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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