FINDING PROPERTIES OF PRISMS AND PYRAMIDS WHILE SOLVING STEREOMETRIC PROBLEMS

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FACULTY OF NATURAL S
CIENCES

CONSTANTINE THE P
HILOSOPHER UNIVERSIT
Y NITRA

ACTA MATHEMATICA 12





FINDING PROPERTIES OF PRISMS AND PYRAMIDS

WHILE SOLVING STEREOMETRIC PROBLEMS


J
OANNA
M
AJOR
,

Z
BIGNIEW
P
OWĄZKA

A
BSTRACT
.

In this article authors give some examples of stereometric tasks in which
afterthought on gained resolve leads to formulate mathematic
al theorem. These theorems are
often unknown to students and they exceed teaching curriculum.



1. Introduction

An important part of education in secondary schools is teaching the elements of flat
and spatial geometry with usage trigonometric functions. I
t creates possibility to practice
computational techniques and develope spatial imagination. There are known various d
i-
dactical works relevant to this topic.
(
vide

e.g. [1]).



Taking a look at stereometric task we mostly find tasks of designating area an
d volume
of the solid, often with usage values of angles between diagonals, edges or walls of the
solid.
The solution of these tasks is often adequate formula. However, it turns out that
discussion of conditions which make the task feasible leads to the pe
culiar conclusions
which describes interesting properties of these solids.

In this article authors give some examples of stereometric tasks in which afterthought
on gained resolve leads to formulate mathematical theorem. These theorems are often
unknown t
o students and they exceed teaching curriculum.

We deem that solving mentioned type of tasks is very important in mathematical ed
u-
cation because they create opportunity to develop mathematical creativity.
The discussion
on gained solutions pursues Polya’s
recommendation to “look back” (see [7]).

To start with, it is worth explaining the legitimacy of our using the word “find” in the
title of the paper. In the [8] we can read the following definitions of this term: “find” 1. to
trace or track down somebody

or something hidden, 2. to discover somebody or something
by searching, 3. to di
s
cover by chance, 4. to state that something exists.

Z. Krygowska mentions four situations in which solving an appropriately selected
problem leads to the formulation of a th
eorem. In the first situation, the st
u
dent solves a
problem presented to him or her a priori. The second case involves the student’s solving a
more open
-
ended problem; one that is clearly geared t
o
wards formulating and then proving
a new theorem. In the
third situation, it is the student who asks questions and tries to a
n-
swer them, either by himself or with the help of the teacher. Finally, it may well be that the
student himself formulates a theorem in the form a hypothesis, based either on his observ
a-
ti
ons of some regular properties in many cases, or on his intuitive conviction that such
connections exist. The question that the student asks at this point, is about the general “a
c-
curacy” of “what he has noticed” (see [2]).

The term “to discover” a theorem

is commonly used to denote a practice opposite to
simply presenting the student with a ready
-
made one (compare [5]). Throughout our p
a
per,
we will use this term to mean “finding” the properties of a mathematical object, as a result
JOANNA

MAJOR,

ZBIGNIEW

POWĄZKA





of a mathematical activ
ity geared towards solving the problem presented, followed by r
e-
flecting on the solution.

Tasks which are presented below do not allow on full realisation of situations pointed
by Krygowska. There are formulated theorems in proposed tasks, which are rel
evant to
some mathematical objects, as the result of reflection on gained s
o
lutions. The essential
fact is that mathematical commands contained in tasks are relevant to different object than
objects mentioned in theorems. For example, the task goes for de
s
ignate volume of cuboid,
while formulated theorem is relevant to dependence between values of angles which are
built by diagonal and edges of cuboid. A student who works with proposed tasks can't fo
r-
mulate adequate theorem
a priori

(before the task is reso
lved). In presented situations
proving of the theorem succeed before formulating it.

This form of mathematical activity requires from student to make advanced usage of
many concepts which could be asociate with discussed topic, in this case with prism or
pyramid. The proposal of this sort of tasks is very important element of mathematical
education because


as it is shown by our researches


operative using of concepts gives a
lot of difficulties to students (see [3] and [4]).


2. The propositions of task
s which lead to discovering theorems

Example 1.

Calculate the volume of the cuboid if there is known that diagonal which length amounts
d:

a)

builds with edges of this cuboid angles which amount is
α
,
β
,
γ
,
(
cf

fig. 1),

b)

builds with faces of this cuboid ang
les which amount is
α
,
β
,
γ
,

(
cf

fig. 2),

c)

builds with the base plane the angle
γ

and diagonals of spatial faces ou
t-
g
o
ing from the same vertex build with the base plane angles
α, β

(
cf

fig. 3).

Resolving of part a) of example 1 leads to discover theorem
1 and conclusion 1.

Theorem 1.

Let
a, b, c

be lengths of cuboid's edges and let
d

be the length of its diagonal.
We
designate as
α
,
β
,
γ
, amounts of angles which are built


by this diagonal and, in s
e-
quence, edges which lengths amount
a, b, c.
Then



Figure
1
.

FINDING PROPERTIES OF PRISMS AND PYRAMIMIDS…



As for the cuboid with edges
a
,
b
,
c

and diagonal
d

equation

is
satisfied, the consequence of theorem 1 is following conclusion.

Corollary 1.

