between the entrance and the exit?

c) The door makes 4 complete rotations in a minute. There is room for a maximum of two people

in each of the three door sectors. What is the maximum number of people that can enter the

building through the door in 30 minutes?


Item a)

To know the angle θ formed by two door wings, we must divide the total 360°, corresponding to

the circular space, by 3, which is the number of compartments. Thus, this angle is 120°, as

calculated below and indicated in the following picture.

1203

360

Item b)

We know that the two door openings, indicated by the arcs dotted in the diagram below, are of

the same size. We are looking for the maximum length L (in cm) of the arc that each door

opening may have, so that the air never flows freely between the inside and the outside.


92

The maximum length L of the arc of each door opening is half the arc length of each

compartment. Each compartment corresponds to one third (1/3) of the total length C of the

circumference.

Thus, half the arc length of each compartment is half (1/2) of a third (1/3) of the total length C

of the circumference.

We conclude that the length L is one sixth (1/6) of the total length C of the circumference, as half

(1/2) times a third (1/3) is 1/6.

The total length C of the circumference is twice its "pi" times the radius R. So:

RC 2

Thus, the maximum length L of the arc of each door opening is calculated by:

RLRL 3

12

6

1

We know that the diameter of the circumference is 200 centimeters. So its radius R is 100

centimeters, because the radius is half the diameter. Thus:

1003

1 L

3

100L

If we use 3.14 for "pi", we have that the maximum length L is about 105cm.

Item c)

The door does 4 full revolutions in one minute. In 30 minutes, it will do 120 revolutions, because:

º 30min 4 30(min) 120min

rotationsN rotations rotations

There is a maximum of 2 people in each of the 3 sections of the door. Therefore, in the 3 sectors

of the door, there is a maximum of 6 people. That is, in each rotation of the door, we have a

maximum of 6 people in their internal space.


93

Within 30 minutes of operation, there are 120 rotations. In this interval, at most, 720 people

were in the inner door space, because:

º 30min 120 6 720N people people

Note. We are thinking that one person only used the door once, as we are imagining 720

different people.



calculate angles corresponding to arcs of circumference;

calculate lengths of arcs of circumference;

understand fractions of circumferences;

apply concepts related to proportions of values.

QUESTION 4.7.

Theme. Estimate of apartment area by its floor plant.

Question 4.7 (Pisa - with adaptations). The following picture shows the floor plan of the

apartment that George's parents want to buy from a real estate agency.


94

To estimate the total floor area of the apartment (including the terrace and the walls), you can

measure the size of each room, calculate the area of each one and add all the areas together.

However, there is a more efficient method to estimate the total floor area where you only need to

measure 4 lengths. Mark on the plan the four lengths that are needed to estimate the total floor

area of the apartment.


One possibility to calculate the total area of the apartment is through measures A, B, C and D

indicated in the following picture.

In this case, the total area S of the apartment is the sum of the areas S1 and S2 indicated in the

picture below.

The area S1 is the area of a rectangle with base D and height C. The area S2 is the area of a

rectangle with base (B-D) and height A. Thus:

21 SSS

ADBCDS

Let us imagine that we measure what is indicated in the following floor plan.


95

For this floor plant, the rectangle of area S1 has base equal 6.8 meters and height equal 9.7

meters. The rectangle of area S2 has base equal 2 meters, as 8.8-6.8 = 2.0, and height equal 5.3

meters.

Therefore, the area S of the apartment is equal to 76.56m2, because:

256,763,527,98,8 mS



realize that the total area of an apartment is the sum of the areas of two rectangles;

check the minimum number of measures necessary to calculate the total area of an

apartment;

calculate rectangle areas.

QUESTION 4.8.

Theme. Comparison of prices of small pizza and large pizza.

Question 4.8 (Pisa). A pizzeria serves two round pizzas of the same thickness, but in different

sizes. The smaller pizza, called "small pizza", has diameter equal 30cm and costs 30 zeds. The


96

larger pizza, called "big pizza", has a diameter equal 40cm and costs 40 zeds. Which of the pizzas

has the most advantageous price? Show your reasoning.


In order to formulate the problem, we assumed that the pizza has the geometry of a cylinder of

height h and radius R. Thus, the volume V of the round pizza is given by:

hRV 2

If P is the total price of the pizza, its price X per unit volume is given by:

V

PX

hR

PX

2

We will calculate the price per unit volume of the small pizza and the price per unit volume of the

large pizza. We should remember that both pizzas are circular disks of the same height h.

The price of the small pizza is 30 zeds and its radius is 30cm. Thus, the price Xp per unit volume

of the small pizza is:

hX P 2)30(

30

hX P

30

1

hX P

1

30

1

The price of the large pizza is 40 zeds and its radius is 40cm. Thus, the price XG per unit volume

of the large pizza is:


97

hXG 2)40(

40

hX G

40

1

hX G

1

40

1

The results obtained using the mathematical formulation to solve the problem allow the

processes of interpretation and evaluation, as it follows.

The price per unit volume of the small pizza is greater than the price per unit volume of the small

pizza (for the small pizza, this value is given by the multiplication of

301 by

h1 , and, for large

pizza, is given by the multiplication of 40

1 by h

1 ). Therefore, for the consumer, it is more

advantageous to buy the large pizza.



realize that the increasement in the price of the pizza is not directly proportional to the

increasement in the radius of the pizza;

assume that the pizza has the geometry of a cylinder of height h and of radius R;

understand relations involving fractions;

evaluate which pizza option has more advantageous price for the consumer.


98

CHAPTER 5. Rule application for problem solving.

You may have seen problems like the one in the following picture.

What is the result?

8 » 56

7 » 42

6 » 30

5 » 20

3 » ?

Are the "transformations" from 8 to 56, from 7 to 42, from 6 to 30, and from 5 to 20

random or do they follow a rule?

It seems that these "transformations" occurred according to the following rule: multiply "a

number" by "that number minus 1". Let's apply it to see if we get the results shown in the

picture.

• If the number is 8, the result is 56, because the multiplication of 8 by 7 results in 56.




So, if the number is 3, the result is 6, because the multiplication of 3 by 2 results in 6.

There are several other examples where results are obtained by the application of rules,

such as those developed in questions 5.1 to 5.6, extracted from Pisa and translated with some

adaptations.

QUESTION 5.1.

Theme. Numbers in faces of six dice.

Question 5.1 (Pisa). In the photo, there are six dice, identified by (a), (b), (c), (d) and (f).


99

For all dice, there is a rule: “the total number of points on two opposite faces of each die is

always seven (7)”.

Write in each box the number of dots on the bottom face of the dice corresponding to the

photograph


We know the following rule: the total number of points on two opposite faces of the die is always

equal to 7 points.

So, we have what follows.

• Face (a) has a 6-point mark. Therefore, the face opposite to face (a) has a 1-point mark,

because the sum of 6 points with 1 point results in 7 points.

• Face (b) has a 2-point mark. Therefore, the face opposite the face (b) has a 5-point mark,

because the sum of 2 points with 5 points results in 7 points.

• Face (c) has a 3-point mark. Therefore, the face opposite the face (c) has a 4-point mark,


• Face (d) has a 5-point mark. Therefore, the face opposite the face (d) has a 2-point mark,


• Face (e) has a 1-point mark. Thus, the face opposite the face (e) has a 6-point mark, because

the sum of 1 point with 6 points results in 7 points.

• Face (f) has a 2-point mark. Therefore, the face opposite the face (f) has a 5-point mark,



100

Therefore, we indicate this conclusion in the following picture.



understand that the total number of points on two opposite faces of the die is always equal to

7 points;

realize that the dots on a face of a dice are associated to points (like 7 dots are equivalent to

7 points);

observe the marked scores on data faces and calculate the scores on the faces opposite the

faces shown.

QUESTION 5.2.

Theme. Construction of die following the rule of "sum 7 on opposite faces".

Question 5.2 (Pisa). The following picture illustrates two dice.

Dice are special cubes for which the following rule applies: the total number of points on two

opposite faces is always seven (7).

You can make a simple cube by cutting, folding and gluing cardboard. This can be done in many

ways. In the following picture, we can see four cuttings (I, II, III and IV) that can be used to

make cubes, with dots on the sides.


101

Which of the following shapes can be folded together to form a cube that obeys the rule that the

sum of opposite faces is 7? For each shape, indicate either “Yes” or “No” in the table below.

Shape Does it obey the rule of sum 7 on opposite?

faces? I “Yes”/“No”

II “Yes”/“No”

III “Yes”/“No”

IV “Yes”/“No”


Let's look at each proposition separately.

PROPOSITION I.

In proposition I, we have an example of a situation in which the total of points on opposite faces

does not result in 7 points: the circled faces are opposite, but the sum of their points is 6, not 7,

because 1 point + 5 points = 6 points.

Besides this, there are other situations in proposition I in which the sum of points on opposite

faces does not result in 7.

Therefore, proposition I can not be used to make a die.


102

PROPOSITION II.

In proposition II, in all situations the total of points in opposite faces results in 7 points.

Therefore, proposition II can be used to make a die.

PROPOSITION III.

In proposition III, in all situations the total of points in opposite faces results in 7 points.

Therefore, proposition III can be used to make a die.

PROPOSITION IV.

In proposition IV, we have an example of a situation where the total of points on opposite faces

does not result in 7 points: the circled faces are opposite, but the sum of their points is 4, not 7,

because 1 point + 3 points = 4 points.

Besides this, there are other situations in proposition IV in which the sum of points in opposite

faces does not result in 7.

Therefore, proposition IV can not be used to make a die.

According to the analyzes, we have the following table.


103

Shape Does it obey the rule of sum 7 on opposite?

faces? I “Yes”/“No”

II “Yes”/“No”

III “Yes”/“No”

IV “Yes”/“No”



understand that the total number of points on two opposite faces of the dice is always equal

7 points.

check propositions that obey the rule to make a die.

QUESTION 5.3.

Theme. Instructions for making an 8-page booklet.

Question 5.3 (Pisa). Picture 1 shows how to make a small booklet.

Picture 1. Small booklet.

The instructions for making the booklet are given below.

Take a small piece of paper and fold it twice.

Staple edge a.

Cut open two edges at b.

So the result is a small 8-page booklet.

Figure 2 shows one side of a piece of paper that is used to make such a booklet. The page

numbers have been put on the paper in advance.

The thick line indicates where the paper will be cut after folding.


104

Picture 2. One side of a piece of paper that is used to make such a booklet.

Write the numbers 1, 4, 5 and 8 in the correct boxes in the following diagram to show which

page number is directly behind each of the page numbers 2, 3, 6 and 7.


If we bend picture 2, making the top part over the bottom, 2 will be on top of 3, and 7 will be on

top of 6.

