Geometry for the First Cycle of Primary Education

MATHS & ART WORKSHOP - 1

Presentation for teachers

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Description The task of linking Art and

Mathematics may seem at first glance somewhat artificial and forced, but it is exactly the opposite. Maths is essential, in an obvious way in Architecture, but also in Sculpture and often in Painting and other Arts, which sometimes are known as "minor arts", but as we must not forget the audience that the workshops are directed to, I am going to focus on the part of mathematics whose relationship to art and artistic expression is more tangible and therefore easier to visualize and understand by children: that is Geometry.

This first workshop, which I have entitled: MATHS & ART WORKSHOP, is firstly aimed at students in the first cycle of primary education, working the foundations of Geometry, focusing on flat shapes: Point, Line and Plane, and within the latter, in Triangle, Circle, Rectangle and Square.

The way I am going to use to present these contents, can be a little diffent to that in which they use to appear in textbooks, but always encouraging children’s creativity. I want to highlight that the objective is not to replace that teaching, more academic, but to complement it. On this way, the workshops will be appropriate to reinforce some knowledges, to help to understand them if there are any doubt, and to reach a more motivating learning.

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The different chapters and sessions of this workshop, will be always based in the same structure:Presentation.Geometric objects in our environment.Geometric objects in Art.Resources and activities of comprehension and

representation of geometric objects.

In the original blog: http://artedesermaestro.blogspot.com.es/ there are some videos included with certain advices, that can help teachers and pupils to improve the process of drawing of some geometric elements.

Workshop Structure

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1. Linguistic communication competence: With the use of language, for listening and understanding contents to communicate between themshelves and the teacher along the development of the activities.

2. Mathematics competence: It is the most obvious of all of them in this workshop, where Geometry, as an important part of Mathematics, takes the main character.

3. Competence in Knowledge and interaction with the physical world: With the comprehension of events, and the prediction of consequences. It is also referred to the progressive hability to investigate and scientific inquiry: Identifying and posing problems, making observations, asking questions and contrast hypothesis.

4. Dealing with information and digital competence: With the use of TICs to present the workshop and to develop several activities.

Basic Competences - 1

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5. Social and civic competence: With respect among all the people, along the workshop and the group activities.

6. Cultural and artistic competence: Being conscious of the artistic heritage in their environment, and about significant masterpieces, as a reference and inspiration for their own creations.

7. Learning to learn competence: With the purpose of learning, enjoying in an effective way, with curiosity and motivation to learn, responsability and personal compromise, accepting mistakes and learning from and with the others.

8. Sense of initiative and entrepreneurship: Which is needed to approach several individual activities, and taking part activily in group activities.

Basic Competences - 2

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To understand and express basic concepts about Geometry.

To know and value the presence of Geometry in Art and our enviroment.

To recognize the creative possibilities of Geometry, as well as its contribution to all kind of disciplines.

To know and understand the next basic elements of Geometry: Point and straight line.Triangle.Rectangle and square.Circle and circumference.

General Objectives

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To identify and understand the presence of the mentioned basic geometric elements in our inmediate environment.

To identify and value the basic geometric shapes in all kind of masterpieces.

To improve the skill in drawing or tracing the basic geometric shapes mentioned before.

To make different activities from basic geometric shapes, using traditional materials and computer programmes.

Specific Objectives

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Point and straight line.Point and straight line in our environment.Point and straight line in Art.To improve the drawing of straight lines. Resources and activities of comprehension and

representation of points and straight lines.Triangle.

Triangle in our environment.Triangle in Art.Resources and activities of comprehension and

representation of triangles.

Contents: Knowledges - 1

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Square and rectangle.Square and rectangle in our environment.Square and rectangle in Art.Resources and activities of comprehension and

representation of squares and rectangles.Circle and circumference.

Circle and circumference in our environment.Circle and circumference in Art.Drawing circles without compass and freehand

drawing.Resources and activities of comprehension and

representation of circles and circumferences.

