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1. Mathematics Project Overview Learning Areas Levels 16-17-18 year olds Objectives To apply and master different mathematical concepts in relation to art Drawings Description The Art of Drawing requires creative and intrinsic thinking processes. Once a drawing is dissected into its various constituent parts; i.e. segments comprising straight lines, parabolas, circles, areas, etc., parts of a whole can be observed. Henceforth, springs a visualising, an approach into the world of Mathematics. In other words, a clear relationship between the world of mathematics and the world of art can be established. Drawings are represented according to their function and regions of inequalities within a restricted range or domain. Therefore, it is possible to create an image, a creative drawing, using various mathematical functions. This pragmatic activity presupposes a high intellectual challenge. This cognitive reflection allows the opportunity to experience the relationship between two fields that have been considered poles apart: Mathematics and Art. This multi-disciplinary project provides an opportunity for students to express themselves visually using mathematical skills. Being able to make use of various mathematical functions, applying them, translating them vertically and horizontally, positioning these lines and curves at a desired, precise location, enables students, once the individual graphs are connected together, to create a work of art. The use of computers enables us to observe the fusion of these two distinct worlds. Authors Robert Levesque, DSS, B.Sc., B.Ed., M.Ed. Title: The Art of Mathematics

2. Software Keywords GrapheEasy; MS Office; Desire2Learn (content platform for distance learning); Interwise (communication platform for distance learning) Drawings, graphs, grapheEasy, functions, horizontal translation, vertical translation, Art, constructive learning, mathematics, project, mathematical parameter Title: The Art of Mathematics Teacher Planning and Management The Cité des Jeunes A.M. Sormany is a Comprehensive Francophone High School, opened in Edmundston, New Brunswick, Canada in 1972.  Student enrolment originally reached around 2000 students a year from grades 10 to 12 with 600/700 students graduating each year.  Presently, there are around 1350 students from grades 9 to 12, including students with special needs. With a wide, diversified curriculum, the school is considered one of the best in the province. Equipped with a modern communication system, it allows students and teaching staff to link up with other learning institutions. The school's aims and objectives are to provide students with a learning environment, where, within an atmosphere of mutual respect between students and staff, they are able to realize their full potential. The school's fundamental values are based on self-reliance, respect and social responsibility. The school strives to guide students through their intellectual, creative and social pursuits so as to enable them to play their full, positive role in an ever-changing society. The school also encourages students to have a sense of pride in their francophone identity. 

3. Title: The Art of Mathematics Teacher Planning and Management The project I am presenting to you now, has been used for the past three years, both in the classroom environment and via the internet, throughout the province. I have found this project encourages and motivates students. It allows them to explore numerous possibilities offered by this approach and to develop their artistic abilities. It also facilitates them to gain a better understanding of various mathematical concepts. By using this approach, I came to realize that students were able to master mathematical concepts related to the project. They showed little or no difficulties during the review period. The effectiveness of this approach is also evident from the results of the final examination. All the schools using this programme via the internet also have access to a software called “GrapheEasy”. This makes the use of the programme a lot easier.

4. Title: The Art of Mathematics Teacher Planning and Management This project meets the requirements of the Department of Education for the Province of New-Brunswick as to the mathematical content of its curriculum. Students can apply their knowledge pertaining to the graphing of functions such as Linear, Constant, Quadratic, Square Root, Absolute value, Exponential, Logarithmic and Trigonometric (sinus, cosine). More importantly, students learn how to modify these accordingly to certain parameters. This project allows students not only to learn mathematical concepts, but to apply and use them and master them while building a challenging drawing. Applying mathematical concepts to a practical, visual project is a very challenging and gratifying experience. This is not a new concept and has existed for a long time. Very often, teachers would decline this initiative because it is time consuming and very tedious to correct. With the arrival of computers and the availability of easy to use software, we can now make drawings with a great deal of precision. Also, this approach necessitates few corrections since an error can be readily observed with the software.

