1

Cross-curricular Approach of Mathematics

Methodological Guide

“QED – Quality in Europe's Diversity”

Comenius Multilateral Partnership 2011 - 2013

11-PM-75-BT-RO

2

Chapters:

I. Applications of Mathematics in Science ………………..page 3

QED – team of Borgarholtsskóli from Reykjavik, Iceland

II, Maths in cultural and social life ………………..page 12

QED – team of Geniko Likeio from Kissamos, Greece

III. Mathematics &Economy ………………..page 24

QED – team of Ahmet Eren Anadolu Lisesi from Kayseri Turkey

IV. ICT applications in math teaching ………………..page 28

QED – teams of Goetheshule Wetzlar, Germany, and CETCP Botosani,

Romania

External:

http://www.geogebratube.org/collection/show/id/3309

http://www.slideshare.net/effiefil/maths-in-culturar-life-new

http://new-twinspace.etwinning.net/c/portal/layout

3

I. Applications of Mathematics in Science

The QED – team from Borgarholtsskóli, Iceland

Natural sciences are important for daily life. It is important that students choose to study

natural sciences, health sciences and engineering. In Iceland today there is a lack of students

who study these subjects. Students seem to choose something else.

Mathematics is closely related to science. Natural sciences can not be without

mathematics. For students, natural sciences and mathematics, tend to be difficult and they are

intimidated by them. For those who like to study these subjects it is often difficult to transfer the

knowledge of them to other subjects.

The exercises in this pamphlet are based on to link together mathematics and physics

through assignments which are done „manually! “ The goal by combining mathematics and

physics is to integrate natural sciences with mathematics. By using practical and „manual“

exercises the students should increase their understanding and also they might be more positive

towards natural sciences. In addition these exercises are important for physics.

The exercises have been used in the first and second level in physics in

Borgarholtsskóli, Iceland. The uniqueness of the school is that the background of the students is

very different and that means that the teaching is very individually based.

The exercises can easily be changed, that is, their focus can be changed, they can be

made more difficult, they can be less time consuming or more time consuming.

Exercise 1: Errors in measurements

Purpose:

When doing measurements in science there is never a 100% certainty in the numbers we

measure. There is always some minimum resolution we can get, determined by the

4

instruments we use or by the users doing the measurements. In this exercise student will

practice measuring objects using simple tools, determining the uncertainty or error in the

measurements and then use errors in simple calculations. As a prerequisite, students need

to know how to determine errors in measurements and to do calculations using errors.

Materials

Stopwatch

Ruler or caliper

A few pencils or sticks

Procedure

Part 1: Sticks, length, addition

Groups get one stick/pencil for each member of the group

a) Find an empty page in your workbook. Write down the name of the exercise, the date

and the names of the students in your group

b) Give each stick a „name“ or make sure to be able to distinguish between them.

c) Each student measures the length of one stick and determines the uncertainty/error in

the measurements. Write all the results down, including the name of the person doing

each measurement

d) Calculate the total length of the sticks and determine the total error in that

determination

e) Calculate the average length of all the sticks and the error in that value using the

MIN/MAX method of determining total errors. Make sure to document everything in

your workbook

Part 2: Box, volume, multiplication, % error

Each group of students gets one small box

a) Write down a name for the box, draw a picture and mark each edge so it´s certain

what side is what

b) Measure the length of the edges and determine the error in the measurements.

5

c) Calculate the volume of the box and determine the errors in that using the

MIN/MAX method

d) Calculate percentage error which is the error divided by volume

Part 3: Measuring time, % error, instrument error, user error

Each group of students gets a stopwatch

a) Each student tries 5 times to start a stopwatch and stop at exactly 10:00 sec. Each

student writes down his numbers

b) Each student calculates the average time and determines the error. Also the % error

which is error divided by average time

There are 2 main ways to determine the errors in this step

i) Standard deviation: Use a spreadsheet or a calculator to calculate average time and

standard deviation. The error is then the standard deviation

ii) Min/max variation:

6

Exercise 2: Graphs and lines

Purpose

The purpose of this exercise is for students to practice matching curves or lines to specific

equations. Once that is done, it´s possible to match the curves and values from the curves

to certain physical laws or rules.