After this, if we do a new folding, from left to right, we will have the numbers of the following

picture.


105



understand the instructions for making a booklet;

apply instructions to get the correct insertion of page numbers into a booklet.

QUESTION 5.4.

Theme. Numbers of faces of a tower that can be seen from certain positions.

Question 5.4 (Pisa). In pictures 1 and 2 below, you see two drawings of the same tower. In

picture 1, you see three faces of the roof of the tower. In picture 2, you see four faces.


106

In the following diagram, the view of the roof of the tower, from above, is shown. Five positions

are shown on the diagram. Each one is marked with a cross (X) and they are labeled P1, P2, P3,

P4 and P5.

From each of these positions, a person viewing the tower would be able to see a number of faces

of the roof of the tower.

In the table below, indicate the number of faces that could be seen from each of these positions.

Position Number of faces that could be seen

from that position

P1 1 2 3 4 More than 4






In the following picture, the faces of the tower are identified by F1, F2, F3, F4, F5, F6, F7 and F8.


107

In the following pictures, we have lines that delimit the faces that can be seen from the positions

P1, P2, P3, P4 and P6, constructed according to the rule:

From a selected position, draw a line that starts from it to the farthest edge on the right side. Go

back to the selected position, and do the same on the left side.

• From the P1 position, 4 faces can be seen (F1, F2, F3 and F4).

• From the P2 position, 3 faces can be seen (F3, F4 and F5).

• From the P3 position, 1 face can be seen (F3).

• From the P4 position, 2 faces can be seen (F7 and F8).

• From the P5 position, 2 faces can be seen (F6 and F7).

The 4 faces that can be seen from the position P1 are shown below.


108

The 3 faces that can be seen from position P2 are shown below.

The face that can be seen from the position P3 is shown below.

The two faces that can be seen from position P4 are shown below.


109

The two faces that can be seen from position P5 are shown below.

Thus, we can indicate, in the table, the numbers of tower faces that can be seen from each of

the positions.

Position Number of faces that could be seen

from that position








read a drawing containing the floor plan of an 8-sided tower;

understand the possibilities of sightings of the roof faces according to the position of the

observer.

QUESTION 5.5.

Theme. Construction of stages patterns with squares.

Question 5.5 (Pisa). Robert constructs a pattern of stages using squares. In the picture below,

we can see the first three stages of this construction.


110

Robert uses one square for stage 1, three squares for stage 2 and six squares for stage 3. How

many squares should Robert use for stage 4?


In the table below, the number of "steps" and the number of squares of the picture, which

resemble a staircase, are indicated for each stage.

Stage Number of steps Number of squares Picture

1 1 1

2 2 3

3 3 6

4 4 10

We conclude that Robert should use 10 squares for stage 4.



recognize the pattern used to generate staircase-like pictures;

propose the fourth stage of a particular construction pattern.


111

QUESTION 5.6.

Theme. "Design by Numbers", a drawing tool for generating graphics on computers.

Question 5.6 (Pisa). "Design by Numbers" is a drawing tool for generating graphics on

computers. Pictures can be generated by giving a series of commands to the program. Study

carefully the command examples in the picture below and their respective pictures before

responding to items a, b, and c.

a) Which command generates the graph below?

A. Paper 0 B. Paper 20 C. Paper 50 D. Paper 75

b) Which set of commands generates the graph below?


112

A. Paper 100 Pen 0 Line 80 20 80 60

B. Paper 0 Pen 100 Line 80 20 60 80

C. Paper 100 Pen 0 Line 20 80 80 60

D. Paper 0 Pen 100 Line 20 80 80 60

c) The following image shows an example of using the "Repeat" command.

"Repeat A 50 80" command tells the program to repeat the actions in parentheses {} for

successive A values, from A = 50 to A = 80.

Based on this, type commands to generate the following picture.


Initially, we will study commands "Paper", "Pen" and "Line" indicated in the image of the

statement.

"Paper" command.

By the following picture, we will study the influence of "Paper" command on the graph to be

constructed with the "Design by Numbers" tool.


113

We see that "Paper" command is responsible for the grayscale color of the chart background.

Reading the picture, we notice that "Paper 0" corresponds to white; "Paper 50" corresponds to

intermediate gray; and "Paper 100" corresponds to black. That is, in "Paper X", X ranges from 0

to 100; and the highest the number, the darker the background.

"Pen" command.

By the following picture, we will study the influence of "Pen" command on the graph to be

constructed with the "Design by Numbers" tool.

We see that "Pen" command is responsible for the color of the chart line. Reading the picture, we

notice that "Pen 100" generates a black line graph; and "Pen 0" generates a white line graph.

“Line” command.

By the following picture, we will study the influence of "Line" command in the graph to be

constructed with the tool "Design by Numbers".

We see that "Line" command is responsible for the graph line itself.


114

Let's start with the first chart, including the highlights of the following picture.

Reading the picture, we observe that "Line 20 0 80 60" generates a segment of line that starts at

P1 = (20,0), point with abscissa 20 and ordinate 0; and ends in P2 = (80,60), point with abscissa

80 and ordinate 60. The background is white, because we have "Paper 0". The graph line is

black, because we have "Pen 100".

Now, let's look at the second chart, including the highlights of the following picture.

Reading the picture, we see that set of commands "Line 20 20 80 20", "Line 80 20 50 80" and

"Line 50 80 20 20" generates a triangle whose vertices are P1 = (20,20), point with abscissa 20

and ordinate 20, P2 = (80.20), point with abscissa 80 and ordinate 20, and P3 = (50,80), point

with abscissa 50 and ordinate 80. The graph line is white, because we have "Paper 100". The

background is black, because we have "Pen 0".

Considering these preliminary remarks, we will answer the items of the question.

Item a)

The background of the image of this item is “gray intermediate”, between white, which

corresponds to "Paper 0", and gray, which corresponds to "Paper 50". Thus, this background

corresponds to "Paper X" command, with X between 0 and 50. The only alternative that obeys

these criteria is alternative B, which indicates "Paper 20".


115

Item b)

Let's see, in the picture given in this item, details of its elements.

As the background is white, we have "Paper 0".

As the graph line is black, we have "Pen 100".

As the graph is a line segment that starts at P1 = (20.80), point with abscissa 20 and ordinate 0,

and ends at P2 = (80,60), point with abscissa 80 and ordinate 60, we have "Line 20 80 80 60 ".

Therefore, the correct alternative is D.

Item c)

Let's see, in the picture given in this item, details of its elements.

As the background is white, we have "Paper 0".

As the picture fill is black, we have "Pen 100".

As the horizontal range of the picture ranges from abscissa 20 to abscissa 60, we have, in this

direction, A from 20 to 60.

As the vertical range of the picture ranges from 0 to 40, we have A in this direction from 0 to 40.

Based on this, the commands to generate the picture under study are the ones that follow.

Paper 0 Pen 100 Repeat A 0 40 { Line 20 A 60 A }


116

CHAPTER 6. Use of probability and measures of central tendency.

6.1. Probability.

We can calculate the probability P of success in some event, such as, for example, picking

the single strawberry toffee in a bag with 20 toffees, by dividing the "number of cases favorable

to our success" by the "total number of possible cases". So:

( )Number of cases favorable to our success

Total number of possiP P s

bleucess

cases

In the case, as there is only one strawberry toffee in the bag with the total of 20 toffees,

we have one (1) favorable case in 20 possible cases. Therefore, the probability of a person

picking this single strawberry toffee is equal to 0.05 because:

1( ) 0.05

20strawberry tof eP fe

As the number of favorable cases is always less than or equal to the total number of

possible cases, the probability P is a real number ranging from 0 to 1 (0P1). The probability of

something which is certain to happen is 1. The probability of something which is impossible to

happen is 0.

Often, the probability P is multiplied by 100% to express the result in percentage (%).

Thus, for the example, the probability of a person picking the strawberry toffee can be

expressed in percentages by 5%, as 0.05 multiplied by 100% results in 5%.

As 5% is less than 50%, there is more chance of a no strawberry toffee to be picked than a

strawberry one.

Another example commonly used in probabilities is the roll of a die, with faces numbered

1 to 6. In this case, what is the probability of getting an odd face?

There are the total number of 6 possible cases, because the possible results in the roll of

a die are: 1, 2, 3, 4, 5 or 6.

The set of the possible results of the experiment is defined as the sample space E. Then,

in our example, E = 1, 2, 3, 4, 5, 6.

From the 6 possible results (the entire sample space E), the success corresponds to

obtaining an odd face, that is, results 1, 3 or 5. We conclude that of the six (6) possible

outcomes, only three (3) of them are favorable.


117

Thus, the probability that the single face of the die is odd is "3 in 6", that is, 0.5 (or 50%),

because:

3 1( ) 0,5

6 2

Number of favorable cases

Total number of possible casesP odd face

6.2. Measures of central tendency: mean, median and mode

The measures of central tendency of a set of numerical data are mean (average), median

and mode. Each of these quantities is calculated differently and has different meanings, as we

shall see in the following items.

6.2.1. Mean (average).

Let's suppose you want to average the following set of numbers:

2 2 1 3 116

The mean (average) of a set of numbers is the sum of all values, with this sum divided by

the number of sums added. So, for the example under study, we must add the numbers 2, 2, 1,

3 and 116 and divide this sum by 5. The result is 24.8, because:

2 2 1 3 116 12424.8

5 5Mean

We see that in this case the mean value of 24.8 does not correspond to any value of the

data set. Besides, it is greatly affected by an extreme value (116).

The mean does not necessarily portray the reality of the facts. For example, if I eat two

bars of chocolate and you do not eat any bar, on average each of us ate a bar of chocolate. But

you did not even taste that delicacy...

6.2.2. Median.

Now let's suppose you want to calculate the median of the following set of numbers:

2 2 1 3 116


118

As the median is the central value of the numerical distribution, we first need to put the

given values in ascending order, as it follows.

1 2 2 3 116

We see that, in the case of 5 values, the central value is the one that occupies the third

position. That is, it is 2. Thus, the median of this set is 2.

We observe that the median is not affected by an extreme value (116). If the data set has

an odd number of elements, the median is the mean of the two core values, as in the following

example.

10 12 12 20 21 30

In this case, the median is 16, because it is the mean between the central values (12 and

20), as we calculated below.

12 20 3216

2 2Median

6.2.3. Mode.

Finally, let's determine the mode of the following set of numbers:

2 2 1 3 116

As mode is the most frequent value (value that appears more often), in the case under

analysis, it is 2. If the data set does not have a more frequent value, it has no mode.

Let us apply the concepts studied in the resolution of questions 6.1 to 6.6, extracted from

Pisa and translated with some adaptations.

QUESTION 6.1.

Theme. Picking one candy from a bag with 30 candies.

