authorsrobert levesque, dss, b.sc., b.ed., m.ed. to apply and master different

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Authors Robert Levesque, DSS, B.Sc., B.Ed., M.Ed.

To apply and master different mathematical concepts in relation to art Drawings

Objectives

DescriptionThe 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.

Learning Areas Mathematics

Levels 16-17-18 year olds

Project Overview

Title: The Art of Mathematics

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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.


GrapheEasy; MS Office; Desire2Learn (content platform for distance learning); Interwise (communication platform for distance learning)

Software

Drawings, graphs, grapheEasy, functions, horizontal translation, vertical translation, Art, constructive learning, mathematics, project, mathematical parameter

Keywords

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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.


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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.


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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.


Graphical representation of absolute values, square roots, rational expressions.


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.


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.


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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


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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..


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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.


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Work Samples, Teacher and Student Reflection

Student: Julie Leblanc.

Approximately 135 mathematical equations


sinus function

quadratic functionlogarithmic function

sinus function

Logarithmic function

circle

linear

quadratic function

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Student: Megan



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Student: Valérie Lang



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Student: Tristan Martin



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Student: Sophie Chiasson



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Student: Stacey Morris



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Student: Billy Nowlan



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Student: François Laplante



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Student : Chantal Richard



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Student: Stéphanie Turner.



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Student: Gisèle Doiron.



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Student: Clément Savoier.



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Student: Stéphanie Caissie.



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Student: Clément Savoie.



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Student: Joline Poirier.



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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.


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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.


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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


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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 curves 0 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.


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Annexe A

<Information about school and teacher>

Students write their

mathematical

equations here The graphs related to

their equations

appear here.


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Annexe A

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Step 1 : Click here in order to write

your first equation

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.

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Assessment and Standards


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 Step 4 : Choose the desired

color “blue” and a thicker

curve. Click end.

Your first equation.

Click on the « + » sign in order

to have more information on

the equation and curve