Materials

Graphing paper and a ruler

Procedure

Part 1: Calculating a trend line

Draw the points/coordinates in table 2 on a graph. The v

[m/s ] is on the y-axis and t [s] on the x-axis and make sure

to mark clearly the axis t or v. Draw a straight line (using a

ruler) that intersects all the points on the graph and calculate

the equation for the line. The equation is on the form y = kx

+ c.

Part 2: Graph matching

Draw the data in table 2 on a graph and try to determine

which type of a line/curve fits the data. Then determine the

equation for the line. I.e. calculate a value for k or a, b and

c.

Table 1

t [s] v [m/s]

0 1

1 4

2 7

3 10

4 13

Table 2

x-axis y-axis

0 0

1 2

2 8

3 18

4 32

7

Suggestions to try out:

Linear equation

8

x. The force constant (k) of each spring, determines how hard it is to stretch or compress it. A

stiff spring has a large k but a loose spring has a low k. The purpose of this experiment is to

determine the force constant in a spring by drawing a graph and calculating the trend line. An

experiment was done where force was applied to the spring and the stretching of the spring was

measured.

Materials

Graphing paper and a ruler

Procedure

Table 1 contains data from a physics experiment. Force was applied to a spring and the length

that the spring stretched was measured.

1. Draw a graph and each point on the graph. Mark the axis

clearly as y-axis Force, F, [N] and x-axis as

expansion/stretch of the spring, x, [m]. Make sure to keep

the scale on the graph such that it´s is detailed. Try to use

whole page in your book.

2. Draw the error bars on each point. The errors in F will

be shown as small lines extending up and down from each

point. The error in x will be shown from left to right. When

all is done, the graph should look like it has six small

crosses.

3. Use a ruler and try to find out what you feel is the „best“ straight line that intersects the group

of points the best. Do not connect the dots, the end results should be a graph showing small

crosses and one straight line intersecting most of them.

4. Determine the equation for the line you have drawn. Make sure to pick points that are

actually on the line, to do the calculations. The equation should be on the form y=kx+c, (F=kx)

where c is 0.

5. Compare the equation and Hooke´s law. What is the value of the spring´s force constant (k)?

Table 1

x-axis y-axis

X [m] +/-

0,03

F [N] +/-

0,5

0,00 0,0

0,30 10,0

0,43 15,0

0,59 20,0

0,71 25,0

0,90 30,0

9

6. Use the equation to forecast how much you would expect the spring to stretch if a force of 23

N was applied to the spring.

Exercise 4: Graph matching

Purpose

Scientist often do experiments to test hypothesis. A large number of data points are then

measured and then the scientist tries to make sense of it all and to find the mathematical

correlation between variables. In this exercise data from physics experiments is collected in

tables. The data must be entered into graphing software, the relevant graph plotted and matched

to mathematical functions. When the equations are determined the software shows the relevant

constants that can then be matched to the variables that are unknown in each physics equation.

Materials

Graphic calculator or graphing software

Method

Input the data in tables 1-4 in to a graphics calculator or graphing software one at a time. Then

match each curve to the correct physics equations. I.e. find the equation that matches each table.

Use the equations for the curves to calculate or determine the relevant constants in each equation.

Table 1

X y

1 66,70

2 16,68

3 7,41

4 4,17

5 2,67

6 1,85

7 1,36

8 1,04

10

Equation 1: Position (s) in relation to time lapsed (t)

11

This equation is called the ideal “gas” equation. It is derived by many famous scientists

who each contributed greatly to physics and chemistry studies in the past like: Clapeyron,

Boyle and Charles. This equation has as a limit that it only applies to an “ideal” gas which

is a hypothetical gas where there are no interactions between the gas molecules. However it

is often a good approximation to real gas.