Question 6.1 (Pisa). Robert’s mother allows him pick one candy from a bag. He can’t see the

candies. The number of candies of each color in the bag is shown in the following graph.


119

How likely is it that Robert gets a red candy? Or, in other words, what is the probability that

Robert will pick a red candy?


We can see, in the table below, the reading of the graph of the question. In this table, we

organize the quantities of candies of each color.

Candy calor Amount

Red 6

Orange 5

Yellow 3

Green 3

Blue 2

Pink 4

Purple 2

Brown 5

The probability P of success in an event is the division of the "number of cases favorable to our

success" by the "total number of possible cases":

( )Number of cases favorable to our success

Total number of possiP P s

bleucess

cases


120

Thus, the probability that Robert picks a red candy is the division of the number of red candies

(6) by the total number of candies (30), as calculated below.

6) 0.2

30

Number of red candiesP P

Totalrobability

numberof can(pick a red candy

dies

If we multiply the probability of 0.2 per 100%, we see that, in percentage, the probability that

Robert picks a red candy is 20%.



read, on a graph, the quantities of candies of each of the colors;

apply a definition of probability.

QUESTION 6.2.

Theme. Probability of an earthquake.

Question 6.2 (Pisa). A documentary was broadcast about earthquakes and how often

earthquakes occur. It included a discussion about the predictability of earthquakes. A geologist

stated what is shown below.

Which of the following best reflects the meaning of the geologist’s statement?

A. 13.3203

2 , so between 13 and 14 years from now there will be an earthquake in Zed City.

B. 3

2 is more than

2

1, so you can be sure there will be an earthquake in Zed City at some time

during the next 20 years.

C. The likelihood that there will be an earthquake in Zed City at some time during the next 20

years is higher than the likelihood of no earthquake.

D. You cannot tell what will happen, because nobody can be sure when an earthquake will

occur.

In the next twenty years, the chance that an earthquake will occur in Zed City is two out of three.


121


According to the geologist, "in the next twenty (20) years, the chance that an earthquake will

occur in Zed City is two (2) out of three (3)”. This means that the likelihood of an earthquake in

Zed in the next 20 years is 2/3 (about 0.67 or 67%).

Based on this interpretation, we will analyze the alternatives presented in the question.

Alternatives A and B.

Alternatives A and B are incorrect, as probability equals 2/3 provides the chance of an earthquake

happening in Zed in the next 20 years. This value does not give the certainty that it will occur.

Alternative C.

Alternative C is correct, as the probability of an earthquake occurring in the city of Zed any time

in the next 20 years, of 2/3, is greater than the probability that it will not occur, because two

thirds is larger than onde thirds (2/3 > 1/3).

Alternative D.

Alternative D is incorrect because the probability equals 2/3 indicates that there is a significant

chance (about 67%) of an earthquake occurring in the town of Zed sometime in the next 20

years.



understand that the chance of two (2) in three (3) equals the probability of 2/3 (or 67%);

realize the concept of probability;

observe that words and expressions like “there will be” and “can be sure” are not related to

pobabilities.

QUESTION 6.3.

Theme. Rainfall forecast.

Question 6.3 (Pisa). On a particular day, the weather forecast predicts that from 12 noon to

6 pm the chance of rainfall is 30%.

Which of the following statements is the best interpretation of this forecast?

A. 30% of the land in the forecast area will get rain.


122

B. 30% of the 6 hours (a total of 108 minutes) will have rain.

C. For the people in that area, 30 out of every 100 people will experience rain.

D. If the same prediction was given for 100 days, then about 30 days out of the 100 days will

have rain.

E. The amount of rain will be 30% of a heavy rainfall (as measured by rainfall per unit time).


In the statement, it was said that for a given day, the weather forecast predicts that from

12 noon to 6 p.m. the chance of rainfall is 30%.

Based on this information, we will analyze the alternatives of the question.

Alternative A.

Alternative A is incorrect, because the weather forecast is not related to the area in which rainfall

may occur.

Alternative B.

Alternative B is incorrect, because the weather forecast is not related to how long the rainfall may

occur.

Alternative C.

Alternative C is incorrect, because the weather forecast does not refer to the number of people

who will catch rainfall.

Alternative D.

The alternative D is correct, because the chance of rainfall equals 30% allows us to inferr that if

the same forecast were given for 100 days, in 30 of them there would be rainfall.

Alternative E.

The alternative C is incorrect, because the weather forecast does not refer to the precipitation

intensity.


To solve question 6.3, the student should:


123

understand that the weather forecast of 30% of rainfall is equivalent to the prediction that,

for 100 days, in 30 of them there would be rain;

realize the concept of probability.

QUESTION 6.4.

Theme. Mei Ling average on five science tests.

Question 6.4 (Pisa). In Mei Lin’s school, her science teacher gives tests that are marked out of

100 points. Mei Lin has an average of 60 points on her first four Science tests. On the fifth test

she got 80 points.

What is the average of Mei Lin’s points in Science after all five tests?


In the table below, we have a summary of Mei Ling's scores in the five tests.

Test Mei Ling's scores (points)

First test N1

Second test N2

Third test N3

Fourth test N4

Fifth test 80

Mei Ling scored average M1 of 60 points in her first four tests. So:

4

43211

NNNNM

4

432160

NNNN

4321460 NNNN

2404321 NNNN

In the fifth test, Mei Ling reached 80 points. So, her average M2 after doing all five tests is 64

points, because:


124

5

543212

NNNNNM

We saw that 2404321 NNNN and that N5 equals 80 points. So:

240 80 3202

5 5M

2 64M pontos



understand the scores obtained by Mei Ling in science tests;

apply the expression for Mei Ling's average score calculation;

realize that we need only the sum results to calculate the avarage of a set of values.

QUESTION 6.5.

Theme. Average height of 25 girls in a classroom.

Question 6.5 (Pisa). There are 25 girls in a classroom. The average height of these girls is

130cm. Based on the exposed situation, answer the following items.

a) Explain how this average height of 130cm can be obtained.

b) Identify the statements given in the following table as "True" or "False".


125

Statement "True" or "False"?

If there is a girl with height equals 132cm in this room, there must also

be a girl with height equals 128cm.

“True”/“False”

Most girls in the room must have height equals 130cm. “True”/“False”

If the girls in the room are ordered from the lowest to the highest, the

middle girl must have height equals 130cm.


Half the girls in the room must be lower than 130cm and half the girls

in the room must be taller than 130cm.


c) Imagine that an error has been found as far as a girl's height is concerned. This height was

given as 120cm, but in reality it is 145cm. Made this correction, what is the true average height

of the girls in the room?


Item a)

The average height of 130cm can be obtained by adding the heights of each of the 25 girls and

dividing that result by 25.

Item b)

We will analyze each statement in the table.

First statement.

The first statement is false, because if there is a girl with height equals 132cm, there should not

necessarily have another girl with height equals 128cm, because 130cm is the average height of

the group of 25 girls. This mean (130cm) is not the average height of not only 2 girls, one with

132cm and another with 128cm.

Second statement.

The second statement is false, because not necessarily most girls in the room should have height

equals 130cm. As 130cm is the average height of the group of 25 girls, there may be girls with

heights taller than 130cm who "compensate" girls with heights less than 130cm.


126

Third statement.

The third statement is false, because not necessarily if the girls in the room are ordered from the

shortest to the tallest, the middle girl has to have height equals 130cm. This situation refers to

the determination of the median of heights, not the mean height.

Fourth statement.

The fourth statement is false, because it is not necessary that half of the girls in the room have

height less than 130cm and half of the girls in the room have height more than 130cm so that

the average height is equal to 130cm. In addition, this situation would not be possible because

the room has an odd number of students.

The results of the analysis are shown in the following table.

Statement "True" or "False"?

If there is a girl with height equals 132cm in this room, there must also

be a girl with height equals 128cm.


Most girls in the room must have height equals 130cm. “True”/“False”

If the girls in the room are ordered from the lowest to the highest, the

middle girl must have height equals 130cm.


Half the girls in the room must be lower than 130cm and half the girls

in the room must be taller than 130cm.


Item c)

Initially, the mean height M1 of the group of 25 girls was calculated at 130cm. If we call S1 the

sum of the heights of each of the girls, we have the following:

25

11301

SM

251301 S

1 3,250S

There was an error in the measure of a girl's height: she was given as 145cm, but in reality it is

120cm. That is, the sum S1, equal to 3,250, is 25 more than it should be, which is the difference

between 145 and 120 (145-120 = 25).

Therefore, the true sum S2 of the girls' heights should be 3,225, as:


127

2 1 25 3,250 25 3,225S S

Thus, if the correction is made, the true mean height M2 of the group of 25 girls is equal to

129cm, because:

25

225.3

25

22

SM

cmM 1292



know how to calculate the average height of a group of 25 girls;

understand the mean height significance;

correct values in the calculation of the average height of a group of 25 girls, due to an

erroneous measure.

QUESTION 6.6.

Theme. Surveys to verify support for the president for the next election.

Question 6.6 (Pisa). In Zedland, opinion polls were conducted to find out the level of support

for the President in the forthcoming election. Four newspaper publishers did separate nationwide

polls. The results for the four newspaper polls are shown below.

Newspaper 1: 36.5% (poll conducted on January 6, with a sample of 500 randomly selected

citizens with voting rights).

Newspaper 2: 41.0% (poll conducted on January 20, with a sample of 500 randomly

selected citizens with voting rights).

Newspaper 3: 39.0% (poll conducted on January 20, with a sample of 1000 randomly

selected citizens with voting rights).

Newspaper 4: 44.5% (poll conducted on January 20, with 1000 readers phoning in to vote).

Which newspaper’s result is likely to be the best for predicting the level of support for the

President if the election is held on January 25? Give two reasons to support your answer.


128


To answer the question, let's make some previous comments.

• Poll for newspaper 1 is the least recent, as it was conducted on January 6 and polls for the

other newspapers were conducted on January 20.

• Poll for newspaper 4 was not randomized, while polls for the other newspapers were randomly

made.

• Polls for newspapers 1, 2 and 3 were randomly made. Polls for newspapers 1 and 2 were

conducted with 500 people. Poll for newspaper 4 was conducted with 1,000 people.

Based on these comments, we conclude that the best prediction is the newspaper 3 prediction,

because its poll is the most recent, worked with random selection of participants and had larger

sample size.

Note. The best sample is not necessarily the largest one. In a good sample, the different

components of the population appear in the same proportion as they appear in the population

itself.

Comments on the resolution of the question 6.6.


compare the conditions under which 4 opinion polls were conducted on the level of support to

the president for the next election;

identify the poll that presents the best prediction of results.


129

CHAPTER 7. Critical review of information.

