Topicm4dabf4076f816696_1528449000663_0Topic

Introduction to trigonometry

Levelm4dabf4076f816696_1528449084556_0Level

Third

Core curriculumm4dabf4076f816696_1528449076687_0Core curriculum

VIII. Planimetry.

The basic level. The student:

3) recognises regular polygons and applies their basic properties;

8) applies characteristics of similar triangles.

Timingm4dabf4076f816696_1528449068082_0Timing

45 minutes

General objectivem4dabf4076f816696_1528449523725_0General objective

Using the mathematical language to create mathematical texts, including description of reasoning and justification of conclusions, as well as presenting data.

Specific objectivesm4dabf4076f816696_1528449552113_0Specific objectives

1. Applying characteristics of similar, right‑angled trianglesright‑angled trianglesright‑angled triangles to do text exercises from planimetry.

2. Applying properties of right angled triangles 90°,45°, 45° and 90°,60°, 30° while doing planimetry exercises.

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesm4dabf4076f816696_1528450430307_0Learning outcomes

The student:

- applies characteristics of similar, right‑angled triangles to do text exercises from planimetry,

- applies properties of right angled triangles 90°,45°, 45° and 90°,60°, 30° while doing planimetry exercises.

Methodsm4dabf4076f816696_1528449534267_0Methods

1. Discussion.

2. Wandering posters.

Forms of workm4dabf4076f816696_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm4dabf4076f816696_1528450127855_0Introduction

Students revise information about right‑angled triangles, their properties and theorems about them, while discussing in pairs.

Procedurem4dabf4076f816696_1528446435040_0Procedure

Students work individually, using computers. Their task is to get to know the interactive illustration, that describes the right‑angled triangle.

[Interactive graphics]

The teacher divides students into three groups. Each group prepares a poster about the given subject. After 5 minutes, posters are passed to the net group, whose task is to add missing information. After 5 minutes, posters are passed again. After another five minutes, they go back to initial groups, that have 5 minutes to prepare a presentation about their subject.

Poster I:

Right‑angled trianglesright‑angled trianglesRight‑angled triangles whose angles are 90°,45°, 45°.

Poster II:

Right‑angled triangles whose angles are 90°,60°, 30°.

Poster III:

Right‑triangle similarity.

Students present their work.

Most important information that should be on the poster.

I.

[Illustration 1]

A right‑angled triangle whose angles are 90°,45°, 45° is a half of the square whose side is a.

In each right‑angled triangle that has an acute angleacute angleacute angle equal to 45°, the ratio of any cathetuscathetuscathetus to the hypotenusehypotenusehypotenuse is equal to $\frac{\sqrt{2}}{2}$.

II.

[Illustration 2]

A right‑angled triangle whose angles are 90°,60°, 30°is a half of the equilateral triangle whose side is 2a.

In each right‑angled triangle that has an acute angleacute angleacute angle equal to 30°,

the ratio of the shorter cathetus to the hypotenusehypotenusehypotenuse is equal to $\frac{1}{2}$, and the ratio of the shorter cathetuscathetuscathetus to the longer cathetus is equal to $\frac{\sqrt{3}}{3}$.

III.

[Illustration 3]

Right‑angled triangles are similar if one of their acute angles is congruent or the ratio of two respective sides is the same.m4dabf4076f816696_1527752256679_0Right‑angled triangles are similar if one of their acute angles is congruent or the ratio of two respective sides is the same.

The teacher evaluates students’ work and clarifies doubts.

An extra task:

In the ABC triangle, the angle at the vertex A is equal to 60°, and the angle at the vertex C is equal to 75°. The altitude CD is equal to 6 cm. Calculate the area and the perimeter of this triangle.

Lesson summarym4dabf4076f816696_1528450119332_0Lesson summary

Students do the revision exercises. Then together they sum‑up the classes, by formulating the conclusions to memorise.

- In each right‑angled triangle that has an acute angle equal to 45°, the ratio of any cathetus to the hypotenuse is equal to $\frac{\sqrt{2}}{2}$.
- In each right‑angled triangle that has an acute angle equal to 30°, the ratio of the shorter cathetus to the hypotenuse is equal $\frac{1}{2}$
- Right‑angled triangles are similar if one of their acute angles is congruent or the ratio of two respective sides is the same.m4dabf4076f816696_1527752263647_0- In each right‑angled triangle that has an acute angle equal to 45°, the ratio of any cathetus to the hypotenuse is equal to $\frac{\sqrt{2}}{2}$.
- In each right‑angled triangle that has an acute angle equal to 30°, the ratio of the shorter cathetus to the hypotenuse is equal $\frac{1}{2}$
- Right‑angled triangles are similar if one of their acute angles is congruent or the ratio of two respective sides is the same.

Selected words and expressions used in the lesson plan

acute angleacute angleacute angle

cathetuscathetuscathetus

characteristics of similar right‑angled trianglescharacteristics of similar right‑angled trianglescharacteristics of similar right‑angled triangles

hypotenusehypotenusehypotenuse

properties of trianglesproperties of trianglesproperties of triangles

ratio of the cathetus to the hypotenuseratio of the cathetus to the hypotenuseratio of the cathetus to the hypotenuse

right angleright angleright angle

right‑angled trianglesright‑angled trianglesright‑angled triangles

similar trianglessimilar trianglessimilar triangles