If diagonal of the cuboid builds w
ith edges of this cuboid angles which amounts are
α
,
β
,
γ
,
then


Resolving of the part b) of example 1 leads to discover theorem 2 and conclusions 2 and 3.

Theorem 2.

Let
a, b, c

be length of cuboid's edges,
d

length of its diagonal and
α
,
β
,
γ
, amounts of
a
n-
gles which are built by diagonal and faces of cuboid. Then



Figure
2
.

In this case there is true the theorem analogous to the conclusion 1.

Corollary 2
.

If diagonal of cuboid builds with edges of this c
uboid angles which amounts are
α
,
β
,
γ
,
then


If we assumpt that the cuboid is inscribed into sphere which radius amounts
R
, then, d
i
a
g-
onal of this cuboid is the radius of the sphere. As consequence of theorem 2 we can r
e
mark
following fact
.

Corollary 3.

If cuboid which edges amount
a
,
b

and
c

is inscribed into sphere with radius
R,

then


where
α
,
β
,
γ

are amounts of angles which are built by radius of this sphere (connecting
two o
p
posite vertexes of the cuboid) and
adequate edges of this cuboid.

Forthcoming formula is analogous to law of sines on plane.

Resolving of part c) of example 1 leads to discover theorem 3.

JOANNA

MAJOR,

ZBIGNIEW

POWĄZKA





Theorem 3
.

Let

a, b, c

be lengths of cuboid’s edges, let
α
,
β

be amounts of angles which are built by
d
i
agonals of lateral faces and base’s plane and let
γ

be the amount of the angle created by
diagonal of the cuboid and d
i
agonal of the base. Then



Figure
3
.

This leads us to notice f
ollowing relation between angles from the theorem 3.

Corollary 4.

If


in the cuboid
α, β
are amounts of angles which are built by diagonals of lateral faces and
base’s plane and
γ

is the amount of the angle created by diagonal of the cuboid and diag
o-
nal o
f the base, then



Example 2.


Calculate the volume of tetragonal right regular prism in which diagonal of the base
amounts
d

and it builds angle
α

with lateral face’s diagonal which outgoing from the same
vertex.




Figure
4
.

FINDING PROPERTIES OF PRISMS AND PYRAMIMIDS…



We take designations as at fig. 4.


We have



.

It follows that

hence
. As the triangle
ACH
is acute
-
angled and isosceles simultaneously, thus

Therefore


Now then we obtain theorem 4.

Theorem 4.

At every right regular tetragonal prism amount of angle between lateral face’s diagonal and
base’s diagonal which outgoing from the same vertex belongs to bracket
.

Example 3.

At right regular tetragonal prism edge of the base amounts
a
. Calculate the volume of this
prism if there is known that amount of angle built by lateral edge and base’s edge outgoing
from the same vertex is
α
.


Figure
5
.

We take designations as at fig. 5.



The volume of prism amounts

.

The angle
α

is acute angle so we obtain that the task has solution if and only if

The consequesnce of the task solu
tion is theorem 5.

Theorem 5.

At every right regular tetragonal prism amount of acute angle at base of lateral face
belongs to bracket
.

The problems which are presented above could be extended and they should be treated
as patterns
of building several sets of tasks. The usage of this kind of examples requires
knowledge and skills relevant to properties of trigonometric functions and creates oppo
r-
t
u
nity to expand knowledge about amounts of angles in solids. These students who work
wit
h described tasks obtain the chance to self
-
reliant finding and formulating theorems.
JOANNA

MAJOR,

ZBIGNIEW

POWĄZKA





Lastly, it is worth to mark that solving of the task is in fact searching of undisclosed the
o-
rem and its proof.



Bibliography

[1.]

Kartasiński S.,
Nauczanie trygonometrii
,
Państwowe Zakłady Wyda
w
nictw
Szkolnych, Warszawa 1960.

[2.]

Krygowska Z.,


Zarys dydaktyki matematyki, cz. 3, WSiP, Warsz
a
wa 1977.

[3.]

Major, J.,
Rola zadań i problemów w kształtowaniu pojęć matematycznych na
przykładzie bezwzględnej wartości liczby rzeczywistej
,
Roczniki PTM, s
e
ria
V, Dydaktyka Matematyki 29
, 2006, 297
-

310.

[4.]

Major, J., Powązka, Z.,
Pewne problemy dydaktyczne związane z pojęciem
wartości bezwzględnej
,
Annales Academiae Pedagogice Cracoviensis St
u
dia
Ad Didacticam Mathematicae Pertinentia

I, Kraków

2006, 163
-

185.

[5.]

Nowak W.

, Konwersatorium z dydaktyki matematyki, PWN, Wa
r
szawa 1989.

[6.]

Nowosiłow S., Sp
ecjalny wykład trygonometrii,

Państwowe Wyda
w
nictwo
Naukowe, Warszawa 1956.

[7.]

Polya G., Jak to rozwiązać, Wydawnictwo Naukowe PWN, Wa
r
szawa 1993.

[8.]

Mały Sł
ownik Języka Polskiego
, red. Skorupka, S., Auderska, H., Łempi
c
ka,
Z., PWN, Warszawa 1969.

Joanna Major & Zbigniew Powązka

Pedagogical University of Cracow

Institute of Mathematics

Podchor
ążych 2

PL
-

30
-
084 Cracow

e
-
mail:
jmajor@up.krakow.pl


e
-
mail:
powazka@up.krakow.pl