When we read a text, we must analyze critically the information that it contains. The first

step for doing this is to understand the transmitted message, such as in the following image.

In the picture, there are two people, with emphatic expressions, proclaiming different

numbers for the same representation: one says "six" and the other says "nine." Who is right?

What are you trying to convey to the reader in this situation?

We see this is a situation that does not admit an absolute verdict of right or wrong:

depending on the position, the person sees a "six" or the person sees a "nine." That is to say, it is

intended to convey the message that right or wrong are associated with the position of the

observer.

Now, let's take a critical reading of a text published in a Brazilian newspaper in 1990.

(...) Thus, the vicious circle of pessimism is generated. Things are not going well because nobody trusts in government. And because nobody trusts in government, things are not going well.

Gilberto Dimenstein, Folha de S. Paulo, 22.11.90.

The intention of the statement was to show a relation of cause and consequence.

However, the author only inverted the order of the prayers, which did not alter the relation of

cause and consequence. There was only a repetition of the statement "because nobody trusts in

government", without the occurrence of mutual causality between "things are not going well" and

"nobody trusts in government".

In order to achieve the real objective, the text should have been written as it follows. (...) Thus, the vicious circle of pessimism is generated. Things are not going very well because nobody trusts in government, and nobody trusts in government because things are not going very well.


130

Let's analyze more situations that require critical analysis of information in questions 7.1

to 7.16, extracted from Pisa and translated with some adaptations.

QUESTION 7.1.

Theme. Recognition of priority tasks according to budget.

Question 7.1 (Pisa). Claire and her friends are going to rent a house, and they

have been working for two months;

have no money savings;

earn monthly payments and have just earned their salaries;

have done the task list shown below.

Task

To install cable TV.

To pay the rent.

To buy furniture for the outside of the house.

Indicate "Yes" in the table below for the task that probably needs more readiness from Claire and

her friends. For other tasks, indicate “No".

Task Does task need more readiness?

To install cable TV. “Yes”/“No”

To pay the rent. “Yes”/“No”

To buy furniture for the outside of the house. “Yes”/“No”


To resolve this question, we need to recognize that, because of the budget constraints of Claire

and her friends, who work have been working for few months and do not have cash savings, the

priority is rent. The installation of cable TV and the purchase of furniture for the outside of the

house are tasks that can be postponed, but non-payment of rent implies fines or even forced

eviction of the property. So we have the situation in the following table.


131

Task Does task need more readiness?

To install cable TV. “Yes”/“No”

To pay the rent. “Yes”/“No”

To buy furniture for the outside of the house. “Yes”/“No”

Comments on the resolution of the question 7.1.


evaluate the budgetary conditions of Claire and her friends;

identify the task that needs ready care, considering the budget constraints of Claire and her

friends.

QUESTION 7.2.

Theme. Identification of bank deposit related to payment of monthly salary.

Question 7.2 (Pisa). Every month, Jane Green's salary is deposited into her bank account. The

following table shows the summary of this payment in July, held on the last day of the month.

Employee name: Jane Green

Position: from July 1 to July 31

Gross salary: 2,800 zeds

Deductions: 300 zeds

Net wage: 2,500 zeds

Gross salary accumulated so far: 19,600 zeds

What was the amount that the employer deposited into Jane's bank account on July 31?


The amount effectively deposited by the employer in Jane's bank account on July 31 corresponds

to her net wage (net salary). Therefore, this value was 2,500 zeds.

Note that net wage (2,500 zeds) is the gross salary (2,800 zeds) minus the deductions (300

zeds). The gross salary accumulated (19,600 zeds) is not the amount deposited in a given month,

but the sum of several months of gross salary.




132

distinguish gross salary, net salary and accumulated gross salary;

read data from a posting and payment table;

identify the amount effectively deposited into Jane's account on July 31.

QUESTION 7.3.

Theme. Assessment of factors that influence the price of motorcycle insurance.

Question 7.3 (Pisa). Last year, Steve's motorcycle was handled by the company PINSURA. The

insurance policy included coverage for motorcycle damage resulting from accidents and coverage

for theft. Steve plans to renew his insurance with PINSURA this year, but a number of situations

in his life has changed.

In the table below, we find the factors in Steve's life that have changed.

Indicate, in this table, the correct answers to the following question: "how does the factor

probably affect the cost of Steve's insurance?"

Factor How does the factor probably affect the cost of

Steve's insurance?

Steve traded his old motorcycle for a much more powerful model.

"Increases cost" / "Reduces cost" / "Does not affect cost"

Steve painted his motorcycle with a different color from the original color.


Steve was responsible for two road accidents last year.



To resolve this question, we need to recognize that there are factors that increase the chance of

traffic accidents and, as consequence, make a motorcycle's insurance value more expensive, such

as its exchange for a more powerful model and the owner's history of previous road accidents.

So we have the situation in the following table.

Factor How does the factor probably affect the cost of

Steve's insurance?

Steve traded his old motorcycle for a much more powerful model.

"Increases cost" / "Reduces cost" / "Does not affect cost".

Steve painted his motorcycle with a different color from the original color.


Steve was responsible for two road accidents last year.



133



identify factors that generate cost increases in motorcycle insurance;

identify factors that don’t generate cost increases or decreases in motorcycle insurance.

QUESTION 7.4.

Theme. Immunization against the flu at ACOL.

Question 7.4 (Pisa). Read the text below.

ACOL VOLUNTARY FLU IMMUNIZATION PROGRAM

As you are no doubt aware, the flu can strike rapidly and extensively during winter. It can leave

its victims ill for weeks.

The best way to fight the virus is to have a fit and healthy body. Daily exercise and a diet

including plenty of fruit and vegetables are highly recommended to assist the immune system to

fight this invading virus.

ACOL has decided to offer its staff the opportunity to be immunized against the flu as an

additional way to prevent this insidious virus from spreading amongst us. ACOL has arranged for

a nurse to administer the immunizations at ACOL, during a half-day session in work hours in the

week of May 17. This program is free and available to all members of the staff.

Participation is voluntary. Staff taking up the option will be asked to sign a consent form

indicating that they do not have any allergies, and that they understand they may experience

minor side effects.

Medical advice indicates that the immunization does not produce influenza. However, it may

cause some side effects such as fatigue, mild fever and tenderness of the arm.

Fiona McSweeney, the personnel officer at a company called ACOL, prepared the following

information sheet for ACOL staff.


134

WHO SHOULD BE IMMUNIZED?

Anyone interested in being protected against the virus.

This immunization is especially recommended for people over the age of 65. But regardless of

age, ANYONE who has a chronic debilitating disease, especially cardiac, pulmonary, bronchial or

diabetic conditions.

In an office environment ALL staff are at risk of catching the flu.

WHO SHOULD NOT BE IMMUNIZED?

Individuals hypersensitive to eggs, people suffering from an acute feverish illness and pregnant

women.

Check with your doctor if you are taking any medication or have had a previous reaction to a flu

injection.

You would like to be immunized in the week of May 17 please advise the personnel officer, Fiona

McSweeney, by Friday May 7. The date and time will be set according to the availability of the

nurse, the number of participants and the time convenient for most staff. If you would like to be

immunized for this winter, but cannot attend at the arranged time, please let Fiona know. An

alternative session may be arranged if there are sufficient numbers.

For further information, please, contact Fiona on ext. 5577


a) Which one of the following describes a feature of the ACOL flu immunization program?

A. Daily exercise classes will be run during the winter.

B. Immunizations will be given during working hours.

C. A small bonus will be offered to participants.

D. A doctor will give the injections.

b) The information sheet suggests that if you want to protect yourself against the influenza virus,

you have to take the vaccination. This information sheet suggests that if you want to protect

yourself against the flu virus, a flu injection is

A. more effective than exercise and a healthy diet, but riskier.

B. a good idea, but not a substitute for exercise and a healthy diet.

C. as effective as exercise and a healthy diet, and less troublesome.

D. not worth considering if you have plenty of exercise and a healthy diet.


135

c) Consider the following excerpt from the information sheet.

WHO SHOULD BE IMMUNIZED?

Anyone interested in being protected against the virus.

After Fiona had circulated the information sheet, a colleague told her that she should have left

out the words “Anyone interested in being protected against the virus” because they were

misleading. Do you agree that these words are misleading and should have been left out?

Explain your answer.

d) According to the information sheet, which one of these staff members should contact Fiona?

A. Steve, from the store, who does not want to be immunized because he would rather rely on

his natural immunity.

B. Julie, from sales, who wants to know if the immunization program is compulsory.

C. Alice, from the mailroom, who would like to be immunized this winter but is having a baby in

two months.

D. Michael, from accounts, who would like to be immunized but will be on leave in the week of

May 17.


Item a)

Based on the text, we will analyze the alternatives in this item.

Alternatives A and C.

Alternatives A and C are incorrect, because ACOL did not provide a daily exercise program in the

winter and did not offer a small bonus to the participants in the immunization program.

Alternative B.

Alternative B is correct, because, according to the text, "ACOL has arranged for a nurse to

administer the immunizations at ACOL, during a half-day session in work hours in the week of

May 17".

Alternative D.

Alternative D is incorrect, because a nurse will give the injections in the immunization program,

not a doctor.


136

Item b)

Based on the text, we will analyze the alternatives in this item.

Alternatives A and C.

Alternatives A and C are incorrect, because the text does not say that vaccination is more

effective than (or as effective as) physical exercise and a healthy diet to protect a person against

the flu virus.

Alternative B.

Alternative B is correct because, according to the text, "daily exercise and a diet including plenty

of fruit and vegetables are highly recommended to assist the immune system to fight this

invading virus". Besides, a flu injection cannot be a substitute for exercise and a healthy diet.

Alternative D.

Alternative D is incorrect, because the fact that a person does exercises and has a healthy diet

does not exclude the vaccination.

Item c)

According to the information sheet, "individuals hypersensitive to eggs, people suffering from an

acute feverish illness and pregnant women" should not be vaccinated. Therefore, it would be

incorrect to say that "anyone interested in being protected against the virus" can be immunized.

However, it seems to be an exaggeration to classify the message as "misleading."

Item d)

In the information sheet, it is said that "if you would like to be immunized for this winter, but

cannot attend at the arranged time, please let Fiona know". The staff member who fits this

situation is Michael, "from accounts, who would like to be immunized but will be on leave in the

week of May 17”.

Therefore, the correct alternative is D.



read and interpret a text about an immunization program against influenza proposed by

ACOL;

select alternatives that are consistent with the given information;


137

analyze the discourse used in the information sheet intended for employees of the ACOL

company in order to assess whether it contains a misleading message or not.

QUESTION 7.5.

Theme. Use of cell phones and possible damage to health.

Question 7.5 (Pisa). Consider the following texts.

Are cell phones dangerous?