Contents: Knowledges - 2

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We observe the basic geometric elements (point, straight line, triangle, rectangle, square, circle and circumference) in our environment.

We identify the basic geometric elements in different masterpieces.

We draw straight lines, triangles, rectangles and squares without tools and with computer programmes.

We draw circumferences with computer programmes, by hand and with auxiliary elements different to the compass (we learn to built and to use an auxiliary tool for this purpose).

We develop our own artistic works, by hand and by computer, from basic geometric shapes.

We perform activities, traditional and computer games, based on basic geometric shapes.

Contents: Procedures

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To show interest in the presence of basic geometric elements, in our environment and taking part of masterpieces.

To respect and value our artistic and cultural heritage.

Interest in the performing of our own artistic compositions, based on the use of basic geometric shapes, both by hand and by computer.

To respect and value the work and creations of our classmates.

Contents: Attitudes

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Each of these four parts can be developing in a work session along an hour and a half or two hours. In all the cases, we will expend among five and ten minutes to show and to talk the whole group the examples of geometric shapes in our environment; and we will do the same about Art. So, it will be about eighty minutes to develop the activities, preferibly in the order that I state here, but with total flexibility, depending on the circumstances of the class.

Total estimated time for the workshop : 8 hours (4 work sessions).

Timing

Educational levels This workshop is initially thought for the First Cycle of

Primary Education, over all for pupils in the 2º year; but with little corrections, especially about the activities, it could be equally appropriate for the rest of the cycles of this stage. 13

Evaluation will be based on the continuous and systematic observation of pupils’ work, and the interest shown along the exposition of contents and in the performing of the different activities.

In this sense, many activities will leave a physical record as an assessable material, such as drawings and artistic compositions on paper, or computer files with their digital works.

Nevertheless, we mustn’t forget that the educational possibilities of this workshop, are mainly based on pupils perceiving it as a funny and less monotonous way for acquiring knowledges.

Evaluation

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Geometry for the First Cycle of Primary Education

MATHS & ART WORKSHOP - 1

Workshop developed to be presented in the classroom. The introductory texts can be appropriate as a help or

reference for the teacher, adapting the language to pupils’ level if necessary. You can see more information:

http://artedesermaestro.blogspot.com.es/

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1. Point and straight lineThe original base of Geometry is the point; in anisolated

way it is nothing, it isn’t a physical object, but it is necessary to form the rest of the basic geometric elements that we know, everything around us. What is a straight line, but a sequence of points all together in the same direction?  and is it not the point a way of representing the intersection between two straight lines at the same time? so, the straight lines and points are two of the main elements of Geometry, as well as the plane, and these two first elements are precisely those we are going to talk about in this chapter.

TIMING: 1 session=2 hours.You can see more complete information about this chapter

in: http://artedesermaestro.blogspot.com.es/2012/04/geometria-para-primer-ciclo-de.html

1.1. Point and straight line in our environment- 1

We can see the sun and the full moon as a point; and the skyline, as a straight line between the sky and the sea. 17

1.1. Point and straight line in our environment- 2

The spectators at a sports stadium are also perceived as small points. Sometimes, wonderful colourful mosaics are formed. 18

1.1. Point and straight line in our environment- 3

Many times, the parallel train tracks continue in a straight line for several kilometers until they disappear in the distance. 19

1.1. Point and straight line in our environment- 4

Who doesn’t know a picture like this? Can you see the points and the straight line? What other points we can distinguish on the picture? 20

1.1. Point and straight line in our environment- 5

We only need to open one of our notebooks to see structures based on points and straight lines. The holes of the notebooks can be seen as a

sequence of aligned points. 21

1.2. Point and straight line in Art- 1

Wassily Kandinsky – “Composition VIII” 22

1.2. Point and straight line in Art- 2

Joan Miró – “Maternity” 23

1.2. Point and straight line in Art- 3

Piet Mondrian – “Composition with red, yellow and blue” 24

1.2. Point and straight line in Art- 4

Roman mosaic 25

1.2. Point and straight line in Art- 5

Georges Pierre Seurat – “The circus” 26

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1.3. Advices to improve the drawing of straight lines

In this video, recorded in Spanish, I show several advices, which can be useful to transmit to your pupils how to improve the accuracy of the drawing by hand. The video is directed to teachers, in order to select what they consider appropriate, and adapt it to the resources they have in the classroom (traditional blackboard, interactive whiteboard, etc.)

