maths course exercises

Liceo Scientifico Isaac Newton – Roma

polyhedra

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

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polyhedra exercises

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

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polyhedra exercises

1

Strategies Before

Prerequisites

Geometric

transformations

Polyhedra

Plane

geometry Geometry in

space

Trigonometry

Basic concept of

Euclidean

geometry

Pythagoras’

theorem and

Euclid’s theorem

Areas of

polygons

Straight-

lines,

planes and

angle in

space

Central symmetry

Axial symmetry

Orthogonal symmetry

The first and the

second theorem

of trigonometry

for right-angle triangles

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polyhedra exercises

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?

• Do you know the theorem of three perpendiculars?

• 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?

• What is a dihedral angle?

• What is a solid angle?

• What is the measure of the sum of the inner angles of a polygon having n

sides?

• What is the maximum value of the sum of the faces of a solid angle?

• Do you know the first and the second theorem of trigonometry for right-

angle triangles?

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polyhedra exercises

3

Strategies During

Italian / English Glossary

altezza Height

ampiezza Width

angolo angle

angolo giro round angle

angolo piatto straight angle

angolo retto right-angle

angoloide solid angle

apotema apothem

base base

circonferenza circumference

circumscrittibile circumscribable

convesso convex

diagonale diagonal

diedro dihedral angle

dodecaedro dodecahedron

duale dual

esaedro (cubo) hexahedron (cube)

faccia face

icosaedro icosahedron

ottaedro octahedron

pentagono pentagon

piano plane

piramide pyramid

poliedro polyhedron ( pl polyhedra)

poligono polygon

prisma prism

quadrato square

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polyhedra exercises

4

retta straight line

solido solid

spigolo edge

superficie surface

sviluppo piano development plane

tetraedro tetrahedron

triangolo triangle

vertice vertex (pl vertices)

volume volume

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polyhedra exercises

Strategies During

Keywords

1) Circle which of the following solids are platonic:

pyramid – sphere – tetrahedron – cube – prism right – octahedron –

parallelepiped – icosahedron - cylinder – cone - dodecahedron

2) Circle the odd one out

edge – apothem – height – vertex – diagonal – face – polyhedron convex –

prism – function – Euler’s relation – vector –dihedral angle – solid angle

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polyhedra exercises

5

Strategies During

Conceptual Map

Complete the conceptual map using the following words:

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polyhedra

Platonics

Prism

Dual

Dual

Pyramid

base polygon

circumscribes the

circle

dual

base regular

polygon

tetrahedron right parallelogram dodecahedron

hexahedron icosxahedron regular

Side faces

octahedron

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polyhedra exercises

Strategies After Multiple Choice

1) A plane intersects a parallelepiped; the polygon which represents the section

is always

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

2) A prism is right if:

a. two consecutive side faces form a right dihedral angle b. the height is perpendicular to the planes of the base c. the side faces form right dihedral angle whit the planes of the base d. none of these

3) Which of the following polyhedron have no triangular faces:

a. octahedron b. dodecahedron c. tetrahedron d. icosahedron

4) Consider the extremes of the edges of a cube converging in a vertex; the

triangle which has as vertices these three points is:

a. a right-angled triangle b. an isosceles triangle c. an equilateral triangle d. none of these

5) A pyramid has a rectangle as base and one of its lateral edge is perpendicular

to the base. The lateral faces are:

a. all right-angled triangles b. all equilateral triangles c. only two are right-angled triangles d. all are isosceles triangles

6) Correct the sentences where necessary:

a. the regular tetrahedron has four vertices b. the regular hexahedron has six faces c. in the icosahedron the sum of the faces of each solid angle is equal to 300

degrees d. in the dodecahedron the sum of the faces of each solid angle is equal to

330 degrees

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polyhedra exercises

7) Correct the sentences where necessary:

a. The sections of a prism made by two parallel planes are congruent b. If we intersect a pyramid by a plane parallel to the base and not passing

through the vertex, the section is congruent to the base c. In a right pyramid, its base polygon circumscribes a circle and the base

point of the height is the centre of the circle d. The side faces of a pyramid are triangles, having as vertices, the vertex of

a pyramid and two consecutive vertices of the base polygon

8) Consider a pyramid of unknown height and a 12 x 12 meter square base. If

the height is increased by 2 meters, the lateral surface area is increased by

24 square meters. How high is the original pyramid? (The lateral surface

does not include the pyramid’s base)

a. 2,5 meters b. 4,5 meters

c. 4

53 meters

d. 5,625 meters e. None of these

(USA North Carolina State High School Mathematics Contest Finals 2004)