Yes No

Key point

Conflicting reports about the health risks of cell phones appeared in the late 1990’s.

Key point

Millions of dollars have now been invested in scientific research to investigate the effects of cell phones.

1. Radio waves given off by cell phones can heat up body tissue, having damaging effects.

Radio waves are not powerful enough to cause heat damage to the body.

2. Magnetic fields created by cell phones can affect the way that your body cells work.

The magnetic fields are incredibly weak, and so unlikely to affect cells in our body.

3. People who make long cell phone calls sometimes complain of fatigue, headaches, and loss of concentration.

These effects have never been observed under laboratory conditions and may be due to other factors in modern lifestyles.

4. Cell phone users are 2.5 times more likely to develop cancer in areas of the brain adjacent to their phone ears.

Researchers admit it's unclear this increase is linked to using cell phones.

5. The International Agency for Research on Cancer found a link between childhood cancer and power lines. Like cell phones, power lines also emit radiation.

The radiation produced by power lines is a different kind of radiation, with much more energy than that coming from cell phones.

6. Radio frequency waves similar to those in cell phones altered the gene expression in nematode worms.

Worms are not humans, so there is no guarantee that our brain cells will react in the same way.

http://www.bbc.co.uk/science/hottopics/mobilephones/safety.shtml



138

Key point

Given the immense numbers of cell phone users, even small adverse effects on health could have major public health implications.

Key point

In 2000, the Stewart Report (a British report) found no known health problems caused by cell phones, but advised caution, especially among the young, until more research was carried out. A further report in 2004 backed this up.

If you use a cell phone …

Do Don’t

Keep the calls short. Don't use your cell phone when the reception is weak, as the phone needs more power to communicate with the base station, and so the radio-wave emissions are higher.

Carry the cell phone away from your body when it is on standby.

Don't buy a cell phone with a high “SAR” value. This means that it emits more radiation.

Buy a cell phone with a long “talk time”. It is more efficient, and has less powerful emissions.

Don't buy protective gadgets unless they have been independently tested.

Observation. SAR (rate) is a measurement of how mh electromagnetic radiation is absorved by body tissue

while using a cell phone.


a) What is the purpose of the key points?

A. To describe the dangers of using cell phones.

B. To suggest that debate about cell phone safety is ongoing.

C. To describe the precautions that people who use cell phones should take.

D. To suggest that there are no known health problems caused by cell phones.

b) “It is difficult to prove that one thing has definitely caused another”.

Choose the correct relationship of this piece of information to the point 4 - Yes and No

statements in the table “are cell phones dangerous?”

A. It supports the Yes argument, but does not prove it.

B. It proves the Yes argument.

C. It supports the No argument, but does not prove it.

D. It shows that the No argument is wrong.

http://www.bbc.co.uk/science/hottopics/mobilephones/safety.shtml#stewart



139

c) Look at point 3 in the No column of the table. In this context, what might one of these “other

factors” be? Give a reason for your answer.

d) Look at the table with the heading “if you use a cell phone …“

Which of these ideas is the table based on?

A. There is no danger involved in using cell phones.

B. There is a proven risk involved in using cell phones.

C. There may or may not be danger involved in using cell phones, but it is worth taking

precautions.

D. There may or may not be danger involved in using cell phones, but they should not be used

until we know for sure.

E. The Do instructions are for those who take the threat seriously, and the Don’t instructions are

for everyone else.


Item a)

According to the texts, there are "conflicting reports about the health risks of cell phones

appeared in the late 1990’s"; and "in 2000, the Stewart Report (a British report) found no known

health problems caused by cell phones, but advised caution, especially among the young, until

more research was carried out”. This suggests that “debate about cell phone safety is ongoing",

as set out in Alternative B. In addition, there are no precise answers about the conditions under

which whether or not cell phones are dangerous.

Item b)

In point 4 of the first table, it is said that "cell phone users are 2.5 times more likely to develop

cancer in areas of the brain adjacent to their phone ears", but no evidence is provided to support

this claim. So, the answer is "it supports the Yes argument, but does not prove it”.

Item c)

In point 3 of the first table, it is said that "people who make long cell phone calls sometimes

complain of fatigue, headaches, and loss of concentration". At the same point, this statement is

refuted by "These effects have never been observed under laboratory conditions and may be due

to other factors in modern lifestyles".




140

Item d)

In the table with the heading “if you use a cell phone…“, there are recommendations for the

use of cell phones with caution, even if their harmful effects on health have not been proven yet,

as set out in alternative C.



read and interpret information about the possible damages that the use of cell phones can

make to health;

check whether there are or there are not evidences on statements concerning to the harmful

effects of the use of cell phones;

understand the recommendation to use of cell phones with caution, even before its possible

deleterious effects on health have been proven.

QUESTION 7.6.

Theme. Advantages and disadvantages of telecommuting.

Question 7.6 (Pisa - with adaptations). Read the following texts.

The way of the future

Just imagine how wonderful it would be to “telecommute”1 to work on the electronic highway,

with all your work done on a computer or by phone! No longer would you have to jam your body

into crowded buses or trains or waste hours and hours travelling to and from work. You could

work wherever you want to – just think of all the job opportunities this would open up!

Molly

Disaster in the making

Cutting down on commuting hours and reducing the energy consumption involved is obviously a

good idea. But such goal should be accomplished by improving public transportation or by

ensuring that workplaces are located near where people live. The ambitious idea that

telecommuting should be part of everyone’s way of life will only lead people to become more and

more self-absorbed. Do we really want our sense of being part of a community to deteriorate

even further?

Richard

“Telecommuting” is a term coined by Jack Nilles in the early 1970’s to describe a situation in which workers work on a computer away from a central office (for example, at home) and transmit data and documents to the central office via telephone lines.


141


a) What is the relationship between “The way of the future” and “Disaster in the making”?

A. They use different arguments to reach the same general conclusion.

B. They are written in the same style but they are about completely different topics.

C. They express the same general point of view, but reach to different conclusions.

D. They express opposing points of view on the same topic.

b) What is one kind of work for which it would be difficult to telecommute? Justify your answer.

c) Which statement would both Molly and Richard agree with?

A. People should be allowed to work for as many hours as they want to.

B. It is not a good idea for people to spend too much time getting to work.

C. Telecommuting would not work for everyone.

D. Forming social relationships is the most important part of work.


Item a)

The text "The way of the future" defends telework unreservedly, while the text "What we really

want" suggests that teleworking further deteriorates our sense of being part of a community.

Therefore, these texts express opposing points of view on the same topic, as established in

alternative D.

Item b)

Some types of work are difficult or impossible to do remotely. For example:

nurses cannot make distant drug administrations;

surgeons cannot operate patients with severe illnesses in their own homes;

masters of construction workers cannot make bricks for the building of a house on the

internet;

volleyball players and athletes in general are unable to perform their duties in the mode of

telecommuting.

Item c)

Both Molly and Richard do not defend the ideia that people spend excessive time working, as set

out in alternative B.


142



read and interpret two texts about “teleworking”;

understand different views on distance work.

QUESTION 7.7.

Theme. Semmelweis and death of women when giving birth by puerperal fever.

Question 7.7 (Pisa). Read the following texts.

Text 1 - SEMMELWEIS’ DIARY

July 1846. Next week I will take up a position as “Herr Doktor” at the First Ward of the

maternity clinic of the Vienna General Hospital. I was frightened when I heard about the

percentage of patients who die in this clinic. This month not less than 36 of the 208 mothers died

there, all from puerperal fever. Giving birth to a child is as dangerous as first-degree pneumonia.

These lines from the diary of Ignaz Semmelweis (1818-1865) illustrate the devastating effects of

puerperal fever, a contagious disease that killed many women after childbirth. Semmelweis

collected data about the number of deaths from puerperal fever in both the First and the Second

Wards (see diagram).


143

Physicians, among them Semmelweis, were completely in the dark about the cause of puerperal

fever. Semmelweis’ diary again:

December 1846. Why do so many women die from this fever after giving birth without any

problems? For centuries science has told us that it is an invisible epidemic that kills mothers.

Causes may be changes in the air or some extraterrestrial influence or a movement of the earth

itself, an earthquake.

Nowadays not many people would consider extraterrestrial influence or an earthquake as possible

causes of fever. We now know it has to do with hygienic conditions. But in the time Semmelweis

lived, many people, even scientists, did! Semmelweis knew that it was unlikely that fever could

be caused by extraterrestrial influence or an earthquake. He pointed at the data he collected (see

diagram) and used this to try to persuade his colleagues.

Text 2 - SEMMELWEIS’ DIARY

Part of the research in the hospital was dissection. The body of a deceased person was cut open

to find a cause of death. Semmelweis recorded that the students working on the First Ward

usually took part in dissections on women who died the previous day, before they examined

women who had just given birth. They did not pay much attention to cleaning themselves after

the dissections. Some were even proud of the fact that you could tell by their smell that they had

been working in the mortuary, as this showed how industrious they were!


144

One of Semmelweis’ friends died after having cut himself during such a dissection. Dissection of

his body showed he had the same symptoms as mothers who died from puerperal fever. This

gave Semmelweis a new idea.

Based on the situation described above, answer the following.

a) Suppose you were Semmelweis. Give a reason (based on the data Semmelweis collected) why

puerperal fever is unlikely to be caused by earthquakes.

b) Semmelweis’ new idea had to do with the high percentage of women dying in the maternity

wards and the students’ behavior.

What was this idea?

A. Having students clean themselves after dissections should lead to a decrease of puerperal

fever.

B. Students should not take part in dissections because they may cut themselves.

C. Students smell because they do not clean themselves after a dissection.

D. Students want to show that they are industrious, which makes them careless when they

examine the women.

c) Semmelweis succeeded in his attempts to reduce the number of deaths due to puerperal

fever. But puerperal fever even today remains a disease that is difficult to eliminate.

Fevers that are difficult to cure are still a problem in hospitals. Many routine measures serve to

control this problem. Among those measures are washing sheets at high temperatures.

Explain why high temperature (while washing sheets) helps to reduce the risk that patients will

contract a fever.

d) Many diseases may be cured by using antibiotics. However, the success of some antibiotics

against puerperal fever has diminished in recent years.

What is the reason for this?

A. Once produced, antibiotics gradually lose their activity.

B. Bacteria become resistant to antibiotics.

C. These antibiotics only help against puerperal fever, but not against other diseases.

D. The need for these antibiotics has been reduced because public health conditions have

improved considerably in recent years.


145


Item a)

If an earthquake occured, both the First Ward and the Second Ward of the hospital would be

affected in the same way. With both wards equally hit by the earthquake, if it were the cause of

the women's deaths, the mortality rate should be the same in the First and in the Second Wards.

