Alternatives for Innovative Math Study TEACHING GUIDE

COMENIUS – ETWINNING PROJECT 2013 – 2015 Disclaimer This project has been funded with support from the European Commission. This publication reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

Authors: Irina Vasilescu, Nicolas Houpert, Maria Teresa Asprella – Libonati, Erik Atsma, Valentina Cuadrado Marcos, Katarzyna Pietrzak, Anthoula Sofianopoulou-Karipidou and Lazaros Kartalas

Co-authors and participating teams: Scoala Gimnaziala “Hamburg” Teachers: Irina Vasilescu, Maria Petrescu, Monica Corbus, Doina Militaru, Stefania Voicu , Radu Cihodariu, Ioana Nitu Students: 2013-2014: students from classes 6C, 7A, 7C, 8B, 8C 2014-2015: students from classes 6D, 7C, 8C

Lycée “Francois Bazin” Teachers: Nicolas Houpert, Olivier Trussy Students: 2013-2014: students from classes première and seconde Européenne 2014-2015: students from classes première and seconde Européenne

Liceo Classico “Emanuele Duni” Teachers: Maria Teresa Asprella Libonati, Rosanna Russo Students: 2013-2014: students from classes 1A, 1B, 1C, 1D 2014-2015: students from classes 2A, 2B, 2C, 2D

Hervormd Lyceum West Teachers: Erik Atsma, Bob Naber Students: 2013-2014: students from classes 2vT and 3vT 2014-2015: students from classes 3vT, 3h, 4h, 4v, 5v

Instituto de Ensenanza Secundaria Alonso de Madrigal Teachers: Valentina Cuadrado, Fernando Galeano, Luis Vaquerizo, José Ramón Ladrero, Lorenzo Piera, María Jesús Terrón, Ricardo González-Tablas Students: 2013-2014: students from English Bilingual Section. 4th, 3rd and 2nd Secondary Education. 2014-2015: students from English Bilingual Section. 4th, 3rd and 2nd Secondary Education.

Gimnazjum nr 3 im.Marszalka Jozefa Pilsudskiego w Myślenicach Teachers: Katarzyna Pietrzak, Magdalena Gołąb, Agata Branewska, Dorota Płatek Students 2013-2014: students from classes : 1b, 1c, 2a, 2b, 2c 2014-2015: students from classes : 2b, 2c, 3a, 3b, 3c

4o Geniko Lykeio Stavroupolis Teachers: Lazaros Kartalas, Anastasia Sivri, Ioanna Kalafatidou, Polixeni Argyrou, Fotios Xatzivasiliou, Stavros Nikou, Georgia Karavasili, Efstathios Kozalakis, Anthoula Sofianopoulou Students 2013-2014: students from classes : A, B and C 2014-2015: students from classes : A, B and C

ISBN 978-973-0-19377-0 Bucharest 2015

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Partner schools Scoala Gimnaziala “Hamburg”, Bucureşti - Romania. Local coordinator: Irina Vasilescu

Lycée “Francois Bazin”, Charleville Mezieres - France. Local coordinator: Nicolas Houpert

Liceo Classico “Emanuele Duni”, Matera - Italy. Local coordinator: Maria Teresa Asprella - Libonati

Hervormd Lyceum West, Amsterdam - the Netherlands. Local coordinator: Erik Atsma

Instituto de Ensenanza Secundaria Alonso de Madrigal, Ávila - Spain. Local coordinator: Valentina Cuadrado Marcos

Gimnazjum nr 3 im.Marszalka Jozefa Pilsudskiego w Myślenicach, Myślenice - Poland. Local coordinator: Katarzyna Pietrzak

4o Geniko Lykeio Stavroupolis, Thessaloniki - Greece. Local coordinators: Anthoula Sofianopoulou-Karipidou / Lazaros Kartalas

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Contents 1. Introduction

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2. Why MI in Maths?

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3. Gardner's Multiple Intelligences

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4. Multiple Intelligences Test

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5. Interpersonal and intrapersonal intelligence

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Activity Summaries

14

6. Musical intelligence

31

Activity Summaries

32

7. Linguistic intelligence

46

Activity Summaries

47

8. Bodily Kinestetic intelligence

65

Activity Summaries

66

9. Spatial intelligence

88

Activity Summaries

89

10. Logical intelligence

108

Activity Summaries

109

11. Naturalistic intelligence

123

Activity Summaries

124

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1 - Introduction The project aims at increasing students’ motivation and interest for the study of Maths, with special focus on low-achievers. The participating teachers used the Multiple Intelligences (MI) theory to discover and develop each student’s particular skills, needs and interests, differentiate and adapt the teaching methods to them. Students actively created their own learning materials for each type of MI and used motivating ICT tools. They used collaborative work, very uncommon in traditional Maths classes. Another goal was to stimulate pupils’ curiosity and investigative spirit by showing them the connection between Maths and real life. We wanted to answer their permanent question: ”Why do we have to study this?” by showing them that Maths can be interesting, useful and can be found anywhere. The final product of the project is a pedagogical kit with two parts:

The first part is a pool of materials for each MI appropriate for motivating pupils, especially the less gifted ones, stimulate their interest in Maths. The materials are published both in a blog to be found at http://aimscomenius2013.wordpress.com/ and in a material form (DVD). They are especially useful in CLIL /bilingual or school curriculum (optional) classes. The entire work process is visible on the eTwinning portal at http://new-twinspace.etwinning.net/web/p97560/welcome. The second part is this guide, meant for teachers who are curious to know more about this theory, willing to experiment new strategies and who really care for their students’ effective learning.

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2 - Why MI in Maths? Mathematics is not a popular subject. Pupils tend to dislike it, especially when they fail to obtain the desired academic results, and it can cause anxiety and even phobia. The difficulties they find are not only due to insufficient knowledge, but also to the inability to transfer knowledge in order to face different situations successfully. There are high rates of school failure, mostly in the compulsory years at the secondary level. In order to overcome these phenomena, it is necessary for teachers to discover new teaching activities and procedures, which have to be liberating and promotional, rich, stimulating, challenging, informal, producing participation among the students. The classical approach in teaching Maths creates passive learners; while it’s important to engage pupils in order to have them take an active role (Bednar, Coughlin, Evans, & Sievers, 2002). Image source: http://goo.gl/iGQjnf Another important aspect is the students’ mind-set, as a progress factor. A growth mind-set, that is the understanding that the abilities and intelligence can be developed (as opposed to a fixed mind-set, the idea that they are fixed and cannot be influenced) has been shown by researchers to have powerful effects on students’ motivation and learning. Stanford professor and author Carole Dwek has been carrying on research on the mind-sets and their importance in school success. These goals can be achieved by teaching with multiple intelligences. In order to help all the students use their full thinking potential, it is necessary not only to teach them what a good mind-set is, but to find ways to convince students of the value of using thinking strategies that may seem strange and uncomfortable at first. A good use of this approach includes both the type of activities and the tools that are most suitable for each of our students. In a time when the students’ motivation for studying STEM subjects dramatically needs to be increased all over Europe (only 12% of the European students get a STEM degree, compared to 45% in China), we want to find methods to teach a new, more friendly, enjoyable, useful, a less abstract Mathematics, a subject for each of our students, adapted to their particular skills and competences, a subject that can be more than a tedious hour in the timetable, a vehicle for mutual understanding and for communication. Any educator knows that the mere scientific competence of a teacher offers no guarantee for his/her successful classroom activity. According to Gardner, the author of the Multiple Intelligences theory, the aim of education “should be to develop intelligences and to help people reach vocational and a-vocational goals that are appropriate to their particular spectrum of intelligences. People who are helped to do so, he believes, feel more engaged and competent and therefore more inclined to serve society in a constructive way." The theory he proposed in his 1983 book Frames of Mind: The Theory of Multiple Intelligences 6

differentiates intelligence into eight specific (primarily sensory) "modalities": verballinguistic, logical-mathematical, spatial-visual, bodily-kinaesthetic, musical-rhythmic, interpersonal, intrapersonal and naturalistic, rather than seeing it as dominated by a single general ability. In more recent works, a ninth type, the existentialist intelligence, has been added. Although all these types are present in any person, one or several are dominant for each of us. They are located in different parts of the brain and can work together, as well as independently. All of them can be improved throughout our life, provided we have a growth mind-set, not a fixed one. This theory explains why some pupils remember best what they have seen, while others are good with words, or at building things, some are very creative but find it hard to remember formulas or work in formal Mathematics. In order to get the best results, a teacher has to meet the learning needs of the students to find ways of accommodating their individual ways of learning to his/her teaching. The Multiple Intelligences theory states that pupils will benefit more from a broader vision of education, that would drive teachers to use different methodologies, exercises and activities to reach all students, not just those who excel at linguistic and logical intelligence and challenge them to discover "ways that will work for this student learning this topic". According to Gardner’s theory, students have different types of dominant intelligence and they can be reached more effectively by using a wider array of approaches. “Pupil engagement is a multi-faceted construct that includes affective, behavioural and cognitive dimensions” (Fredricks, Blumenfeld and Paris, 2004). While the teacher can choose the approach for presenting a certain notion or task, it’s important also that the students learn not only to understand and value their own approach to successful Mathematics learning, but also the conditions under which they learn best. In this way they will broaden their approach to learning and, at the same time, they will learn to value their peers. Teachers can encourage students to reflect on how they grasp mathematical ideas best, as well as realize that, although students learn in different ways, they can still be equally effective as learners and learn the same ideas.

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3 - Gardner's Multiple Intelligences: descriptions, preferences, personal potential, related tasks and tests Intelligence type

Typical roles, preferences, potential 1. Perception of other Therapists, HR Interpersonal people's feelings; professionals, and ability to relate to mediators, leaders, intrapersonal others; interpretation counsellors, of behaviour and politicians, communications; educators, understanding the salespeople, clergy, relationships between psychologists, people and their teachers, doctors, situations, including healers, organisers, people carers, advertising professionals, coaches and mentors; (there is a clear association between this type of intelligence and what is now termed 'Emotional Intelligence' or EQ) 2. Musical ability, Musicians, singers, Musical/Artist awareness, composers, DJs, ic appreciation and use music producers, of sound; recognition piano tuners, of tonal and rhythmic acoustic engineers, patterns, entertainers, partyunderstanding the planners, relationship between environment and sound and feeling noise advisors, voice coaches

3. Linguistic

Intelligence description

Related tasks, activities or tests

Preferred learning style Interpreting moods Human from facial contact, expressions; communicati demonstrating ons, feelings through cooperation, body language; teamwork affecting the feelings of others in a planned way; coaching or counselling another person

Performing a musical piece; singing a song; reviewing a musical work; coaching someone to play a musical instrument; specifying mood music for telephone systems and receptions Words and language, writers, lawyers, Writing a set of written and spoken; journalists, instructions; retention, speakers, trainers, speaking on a interpretation and copywriters, subject; editing a explanation of ideas English teachers, written piece or and information via poets, editors, work; writing a language, linguists, speech; understanding the translators, PR commentating on relationship between consultants, media an event; applying communication and consultants, TV and positive or meaning radio presenters, negative 'spin' to a voice-over artists story

Music, sounds, rhythm

Words and language

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4. Bodily kinaesthetic

5. Spatial visual

6. Logical mathematical

Body movement control, manual dexterity, physical agility and balance; eye and body coordination

Dancers, demonstrators, actors, athletes, divers, sportspeople, soldiers, fire fighters, PTIs, performance artists; ergonomists, osteopaths, fishermen, drivers, craftspeople; gardeners, chefs, acupuncturists, healers, adventurers Visual and spatial Artists, designers, perception; cartoonists, story interpretation and boarders, architects, creation of visual photographers, images; pictorial sculptors, townimagination and planners, expression; visionaries, understanding the inventors, relationship between engineers, cosmetics images and meanings, and beauty and between space consultants and effect Logical thinking, Scientists, detecting patterns, engineers, computer scientific reasoning experts, and deduction; accountants, analysing problems, statisticians, performing researchers, mathematical analysts, traders, calculations, bankers understanding the bookmakers, relationship between insurance brokers, cause and effect negotiators, towards a tangible dealmakers, troubleoutcome or result shooters, directors

Juggling; demonstrating a sports technique; flipping a beermat; creating a mime to explain something; tossing a pancake; flying a kite; coaching workplace posture, assessing workstation ergonomics

Physical experience and movement, touch and feel

Designing a costume; interpreting a painting; creating a room layout; creating a corporate logo; designing a building; packing a suitcase or the boot of a car

Pictures, shapes, images, 3D space

Performing a Numbers mental arithmetic and logic calculation; creating a process to measure something difficult; analysing how a machine works; creating a process; devising a strategy to achieve an aim; assessing the value of a business or a proposition

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7. Naturalistic Ability to discriminate among living things (plants, animals) as well as sensitivity to other features of the natural world (clouds, rock configurations). It is also speculated that much of our consumer society exploits the naturalist intelligence, which can be mobilized in the discrimination among cars, sneakers, kinds of makeup, and the like.

In our evolutionary past this ability was present in hunters, gatherers, and farmers; it continues to be central in such roles as botanist or chef.

Examples include classifying natural forms such as animal and plant species and rocks and mountain types.

Inquirybased learning and direct instruction techniques

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4 - Multiple Intelligences Test based on Howard Gardner's MI Model Score the statements: 1- Mostly Disagree, 2 = Slightly Disagree, 3 = Slightly Agree, 4 = Mostly Agree Statement The world of plants and animals is important to me I can play a musical instrument I find it easiest to solve problems when I am doing something physical I often have a song or piece of music in my head I find budgeting and managing my money easy I find it easy to make up stories I have always been very coordinated When talking to someone, I tend to listen to the words they use not just what they mean I enjoy crosswords, word searches or other puzzles I don't like ambiguity, I like things to be clear I enjoy logic puzzles such as 'sudoku' I enjoy (my) pets Music is very important to me I am a convincing liar I play a sport or dance I like learning about nature People behaving irrationally annoy me I find that the music that appeals to me is often based on how I feel emotionally I am a very social person and like being with other people I like to be systematic and thorough I find graphs and charts easy to understand I can throw things well - darts, skimming pebbles, frisbees, etc. I find it easy to remember quotes or phrases I can always recognise places that I have been before, even when I was very young I enjoy a wide variety of musical styles When I am concentrating I tend to doodle I could manipulate people if I choose to I enjoy taking care of (my) house plants I find mental arithmetic easy I can identify most sounds without seeing what causes them At school one of my favourite subjects is / was English I like to think through a problem carefully, considering all the consequences I enjoy debates and discussions I love adrenaline sports and scary rides I enjoy hunting and fishing I care about how those around me feel

Score

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My house is full of pictures and photographs I enjoy and am good at making things - I'm good with my hands I like having music on in the background I find it easy to remember telephone numbers I enjoy hiking in natural places I am a very tactile person I can tell easily whether someone likes me or dislikes me I can easily imagine how an object would look from another perspective I never use instructions for flat-pack furniture I find it easy to talk to new people To learn something new, I need to just get on and try it I often see clear images when I close my eyes I don’t use my fingers when I count I often talk to myself – out loud or in my head At school I loved / love music lessons When I am abroad, I find it easy to pick up the basics of another language I find ball games easy and enjoyable My favourite subject at school is / was Maths I look forward to visiting the zoo I enjoy gardening outdoors I like horticultures and cooking I am very aware of other people’s body language My favourite subject at school was / is art I find pleasure in reading I can read a map easily It upsets me to see someone cry and not be able to help I am good at solving disputes between others I have always dreamed of being a musician or singer I prefer team sports Singing makes me feel happy I never get lost when I am on my own in a new place If I am learning how to do something, I like to see drawings and diagrams of how it works I regularly check weather reports My friends always come to me for emotional support and advice

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5 - Interpersonal and intrapersonal intelligence Although Gardner classifies interpersonal and intrapersonal intelligences separately, there is a lot of interplay between the two and they are often grouped together. These types of intelligence are connected to interaction with peers/ with oneself. Interpersonal intelligence is the core capacity to notice distinctions among others, particularly contrasts in their moods, temperaments, motivations and intentions (Gardner, 1993, p 42). Interpersonal intelligence is often confused with being extroverted. Students with this type of intelligence communicate effectively and empathize easily with others, and may be either leaders or followers. They have the ability to interpret and respond to the moods, emotions, motivations, and actions of others, as well as good interaction skills. They typically learn best by working in collaboration and enjoy discussion and debate, seem to know exactly what to do or say in any given situation, they are great at organizing. They enjoy solving conflicts, comparing, relating, sharing, interviewing, cooperating and perform better in social environments such as learning circles or groups, social gatherings, community events, clubs, apprenticeships. Teachers can encourage the growth of interpersonal intelligence by designing lessons that include group work, peer-teaching, role-playing and by planning collaborative learning activities. This type of learner also benefits from understanding learning in terms of historical, social, cultural or religious knowledge, comparing learning outcomes from a cultural perspective, for example, analysing how students from different cultures, when exposed to the same mathematics teaching, may attempt to learn it in different ways, or how gender stereotypes can affect girlsâ€™ performance in Maths. Intrapersonal intelligence has to do with introspective and self-reflective capacities, as well as with having a deep understanding of the self. It can be seen as the internalized version of the interpersonal intelligence. Such students are effective self-directed learners and are aware of their own strengths and weaknesses. They have access to their own emotions, as well as the ability to distinguish, label and use them as a means of understanding and guiding their behaviour (p. 44). It should be emphasized that this intelligence involves the use of all the others. Students should tap into their other intelligences to completely express their intrapersonal intelligence. Drills and skill games as well as self-guided projects are great ways to stimulate intrapersonal learners. They like to work alone, at their own pace, make their own choices and reflect on their own work, to keep records of their activities - scrapbooks, journals, photo albums etc., as well as to talk about themselves and their feelings. They have a strong sense of fairness. In order to involve them, teachers should connect everything they learn to their own life, give them time to analyse what they did or learnt, connect concepts to real life (for example explain how geometry helps create the building they live in), ask them to compare and contrast various ways of solving equations.