Equation 3: Describes the force of gravity (F) between two objects as function of the

distance between them (r)

12

II, Maths in cultural and social life

QED – team of Geniko Likeio from Kissamos, Greece

13

14

15

16

17

18

19

20

21

22

23

24

III. Mathematics & Economy

QED – team of Ahmet Eren Anadolu Lisesi from Kayseri Turkey

John Nash when he visited our country, according to OECD countries are far below average

in mathematics to learn mathematics with a weak justice system is weak in countries like

writing a sentence written on the corner a lot of comments made, he said, the issue from a

different perspective, a perspective essential to the economy and financial mathematics was.

Economy and finance work on some math before.

A direct correlation between mathematics and the justice system and Nash's work on this issue of

the existence of a concrete links to the first page of google is not coming out from across the

academic studies of the relationship between mathematics and social justice * was able to

achieve. The OECD's PISA (Programme for International Student Assessment) the name of the

member states of the research being done for many years. Accordingly, the sin of the neck of

PISA, Turkey not only the math behind the OECD average. The other two titles in the field of

Science and Reading below the average of OECD member countries. We're not the last or second

to last ** But, on the bright side of this.

OECD and Nash do not know, but for a long time in a generation, "He wears the head teachers"

believing in the existence of (except for exceptions) grew paranoid schizophrenic behavior and

we all know that a thought. One way or already in this world, no friends other than Turks do not.

Fitting your head, so this is usually the title of the positive sciences of mathematics, physics and

chemistry in the equation, the theory was hosting the topic. If the reason is unknown, but is it a

result of this, the number of Facebook users around the world 5 Taking Ranked assume that the

social sciences and perhaps even be able to say a society more prone to socializing. However,

team games, participate in the Olympics for the first time that the family of the concept of

solidarity is so strong, so socialization imecenin and social media in a separate contradiction.

Ready yakalamışken Nash Is there a correlation between them sorulsaydı question he could have

been better.

High school years. Table of elements in chemistry knows almost by heart, how many protons,

neutrons have a few now learn a curiosity. Higgs Bozonu'na luckily we did not have to memorize

them a newer reached. In mathematics, the axioms and theorems do not remember many of the

25

functions, solutions and parabollerini you draw three unknowns. It is impossible to forget

parabola, even after many years. Middle and high school during my college level 3 the largest

contribution to the integral degree of learning professional life to me, was that it provided

analytical and strategic thinking competencies. As well as the "Age of Enlightenment",

especially French literatures, Montesquieu, Rousseau, Moliere, read the works of authors such as

Balzac, all the best and forward-questioning and integrating the social dimension of the positive

sciences, this learning process to be a world citizen who loves his country was a very significant

contribution. Eşdeneği Baccalauréat in France diplomamda accepted these reasons, I had to give

the right of mathematics. It is not easy in the curriculum and establishment of the Turkish

Literature, and French Literature and Grammaire'ine to have. Moreover, the two-stage OSS and

IMS prepares to enter college, university-level education in the sciences, Plasticity positive. 2. 4

hours per week English as a language of the bargain. And of course in the college preparatory

courses for those years were infallible system. Year of high school graduation, in 1990 and

again, for those wondering, though I read it eight years.

Our teachers of French origin which is usually higher education in their own countries and their

military service in developing countries were French citizens who exercised their areas of

expertise to continue. The mid-2nd grade surname Fau (faux French as the probe over an x letter

word that can go wrong), French accent, a French teacher because my friends were having

difficulty understanding, including math courses, one of the greatest lessons life taught me.

Appreciation and thanks too close to, or who want to ensure that as many students who pass the

course, or, at the end of the year I wanted to support him in order to pass the class.

Replenishment of course I do not want to poison a single holiday in adhering to pass him in

class, giving me 5 annotations, as a bonus, all right I said I wanted to support. But there was a

condition. My eyes flashed, I said yes. Note book, opened it and showed me the notes that all

year round. 5 If you show me one and I'll give you 5 to spend and the class said. I looked at all

the notes in both periods was 4. That day, I noticed that there was the fact that teachers who wear

us, on our part, we have not had enough. Mr. Fau French-speaking dialects, ie "Monsieur wrong"

we wear the name of the teacher and gave me a great lesson on the truth taught to qualify. Did

not know mathematics today, but that does not excuse the bottom of the economies of the

advanced mathematics of derivative products, ie, the dose due to kidnapping and being more

26

prescriptive system-dependent and can easily esneyemediklerinden. So some features are not in

the know, our work does not. Practical intelligence, it works in a lot of places.