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polyhedra exercises

7

Strategies After Matching

Match the polyhedron on the left with the correct tern (n° faces, n°

edges, n° vertex) on the right:

1) Hexahedron a) (20, 30, 12)

2) Dodecahedron b) (4, 6, 4)

3) Icosahedron c) (12, 30, 20)

4) Tetrahedron d) (8, 12, 6)

5) Octahedron e) (6, 12, 8)

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

1) Line segments connecting the

vertices that do not belong to

the same face

2) Tetrahedron

3) Octahedron

4) Hexahedron

5) Part of solid angle delimited

by a plane

6) Icosahedron

7) Polyhedron bounded by two

bases congruent and parallel

a) Fire

b) Prism

c) Pyramid

d) Water

e) Earth

f) Diagonal

g) Air

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polyhedra exercises

8

Strategies After True or False

State if the sentences are true or false.

1. The sum of the amplitudes of the faces of an solid angle is less than straight-

angle.

2. A pyramid is called regular if its base is a regular polygon.

3. The apothem of a right pyramid is the height of every side face.

4. A prism is right if the side faces are perpendicular to the planes of the bases.

5. A polyhedron is regular if its faces are regular and congruent polygons.

6. The diagonals of a polyhedron are the line segments connecting the opposite

vertices of a same face.

7. The surface area of a pyramid, having triangular base, is equal to the sum of

the areas of five triangles.

8. A pyramid is equivalent to the third part of a prism having the same base

and the same height.

9. The centres of the faces of a cube are the vertices of a regular octahedron.

10. According to the Euler’s theorem, in a polyhedron, the number of the edges

minus the number of the vertices is equal to the number of the faces plus

two.

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polyhedra exercises

9

Strategies After

Cloze

Complete the text.

A … [1] is the part of space bounded by n polygons belonging to different planes that have two by two a ….[2] in common.

The polygons are the … [3], the sides of the polygons are the …[4], the vertices of the polygon are the ….[5] of the polyhedron.

A polyhedron is ….[6] if its faces are regular and congruent ….[7] and its dihedral angles and solid angles are also ….[8].

There are only….[9] regular polyhedra .

A prism is a polyhedron bounded two congruent …[10] placed on ….[11].

The …… [12] is a solid angle with vertex V and delimited by a plane α not passing through V.

The total area of a polyhedron is equal to the …[13] of the areas of the polygons that are its …[14].

Polyhedra having the same volume are called ….[15].

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polyhedra exercises

10

Strategies After

Flow Chart

Complete the flow chart referring to prisms. You can use the terms

listed below: right prism - regular prism - oblique prism.

false

start

Side edges

perpendicu

lar to the

bases

true

Prism input

false

The base is

a regular

polygon

true

output

output

end

output

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polyhedra exercises

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 polyhedra;

B) the student is preparing for an interview on a local TV about polyhedra.

The student should:

1) Choose one of the following topics:

• Mastering an archimedean polyhedra or Catalan’s polyhedra • Euler's relation and connection to planar graph • Crystals and polyhedra • Polyhedra in the history of art • Polyhedra in the Renaissance

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.

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polyhedra exercises

12

Strategies After

Summary

A convex polyhedron is the part of space bounded by n polygons belonging to

different planes that have two by two a side in common. The polygons are the

faces, the sides of the polygons are the edges, the vertices of the polygon are

the vertices of the polyhedron.

Every polyhedron satisfies Euler’s relation, according to which, the sum of the

number of the faces and the number of the vertices, minus the number of the

edges is equal to two.

A polyhedron is regular if its faces are regular and congruent polygons and its

dihedral angles and solid angles are also congruent.

There are only five regular polyhedra, because the number of the faces

converging in one vertex must be equal to three and the sum of the angles

going out from one vertex must be less than the round angle.

They are:

• Tetrahedron, having four faces, that are equilateral triangles;

• Octahedron, having eight faces, that are equilateral triangles;

• Hexahedron, having six faces, that are equilateral squares;

• Dodecahedron, having twelve faces, that are pentagons;

• Icosahedron, having twenty faces, that are equilateral triangles.

Two polyhedra are called dual if the number of the faces, of the vertices and of

the edges of the first polyhedron are equal to the number of the vertices, of the

faces and of the edges of the second.

