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Title of the Experience

Goniometric functions in a right sided triangle

Name of the teacher

Viera Kolbaská

Country where it took place

Slovakia

School typology

High Secondary School

Thematic Area

Maths

Experience typology

Teaching in class

Type of contact

Direct

Description of the Experience

The aim of this course unit is to connect students’ knowledge on the calculation of the missing lengths of sides and heights from primary school in different dimensional figures using Pythagoras' theorem and the calculation of the angles in those shapes using trigonometric functions. This way it also gives pupils the opportunity to critically assess the benefit of the use of trigonometric functions when calculating the lengths of sides, respectively heights in planar shapes.

Thematic unit of Goniometric functions in a right sided triangle follows these teaching thematic units:

1. Triangles - essential elements of triangles and sorting of triangles (repetition of the primary school curriculum in a form of brainstorming and conceptual map)

2. Right sided triangle, isosceles triangle, equilateral triangle - naming their sides and using the Pythagoras' theorem in calculating the length of one side of the triangle or altitude (repetition of the primary school curriculum, problem solving worksheet)

3. Surface shapes (quadrilaterals - square, rectangle and trapezoid) - calculate the lengths of one side or the length of the height (repetition of the primary school curriculum, solving the tasks from the mathematical task collection and text books)

After these steps follow:

1. Definition of goniometric functions in the right sided triangle (1st grade secondary school curriculum) - frontal teaching and conversation with the students, activities with calculator

2. The calculation of goniometric functions in the various types of right sided triangles – used problem solving worksheet

3. Calculation of the lengths and sizes of angles in the right sided triangle - solving problems from the mathematical task collection textbook

4. Calculation of the missing elements in the application tasks using goniometric functions and the Pythagoras' theorem – work with the calculator, mathematical tables, worksheets and mathematical task collection textbook

5. Setting the homework as a preparation for the review test by a set of tasks from the website, consultations in extracurricular time to the topic of websites tasks

6. Written verification of acquired knowledge, written test

Solving problems using the goniometric functions is a curriculum that is taught at the end of the second semester of the first year of secondary school. This curriculum was according to the need operationally moved before the subject matter of the linear equation, because it required previous instruction from the subject of physics.

The problem for the successful implementation of the curriculum of this course topic is:

1. Small knowledge of working with the mathematical tables (I assumed students were familiar with tables of similar kind from the primary school, but they have not worked with them previously)

2. Insecure curriculum knowledge of plane figures and the Pythagoras theorem that pupils usually know to apply only in the case of a right sided triangle ABC with the right angle at the top of C (i.e. lack of understanding of the relationship square - hypotenuse = squares - cathetuses)

3. Students do not receive calculators for their use at school and can not use their own calculators properly (it is difficult for the teacher to explain to each student how to work with his or her own calculator and show them the procedure for determining the values of goniometric functions or calculating angles in a class of more than 20 pupils, the dynamism of the lesson is lost quickly)

4. Pupils have insufficient knowledge and skills from the primary school in solving design tasks and sketching (small range of subject matter lessons within the State Education Program, extended to the least within the schools in the school educational programs)

Recommended website: www.kolbi.wbl.sk.

I must say that pupils in small scale used the website. Either they did not bring teaching aids (calculators, spreadsheets) to the classroom.

The most effective solution to problems was using the worksheets either individually, in pairs or in groups.

The best motivation for most students was receiving additional points to their summative evaluation.

Students positively evaluated solving tasks aiming at calculating the lengths of sides of plane figures using trigonometric functions. They stated identifying appropriate data and calculating data is quicker and more accurate at the same time.

Thematic unit of Goniometric functions in a right sided triangle follows these teaching thematic units:

1. Triangles - essential elements of triangles and sorting of triangles (repetition of the primary school curriculum in a form of brainstorming and conceptual map)

2. Right sided triangle, isosceles triangle, equilateral triangle - naming their sides and using the Pythagoras' theorem in calculating the length of one side of the triangle or altitude (repetition of the primary school curriculum, problem solving worksheet)

3. Surface shapes (quadrilaterals - square, rectangle and trapezoid) - calculate the lengths of one side or the length of the height (repetition of the primary school curriculum, solving the tasks from the mathematical task collection and text books)

After these steps follow:

1. Definition of goniometric functions in the right sided triangle (1st grade secondary school curriculum) - frontal teaching and conversation with the students, activities with calculator

2. The calculation of goniometric functions in the various types of right sided triangles – used problem solving worksheet

3. Calculation of the lengths and sizes of angles in the right sided triangle - solving problems from the mathematical task collection textbook

4. Calculation of the missing elements in the application tasks using goniometric functions and the Pythagoras' theorem – work with the calculator, mathematical tables, worksheets and mathematical task collection textbook

5. Setting the homework as a preparation for the review test by a set of tasks from the website, consultations in extracurricular time to the topic of websites tasks

6. Written verification of acquired knowledge, written test

Solving problems using the goniometric functions is a curriculum that is taught at the end of the second semester of the first year of secondary school. This curriculum was according to the need operationally moved before the subject matter of the linear equation, because it required previous instruction from the subject of physics.

The problem for the successful implementation of the curriculum of this course topic is:

1. Small knowledge of working with the mathematical tables (I assumed students were familiar with tables of similar kind from the primary school, but they have not worked with them previously)

2. Insecure curriculum knowledge of plane figures and the Pythagoras theorem that pupils usually know to apply only in the case of a right sided triangle ABC with the right angle at the top of C (i.e. lack of understanding of the relationship square - hypotenuse = squares - cathetuses)

3. Students do not receive calculators for their use at school and can not use their own calculators properly (it is difficult for the teacher to explain to each student how to work with his or her own calculator and show them the procedure for determining the values of goniometric functions or calculating angles in a class of more than 20 pupils, the dynamism of the lesson is lost quickly)

4. Pupils have insufficient knowledge and skills from the primary school in solving design tasks and sketching (small range of subject matter lessons within the State Education Program, extended to the least within the schools in the school educational programs)

Recommended website: www.kolbi.wbl.sk.

I must say that pupils in small scale used the website. Either they did not bring teaching aids (calculators, spreadsheets) to the classroom.

The most effective solution to problems was using the worksheets either individually, in pairs or in groups.

The best motivation for most students was receiving additional points to their summative evaluation.

Students positively evaluated solving tasks aiming at calculating the lengths of sides of plane figures using trigonometric functions. They stated identifying appropriate data and calculating data is quicker and more accurate at the same time.

Comments on this Teachers Experience

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Date: 2014.11.09

Posted by **Darina Berčakova** (*Slovakia*)

When teaching trigonometric functions of a right angled triangle the motivation of students is very important. At our school – vocational school – I very often face the question “When will I need it?” when teaching.

Ms. Kolbaská experience is inspiring for me mostly because of the way how she uses ICT in achieving the educational objectives and because she is also leading the students to work individually. I also use the internet during my hours and I'm trying to lead the pupils to work individually

Date: 2014.10.28

Posted by **Darina Berčáková** (*Slovakia*)

Pri vyučovaní goniometrických funkcií pravouhlého trojuholníka je dôležitá motivácia žiakov. Často sa stretávam pri vyučovaní na našej škole - Strednej odbornej škole s otázkou " na čo to budem potrebovať?"

Skúsenosť pani Kolbaskej je pre mňa inšpirujúca najmä v tom, že využíva IKT pri získavaní výchovno-vzdelávacieho cieľa a zároveň vedie žiakov k samostatnosti.

Ja tiež na hodinách využívam internet a snažím sa žiakov viesť k samostatnosti.

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