maths course exercises

Liceo Scientifico Isaac Newton - Roma

solids of revolution

in accordo con il

Ministero dell’Istruzione, Università, Ricerca e sulla base delle

Politiche Linguistiche della Commissione Europea

percorso formativo a carattere tematico-linguistico-didattico-metodologico

scuola secondaria di secondo grado

professor Tiziana De Santis

2

solid of revolution

Indice Modulo

Strategies - Before

• Prerequisites

• Linking to Previous Knowledge and Predicting con questionari basati su stimoli relativi alle conoscenze pregresse e alle ipotesi riguardanti i contenuti da affrontare

• Italian/English Glossary

Strategies – During

• Video con scheda grafica • Keywords riferite al video attraverso esercitazioni mirate • Conceptual Map

Strategies - After

• Esercizi: � Multiple Choice � Matching

� True or False � Cloze o Completion � Flow Chart

� Think and Discuss

• Summary per abstract e/o esercizi orali o scritti basati su un questionario e per esercizi quali traduzione e/o dettato

• Web References di approfondimento come input interattivi per test orali e scritti e per esercitazioni basate sul Problem Solving

Answer Sheets

3

solid of revolution

1

Strategies Before Prerequisites

Geometric

transformations

Solids of revolution

Plane

geometry Geometry in

space

Basic concept of Euclidean

geometry

Pythagoras’ theorem and

Euclid’s theorem

Areas of polygons

Barycentre of a curve

Barycentre of a plane figure

Straight-lines, planes

and angle in space

Surface and volume

Cavalieri’s principle

Barycentre of a solid

Central symmetry

Axial symmetry

Orthogonal symmetry

4

solid of revolution

2

Strategies Before

Linking to Previous Knowledge and Predicting

• Do you know the conditions of perpendicularity and parallelism between two

straight-lines in the plane?

• Do you know the conditions of perpendicularity and parallelism between two

straight-lines in space?

• Are you able to calculate the area and the perimeter of a plane figure?

• Are you familiar with the concept of central symmetry?

• Are you familiar with the concept of axial symmetry?

• Are you familiar with the concept of orthogonal symmetry?

• Do you know the concept of surface area of a solid?

• Do you know the concept of volume of a solid?

• Do you know Cavalieri’s principle?

• Do you know how to find the barycentre of a triangle, a rectangle and a

circle?

5

solid of revolution

3

Strategies Before

Italian/English Glossary

altezza height

angolo angle

anticlessidra anti-clepsydra

apotema apothem

asse axis

baricentro (centroide) barycentre (geometric centroid)

base base

cateto cathetus (pl. catheti)

cerchio (semicerchio) circle (half-circle)

cilindro (indefinito) cylinder (infinite)

circonferenza (semicirconferenza) circumference (half-circumference)

circumscrittibile circumscribable

cono (indefinito) cone (infinite)

corona circolare annulus (pl annuli)

diagonale diagonal

diametro diameter

equilatero equilateral

esterno external

inscritto inscribed

lato side

parallelo parallel

perpendicolare perpendicular

piano plane

quadrato square

6

solid of revolution

raggio radius (pl. radii)

retta straight line

rettangolo rectangle

retto right

rotazione revolution (rotation)

scodella bowl

secante secant

sfera (semisfera) sphere (half-sphere)

simmetria symmetry

solido solid

Superficie surface

tangente tangent

toro torus

trapezio trapezium

triangolo triangle

tronco di cono truncated cone

vertice vertex (pl vertices)

volume volume

7

solid of revolution

4

Strategies During

Keywords

1) Circle the solids of revolution:

pyramid – sphere – right prism – torus– octahedron – circumference –

truncated cone – annulus - cylinder – cone

2) Completion:

• The volume of a sphere is equivalent to that of the ……………………….

• The surface area of a ………..is equivalent to that of the cylinder

circumscribes it.

• The …….of the cylinder are obtained from the complete rotation of the

radii of the base.