Now let's Game theory and mathematics. Game theory briefly human behavior, math, strategy

includes the titles of every individual in the group's strategic choices and shows the status of the

whole group. So the relationship between social existence and consequences of human behavior

examines the social order. Personal interests to co-operate with the public interest or individual

acting reveal potential conflicts between the theories of game one of the best-known "prisoners

dilemma". This game is described with the aid of theories *** Matrices, taxes, "Prisoners

Dilemma" and we consider, as a result, the picture is as follows. Each individual does not tax its

own interests in mind, or kaçındığında everyone the full tax situation occurs when a much worse

outcome. "Zero sum game" at the end of the day, the fact that the overall game theory there is no

winner. So the fact that the individual who thinks he has won not pay any tax on the side of the

losers of the game as a whole takes place. In mathematics, we have an individual tax planning as

well as the fact that the social economy as a whole in the fact that the results were inconclusive

in this sense goes back to the primary order. Informality, transparency, the same thing occurs

titles such as tax havens. Nash equilibrium theory of game theory and game theory, including

here it is gaining importance, ie, the maximum recovery of lost or at least get everyone's

determination of the equilibrium point of the situation.

The other is the title of our article, game theory, which is critical in relation to mathematics, if

not the direct subject of the economy, a sub-title finance. Retired Teacher and against Saleh Bu

neighbor aunt wants to assess knowledge, you do not know the relationship between nominal and

real interest rates and inflation will return and the present financial mathematics. A lack of cases

in the calculation of simple interest and compound interest is felt in many areas. 91-day treasury

bill discount rate (yield) and the expression of sex-month-year deposits (for example, 9% per

month interest rate) but thinks Saleh Bu Aunt, Uncle, and fortunately does not know the

definition of the opportunity cost of an alternative. In fact, investment specialists and portfolio

managers trust the situation is not much different from them most of the time. Branches of

current interest in favor of the working principle of state sovereignty Aunt Ayşe eşitlese

sometimes, the need for double-sided game theory when you go towards the financial math, in

fact, the country's deposit base and the maturity structure of the "Zero Sum Game" will support

tends to argue. This vicious cycle, a lot of apples and oranges compared to the investment

27

decision, and foreign exchange, gold, stock market investments, the financial center of the

triangle is limited to the spiral of a candidate country. Taxi plate, pick-line and real estate is still

the most popular investment instrument because it is easy to account. The owner of the plate than

the father gained a lot of investors and investment specialist with over years. It also does not

require knowledge of financial mathematics. Market mathematics is sufficient. I bought one

years after the sentence has doubled to 1 second, although exponential function to create a

parabola, though, and easy to understand words, the wealthy If the property'm broke his jaw.

Credit cards and consumer loans using cash loan debt by paying the minimum each month to

skip to cause a social disaster in the name of the point where we actually a summary of financial

mathematics. A similar situation without cash budget, creating a company manage the pro-forma

cash flow statement is true for a lot of running and managing the boss. The machine will park as

well as the net present value of investment and production plans take account of the break-even

point, cash management, balance sheet appears planlamayarak (and invisible) said that an SME

will pay somehow looking at stock values, rather than long-term financing overdraft or revolving

credit account with the use of work to finance investment in the financial mathematics is the

ultimate. The last 20 years on behalf of entrepreneurship is about 3 times the number of SMEs

are micro enterprises at a rate of 99.3%, but still be / remain institutionalized and growing capital

base as an indicator of distress spreading a concrete sign of our need of financial mathematics.

