Lesson plan

Topicm501e607564c9c594_1528449000663_0Topic

A circular sector

Levelm501e607564c9c594_1528449084556_0Level

Third

Core curriculumm501e607564c9c594_1528449076687_0Core curriculum

VIII. Planimetry. The student:

6) uses the formulas for the area of a circular sector and the length of the arc.

Timingm501e607564c9c594_1528449068082_0Timing

45 minutes

General objectivem501e607564c9c594_1528449523725_0General objective

Creating mathematical objects based on existing ones, in order to help to make a reasoning or solve a task.

Specific objectivesm501e607564c9c594_1528449552113_0Specific objectives

1. Calculating the area of a circular sector and the length of its arc.

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

Learning outcomesm501e607564c9c594_1528450430307_0Learning outcomes

The student:

- calculates the area of a circular sector and the length of its arc.

Methodsm501e607564c9c594_1528449534267_0Methods

1. Discussion.

2. Brainstorming.

Forms of workm501e607564c9c594_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm501e607564c9c594_1528450127855_0Introduction

The teacher introduces the subject of the lesson - calculating the area of a circular sector and the length of its arc.

Procedurem501e607564c9c594_1528446435040_0Procedure

Discussion – which part of the circlecirclecircle is the circular sectorcircular sectorcircular sector?

Students formulate the definition of the circular sector.

Definition

- The circular sector is a part of a circle enclosed by two arms of the interior angle and an arc.m501e607564c9c594_1527752263647_0- The circular sector is a part of a circle enclosed by two arms of the interior angle and an arc.

[Illustration 1]

Brainstorming.

Students are divided into groups and discuss ways of calculating the area of a circular sector, defined by the interior angle. Together they notice that the circular sector is the same part of the circle as the interior angle is of the full angle

Conclusion:

- The area of the circular sectorcircular sectorcircular sector defined in a circlecirclecircle whose ray is r and the interior angle is α is equal to:

P_{w} = \frac{α}{360^{\circ}} \cdot π \cdot r^{2}

- The length of the arclength of the arclength of the arc $L_{w}$ defined in a circle whose ray is r and the interior angle is α is equal to:

L_{w} = \frac{α}{360^{\circ}} \cdot 2 π \cdot r

Students calculate the area of the circular sectorarea of the circular sectorarea of the circular sector and the length of the arclength of the arclength of the arc of the circumference whose ray is 3 cm, defined by the interior circlecirclecircle equal to 60°.

First they notice that the angle 60° is $\frac{60 °}{360 °} = \frac{1}{6}$ of the full angle.

Therefore:

P_{w} = \frac{1}{6} π \cdot r^{2}

P_{w} = \frac{1}{6} π \cdot 3^{2}

P_{w} = \frac{3}{2} π

The area of the circular sectorarea of the circular sectorarea of the circular sector is equal to $\frac{2}{3} π .$

L_{w} = \frac{α}{360^{\circ}} \cdot 2 π \cdot r

L_{w} = \frac{1}{6} \cdot 2 π \cdot 3

L_{w} = π

The length of the arclength of the arclength of the arc is equal to $π$ .

Task
Students work individually, using computer. Their task is to observe the triangular lunula, that is an object in the shape of the moon. Students compare the area of the moon and the area of the triangle.

[Geogebra aplet]

Based on the previous description, students make the construction of the triangular lunula on their own and calculate its area. Then they compare the area of the moon with the area of a right‑angled, isosceles triangle.

Students use obtained information in the exercises.

Task
Calculate the area of the circular sectorcircular sectorcircular sector defined by the interior angle 45°.

Task

[Illustration 2]

A circle whose centre is point S and radius is 3 was divided into 6 equal parts. Calculate the area of the coloured part.m501e607564c9c594_1527752256679_0A circle whose centre is point S and radius is 3 was divided into 6 equal parts. Calculate the area of the coloured part.

Task
Calculate the area of the circlecirclecircle knowing that the circular sector defined by the interior angle 60° has the area equal to $4 π .$

Task
In the circle whose radiusradiusradius is 6 cm was marked an arc defined by the interior angle 330°. Calculate the length of the arclength of the arclength of the arc.

Task
Draw a circumference whose ray is 5 cm. Mark $\frac{1}{3}$ of the length of the arc.

An extra task:
Calculate the radiusradiusradius of the circlecirclecircle in which the interior angle of 120° corresponds to the circular sector whose area is $9 π .$

Lesson summarym501e607564c9c594_1528450119332_0Lesson summary

Students do the revision exercises.

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

Information to remember:

- The circular sectorcircular sectorcircular sector is a part of a circle enclosed by two arms of the interior angle and an arc.

- The area of the circular sectorarea of the circular sectorarea of the circular sector defined in a circle whose ray is r and the interior angle is α is equal to:

P_{w} = \frac{α}{360^{\circ}} \cdot π \cdot r^{2}

- The length of the arc $L_{w}$ defined in a circle whose ray is r and the interior angle is α is equal to:

L_{w} = \frac{α}{360^{\circ}} \cdot 2 π \cdot r

Selected words and expressions used in the lesson plan

area of the circular sectorarea of the circular sectorarea of the circular sector

circlecirclecircle

circular sectorcircular sectorcircular sector

diameterdiameterdiameter

length of the arclength of the arclength of the arc

number $π$ number $π$ number $π$

radiusradiusradius

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circle1

circular sector1

length of the arc1

area of the circular sector1

radius1

diameter1

number

π

number

π

liczba $π$

RFTkKBaxGrUp41

Source: GroMar, licencja: CC BY 3.0.

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Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: number

Scenariusz

Topicm501e607564c9c594_1528449000663_0Topic

Lesson stages

Selected words and expressions used in the lesson plan