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Activity Summaries 1. Math Curriculum in the Netherlands Title of the activity

Math Curriculum in the Netherlands

General description

Students had to describe the topics given in each year for Mathematics in pre-university education.

Students’ age

13-15

Type of intelligence addressed

Interpersonal intelligence

Educational goals

Expected results

Methodology

Required tools Evaluation

To learn about the Maths curricula at pre-university level To cooperate with classmates to make the file with the mathematical subjects of the Maths curriculum To discuss about the different Maths curricula (in the fourth grade, students have to choose between three different kinds of Maths curricula in the Netherlands: Maths A, B or C. Because Maths C is a part of Maths A the students described only Maths A and B) A total overview of the topics from year 1 till year 6 A deeper knowledge about the Maths curriculum in the higher classes In small groups students have to describe the Maths contents of each year (in the higher grades they have to do it only for Maths A or B). After that they have to make one file out of it and explain to each other what the contents are in the different classes.

Maths books for all pre-university level years Word/Excel to make a schedule The students could learn a lot about Maths and the differences in the curricula they had to choose from in the higher classes. The cooperation in small groups and finally in the whole class was something they could learn from as well. It was also nice to see some of the reactions from the partners as they had read the document.

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Teachersâ€™ recommendations

Starting in small groups and giving students only one year to describe is the best you can do. It isnâ€™t handy if they have to do too much research. The discussion afterwards is essential, but also a bit difficult: talking about Maths contents from the higher classes is not easy, so the role of the teacher is necessary. The teacher can explain what the topics are more or less about and what the differences among Maths curricula are.

Link(s)

http://bit.ly/1uVUqB3 (Math curriculum NL)

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2. MI test

Title of the activity

MI test

General description

At the start of our project students from all countries had to fill in a questionnaire about the Multiple Intelligence system. This produced an outcome where students could see how they scored at the different parts of the intelligence system.

Students’ age

13-15

Type of intelligence addressed

Not one specifically, but reviewing the results in groups addressed the interpersonal intelligence

Educational goals

To make students aware of their own way of learning To compare their way of learning to their fellow students or to students from other countries To compare the results of all countries and explain the differences

Expected results

Completed tests by all students of each country An overall document with the combined results of all countries

Methodology

First students fill in the tests individually. Then they compare the results of classmates in groups. Finally they compare the results of all the project participants as a class.

Required tools

Computer, Excel

Evaluation

Comparing national results and international results was a great way for students to get to know their own learning styles and to get a good start with this project.

Teachers’ recommendations

If you want to do something with Multiple Intelligences, it is essential to do a test like this one. Students know about their learning styles. Also comparing the results on both national and international level is great for them. They understand the differences among the different MIs better, but they also get an awareness about other cultures due to the discussions on international results.

Link(s)

http://bit.ly/1pIj6ej (MI test) http://bit.ly/1xlQSWk (Overall results)

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Poland

Spain

Romania

The Netherlands

Greece

Mean

France

Intelligence type

Italy

Overall results for the MI tests

Linguistic 27 31 26 27 26 23,2 29 Logical-Mathematical 30 31 28 28 28 33 29 Musical 30 33 29 29 24 35,4 31 Bodily-Kinesthetic 28 28 27 28 28 26,6 30 Spatial-Visual 28 30 24 28 26 28,9 30 Interpersonal 29 34 27 31 28 27,9 30 Naturalistic 25 26 23 25 24 20,2 31 Table 1: the averages of the separate countries together with the mean of all countries together

Mean 30 28 26 24 22

Graph 1: The mean score of all countries together

Mean

40 30

Linguistic Logical-Mathematical

20 10 0

28 30 28 29 29 26 25

Graph 2: The average scores of all countries together in one graph

Musical Naturalistic

Bodily-Kinesthetic Bodily-Kinesthetic Spatial-Visual Linguistic Interpersonal Naturalistic

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3. Survey of students’ preferences Title of the activity

Survey of students’ preferences

General description

Students made a survey. The following information was collected by 8 teams. The questions were asked to 7-10 and 11-14 year-old children: 1. Have you ever been in one of the partner countries? 2. How do you spend your spare time? 3. What is favourite study subject? 4. What is your favourite sport? 5. What is your favourite video game type? 6. What is your favourite type of music? 7. What landmarks have you visited in Romania? 8. What do you wish to become in the future? Students registered their colleagues' responses to the 8 questions, organized the collected data and then presented them as graphs.

Students’ age

13

Type of intelligence addressed

Interpersonal intelligence

Educational goals

To develop the capacity for knowledge To understand the surrounding context To stimulate curiosity for investigation in connection with certain themes To develop human relationships through teamwork To create interdisciplinary connections

Expected results

Developing initiative and willingness to tackle a variety of tasks as well as collaborating with students of different ages Observing the students’ interest in the activity Using the knowledge acquired in Mathematics to organize recorded data in order to create tables, charts and graphs in Excel and PowerPoint presentations

Methodology

Project-based learning Investigation Cooperative Learning Problem-solving Heuristic conversation Debate Use of ICT

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Required tools Evaluation

Paper sheets, pencils, ruler, drawings, watercolours computers, video camera

Systematic observation of students' behaviour, their interest for the survey both at the time of the data collection and during processing and presenting the information Assessment of the materials as well as of the work process Quality of statistical summaries and graphs, emphasizing originality, team spirit and cooperation Involvement in the activities, highlighting the findings and explanations given.

Teachers’ recommendations

Achievement of several themes and group projects Monitoring of how tasks are distributed to each member of the group according to their possibilities Differential treatment of students in the research topic.

Link(s)

Statistics presentation in the TS http://goo.gl/BDc6S1 Slideshare presentation http://slidesha.re/1xIkjkR Students’ videos http://www.youtube.com/watch?v=q9YiaqNPO2s http://www.youtube.com/watch?v=Fpwymo4XwJU http://www.youtube.com/watch?v=2rKBMi5WCEU

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4. The Survey. (Descriptive Statistics) Title of the activity

The Survey. (Descriptive Statistics)

General description

Presentation of the work done for the unit Descriptive Statistics in 4th Compulsory Secondary Education.

Students’ age

14 – 15 year-old students in 4th grade.

Type of intelligence addressed

Interpersonal and Intrapersonal intelligence

Educational goals

Expected results

To understand the concept of survey in all the stages: design, data collection as well as processing and presentation of statistical results To promote teamwork and the use of ICT: Excel, Powerpoint, Slideshare Practical knowledge of Descriptive Statistics Teamwork strategies development Communication abilities improvement Awareness of the role of statistics in society

Methodology

Students are provided with information on survey design. Thematic blocks are chosen. Students are divided into groups and given roles. Questions are written and teams make the survey in different groups in school. After data collection, graphics with results are created to be added in a presentation.

Required tools

Computers, Internet, Excel. Slideshare account

Evaluation

Active participation and commitment Punctuality Quality of work Team abilities

Teachers’ recommendations

Students should be encouraged not to be too demanding when making the questions Questions should be very well selected Teamwork attitudes should be reinforced, allowing students to choose the task they feel more comfortable in

Link(s)

http://bit.ly/1S3MXHM (slideshow of the statistics)

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5. Thessaloniki, the city of Greek team Title of the activity

Thessaloniki, the city of Greek team

General description

Presentation of our city Thessaloniki

Students’ age

15-16

Type of intelligence addressed

Interpersonal intelligence

Educational goals

To become familiar with the geometrical shapes of the typical monument of Thessaloniki such as Aristotelous square, Byzantine churches, Roman monuments. To develop analysis and synthesis by distinguishing geometrical shapes on an entire construction. To be aware of the role of culture as a motivation of attracting visitors to our city and in this way improving communication. To learn to cooperate by developing team spirit and getting to know the culture of the city.

Expected results

Becoming familiar with one’s own hometown Presenting one’s hometown to the students of the partner schools Using the digital tools in a creative way

Methodology

Creation of teams of different abilities and interests Creation of emails and use of Google docs Research work using websites and books Collections of photographs

Required tools

Computers, cameras, the Internet

Evaluation

The activity is suitable to understand and revise the cultural value as a means of self-knowledge, but also to communicate with other people.

Teachers’ recommendations

Exploration of the immediate surroundings of the town (field study) is a priority.

Link(s)

http://bit.ly/1GSXn8v (slideshow Thessaloniki)

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6. Timeline of Greek Mathematicians Title of the activity

Timeline of Greek Mathematicians

General description

Presentation of the biographies of Greek Mathematicians on a timeline

Students’ age

15-16

Type of intelligence addressed

Interpersonal intelligence

Educational goals

To raise awareness of the value of social media To develop the cooperative method To reinforce a scientific attitude To evaluate information and its sources critically To activate students

Expected results

Getting to know about Greek Mathematicians Enhancing mathematical knowledge Interpreting and using Maths to create a quiz Disseminating research to the network

Methodology

First students are divided into teams. Each team chooses two mathematicians, an ancient one and a contemporary one. Students research on them using the Internet or books, gather information and present the work to the other teams. They use digital tools and Web2.0.

Required tools

Maths books, computers, the Internet

Evaluation

Effective, original and suitable activity to arouse students’ curiosity, interest and motivation. It brings out students’ command of Maths

Teachers’ recommendations

It is suitable for students of middle school

Link(s)

http://bit.ly/1S3MXHM (timeline)

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7. The great Mathematicians Title of the activity

The great Mathematicians

General description

Pupils research about the lives of famous Mathematicians in their countries. They focus on interesting biographical events and important Maths outcomes (works, discoveries, formulas, etc.). Then they prepare a presentation choosing an ICT tool they like.

Students’ age

13-14

Type of intelligence addressed

Interpersonal intelligence

Educational goals

To enliven Maths linking this subject, which is commonly felt as

Expected results

aseptic and cold, to the lives of real people living in real contexts and having real problems To contextualize the study of Maths To implement a “humanistic” and historical approach to the study of Maths To use L2 (English) to present a content (CLIL) To work in teams for getting and selecting meaningful information, and presenting results.

Research work and presentations of great national mathematicians Raised awareness of the historical and human implications of Maths

Methodology

Team work for the whole task Dividing tasks in the team according to pupils’ abilities (collecting information, translating and editing the English version, choosing an ICT tool and studying its use for the presentation, making the presentation)

Required tools Evaluation

Teachers’ recommendations

Computers, research instruments (the Internet, books, magazines, etc.), a video projector for presentations. Discussion Comparing the results of the teams from a historical, scientific and technological point of view It is important that pupils are given tasks in the team according to the abilities of each of them. This helps to their MI awareness

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Link(s)

http://youtu.be/vDVYCbSexHI (video about Fibonacci) http://bit.ly/1yCNCFz (Fibonacci on the Twinspace) http://bit.ly/1v6NMIf

(slideshare famous mathematicians)

http://bit.ly/1wt1v6N (Twinspace famous mathematicians) http://bit.ly/1zmPtPW (slideshare Maria Gaitana)

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8. A Piece of Pie in ... Title of the activity

A Piece of Pie in ...

General description

Pupils receive the recipe of a pie and a set of tasks. They have to calculate quantities of ingredients, costs, to transform Celsius degrees into Fahrenheit degrees, and finally to calculate the cost of the same piece of pie in the currencies of the partner countries.

Students’ age

12 – 13

Type of intelligence addressed

Interpersonal intelligence

Educational goals

To apply mathematical knowledge to a practical situation To work in teams in order to get information, create and solve the tasks, present the results

Expected results

Each team answers the tasks of the activity and makes a presentation of the results The teams compare their results to the results of their colleagues

Methodology

Team work for the whole task Dividing tasks in the team according to pupils’ abilities (there are different tasks – collecting information, making calculations, presenting results, possibly even baking a pie – so different abilities are necessary)

Required tools

Paper, calculators, if possible at least one computer and a video projector for presentations.

Evaluation

Discussions, comparing the results of the teams from the same country, comparing the results of the teams from different countries.

Teachers’ recommendations

Pupils should be given the tasks in the team according to the abilities of each of them.

Link(s)

Work process description http://slidesha.re/1xfdvOE Prezi presentation http://bit.ly/1xpSwpZ Students’ video http://youtu.be/TqXvRarUnys TS page http://bit.ly/1qCzVSK

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9. Playing with numbers Title of the activity

Playing with numbers

General description

Students create “intelligent sequences of numbers” resembling Fibonacci sequence. In groups they form the sets of puzzles with these sequences and then exchange the tasks between groups. They have to find missing numbers in sequence or the sum of several terms. Before that, they have to find the rule. Finally they create an online quiz for other students.

Students’ age

13 -15

Type of intelligence addressed

Interpersonal intelligence

Educational goals

To learn about sequences To learn logical thinking To learn how to create a quiz online To improve efficient counting To create, solve and present problems in groups

Expected results

Students create the sets of the puzzles online. Students from other countries solve the quiz.

Methodology

Collaborative work in groups for the whole task. Team work.

Required tools

Maths books, paper, computer, online quiz.

Evaluation

Discussion about what was the most difficult part in creating and solving the tasks. For most students solving the quiz was enlightening, they learned to cooperate with others and they had also fun.

Teachers’ recommendations

Good way to introduce the concept of sequence, very interesting and exciting for students, even for the younger ones.

Link(s)

Quiz on the Twinspace http://goo.gl/4de3LD Quiz on the website of the school http://goo.gl/ZuUXqN

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10. The Maths Quiz Game Title of the activity

The Maths Quiz Game 1st step : Slideshow Pupils work on the biography of mathematicians and create a few questions about them. They also create questions using riddles. They make videos about calculations and equations.

General description

2nd step : Online Pupils log in the quizrevolution.com. They create questions, adding 4 different answers, and either image or video to illustrate it. 3rd step : The buzzer Pupils create a buzzer in the electronic lab. Then they will be able to play one team vs. another, using this buzzer and two joysticks. 4th step : The game All students from different countries can play the quiz, answering questions about Maths, winning points, and challenging themselves and each other.

Students’ age Type of intelligence addressed

12 – 15 Interpersonal intelligence To learn about mathematicians and Maths topics

Educational goals

To learn how to create a quiz online (web 2.0 tool) To collaborate to a common work with European students To share skills when drilling and soldering together on the buzzer To work in teams when playing the quiz all together Each team creates questions in the quiz online (more than 150

Expected results

questions written and illustrated) A buzzer to enable students to play easily A final challenge: The European teams play the game in France

Methodology

Pair work for the research work and the creation of the buzzer Collaborative work with all students from each country Teamwork when playing the quiz in France

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Required tools

Evaluation

Powerpoint : pre-work about mathematicians Video to illustrate the quiz Quiz online Electronic tools to create the buzzer The final game in France played by teams gave us a good evaluation since students played it with joy and motivation!

Teachers’ recommendations

The students must find a way to collaborate with everybody, even if they are not friends at all. The group work must create links among students for one goal: working!

Link(s)

The wiki page : http://aims-comenius.wikispaces.com/MATHEMATICIANS An example of a calculation in the quiz : https://www.youtube.com/watch?v=812hCLidbYg An example of an equation in the quiz : https://www.youtube.com/watch?v=_-zxIfEl8do A video showing how to play the quiz in the meeting : https://www.youtube.com/watch?v=efq3xt18Rko A Slideshow about the quiz and the buzzer : http://bit.ly/1zmPtPW The activity about the buzzer : http://bit.ly/1f2lM1P Photos about the quiz : http://bit.ly/1f2lM1P Photos about the buzzer and joysticks : http://bit.ly/1dvPITc

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11. Spanish Mathematicians Title of the activity

Spanish Mathematicians

General description

Use of Glossi, a digital magazine, to publish Spanish Mathematicians’ biographies, interviews and other works related to Mathematics at school

Students’ age

12 -15 year-old students from 2nd, 3rd and 4th Compulsory Secondary Education doing Bilingual Studies in Mathematics, Technology and Science

Type of intelligence addressed

Interpersonal and Intrapersonal intelligence

Educational goals

To gain knowledge of Spanish Mathematicians’ lives To work on the strategies useful to write biographies and to make interviews To become aware of the role of Mathematics in Science and common life To use new ICT tools

Motivation and participation Involvement of different subjects: Maths, History, Geography Knowledge of Spanish Mathematicians’ works Improvement in writing and speaking strategies Coordination among the groups Interest to go on working with the tool

Expected results

Methodology

A Glossi (online magazine using a free digital platform) is started and students are taught how to use it. Students in 4th and 3rd classes are given a list of Mathematicians and sites to do research. Biographies are written and sent to editors. Images are added. Students in 2nd class are divided into two different groups to write interviews on the topic: “Living with a Mathematician”. The interview is shown as a TV program in a video. Photos are taken from each activity related to Mathematics at school to be published in the magazine section: “Maths at School”.