Market, grocery store scales, the use of fraudulent, taxi tourists to walk the city as an indicator of

hospitality, the amount of iron used in construction of earthquake country, treasure land illegally

erected a small shanty illegal electricity accounts to the use of mathematics, game theory that

dominated in recent years have reached the most critical point. Used in all cases that are not our

business "Let's see the size of NAS to NAS" approach is no longer the period of Nash's high pass

approach. If you are going to be one of the 10 largest economies in the world by 2023, we need

to know that the teachers will support us, no matter which branch of positive science does not

wear, say the truth in general. Mathematics and statistics acceptable confidence interval and

standard (non-standard) except for the deviations is always, of course.

28

IV. ICT applications in math teaching

QED – teams of Goetheshule Wetzlar, Germany, and CETCP Botosani,

Romania

1. Teaching Maths

and “New Media“ at German schools

First we have to define „what is meant by new media“

The term „new media“ in the following refers to the use of pc systems and / or calculators that

can handle graphics by students as well as presentations by teachers using either beamers of

interactive whiteboards

The curriculum

New media have been introduced more and more to German schools in recent years. The

curriculum for Mathematics demands the use of specified software at school hence programs

such as excel are introduced and used already in 5th grade. Further advanced student do regularly

rely on this when using specified algorithms (HERON / GAUSS). The use of a dynamical

software for geometries (dyna-geo) is also proposed. The competence orientated curricula refer

to this more openly as “media competence” but demand a critical use of software and pc s.

Availability of new media at school

Most German schools have special rooms with pcs for student use. There are also some ordinary

classrooms which are equipped with computers. Currently great efforts are being made to install

beamers or interactive whiteboards in as many classrooms as possible hence the use of new

media in teaching will be promoted. Yet a regular pc workstation for every single student will

remain a future vision.

GeoGebra, Derive, GeoNext, Archimedes Geo3D und Co.

A broad variety of software is introduced, this is mostly specified by the curricula of the

individual school. Software which was licenced by the state for all schools is being used as well

29

as software which each school has to licence individually through its own funds. The curriculum

specifies the categories of the software but not a specific software to be used in class.

How and to what means do we use new media in teaching maths?

New media are not being used for their own sake but only to support and forward lessons. A

dynamical geometric software enables students to insights that simply would have been

impossible when limited to the use of compass, straight edge and pencil. Software such as excel

solve repeating and easy calculations for the student thus helping him to concentrate and the

more essential questions

This changing attitude also influences A-level tests. Student can decide whether they want to use

a computer algebra system. Parts of these tests are then specially designed for the use of such

programs and do demand a deeper insight into the topics.

Prospect

Hence new media are a support and an effective tool for teaching but can even become more.

They can enable students to learn and experiment more individually at their own speed and

responsibility and do thus support aims which reach beyond a single subject.

30

Maths lesson

Unit: “Differential Calculus”

Topic: Repetition and consolidation of

the derivation its applications

31

Inhalt

General didactic considerations concerning the unit 31

Didactics and methods applied in the unit 31

Conclusion of the unit 31

The lesson 31

Quiz questions 31

General didactic considerations concerning the unit

The pupils (p) get in contact with differential calculus during their introductory phase of

secondary education. It is here, that they learn about examining functions. The teacher focuses

on helping the pupils to understand the applications of these mathematical procedures and their

consequences. Classical curve sketching is used as a tool, but should not replace actual

understanding of the application of knowledge. Pupils have to be enabled to combine different

aspects of the examination of functions.

The following lesson is an example of the integration of new media into such a lesson.

During the stay of the Comenius Project in Wetzlar, this lesson was conducted with international

participants in a workshop using the English language. The next day it was taught as a regular

school lesson in German. This enabled the Comenius participants to follow the course of the

lesson without language barriers.

Didactics and methods applied in the unit

The lesson portrayed here is the unit finale. The pupils are to apply all the content learned in a

quiz and thus show their understanding. The quiz is derived from the basic principles of the show

“Who wants to be a millionaire?”, which has been popular in Germany since 1999 and is

therefore well-known to the pupils.

32

The idea to use this game in class was first published in „Mathematik Methodik“ by Barzel,

Büchter und Leuders (Cornelson Scriptor 2007, ISBN 978-3-589-22378-7) and has been adapted

for this lesson.