• A cone is called ……….. if its apothem is congruent to the diameter of the

base.

• The ……………………………. is also obtained from the rotation of a right

triangle around one of its catheti.

________________________________________________________________

sphere, equilateral, anti-clepsydra, right circular cone, bases

8

solid of revolution

5

Strategies During

Conceptual Map

Complete the conceptual map using the following words:

volume

Revolution

solids

Obtained by rotating

rectangle right triangle half-circle circle

lateral surface

volume

V= π r2h

lateral surface

volume

V=(ππππ r2h)/3

S =ππππ r √√√√ (h2+ Position of Straigth-

line/plane

external

surface

S =2 π r h

V= 4πr3/3

equivalent

cone sphere cylinder anti-clepsydra

secant tangent S=4 π r2 torus

9

solid of revolution

6

Strategies After

Multiple Choice

1) A plane intersects a sphere; the polygon which represents the section is always

a. a rectangle b. a square c. a circle d. none of these

2) The cone can be obtained from a complete rotation of a. a square b. a triangle c. a trapezium rectangle d. none of these

3) A truncated cone can be obtained from a complete rotation of

a. a square b. a triangle c. a trapezium rectangle d. none of these

4) Indicate which of the following statement is correct:

a. the sphere is equivalent to 1/3 of the cylinder circumscribed b. the cone inscribed in a cylinder whit base radium 2r and height 2r is

equivalent to Galileo’s bowl c. the cone is equivalent to the anti-clepsydra d. none of these

5) The surface area of a sphere measures S=16π, the surface area of the

circumscribed cylinder is:

a. Greater than 16π b. Less than 16π c. 16π d. none of these

10

solid of revolution

6) The surface area of a sphere measures 3π, its volume is:

a. (π√3)/3 b. 4π/3 c. (π√3)/2 d. 4√3 π/3

7) The volume of anti-clepsydra is equivalent inscribed in a cylinder equilater of

height 2r is:

a. π r3 b. (π r3)/3 c. 2π2

r2

d. 4πr3/3 8) Galileo’s bowl is equivalent to:

a. half-sphere b. cone inscribed c. cylinder d. none of these

11

solid of revolution

7

Strategies After

Matching

Match the words on the right with the correct definition on the left:

1. Cone

2. Galileo’s bowl

3. Anti-clepsydra

4. Cylinder

5. Sphere

6. Torus

a. Solid generated by rotating

the circle

b. Solid generated by rotating

the half-circle

c. Complementary double cone

solid circumscribed in a

cylinder

d. Cylinder minus the inscribed

half-sphere

e. Solid generated by rotating

the rectangle

f. Solid generated by rotating

the triangle rectangle

12

solid of revolution

8

Strategies After

True or False

State if the sentences are true or false.

1. The points that do not belong to the surface of the cylinder and having

distance from the axis smaller than the radius are internal to the surface.

2. The surface area of the cylinder is equivalent to that of the inscribed sphere.

3. A right circular cylinder can be obtained from the rotation of a rectangle

triangle around one of its catheti.

4. The intersection figures of two planes perpendicular to the rotation axis of an

indefinite cylinder are two congruent circles.

5. In a circular right cone the plane of the base is perpendicular to the

generatrix of the cone.

6. The apotheme of the cone is any segment having as extremes the vertex of

the cone and one of the points of the circumference of base.

7. An equilateral cone sectioned by a plane passing through the axis of the

cone is an equilateral triangle.

8. The height of a truncated cone is the distance between the two bases.

9. A plane α is secant a spherical surface S if it has a segment in common with

the surface S.

10. A plane tangent to a sphere S has a circumference in common with the

solid.

13

solid of revolution

9

Strategies After

Cloze

Complete the text.

The solids of revolution are generated by the rotation of a…….[1] around a

straight line. In particular:

the cone is generated by the complete rotation of a …. [2] around one of its

catheti, the sphere is generated by the complete rotation of a …[3] around its …

[4], the …. [5] is generated by the complete rotation of a rectangle around one

of its sides.