1.4. Resources and activities of comprehension and representation of points and straight lines - 1

1. MOSAICS GAME. There are a wide range of this kind of games in the specialized shops, and it is a wonderful material for our pupils. 28

1.4. Resources and activities of comprehension and representation of points and straight lines - 2

2. CONNECT POINTS WITH STRAIGHT LINES. From a template prepared on a standard sheet, this is a good exercise to develop the drawing skill. 29

1.4. Resources and activities of comprehension and representation of points and straight lines - 3

3. PLAYING TO “NOUGHTS AND CROSSES”. It is a classical game which rarerly need to be explained to start the game, and we can play with it even

with simple stones. 30

1.4. Resources and activities of comprehension and representation of points and straight lines - 4

4. WORKING WITH POINTS AND STRAIGHT LINES USING COMPUTER PROGRAMMES. The software “Paint” by Windows, is a good choice to

start. 31

1.4. Resources and activities of comprehension and representation of points and straight lines - 5

5. PLAYING TO “FOUR IN A ROW”. On-line game: http://www.pequejuegos.com/juego-cuatro-en-raya.html 32

1.4. Resources and activities of comprehension and representation of points and straight lines - 6

6. PAINTING LIKE SEURAT. With the software “Tux Paint”: Downloadable for free at : http://www.tuxpaint.org 33

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2. TriangleAfter finishing the first part of the workshop, about

points and straight lines, we are going to see now another basic element for Geometry, such as the triangle. As the rest of geometric shapes we will see along this workshop, triangles are formed by points (vertices) and straight lines (rectangle’s sides), and if we link properly these concepts from the very beginning, children will understand perfectly why we are following this order.

TIMING: 1 session=2 hours.You can see more complete information about this

chapter in: http://artedesermaestro.blogspot.com.es/2012/05/geometria-para-primer-ciclo-de.html

2.1. Triangle in our environment- 1

Children can easily recognice the triangle structure on the base or composition of many infant swings. 35

2.1. Triangle in our environment- 2

The square and the bevel that pupils have or are going to have soon, into their school kits, are another clear example of the presence of triangles

without leaving their school bag. 36

2.1. Triangle in our environment- 3

It is very important that children get used to the meaning of traffic signals, as soon as possible. Triangle is one of the most common shapes of them. 37

2.1. Triangle in our environment- 4

When we were children, most of us have drawn some mountains with a shape based in the triangle. 38

2.1. Triangle in our environment- 5

Although it is not an exact triangle, a slice of pizza or a wedge, could be perfectly in this section. 39

2.2. Triangle in Art - 1

Wassily Kandinsky – “Black and Violet” 40

2.2. Triangle in Art - 2

In this case, as an architectural reference, we can mention how the pyramids, which children usually like, are formed by four triangular faces. 41

2.2. Triangle in Art - 3

Paul Klee– “The Castle and the Sun” 42

2.2. Triangle in Art - 4

We have another architectural reference again, with the huge triangular tympanum of the Agripas’ Pantheon in Rome. 43

2.2. Triangle in Art - 5

We can finish showing a simple musical instrument such as the triangle to highlight the importance of Music as Art for children. 44

2.3. Resources and activities of comprehension and representation of triangles - 1

1. RUBIK’S SNAKE. We can let pupils play some minutes with it, and highlight the presence of the triangle as the basic form of this game. 45

