Topicm939b7e73d42d641e_1528449000663_0Topic

The diagonals of the polygon.

Levelm939b7e73d42d641e_1528449084556_0Level

Third

Core curriculumm939b7e73d42d641e_1528449076687_0Core curriculum

VIII. Planimetrics. The student:

4) applies the properties of angles and diagonals in rectangles, parallelograms, rhombuses and trapezoids.

Timingm939b7e73d42d641e_1528449068082_0Timing

45 minutes

General objectivem939b7e73d42d641e_1528449523725_0General objective

Interpretation and the use of information presented both in a mathematical and popular science texts also using graphs, diagrams and tables.

Specific objectivesm939b7e73d42d641e_1528449552113_0Specific objectives

1. Communication in English, developing mathematical, IT and basic scientific and technical competence, developing learning skills.

2. Getting to know the concept and properties of diagonals in a polygonpolygonpolygon.

3. Getting to know the theorem about diagonals.

Learning outcomesm939b7e73d42d641e_1528450430307_0Learning outcomes

The student:

- gets to know the concept and properties of diagonals in a polygonpolygonpolygon,

- applies the theorem about diagonals.

Methodsm939b7e73d42d641e_1528449534267_0Methods

1. Open ear.

2. Situational analysis.

3. Problem discussion.

Forms of workm939b7e73d42d641e_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm939b7e73d42d641e_1528450127855_0Introduction

The students use the „open ear” technique to revise the information about types and properties of polygons.

Procedurem939b7e73d42d641e_1528446435040_0Procedure

The teacher informs the students that the aim of the class is getting to know the concept and properties of the diagonal in a polygonpolygonpolygon.

In a polygonpolygonpolygon, the students draw line segments, whose ends are the vertices of these polygons. They indicate the ones which are not the sides of the polygons. The teacher informs the students that these line segments are diagonals.

The definition

The diagonal of the polygon is the line segment joining two vertices and not being the side of the polygon.m939b7e73d42d641e_1527752263647_0The diagonal of the polygon is the line segment joining two vertices and not being the side of the polygon.

Discussion – does every polygonpolygonpolygon have the diagonals? What is the minimum number of vertices in a polygonpolygonpolygon to have the diagonals? The students formulate hypotheses, check them and formulate the conclusion.

The conclusion

The polygon with the smallest number of vertices which has the diagonals is a quadrangle.m939b7e73d42d641e_1527752256679_0The polygon with the smallest number of vertices which has the diagonals is a quadrangle.

The students cooperate to find out how many diagonals can be drawn from one vertex of a decagon, a 100‑gon, an n‑gon.

The conclusion

You can draw $(n-3)$ diagonals from one vertex of a polygonpolygonpolygon.

The students work in groups to find the number of diagonals in any polygonpolygonpolygon. The students formulate hypotheses and conclusions.

[Illustration interactive]

The formula that the students should derive.

The number of the diagonals in $(n\in N,n>3)$ is expressed with formula $\frac{n(n-3)}{2}$.

The students use the information to solve the tasks.

Task 1
Find the number of diagonals in a:

a. dodecagon (12 gon),

b. icosagon (20 gon),

c. hectogon (100 gon).

Answer: a)54, b) 170, c)4850.

Task 2
In which polygonpolygonpolygon the number of diagonals is:

a. the same as the number of sides,

b. five times larger than the number of sides,

c. fifteen times larger than the number of sides.

Asnwer: a) in a pentagon, b) in a 13 gon, c) in a 33 gon.

Task 3
The difference of the number of sides in two polygons equals 1, and the difference of the number of diagonals equals 12. What are these polygons?

Answer: a 13 gon and a 14 gon.

Task 4
25 people met at a party. How many welcomes were there if every person greeted everyone else.

Answer: 300.

Having finished all the tasks, the students present their results. The teacher assesses the students’ work and explains any doubts.

An extra task:
10 fishermen lived at the lakeshore. During a frosty winter a thick layer of ice covered the lake. The fishermen made footpaths so that the houses of any two fishermen were linked by a footpath. How many footpaths were there?

Answer: 45.

Lesson summarym939b7e73d42d641e_1528450119332_0Lesson summary

The students do the consolidation tasks. They formulate the conclusions that they need to remember.

- The diagonal of the polygondiagonal of the polygondiagonal of the polygon is the line segment joining two vertices and not being the side of the polygonpolygonpolygon.

- The polygonpolygonpolygon with the smallest number of vertices which has the diagonals is a quadrangle.

- The number of the diagonals in n –gon $(n\in N,n>3)$ is expressed with formula $\frac{n(n-3)}{2}$.

Selected words and expressions used in the lesson plan

diagonal of the polygondiagonal of the polygondiagonal of the polygon,

hexagonhexagonhexagon,

number of the diagonals in the polygonnumber of the diagonals in the polygonnumber of the diagonals in the polygon,

penatgonpenatgonpenatgon,

polygonpolygonpolygon,

vertex of the polygonvertex of the polygonvertex of the polygon