Topicm68b3edf20ee970db_1528449000663_0Topic

ConeconeCone cross‑sections

Levelm68b3edf20ee970db_1528449084556_0Level

Third

Core curriculumm68b3edf20ee970db_1528449076687_0Core curriculum

X. Stereometry. The student:

4) identifies the angle between line segments and the angle between line segments and planes (for example the angle of opening of a coneconecone, the angle between the lateral and the base), calculates these angles;

6) calculates volumes and surface areas of prisms, pyramids, cones, spheres, also using trigonometry and learnt theorems.

Timingm68b3edf20ee970db_1528449068082_0Timing

45 minutes

General objectivem68b3edf20ee970db_1528449523725_0General objective

Interpreting and operating information presented in the text, both mathematical and popular science texts, as well as in the form of graphs, diagrams, tables.

Specific objectivesm68b3edf20ee970db_1528449552113_0Specific objectives

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

2. Obtaining cones and their properties.

3. Calculating elements of the coneconecone using sections.

Learning outcomesm68b3edf20ee970db_1528450430307_0Learning outcomes

The student:

- obtains cones and their properties,

- calculates elements of the coneconecone using sections.

Methodsm68b3edf20ee970db_1528449534267_0Methods

1. Diamond classification.

2. Situational analysis.

Forms of workm68b3edf20ee970db_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm68b3edf20ee970db_1528450127855_0Introduction

Students work in groups and use diamond classification to order previously learnt information about the coneconecone. After having finished the work, they present their posters.

Procedurem68b3edf20ee970db_1528446435040_0Procedure

The teacher introduces the subject of the lesson – learning about conic cross‑sections and calculating elements of the coneconecone using cross‑sections.

Task
Students work in groups and analyse material presented in the slideshow. They pay attention to figures they obtain depending on the angle of inclination of the section of a coneconecone with a plane non‑parallel to the cone’s axis. They make theories and draw conclusions.

[Slideshow]

Conclusions:

If the angle of inclination of the plane of the section to the axis of the cones is:

$0°<\alpha <45°$ then hyperbolahyperbolahyperbola,

$\alpha =45°$ - parabolaparabolaparabola,

$45°<\alpha <90°$ - ellipseellipseellipse,

$\alpha =45°$ - circlecirclecircle,

$90°<\alpha <135°$ - ellipseellipseellipse,

$\alpha =135°$ - parabola,

$135°<\alpha <180°$ - hyperbolahyperbolahyperbola,

$\alpha =180°$ - two intersecting lines.

The teacher says that figures that are parts of the cutting plane and the lateral surface of a coneconecone are conic sectionsconic sectionsconic sections.

Task
Look for information about conic sections in available sources. Write proper definitions.

Students should find following information.

There are following conic sectionsconic sectionsconic sections, depending on the angle made by the cutting plane with the coneconecone axis and angle between the lateral and the cone axis.

[Illustration 1]

Definition
An ellipse – is obtained when the angle between the cutting plane and the axis of the cone is greater than the angle between the lateral and the cone axis.m68b3edf20ee970db_1527752263647_0An ellipse – is obtained when the angle between the cutting plane and the axis of the cone is greater than the angle between the lateral and the cone axis.

Definition
A circlecirclecircle – is obtained when the cutting plane is perpendicular to the cone axis.

Definition
A parabolaparabolaparabola – is obtained when the cutting plane is parallel to the lateral.

Definition
A hyperbola – is obtained when the angle between the cutting plane and the cone axis is smaller than the angle between the cone axis and its lateral, or when the cutting plane is parallel to the cone axis but does not contain this axis.m68b3edf20ee970db_1527752256679_0A hyperbola – is obtained when the angle between the cutting plane and the cone axis is smaller than the angle between the cone axis and its lateral, or when the cutting plane is parallel to the cone axis but does not contain this axis.

Students use obtained information and do exercises on their own.

Task
Draw the axial section of a coneconecone whose angle of opening is 160° and the lateral is 8 cm. Calculate the radius of the base.

Answer:
$r\approx 7,9$ cm.

Task
The cross‑section of a cone is a circlecirclecircle whose radius is smaller than the radius of the cone’s base. Calculate the area of the cross‑section, knowing that the height of the coneconecone is equal to 18 cm and the diameter of the base is 20 cm. The cone was cut with a plane parallel to the base, 9 cm from the apex of the cone.

Answer:
$25·\mathrm{\pi }$ cmIndeks górny 2².

Task
The axial section of a coneconecone is an isosceles triangle in which the angle between arms is 80°. The arm of this triangle is 12 cm long. Calculate the height and the radius of the base of the cone.

Answer:
$H\approx 9,2$ cm, $r\approx 7,7$ cm..

Task
The angle between the lateral and the height of a cone is 75°. The radius of the base of the cone is equal to 7 cm. Calculate the height of the coneconecone.

Answer:
Around 1,9 cm.

Task
Calculate the angle of opening of a cone whose lateral is equal to $6\sqrt{2}$ cm and the height is 5 cm.

Answer:
107°40’.

An extra task
Calculate the area of the axial section of a coneconecone whose height is 20 cm and $\mathrm{tg} \alpha =4$ ( $\alpha$ - angle between the cone’s lateral and its radius).

Answer:
100 cmIndeks górny 2².

Lesson summarym68b3edf20ee970db_1528450119332_0Lesson summary

Students do the revision exercises.

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

A figure that is the cross‑section of the cone depends on the angle of inclination of the plane of the section to the cone axis:
- If $\alpha =0 \mathrm{or} \alpha =180°$, then we obtain two intersecting lines.
- If $\alpha \in (0°,45°) \mathrm{or} \alpha \in (135°,180°)$, then we obtain a hyperbola.
- If $\alpha \in (45°,90°) \mathrm{or} \alpha \in (90°,135°)$, then we obtain an ellipse.
- If $\alpha =45° \mathrm{or} \alpha =135°$,then we obtain a parabola.
- If $\alpha =90°$,then we obtain a circle.m68b3edf20ee970db_1527712094602_0A figure that is the cross‑section of the cone depends on the angle of inclination of the plane of the section to the cone axis:
- If $\alpha =0 \mathrm{or} \alpha =180°$, then we obtain two intersecting lines.
- If $\alpha \in (0°,45°) \mathrm{or} \alpha \in (135°,180°)$, then we obtain a hyperbola.
- If $\alpha \in (45°,90°) \mathrm{or} \alpha \in (90°,135°)$, then we obtain an ellipse.
- If $\alpha =45° \mathrm{or} \alpha =135°$,then we obtain a parabola.
- If $\alpha =90°$,then we obtain a circle.

Selected words and expressions used in the lesson plan

circlecirclecircle

coneconecone

cone’s axial sectioncone’s axial sectioncone’s axial section

cone’s cross‑sectioncone’s cross‑sectioncone’s cross‑section

conic sectionsconic sectionsconic sections

ellipseellipseellipse

hyperbolahyperbolahyperbola

parabolaparabolaparabola