planning a clil project -...
TRANSCRIPT
Planning a CLIL Project
“Figure Out My Strange Triangle VCK”
Annalisa Canarini Maths Teacher
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CONTENTS
Introduction p. 3
The context and the activities p. 3
Planning a CLIL Project: “Figure Out My Strange Triangle VCK” p. 5
Lesson planning p. 8
Follow-up p. 15
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Introduction
This paper deals with CLIL activities for a class of 11th
grade of students (3rd
year of
High School), 3D with a scientific curriculum.
It refers to the implementation of the CLIL approach in teaching/learning Maths
through English in the context of Liceo Rinaldo Corso during the a.y. 2014/2015.
The Scientific Studies provides four teaching periods a week for maths. It combines
humanities and science while offering a truly balanced approach that grants thorough
and complete scientific knowledge and competence along with a development of
languages and literature.
The CLIL practice described in this paper was implemented in the framework of the 3-
years project “Figure Out My Country”, a partnership which involves three high schools
with scientific courses: Liceo Rinaldo Corso (Correggio, Italy), Maurick College
(Vught, The Netherlands) and Karhulan Lukio (Kotka, Finland).
The project aims at representing our countries through Maths (using Maths concepts,
procedures, tools, calculations, models, graphs to describe, interpret and predict
phenomena), reconsidering and discovering our countries through Maths (by using a
multi-subject approach) and through mobility.
The activities of the project deal with exchange of practices, implementation and
development of an innovative approach that will be verified and tested, with
transnational initiatives and mobility that may make students gain awareness of
European diversity and may stimulate them to use the competences acquired in Maths,
Sciences, English and ICT to face their future challenges in life, study and work. Pupils
and teachers are involved in the mobility activities as well and will have the opportunity
to work together and to acquire and improve skills related to the topic which is the focus
of the mobility and will also improve team working, intercultural learning, ICT and
language competences.
The context and the activities
In order to implement CLIL in Liceo Rinaldo Corso, a group of teachers from this
school collaborated to design cross-curricular activities. We decided to pilot CLIL
inside the project FOMyC in a class with a scientific curriculum where maths is an
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important subject. In all the schools involved in the project, the second language taught
is English so we had no possibility to use a different foreign language. We chose to plan
CLIL and FOMyC activities for a school grade that allowed which ensured enough
autonomy by students to produce mathematical statements in English. The maths
curriculum in this grade is based on trigonometry and analytical geometry. During the
first term students are introduced to the trigonometry, working with angles, triangles
and trigonometric functions, whereas the second term is completely devoted to the study
of the conic sections. Last year the second term was chosen to develop the project
activities. This made it possible to work with completely new concepts, and to refer the
basic concepts already used in the first period.
The class is composed of 20 students, out of whom 2 are considered low-achievers in
maths and 3 low-achievers in English.
About the language level, the group involved in the project is made up of quite
independent users who can understand the main points of a clear and standard input on
familiar matters (school, work, leisure, environment, city life…) both in the spoken
interaction and in written passages. They can deal with most situations that may occur
when they are travelling to areas where English is spoken. A few of these students are
developing a more confident understanding that may allow them to deal with more
complex contexts, not always familiar, and abstract ideas and discussions. These few
students can communicate with a higher degree of fluency and spontaneity that may
allow them to carry out a natural-like interaction with a native speaker without strain for
either party and are developing an accurate and flexible range of vocabulary.
As far as written production is concerned, most of the students can describe experiences
and produce texts on topics which are familiar or are of personal interest (leisure time,
school, friends, descriptions of people…), they can produce texts about their dreams and
hopes, their past, can clearly explain their opinions and plans. A few of the students
involved in the project can master the written language more independently and produce
detailed written passages fit to different purposes and explaining viewpoints, they can
produce texts on familiar topics dealing with advantages and disadvantages, dealing
with personal interests and experiences. They can use accurate and precise vocabulary
and can organize their texts effectively.
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Finally, the majority of the students is B1+ (Threshold) whereas a few of them are
confident B2 (going toward Vantage) in most skills.
For the a.y.2014/2015 the project FOMyC focused on these cross-curricular activities
and the related topics (see Appendix 1):
Geography-Science & Maths: “Figure Out My Strange Triangle VCK”
(Topics: Geographic Coordinate, Trigonometry, Non-Euclidean Geometry on the Sphere)
Art & Maths: “My Maths Stroll in Florence”
(Topics: Plane Geometry, Transformations, Golden Ratio, Tessellation)
Citizenship & Maths: “Italy in figures”
(Topics: Economic data, Graphs)
Chemistry, Art, Physics & Maths: “MaTH – MathsTreasure Hunt”
(Topics: discovering Correggio with problem solving)
Exchange week in Italy: 27 May-2 June 2015
(Host student, workshop, visits…)
Planning a CLIL Project: “Figure Out My Strange Triangle VCK”
Content: distance and angle sum of a triangle in Spherical Geometry.