Required tools

Computers, digital cameras. Glossi account to get the magazine.

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Evaluation

Involvement and active participation in tasks Punctuality to hand the work Responsibility in the groups Work quality

Teachers’ recommendations

Roles in the groups should be changed from time to time in order to give everybody a chance to learn the new tool. Students’ capacities should be taken into account. The magazine should be kept alive and students should be encouraged to start new ones for different topics.

Link(s)

http://glossi.com/alonsoinenglish/61274-spanish-mathematicians

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6 - Musical/Artistic intelligence This type of intelligence has to do with sensitivity to sounds, rhythms and music. Students with high musical intelligence are able to sing, play musical instruments, and compose music, they are also able to think in rhythms and patterns, as well as to recognize and manipulate them. They will sometimes use songs or rhythms to learn. They learn best through sounds, repetitive patterns or memory rhymes. Some may learn best via lecture or while listening to background music. These students can prefer learning ideas by rote or by chanting, for example, in order to get the times table 'by heart'. They listen and respond with interest to a variety of sounds: human voice, music, environmental sounds. They use vocabulary and notations of music, respond to music kinaesthetically by conducting, performing, creating, dancing, recognize different musical styles, genres, develop a personal frame of reference for listening to music, enjoy improvising and playing with sounds and have the ability to interpret meaning from music. They are holistic learners: the acquired material is a whole image or a complete episode; the component parts can only be retrieved by taking apart the complete episode and retrieving logical patterns among the items, where these exist. Students use this format when they recall an idea by embedding it in a rhythm. Teachers can nurture this type of intelligence by integrating activities into their lessons that encourage students' musical intelligence by playing music for the class and assigning tasks that involve students creating songs about the material being taught or creating a musical about an episode in the history of Maths, playing background music for various activities and different moods in the classroom. Although H. Gardner defines this intelligence specifically “musical”, we have decided to expand it to other artistic forms, in respect of the preferences, creativity and skills of the different teams. For this reason some of the activities carried out by the partner schools are not strictly “musical”, but have to do with other artistic expressions, for ex. visual.

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Activity Summaries 12. Funny riddles Title of the activity

Funny riddles

General description

Pupils search and record funny Maths riddles for their classmates to solve using their listening skills.

Students’ age

13-14

Type of intelligence addressed

Musical intelligence

Educational goals

Expected results

Present and solve Maths riddles through an audiosource in a challenging context

Methodology

The teacher assigns the task of searching for funny riddles to a group of 3-4 students, possibly demotivated and not very good at Maths. A couple of students are assigned the translation task. Students with a good pronunciation in L2 record the riddles chosen by their classmates on Soundcloud. Students with good ICT skills work with the tool Thinglink for the audio presentation. The riddles are presented to the class divided into groups and the students are asked to listen carefully and solve them. The team solving most riddles is the winner. The teacher divides tasks according to pupils’ abilities/weaknesses (choosing the riddles, translating and editing the English version, choosing adequate ICT tools to record and present riddles).

Required tools

To enhance peer education To empower problem solving and logic through listening To present Maths problems in a funny way To enhance Maths learning through the musical intelligence To use L2 (English) to present a content (CLIL) To improve L2 intonation and pronunciation

Computers, ICT audio tools (Soundcloud), presentation tools (Thinglink)

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Evaluation

Evaluation must consider the following points: Language skills (pronunciation, intonation, translation) ICT skills (use of tools) Logic skills (number of riddles solved by students) Students’ involvement and motivation

Teachers’ recommendations

It is important that pupils are assigned the task according to their abilities. This helps to their MI awareness.

Link(s)

http://bit.ly/1QLlb08 (Thinglink with riddles) http://goodriddlesnow.com/math-riddles (website with riddles) http://bit.ly/1I1djTC (website with riddles)

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13. Mondriaan Christmas Cards Title of the activity

Mondriaan Christmas Cards

General description

In this activity students can create differently coloured cards following certain rules. The mathematical part in it is trying to get the lowest Mondriaan value.

Studentsâ€™ age

12-15

Type of intelligence addressed

Musical/Artistic (more Artistic than musical)

Educational goals

To learn how to reach the ultimate value and not just try to get a card, but the best card.

Expected results

Many different cards, maybe even multiple cards by the same student trying to reach a better value.

Methodology

Students worked individually, trying to better the scores of their classmates.

Required tools

Paint is very handy here because it saves a lot of time, but it can be done just on paper as well.

Evaluation

Students loved this task and were really trying to get better values every time, some by a mathematical process, some by just trial and error. It was nice to see cards from the partner countries as well.

Teachersâ€™ recommendations

If possible use Paint in a computer room or at home, but all together in one room created kind of a competition to get the best card. Doing it on paper is more difficult, especially if you make mistakes.

Link(s)

Activity: http://bit.ly/10QiZS6 Solution: http://bit.ly/1xlRKKS Examples: http://bit.ly/1uVWQ2s

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14. Famous Mathematician videos Title of the activity

Famous Mathematician videos

General description

It is a video about a famous mathematician, Stefan Banach, his love for Maths and not only. We can see the story of his inspiration to create one of his theorems, which was impossible to use in real life and now it is known as the “ Banach- Tarski paradox”. It is not a real love story, but a very probable version.

Students’ age

14-15

Type of intelligence addressed

Musical-artistic intelligence

Educational goals

To teach students a mathematical theorem To practice listening skills in English To show students alternative ways of dealing with problem-solving tasks

Expected results

Students will get to know the figure of Stefan Banach, they will activate their listening skills, and get more interested in Maths.

Methodology

Role plays, drama

Required tools

A script, a video recorder, a software for making movies

Evaluation

Students did very well, they could boost their talents at acting as well as filming and directing. They became more interested in Maths and improved their English skills.

Teachers’ recommendations

Although it seems to be time and energy consuming, it is for sure worth recommending.

Link(s)

Video Banach http://goo.gl/8fjDTF

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15. MATHS FLIPPED LESSONS Title of the activity

MATHS FLIPPED LESSONS 1st step : Brainstorming about Maths At home or at school, the students think of how to introduce a Maths topic using a video. The Maths topic could be found in the curriculum or in the Maths history.

General description

2nd step : Creation of the video Using iPads, video cameras, or smartphones, the students record their videos about a given topic. They need to act like a teacher, that is to say, they have to explain verbally and to show examples about the given topic. 3rd step : Upload Students upload their videos on the internet and share them using a Thinglink interface. Students from different countries involved in this task have to upload their videos in order to make this activity collaborative. 4th step : The flipped lesson and the questionnaire All students in different countries watch the flipped lessons and have to answer a questionnaire about it. So they need to understand first, and then to apply what is shown in each tutorial.

Students’ age Type of intelligence addressed

Educational goals

12 – 15 Musical-artistic intelligence To learn about the difference of Maths curricula in different countries To learn how to create and upload a video To collaborate to a common work with European/other students To explain Mathematics verbally as a teacher does in English To listen to Mathematics explained with different accent To study Maths curricula from different parts of Europe

Expected results

Teams involved create flipped lessons shared online Final questionnaire about these lessons created collaboratively among European partners

Methodology

Individual work or team work when creating the video Collaborative work with all students from each country Individual work when answering the questionnaire

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Required tools

Google presentation: online questionnaire Video : Flipped lessons using iPads, Smartphones, and video cameras Thinglink sharing interface

Evaluation

The questionnaire filled by students will be evaluated in order to check whether the students managed to understand the flipped lessons correctly.

Teachers’ recommendations

To explain a Maths topic, the student must speak slowly and pay attention to pronunciation. Due to a huge range of accents from different countries, students have to listen carefully to tutorials. The teachers have to check if the wrong answer in the questionnaire is due to a problem in understanding English or Mathematics.

Link(s)

The wiki page : http://bit.ly/1L15zr5 The Twinspace : http://bit.ly/1S3SYnR A few examples of a tutorial from a French student using iPad : http://www.showme.com/sh/?h=D9fDU12 http://www.showme.com/sh/?h=OyqwmcS (very detailed one) An example of a tutorial from a French student using paper : http://bit.ly/1B1QEtA An example of a tutorial from a Romanian student : http://bit.ly/1Gy66j3 An example of a tutorial from a Romanian student using geogebra : http://bit.ly/1QLlRTw

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16. Narrable: Maths in Art and Life Title of the activity

Narrable: Maths in Art and Life

General description

Students from the Comenius participating schools took photos during the visit in Matera to find Maths in art and life and a collaborative tool was chosen to show their contributions and their voices. 13-16 Musical-Artistic Intelligence

Students’ age Type of intelligence addressed Educational goals

To promote observation skills To encourage students to use their imagination to join Maths concepts with art works and nature elements To develop narration skills to explain concepts and feelings from a photography To encourage students to participate in a common product To make students aware of the importance of investigating new strategies when facing a problem

Expected results

Helping students gain confidence to express concepts and feelings in

Methodology

Concept investigation when using photography Oral practice Use of ICT: digital cameras or smartphones

Required tools Evaluation

voice files Using, systematizing and consolidating mathematical concepts in the covered curriculum Learning to learn skills development Improving Maths vocabulary to tell a story through photography Understanding the importance of visual support when providing knowledge Developing strategies to make decisions concerning unexpected problems.

Digital cameras, smartphones, computers.

Quality of pictures Correct English use to transfer Maths concepts and feelings Direct link between narration and pictures Involvement to provide pictures and narrations

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Teachersâ€™ recommendations Link(s)

Suggested for a variety of subjects where narration is implied. Giving responsibilities to different students in groups according to their capacities. https://old.narrable.com/app#narrables/2nyezf (Narrable)

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17. Maths Advent Sound Calendar Title of the activity

Maths Advent Sound Calendar

General description

It is an activity that would benefit learners who understand better by listening and using spoken instructions and tasks, as well as rhythm, patterns or music. The activity consists of an Advent calendar that has an audio file with a Maths question attached to each day. It is intended as a collaborative task, and as a challenge to project partners at the same time.

Students’ age

14, but it can be used with younger students too.

Type of intelligence addressed Educational goals

Musical intelligence

To learn by hearing To understand spoken tasks To work collaboratively To choose and express Maths questions in a clear and correct form To use ICT tools

Expected results

As a tangible result, the calendar and the recorded questions, with the respective answers given by other students. As a nonmeasurable result, improved skills and better understanding of audible materials.

Methodology

Students worked in groups of four. One of the students created the picture for the final calendar. The questions were chosen by the members of the group, jointly. In each group, a student was responsible for the translation, two for recording and one checked that the question was clear, simple, general and correct.

Required tools

The picture can be created anyway the students prefer, the audio files can be recorded with a sound recording tool such as Audacity but also using a mobile phone. The only required tools are Soundcloud (https://soundcloud.com ) for storing the audio files online and Thinglink http://www.thinglink.com/ for the final product.

Evaluation

Observation of the students’ work process, collaboration and involvement, as well as the quality of the product.

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Teachersâ€™ recommendations

If the questions are meant to be answered by another group of students, the teacher should make sure that they follow the same curriculum as the proposing group.

Link(s)

Thinglink Calendar http://bit.ly/11eAFHR

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18. Geometric paintings Title of the activity

Geometric paintings

General description

Inspired by geometrical shapes and graphs, students create paintings on given themes with Geogebra and other suitable computer programs.

Students’ age

15-16

Type of intelligence addressed

Artistic intelligence

Educational goals

To recognize geometrical shapes in real life To feedback geometrical meanings by using Geogebra To find out and develop artistic abilities

Expected results

Each student will create works of art on their favourite theme Each student will give his/her painting a title A final exhibition of all works

Methodology

Worksheets with examples of how to use Geogebra Students choose the theme and present their ideas to the teams The members of the teams help each other

Required tools Evaluation

Computer Exhibition with the creations of students in the school Discussion of ideas, titles and chosen geometrical shapes

Teachers’ recommendations

Pupils should use papers and coloured pencils instead of computers

Link(s)

http://bit.ly/14iAZGF (Kizoa collection)

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19. Geometric Self-Portraits Title of the activity

Geometric Self-Portraits

General description

Inspired by geometrical shapes or solids, pupils make self-portraits by drawing, painting or using computer programs.

Students’ age

12 – 13

Type of intelligence addressed

Artistic intelligence

Educational goals

To observe mathematical shapes in real life, including students’ own faces and bodies To make a self-portrait in an artistic way, by drawing, painting or using computer programs To develop students’ observation and artistic abilities

Expected results

Each student will analyse and observe shapes and solids that resemble their physical appearance and can be used to present their portrait Each student will make a self-portrait A team of students will create a final product (a presentation) containing all the self-portraits

Methodology

Observation of geometric shapes and solids Realization of the self-portraits with an instrument which is the most comfortable for each student (drawing, painting, computer drawing, etc.) Team work to create the final presentation

Required tools Evaluation

Paper, coloured pens or water colours, computer. Presentation of the self-portraits in a final product Exhibition in the classroom with the self-portraits Discussions

Teachers’ recommendations

Pupils should choose the instruments they are most comfortable with (drawing, painting, computer program, etc.)

Link(s)

Video on students’ creations http://vimeo.com/84808528

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20. Representing Numbers Title of the activity

Representing Numbers

General description

Making a video based on a research about how ancient cultures represented numbers (ancient calligraphies).

Students’ age

14 – 15

Type of intelligence addressed

Musical-Artistic Intelligence

Educational goals

To promote research about ancient cultures To encourage students to use different tools to write To use music as a way to highlight products To use clear instructions to generate knowledge To understand the value of visual art, such as a video, as a way to learn

Expected results

Students are expected to experiment with different writing tools and to learn how to speak in front of a camera in order to be understood. Production of a final video.

Methodology

Students choose the cultures they want to work on and make research in order to write a set of instructions. In pairs, one being the calligrapher and the other the speaker, they rehearse the number representation. Music is chosen among a variety offered by teachers. Short introductions are written. The video is recorded in two sessions. Students learn to follow instructions from the publicity. Students’ workshop.

Required tools

Writing material: special paper for calligraphy, pens, brushes and ink. Video camera with exterior microphone.

Evaluation

The combination of music, art performance, calligraphy and video production is highly motivating. Collaboration and team work skills are promoted. Personal abilities are encouraged, so students get a lot of confidence.

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Teachersâ€™ recommendations

Personal abilities must be taken into account. Before recording, time must be allocated for rehearsal with the new writing tools as students are not used to them. Video recording should be carried out in, at least, two sessions to allow possible repetitions when mistakes are made.

Link(s)

http://youtu.be/aK89_9qNgrM (video about calligraphy)

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7 - Linguistic intelligence This intelligence includes the ability to effectively manipulate language to express oneself rhetorically or poetically. Moreover, this intelligence allows us to use language as a means to remember information. Everyone is thought to possess this intelligence at some level, but poets, authors, attorneys and those with strong rhetorical and oratory skills exhibit strong Linguistic intelligence. Students who have high verbal-linguistic intelligence find it easy to work with words and languages. These students learn and think most easily by discussing, arguing, debating and following spoken explanations. They are typically good at reading, writing, explaining, telling stories and memorizing words along with dates. Stories are especially motivating for them. They understand better when paraphrasing or verbalizing the task and remember best the information that they have converted to a verbal code. Verbal learners absorb information easily from written material and benefit from well-written instructions and lessons. This type of intelligence is used when students convert numerical data to a verbal code, explain to each other or to themselves the meaning of number statements or the content of a particular diagram. When students talk to themselves or to their peers about mathematical ideas, they can use their verbal logic and reasoning more easily. These learners think in words rather than pictures, they frequently need to talk to themselves as they are learning. They often benefit from â€˜process approaches to mathematics learningâ€™. Students who represent ideas in this way sometimes exhibit difficulties when using their knowledge to solve real-life problems and may need time to translate their ideas into actions.

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Activity Summaries 21. CRIME STORIES involving Mathematics Title of the activity

CRIME STORIES involving Mathematics 1st step : Teamwork Organization of the teams made up by two students 2nd step : Brainstorming of the story Students have to think of a story split into 4 parts. The first part is the introduction which describes the characters (suspects and victim), the crime itself, the place where the crime takes place. 3rd step : Writing the story On a word document, write your story adding pictures or maths graphics. Collaborative way to work with partners : The team A starts the story by describing the scene and the study of the first clue. Team A can finish the stories as well. The team B will give the answer if there isn’t an answer given. The team B wrote a story about a different subject, but with the same idea. Students had to learn about encrypted messages and use some quadratic formulas to support this. Students were divided into several groups all with a different role. The team A answered the team B’s crime story. The team B answered the parts where there isn’t an answer already given by the team A. 4th step : Publishing the story With the help of your teacher, publish your story on a common document in the website madmagz.