In the TV show a candidate has to answer 15 consecutive questions of increasing difficulty, each

of which promises an ever higher prize. The candidate is furnished with four alternative answers

and has to pick one or quit the game. In case of difficulties, the candidate has access to three

“lifelines” (help): He is allowed to call somebody, to ask the audience or to eradicate two of the

false answers.

The aforementioned book illustrates how to play this game in class with single pupils, for

example with the students making up questions and quizzing each other. The lifelines can be

applied in class in a similar manner.

In order to activate every single pupil and to motivate them all to involve themselves in the

lesson, we designed the questions ourselves and asked them the whole class.

A variable is the form of working used to solve the questions. In different classes we

experimented with the pupils working in groups of four, in pairs and on their own. Of these three

methods, the work in pairs has brought the best results and seems to be the most appropriate.

This is because of the cooperation between the partners in the question phase which intensifies

the process of learning. It furthermore increases the obligation to get involved in the process,

which is less the case in larger groups where the pressure of time makes the group trust the

allegedly most competent pupil without questioning her or his results.

Another decision was whether to show the questions to the pupils one by one, solving them each

individually, or to show them to them all at once.

The first approach, however, caused a stir, because of the quick sequence of concentration and

verbal exchange between the pupils.

Because of that we decided to ask the questions on after the other in quick succession, leaving

the process of solving and discussing them for after the presentation.

In order to realize the concept, the use of a computer connected to a projector quite helpful. This

enables you to switch quickly and automatically between the questions and makes it possible to

design the visual presentation of the questions to resemble those of the TV show which in turn

enhances the pupils’ association with the same.

33

This is reinforced by the use of an adapted logo in which the word “millionaire” was replaced by

“master of functions”.

The pupils are shown a questions for 30 to 55 seconds and have to find the answer as well as an

explanation for it in that time. This means a lot of stress for the pupils because they have to apply

a whole unit-worth of knowledge to a single question.

The lack of a time limit increases the duration of the quiz but not the pupils’ understanding. The

limits are results of careful testing and make it possible, but not too easy, to solve the questions.

It is advisable to not only ask the pupils to answer but also to explain their results in order to

avoid wild guessing. However, it is quite difficult to write down an elaborate explanation in the

given period of time. Therefore, it is necessary to come up with a suitable regulation to evaluate

the explanations.

The pupils like to know who won the game. A possible way to find that out is to reward each

correct answer with a token. The team which is able to explain their solution on the blackboard is

rewarded with an additional one. This encourages the pupils to explain their solutions publicly.

The selection and design of the questions used in the quiz is of utmost importance if the lesson is

supposed to be more than a lesson of fun and games, however.

It is quite a task to come up with suitable questions: They should be answerable in a short period

of time without the solutions being obvious. Otherwise it is not only not a challenge but quite

boring for the pupils.

It is advisable to take typical pupils’ errors into consideration to provide the opportunity to

present different apparently logical solutions which leads to discussions and enhances

understanding of the topic.

Quite often the design of the questions used here forces the pupils to not only reproduce their

knowledge but also to apply it on new situations. They are, for example, asked to discover the

features of a graph in a function.

Conclusion

A lesson like this one is highly suited to consolidate the knowledge at the end of a unit.

Alternatively the pupils could design their own questions. Thus, they are familiarized with the

difficulty of doing so, if the answers are supposed to be both, plausible and not obvious.

34

If the pupils do so, it is highly likely that imprecise questions make the problems impossible to

solve. In that case there is the opportunity to discuss about it and to transform it into a “good”

question, thus enhancing the class’s overall understanding. While doing so, the pupils have to get

intensively involved in the subject matter.

Concludingly, it has to be noted that the pupils enjoy this kind of lesson immensely.

Although it is impossible to apply this concept to each and every lesson, it is a highly appropriate

way of finishing units and to consolidate and repeat the learned topic.

The effort to design this lesson is justified if this lesson is taught repeatedly in different classes.

