Topicmc85ad4dc2316a9f0_1528449000663_0Topic

Regular polygons

Levelmc85ad4dc2316a9f0_1528449084556_0Level

Second

Core curriculummc85ad4dc2316a9f0_1528449076687_0Core curriculum

IX. Polygons. The student:

1) knows the concept of a regular polygonregular polygonregular polygon.

Timingmc85ad4dc2316a9f0_1528449068082_0Timing

45 minutes

General objectivemc85ad4dc2316a9f0_1528449523725_0General objective

Using simple, well known mathematical objects, interpreting mathematical concepts and manipulating mathematical objects.

Specific objectivesmc85ad4dc2316a9f0_1528449552113_0Specific objectives

1. Introduction of the concept of a regular polygonregular polygonregular polygon.

2. Calculation of the measure of angles of a regular polygon.

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

Learning outcomesmc85ad4dc2316a9f0_1528450430307_0Learning outcomes

The student:

- identifies regular polygons,

- calculates the measure of angles in a regular polygonregular polygonregular polygon.

Methodsmc85ad4dc2316a9f0_1528449534267_0Methods

1. Discussion.

2. Situational analysis.

Forms of workmc85ad4dc2316a9f0_1528449514617_0Forms of work

1. Group work.

2. Individual work.

Lesson stages

Introductionmc85ad4dc2316a9f0_1528450127855_0Introduction

The teacher introduces the topic of the lesson: the concept of the regular polygon and calculation of its angles.

The teacher asks the students:

- What are the common properties of the squaresquaresquare and the rectangle?

- What are the common properties of the rhombusrhombusrhombus and the square?

- What are the common properties of the equilateral triangleequilateral triangleequilateral triangle and the square?

Student discuss the topics. The summary of this part of the lesson is the definition of the regular polygonregular polygonregular polygon.

Proceduremc85ad4dc2316a9f0_1528446435040_0Procedure

Definition – regular polygonregular polygonregular polygon.
A regular polygon is a polygon whose all angles are equal in measure and whose all sides have the same length.mc85ad4dc2316a9f0_1527752263647_0A regular polygon is a polygon whose all angles are equal in measure and whose all sides have the same length.

[Illustration 1]

Students give examples of regular polygon models, which they find around them (e.g. hexagons in a honeycomb).

The teacher explains how to approximately construct a regular polygon.

When we mark points at equal distances on a circle and then we connect them consecutively, we get a regular polygon. If we lead the radii to each of the vertices of the polygon from the center of the circle, we get as many congruent triangles as there are sides in that polygon.mc85ad4dc2316a9f0_1527752256679_0When we mark points at equal distances on a circle and then we connect them consecutively, we get a regular polygon. If we lead the radii to each of the vertices of the polygon from the center of the circle, we get as many congruent triangles as there are sides in that polygon.

Students work individually or in pairs, using computers. They discover the relationship between the angles of the regular polygonregular polygonregular polygon and the angles of the triangles in which this polygonpolygonpolygon is divided.

Task
Open the Geogebra applet Regular n‑gons. Change the number of sides of the n‑gonn‑gonn‑gon and answer the following questions.

- What is the measure of the angleangleangle of the triangle at the vertex that is the center of the circle for n = 4, n = 6, n = 10? And for n = 36? Why?

- What is the sum of measures of two angles: the interior angleinterior angleinterior angle of a polygon and the angle of the triangle at the vertex that is the center of the circle for n = 3, 5, 12? And for n = 40? Why?

- What is the measure of the interior angle of a regular n‑gonn‑gonn‑gon for n = 9? Calculate it and then check result using the applet.

- How does the measure of the interior angleinterior angleinterior angle of the regular polygon change when the number of its sides increases? Can the interior angle of a regular polygon be as wide as 180°?

5. Which regular polygonregular polygonregular polygon has a greater number of sides: the polygonpolygonpolygon with the interior angle of 120° or 162°?

Conclusions:

- The angleangleangle of the triangle at the vertex that is the center of the circle has the measure of:

where: 
n - is the number of sides of that polygon.

- To calculate the interior angleinterior angleinterior angle of a regular polygon, first divide 360° by the number of sides of this polygonpolygonpolygon, and then subtract this value from 180°.

Students work individually, solving the following problems. Having completed the exercises, they present the results and discuss them.

Task
How many sides does a regular polygon whose interior angle is 162° have?

Task
How many sides does a regular polygonregular polygonregular polygon whose sum of angles is 720° have?

An extra task:
Derive the formula for measuring the interior angleinterior angleinterior angle regular n‑gons depending on n.

Lesson summarymc85ad4dc2316a9f0_1528450119332_0Lesson summary

Students do the revision exercises. Then together they summarize the class, by formulating conclusions to memorize.

- A regular polygon is a polygon in whose all angles are equal in measure and whose all sides have the same length.
- To calculate the $\alpha$ - interior angle of a regular polygon, first divide 360° by the number of sides of this polygon, and then subtract this value from 180°. The measure of the angle $\alpha$ can be expressed by the formula:mc85ad4dc2316a9f0_1527712094602_0- A regular polygon is a polygon in whose all angles are equal in measure and whose all sides have the same length.
- To calculate the $\alpha$ - interior angle of a regular polygon, first divide 360° by the number of sides of this polygon, and then subtract this value from 180°. The measure of the angle $\alpha$ can be expressed by the formula:

Selected words and expressions used in the lesson plan

angleangleangle

equilateral triangleequilateral triangleequilateral triangle

hexagonhexagonhexagon

interior angleinterior angleinterior angle

n‑gonn‑gonn‑gon

polygonpolygonpolygon

regular polygonregular polygonregular polygon

rhombusrhombusrhombus

squaresquaresquare