Topicm9d726b006de35786_1528449000663_0Topic

Polygons and their properties

Levelm9d726b006de35786_1528449084556_0Level

Second

Core curriculumm9d726b006de35786_1528449076687_0Core curriculum

IX. Polygons. The student:

2) uses the formulas to calculate the area of a triangle, rectangle, square, parallelogram, rhombus, trapezoid and is able to determine the lengths of line segments in tasks of comparable difficulty:

a) calculate the shortest altitude of a right triangle, whose sides are: 5 cm, 12 cm and 13 cm;

b) The diagonals of a rhombus ABCD are AC = 8 dm i BD = 10 dm. The diagonal BD is prolonged to point E in such a way that the line segment BE is twice as long as this diagonal. Calculate the area of the triangle CDE (there are two possible answers).

Timingm9d726b006de35786_1528449068082_0Timing

45 minutes

General objectivem9d726b006de35786_1528449523725_0General objective

Noticing regularities, similarities and analogies and formulating relevant conclusions.

Specific objectivesm9d726b006de35786_1528449552113_0Specific objectives

1. Defining an n‑gonn‑gonn‑gon, determining the number of diagonals in a polygon.

2. Calculating the sum of the angles in a polygonpolygonpolygon.

3. Communicating in English, developing basic mathematical, computer and scientific competences, shaping the ability to learn.

Learning outcomesm9d726b006de35786_1528450430307_0Learning outcomes

1. Defines the concept of an n‑gonn‑gonn‑gon, checks the number of diagonals in a polygonpolygonpolygon (In Polish and English).

2. Calculates the sum of the angles in an n‑gonn‑gonn‑gon.

Methodsm9d726b006de35786_1528449534267_0Methods

1. Practical exercises.

2. Analysis of the situation.

Forms of workm9d726b006de35786_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm9d726b006de35786_1528450127855_0Introduction

The teacher informs the students that during this class they will find out what an n‑gonn‑gonn‑gon is, how to calculate the number of diagonals of an n‑gonn‑gonn‑gon and the sum of angles of an n‑gonn‑gonn‑gon.

Task
The teacher asks students:

- How do we call a polygon that has three sides?

- How do you call a polygonpolygonpolygon that has seven sides?

- What is the name of a polygon that has n sides?

Procedurem9d726b006de35786_1528446435040_0Procedure

The students give the definition of a polygonpolygonpolygon.

Definition

- A polygon that has n sides is called an n‑gon.m9d726b006de35786_1527752256679_0- A polygon that has n sides is called an n‑gon.

Task
Draw any ABCDEF heptagon.

1. Is the line segment DE a side of a polygon?

2. Is the line segment AC a side of a polygonpolygonpolygon?

3. What is the name of the line segment AC?

4. What is the name of the line segment BD?

The students give the definition of a diagonal.

Definition

- A diagonal is a line segment that links two vertices of a polygon which are not adjacent.m9d726b006de35786_1527752263647_0- A diagonal is a line segment that links two vertices of a polygon which are not adjacent.

Task
Look at the pictures. You can see the polygons with marked diagonals. See how many diagonals each polygonpolygonpolygon has.

[Illustration 1]

Can you determine how many diagonals a heptagon and an octagonoctagonoctagon have?

The teacher and the students write down the formula for the number of diagonals of a polygon.

Theorem

- Let n be a natural number greater than 3. An n‑sided polygonpolygonpolygon has  $\frac{n\left(n-3\right)}{2}$ diagonals.

Task
Calculate the number of diagonals of an 50‑gon.

Task
Draw a pentagon whose adjacent sides are perpendicular. The teacher asks students to check in pairs if the drawing is correct and discuss the possible solutions.

Task
Students work individually, using computers. Their task is to observe how to calculate the internal angles of n‑gons. It is important that the students see that each n‑gonn‑gonn‑gon can be divided into n‑2 triangles.

[Geogebra applet]

Having completed the exercise, they present the results of their observations by answering the questions:

- What is the sum of angles in a triangle?

- Into how many triangles you can divide tetragon, pentagon, etc?

- Do you see the relation between the number of angles in a polygonpolygonpolygon and the number of created triangles?

The teacher and the students write down the formula for the sum of angles in an n‑gonn‑gonn‑gon.

Theorem

Let n be a natural number greater than two. The sum of angles in an n‑gonn‑gonn‑gon is equal to (n - 2) · 180°.

Task
Calculate the sum of angles of a 12‑gon.

An extra task:

Is there a polygonpolygonpolygon whose sum of angles is equal to 1000°?

Lesson summarym9d726b006de35786_1528450119332_0Lesson summary

Students do the revision exercises.

Then, they sum up the lesson by formulating the conclusions to remember.

- A polygon that has n sides is called an n‑gonn‑gonn‑gon.

- A diagonal is a line segment that links two vertices of a polygonpolygonpolygon that are not adjacent.

- If n is a natural number greater than 3, an n‑sided polygon has $\frac{n\left(n-3\right)}{2}$ diagonals.

- If n is a natural number greater than two, the sum of angles in an n‑gonn‑gonn‑gon is equal to (n - 2) · 180°.

Selected words and expressions used in the lesson plan

angleangleangle

diagonal of a polygondiagonal of a polygondiagonal of a polygon

n‑gonn‑gonn‑gon

octagonoctagonoctagon

polygonpolygonpolygon

sum of angles of a polygonsum of angles of a polygonsum of angles of a polygon