Aims: Through exploration, students will gain knowledge of the properties of some
objects in spherical geometry. They will be able to compare and contrast the two-
dimensional objects to the three-dimensional objects and to calculate the great-circle
distance between two points (the shortest distance over the earth’s surface) and the sum
of the angles in any spherical triangle.
Task: “Figure Out My Strange Triangle VCK” (solving a triangle on a sphere)
Class: 3rd
class High School
Time: 15 hours
Teaching objectives:
Content Non-Euclidean Geometry: Spherical Geometry
Antipodal points, Great Circle, Geodesic
Spherical Triangle
Spherical Law of Cosines
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Communication New vocabulary in order to:
Classify and compare (differences between Euclidean
and Non Euclidean Geometry)
Define, describe and explain (defining distance in Non
Euclidean Geometry, describing triangle in different
geometries…)
Conjecture, hypothesise and estimate (predicting the
sum of the angles in triangle in different geometries…)
Discuss ideas (comparing suggestions and hypothesis…)
Present information, make a description of an
experimental procedure (describing the procedure to
estimate distance and measure…)
Present and defend an argument/plan on the topics
Cognition Identify the properties of Non Euclidean Geometry
Compare different Geometries
Predict and reason
Figure out triangle in order to solve real-life problem
Culture Calculate the shortest flight or navigation path
(Spherical Geometry)
Comprehend the art of M.C. Escher (Hyperbolic
Geometry)
GPS (Spherical Geometry)
Learning outcomes:
Know
the definitions of great circle, geodesic, antipodal points
the Law of Cosines in the Spherical Geometry
Be able to construct a great circle and determine that it is the
simplest line in spherical geometry
determine that parallel great circles do not exist
investigate conventional methods for measuring
segment lengths and angles in spherical geometry
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to construct a spherical triangle and to notice the
differences in the properties of planar and spherical
triangles.
to determine the maximum interior angle sum of a
spherical triangle
to construct a spherical triangle containing one, two, and
three right angles
to investigate the Law of Cosines and determine if they
hold true for spherical trigonometry
to utilise the Cinderella tool and make other explorations
and discoveries
Be aware that some real-life problems need different geometric
context
Assessment:
(Based on objectives)
Groups assessment will be performed on the answers to the worksheet questions, on
the discoveries provided and on the completion of the task.
Individual assessment will be performed on participation in groups and whole class
discussions.
Resources:
Balls, Rubber bands, Rulers, Protractors, Atlases, Globes, Computer, Video, Cinderella
Software, Worksheets, Language of cause-effect, Oxford Student’s Mathematics
Dictionary, Peda.net (Finnish Portal tool with English version).
Procedure:
Whole class to activate prior knowledge of Euclidean Geometry and
Trigonometry
Groups to use balls and rubber bands to model great circle and
triangle
to explore if axioms that hold in Euclidean space hold in
spherical space
to determine if Euclid’s fifth postulate holds on the
sphere
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Individually to record observations and conclusions on worksheets
Whole class to feedback ideas
Individually ICT activity: exploration of the properties of some
objects in spherical and hyperbolic geometry through
the Cinderella tool
Whole class to feedback ideas and conclusions
Groups to complete a task: solving a specific triangle on the
sphere
Whole class Feedback: sharing and comparing the solutions
Individually ICT activity: check the solution with distance calculator
for latitude/longitude points
Whole class Final plenary: applications of Non-Euclidean Geometry
Follow-up to set up a Math-Challenge form for Finnish and Dutch
partners Upload the form on Peda.net
to record data through a spreadsheet on Excel
Lesson planning
Lesson 1
Aim: Activating Prior Knowledge and Content – obligatory language
Materials: Oxford Mathematics Dictionary
In Euclidean Geometry students already know how to:
Find the shortest distance between two points
Measure an angle and a line segment
Describe intersecting lines, parallel lines, circles, and a sphere
In Trigonometry students already know how to:
Solve a triangle
Law of cosines
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Activity 1 (groups): vocabulary needed for Euclidean Geometry and Trigonometry
Group 1: Find the key words related to Euclidean Geometry (prior knowledge) on
Oxford Mathematics Dictionary
Group 2: Find the key words related to Trigonometry (prior knowledge) on Oxford
Mathematics Dictionary
Activity 2 (whole class): Group 1 draws on the board key concepts related to the topics
of Group 2 and Group 2 write on the board the corresponding key words.
