Topicm19a47152979a5f4e_1528449000663_0Topic

Polygons and their properties II

Levelm19a47152979a5f4e_1528449084556_0Level

Second

Core curriculumm19a47152979a5f4e_1528449076687_0Core curriculum

IX. Polygons. The student:

2) uses the formulas to calculate the areaareaarea of a triangle, rectanglerectanglerectangle, squaresquaresquare, parallelogramparallelogramparallelogram, rhombus, trapezoid and is able to determine the lengths of line segments in tasks of comparable difficulty:

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

b) The diagonals of a rhombusrhombusrhombus 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 diagonaldiagonaldiagonal. Calculate the area of the triangle CDE (there are two possible answers).

Timingm19a47152979a5f4e_1528449068082_0Timing

45 minutes

General objectivem19a47152979a5f4e_1528449523725_0General objective

Noticing regularities, similarities and analogies and formulating relevant conclusions.

Specific objectivesm19a47152979a5f4e_1528449552113_0Specific objectives

1. Recognising, drawing and using the properties of parallelograms.

2. Determining the areaareaarea and perimeterperimeterperimeter of a parallelogramparallelogramparallelogram.

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

Learning outcomesm19a47152979a5f4e_1528450430307_0Learning outcomes

The student:

- recognises, draw and uses the properties of parallelograms, also in English,

- determines the areaareaarea and perimeterperimeterperimeter of a parallelogramparallelogramparallelogram.

Methodsm19a47152979a5f4e_1528449534267_0Methods

1. Practical exercises.

2.Brainstorming.

Forms of workm19a47152979a5f4e_1528449514617_0Forms of work

1.Individual work.

2.Group work.

Lesson stages

Introductionm19a47152979a5f4e_1528450127855_0Introduction

The teacher informs the students that during this class they will learn to recognise, draw and use the properties of parallelograms. They will also revise the formulas for the areaareaarea and the perimeterperimeterperimeter of parallelograms.

Task
The teacher asks the students:

- How do we calculate the perimeter of a polygon?

- Calculate the perimeterperimeterperimeter of a rectanglerectanglerectangle whose sides are 4 cm and 5 cm.

The students are reminded that the perimeter of a polygon is the sum the lengths of all its sides.

Procedurem19a47152979a5f4e_1528446435040_0Procedure

The students give the definition.

Definition

- A tetragon that has two pairs of parallel sides is called a parallelogram.

[Illustration 1]

The students revise the following definition.

Definition

- If all sides in a parallelogram are equal, the parallelogram is called a rhombus.m19a47152979a5f4e_1527752256679_0- If all sides in a parallelogram are equal, the parallelogram is called a rhombus.

Properties of parallelograms.

Task
Students work individually, using computers. Their task is to change the location of the apexes of the parallelogramparallelogramparallelogram and observe how the lengths of its diagonals change and at what angles the diagonals cross.

[Geogebra applet]

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

- Can you set the location of apex S in such a way that the diagonals are of the same length and the sides are different?

- Can you set the location of apex S in such a way that the sides are of the same length and the diagonals are different?

- Can you set the location of apex S in such a way that the angleangleangle at which the diagonals cross is the right angle and the diagonals are of the same length?

- What is point O for each diagonaldiagonaldiagonal of the parallelogram?

Task
Fill in the blanks.

- Diagonals in a parallelogramparallelogramparallelogram which is not a rectangle are of ......... length.

- If a parallelogram is a rectangle, then its diagonals are of ......... length.

- In a parallelogram whose adjacent sides have different lengths the diagonals ......... perpendicular.

Task
The teacher divides the class into groups of 4‑5 students. Each group must come up with as many properties of parallelograms as possible. The teacher reminds the students not to forget the properties of the rhombusrhombusrhombus, the rectanglerectanglerectangle and the squaresquaresquare.