5. Title: The Art of Mathematics Teacher Planning and Management Programe of studies: This project includes many concepts proposed by the Provincial Department of Education for students of the grades 11 and 12 levels. https://portail.nbed.nb.ca/Topics/Educateurs/Ressources%20pedagogiques%20et%20pro/Mathematiques/Pages/default.aspx Math 30311 (Grade 11) Specific learning skills: Able to solve problems and analyse situations using quadratic functions and their graphs i.e. problems of maximum and minimum values as applied to everyday situations. Math 30321 (Grade 11) Graphical representation of absolute values, square roots, rational expressions. Math 30411 (Grade 12) Graphical representation of trigonometric functions such as sinus and cosine and the ability to use them in problem solving situations concerning the amplitude, the period and phase shift. Math 30421 (Grade 12)  To recognize algebraically and graphically the characteristics of functions: domain, range, use of parameters, the use of symmetry in relations the “x” and “y” axes. Being able to recognize the characteristics and to transform specific functions such as linear, quadratic, cubic, rational, square root, absolute values, exponential, logarithmic and trigonometric.

6. Title: The Art of Mathematics Teacher Planning and Management Canadian Planning and Management: The mathematical concepts previously mentioned can be found in all Canadian provincial curricula. The concepts can be observed at different grade levels such as grade 10, 11 or 12. Details can be observed from the following sites: Québec:, http://www.mels.gouv.qc.ca/DGFJ/dp/programmes_etudes/secondaire/pdf/mat536.pdf Manitoba: http://www.edu.gov.mb.ca/k12/cur/parents/senior/grade12.html#math Nova Scotia: https://sapps.ednet.ns.ca/Cart/items.php?CA=12&UID=20071001163058204.82.241.153 Alberta: http://www.education.gov.ab.ca/french/Math/10-12/Program/Applique/appl.asp Prince Edward Island: http://www.gov.pe.ca/photos/original/ed_sps_0708.pdf Newfoudland and Labrador: http://www.ed.gov.nl.ca/edu/sp/sh/math/math3206.pdf

7. Title: The Art of Mathematics Teacher Planning and Management The Francophone section of the Department of Education purchased the license of the software ” GrapheEasy” allowing its installation in each of the French schools throughout New-Brunswick. When teaching on line, all of my students have access to computers furnished by the schools. This allows them to work independently on their project using the software GrapheEasy. When teaching in a classroom situation, students are invited to produce their choice of a drawing using the two computers available in the classroom. Also available to them are computer laboratories where some 30 computers are installed. They have access to these during lunch period, after school or during their laboratory courses. In order to get started on their project, brief examples of several graphs are available through tutorials, (see annexe A). The purchase of the license also allows the teachers to download the software at their home for their personal use (software available in English and French). This allows them to familiarize themselves and to master the content of the software. This is a semestrial project offered over a four month period. This approach allows students to work at their own rate on these mathematical concepts and apply the learned knowledge to produce a practical, visual project. The teacher is available to answer questions either on the software or the mathematical concepts. The teacher acts as a guide throughout these learning experiences..

8. Title: The Art of Mathematics Teacher Planning and Management As explained, students can work individually at their computer but can also help each other. For those who have a computer and the internet at home, they can download a temporary version of the software in order to experiment with various functions. However, this version will not allow them to save their work. In the following pages, you will find drawings made by my students. With each drawing, you will find the name of the students and also the numbers of equations required in order to construct their drawings. It must not be forgotten that the drawings are made of segments of straight lines, curves and areas completely defined by the students. All equations and functions are represented by a limited domain and range that students have to define in order to arrive at a desired result. Please notice the small details.

9. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Student: Julie Leblanc. Approximately 135 mathematical equations Logarithmic function circle linear sinus function quadratic function quadratic function logarithmic function sinus function

10. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Student: Megan Approximately 135 mathematical equations

11. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Student: Valérie Lang Approximately 225 mathematical equations

12. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Student: Tristan Martin Approximately 210 mathematical equations

13. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Student: Sophie Chiasson Approximately 135 mathematical equations

14. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Student: Stacey Morris Approximately 225 mathematical equations

15. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Student: Billy Nowlan Approximately 105 mathematical equations

16. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Student: François Laplante Approximately 270 mathematical equations

17. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Student : Chantal Richard Approximately 90 mathematical equations

18. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Student: Stéphanie Turner. Approximately 120 mathematical equations

19. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Student: Gisèle Doiron. Approximately 165 mathematical equations

20. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Student: Clément Savoier. Approximately 75 mathematical equations

21. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Student: Stéphanie Caissie. Approximately 255 mathematical equations

22. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Student: Clément Savoie. Approximately 75 mathematical equations

23. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Student: Joline Poirier. Approximately 120 mathematical equations

24. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection Most of the time, when the project is completed, the students’ own expectations are exceeded. Teachers and students are fascinated by the ideas, creativity and the complexity of the chosen equations which make up the drawing. At the beginning, students will often ask the number of equations required in order to meet the teacher’s objectives. I have never required a minimum number of equations. Through self-motivation and interest toward their project, students often surpass themselves and produce very creative work indeed. The sharing of the projects at the end of the semester, is much appreciated by all; students and their peers show a keen and intelligent interest in each other’s work. Once they are involved in their project, they will often ask how they could improve their drawing by making use of other mathematical functions. This challenge brings them to explore mathematical concepts that are not taught at their grade level (for example, application of the integral calculus). After a few explanations on the teacher’s behalf, they apply these new applications to their drawings thereby exceeding the course outlines and the objectives set for that grade level. Clearly, without the technological advances now available, all this would be impossible. I presented this approach in 2005 and also in 2007 at an Atlantic convention called APTICA (Pedagogical Advancement of Technologies and Communications in the Atlantic Provinces). The enthusiasm that I received was very reassuring.

25. Title: The Art of Mathematics Work Samples, Teacher and Student Reflection The success of the students depends very much on the time spent working on his project. To help the student, it is essential that the teacher requires a rough sketch by mid-semester in order that the student does not undertake his project at the last moment. This is a semestrial project and without establishing this deadline of mid-semester, many students might postpone starting their project until the end, thus producing an inferior drawing with less mathematical content. Generally, students are very enthusiastic to work on the project. When teaching mathematics, a teacher often hears the following comments: “Why are we learning this and what use will it be?” Using this approach, I have never heard that remark when studying vertical or horizontal translations. The necessity of these concepts is indispensable for their project and gives meaning to their learning experience. After offering a few explanations and examples, students demonstrate little difficulty initiating their project. Naturally, throughout the semester, certain students will ask precise questions pertaining to the use of the software.

26. Title: The Art of Mathematics Teaching Resources • Student Project Overview: • Tasks Required: • Explain the project at the beginning of the year when presenting the course outline. • Ensure all students have access to a computer and the software. • Teach the students the necessary mathematical concepts and familiarize them with the software, its environment and applications. • Use the software for the graphing of equations and inequalities. • Specify a date, at mid-semester, for their submission of a sketch of their proposed drawing. • Specify the method of evaluation in order that students are aware of the criteria of assessment. Documents

27. Title: The Art of Mathematics Assessment and Standards Assessment Rubrics: Grading procedures will vary with different teachers. For example, I offer the following suggestion. The ”weight” and the evaluation of the project are based on the following criteria's : (calculated on a value of 40) - Creativity of the drawing 0 2 4 6 8 10 points - Level of difficulty of equations 0 2 4 6 8 10 points - Variety of equations; linear, cubic, absolute value, inequalities, circle, quadratic, logarithmic, exponential, trigonometric.. 0 2 4 6 8 10 points - Appearance: color, design, thickness of curves0 2 4 6 8 10 points Mapping the Standards: This project allows students to explore and master mathematical concepts as prescribed by the Department of Education in all Canadian Provinces. We can expect learning experiences that are reliable, lasting and transferable.

28. Title: The Art of Mathematics Annexe A <Information about school and teacher> Students write their mathematical equations here The graphs related to their equations appear here.

29. How to work with GrapheEasy ? Title: The Art of Mathematics Annexe A Step 1 : Click here in order to write your first equation <Information about school and teacher> Step 2 : What form do you want? Click on the quadratic form and the software will propose different type of equations for the parabola. Choose the first form i.e. the standard form A(x-B)2 + C. Click next.

30. Then… Title: The Art of Mathematics Step 3 : Choose 2 for the value of A, B and C, that is, the equation of the form y (x)= 2(x-2)2+2 Assessment and Standards Step 4 : Choose the desired color “blue” and a thicker curve. Click end. Click on the « + » sign in order to have more information on the equation and curve Your first equation.