A prism is a polyhedron bounded two congruent bases placed on parallel planes;

its side faces are parallelograms and the distance between the planes is the

height of the prism. If the side faces are perpendicular to the planes of the

bases, the prism is called a regular prism. A parallelepiped is a prism having six

parallelograms as faces.

The pyramid as a solid angle with vertex V and delimited by a plane α not

passing through V.

It is called right if its base polygon circumscribes a circle and the base of its

height coincides with the centre of the circle. A pyramid is called regular if its is

right and the base polygon is a regular polygon. The apothem is the height of

one of its faces.

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polyhedra exercises

The total area of a polyhedron is equal to the sum of the areas of the polygons

that are its faces.

The polyhedra, having the same volume, are called equivalent.

According to the Cavalieri’s Principle, if parallel planes intersect two solids so

that each plane defines an equivalent section, then the two solids are equivalent,

that is the volumes of the two solids are equal.

The volume of a prism is the product of the area of its base and its height.

The volume of a pyramid is the third part of the volume of a prism having the

same base and the same height.

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

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

a. What is the definition of a convex polyhedron?

b. Talk about Euler’s relation.

c. What is meant by dual polyhedra?

d. What are the symmetries of the tetrahedron?

e. What is the definition of prism?

f. What is the definition of pyramid?

g. Illustrate by examples Cavalieri’s Principle.

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

the main points of the video.

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polyhedra exercises

Web References

The history of polyhedra and polyhedra in art

http://www.georgehart.com/virtual-polyhedra/art.html

Virtual Polyhedra-The Encyclopedia of Polyhedra By George W. Hart

http://www.georgehart.com/virtual-polyhedra/vp.html

The site deals with ‘Symmetry, Crystals and Polyhedra’ di Steven Dutch, Natural

and Applied Sciences, University of Wisconsin – Green Bay

http://www.uwgb.edu/dutchs/symmetry/symmetry.htm

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

formulas, pictures, diagrams, tables, and examples

http://www.mathwords.com

Encyclopedia of mathematics

http://mathworld.wolfram.com

Paper models of polyhedra

http://www.korthalsaltes.com/

Encyclopedia of polyhedra containing tables of the characteristics of the principal

polyhedra

http://polyhedra.mathmos.net/

A video clip investigating planar graphs and Euler's relationship

http://www.waldomaths.com/wmv/Planar01v.jsp

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polyhedra exercises

13

Activities Based on Problem Solving

a. Calculate the total area of the five platonic solids.

b. The centres of the faces of a cube are the vertices of a octahedron. Is it regular? What is the quotient between the volumes of two solids?

(da Esame di Stato Liceo Scientifico, 2006)

c. The regular polyhedra are called platonic solids and are five. Are you able to demonstrate it?

(da Esame di Stato Liceo Scientifico, 2006)

d. The height of a square regular pyramid is double the edge of the base. Calculate the quotient between the volume of the cube inscribed in the pyramid and the volume of the same pyramid.

(da Esame di Stato Liceo Scientifico, suppletiva del 2006)

e. Calcolate the volume of a regular octahedron, knowing that the length of its edge is ‘s’.

(da Esame di Stato all’estero 2002)

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polyhedra exercises

Answer Sheets

Keywords:

1) tetrahedron – cube – octahedron – icosahedron - dodecahedron

2) function, vector

Conceptual map:

Multiple Choice:

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

Matching:

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

1F,2A,3G,4E,5C,6D,7B

polyhedra

Platonics

Dodecahedron

Icosxahedron

Hexahedron

Prism

octahedron Dual

Tetrhahedron

Dual

Pyramid

base polygon

circumscribes the

circle

right

dual

parallelogram

base regular

polygon

regular

Side faces

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polyhedra exercises

True or False:

1 false, 2 false, 3 true, 4 true, 5 true, 6 false, 7 false, 8 true, 9 true, 10 true.

Cloze:

[1] convex polyhedron [2] side [3]faces [4] edges [5]vertices [6] regular

[7] polygons [8] congruent [9] five [10] base [11] parallel planes [12] pyramid

[13] sum [14] faces [15] equivalent

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polyhedra exercises

Flow Chart:

Problem Solving:

b) 1/6 d) 4/9 e) V=s3√2/3

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

start

Side edges

perpendicu

lar to the

bases

true

Oblique

prism

Prism input

false

Right prism

The base is

a regular

polygon

Regular

prism

true

output

output

end

output