General description

Students’ age

12 – 15

Type of intelligence addressed

Educational goals

Linguistic intelligence

To learn about how to apply Mathematics to a concrete situation To learn how to create a book online To collaborate to a common work with European students To explain a crime using a specific vocabulary linked to Mathematics To write a story following rules for the presentation To become aware of different Maths curricula in Europe

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Expected results

Methodology

Required tools

Teams involved create a book online The best stories are printed and put together to create a booklet full of crime stories A part of a collaborative Madmagz document filled by all partners Peer work when thinking of the story Collaborative work with different groups from different countries Word Madmagz Image editor : Paint

Evaluation

The students were really involved in writing the stories and working together. After that they answered the task from the other country and they liked it. The result is in a common magazine including also other kinds of stories.

Teachers’ recommendations

The mathematical theorems must be explained in detail before. For example, the teacher needs to explain the graph theory or encrypted messages if these topics are not in the class curriculum.

Link(s)

The wiki page : http://bit.ly/1uVXMEd The Twinspace : http://bit.ly/1oC7vfS The Madmagz : http://bit.ly/1ty3GEh

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22. PICOMIX Title of the activity

PICOMIX

General description

Students invent stories about PI and present them in the illustrated visual art of the comic medium

Students’ age

13-14

Type of intelligence addressed

Linguistic intelligence

Educational goals

To approach a Maths content in a practical and historical way To reflect on a Maths topic using creativity and imagination To improve Maths knowledge encouraging research To strengthen writing skills in L2 (English) To use L2 (English) to present a Maths content (CLIL: content and language integrated learning) To learn the basics of the comics medium To use innovative presentation tools

Expected results

Students will have a better understanding of the Maths role of PI through their creative verbal effort to “tell stories” their way.

Methodology

1. WARMING UP The teacher introduces the concept of PI, presenting and solving a practical problem. Then assigns students riddles/problems similar to the solved one, and has them work in teams of 3-4. 2. RESEARCH ACTIVITY Teams with mixed skills are formed (ICT, Maths and English experts). They have to collect Maths and historical information about PI and create stories. 3. PIXTON ACTIVITY Teams must adapt scripts to the PIXTON tool (image, language, dialogue, balloons, etc.)

Required tools

Computers, Internet, presentation tools (Pixton www.pixton.com/)

Evaluation

Evaluation must consider the following points: Precision and reliability of Maths and historical information ICT skills (use of tools) L2 skills (writing, spelling, grammar, dialogue)

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Teachersâ€™ recommendations

It is important that teams are well-mixed and pupils are assigned the task according to their motivation and skills. This helps to their MI awareness

Link(s)

http://bit.ly/1GrD8zs (Picomix on issuu) http://bit.ly/1oC7vfS (Picomix on the Twinspace)

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23. A world without Maths Title of the activity

A world without Maths

General description

Students invent stories about what the world would be like if Maths disappeared

Students’ age

13-14

Type of intelligence addressed

Linguistic intelligence

Educational goals

To reflect on a Maths topic using creativity and imagination To understand the practical role of Maths in everyday life To strengthen writing skills in L2 (English) To use L2 (English) to present a Maths content (CLIL: content and language integrated learning) To use innovative presentation tools

Expected results

Students will have a better understanding of the role of Maths in their life using imagination and paradox and creating dystopian or utopian worlds

Methodology

1. WARMING UP The teacher brain-storms students about the role of Maths in everyday life. A map of ideas is drawn on the board. 2. WRITING Students in pairs invent short stories, find images or draw sketches about the story 3. MADMAGZ Students adapt scripts to the tool. The work is part of a collaborative online magazine made with Madmagz with contributions from all partners.

Required tools

Computers, Internet, presentation tools (Madmagz)

Evaluation

Evaluation must consider the following points: Reliability of Maths concepts ICT skills (use of tools) L2 skills (writing, spelling, grammar, dialogue)

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Teachersâ€™ recommendations

It is important that pairs are well-matched and pupils are assigned the task according to their motivation and skills (L2, ICT). This helps to their MI awareness

Link(s)

http://newtwinspace.etwinning.net/c/portal/layout?p_l_id=31618113 http://madmagz.com/magazine/337636

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24. Crosswords–puzzle Title of the activity

Crosswords–puzzle

General description

The activity starts with a presentation about mind games, effective thinking, intelligence, intuition and inventiveness. Several strategies for creating mathematical games are taken into account such as: crosswords, puzzles, etc. Then the students choose the type of game to create and to solve in English, French or Italian.

Students’ age

12 - 13

Type of intelligence addressed

Linguistic intelligence

Educational goals

To stimulate imagination and creativity To improve intuition To encourage students to create and solve logical games in order to empower latent creativity and to form logical thinking skills To stimulate mind development through cooperation and competition, exploration, innovation and winning strategies

Expected results

Helping students gain and deepen knowledge Using, systematizing and consolidating mathematical concepts in the covered curriculum Translating mathematical definitions in English, French or Italian accurately. Capturing pupils' interest in mathematical and logical games, by using attractive challenges that motivate students to strive for solutions Improving creativity and memory, as well as the proper use of specific terms of Mathematics.

Methodology

Required tools

Learning through playing Investigation Problem-solving Heuristic conversation Use of ICT Sheets of paper, pencils, geometry kit, watercolours Computers, video projector, video camera

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Evaluation

Systematic observation of the students’ behaviour and interest in creating various games Analysis of the materials they created, observation of the mathematical terms used, as well as the definitions they formulated Appreciation of the look and presentation of the games (drawing and handwritten or created on the computer using various ICT tools). Originality of the games and accuracy of the suggested solutions.

Teachers’ recommendations

Suggest various types of games, according to students’ age and preferences. Use the tools that suit their expertise and fit the equipment of the class/ school, so that the activities performed are as interesting and interactive as possible.

Link(s)

Crosswords document http://bit.ly/1qCAIDg

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25. Thinglink: we explain Maths Title of the activity

Thinglink: we explain Maths

General description

Students are told about the importance of getting skills to explain their opinions and their work in a linguistic way. The video is chosen as a way to practice oral linguistic skills. 13-15 years Linguistic intelligence

Students’ age Type of intelligence addressed Educational goals

To promote linguistic research for the best Maths terms. To encourage students to use a variety of resources to provide explanations. To get used to speaking aloud for an audience in order to convince and transfer knowledge

Expected results

Helping students gain confidence Using, systematizing and consolidating mathematical concepts in the

Methodology

Required tools Evaluation

Teachers’ recommendations Link(s)

covered curriculum Learning to learn; developing skills Improving the vocabulary of Maths concepts Developing strategies to make decisions concerning unexpected questions Concept investigation Oral activity practice Use of ICT: video, photography White Board, blackboard, chalk, computer, smartphone or video camera Direct observation of the students while lecturing in front of the group Correct English use to transfer Maths concepts Observation of the group reaction: interest, possible questions Suggest a variety of Maths concepts, according to students’ age. Natural use of ICT tools. Open time task to be done when students have something ready to be explained. http://bit.ly/1Kpd89F (Thinglink)

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26. Graph stories Title of the activity

Graph stories

General description

Students are given a function graph and asked to interpret it and write a story to describe what is depicted in the graph. Every written problem begins with a story which needs to be interpreted and solved. It is easier to understand a picture (a graph for example) than a set of data. The advantage of a graph is that one can see the whole story at a glance.

Students’ age Type of intelligence addressed Educational goals

13-16 Linguistic intelligence

To connect Maths knowledge to real context To interpret graphs and understand some of their features To use language to communicate mathematical knowledge To help students represent given data in various ways and understand there are more ways to present the same situation

Expected results

Stories to interpret the given graphs. If the story is correct, someone reading it must be able to recreate the graph based solely on the story.

Methodology

The activity can be used with any kind of functions, starting from linear ones. Students can work in groups or individually. After writing the story, they can illustrate it by their own drawings or find a new graph to challenge their colleagues. After they finish, students can present their graph and stories to the class.

Required tools

Worksheets, pen and paper, computers. Optionally, Geogebra or some other similar software.

Evaluation

The teacher can observe and assess the involvement of the students and the oral story presentation. Some appropriate evaluation criteria for this activity would be the accuracy of the Mathematics involved, the accuracy of the of story in relation to the graph, the delivery of presentation and followed directions.

Teacher’s recommendations

Higher level students will need very little guidance while lower level or younger children will need a review or a given example story. Students can be allowed to discuss and exchange ideas and check each other’s work by trying to recreate the graph from the story. If students work individually, a greater variety of stories can be produced, while working in groups may narrow the range of ideas. The activity can be further used in the opposite direction, by randomly handing students the stories and asking them to re-create the graph.

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Link(s)

Stories created from graphs http://www.scribd.com/doc/220413821/The-Meeting-Point http://www.scribd.com/doc/211867100/The-Contest

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27. The axioms of Euclid Title of the activity

The axioms of Euclid

General description

A try to learn the 5 axioms of Euclid

Students’ age

16-17

Type of intelligence addressed

Linguistic intelligence

Educational goals

Expected results

To learn and understand the axioms of Euclid To read, learn and understand them in ancient Greek To hear them in Erasmus pronunciation To observe and find the differences between ancient Greek and the Erasmus pronunciation To translate the axioms into English To learn to cooperate To develop team spirit To think critically To develop descriptive and synthetic strategies To understand the role of culture in communication

Becoming familiar with the axioms Presenting the axioms in ancient Greek, modern Greek, Italian and English Becoming aware of the relationship between geometric shapes and the correct terminology of geometric concepts in the languages above Using digital tools in a creative way

Methodology

The students were divided into 5 groups, they chose an axiom and decided how to present it. Using cardboard and crayons they wrote and drew the axioms in Greek, ancient Greek and English. Students read and recorded the axioms in three different languages using suitable tools on the Internet for recording.

Required tools

Papers, cardboards, crayons, computers, cameras, the Internet

Evaluation

The activity is suitable not only to become familiar with the axioms of Euclid by working on them, but also to gain experience translating and pronouncing them in ancient Greek, English and the Erasmus language.

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Teachersâ€™ recommendations

It is suitable for the age of our students. This activity contributes to the cooperation of teachers of three different subjects, e.g. Maths teacher, Ancient Language teacher and English teacher.

Link(s)

http://bit.ly/1FIHkae (Axioms on Thinglink)

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28. Madmagz. Mathward Bound Title of the activity

Madmagz. Mathward Bound

General description

A digital magazine to publish common products created by all the participating schools in AIMS about the use of language related to mathematical concepts. 14-16 Linguistic Intelligence

Students’ age Type of intelligence addressed Educational goals

To promote reading and writing skills when dealing with Maths To encourage students to use their imagination to join Maths concepts with literature, both reading and writing To develop narration skills to explain graphs To encourage students to participate in a common product To get aware of the importance of the diversity in written works connected with Maths

Expected results

Helping students develop their linguistic abilities in different ways Using, systematizing and consolidating mathematical concepts in

Methodology

Getting a graph and providing a story to explain it Reading and finding out links with Maths in national literature works Writing about mathematical concepts Using plans with Maths data to create a story Using ICT tools, computers, digital cameras, books

Required tools Evaluation

the covered curriculum Learning to learn skills development Improving vocabulary about Maths concepts to explain graphs, riddles, short stories about Maths Understanding the importance of observation skills Promoting reading and writing Improving team work skills

Digital cameras, smartphones, computers, books.

Quality of stories. Correct English use to transfer Maths concepts Product originality Topic variety Involvement in team tasks

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Teachersâ€™ recommendations

Link(s)

Suggested for a variety of topics to promote active reading and writing. Give responsibilities to different students in groups according to their capacities. Accept even the smallest contributions in order to promote real team work. Encourage students to provide their own photos to support texts. http://madmagz.com/magazine/337636 (Magazine on Madmagz)

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29. Editing Aims Magazine Title of the activity

Editing Aims Magazine

General description

The students’ task is to edit a weekly magazine devoted to Maths.

Students’ age Type of intelligence addressed Educational goals

They should: prepare an interview with a famous mathematician illustrate a mathematical motto of the day either with one or more pictures (they can even prepare a mini comic book) create a Maths crossword, rebus or magic square prepare a Mathematical Curiosities section. explain the meaning of the number, which is the number of magazine’s page write a story expressing the given equation. 13-16 Linguistic intelligence

To use words as a primary way of thinking and solving Maths problems To revise Mathematics concepts through crossword puzzles To improve the ability of choosing the most important information and writing a short dialogue based on a long text To interpret a Maths equation and connect it to real life creating a story to it To improve group work and the ability of sharing tasks

Expected results

Aims Magazine in paper form, and if possible, an online presentation.

Methodology

The students should work in groups of four. Each group should make one page of the magazine using materials prepared before, then combine the pages together.

Required tools

Paper, pens, pencils, materials about mathematicians and numbers prepared in advance.

Evaluation

Assessment by teachers. Happening/dissemination – to show the results to others - it is nice to see reactions from the students. The diversity of ideas makes the magazine very interesting and extremely colourful.

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Teachersâ€™ recommendations

Suggest a variety of Maths concepts, according to studentsâ€™ age. Natural use of ICT tools. Open time task to be done when students have something ready to be explained.

Link(s)

Aims magazine on Issuu http://goo.gl/YVdQ6l Aims magazine on Polish website http://goo.gl/FQMSVx

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30. Pilish Literature Title of the activity

Pilish Literature

General description

The Pilish Literature book is created collaboratively and comprises several types of works inspired by the number PI: piems (poems having the number of words on each line or the number of letters in each word according to the digits in the number PI), limericks, a sketch. 13-16 Linguistic intelligence

Students’ age Type of intelligence addressed Educational goals

Expected results Methodology

Required tools Evaluation

Teachers’ recommendations

Link(s)

To use language to communicate mathematical knowledge To help students use mnemonics as a tool To empower creativity, originality, imagination and thinking “outside the box” To encourage students to express themselves and their feelings towards Maths in various ways To understand more about Pi and irrational numbers To ignite curiosity Collaborative eBook including various types of works The piems are bilingual, in English and the mother tongue of the creator. Both versions follow the digits in PI. The limericks and the sketch are only in English. The students can work either individually or in groups. Some students will find it easier to write in their native language first, and translate afterwards. Any online tool that allows the collaborative creation of a book, such as Mixbook: http://www.mixbook.com/ The teacher can observe and assess the involvement of the students. Some more appropriate evaluation criteria for this activity would be the accuracy of the poem in relation to the digits of PI, the originality of the work and the quality of the translation. It is useful to write a group poem first or present students an example. It is a type of activity that students enjoy, even the low achievers in Maths. It can be very suitable for Pi Day celebrations. Even if it will not make every student remember many digits of PI, they will get a better grasp of some of its features. Online collaborative book http://bit.ly/1wPBF0K

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8 - Bodily-Kinesthetic intelligence This is â€œthe ability to use one's mental abilities to coordinate one's own bodily movements. This intelligence challenges the popular belief that mental and physical activity are unrelatedâ€? (ERIC, 1996, p. 2). The main elements of the kinesthetic intelligence are the control of one's bodily motions and the capacity to handle objects skillfully, as well as the ability to process information through the sensations in the bodies. Students with this type of intelligence tend to learn better by involving muscular movement (e.g. getting up and moving around), and are generally good at physical activities such as sports, dance or building things. These students enjoy all types of sports and physical activities, they need to touch, move or manipulate objects, move around and act. For these learners, performing physical actions can lead to learning mental actions and operations, so the mathematical notions are represented by actions. Such students often express themselves through dance. They prefer using solid or pictorial models wherever possible and act on these, by moving parts around, while sometimes talking to themselves during this process. When these students remember ideas, they can think of the actions that they did or do the actions with their hands, for example running fingers along the sides of a triangle as a help to remember a theorem or formula. When they hear things they can focus on the actions that are being done and try to anticipate the outcomes of actions. As far as their difficulties are concerned, they may find it hard to remember the names and would need time to translate their action knowledge into alternative forms of expression, such as symbols or words. The methods to empower this type of intelligence would be the use of touching, feeling, movement, improvisation, "hands-on" activities, using mime and facial expressions and physical relaxation exercises.

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Activity Summaries 31. Practical Proofs Title of the activity

Practical Proofs

General description

Students do proofs of mathematical theorems and properties in a practical way, by object manipulation, preparing self-made models, doing experiments and observations. They use different materials and auxiliary tools.

Students’ age

11 – 16

Type of intelligence addressed

Bodily - kinesthetic

Educational goals

To understand mathematical abstract theorems or properties by proving them in a practical way To make real models to prove abstract theorems, using different materials To do experiments or proofs with the self-made models and present them to other classmates

Expected results

Better understanding of abstract mathematical theorems and properties Improvement in remembering and using mathematical abstract concepts in the future

Methodology

Students are divided into groups and given tasks according to their skills. They choose a theorem and discuss how to prove it practically (what materials to use, how to work, how to present the experiment, etc.). Students present their ideas to the teacher and share the final plan. Students carry out the practical proof and present it to the classmates (as a short video, a direct experiment or in another way). It is a good idea to do this task in a collaborative way, either in groups from the same class or in transnational groups.

Required tools

(Coloured) paper, cardboard, scissors, plastic solids, a scale, pins, and other materials, according to the students' imagination. A camera and ICT tools such as Thinglink.

