Topicmcf71f7eb27429e35_1528449000663_0Topic

Solids of revolution

Levelmcf71f7eb27429e35_1528449084556_0Level

Third

Core curriculummcf71f7eb27429e35_1528449076687_0Core curriculum

X. Solid Geometry (Stereometry). The student:

6) calculates the volume and surface area of prisms, pyramids, cylinder, cone and sphere, also with the use of trigonometry and theorems learned.

Timingmcf71f7eb27429e35_1528449068082_0Timing

45 minutes

General objectivemcf71f7eb27429e35_1528449523725_0General objective

Applying mathematical objects and manipulating them, interpreting mathematical concepts.

Specific objectivesmcf71f7eb27429e35_1528449552113_0Specific objectives

1. Drawing of solids of revolution.

2. Identification and drawing of axial and cross sections of solids of revolution.

3. Communicating in English, developing competences in mathematics, science, technology and IT, developing learning skills.

Learning outcomesmcf71f7eb27429e35_1528450430307_0Learning outcomes

1. Student draws solids of revolution.

2. Student identifies and draws axial and perpendicular cross sections of solids of revolution.

Methodsmcf71f7eb27429e35_1528449534267_0Methods

1. Discussion.

2. Questions to the expert.

Forms of workmcf71f7eb27429e35_1528449514617_0Forms of work

1. Group work.

2. Whole class.

Lesson stages

Introductionmcf71f7eb27429e35_1528450127855_0Introduction

The teacher asks three students to prepare at home the information about drawing solids of revolution and their sections. Their task is also to get acquainted with the information concerning the most important properties of solids of revolution.

The teacher informs students that in the lesson they will draw solids of revolution. They will also draw and recognize the axial and cross sections of these solids.

Proceduremcf71f7eb27429e35_1528446435040_0Procedure

The teacher divides the class into 3 groups. Each of them works under the guidance of an expert. An expert is a student who prepared the material indicated by the teacher at home.

Task
Students use computers. Their task is to observe the method of creating solids of revolution. Then, each expert will familiarize his/her group with the information obtained. They can illustrate the information learnt at home with a presentation or support it with the internet resources. Students ask questions, exploring the topic of the lesson.

[Geogebra applet]

The summary of this part of the lesson may be creating a poster for each group containing the most important information to remember.

Conclusions:

The data is: the straight line p and the 2D figure F, lying in the same plane.

- By rotating the figure F around the line p, we get a surface that limits the figure, called the solid of revolution.mcf71f7eb27429e35_1527752263647_0By rotating the figure F around the line p, we get a surface that limits the figure, called the solid of revolution.

- The straight line p is the axis of revolution. It is the axis of symmetry of the solid of revolution.mcf71f7eb27429e35_1527752256679_0The straight line p is the axis of revolution. It is the axis of symmetry of the solid of revolution.

- A solid section is a figure that will be created after slicing a solid with any plane.

- The axial section of a solid of revolutiona solid of revolutiona solid of revolution is the common part of this solid with a plane containing the axis of revolution.

- The cross section of a solid of revolutiona solid of revolutiona solid of revolution is the common part of this solid with a plane perpendicular to the axis of revolution.

Students use acquired knowledge to solve word problems.

Task 
Draw a solid that will be created as a result of revolution of:

a) a rectangle around a line containing the shorter side of the rectangle,

b) a right‑angled triangle around a line that includes a longer leg.

Task 
The figure shows the axial and  cross sections of the solid.

[illlustration 1]

[illustrationi 2]

What geometrical figures are the cross sections of the selected solids?

Task
Draw a solid of revolutiona solid of revolutiona solid of revolution, in which the axial section is an equilateral triangle.

Task
Draw a solid of revolutiona solid of revolutiona solid of revolution, which will be created as a result of the revolution of the rectangular trapezoid around the shorter arm.

Task
A cylinder was created as a result of the revolution of the rectangle around one of its sides. The diagonal of the rectangle, whose length is 3, is inclined to the side at an angle of 45Indeks górny o^o.
Calculate the height of the cylinder. Provide all possible solutions.

An extra task:
A right‑angled triangle with a hypotenuse length of $3\sqrt{3}$ rotates around the side. Calculate the area of the axial section of the resulting solid.

Lesson summarymcf71f7eb27429e35_1528450119332_0Lesson summary

Students perform additional exercises.

Then they summarize the lesson together, formulating conclusions to remember.

The data is: the straight line p and the 2D figure F, lying in the same plane.

- By rotating the figure F around the line p, we get a surface that limits the figure, called the solid of revolution.

- The straight line p is the axis of revolution. It is the axis of symmetry of the solid of revolution.

- A solid section is a figure that will be created after slicing a solid with any plane.

- The axial section of a solid of revolutiona solid of revolutiona solid of revolution is the common part of this solid with a plane containing the axis of revolution.

- The cross section of a solid of revolutiona solid of revolutiona solid of revolution is the common part of this solid with a plane perpendicular to the axis of revolution.

Selected words and expressions used in the lesson plan

an axial cross sectionan axial cross sectionan axial cross section

an axis of revolutionan axis of revolutionan axis of revolution

a cross sectiona cross sectiona cross section

a section of a solid of revolutiona section of a solid of revolutiona section of a solid of revolution

a solid of revolutiona solid of revolutiona solid of revolution