2.3. Resources and activities of comprehension and representation of triangles - 2

2. BREAKING GEOMETRIC SHAPES INTO TRIANGLES. From a sheet of paper with several shapes such as those of this example. They can also do this

exercise by computer, with the software “Paint” or “Tux Paint”. 46

2.3. Resources and activities of comprehension and representation of triangles - 3

3. FIND THE TRIANGLES. From a decolored and soft image of Kandinsky’s painting we have seen previously. It is an appropriate activity to be done on the digital whiteboard over a projected image. You can use the next screen. 47

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2.3. Resources and activities of comprehension and representation of triangles - 4

4. PAINTING LIKE KANDINSKY. From the painting of Kandinsky, orKlee , or both of them, we are going to propose pupils do their own masterpieces.49

2.3. Resources and activities of comprehension and representation of triangles - 5

5. A PAPER AIRPLANE. There are many ways to do it, and in this video there is a good example where triangles are the main shapes used. 50

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3. Rectangles and SquaresThis new chapter about rectangles and

squares, could be developed before or after that about triangles, without affecting the essence of this workshop. As well as points and lines as constitutive elements of all geometric shapes, there is no doubt about that the most familiar ways for children, or at least in a more prominent way, are rectangles and squares, as we can see in the next examples, into the school context.

TIMING: 1 session=2 hours.You can see more complete information about this

chapter in: http://artedesermaestro.blogspot.com.es/2012/05/geometria-para-primer-ciclo-de-primaria.html

3.1. Squares and rectangles in our environment - 1

Most pages of our textbooks are full of squares and rectangles, which help to form and distribute the contents. 52

3.1. Squares and rectangles in our environment - 2

And, if we look around us, without leaving the classroom, the predominant ways are the square and the rectangle again. Can you see them? How many

others you can say? 53

3.2. Squares and rectangles in Art - 1

Piet Mondrian – “Composition in Red, Yellow, Black, Grey and Blue”54

3.2. Squares and rectangles in Art - 2

Gerrit Rietveld. “Red and Blue Chair" 55

3.2. Squares and rectangles in Art - 3

Gerrit Rietveld. "Rietveld Schröder’s House“. It was designed and built in 1942, and it still seems a completely modern building. 56

3.2. Squares and rectangles in Art - 4

Wassily Kandinsky. "Thirteen Rectangles". 57

3.2. Squares and rectangles in Art - 5

Hans Hoffman. “Cathedral”. 58

3.3. Resources and activities of comprehension and representation of rectangles and squares - 1

1. PLAYING DOMINOES. Pwe can start with a classic game, very appropriate to help students to develop the mathematical thinking. 59

3.3. Resources and activities of comprehension and representation of rectangles and squares - 2

2. DRAWING WITH “PAINT”. Pupils have to draw with this popular computer programme, trying to use only rectangles and squares. 60

3.3. Resources and activities of comprehension and representation of rectangles and squares - 3

3. PAINTING WITH TEMPERA. From the paintings we have seen and talked about in the previous section, pupils will have to do their own

creative works. 61

3.3. Resources and activities of comprehension and representation of rectangles and squares - 4

4. PLAYING TO TETRIX. This is another on-line game, with a lot of versions on the internet. Here it is one of them:

http://www.pequejuegos.com/juego-tetrix.html 62

3.3. Resources and activities of comprehension and representation of rectangles and squares - 5

5. COLLAGE WITH COLORED POST-TIPS. This is an appropriate activity to be done in small groups, among 2 to 4 pupils, using a usual 50x70 carboard

as base. 63

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4. Circles and circumferencesWith this fourth chapter, about circles and

circumferences, the whole contents and activities of this workshop will be completed. Talking about circles and circumferences, we find basic geometric shapes again, which are usually present in pupils’ environment, and that undoubtedly they know since their first school years. We are going to see this with some simple examples.