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Evaluation

In groups students discuss the stages of their work, ask questions, look for solutions and analyse the outcome of their collaboration. Students can also make a video or do an exhibition displaying the practical models they have built.

Teachersâ€™ recommendations

Students should use simple, cheap materials; the most important thing is the idea of the practical proof and its understanding by the others, not the beauty or the value of the materials. Students should let their imagination free in using unconventional materials (cake or pizza dough, straws, fruit, sweets, etc.)

Link(s)

http://bit.ly/1B9Cxl3 (Practical proofs on Thinglink)

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32. Right Angles Are Always Right Title of the activity

Right Angles Are Always Right

General description

Constructions involving right angles using practical methods and unconventional tools.

Students’ age

14-15

Type of intelligence addressed

Bodily - kinesthetic

Educational goals

To help students think “out-of-the-box” To empower their imagination and practical skills To let them get a more hands-on perception on the way in which the parts of a Geometric figure relate to each other To use collaboration, role playing and ICT tools To connect Math with History

Expected results

A better ability to learn by manipulating objects Better understanding of the involved Math content Students creations to be further used as learning materials

Methodology

Students are offered worksheets and/or visual instructions on the construction process. They can work in groups, deciding on their own roles. The construction process is recorded with a video that they can later edit. At the end, they are asked to explain the Math content under the construction protocol (Thales theorem, Pythagoras theorem, the properties of the perpendicular bisector) and how it proves that the construction is correct. They can be asked to think of more similar examples.

Required tools

For the practical part: construction materials such as string, pins, a rubber band, a wooden spatula etc., as well as clear instructions. For recording and documenting the process: a camera, video editing tools, ICT tools for a common product to display the creations, such as www.thinglink.com

Evaluation

The students should be evaluated regarding both the participation in the activity as well as the adequate use and comprehension of the Maths involved.

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Teachersâ€™ recommendations

Clear instructions are required. Students can work individually or in groups, but in this second case, they should take turns (manipulating objects, recording, directing, editing the videos).

Link(s)

Worksheet on ancient construction methods http://bit.ly/1qE2wqI Worksheet on Theorem of Thales http://bit.ly/1zMD4rB Thnglink interface with all the videos http://bit.ly/1B9Cxl3

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33. Do you look like the Vitruvian man ? Title of the activity

Do you look like the Vitruvian man ?

General description

The purpose of this activity is to compare several measurements of students’ bodies with Leonardo’s measurements on his Vitruvian man. Students are to find the gold number 1.6 as well.

Students’ age

12-17

Type of intelligence addressed

Bodily-kinesthetic

Educational goals

To learn maths by making measurements on the body To find the link between our body and the gold number through the Vitruvian man To collaborate with students from other countries

Expected results

A set of measurements from all students A global mean of all measurements and its link with the gold number

Methodology

Required tools

Evaluation

Students work in groups. Each group receives a measuring tape. The photo of the Vitruvian man is shown with a video projector, so that every student can see it clearly. The parts of the body students have to measure are in colours. Next the students do their measurements and fill an online table with all their data. The mean is automatically computed thanks to a formula. A spreadsheet shared online Measuring tape The photo of the Vitruvian man

The students have to work out the mean of their measurements and check if the mean is approximately equal to the gold ratio 1.6. The more measurements they have, the better! They can compare their own measurements with the gold ratio individually as well.

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Teachersâ€™ recommendations

Collaboration, in this activity, helps gathering more measurements in order to get a very good approximation of the ratio height/navel. The more, the better!

Link(s)

http://bit.ly/1JJH141 (Wikipage famous numbers) http://bit.ly/1FIL4bR (famous numbers on the Twinspace)

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34. The Spacecraft Race Title of the activity

The Spacecraft Race

General description

Students build a vehicle propelled by a balloon and study its movement.

Students’ age

13-14

Type of intelligence addressed

Bodily-kinesthetic

Educational goals

To empower students’ practical thinking and creativity in relation to Math To help students grasp connections between Math and Science To help students think “out-of-the-box”

Expected results

A better ability to learn by building and manipulating objects Better understanding of motion phenomena and its laws Students’ creations to be further used as learning materials

Methodology

The activity has two steps: 1. building a vehicle of lightweight materials, pushed up by the air released from a balloon; 2. completing worksheets with the results of the measurements and calculations to find the speed of the vehicles. This is an experimental activity. The vehicle moves on the rocket principle. The air is suddenly released from the balloon, producing an impulse which makes the vehicle move.

Required tools

Polystyrene trays, flexible straws, toothpicks, coloured balloons, threads, wires, polystyrene cylinders for wheels, sandpaper, geometry instruments and markers.

Evaluation

The evaluation is focused on the students’ involvement as well as the accuracy of their worksheet results and observations.

Teachers’ recommendations

In order to make the vehicle move faster, you can cut the unnecessary parts from the polystyrene tray. By becoming lighter, its speed gets higher.

Link(s)

Slide share presentation http://bit.ly/1vchwQX

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35. Ethnic motives

Title of the activity

Ethnic motives

General description

Students try to draw and construct geometrical ethnic motives.

Students’ age

15-17

Type of intelligence addressed

Bodily-Kinesthetic

Educational goals

To distinguish geometric shapes in traditional textiles and ceramics of students’ own country. To learn how to draw traditional geometric shapes using pencils, pens, GeoGebra, etc. To learn how to embroider traditional geometric shapes, e.g. on aprons. To learn how to construct mosaic using beans, lentils and rice.

Expected results

Presentation of traditional textiles and ceramics in which students underline geometric shapes Students’ drawings with geometric shapes made with the use of pencils, pens, GeoGebra, etc. Aprons decorated by the students with traditional geometric shapes. Mosaics with geometric shapes using beans, lentils and rice.

Methodology

Students work in groups of four or five. They choose one subject and find traditional textiles and ceramics in books, magazines, internet or even in their house. They try to recognize geometric shapes and draw them with pens, pencils, GeoGebra, etc. Some of them try to embroider geometric shapes on aprons while the others try to construct geometric shapes with beans, lentils and rice.

Required tools

Internet, presentations, pens, pencils, paper, textiles, needles, coloured thread, beans, lentils, rice and glue.

Evaluation

The students learn more about the traditions of their own country and find relations between tradition and geometry. They acquire skills such as learning to design, embroider or construct mosaics. They understand the space around them and they learn to reproduce it as well as possible.

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Teachersâ€™ recommendations

Link(s)

Teachers should instruct students how to begin drawing, embroidering and constructing mosaics and how to use needles with safety. http://bit.ly/1FnmsVx (Drawing motives on Youtube) http://bit.ly/1xRHT0T (Traditional Greek motives on Youtube) http://bit.ly/1C9FD97 (Greek motives)

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36. The Handshaking Problem Title of the activity

The Handshaking Problem

General description

Students use physical things and movement to figure out the number of handshakes in a group.

Students’ age

12-15

Type of intelligence addressed

Bodily-kinesthetic

Educational goals

Expected results

To learn about the theory behind the handshaking problem To link the use of rope to the handshaking problem To physically use rope and see the number of handshakes grow rapidly To find a formula for the number of handshakes To use two different types of Mathematics to calculate the number of handshakes To remember the formula better due to the use of physical proof Better understanding of the formula to calculate the number of handshakes Better recollection of the formula due to the use of physical movements Higher motivation to study mathematics

Methodology

Students are divided in small groups first and then groups have to be combined to make sure to get bigger groups for larger amounts of handshakes. After the first counting, tables are made and a formula must be found in two different ways.

Required tools

Only some kind of rope, thread or wool to represent a handshake.

Evaluation

Students enjoy this assignment a lot, they have to use a lot of different things to know the number of handshakes. Writing the results of the number of handshakes on the whiteboard is a way to keep everybody busy as well.

Teachers’ recommendations

You need enough rope to do this assignment, especially if you want to have a big group. For bigger groups you need larger pieces of rope, which you don’t need in the small groups. It could be handy to have a lot of strings of different lengths ready at the start to speed up the process a little bit. Letting students take pictures themselves helps motivation.

Link(s)

Assignment http://bit.ly/1v2cq9V PowerPoint http://slidesha.re/1FiLsPz 75

37. Hands on Maths Fair Title of the activity

Hands on Maths Fair

General description

A selection of Maths-related activities, based on crafts such as wire and leather jewellery, string art and crochet, for students to choose.

Students’ age

14 - 16

Type of intelligence addressed

Bodily-Kinesthetic

Educational goals

Expected results

Improvement of students’ manual abilities to work with wire, thread and needles. Better knowledge of daily tasks New creative activities for students’ spare time related with Maths. Creation of artistic objects by connecting mathematical knowledge with crafts. Production of home decoration ornaments and personal jewels.

To experiment craftsmanship in order to work on Maths concepts. To develop manual and visual work To connect Maths concepts with jewellery craft To enhance cooperation skills in a group To carry out activities regardless of gender To explore fractals, Peano and Hilbert curves in a creative way To work on rotation, symmetry, similarity and segments.

Methodology

There are several working stations to create small pendants following a Hilbert curve pattern, leather bracelets with Peano curve patterns, string art and crochet productions with polygons patterns. Students move from table to table to choose the activities they like. The teacher moves around giving instructions and helping students to work on different tasks. Maths concepts are mentioned so that students can connect them with what they do.

Required tools

Cork pieces, nails, wire, thread, yarn, needles, leather, cardboard, scissors, pliers, digital cameras to record the activities and a computer to edit and publish the videos.

Evaluation

Participation, interest, commitment to the work, cooperation, helping attitudes Quality of final products (but it is not the most important issue)

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Teachersâ€™ recommendations

A big room where the tables can be spread is needed. Space to allow teachers and students to move is also required. Materialsâ€™ preparation, such as cutting pieces of cork and leather, nailing, printing instruction sheets, takes its time. So, it should be done in advance. Activities such as sewing and crocheting are not common nowadays, so students will have to be given the most basic instructions as well as time, patience and encouragement.

Link

http://bit.ly/17IcpAS Vimeo with a video created with pictures. http://bit.ly/1ALE9gb VISHUB with explanations and questions

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38. Maths Origami Workshop Title of the activity

Maths Origami Workshop

General description

Origami, a traditional Japanese technique, is the art of paper folding. In this activity students need to match their manual skills to Maths and use their hands to „make” Maths shapes and objects. In this way geometrical rules, axioms and operations are „physically” elaborated and stimulate kinaesthetic learning.

Students’ age

15

Type of intelligence addressed

Bodily-kinesthetic

Educational goals

To develop fundamental geometrical ideas and spatial intuition To experience geometry by folding paper To know geometrical solids To study Mathematical vocabulary To increase understanding of symmetry, angle, congruence and similarity To make objects by manipulation To practice fractions, ratios, proportions, and measurement To develop problem solving and critical thinking skills

Expected results

The students are expected to test their kinesthetic intelligence by creating geometrical objects. At the end they will be able to say they are/are not kinesthetic learners. This activity can be defined as a metacognitive experience. Final products: paper geometrical objects like regular and starred polyhedra

Methodology

Workshop 1 (2 hours) The expert shows the basics of the origami technique. Students get acquainted with folding and manipulating. They produce simple geometrical objects like cubes. Workshop 2 (2 hours) Students are shown how to make more complex geometrical solids like starred polyhedra compositions. They have their try. Workshop 3 (2 hours) The students try with other solids like octahedra, tetrahedra and dodecahedra.

Required tools

Square sheets of coloured paper, scissors, glue and ruler.

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Evaluation

Evaluation must consider the following points: Quantity and quality of products Involvement of students Raised awareness of kinesthetic learning (this can be done with a final brain storm or debate).

Teachersâ€™ recommendations

You cannot improvise, so you need to learn the procedure of folding on your own beforehand or otherwise involve an expert.

Link(s)

http://youtu.be/IVo5vtirAS4 Video on Maths Origami Workshop http://youtu.be/J9gvXmu_1BE Video Origami cube construction http://www.mathigon.org/origami/ Site on mathematical origami

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39. Paper Maths Title of the activity

Paper Maths

General description

The students are encouraged to use their kinesthetic intelligence to make geometrical paper objects and carry out simple proofs of their geometrical properties through paper folding. Students will also use measurement and manipulation. In this way they will understand relations, rules and theorems in a kinesthetic way.

Students’ age

14

Type of intelligence addressed

Bodily-kinesthetic

Educational goals

Expected results

To find out Maths relations in everyday situations To find motivating inputs for Maths study To understand and apply geometrical properties in various contexts To experience geometry by folding paper To learn Mathematical vocabulary To make objects by manipulation To develop problem solving and critical thinking skills Part 1 The students are expected to analyse the Maths aspects of an everyday situation by manipulating paper. They will be able to do this by measuring, folding the sheet, keeping together a number of A4 sheets in order to get larger sizes. Part 2 The students will make equilateral triangles by folding the paper and will analyse their geometrical properties through manipulation. For both activities the final product will be a Blendspace presentation with worksheets, videos, photos, tutorials and questions for the students (useful for follow-up activities).

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Methodology

Part 1 Students start from the observation of the different paper sizes usually used for photocopying, measuring the length of the sides of different rectangles corresponding to the various paper sizes. Students then calculate the ratio between the sizes of the sides in order to find out that this is constant. The paper sizes are chosen so as to keep the proportions when making the pages larger ( e.g. A4 A3, A3 A2) or smaller ( e. g. A3 A4, A4 A5) and, therefore, without paper waste. This is possible only if all the sizes are similar. The obtained ratio introduces the irrational number √2. In this way we can move from the practical kinesthetic phase of the experience (measuring and manipulating) to the theoretical one (proof of the property found out through experience). Part 2 At first the students make a triangle by folding an A4 page. Then they check that the triangles are equilateral by manipulating and measuring them. Finally they verify some theorems on equilateral triangles by folding the paper.

Required tools

A4 and A3 sheets of paper

Evaluation

Evaluation must consider the following points: Involvement of students Worksheets with measures, calculations and conclusions produced by the students Raised awareness of kinesthetic learning (this can be done with a final brain storming or debate).

Teachers’ recommendations

The first activity is the starting point to introduce the concept of similarity in plane figures for the younger students and the concept of irrational numbers for the older ones. The second activity is a good start for the study of equilateral triangles and their properties.

Link(s)

http://blnds.co/1tse4AF - Maths with an A4 page. First partBlendspace presentation http://blnds.co/1xQ4zzR - Maths with an A4 page. Second part Blendspace presentation

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40. Platonic Solids. Origami Workshop Title of the activity

Platonic Solids. Origami Workshop

General description

Practical lessons to construct 3D polyhedra over a period of 55 minutes.

Students’ age

14 -15

Type of intelligence

Bodily-kinesthetic

Educational goals

Expected results

To improve manual abilities with paper. To promote a contact with the classical aesthetic theory of regularity concerning perfection. To create 3D polyhedra while revising geometry topics. To increase students’ self-esteem after finishing the constructions on their own. To revise English vocabulary

To manipulate paper to construct Platonic Solids To revise geometry concepts: angles, edges, vertices and sides. To prove in a practical way why there can only be 5 Platonic Solids To follow oral instructions To get acquainted with Origami vocabulary and basic folds

Methodology

Students are offered pieces of paper in different colours. An expert gives them basic knowledge about Origami and presents the instructions slowly about each fold. He moves from table to table to check whether his instructions are understood and corrects possible mistakes in students' work. Students ask questions while continuing their work. The expert gives participants instruction sheets at the end of the workshop to go on practicing at home.

Required tools

Origami paper already cut. Instruction sheets to be used after the workshop. Blackboard and chalk. A digital camera to record the session and a computer to edit and publish the results.

Evaluation

Active participation, interest, cooperation with partners, when help is needed, are the main criteria to evaluate the activity as well as the participants' positive attitude towards corrections when failing. The quality of the final products will also be assessed.

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Teachersâ€™ recommendations

An expert, both in drawing and Origami, is needed. Students should be sitting in such a way that they can see and hear the expert's instructions and should have enough room to move their hands. The classroom should allow the expert to move easily from table to table to check participants' work. Instruction sheets should be given at the end of the workshop to prevent students from going through them during the session. In this way their attention to the expert is granted. Time and patience to carry out the whole activity is an important issue.

Link(s)

http://bit.ly/1wVEevh Thinglink with two videos from the workshop sessions.

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41. Mathematical Charades Title of the activity

Mathematical Charades

General description

Active learning based on competition. Students learn and revise mathematical concepts by playing. They use their knowledge of Maths in practice.

Students’ age

11-16

Type of intelligence addressed

Bodily - kinesthetic

Educational goals

Expected results

To facilitate understanding of basic mathematical concepts To improve the students’ ability to think abstractly and enrich their imagination To engage both mind and body in Maths studying. To measure students’ knowledge and understanding of mathematical concepts Better understanding of mathematical concepts Increase of students’ participation in Maths classes.

Methodology The teacher or students prepare a list of mathematical concepts (e.g. theorems, definitions, names of mathematical objects or famous mathematicians, etc.) and write them down on separate cards. Then the students are divided into two teams competing against each other. The representative of a given team draws a card and presents his/her team its contents with the use of gestures, movement or nearby objects. If his/her team guesses the answer, it scores a point. The winner is the team which has more points at the end of the game. Required tools

Index cards or paper torn/cut into pieces.