The surface area of the ….[6] is equivalent to the surface area of the cylinder

that is circumscribed it

The volume of the sphere is equivalent to …. [7] 2/3 of the cylinder’s volume

that is …. [8]

The volume of the cylinder having radius r and height 2r is the sum of the

volume of the …..[9] having radius r and the one of the …. [10] having base

radius r and height 2r

The …. [11] sphere volume is equivalent to that of the anti-clepsydra.

14

solid of revolution

10

Strategies After

Flow Chart

Complete the flow chart referring to the position of a plane in relation to

a spheric surface. You can use the terms listed below: secant- tangent-

external

false

start

points in

common

true

Sphere,

plane input

false

One

point

true

output

end

output

15

solid of revolution

11

Strategies After

Think and Discuss

The following activity can be performed in a written or oral form. The teacher

will choose the modality, depending on the ability (writing or speaking) that

needs to be developed.

The contexts in which the task will be presented to the students are:

A) the student is writing an article about solids of revolution

B) the student is preparing for an interview on a local TV about solids of

revolution

The student should:

1) Choose one of the following topics:

• The parts of the sphere • “On the Sphere and Cylinder” by Archimedes • Theorems of Pappus and Guldin

2) Prepare an article or a debate, outlining the main points of the argument, on

the basis of what has been studied.

3) If the written activity is the modality chosen by the teacher, the student

should provide a written article, indicating the target of readers to whom the

article is addressed and the type of magazine / newspaper / school magazine

where the article would be published.

4) If the oral activity is the modality chosen by the teacher, the student should

present his point of view on the topics to the whole class and a debate could

start at the end of his presentation.

16

solid of revolution

12

Strategies After

Summary

The solids of revolution are generated by the rotation of a plane figure around a

straight line. In particular:

• The cylinder is generated by the complete rotation of a rectangle around

one of its sides;

• The cone is generated by the complete rotation of a right triangle around

one of its catheti;

• The sphere is generated by the complete rotation of a half-circle around its

diameter;

• The torus is generated by rotating a circle around an external coplanar

straight line.

Pappus-Guldin's theorems make it possible to determine the surface area

and volume of solids of revolution. The first theorem states that:

The measure of the area of the surface generated by the rotation of an arc of a

curve around an axis, is equal to the product between the length l of the arc and

the measure of the circumference described by its geometric centroid: S=2 π dl

Thus, for the cylinder and cone the following lateral surfaces are obtained

respectively: Scone = π r √( h2+ r2) Scyl = 2 π r h.

For the sphere and torus the total areas are: Storus=4 π2rR Ssphere=4 π r2.

The second theorem states that:

The volume of a solid of revolution generated by rotating a plane figure F

around an external axis is equal to the product of the area A of F and the

length of the circumference of radius d equal to the distance between the axis

and the geometric centroid: V = 2 π d A

The following formulas are thus obtained:

Vcone=(π r2h)/3 Vcyl = π r2h Vtorus = 2π2r2R Vsphere = 4πr3/3

Archimedes' "On the Sphere and Cylinder" contains significant results achieved

by the mathematician of Syracuse on the solid rotation, as, for example:

• The surface area of the sphere is equivalent to the surface area of the

cylinder that is circumscribed it

• The volume of the sphere is equivalent to 2/3 of the cylinder’s volume that

is circumscribed it

17

solid of revolution

• The volume of the cylinder having radius r and height 2r is the sum of the

volume of the sphere having radius r and the one of the cone having base

radius r and height 2r

Using Cavalieri's principle it can be shown that:

The volume of Galileo’s bowl is equivalent to the volume of the cone inscribed in

the same cylinder, as the cylinder volume is given by the sum of the total

volume of the half-sphere and the bowl, the volume of the half-sphere is given

by the difference between the volume of the cylinder and the volume of the

cone.

The sphere volume is equivalent to that of the anti-clepsydra.

