methodological guide

Cross-curricular Approach of Mathematics Methodological Guide

“QED – Quality in Europe's Diversity” Comenius Multilateral Partnership 2011 - 2013 11-PM-75-BT-RO

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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

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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 3

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.

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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:

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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 Table 1

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

t [s]

v [m/s]

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

0

1

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

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

2

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+ c.

3

10

4

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Part 2: Graph matching Table 2

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

x-axis

y-axis

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

0

0

c.

1

2

2

8

3

18

4

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Suggestions to try out: Linear equation

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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.

Table 1

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

clearly as y-axis Force, F, [N] and x-axis as expansion/stretch of the spring, x, [m]. Make sure to keep

X [m]

+/-

y-axis F [N]

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

0,03

0,5

whole page in your book.

0,00

0,0

0,30

10,0

0,43

15,0

0,59

20,0

0,71

25,0

0,90

30,0

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)?

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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 9

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

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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)

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II, Maths in cultural and social life QED – team of Geniko Likeio from Kissamos, Greece

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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 24

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. EĹ&#x;deneÄ&#x;i BaccalaurĂŠat 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

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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 26

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.

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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

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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.

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Maths lesson

Unit: “Differential Calculus�

Topic: Repetition and consolidation of the derivation its applications

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Inhalt General didactic considerations concerning the unit

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Didactics and methods applied in the unit

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Conclusion of the unit

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The lesson

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Quiz questions

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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.

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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.

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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.

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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:

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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.

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ICT applied in mathematics - optional course program QED – team of CETCP Botosani, Romania

Pre-requisites

- 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.

Specific competences

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

Content and learning activities

No. of hours

1) GeoGebra basic features.

Basic constructions and measurement. Coordinates: Cartesians, polar or complex numbers.

1

2) Investigating and sharing information on geogebratube.org

1

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.

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Methods and means of learning

Methods:

- heuristic conversation - explanation

Evaluation

- frontal appreciations

- autoevaluation

- investigation - observation 1

- modeling - discovering - problem solving

1

- individual evaluation through project method.

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.

2

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

- computers network - internet connection - Java, GeoGebra

4

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Means:

ICT applied in mathematics – applications of the quadratic functions QED – team of CETCP Botosani, Romania

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