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Poland, 24-28 april 2012
Comenius Project
2011- 2013
“The two highways of the life: maths and English”
About the meaning of angles
Introduction
All it takes is a quick glance
around a room, any room,
to find angles everywhere.
Every corner in the room
makes an angle.
Anywhere two lines come
together to form a point, an
angle is created.
θ
Introduction
In geometry, an angle is the figure formed by two rays
(or half-lines), called the sides of the angle, sharing a
common endpoint, called the vertex of the angle.
(http://en.wikipedia.org/wiki/Angle)
Euclid defines a plane angle as the inclination to each
other, in a plane, of two lines which meet each other,
and do not lie straight with respect to each other.
Eudemus regarded an angle as a
deviation from a straight line;
Carpus of Antioch, regarded it as the
interval of space between the
intersecting lines.
Euclid in Raphael's
“School of Athens”
Introduction
The angle is one of the first
concepts of geometry that
pupils study at school but
understanding what angles are
is not so simple as we can
think.
Talking
about
angles
is
important to understand that:
- the measure of an angle does
not vary with the length of
the sides;
- an angle is an open figure;
- the arc and the measure of
an
angle
are
differents
things.
Some questions…
Look at these angles:
-
which of these angles has
a greater amplitude?
α
β
The angle α. The amplitude of an angle is a measure of the inclination (or
rotation) of a ray respect to the other and it is not a consequence of the length of
the sides.
Look at the plotted points:
-
which of these points are
drawn inside the angle?
-
which of these points are
drawn outside the angle?
.B
.
A
δ
.C
Points A and C are plotted inside the angle. Point B is plotted outside. The part of
the plane between the two sides of an angle is extended infinitely because the
sides of the angle are two half-lines.
Playing with a cake
Julia made a cake for her friends and
everyone, except Mark, has already
eaten a slice.
Can you calculate how many people
are invited?
First you need to measure the angle
of a slice.
30°
Then you divide 360° (the
whole pie) by the degrees that
correspond to a slice.
360°/30° = 12 friends
30°
Playing with a clock
Look at these clocks. Only one between A and B works
correctly, which one?
A
B
The clock B. You can see that in the clock B, while the minute hand marks half
an hour, the hour hand arrives in the middle between 4 and 5, and this is
correct. The rotation speed of the hands, forming different angles respect to
the initial position of 12 o'clock, varies in the case of the minute hand and the
hour hand.
Playing with a clock
Consider different positions that the clock hands can have and the
angles that they form. How many degrees do they measure?
90°
30°
180°
120°
270°
360°
210°
330°
Playing with a clock
Which of these angles are acute? Obtuse? Concave? Convex?
Right? Straight? Turn?
right
straight
Acute convex Obtuse convex
concave
concave
turn
concave
Playing with a clock
How many degrees is the angle that the minute hand forms
in a minute?
6°
Playing with a clock
According to this, what time is associated with the angle of 1°?
A minute divided by 6 is 10 seconds.
But if the rotation speed of the second hand is equal to that
of the minute hand we would not be able to see the time
because the angle of one degree is very small!
In order to distinguish hours,
minutes and seconds in a clock
it is necessary that the different
hands have increasing rotation
speed relative to each other.
According to this, the minute
hand has a rotation speed of 1°
every 10 seconds and the
second hand has a rotation
speed of 60° every 10 seconds.
θ
Relations between the measure of
an angle and the length of the sides
An angle of one degree is very small ... which objects may be
contained in it?
a microbe?
a bug?
a tennis ball?
a house model?
a real house?
a whole planet?
Create an angle of 1° with a goniometer and a wool thread…
θ
Relations between the measure of
an angle and the length of the sides
Because the sides of an angle are two rays, the
part of the plane between them never ends.
Within an angle of one degree
can also be a whole planet, but it
is necessary get far enough
away from the vertex!
The arcs that we can draw within the same angle are infinite
in number and their length grows going away from the
vertex.
Relations between the measure of
an angle and the length of the arc
How long must the radius of a circle be so that the arc of an
angle of 1° is 1 m long?
C = 2πr
The arc of an angle of 360° is the whole circumference. We need to calculate the
lenght of the radius of a circumference where an arc of 1 m correspond to an
angle of 1°, so the circumference measures 360 m.
r = C/2π = 360 m/2*3.14 ≅ 57.4 m
δ = 1°
Arc length = 1m
Radius length = 57,32 m
As a consequence, an
object,
observed
by a
distance of about 57.4
times its diameter, has an
angular size of about 1°.
Relations between the measure of
an angle and the length of the arc
Accidentally, the angular diameter of the sun from the earth
is equal to that of the moon.
Sun is about 400 times farther than moon from the earth but
its diameter is also 400 times bigger. This fact implies that in
the sky they appear to have similar dimensions.
This particular coincidence makes possible exciting eclipses
of the sun.
Conclusions
Angles are a difficult subject to teach because they contain
the infinite. Since we usually teach them as imprisoned in a
spreadsheet, it is difficult for a student to reach a complete
comprehension about this matter.
Maybe it could be better to
let angles go out from the
sheet, out from the school,
far away, opening our
imaginations.
References


Anichini G, Arzarello F., Bartolotti C., Ciarrapico L., Robutti O., 2004. “La
matematica per il cittadino”. MIUR, UMI, SIS. Link: http://umi.dm.unibo.it/scuola-99.html
Arzarello F., Ciarrapico L., Camizzi L., Mosa E., 2006. “Progetto [email protected] –
Matematica. Apprendimenti di base con e-learning”. Documento di base 4 aprile 2006.
MIUR.

Battisti R., Spinelli F, Brunelli F. “L’orologio”. Attività matabel inserita nella sezione
GEOMETRIA del PON 2007-2013 (http://www.indire.it/ponmatematicacorso1/)

AAVV, 2011. “Matematica: didattica, esperienze e tecnologie”. RCS Libri Education
S.p.A. http://www.educationduepuntozero.it/speciali/pdf/specialemarzo11.pdf
Web sites:
http://en.wikipedia.org
http://www.indire.it/ponmatematicacorso1/
http://library.thinkquest.org/2647/geometry/glossary.htm