But that is not what happened: reading the diagram, we can see that the death rate in the First

Ward is always greater than the death rate in the Second Ward. Therefore, a possible earthquake

would not be the cause of these deaths.

Item b)

Semmelweis proposed a hypothesis to explain the high death rate of women in the maternity

ward, related to the behavior of medical students. According to text 2, Semmelweis “recorded

that the students working on the First Wward usually took part in dissections on women who died

the previous day, before they examined women who had just given birth”. They did not pay

much attention to cleaning after dissections. Based on this observation, the new idea of

Semmelweis was as it follows: if students clean themselves after dissections, this should lead to a

decrease in puerperal fever, as established in alternative A.

Item c)

The washing of the sheets at high temperatures helps to reduce the risk of patients contracting

fever, because it is a measure aimed at the removal and/or death of bacteria, germs or virus.

Item d)

The success and the effectiveness of some antibiotics against puerperal fever have declined in

recent years, because the microorganisms that cause diseases have become resistant to

antibiotics, as set out in alternative B.



read and interpret texts;

compare charts data to reject a hypothesis of the cause of mortality of women in maternity;

evaluate the negative effects of indiscriminate use of antibiotics.


146

QUESTION 7.8.

Theme. Depletion of the ozone layer.

Question 7.8 (Pisa). Read the following section of an article about the ozone layer.

The atmosphere is an ocean of air and a precious natural resource for sustaining life on the

Earth. Unfortunately, human activities based on national/personal interests are causing harm to

this common resource, notably by depleting the fragile ozone layer, which acts as a protective

shield for life on the Earth.

Ozone molecules consist of three oxygen atoms, as opposed to oxygen molecules which consist

of two oxygen atoms. Ozone molecules are exceedingly rare: fewer than ten in every million

molecules of air. However, for nearly a billion years, their presence in the atmosphere has played

a vital role in safeguarding life on Earth. Depending on where it is located, ozone can either

protect or harm life on Earth. The ozone in the troposphere (up to 10 kilometres above the

Earth’s surface) is “bad” ozone which can damage lung tissues and plants. But about 90 percent

of ozone found in the stratosphere (between 10 and 40 kilometres above the Earth’s surface) is

“good” ozone which plays a beneficial role by absorbing dangerous ultraviolet (UV-B) radiation

from the Sun.

Without this beneficial ozone layer, humans would be more susceptible to certain diseases due to

the increased incidence of ultra-violet rays from the Sun. In the last decades the amount of

ozone has decreased. In 1974, it was hypothesised that chlorofluorocarbons (CFCs) could be a

cause for this. Until 1987, scientific assessment of the cause-effect relationship was not

convincing enough to implicate CFCs. However, in September 1987, diplomats from around the

world met in Montreal (Canada) and agreed to set sharp limits to the use of CFCs.

Source. Connect, UNESCO International Science, Technology & Environmental Education Newsletter,

section from an article entitled “The Chemistry of Atmospheric policy”, Vol XXII, Nº 2, 19978 (spelling

adapted).

In the text above nothing is mentioned about the way ozone is formed in the atmosphere. In

fact, each day some ozone is formed and some other ozone disappears. The way ozone is formed

is illustrated in the following comic strip.


147


a) Suppose you have an uncle who tries to understand the meaning of this strip. However, he did

not get any science education at school and he doesn’t understand what the author of the strip is

explaining. He knows that there are no little fellows in the atmosphere but he wonders what

those little fellows in the strip stand for, what those strange notations O2 and O3 mean and which

processes the strip represents. He asks you to explain the strip.

Assume that your uncle knows:

that O is the symbol for oxygen;

what atoms and molecules are.

Write an explanation of the comic strip for your uncle. In your explanation, use the words atoms

and molecules in the way they are used in the text.

b) Ozone is also formed during thunderstorms. It causes the typical smell after such a storm. In

the text, the author distinguishes between “bad ozone” and “good ozone”.

In terms of the article, is the ozone that is formed during thunderstorms “bad ozone” or “good

ozone”?

Choose the answer and the explanation that is supported by the text.

Bad ozone or good ozone? Explanation

A Bad It is formed during bad weather.

B Bad It is formed in the troposphere.

C Good It is formed in the stratosphere.

D Good It smells good.

c) In the text, it is stated: “without this beneficial ozone layer, humans would be more

susceptible to certain diseases due to the increased incidence of ultra-violet rays from the Sun”.

Name one of these specific diseases.

d) At the end of the text, an international meeting in Montreal is mentioned. At that meeting lots

of questions in relation to the possible depletion of the ozone layer were discussed. Two of those

questions are shown in the table below.

Can the questions listed below be answered by scientific research?

Indicate "Yes" or "No" for each of the questions.


148

Question Answerable by scientific

research?

Should the scientific uncertainties about the influence of CFCs on the

ozone layer be a reason for governments to take no action? “Yes” / “No”

What would the concentration of CFCs be in the atmosphere in the year

2002 if the release of CFCs into the atmosphere takes place at the same

rate as it does now?

“Yes” / “No”


Item a)

The sequence of explanations of the strip for your uncle may be as shown below.

• An oxygen molecule (O2) is divided into two oxygen atoms (O) by the action of sunlight, as

shown in the first image of the strip.

• A "free" oxygen atom (O) combines with an oxygen molecule (O2) to form an ozone molecule

(O3), as shown in the second and in the third images of the strip.

Item b)

Ozone formed during thunderstorms, which causes the typical smell of post-storm, is a "bad

ozone" formed in the troposphere, as established in situation B of the following table.

Bad ozone or good ozone? Explanation

A Bad It is formed during bad weather.

B Bad It is formed in the troposphere.

C Good It is formed in the stratosphere.

D Good It smells good.

Item c)

Without the ozone layer, humans would be more likely to develop skin cancer, because of the

increased incidence of ultraviolet rays from the sun.

Item d)

In the table below, we indicate the responses requested by this item.


149

Question Answerable by scientific

research?

Should the scientific uncertainties about the influence of CFCs on the

ozone layer be a reason for governments to take no action? “Yes” / “No”

What would the concentration of CFCs be in the atmosphere in the year

2002 if the release of CFCs into the atmosphere takes place at the same

rate as it does now?

“Yes” / “No”



read and interpret a scientific text;

assess the possible consequences of the deterioration of the ozone layer;

understand and explain the mechanism of ozone formation, which involves the "breaking" of

oxygen molecules by the action of solar radiation.

QUESTION 7.9.

Theme. Cultures of genetically modified (GM) corn.

Question 7.9 (Pisa). Read the article and answer the following questions.

GM CORN SHOULD BE BANNED

Wildlife conservation groups are demanding that a new genetically modified (GM) corn

to be banned.

This GM corn is designed to be unaffected by a powerful new herbicide that kills

conventional corn plants. This new herbicide will kill most of the weeds that grow in

cornfields.

The conservationists say that because these weeds are fed by small animals, especially insects,

the use of the new herbicide with the GM corn will be bad for the environment. Supporters of the

use of the GM corn say that a scientific study has shown that this will not happen.


150

Here are details of the scientific study mentioned in the article above.

Corn was planted in 200 fields across the country.

Each field was divided into two. The genetically modified (GM) corn treated with the powerful

new herbicide was grown in one half, and the conventional corn treated with a conventional

herbicide was grown in the other half.

The number of insects found in the GM corn, treated with the new herbicide, was about the

same as the number of insects in the conventional corn, treated with the conventional

herbicide.

Based on the situation described above, answer the following items.

a) What factors were deliberately varied in the scientific study mentioned in the article? Indicate

“Yes” or “No” for each of the following factors.

Was this factor deliberately varied in the study? “Yes” or “No”?

The number of insects in the environment. “Yes” / “No”

The types of herbicide used. “Yes” / “No”

b) Corn was planted in 200 fields across the country. Why did the scientists use more than one

site?

A. So that many farmers could try the new GM corn.

B. To see how much GM corn they could grow.

C. To cover as much land as possible with the GM crop.

D. To include various growth conditions for corn.


Item a)

The types of herbicide used are controlled variables in the scientific study mentioned in the

article. However, the number of insects in the environment is not a variable that can be

controlled in the experiment, that is, it is not a deliberately varied factor.

Therefore, the answers requested are in the following table.


151

Was this factor deliberately varied in the study? “Yes” or “No”?

The number of insects in the environment. “Yes” / “No”

The types of herbicide used. “Yes” / “No”

Item b)

Corn was planted in 200 fields across the country in order to be able to study several different

growth conditions, as established in alternative D.



read and interpret a text about genetically modified (GM) corn;

evaluate controlled and uncontrolled variables in a scientific study about genetically modified

(GM) corn;

identify the goal of planting corn in 200 fields in different parts of the country.

QUESTION 7.10.

Theme. Grand Canyon National Park, USA.

Question 7.10 (Pisa). The Grand Canyon is located in a desert in the USA. It is a very large

and deep canyon containing many layers of rock. Some time in the past, movements in the

Earth’s crust lifted these layers up. The Grand Canyon is now 1.6 km deep in parts. The Colorado

River runs through the bottom of the canyon.

See the picture below of the Grand Canyon taken from its south rim. Several different layers of

rock can be seen on the walls of the canyon.


152


a) About five million people visit the Grand Canyon national park every year. There is concern

about the damage that is being caused to the park by so many visitors.

Can the following questions be answered by scientific investigation? Indicate “Yes” or “No” for

each question.

Can this question be answered by scientific investigation? “Yes” or “No”?

How much erosion is caused by use of the walking tracks? “Yes” / “No”

Is the park area as beautiful as it was 100 years ago? “Yes” / “No”

b) The temperature in the Grand Canyon ranges from below 0oC to over 40oC. Although it is a

desert area, cracks in the rocks sometimes contain water. How do these temperature changes

and the water in rock cracks help to speed up the breakdown of rocks?

A. Freezing water dissolves warm rocks.

B. Water cements rocks together.

C. Ice smoothes the surface of rocks.

D. Freezing water expands in the rock cracks.


153

c) There are many fossils of marine animals, such as clams, fish and corals, in the Limestone A

layer of the Grand Canyon. What happened millions of years ago that explains why such fossils

are found there?

A. In ancient times, people brought seafood to the area from the ocean.

B. Oceans were once much rougher and sea life washed inland on giant waves.

C. An ocean covered this area at that time and then receded later.

D. Some sea animals once lived on land before migrating to the sea.


Item a)

Estimating how much erosion of the Grand Canyon is caused by using walking tracks is

something that can be answered by scientific research, with calculations and comparasions.

However, assessing how beautiful the park was 100 years ago is a subjective issue, which can

not be answered by scientific research. Therefore, the answers requested are in the following

table.

Can this question be answered by scientific investigation? “Yes” or “No”?