Evaluation

Watching the video of charades followed by a discussion about the choice of ways to present particular concepts. Brainstorming about possible alternatives in concept presentation.

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Teachersâ€™ recommendations

Mathematical charades can be used as a warm-up activity at the beginning of the lesson or as a revision device. Itâ€™s an efficient tool for younger learners to practice geometrical concepts such as acute or obtuse angles, symmetry, perimeter, etc. and for older students to learn transformations of functions.

Link(s)

https://vimeo.com/113144232 (Mathematical charades on Vimeo)

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42. Practical Proof of Thales’ Theorem Title of the activity

Practical Proof of Thales’ Theorem

General description

Students build and use a wooden construction to prove the theorem of Thales

Students’ age

12-15

Type of intelligence addressed

Bodily-kinesthetic

Educational goals

To learn the theorem of Thales To link the building of the construction to the theorem To physically use the construction to prove the theorem To remember the theorem better than in a normal class situation

Expected results

Higher motivation in studying Mathematics Better understanding of the theorem Better recollection of the theorem

Methodology

Students are divided into groups of two (or three). After watching the instruction video, the students start building the construction. The use of the construction is a physical proof. However, students have to give the theoretical proof as well.

Required tools

A small piece of wood (not too thin) of approximately 10 by 15 cm. A spatula, 4 thumbtacks (drawing pins) and a rubber band.

Evaluation

Students really like this assignment due to the combination of building something and using that for Mathematics. The aim is that when they study this subject in the curriculum, they will remember the theorem through this practical experience.

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Teachersâ€™ recommendations

It is really handy to have a computer room available for showing the instruction video. Another option is to print the PowerPoint presentation and use that information to build the construction. This allows every group to work at their own pace. Devote at least 30 minutes for this assignment, maybe more if you want to include the theoretical proof at the same time.

Link(s)

Instruction video: http://youtu.be/ZJsXr5hMJpE PowerPoint: http://slidesha.re/1uvDeLS Assignment: http://bit.ly/1zjcD9h

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9 - Spatial-visual intelligence This type of intelligence deals with spatial judgment and the ability to visualize things and notions with the mind's eye. This can be included in the right-brain activities category. Itâ€™s the ability to manipulate and create real or mental nonverbal images to solve problems or to understand ideas. This intelligence is not limited to visual domains - Gardner notes that spatial intelligence develops in children who cannot see, as well. Understanding takes place best when using or creating images, graphics. The information is translated into visual codes, spatial relationships, patterns and properties. This representational form allows students to interconnect ideas in relation to their spatial or temporal proximity, rather than using their logical or linguistic relationships. According to studies, it is very important to Math learning. Students with this type of intelligence would draw pictures of the notions taught to them, while learning those notions. It is the case of a pupil who would represent the number 34 by using three bundles of ten sticks and four loose sticks. These students usually assemble specific pieces of information into a bigger mental picture that offers visual clues. They recall ideas by imagining what they look like and by using visual mnemonics. They organize key ideas by using pictures or schematic maps. They also manipulate and reposition parts of the image mentally in logical ways. They benefit of a visual approach or visual tools. Students with strong visual intelligence depend on visual thinking and are very imaginative. They like drawing, painting, or sculpting their ideas and often express their feelings and moods through art. They also enjoy daydreaming, imagining and pretending. They excel at reading diagrams and maps and enjoy solving mazes and jigsaw puzzles. The best instruments for working with them are movies, pictures, videos, charts, graphs, diagrams, graphic organizers, art activities, doodling, microscopes, computer graphics software and demonstrations using models and props.

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Activity Summaries 43.Tessellations Title of the activity

Tessellations

General description

Students study and create tessellations using different techniques and points of view: e. g. hand drawing, geometric transformations or ICT tools. They also try to find tessellations in real life and in their daily environment.

Students’ age

12-16

Type of intelligence addressed

Spatial-visual

Educational goals

To develop students’ visual intelligence To learn about geometrical transformations To find out about mathematical aspects of art, such as Escher’s creations To collaborate on a common topic from different perspectives

Expected results

A collection of students’ tessellation works, created both using computers and by traditional methods. A better capacity to perceive mathematical aspects both in art and in the everyday reality Awareness of Maths as a vehicle for art and culture

Methodology

The teacher can present some information about tessellations such as tessellations using regular polygons, Escher’s work, instructions on how to create tessellations using geometrical transformations or other tools. Students can work in groups on different tasks: creating tessellations by drawing, using ICT tools such as Paint, GeoGebra, web-based tools or mobile apps or taking photos of tessellations in real life.

Required tools

Depending on the methods chosen, the tools can include drawing tools, computers, tablets, mobile apps, but also a camera and a photo editing tool.

Evaluation

Students’ work will be assessed from the point of view of originality and accuracy. The activity is creative and motivating, even for younger students. These can be asked to find animal shapes or similar things in tessellations.

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Teachersâ€™ recommendations

The activity has to be adapted to studentsâ€™ age and skills. For older students, the teacher can emphasise the topics related to geometric transformations.

Link(s)

http://bit.ly/1DwenSf

(Tessellations on Thinglink)

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44.Tessellations in FRANCE Title of the activity

Tessellations in FRANCE

General description

This activity is aimed at working at tessellations using either a computer or student’s hands !

Students’ age

16-17

Type of intelligence addressed

Spatial-visual and Logical

Educational goals

Expected results

Methodology

Required tools

Evaluation

To understand the way of building any shapes from a rectangle in order to create a tessellation To program a tessellation using the software Python (turtle package) Original handmade tessellations from students following their creativity Easy tessellations (triangles, squares, etc.) programmed with the Python software (turtle package). The teacher introduces tessellations using the website : http://bit.ly/1dwahyN Students study a video about the presence of tessellations in nature: Bees and tessellations: http://bit.ly/1QoiIxX Each student focuses on their individual work to create a tessellation of their own design with scissors and pencils using rectangle-shaped pieces of paper; Students engage in peer work to program the tessellation on the computer Sheets of paper, scissors and pencils. Python software with the turtle package

Students’ work will be assessed from the point of view of originality and accuracy.

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Teachers’ recommendations

At the beginning, students must understand what a tessellation is, first with very easy shapes, like square tiles. Then they have to guess if they can make tessellations with any kind of shapes. That’s why we have to show them various examples, some leading to a tessellation, others not! The handmade tessellation must be very precise to be matched together. Students must use colours to underline the edges of the shape. They can decorate the shapes to understand how they must be repeated. If we want to go further, we can ask the students to program an easy tessellation, using for example Python. It is a great way to use the loop “for”, and it is a good connection with the logical intelligence as well.

Link(s)

http://bit.ly/1S47Cvy (wikipage about the spatial intelligence) http://bit.ly/1MIeiwc (Animoto movie about tesselations) http://bit.ly/1dwbhmD (Tessellations on the Twinspace)

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45. Visual Proofs using GeoGebra Title of the activity General description

Visual Proofs using GeoGebra Students do visual proofs of mathematical theorems, using the dynamic mathematics software GeoGebra.

Students’ age Type of intelligence addressed Educational goals

11 – 16 Spatial-visual

Expected results

Methodology

To understand better and more about abstract mathematical theorems and properties by visualising them in a dynamic way To make visual representations of proofs To use Geogebra software

Better understanding of the abstract mathematical theorems and properties. Using spatial intelligence to study abstract theorems Presentations and explanations about these visual proofs to other classmates. Students, working in groups, choose a theorem that they want to prove. Then they discuss the necessary steps that need to be taken in order to build its visual representation with GeoGebra Students share tasks, so each of them carries out a part of the representation. One team member presents and explains the outcome of their work to the rest of the classmates.

Required tools Evaluation

Computers, Internet connection and a video projector. Discussions with the students (presentation, observations, etc.)

questions,

Teachers’ recommendations

A final product (for ex.Thinglink) with all the visual proofs created. Students should know how to use the geometric tools from GeoGebra. It is recommended that students make some simple geometrical constructions to get familiar with the GeoGebra tools before doing this activity.

Link(s)

http://bit.ly/1zPXM5C (Thinglink with visual proofs)

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46. Maths Cube Puzzle

Title of the activity

Maths Cube Puzzle

General description

Creating a puzzle with cubes where Maths contents and pictures with geometry elements are seen

Students’ age

14-15

Type of intelligence addressed

Visual-Spatial Intelligence

Educational goals

To develop students' skills to perceive Maths contents around them To encourage students to find new work strategies. To stimulate creativity. To promote team work. To make students aware of the importance of the game in learning.

Expected results

Students are expected to agree on the contents to show. Students are supposed to manage proportion to split the contents Groups are expected to collaborate manually to create the puzzle A final puzzle with twelve cubes will be produced

Methodology

Students are divided into three groups. Each group is in charge of two faces for the cube creation A problem, an equation and their solutions and two photos from the school are agreed on. Students look for places at school where Geometrical shapes are clearly seen. Two photos are taken. Students choose a problem and an equation they are working on and solve them. Groups decide on the size for the cube. Affinity Designer or similar software is used to write on the faces. Files with the cubes are printed and cut to produce the cubes.

Digital cameras or smartphone. Computers and printer 160 gram paper. Scissors and glue. A cardboard box Affinity Designer, Adobe InDesign or Gimp

Required tools

Evaluation

Observation during the process, accuracy and relevance in contents as well as active collaboration will make part of the evaluation criteria.

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Teachers’ recommendations

Links

Groups shouldn't be very big. 3 or 4 students per group. Software installation must be checked in advance. Four lessons should be dedicated to the whole process. Paper should be ready to start printing. The puzzle should be taken into different classrooms to check its use. https://vimeo.com/117517451 (Cube puzzle on Vimeo)

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47. Mosaic Construction Title of the activity

Mosaic Construction

General description

How a mosaic is constructed with the direct method (by gluing the individual pieces on the surface), for example the flag of a country.

Students’ age

15-17

Type of intelligence addressed

Spatial-visual

Educational goals

to develop students’ ability to perceive the space around them in order to carry out their construction to realize the size and the shape of puzzle pieces so that their construction is aesthetically and symmetrically successful to become familiar with the use of materials such as marble, glass or ceramics to become familiar with the use of tools such as special scissors, tongs or different kinds of glue to learn how to work in groups in order to achieve common aims

Expected results

Production of mosaics representing the flags of different countries Use of special tools and materials for mosaic construction

Methodology

Students prepare the supporting surface on which the mosaic is going to be constructed (this specific surface is made of wood that has been burnished and painted in oil colour). Then they draw the shape they want the mosaic to be constructed on. Next, the students put the puzzle pieces on the surface just to check how to make the best use of the working space. When the students are ready having found the correct place for the pieces, they glue them on the wooden surface.

Required tools

Supporting surface made of wood, small pieces of different materials (marble, glass, ceramics) and colours, special scissors for cutting tesserae, tongs and glue.

Evaluation

The assessment of this activity depends on the accuracy of the mosaic construction and mainly on the final aesthetic result.

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Teachersâ€™ recommendations

Students must be very careful while using the special scissors because the tesserae are very hard and cannot be cut very easily. Moreover, the supporting surface must be thoroughly dried after burnishing and painting.

Link(s)

http://bitly.com/1vL9TVf

(Mosaic construction on Youtube)

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48. Calculation of areas with Geoboard and Pick’s formula Title of the activity

Calculation of areas with Geoboard and Pick’s formula

General description

Students calculate the area of a shape with Geoboard Pick’s formula

Students’ age

15-17

Type of intelligence addressed

Spatial-visual

Educational goals to learn how to use Pick’s formula in order to calculate the area of flat shapes to distinguish the knobs which are appropriate for the calculation of areas to realise when to use Pick’s formula Expected results The ability to calculate the area of any flat shape with Pick’s formula Distinguish the knobs of a Geoboard which are included in a flat shape Methodology

Required tools

Students are given a photocopy of the Geoboard and they are asked to draw a random flat shape on it. They count the knobs both on the perimeter and in the interior of the shape and calculate the area with Pick’s formula: E=λ+1/2κ-1, where κ is the number of knobs which are on the sides of the flat shape, λ is the number of the flat shape and λ is the number of knobs which are in the interior of the shape. Photocopies, geoboards, pencils and rulers.

Evaluation

Correctness of the calculation of the area of random flat shapes with Pick’s formula

Teachers’ recommendations

Most importantly, the knobs on the geoboard must be correctly counted when calculating the area of flat shapes

Link(s)

http://bit.ly/1AjuvAF (Geobord activities on Scribd)

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49.Visual Maths Title of the activity

Visual Maths

General description

Students are stimulated to find Maths in everyday life and in the objects around them in order to find the link that closely unites concrete facts to abstract ideas.

Students’ age

15

Type of intelligence addressed

Spatial-visual

Educational goals

To fight the stereotype that Maths is an abstract and bookish subject To find Maths rules and geometrical concepts in everyday life To empower the observation skills To relate concrete facts to Maths and Geometry patterns and properties To use visual mnemonics To develop induction

Expected results

The students are expected to do a research work looking for Maths patterns and rules in real life: in their houses, in their hometown, in architecture, in art, in music and even in their bodies. They take photographs of the aspects they are interested in and, in teams, they make presentations using Web 2.0 resources.

Methodology

The Maths teacher brainstorms students about possible aspects of Maths presence in real life, gives examples and shows pictures. In some cases he/she may need to explain Maths rules or principles (the golden ratio, tessellation, geometrical shapes, etc.) as a prerequisite. The students are then divided into teams of three and decide their research focus, for ex.: arches the golden ratio in art, architecture, music and human body parallel and perpendicular lines in the hometown geometry in modern architecture all over the world tessellations in real life perspective shapes in Nature

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Students are given a time to do their research and take their photographs. After that, they produce the presentations.

Required tools

A camera, a smartphone and web 2.0 presentation resources (Prezi, Kizoa)

Evaluation

Evaluation must consider the following elements: Involvement of students Quality of products Respect of time and task Raised awareness of visual-spatial learning (this could be done with a final brainstorming or debate)

Teachers’ recommendations

Be careful not to make the students use Internet pictures. The activity is much more effective if they do the “real thing”.

Link(s)

http://bit.ly/1cLAYyC (Thinglink visual maths)

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50. The ‘Christmas’ Fractal tree Title of the activity

The ‘Christmas’ Fractal tree

General description

Students make a kind of Sierpinski triangle in 3D, so they can get a grasp of what fractals are.

Students’ age

12-15

Type of intelligence addressed

Spatial-visual

Educational goals

To let students get an idea of the concept of fractals. To use this 3D model in Mathematics to see the link with the Sierpinski triangle. To work with your hands to build a real 3D model in order to visualise a real fractal.

Expected results

Different stages of fractals according to students’ different skills. Better understanding of the concept of fractals.

Methodology

Students get a worksheet with all the steps to take in order to get to the 3D fractal. Students work in pairs and help each other with folding and cutting.

Required tools

A pair of scissors Two A3-sized pieces of paper (one to fold and cut and the other one as background)

Evaluation

Some of the groups had some problems when they reached higher iterations, mostly because of poor pairs of scissors. It is nice to see that different groups will have a different final result. This way you can show the consecutive iterations.

Teachers’ recommendations

Link(s)

It might be wise to show them the final result first, because they will have a better idea what to do. Groups of two are ideal, bigger is not handy, because then at least one of them won’t be working. http://bit.ly/1MIfwaJ (Christmas trees on slideshare)

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51. Constructing Geometric Solids Title of the activity

Constructing Geometric Solids

General description

Students construct original solids such as Stella Octangula, Platonic Solids or Archimedean Solids. Then they make an exhibition in school.

Students’ age

14 -16

Type of intelligence addressed

Spatial-visual

Educational goals

To learn about various geometric solids and their properties. To learn about the relationship between the number of faces, vertices, and edges in any polyhedron To improve spatial intelligence by constructing physical models of geometric solids To improve Maths vocabulary in English

Expected results

improvement of students’ manual abilities improvement of knowledge related to geometry. revision of English vocabulary. school exhibition

Methodology

Students choose the geometric solid they want to make, download and print its pattern. They find a design that prints in colour, select a black & white pattern to print on coloured card stock, or use their creativity to decorate the solid. Then students colour the pattern template as they like with markers, coloured pencils or crayons. They cut around the polyhedron pattern, fold, form the solid into its shape and glue it. While working students make notes about the solid and discuss its properties.

Required tools

Paper - template with pattern, scissors and glue.

Evaluation

Presentation of the geometric solids as a final product. Exhibition of solids in school. The quality of the final products is assessed by students. Discussion.

Teachers’ recommendations

Students prepare basic information about the solid of their choice before the activity. They should print its pattern out.

Link(s)

http://goo.gl/VAHrAV (Geometric solids on the Twinspace)

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52. TANGRAM

Title of the activity

TANGRAM

General description

Forming flat shapes using seven given geometric shapes. Tangram is a traditional Chinese game/puzzle that consists in putting tohether seven given polygons to form a wide aray of shapes and images.