1) Answer to the following questions. The questions could be answered

in a written or oral form, depending on the teacher’s objectives.

a. How do you obtain a solid of revolution?

b. What is the difference between a solid and solid surface?

c. How do you obtain a right cylinder and right cone

d. Which symmetries does a sphere have?

e. What is the relationship between the surface of the cylinder and that of a

sphere?

f. What is the relationship between the volume of the cylinder and that of a

sphere?

g. Illustrate the first theorem of Pappus-Guldin.

h. Illustrate the second theorem of Pappus-Guldin.

2) Write a short abstract of the summary (max 150 words) highlighting

the main points of the video.

18

solid of revolution

Web References

http://www.mathwords.com

An interactive math dictionary with many math words, math terms, math

formulas, pictures, diagrams, tables, and examples

http://mathworld.wolfram.com

Encyclopedia of mathematics

http://www.britannica.com/EBchecked/topic/428841/On-the-Sphere-

and-Cylinder.

http://en.wikipedia.org/wiki/Archimedes

19

solid of revolution

13

Activities Based on Problem Solving

a) A sphere whose surface is 100 π cm2 is cut by a plane far from the center of

the 3 / 5 of its radius. Determine the relationship between the lateral areas

of the two cones having the circle as common base for the top section and as

vertices the extremes of the diameter perpendicular to the secant plane.

b) The base radius of a right cylinder is 6 cm and height is 9 cm. Determine a

point on the axis V such that the ratio of 4 volumes of two cones having as

bases the bases of the cylinder, and as vertex the point V.

c) In a right circular cylinder the lateral surface is equivalent to 4 / 7 of the

total. Knowing that the total height of the cylinder is 12 cm determine the

volume of the sphere which has radius congruent to half the radius of the

cylinder base.

d) In the Discourses and Mathematical Demonstrations Concerning Two New

Sciences, Galileo Galilei describes the construction of a solid that is

called a bowl considering a hemisphere of radius r and the cylinder

circumscribed on it. The bowl is achieved by removing the hemisphere

from the cylinder. Prove, using Cavalieri’s principle, that the

bowl has volume equal to the cone inscribed in the cylinder.

(Esame di Stato 2009 liceo scientifico sperimentale - PNI)

e) Prove that the proportion of the total area of an equilateral cylinder to the

surface of the circumscribed sphere is 3 to 4.

(Esame di Stato 2004 liceo scientifico sperimentale - PNI)

f) Demonstrate the equivalence between the volume of the sphere inscribed in

a cylinder and the volume of the anti-clepsydra.

20

solid of revolution

Answer Sheets

Keywords:

1) sphere – torus– truncated cone – cylinder – cone

2) anti-clepsydra, sphere, bases, equilateral, right circular cone

Conceptual Map:

volume

Revolution

solids

cylinder sphere cone torus

Obtained by rotating

rectangle rigth triangle half-circle circle

lateral surface

volume

V= π r2h

lateral surface

volume

V=(ππππ r2h)/3

S =ππππ r √√√√ (h2+ Position of Straigth-

line/plane

tangent

external

surface

S=4 π r2

secant

S =2 π r h

V= 4πr3/3

equivalent

anti-clepsydra

21

solid of revolution

Multiple Choice:

1C, 2B, 3C, 4B, 5C, 6C, 7D, 8B

Matching:

1F, 2D, 3C, 4E, 5B, 6A

True or False:

T, T, F, T, F, T, T, T, F, F

Cloze:

[1] plane figure [2] right triangle [3] half-circle [4] diameter [5] cylinder

[6] sphere [7] 2/3 [8] circumscribed it [9] sphere [10] cone

[11] sphere

22

solid of revolution

Flow Chart

false

start

points in

common true

external

Sphere,

plane input

false

One

point

tangent

true

output

end

secant

23

solid of revolution

Activities Based on Problem Solving

a. 2

b. 36/5 cm or 9/5 cm

c. 243 π /2 cm3

Materiale sviluppato da eniscuola nell’ambito del protocollo d’intesa con il MIUR