How much erosion is caused by use of the walking tracks? “Yes” / “No”

Is the park area as beautiful as it was 100 years ago? “Yes” / “No”

Item b)

The temperature in the Grand Canyon ranges from below 0°C to above 40°C, and sometimes the

cracks in the rocks contain water. As the ice water, when heated, expands in the cracks of the

rock, this can accelerate the breaking of rocks, as established in alternative D.

Item c)

The fact that there are fossils of marine animals in the Limestone A of the Grand Canyon can be

justified by the hypothesis that an ocean has covered that area in the past and has retreated

later as set forth in alternative C.



evaluate issues that can and cannot be answered by scientific research;

apply knowledge of thermal expansion to explain rock breaking in the Grand Canyon;


154

identify a hypothesis to justify the fact that there are fossils of marine animals in the

Limestone A of the Grand Canyon.

QUESTION 7.11.

Theme. Action of acid rain on marble.

Question 7.11 (Pisa). Below is a photo of statues called Caryatids that were built on the

Acropolis in Athens more than 2500 years ago. The statues are made of a type of rock called

marble. Marble is composed of calcium carbonate.

In 1980, the original statues were transferred inside the museum of the Acropolis and were

replaced by replicas. The original statues were being eaten away by acid rain.


a) Normal rain is slightly acidic because it has absorbed some carbon dioxide from the air. Acid

rain is more acidic than normal rain because it has absorbed gases like sulphur oxides and

nitrogen oxides as well. Where do these sulphur oxides and nitrogen oxides in the air come from?

b) The effect of acid rain on marble can be modelled by placing chips of marble in vinegar

overnight. Vinegar and acid rain have about the same acidity level. When a marble chip is placed

in vinegar, bubbles of gas are formed. The mass of the dry marble chip can be found before and

after the experiment.

A marble chip has a mass of 2.0 grams before being immersed in vinegar overnight. The chip is

removed and dried the next day. What will the mass of the dried marble chip be?

A. Less than 2.0 grams.


155

B. Exactly 2.0 grams.

C. Between 2.0 and 2.4 grams.

D. More than 2.4 grams.

c) Students who did this experiment also placed marble chips in pure (distilled) water overnight.

Explain why the students include this step in their experiment.


Item a)

Emission of factory chimneys, products from the exhausting of cars and the burning of fossil fuels

are examples of situations that generate polluting gases in the atmosphere, such as sulfur dioxide

and nitrogen oxide. The combination of these oxides with the water vapor present in the

atmosphere is responsible for the formation of acid rain, which destroys marble statues.

Item b)

A marble sample has 2.0 grams before being immersed in vinegar overnight. The sample is

removed from the immersion and dried the next day. As the vinegar is composed of acetic acid,

part of the marble of the sample reacts with this acid and forms a salt soluble in water.

Therefore, there is "loss" of marble and its final mass is lower than its initial mass. That is, the

final marble mass is less than 2.0 grams, as established in alternative A.

Item c)

Students who did this type of experiment also placed marble samples in pure (distilled) water

throughout the night. The students adopted this procedure to show that the marble does not

react with pure water: the presence of an acid (in this case, the acetic acid constituent of the

vinegar) is necessary for the occurrence of the chemical reaction under study.



understand the effects of acid rain on marble;

identify agents responsible for the formation of acid rain;

justify procedures adopted in an experiment on the transformation of marble into a water-

soluble salt.


156

QUESTION 7.12.

Theme. Benefits of regular and moderate exercise.

Question 7.12 (Pisa). Regular but moderate exercise is good for our health.

Based on the above, answer the following items.

a) What are the advantages of regular physical exercise? Indicate "Yes" or "No" for each

statement in the table below.

Is this an advantage of regular physical exercise? “Yes” or “No”?

Physical exercise helps prevent heart and circulation illnesses. “Yes” / “No”

Physical exercise leads to a healthy diet. “Yes” / “No”

Physical exercise helps to avoid becoming overweight. “Yes” / “No”

b) Why do you have to breathe more heavily when you’re doing physical exercise than when

your body is resting?


Item a)

Performing physical exercise can aid in the prevention of coronary and circulatory diseases and in

weight reduction, but it is not the cause of a healthy diet.

Therefore, the answers requested are in the following table.


157

Is this an advantage of regular physical exercise? “Yes” or “No”?

Physical exercise helps prevent heart and circulation illnesses. “Yes” / “No”

Physical exercise leads to a healthy diet. “Yes” / “No”

Physical exercise helps to avoid becoming overweight. “Yes” / “No”

Item b)

You have to breathe more intensively when you’re doing physical exercise than when your body

is resting in order to avoid high levels of carbon dioxide and provide more oxygen to the body.



analyze correctly the positive consequences of performing physical exercises;

justify the fact that we have to breathe more intensely when we are doing physical exercise

than when we are resting.

QUESTION 7.13.

Theme. Level of sun protection of a photoprotective product.

Question 7.13 (Pisa). Mimi and Dean wondered which sunscreen product provides the best

protection for their skin. Sunscreen products have a Sun Protection Factor (SPF) that shows how

well each product absorbs the ultraviolet radiation component of sunlight. A high SPF sunscreen

protects skin for longer than a low SPF sunscreen.

Mimi thought of a way to compare some different sunscreen products. She and Dean collected

the following:

two sheets of clear plastic that do not absorb sunlight;

one sheet of light-sensitive paper;

mineral oil (M) and a cream containing zinc oxide (ZnO); and

four different sunscreens that they called S1, S2, S3, and S4.

Mimi and Dean included mineral oil because it allows most of the sunlight go through, and zinc

oxide because it almost completely blocks sunlight.

Dean placed a drop of each substance inside a circle marked on one sheet of plastic, and then

put the second plastic sheet over the top. He placed a large book on top of both sheets and

pressed it down.

This procedure is illustrated below.


158

Mimi then put the plastic sheets on top of the sheet of light-sensitive paper. Light-sensitive paper

changes from dark gray to white (or very light gray), depending on how long it is exposed to

sunlight.

Finally, Dean placed the sheets in a sunny place.

Based on the above, answer the following items.

a) Which one of these statements is a scientific description of the role of the mineral oil and the

zinc oxide in comparing the effectiveness of the sunscreens?

A. Mineral oil and zinc oxide are both factors being tested.

B. Mineral oil is a factor being tested and zinc oxide is a reference substance.

C. Mineral oil is a reference substance and zinc oxide is a factor being tested.

D. Mineral oil and zinc oxide are both reference substances.

b) Which one of these questions were Mimi and Dean trying to answer?

A. How does the protection for each sunscreen compare to the others?

B. How do sunscreens protect your skin from ultraviolet radiation?

C. Is there any sunscreen that gives less protection than mineral oil?

D. Is there any sunscreen that gives more protection than zinc oxide?

c) Why was the second sheet of plastic pressed down?

A. To stop the drops from drying out.

B. To spread the drops out as far as possible.


159

C. To keep the drops inside the marked circles.

D. To make the drops the same thickness.

d) The light-sensitive paper is dark gray and fades to a lighter gray when it is exposed to some

sunlight, and to white when exposed to a lot of sunlight.

Which one of these diagrams shows a pattern that might occur? Explain why you chose it.


Item a)

Mineral oil and zinc oxide are both reference substances, as set forth in alternative D. This can be

verified by the following excerpt from the heading: "Mimi and Dean included mineral oil because

it allows most of the sunlight go through, and zinc oxide because it almost completely blocks

sunlight".

Item b)

Mimi and Dean were trying to find out "how the protection of sunscreen data compares to the

protection of others" as set out in alternative A. This can be verified by the following excerpt from

the heading: "Mimi and Dean wondered which sunscreen product provides the best protection for

their skin".

Item c)

The second plastic sheet was pressed to make drops of the same thickness, as established in

alternative D.

Item d)


160

On light-sensitive paper, dark gray becomes lighter gray when there is little exposure to sunlight

and white when there is much exposure to sunlight. In this case, ZnO spot must have remained

gray (because zinc oxide blocks sunlight) and spot M must have gone white (because mineral oil

absorbs little sunlight), as shown in the picture in alternative A.



understand the steps and objectives of an experimental procedure on the determination of

the sun protection factor of a substance;

distinguish "reference substance" and "substance to be tested";

predict a result in a scientific methodology.

QUESTION 7.14.

Theme. History and indication of vaccination.

Question 7.14 (Pisa). Read the following newspaper article and answer the questions that

follow.

THE HISTORY OF VACCINATION

Mary Montagu was a beautiful woman. She survived an attack of smallpox in 1715 but she was

left covered with scars. While living in Turkey in 1717, she observed a method called inoculation

that was commonly used there. This treatment involved scratching a weak type of smallpox virus

into the skin of healthy young people who then became sick, but in most cases only with a mild

form of the disease.

Mary Montagu was so convinced of the safety of these inoculations that she allowed her son and

daughter to be inoculated.

In 1796, Edward Jenner used inoculations of a related disease, cowpox, to produce antibodies

against smallpox. Compared to the inoculation of smallpox, this treatment had less side effects

and the treated person could not infect others. The treatment became known as vaccination.


161

Available from <https://www.poetryfoundation.org/poets/lady-mary-wortley-montagu>. Access Jun. 01 2018.


a) What kinds of diseases can people be vaccinated against?

A. Inherited diseases like haemophilia.

B. Diseases that are caused by viruses, like polio.

C. Diseases from the malfunctioning of the body, like diabetes.

D. Any sort of disease that has no cure.

b) If animals or humans become sick with an infectious bacterial disease and then recover, the

type of bacteria that caused the disease does not usually make them sick again.

What is the reason for this?

A. The body has killed all bacteria that may cause the same kind of disease.

B. The body has made antibodies that kill this type of bacteria before they multiply.

C. The red blood cells kill all bacteria that may cause the same kind of disease.

D. The red blood cells capture and get rid of this type of bacteria from the body.

c) Give one reason why it is recommended that young children and old people, in particular,

should be vaccinated against influenza (flu).


162


Item a)

People should be vaccinated against diseases caused by viruses, such as poliomyelitis (polio), as

set out in alternative B. This is due to happen because vaccination sensitizes the body's immune

system, which prevents the occurrence of diseases caused by specific bacteria and virus.

Item b)

Vaccination is the introduction, in a person or an animal, of a substance capable of creating

immunity to a particular disease caused by virus or bacteria. The immunity generated by the

vaccine is based on the ability of the body to react to infectious agents through the production of

antibodies. Therefore, the correct alternative is B.

Item c)

Young children (aged 6 months to 5 years) and elderly people (60 years old or older) are in the

called "priority groups" for receiving the influenza vaccine. This is because these groups are

vulnerable to contracting the most serious form of the flu, which can progress to pneumonia or

death, because they have the "weakest" immune system. Therefore, vaccination of children and

the elderly is an important preventive measure in the public health area.



assess the types of diseases that can be prevented by vaccination;

understand the mechanism by which vaccination is carried out;

justify the priority vaccination of young children and old people against influenza.