Activity 3 (pairs): Language of “cause-effect” in Maths
Using the language of “cause-effect” (see Appendix 1), rewrite three theorems, axioms or
statements connected to the topics above.
Activity 4 (whole class): sharing and comparing the statements.
Lesson 2
Aim: To introduce students to the concept of spherical geometry.
Materials: Each student needs a ball, rubber bands, strings.
1) Explain to the students that we will be studying Spherical Geometry.
(Spherical Geometry – the branch of geometry which deals with a system of points, great circles (lines),
and spheres (planes)).
2) Let students watch the video “Assignment Discovery: Spherical Geometry” by the
Discovery Channel which can be found at the following address:
http://videos.howstuffworks.com/discovery/28013-assignment-discovery-spherical-geometry-video.htm.
3) Open the class with a discussion. Ask the following questions:
What are the limits of planar geometry? Possible answers: lines are straight, surfaces are flat
If we want to travel by air, what type of geometry do you think we should use? Possible
answers: non-Euclidean geometry, spherical geometry. Since the earth is a sphere, there is a branch of
geometry known as spherical geometry.
What are some of the applications of spherical geometry? Possible answers: navigation,
astronomy, GPS
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4) Ask students what a sphere is and to compare it to a plane.
Definition: Sphere - a set of points in 3-dimensional space that are equidistant from one fixed point (the
center).
5) Show students a ball. Explain that this is a sphere. The set of points in the definition
above describes all points on the surface of the sphere.
6) Hand out the materials to the students. Have them locate any two points A and B on
their sphere. Ask how many paths there are from A to B?
7) Explain how the flat lines that we are use to, cannot happen on a sphere.
Use the string to show several curved paths from A to B. Measure the length of each
string to the nearest millimeter.
8) Make a conjecture about the shortest distance between two points in spherical space.
Wrap your string from A through B around the sphere until you return to point A. What
appears to be true about the shortest path from point A to point B? What is true about
the relationship between the sphere and the string? (The string forms a great circle that divides
the sphere in half)
9) Show the students that a rubber band is the best method of showing the shortest path
because it hugs the sphere.
Give students a few minutes to make their own great circles on their individual balls.
Ask students if they know what lines on a sphere are called. Lines are called geodesics,
which are great circles, like the equator.
Definition: Great Circle – When a plane passes through the center of a sphere, cutting the sphere in half.
Considering the Earth, a great circle would be the equator. The parallels of latitude are small circles, and
lines of longitude are great circles.
Definition: Geodesics: are the shortest distances between two points on the sphere. They are line
segments along a great circle.
10) Chooses two new points C and D so that they appear to be directly opposite each
other on your sphere. These are called antipodal points on of the sphere. (Like the North
and South pole)
Definition: Antipodal Points – are opposite points on the sphere. A good example of antipodal points is
the north and south poles on the earth.
11) Draw a figure showing the sphere and points C and D. In how many different ways
can you measure the shortest path from C to D? (Infinitely many great circles pass through
polar points)
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Ask students if parallel lines exist on a sphere. Remind them that lines continue all the
way around the sphere back to its’ starting point. Since all great circles intersect all
other great circles, parallel lines do not exist on a sphere.
In Groups, students should work on the following handout. (see Appendix 2)
Extended Practice (homework):
Have students revise the definitions given in Spherical Geometry. (see Appendix 3)
Lesson 3
Aim: To explore axioms that hold in Euclidean space and determine if they hold in
spherical space.
Materials: Balls, rubber bands
1) Go over worksheet from Lesson 2.
2) When you first studied geometry in high school, you were told that you needed to
accept certain postulates without proof. We will work with postulates or axioms. An
axiom is a known truth.
For example, in Euclidean space, we accept the following axiom:
Axiom 1: Given any line l, then exist at least two distinct points on l. (Draw a line on the
board and demonstrate that you can find two distinct points).
3) Ask: Is Axiom 1 true on the sphere? (Students can demonstrate with a ball and rubber bands, or
with a sketch that Axiom 1 does hold in spherical space)
Axiom 2: Given two distinct points P, Q, there exists a unique line incident with P, Q.
4) As a class determine whether Axiom 2 holds. (Students should be able to recognize that
there are an infinite number of great circles that can pass through two antipodal points. Have a student
draw an example on the board or model this for the class using a ball and rubber bands)
5) Assign the following questions for groups to work on.
a) Recall some of the first postulates you learned in Euclidean space:
There is a line between any two points.