TIMING: 1 session=2 hours.You can see more complete information about this

chapter in: http://artedesermaestro.blogspot.com.es/2012/05/geometria-para-primer-ciclo-de-primaria_18.html

4.1. Circles and circumferences in our environment - 1

Since we get up at first hour in the morning, we can see a lot of circular shapes, even on the table at breakfast. 65

4.1. Circles and circumferences in our environment - 2

Over all, in these first moments of the day, it is very important that we know what the time is to be always punctual. 66

4.1. Circles and circumferences in our environment - 3

When we go out to the street, looking directly at the sun is not a good thing, in fact, it is very dangerous, but we know the shape it has, as round as the

full moon. 67

4.1. Circles and circumferences in our environment - 4

Going to the school by bycicle woud be fun. In any case, it is sure that we can see round wheels in the cars along the road. 68

4.1. Circles and circumferences in our environment - 5

And, if we travel by car, when we arrive to the school, we have to be concentrated in order that our driver is alert and doesn’t park where a sign

like this is. 69

4.2. Circles and circumferences in Art - 1

Dome of the “Ermita de San Antonio de la Florida” in Madrid, painted by Francisco de Goya. 70

4.2. Circles and circumferences in Art - 2

Detail of the front of the Cathedral of León, where you can see the rose window and other circular shapes. 71

4.2. Circles and circumferences in Art - 3

View from inside the Cathedral of Leon, where you can see the rose window of the previous image, illuminated by external light. 72

4.2. Circles and circumferences in Art - 4

Wassily Kandinsky. “Some Circles”. 73

4.2. Circles and circumferences in Art - 5

Robert Delaunay. "Rythm, Joie de Vivre". 74

4.2. Circles and circumferences in Art - 6

Fra Angélico. “The Annunciation”. 75

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4.3. Drawing circles without compass and freehand drawing.

In this video I show several advices, in order to improve the drawing of circles by hand and with auxiliary tools different from compass. At the same time, I show step by step how to built and use a simple tool for doing circumferences with 10 pre-defined radios. As well as the other video we seen previously, this one is directed to teachers, in order that they transmit these advices to their pupils on the way they feel appropriate.

4.4. Resources and activities of comprehension and representation of circles and circumferences - 1

1. NUMERIC MEMORY GAME WITH BOTTLE CAPS. This is a simple game which is useful at the same time to encourage recycling among our pupils. Material: 20 caps for the same game,

with 10 pairs of numbers from 1 to 10, which will be hidden inside the caps. 77

4.4. Resources and activities of comprehension and representation of circles and circumferences - 2

2. PAINTING A WHEEL-ROSETTE. Using an old CD or DVD as a template to draw the circles, paint freely a wheel-rosette. This is only an example. 78

4.4. Resources and activities of comprehension and representation of circles and circumferences - 3

3. ON-LINE GAME: “SHOOTS BALLS”. This is one of the many games we can find on the internet, based on circles and spherical shapes. http

://www.pequejuegos.com/juego-dispara-bolas.html 79

4.4. Resources and activities of comprehension and representation of circles and circumferences - 4

4. COMPOSITION WITH CIRCLES BY COMPUTER. Using the programmes“Paint” or “Tux Paint”, we are going to emulate the great

masters, making our own digital creations based on circles. 80

4.4. Resources and activities of comprehension and representation of circles and circumferences - 5

5. CULTURAL TOUR: THE CIRCLE AND THE CIRCUMFERENCE IN ART. If we are in Madrid, I propose to end with a trip to the “Ermita de San Antonio de la Florida”, and look ing from below the ceiling painted by Goya. In cities with Gothic Cathedrals, visit them to observe their rosettes, would certainly be one of the

most interesting options. 81

Geometry for the First Cycle of Primary Education

MATHS & ART WORKSHOP - 1

In my blog “El Arte de ser Maestro”, you can see an Spanish version of this workshop. http

://artedesermaestro.blogspot.com.es/

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