QUESTION 7.15.

Theme. Intelligent clothes for children with speech disabilities.

Question 7.15 (Pisa). Read the text and answer the questions.


163

CLOTHES TEXT

A team of British scientists is developing “intelligent” clothes that will give disabled children the

power of “speech”. Children wearing waistcoats made of a unique electrotextile, linked to a

speech synthesiser, will be able to make themselves understood simply by tapping on the touch-

sensitive material.

The material is made up of normal cloth and an ingenious mesh of carbon-impregnated fibers

that can conduct electricity. When pressure is applied to the fabric, the pattern of signals that

passes through the conducting fibers is altered and a computer chip can work out where the

cloth has been touched. It then can trigger whatever electronic device is attached to it, which

could be no bigger than two boxes of matches.

“The smart bit is in how we weave the fabric and how we send signals through it – and we can

weave it into existing fabric designs so you cannot see it’s in there”, says one of the scientists.

Without being damaged, the material can be washed, wrapped around objects or scrunched up.

The scientist also claims it can be mass-produced cheaply.

Source. Steve Farrer, “Interactive fabric promises a material gift of the garb”, The Australian, 10 August

1998.


a) Can these claims made in the article be tested through scientific investigation in the

laboratory? Indicate “Yes” or “No” for each.

The material can be

Can the claim be tested through scientific

investigation in the laboratory?

washed without being damaged. “Yes” / “No”

wrapped around objects without being damaged. “Yes” / “No”

scrunched up without being damaged. “Yes” / “No”

mass-produced cheaply. “Yes” / “No”

b) Which piece of laboratory equipment would be among the equipment you would need to

check if the fabric is conducting electricity?

A. Ammeter.

B. Light box.

C. Micrometer.

D. Sound meter.


164


Item a)

Characteristics related to the "physical" behavior of the material can be investigated in the

laboratory. Cost issues, for example, depend on financial and marketing evaluations.

So we have the answers in the following table.

The material can be

Can the claim be tested through scientific

investigation in the laboratory?

washed without being damaged. “Yes” / “No”

wrapped around objects without being damaged. “Yes” / “No”

scrunched up without being damaged. “Yes” / “No”

mass-produced cheaply. “Yes” / “No”

Item b)

The ammeter, indicated in alternative A, is laboratory equipment that measures the intensity of

electric current passing through a conductor.



understand the content of a text about intelligent tissues that can help children with speech

disabilities;

check characteristics relating to a material that can be investigated in the laboratory;

identify a device that measures the intensity of the electric current.

QUESTION 7.16.

Theme. Footwear specially designed for sportsmen.

Question 7.16 (Pisa). Read the following text.


165

Feel good in your runners For 14 years the Sports Medicine Centre of Lyon (France) has been studying the injuries of young

sports players and sports professionals. The study has established that the best course is

prevention … and good shoes.

Knocks, falls, wear and tear...

Eighteen percent of sports players aged 8 to 12 already have heel injuries. The cartilage of a

footballer's ankle does not respond well to shocks, and 25% of professionals have discovered for

themselves that it is an especially weak point. The cartilage of the delicate knee joint can also be

irreparably damaged and if care is not taken right from childhood (10–12 years of age), this can

cause premature osteoarthritis. The hip does not escape damage either and, particularly when

tired, players run the risk of fractures as a result of falls or collisions.

According to the study, footballers who have been playing for more than ten years have bony

outgrowths either on the tibia or on the heel. This is what is known as “"footballer’s foot”, a

deformity caused by shoes with soles and ankle parts that are too flexible.

Protect, support, stabilize, absorb

If a shoe is too rigid, it restricts movement. If it is too flexible, it increases the risk of injuries and

sprains. A good sports shoe should meet four criteria:

Firstly, it must provide exterior protection: resisting knocks from the ball or another player,

coping with unevenness in the ground, and keeping the foot warm and dry even when it is

freezing cold and raining.

It must support the foot, and in particular the ankle joint, to avoid sprains, swelling and other

problems, which may even affect the knee.

It must also provide players with good stability so that they do not slip on a wet ground or skid

on a surface that is too dry.

Finally, it must absorb shocks, especially those suffered by volleyball and basketball players who

are constantly jumping.


166

Dry feet

To avoid minor but painful conditions such as blisters or even splits or athlete’s foot (fungal

infections), the shoe must allow evaporation of perspiration and must prevent outside dampness

from getting in. The ideal material for this is leather, which can be water-proofed to prevent the

shoe from getting soaked the first time it rains.

Based on the article, answer the following items.

a) What does the author intend to show in this text?

A. That the quality of many sports shoes has greatly improved.

B. That it is best not to play football if you are under 12 years of age.

C. That young people are suffering more and more injuries due to their poor physical condition.

D. That it is very important for young sports players to wear good sports shoes.

b) According to the article, why should sports shoes not be too rigid?

c) One part of the article says, “a good sports shoe should meet four criteria.”

What are these criteria?

d) Look at this sentence from near the end of the article. It is presented here in two parts.

First part.

“To avoid minor but painful conditions such as blisters or even splits or athlete’s foot (fungal

infections)”…

Second part.

“…the shoe must allow evaporation of perspiration and must prevent outside dampness from

getting in”.

What is the relationship between the first and second parts of the sentence?

The second part

A. contradicts the first part.

B. repeats the first part.

C. illustrates the problem described in the first part.

D. gives the solution to the problem described in the first part.


Item a)

The author intends to show the importance of young players wearing good sports shoes in order

to avoid injuries and other problems arising from sports. Therefore, the correct alternative is D.


167

Item b)

According to the text, if a shoe is very rigid, it restricts the movements.

Item c)

According to the text, a good sports shoe must meet four criteria, as shown below.

To promote external protection against kicking and to keep the foot warm and dry even when

it is freezing cold and raining.

To provide foot support to prevent sprains, swelling and other problems.

To provide good stability, with the intention that the players do not slip on wet surfaces.

To absorve shocks, especially those suffered by athletes constantly.

Item d)

The second part of the sentence (“…the shoe must allow evaporation of perspiration and must

prevent outside dampness from getting in”) provides the solution to the problem described in the

first part (“to avoid minor but painful conditions such as blisters or even splits or athlete’s foot

(fungal infections)…”).. So the correct alternative is D.

Comments on resolution 7.16.


understand the main idea of a text;

identify the criteria to be met by "a good sports shoe";

understand the relationship between the first part and the second part of a sentence.


168

CHAPTER 8. Bibliographic indications and consulted links.

OECD (2001), Knowledge and Skills for Life – First Results from PISA 2000. Acess in May,

2017, in

https://www.oecd.org/edu/school/programmeforinternationalstudentassessmentpisa/3369162

0.pdf.

OECD (2003), 2003 First PISA Assessment Together with 28 OECD Countries in 2000. Acess

in June, 2017, in

http://www.oecd.org/edu/school/programmeforinternationalstudentassessmentpisa/33690591

.pdf.

OCDE (2005), PISA 2003 Technical Report. Acess em October, 2016, in

https://www.oecd.org/edu/school/programmeforinternationalstudentassessmentpisa/3518857

0.pdf.

OCDE (2007), PISA 2006 Science Competencies for Tomorrow’s World. Acess in October,

2016, in https://www.oecd.org/pisa/pisaproducts/39725224.pdf.

OECD (2010), PISA 2009 Results: What Students Know and Can Do – Student Performance in

Reading, Mathematics and Science. Acess in October, 2016, in

http://www.oecd.org/pisa/pisaproducts/48852548.pdf.

OECD (2014), PISA 2012 Results in Focus: What 15-year-olds Know and What They Can Do

with What They Know. Acess in October, 2016, in http://www.oecd.org/pisa/keyfindings/pisa-

2012-results-overview.pdf.

OCDE (2016), PISA 2015 Results in Focus. Acess em December, 2016, in

https://www.oecd.org/pisa/pisa-2015-results-in-focus.pdf.

<https://nces.ed.gov/surveys/pisa/pdf/items2_reading.pdf>

<https://nces.ed.gov/surveys/pisa/pdf/items_financial.pdf>

<https://nces.ed.gov/surveys/pisa/pdf/items_math.pdf>

<https://nces.ed.gov/surveys/pisa/pdf/items2_math2012.pdf>

<https://nces.ed.gov/surveys/pisa/pdf/items_reading.pdf>

<https://nces.ed.gov/surveys/pisa/pdf/items_science.pdf>

<https://www.oecd.org/pisa/38709418.pdf>

<https://nces.ed.gov/surveys/pisa/pdf/items2_solving.pdf>

https://www.oecd.org/edu/school/programmeforinternationalstudentassessmentpisa/35188570.pdf

https://www.oecd.org/edu/school/programmeforinternationalstudentassessmentpisa/35188570.pdf

http://www.oecd.org/pisa/pisaproducts/48852548.pdf



http://www.oecd.org/pisa/keyfindings/pisa-2012-results-overview.pdf

http://www.oecd.org/pisa/keyfindings/pisa-2012-results-overview.pdf

https://nces.ed.gov/surveys/pisa/pdf/items_math.pdf

https://nces.ed.gov/surveys/pisa/pdf/items2_math2012.pdf

https://nces.ed.gov/surveys/pisa/pdf/items_reading.pdf

https://nces.ed.gov/surveys/pisa/pdf/items_science.pdf

https://eur02.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.oecd.org%2Fpisa%2F38709418.pdf&data=02%7C01%7C%7Caeb71d0db65541e8047808d5b1c40abd%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C636610377905497692&sdata=%2FG4gY8FOK2xuA%2BTjlDfZl4QPUNGS%2BwnVPX2AIjNMFXI%3D&reserved=0

https://eur02.safelinks.protection.outlook.com/?url=https%3A%2F%2Fnces.ed.gov%2Fsurveys%2Fpisa%2Fpdf%2Fitems2_solving.pdf&data=02%7C01%7C%7Caeb71d0db65541e8047808d5b1c40abd%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C636610377905497692&sdata=Ib76eCMGbOjY58ywuGT4YHSB3UAfLLuPzZFTKKHv2gE%3D&reserved=0


169

I thank you for the gracefully gifts received from God.

Thank you for the warmth, support and encouragement received from my family.

Thank you for the constant motivation received from friends.

Thank you for the reading and contributions received from Fabíola Mariana Aguiar Ribeiro,

Jamilson José Alves da Silva, Rogerio Menale and Tiago Guglielmeti Correale.

Christiane Mazur Doi

exercising the abilities of reasoning, calculating and … · 2019-03-26 · exercising the...

Documents