A line segment can be extended indefinitely.
You can draw a circle about every point with any radius.
Tell whether or not they hold on spherical space.
b) If possible, find two perpendicular "lines" on the sphere.
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Lesson 4
Aim: To determine whether Euclid’s fifth postulate holds on the sphere.
1) Go over problems from Lesson 3 handout. Have students demonstrate on a ball that
the 3 postulates will work on the sphere, and explain their thinking.
2) Euclid’s fifth postulate states:
Given a line and a point, there exists a unique line passing through the point parallel to
the first line.
3) Give students time to work on this in their groups. Discuss whether or not this will
work in spherical space.
Students should try to verify this on the sphere and fail due to the fact that there are no parallel lines on
the sphere. Every line on the sphere intersects in at least two points.
4) Have students work on the following questions:
a) In Euclidean space, given a line and a point P on that line, there is exactly one
perpendicular to P. Demonstrate that this does not hold in spherical space.
b) Think of another property that holds in Euclidean space but does not hold in spherical
space.
Extended Practice:
Have students compare and contrast lines in Euclidean Geometry to lines in Spherical
Geometry by completing the worksheet. (see Appendix 4)
Lesson 5
Aim: Assessment
Activity 1 (Individually): True or False? (see Appendix 5)
Activity 2 (Whole class): Revision and discussion
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Lesson 6
(Language: Italian)
CONFERENCE held by Modena and Reggio Emilia University professor on Non
Euclidean Geometry
“La Rivoluzione Non Euclidea”
Lecturer: Professor Bandieri Paola (Unimore)
Lesson 7
(Language: Italian)
LAB-ACTIVITY (3 hours) held by Professor Bandieri Paola c/o University of Modena
and Reggio Emilia in the ICT Lab of Math Dept.
Aim: Students will be able to use the Cinderella Software tool to investigate Spherical
and Hyperbolic trigonometric relationships and make comparisons to planar
trigonometry. Through exploration, students will gain knowledge of the similarities and
differences of trigonometric properties of segments, angles, and triangles in spherical
and hyperbolic geometry.
Lesson 8
Aim: to investigate conventional methods for measuring segment lengths and angles in
spherical geometry and to investigate the Law of Cosines for spherical trigonometry
Activity 1 (whole class): Open the class with an activity for the students to develop
their own methods of measuring angles and segments on a sphere or in spherical
geometry. Select several students to present their methods to the class. From the
presentations, identify the conventional methods for measuring angles and segments in
spherical geometry. (Measuring angles is the same in radians. Measuring segments is usually done
using radians, because the segment is thought of as an arc of the circle resulting from the cross-section of
the sphere)
Activity 2: teacher develops a formula similar to the Euclidean Law of Cosines for
spherical triangles.
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Activity 3 (groups): Have the students form groups of three. Each group should relocate
to a computer and find at least one real world application that can be solved by using the
Law of Cosines of spherical trigonometry.
Lesson 9
Aim: to solve a “specific” triangle through spherical trigonometry
Materials: protractor, string, ruler, atlas, globe
Task: Figure Out My Strange Triangle VCK
Activity 1 (groups): Have the students form 5 groups of four. Using protractor, string,
ruler, atlas, globe students have to estimate the distance between C (=Correggio),
V(=Vught) and K(=Kotka), the cities involved in the partnership FOMyC and the
measures of the spherical angles with vertexes in C, V, K.
Each group have to complete the attached worksheet (see Appendix 6).
Activity 2 (whole class): the whole class will discuss the answers to the worksheet and
each group will present any discoveries that they have found. Each group will
participate in discussing answers.
Lesson 10
Aim: to solve a “specific” triangle through spherical trigonometry
Materials: calculator, computer
Task: Figure Out My Strange Triangle VCK
Activity 1 (groups): Have the students form 5 groups of four. Using calculator and the
Law of Cosines students have to calculate the distance between C (=Correggio),
V(=Vught) and K(=Kotka), the cities involved in the partnership FOMyC and the
measures of the spherical angles with vertexes in C, V, K.
Each group have to complete the attached worksheet (see Appendix 7).
Activity 2 (whole class): the whole class will discuss the answers to the worksheet and
each group will present any discoveries that they have found. Each group will
participate in discussing answers.
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Follow-up
Set up a Math-Challenge form for Finnish and Dutch partners and upload the
form on Peda.net
Record data through a spreadsheet on Excel
Sharing the results with European partner
(see Appendix 8)
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