Students’ age

15-17 years old

Type of intelligence addressed

Spatial-visual

Educational goals

Students must learn to analyse geometric shapes and decompose them in simple polygons to compose geometrical shapes from the given polygons to assess the precision of their own work to think divergently and look for several solutions for the same problem to compute and compare surfaces.

the ability to create flat shapes using seven flat geometric shapes the ability to compare surfaces critical thinking imagination and divergent thinking

Expected results

Methodology

Students can work individually or in pairs 1st step: Cut the seven parts of the tangram with scissors. 2nd step: Put the geometric shapes in order and make figures. 3rd step: Look for a different way to create the same image, if possible 4th step: Look for a new shape than can be created by the same polygons, using Google Images and challenge your colleagues to create it.

Required tools

Board with the seven pieces of tangram, the figures of the tangram and a pair of scissors. Computers with an internet connection.

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Evaluation

Students’ ability to construct the figures of the tangram. The ability to correctly compute and compare surfaces. Students’ involvement in the task, the use of Google Images search tool using the advance options in order to find the ones they are allowed to reuse. Students’ persistence and curiosity.

Teachers’ recommendations

Start with easy figures and continue with more complicated ones.

Link(s)

http://bit.ly/1Ffao8l

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53. Broader Perception and the Flatland

Title of the activity

Broader Perception and the Flatland

General description

Understanding the three dimensions and the differences of perception, in a broader sense. This example comes from Edwin Abbot’s book entitled “The Flatland". In this short story there is a world on a plane surface, with inhabitants which are two dimensional shapes, that is squares, triangles etc. It is obvious that the inhabitants of this country, the Flatland, cannot perceive height. Their world has only two dimensions: length and width. One day a creature of the three dimensional world, a sphere, visits a Flatland mathematician who has the shape of a square, with the view of revealing the secrets of the third dimension to him. The square using its vision and perceive the sphere as a circle since its section (intersection) with the plane is actually a circle and the square can perceive only two dimensions because this is his world. The sphere tries in vain to explain to it that there is a third dimension displaying its qualities. Abbot’s book is a wonderful work of imagination and at the same time a biting social satire. Starting from this idea, the activity challenges the students to represent sections or projections of three-dimensional objects on a plane, using a worksheet created by the teacher.

Students’ age

15-17

Type of intelligence addressed

Visual – spatial

Educational goals

To understand the meaning of the third dimension and therefore the remaining dimensions. To understand the concept of projection to a horizontal plane of various geometric solids. To understand the intersection of a solid to a horizontal plane. To reflect on how the point of view can alter our perception, both from a geometrical point of view, and from a practical one.

Expected results

The ability to draw the object projection to a horizontal plane. The ability to draw the object incision to a horizontal plane. The ability to choose the most suitable projections for the best presentation of the solid. The understanding the differences between two- and threedimensional images and the perspective effect. 105

Methodology

o Introduction: The students are explained the idea of the activity. They can watch a video regarding “Flatland “ or listen to a reading from the short story. 1st step: students draw the correct projections of a cube 2nd step: students draw the correct projections of a cylinder 3rd step: students draw the projections of a rectangular cuboid 4th step: students draw the projections of a geometrical solid. Conclusions: students are asked to reflect and discuss the idea of perception, as used in Geometry but also in the real-life situations.

Required tools

Drawing paper, drawing instruments (ruler, right-angled triangles, compasses), worksheet.

Evaluation

the accurate design of the object projections students’ involvement in the task relevant ideas regarding perception and how we can acknowledge and harmonise different points of view.

Teachers’ recommendations

Teachers should emphasize the importance of the design scale and the accuracy of the design, as well as the idea of different perception, as used in Geometry, but also in the real life situations.

Link(s)

http://bit.ly/1EFEgtO

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54. How to create a map

Title of the activity

How to create a map

General description

Students learn to draw a scale map

Students' age

15-17

Type of intelligence addressed

Spatial

Educational goals

Expected results

to get familiar with the science of cartography to learn how to draw a map to familiarize with the use of Cartesian co-ordinates to conceive the scale of a map to recognise the symbols on a map The students should draw a map to a specific scale

Methodology

Students draw a geoboard (grid). The choice of a map scale depends on the width of the area depicted on the map. They represent the points on the surface where they are going to be drawn with Cartesian co-ordinates. The students the points together depending on what they depict on the surface of the earth (e.g. points of a building or a building site). The students draw symbols for characteristic points of the surface depicted on the map (e.g. rivers, streets, electric current lines, etc.). In the end they draw the titles of the map

Required tools

Special drawing paper, pencils, erasers, sharpeners, an architect’s scale, rulers. Special software suitable for drawing in a computer.

Evaluation

Students’ ability: To draw a map To use appropriate symbols and drawing scales final result which is a complete map drawing.

Teachers' recommendations

The teachers should pay attention to the accuracy of both drawings and the map scale.

Link(s)

http://bit.ly/1Qic5NJ 107

10 - Logical-mathematical intelligence The logical-mathematical intelligence has to do with the ability to detect patterns, to reason deductively and think logically. This intelligence is most often associated with scientific and mathematical thinking. It uses both inductive and deductive logic, abstractions, reasoning, numbers and critical thinking. but it is also connected with the capacity to understand the underlying principles of the causal system. The logical-mathematical intelligence can refer to an individualâ€™s ability to do things with data such as collecting, and organizing, analysing, interpreting, concluding and predicting. It was studied and documented by Piaget. Students with this type of intelligence like working with numbers and patterns, excel at drawing conclusions from gathered data, ask questions and conduct experiments eagerly. They are good at using mathematical and scientific symbols. By using â€˜scientific logicâ€™ or reasoning to link ideas, they look for logic, order and consistency. Such an individual would be a contemplative problem solver; one who likes to play strategy games and to solve mathematical problems, to draw conclusions and formulate hypotheses as well as to apply general rules to particular situations. These pupils are usually analytic learners. They prefer to break mathematical ideas to be learnt into small parts and work on each part at a time. Finally, they reconnect the parts to each other in a logical manner into a mental picture. This type of intelligence often implies a great scientific ability. Teachers can empower this intelligence by encouraging the use of computer programming languages, critical-thinking activities, linear outlining, science-fiction scenarios, logic puzzles and logical/sequential presentation of subject matter. The challenge when teaching these students is to avoid boredom.

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Activity Summaries 55.The book of Encrypted Messages Title of the activity

The book of Encrypted Messages

General description

This activity is about encrypting and decrypting messages using different coding methods e. g. the transposition code, Caesar’s code, Vignere code, ATBASH code, Pigpen code, Affine code, etc. A common booklet is the final product.

Students’ age

12-16

Type of intelligence addressed

Logical-mathematical

Educational goals

To develop students thinking skills and problem-solving capacity To nurture curiosity and imagination To use team work and learning by playing as teaching strategies

Expected results

Encrypted and decrypted messages a common booklet worksheets pictures videos of students at work.

Methodology

Explanations are provided to students regarding the use of codes as well as on some coding and decoding methods. Then, they are asked to encode/decode messages. As a variation, they can challenge their classmates/project partners. The codes can be gathered into a common booklet, a presentation, etc.

Required tools

The activity does not require advanced technology and can be adapted to very diverse learning environments. A worksheet can be the only tool, but any additional information media is a plus. A camera is required in order to capture images and videos of students at work. If a book is created, it can be done by using an online editing tool such as bookemon.com, or just in a document.

Evaluation

The evaluation can focus both on the accuracy of the encryption/decryption and the students’ involvement in the task, teamwork, autonomy and other aspects.

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Teachersâ€™ recommendations

The mathematical content should be adapted to match studentsâ€™ age. In any case, the connection between encryption and the mathematical notions should be emphasised.

Link(s)

http://bit.ly/1Ngo3Rc The online book http://www.scribd.com/doc/257406470/Encrypted-messages Worksheet http://bit.ly/1BqTS2U Animated answers http://bit.ly/19Q1ab2 Video

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56. SMARTPHONE APPS Title of the activity

SMARTPHONE APPS

General description

Students learn how to create a smartphone application to be downloaded on their android device, and shared with friends.

Students’ age

15-16-17

Type of intelligence addressed

Logical-mathematical

Educational goals

To develop a program for a smartphone application To work in an android environment To brainstorm the steps of the program in a logical and efficient way To share an application thanks to a QR code

Expected results

3 applications running on the smartphone

Methodology

1st step - The Burger App for Beginners:The students learn about the AppInventor interface through a very easy app. They create a button and show/hide a photo of a burger once the button is clicked. 2nd step - Robot app: The students discover advanced functions of the AppInventor interface to move the NXT Robot in Bluetooth. They create a design for the app, and set up a Bluetooth connection between the robot and the smartphone. 3rd step : The students are involved in the creation of the AIMS App, the app of the project. They record voices and create links to the TwinSpace using buttons. 4th step : The students share their apps with a QR code.

Required tools

Evaluation

An AppInventor website (Google tool) An android smartphone An emulator downloadable in the AppInventor Website A QR code generator The teacher checks each program, its design and algorithm.

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Teachersâ€™ recommendations

Make the students enter the android world step by step and show them the final application working on your smartphone. This will make them get more motivated to create the app, especially if you use the smartphone as a remote control to move the robot. You do not need an android smartphone, since you can download an emulator.

Link(s)

AppInventor website : http://appinventor.mit.edu/explore/ Emulator : http://bit.ly/1eUrT1Y The burger app/the robot app/the AIMS app : In the wiki : http://bit.ly/1eUrT1Y In the TwinSpace http://bit.ly/1QolhzX The QR code generator https://www.the-qrcode-generator.com/

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57. Numbers Webquest Title of the activity

Numbers Webquest

General description

A webquest activity on interesting categories of numbers.

Students’ age

12-15

Type of intelligence addressed

Logical-mathematical

Educational goals

To develop students’ research abilities and autonomy To nurture their curiosity To encourage teamwork and collaboration To teach students to look for relevant information, as well as how to check and cite the sources they have used.

Expected results

Presentations of the requested categories of interesting numbers, such as Fibonacci numbers, amicable numbers, perfect numbers, figurative numbers, etc.

Methodology

The teacher creates a webquest activity using zunal.com or any other similar tool. The webquest task includes the work process, the recommended web sources and the evaluation grid. Students are split into groups of four and asked to find information about specified categories of numbers, using some predetermined websites as well as their own sources. They create a presentation with the most relevant facts and examples and share it with their colleagues. In each group, every student has a different role and task: mathematician, ICT expert, information expert, language and communication expert. They give account of the work sharing at the end of the presentation.

Required tools

Computers with the internet connection. A web tool such as zunal.com or a similar webquest creator, or just a document.

Evaluation

An evaluation grid included in the webquest task focusing on the main criteria to be assessed, such as mathematical accuracy, clarity, grammar, etc.

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Teachers’ recommendations

It is important to select some relevant sources of information for each group prior to the start of the activity, in order not to have students waste a lot of time searching the web. Presenting the outcomes to classmates is very motivational. The activity can be turned into a contest or a “mini-conference” and the students can acquire some basic presentation skills.

Link(s)

http://bit.ly/1QolhzX (Numbers webquest on Thinglink) http://bit.ly/1KpHA3r (Numbers webquest on Zunal)

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58. Logical Games Title of the activity

Logical Games

General description

A Time Line showing three Logical Games being played by students: The Tower of Hanoi, The Camels and the Gallows.

Students’ age

14 -15

Type of intelligence addressed

Logical-Mathematical

Educational goals

To explore possibilities when solving a problem To find a rule that applies to a pattern To find recursive patterns To predict results for next steps based on previous moves information To enhance critical thinking and logical skills To foster team work

Expected results

Students are supposed to find the recursive patterns in the game in order to find the solution. Players are expected to discuss the steps Participants are expected to teach the whole group how to play.

Methodology

Once the objectives of games are explained, students play on the basis of trial and error. They note down the moves they make in order to find the recursive pattern. Players decide on the way to present the games to their partners. Videos are recorded and a Time Line with Google Spreadsheets is created

Required tools

Foam in different colours to make the discs for the Tower of Hanoi and The Camels, cork, pencils, clothes pegs, cardboard, Lego pieces and small figures. A digital camera or a smartphone and a computer.

Evaluation

Observation during the research done by students will provide data concerning their involvement level. Both team work attitudes and collaborative skills will be evaluated as well as logic procedures.

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Teachers’ recommendations

Link(s)

Groups must be small. Games should be chosen by groups according to their likes or previous experience. Materials for the games must be prepared in advance. Each group should have at least one lesson to play in front of the whole class. https://youtu.be/_QcxY8ibC6g (Tower of Hanoi) https://youtu.be/Dn0ReNhUw5Y (Tower of Hanoi) https://youtu.be/1F1NCq9WgGQ (Tower of Hanoi) https://youtu.be/cYQB-cMwZBY (Tower of Hanoi) https://youtu.be/wgBqE-83zT4 (Tower of Hanoi) https://youtu.be/IthEe74X5lY (The camels) https://youtu.be/IOyibZ_eT1E (The camels) https://youtu.be/bbdZrL4Qt34 (The gallows)

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59. Making Magic Squares Title of the activity

Making Magic Squares

General description

Students learn about magic squares, their properties, the way to create and solve them. Students prepare magic squares for their classmates to solve.

Students’ age

12 -16

Type of intelligence addressed

Logical-mathematical

Educational goals

To explore the history of magic squares To learn about various magic squares and their properties To learn the definition of a magic square To improve the ability to sum several numbers mentally To solve mathematical problems and puzzles To improve the use of algebra for solving mathematical tasks.

Expected results

Presentation of the history of magic squares and their most famous examples Improvement of students’ knowledge related to magic squares Instructions for solving magic squares made by students A booklet of magic squares created by students

Methodology

Students explore the history of magic squares and make a presentation using mixbook or other tools. During the lesson, the teacher discusses magic squares using the presentation prepared by students. The students discuss in pairs the properties of a chosen magic square. The students write down each and every property they have noticed. Then, with the use of these properties, each student creates his/her own magic square. Students in groups of three share their magic squares and complete one each.

Required tools

Internet, ICT tools, worksheet.

Evaluation

Students’ involvement in creating and solving the magic square is assessed.

Teachers’ recommendations

The teacher chooses the level of tasks depending on the age of students. If there is more time you can discuss and use algebra to solve magic squares.

Link(s)

http://goo.gl/nCUcvO presentation about Magic Squares . http://goo.gl/aRZwn1 Making a Magic Square Worksheet.

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60. Games Competition

Title of the activity

Games Competition

General description

In this competition a variety of logical games are used (such as Smartgames, Thinkfun games and more). Students solve one particular task from each game and the time needed is written down for each game. All students play all games and the one with the least amount of seconds will be the winner.

Students’ age

12-15

Type of intelligence addressed

Logical-Mathematical

Educational goals

To increase motivation by doing something else than exercises. To let students use their mathematical skills in a totally different way. To show that mathematics can be both fun and informative

Expected results

Fun in the classroom. Students’ appreciation of this particular lesson More students involved in mathematics

Methodology

Required tools

Evaluation

Half the group should be game host and the other half plays the tasks. The game hosts keep track of the time and explain the games to the players. The player then gets a maximum of 5 minutes to play the game. The number of seconds needed is written down on a sheet of paper with all the games on it. The players proceed from one game to the other and at the end you can count the number of seconds from each student. The least amount of seconds will win a player the game! In another lesson you can reverse the roles of players and game hosts. Enough mathematical games to split your group in two halves (e.g. with a group of 30 students you will need 15 of these games) 15 things to keep track of time (mobile phones or stopwatches). A big timer on the whiteboard to keep track of the 5 minutes per game and to know when to go to the next game. It is a really funny thing to do sometimes with your class. Students really like it and are busy all the time.

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Teachers’ recommendations

It will give a lot of noise in the room, so be prepared for that (maybe warn your colleagues next door…). Don’t make the tasks too difficult or too easy: too difficult will kill motivation, too easy will mean that they will be done within a minute and then have 4 minutes left…

Link(s)

Instructions: http://bit.ly/1GynS5X Form to keep track of the amount of time used: http://bit.ly/1e2WRv4

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61. International Kangaroo competition Title of the activity

International Kangaroo competition

General description

Students take part in the international Kangaroo competition for Mathematics joining 55 different countries with a total amount of participants of over 6 million!

Students’ age

12-15

Type of intelligence addressed

Logical-mathematical

Educational goals

To do a kind of mathematics different from the normal lesson To increase motivation by means of competition To challenge every student to get the most out of themselves.

Expected results

Involvement of every student during this test Increased motivation Fun in discussing the exercises afterwards

Methodology

Required tools

It might be wise to practise with old tests which can be found at the website of the organisation. Tell students in advance that it can be a hard test and that the main goal is to have fun and to score as high as possible. Students will need 75 minutes to complete the test.

Answer sheets provided by the organisation. Pencils for filling in the answer sheets. A timer for keeping track of time. One or more teachers to watch the students during the test.

Evaluation

The test is nice, but it may be a bit difficult. What matters is that students face the activity as a game trying to learn as much as possible but keeping relaxed.

Teachers’ recommendations

Make sure to sign up to this competition (middle of March every year).