Lesson plan

Time Content Method

5 Minutes Introduction of the quiz and its rules lecture

12 Minutes Pupils solve the questions Pupils work in pairs;

presentation via projector

28 Minutes Discussion and explanation of results Conversation on class;

blackboard

Quiz questions:

35

36

37

2. ICT applied in mathematics - methodological suggestions

QED – team of CETCP Botosani, Romania

In this course proposal, we want to reveal the real advantages of using ICT in teaching

and learning mathematics. Though much educational mathematics software exists today, they are

still exploited in a small measure in the classroom, from different reasons. One reason is the lack

of physical resources, namely computers, PC-tablets, smart / white / active boards for extensive

use in every math lesson, but obviously, this is the system duty to contribute. Another reason is

the lack of information and teachers training, though the information exists, it needs to be

disseminated and exploited better in the classroom. In addition of the other efforts to do this, we

want to contribute with innovative methods of using math educational software in teaching and

learning mathematics, proving that these methods are not “time consumers”, but contrary, they

can improve students maths competences, through changing the attitude and mentality about

mathematics, and through facilitate the active learning.

To maximize the impact of using ICT upon mathematics learning, it is recommendable to

put accent on the most suitable methods, which to combine the specific case of mathematics

learning (problem solving, discovery, and modeling) with the advantages of using ICT. From this

point of view, the investigation as a method of exploring the various features of GeoGebra, and

the project as a method of learning and evaluation, favourise a total implication of student in the

learning process, stimulating their creativity, and also their interest for searching or sharing the

information. Anyway, these methods, which are relatively new, are not pure, but include others,

like direct observation, exemplification, discovery, modeling, and problem solving. Method of

investigation is very appropriate in order to learn different aspects or representations of software.

Method of project is the best way to give students the chance to apply in practice all they learnt.

Thus, the both methods will lead to achieving of a true “learning by doing” style of teaching.

From the variety of existent software, IXL, XyAlgebra, GeoGebra, we choose the last

one, because of its larger applicability (algebra, geometry and calculus) and addressability (for

any level students and many languages availability). GeoGebra allows teaching and learning

through the developing of proofs in real time, together with the students, in a dynamic way, and

facilitating cross-curricular applications.

38

ICT applied in mathematics - optional course program

QED – team of CETCP Botosani, Romania

Pre-requisites Specific competences Content and learning activities No. of hours

Methods and means of learning

Evaluation

- Synthetic geometry, medium to advanced;

- Analytic geometry, medium level;

- Vectors, basic level;

- Functions and equations, at least linear, quadratic, sine, cosine, medium;

- Real and complex numbers, medium;

- Basic logics.

1. Using GeoGebra to:

1.1. Describing geometrical configurations in various ways, analytic, synthetic, vector;

1.2. Modeling geometrical configurations in various ways, analytic, synthetic, vector;

1.3. Interpreting algebraic entities (functions, equations, real or complex numbers) in various geometrical contexts;

1.4. Transposition of real situations, like objects movement, in mathematical

1) GeoGebra basic features. Basic constructions and measurement. Coordinates: Cartesians, polar or complex numbers.

2) Investigating and sharing information on geogebratube.org

3) Sliders and movement. Mobile points moving alongside a line, conic or function graph.

4) Special features. Colors, text or other objects appearing based on logical conditions.

4) Construction and geometrical locus problems.

1

1

1

1

Methods:

- heuristic conversation

- explanation

- investigation

- observation

- modeling

- discovering

- problem solving

- frontal appreciations

- auto-evaluation

- individual evaluation through project method.

39

language or configurations;

1.5. Using all information of the mathematical models to solve various problems.

2. Using online specific platforms, like geogebratube.org, to investigate or to share information.

5) Geometrical interpretation of functions properties, equations roots, implicit equations, complex numbers operations.

6) Simulating of real movements or phenomenon in GeoGebra models (rubber pencil illusion, bike movement, wings etc.)

2

2

4

Means:

- computers network

- internet connection

- Java, GeoGebra

40

ICT applied in mathematics – applications of the quadratic functions

QED – team of CETCP Botosani, Romania

41

42