Students provide the properties:
- The diagonals of a parallelogram bisect each other.
- Each diagonal of a parallelogram separates it into two congruent triangles.
- The point of intersection of the diagonals determines the centre of symmetry of the parallelogram.
- Opposite angles are congruent.
- The sum of adjacent angles is 180°.
- If all sides of a parallelogram are equal, the parallelogram is called a rhombus.
- If all angles in a parallelogram are equal, the parallelogram is called a rectangle.
- If a parallelogram has equal sides and angles, the parallelogram is called a square.m19a47152979a5f4e_1527712094602_0- The diagonals of a parallelogram bisect each other.
- Each diagonal of a parallelogram separates it into two congruent triangles.
- The point of intersection of the diagonals determines the centre of symmetry of the parallelogram.
- Opposite angles are congruent.
- The sum of adjacent angles is 180°.
- If all sides of a parallelogram are equal, the parallelogram is called a rhombus.
- If all angles in a parallelogram are equal, the parallelogram is called a rectangle.
- If a parallelogram has equal sides and angles, the parallelogram is called a square.

The teacher asks the students to provide the formulas for the areaareaarea of a parallelogram.

Theorem

- The area of a parallelogram is equal to the product of the base and the altitude perpendicular to this base.m19a47152979a5f4e_1527752263647_0- The area of a parallelogram is equal to the product of the base and the altitude perpendicular to this base.

[Illustration 2]

The area of a square and the areaareaarea of a rhombusrhombusrhombus can be calculated by using their diagonals.

Theorem

- The area of a squaresquaresquare is equal to half of the square of its diagonaldiagonaldiagonal. The areaareaarea of a rhombus is equal to half of the product of its diagonals.

[Illustration 3]

Task
Calculate the area of a parallelogramparallelogramparallelogram whose side is equal to 5 cm and the altitudealtitudealtitude perpendicular to the side is 4 cm.

Task
The sum of the lengths of the diagonals of the rhombusrhombusrhombus is 15 cm. One diagonaldiagonaldiagonal is twice as long as the other. Calculate the areaareaarea of the rhombus.

An extra task:
The ratio of two sides of a rectanglerectanglerectangle is 2 : 5. Calculate the area of the rectangle, knowing that its perimeterperimeterperimeter is equal to 42 cmIndeks górny 2².

Lesson summarym19a47152979a5f4e_1528450119332_0Lesson summary

Students do the revision exercises.

Then together they sum‑up the classes, by formulating the conclusions to remember.

- The perimeterperimeterperimeter of a polygon is the sum of the lengths of all its sides.

- A tetragon that has two pairs of parallel sides is called a parallelogram.

- If all sides in a parallelogramparallelogramparallelogram are equal, the parallelogram is called a rhombusrhombusrhombus.

Properties of a parallelogram:

- The diagonals of a parallelogram bisect each other.

- Each diagonaldiagonaldiagonal of a parallelogram separates it into two congruent triangles.

- The point of intersection of the diagonals determines the centre of symmetry of the parallelogramparallelogramparallelogram.

- Opposite angles are congruent.

- The sum of adjacent angles is 180°.

- If all sides of a parallelogram are equal, the parallelogramparallelogramparallelogram is called a rhombus.

- If all angles in a parallelogram are equal, the parallelogram is called a rectanglerectanglerectangle.

- If a parallelogram has equal sides and angles, the parallelogram is called a squaresquaresquare.

- The area of a parallelogramparallelogramparallelogram is equal to the product of the base and the altitudealtitudealtitude perpendicular to this basebasebase.

- The areaareaarea of a square and the area of a rhombusrhombusrhombus can be calculated by using the diagonals of these figures.

- The area of a squaresquaresquare is equal to half the square of the diagonaldiagonaldiagonal. The areaareaarea of a rhombus is equal to half of the product of its diagonals.

Selected words and expressions used in the lesson plan

altitudealtitudealtitude

angleangleangle

areaareaarea

basebasebase

diagonaldiagonaldiagonal

parallelogramparallelogramparallelogram

perimeterperimeterperimeter

rectanglerectanglerectangle

rhombusrhombusrhombus

squaresquaresquare