Link(s)

Example of WizBrain: http://bit.ly/1f2Ddzt

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62. Treasure Hunt

Title of the activity

Treasure Hunt

General description

The main goal of the activity is to make our students visit the city through maths games or riddles. The students have a city map, a calculator and their brain to follow the path and to discover the town by solving “problems”.

Students’ age

12-17

Type of intelligence addressed

Logical (interpersonal)

Educational goals

To enhance teamwork To share ideas and brainstorm about a problem To discover the architecture and the history of the city through maths To create links between maths and life in the city

Expected results

The students solve maths problems through communication. Beyond the difficulty of the exercises, the students talk together and make friends, and the teachers help them in coordinating their ideas.

Methodology

1st step : The students get the Treasure Hunt Instructions and a map with several points of interest. Each team has to follow the path linking these points in a different order. 2nd step : At each point, the Instructions mention a problem to solve linked with the place where the team is. Then the team has to read the problem and solve it. 4th step : At the end, the students check their answers and the team who has solved most problems is the winner!

Required tools

Evaluation

A calculator (or a smartphone) A pencil Treasure Hunt instructions A map Just check the sheet with all answers, and give a mark.

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Teachersâ€™ recommendations

Beyond the math task and the time running out, the teachers must focus on the communication among the students and make the students work on the activity properly. The teachers must create a situation of communication between smarter students and weaker ones.

Link(s)

Treasure hunt in FRANCE : Photos : http://bit.ly/1d5rZbV Documents : http://bit.ly/1G9ImfI Treasure hunt in ITALY : Photos : http://bit.ly/1Bs0Z1S Documents : http://bit.ly/1LjaUKo Treasure hunt in SPAIN : http://bit.ly/1K1JjLN Treasure hunt in ROMANIA : http://bit.ly/1Bo8R4B Treasure hunt in THE NETHERLANDS http://bit.ly/1BwsbfI

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11 - Naturalistic intelligence This type of intelligence has to do with nurturing and relating the data to oneâ€™s natural surroundings, being aware of the living organisms such as the flora and fauna around. Students who have this type of intelligence as a dominant one can easily recognize and classify plants, animals and minerals and show a mastery of natural and artificial taxonomies. These learners are holistic thinkers, they recognize the particular and value the unusual. They enjoy working in nature, exploring living things, learning about plants and natural events. Teachers can empower this type of intelligence by using environment-related and classification activities, encouraging the study of relationships such as patterns and order, comparing and contrasting sets of groups and finding the connections between real life and science issues. In order to get these students involved during the Math class, the teacher can try to connect mathematical concepts to nature. For example, relate mathematical progressions to how plants grow, relate sets and Venn diagrams to types of flowers and how they share certain characteristics and not share the others, have them read about Pi or the golden ratio, point out mathematical influences in nature, such as symmetry or the various geometrical patterns in natural formations or studying living organisms with a certain shape (a pentagon, for example).

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Activity Summaries 63.Statistic on Leaves Title of the activity

Statistic on Leaves

General description

Practical study on the ratio between the length and the width of leaves of several species of plants. The aim is to prove or disprove the idea that the ratio is constant or close to a constant for a given plant species.

Students’ age

12-13

Type of intelligence addressed

Naturalistic

Educational goals

To connect Math to the natural environment To search for patterns and mathematical rules in real life To look for information online as a way of using real resources: dictionaries, the leaflets and labels in the Botanical Garden, etc. To foster autonomy and responsibility by collaborating on a common document, sharing tasks, taking roles in the group.

Expected results

A common, collaborative document with the data on the plant species, statistical data, results and graphs. Getting acquainted with collaborative tools such as Google Drive and real-time collaboration. Better understanding of mathematical rules and patterns in nature and their limitations.

Methodology

Students split in groups and visit the Botanical Garden. They choose a plant type, make observations and gather information from the mathematical point of view, following the worksheet provided to them by the teacher. They look for symmetry, Fibonacci numbers etc. Then, they measure the length and the width of at least 10 different leaves, fill in the table, calculate the l/w ratio and draw a graph to represent its values. Finally, they draw conclusions.

Required tools

Measurement tools, worksheet, pen, computer with the Internet in order to create the common document, cameras.

Evaluation

The evaluation is focused on mathematical accuracy, teamwork, involvement and meeting the deadlines.

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Teachers’ recommendations

The statistical part should match the students’ age. For very young students, it is a good idea that the teacher prepares the document and assigns the space and the role for each group and group member. If this is the students’ first contact with Google Drive, they should be instructed and supervised during their first work session. Back-up copies should be made regularly.

Link(s)

http://bit.ly/1GmB5Nq (Presentation) http://bit.ly/1GsCh0k (Assignment)

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64. Mathematical Modelling in Hydrology Title of the activity

Mathematical Modelling in Hydrology

General description

Determination of mean precipitation on a river basin using different methods (Arithmetic mean and Thiessenpolygon method). Establishing warnings using codes for a given situation.

Students’ age

12 - 16

Type of intelligence addressed

Naturalistic

Educational goals

To apply mathematical knowledge to a practical situation. To make calculations and geometrical constructions in a given practical situation. To understand better how Math is useful for hydrology. To work in teams for solving tasks and presenting the results.

Expected results

Students understand a way of using Math in hydrology. Students apply Math knowledge to make calculations and geometrical constructions for modelling a hydrological situation. Students are able to present correct conclusions about the establishing of a water warning code

Methodology

Students in groups receive a worksheet with instructions and a map with a river basin They study the given situation, make calculations and geometrical constructions according to the instructions, and conclude over a certain warning code that should be emitted. Students present their conclusions to the class.

Required tools

Paper, pencil, mathematical instruments (ruler and square)

Evaluation

Presentation of the results to the class. Discussions in the class about the correctness of the results. Discussions in the class about the mathematical meaning of the methods.

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Teachersâ€™ recommendations

There are two levels of this activity, so it can be adapted to the studentsâ€™ age.

Link(s)

http://bit.ly/1MKuiNQ (Assignment)

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65. VIRTUAL MUSEUM in AUGMENTED REALITY Title of the activity

VIRTUAL MUSEUM in AUGMENTED REALITY

General description

Students create a slide about a chosen topic linking maths and nature. This poster must be animated by a video in augmented reality, and stuck in the classroom to create a virtual museum.

Students’ age

15-16-17

Type of intelligence addressed

Naturalistic

Educational goals

Expected results

Several slides to be stuck on the hall as paintings in a museum

Methodology

1st step : Create your slide

To find links between Maths and nature To study Maths with a concrete background To create a slide with PowerPoint To record a video explaining the slide To create an aura which combines the video and the slide in augmented reality using AurasmaStudio

Choose a topic linking Maths and nature and extract one information about it. For example, in the topic Maths and insects, you can study the family tree of bees. 2nd step : Record your video with your smartphone According to the topic chosen, record a video explaining or detailing what is shown on your slide. The video must be about 1 minute long, to be effective. 3rd step : Be familiar with AurasmaStudio With your Gmail address, sign up for the AurasmaStudio website Watch the tuto 3’11 to be able to overlay your video on your slide in augmented reality (called an aura) 4th step : Create your aura Follow the steps detailed in the tutorial to create your aura from your slide and your video. 5th step : Share your aura Add a hash tag to your aura before publishing (for example #virtualmuseum) to be found by your future followers Share your aura publicly. 6th step : Create a QR code to give the possibility to the visitors to download the aura.

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Required tools

Evaluation

PowerPoint Smartphone or any video camera AurasmaStudio online App Aurasma from the PlayStore A Qr code generator The teacher asks his students to send him/her the QR code for each aura. The teacher downloads all auras on his smartphone by scanning each student’s QR code. The teacher prints all students’ slides and scan them using the Aurasma app on the smartphone. If the video appears, the job is well done. This is how the teacher could check the student’s work.

Teachers’ recommendations

The teacher must show an example before starting the project. As nobody knows what augmented reality is, the teacher should use at least 30 minutes to create an aura, and to test it in front of students. They need to understand first what they have to create and how easy it is.

Link(s)

Aurasma Studio online : http://studio.aurasma.com/ The wiki with the tutorials http://bit.ly/1N1PoYP In the TwinSpace http://bit.ly/1GYnmOV The QR code generator https://www.unitag.io/fr/qrcode

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66. The Universe in Numbers Title of the activity

The Universe in Numbers

General description

Students search for very small and really large numbers in nature. They collect data expressed in numbers and write them down in Scientific Notation.

Students’ age

12-14

Type of intelligence addressed

Naturalistic

Educational goals

to learn about large and small numbers in the universe to write and interpret large and small numbers using scientific notation to develop an understanding of large and small numbers and use exponential and scientific notation appropriately

Expected results

a presentation concerning large and small numbers in nature a set of varied examples of such numbers the ability to write and interpret large and small numbers using scientific notation.

Methodology

1. Students search websites for data about diverse objects in the universe and their characteristics presented in numbers e.g. their size, weight or temperature. Students can work in 2 teams. Team 1 can deal with finding the information about large objects such as the Sun or the Moon, whereas team 2 can look for information about small objects such as electrons or protons, etc. The measurements they encounter can be presented in many different ways e. g. in decimal or exponential notation. 2. The teacher discusses with the students the reason for using a particular notation to characterise a given object. The presentation of two measurements which differ from one another in size, for ex. the mass of an electron and the distance between the Earth and the Sun, can be a useful hint for the students. 3. The teacher explains to students how to write 1000000 in a shorter form and asks them to think up the definition for an exponential notation. Students practise writing numbers using the exponential notation until the end of the lesson. 4. In the next Math lesson, students prepare a set of numbers to be written down in the exponential notation for their classmates to work on. 130

Required tools

A worksheet, a pen, a computer with the internet access.

Evaluation

By the end of the lesson, students working in pairs write the numbers, given by the teacher, in the exponential notation.

Teachersâ€™ recommendations

If you want students to prepare a set of exercises you need to devote a couple of lessons to this subject.

Link(s)

http://goo.gl/vuTuOO

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67. Statistical research into genetics Title of the activity

Statistical research into genetics

General description

Students develop a questionnaire. This questionnaire is about genetic properties of people. After getting the results back from all the partners the students make a statistical research.

Students’ age

12-15

Type of intelligence addressed

Naturalistic

Educational goals

Expected results

Making a Google form Using Excel to represent the data into tables and/or graphs

Methodology

First the students learn to use Google forms The different questions of the students are combined into one final form The link for the form is sent to all the partners The data are collected The data are represented into tables and graphs

Required tools

A Google account to make the Google forms Excel to represent the data Email to send the link to the partners

To use Excel To use Google forms To make a statistical research To represent the statistical data To get an idea about the world of genetics

Evaluation

The students really learn something useful, such as Google forms, to be used in different school subjects. Excel may be more difficult to use, especially making graphs out of the data. The teacher’s help is required here.

Teachers’ recommendations

Make sure to start early with this subject. The students need more time to explore Excel. Start with easy tables and let the students make graphs out of these tables. This way the assignment is done quicker.

Link(s)

http://bit.ly/1KYsdzq

(Results)

132

68. Green Maths Title of the activity

Green Maths

General description

A website created by students after exploring the three most important gardens in town in order to discover Mathematical concepts: patterns, shapes, symmetry, etc.

Students’ age

14-15

Type of intelligence addressed

Naturalistic

Educational goals

Expected results

To improve students’ observation skills. To connect Mathematics with history, geography and the environment. To get aware of diversity in nature. To encourage ICT use in a sensible way. To promote team work. Students are expected to go to three gardens in town to gather visual material concerning Mathematics. They are encouraged to discuss ICT tools to be used in order to present their findings. They are supposed to include historical, geographical, scientific and Mathematical information in the website created by themselves.

Methodology

Once the students have been split into groups, they choose a garden in town. Options to present their work are shown in the classroom, Weebly is chosen to publish a website. Instructions on how to use it are given during a workshop. Groups agree on a day to visit a garden and take photos to show Mathematical contents following a worksheet. Research is done in a public library and on the Internet to get historical information. ICT tools to be included in the website are discussed by groups. Mixbook, slideshows and videos are chosen. Everything is uploaded and the website is published.

Required tools

Smartphones or digital cameras. Computers. Environment magazines and books. An email account to start the website in Weebly.

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Evaluation

Quality of the information concerning Mathematical knowledge and other relevant elements such as history or mapping in the website, diversity of tools, team work organization to show real collaboration, correct use of English will be taken into account to evaluate the students' work.

Teachersâ€™ recommendations

Clear information about the main aspects concerning the Mathematical contents to be included has to be given in advance through videos or pictures. Students should have a worksheet to be used as a guide. A variety of ICT tools must be shown so that students can get the ones they feel more comfortable with.

Link(s)

http://bit.ly/1KkYjVD http://bit.ly/1dqU4uG http://bit.ly/1G7pfoL http://bit.ly/1KkYKiK http://bit.ly/1BUAdKs

(Worksheet) (Short video) (Short video) (Short video) (Weebly webpage)

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69. About bones and Maths Title of the activity

About bones and Maths

General description

Students make a statistical survey on the length of specific bones in the human body in order to calculate a person’s height. The activity links Maths to anatomy, challenging students’ logical and manual skills at the same time.

Students’ age

15

Type of intelligence addressed

Naturalistic

Educational goals

To link Maths to practical experience and to Anatomy To collect data To organize and analyze the collected data To use formulas in order to “predict” theoretical results/data To compare measurements with the obtained results through the use of the formulas To evaluate the reliability of formulas/measurements To learn the meaning of some statistical indexes To use graphs and tables To use a spreadsheet (Excel)

Expected results

Students will make a small statistical survey on the human body. They will collect and organize the data and analyse them through statistical indexes and graphs. They will use formulas to predict real measurements and compare their results to real data.

Methodology

Step 1 – Anatomy focus: Students must look for the Italian and English name of specific bones of the human body and must be able to find them on an anatomy map. Step 2 – Statistics focus: The teacher starts the survey, giving the students the formulas used by criminologists and archaeologists to estimate a person’s height from his bones’ length. Step 3 – Measurements: The students collect the data, measuring the length of tibias, femurs, radius and humerus of a group of classmates. They record their height as well. Step 4 – Data processing: The students will use manual tools at first and a spreadsheet later. The data are reported in tables and visualized with graphs. The data are analyzed both through the frequency graphs and the calculated statistical indexes. The students will estimate the students’ heights with the formulas 135

and compare their results with the real measurements. They debate the reliability of the formulas and their prediction potential. The work is presented with the Web 2.0 tools. Required tools

Flexible tape, squared/grid paper, spreadsheet (Excel), web 2.0 tools for the activity presentation.

Evaluation

Evaluation must consider the following points: Involvement and motivation of students Accuracy of the worksheets with measures, calculations and conclusions produced Raised awareness of “naturalistic” Maths learning (this could be done with a final brain storming or debate).

Teachers’ recommendations

It is important that students make accurate measurements. It is recommended to make students draw graphs manually before using a digital spreadsheet.

Link(s)

http://bit.ly/1fm1Ze3

(Blendspace website)

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70. Map Challenge

Title of the activity

Map Challenge

General description

This activity aims to connect students’ immediate reality, their daily environment, to mathematical ideas. Students are given a map of their town, where some interest points are highlighted, and asked to find information regarding these places, as well as solve problems about the same places of interest.

Students’ age

12-16

Type of intelligence addressed

Naturalistic

Educational goals

To connect Math to reality and show that mathematical patterns and rules, geometrical shapes, equations etc. are not confined to the classroom, but are a part of our natural environment and can be recognised and used everywhere To allow students to experience Math outdoors and learn beyond the walls of the school To encourage the use of mobile technologies for learning To empower skills such as autonomy, curiosity, research, classification, abstraction etc.

Expected results

Information about the chosen places of interest, that will allow students to better understand and know their own natural environment A better perception about uses of Math in real life Accurate data and solutions to the given problems/questions

Methodology

The task can be done collaboratively or individually. The map for the activity can be created with Google Maps or another similar tool. The chosen points are the vertices of a polygon (a triangle in our case) and also have some significance for the natural or cultural environment. After some research regarding this significance, students have to solve a problem for each point. Some examples: calculating percentages or ratios, a perimeter (length of an itinerary), the surface of a park (or the green area for every inhabitant of the town), the volume of a lake/fountain, the height of a tree etc. The methods should be as diverse and as new as possible, some of them should need to be “rediscovered” by the students.

Required tools

Worksheet, computer/tablet/phone with an internet connection 137

Evaluation

The accuracy of the results is important, but teachers should also pay attention to the research part, to students’ capacity of abstraction, to their collaborative skills.

Teachers’ recommendations

The best way to do this activity would be really taking students outdoors, to see and experience the places described by the task. It is also suitable as a challenge for partners from another town/country, as a way of mutually getting familiar with the place partners live in. The problems need to be adjusted to students’ age. For older students, they can choose their own points of interest, create the problems and challenge their classmates/partners.

Link(s)

http://bit.ly/1ElI9tM

(Assignment)

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ISBN 978-973-0-19377-0 Bucharest 2015 140

Teaching guide AIMS

Published on Jun 24, 2015

A pedagogical guide created in order to help teachers use the Multiple Intelligence Theory in Math classes, for creating student-centered ac...

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