Download Angles

FD 1
LT 90
FD 1
RT 90
FD 3
RT 90
FD 1
LT 90
FD 1
Angles
North
000°
North
000°
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B
180°
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W
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E
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O
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45°
45°
A
The bearing of A from B is 225°
Angles
© 2000 Andrew Harris
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© 2000 Andrew Harris
Contents
Defining Angle
3
Different Conceptual Structures for Angle
3
3
3
Static Experiences
Dynamic Experiences
Progression in Learning about Angle
1. Pre-Measurement Experiences
4
4
2. Non-Standard Units (NNS Years 1 & 2)
(a) Whole and Fractional Turns
(b) Introducing right angles (NNS Years 2 & 3)
5
5
6
3. Standard Units (NNS Year 4)
Introducing Degrees
Application in Work on Direction
Early Measurement of Angles
Application in Work on Shape
9
9
10
10
10
5. Measuring using Standard Devices
Classification of Angles
Application to Work in 2D Shape
Rotation of Shapes
The Sum of the Interior Angles of a Triangle
The Sum of the Angles in Any Quadrilateral
A Simple Clinometer
11
11
12
12
14
14
15
6. Extension Material and Additional Subject Knowledge
More Kinds of Angles
Bearings
The Sum of the Exterior Angles of a Polygon
The Sum of the Interior Angles of a Polygon
Angle in a Semicircle
Constructions of Angles with a Ruler and Pair of Compasses
16
16
16
17
18
19
19
Common Misconceptions and Errors : Angles
21
Resources
23
Angles
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© 2000 Andrew Harris
Defining Angle
Angle is a measure of turn or rotation. The units of angle are called degrees and denoted by the
symbol °.
Different Conceptual Structures for Angle
Static Experiences
These focus on the difference in direction between two lines.
This view of angles allows us to:
• compare,
• order
and
• measure
the sizes of angles with ease, either directly by superimposing one angle on top of the other
or indirectly using a protractor.
For example, this angle
is smaller than this one
.
However, there are disadvantages as well in viewing angles in this way:
• the notion of angle as a measure of rotation is not visually apparent
(though it is possible, and desirable, to encourage children to imagine facing in the
direction defined by one of the lines and turning to face in the direction defined by
the second line);
• this view of angles is limited to angles less than or equal to 360°;
• misconceptions about angles may occur (see page 21).
Dynamic Experiences
These focus on the actual rotation in moving from one orientation to another.
This view of angles:
• gives a sense of the movement by which the angle is created (i.e. is dynamic).
• accommodates angles of any size (i.e. including those greater than 360°)
• allows a distinction between clockwise and anti-clockwise rotations.
However, this view of angles also has some disadvantages:
• the angles described by a dynamic movement are transient;
• the angles produced are less easily compared, ordered and measured.
Because the dynamic view of angles does emphasise angle as a measure of rotation, it is
this aspect of angle work which is introduced to children first.
Angles
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© 2000 Andrew Harris
Progression in Learning about Angle
1. Pre-Measurement Experiences
(NNS: Reception)
As with all measures, children’s initial experiences of angle should be practical and be
rooted in familiar, everyday situations. At the pre-measurement stage, children should be
allowed to explore ideas related to angle through unstructured and structured play. Part of
this ‘learning through play’ should be the development of understanding and knowledge of
relevant mathematical vocabulary.
This learning includes understanding of position and positional/directional vocabulary (in
front, behind, to the side, beside, underneath, below, above, on top etc. ). Many of these can
be developed through P.E. as well as in mathematics. Initially, such positions should be in
relation to the child but, when this is secure, positional relationships between two objects
(other than the child) may be explored as well (refer to the National Numeracy Strategy’s
‘Framework for Teaching Mathematics’ (1999), Section 4, page 27).
Building an understanding of ‘left’ and ‘right’ is also important. Many children take a long
time to distinguish consistently between these terms. Other useful, movement-related
vocabulary includes such terms as ‘along’, ‘around’, ‘over’, ‘under’, ‘through’, ‘across’.
Many of these can be explored through rhymes and stories. Refer to the National Numeracy
Strategy’s ‘Framework for Teaching Mathematics’ (1999), Section 4, page 27 for further
details of typical activities at this stage. It is important to note that many of these terms are
relative. For example, two people facing each other will have different viewpoints regarding
what constitutes a move to the left or the right. This aspect of positional and movementrelated vocabulary is very confusing for the child. A full understanding of left and right
which enables the child to determine left or right directions from a perspective other than
the child’s own usually takes several years to acquire.
(NNS Year 1)
One possible approach to introducing the idea of turning as a form of movement might be
through cross-curricular topics. Collecting objects which turn could form part of interactive
classroom displays or as a play corner feature. Children should be encouraged to discuss
different types of movement exhibited by different ‘real’ objects.
Angles
Objects using Rotation
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© 2000 Andrew Harris
Toys and play equipment often form rich sources of objects that exhibit rotational
movements:
Typical activities:
• Making movements in P.E. in response to instructions;
• Constructing and making things which rotate such as windmills, vehicles, clocks;
• Sorting both real objects (e.g. toys) and mathematical objects (such as shapes)
according to types of movements (rolling, sliding etc.).
These activities use the dynamic aspect of angle.
2. Non-Standard Units (NNS Years 1 & 2)
(a) Whole and Fractional Turns
For angles, the whole turn serves as a non-standard unit. A whole turn is defined to
children as turning on the spot until you are facing the same direction as when you
started. To progress further, children will need to draw upon knowledge of simple
fractions (halves and quarters) so that the whole turn can be subdivided into half-turns
(NNS: in Year 1) and quarter-turns (NNS: in Year 2).
Initially, the notion of whole and fractional turns is best taught by making the child the
object which rotates. Holding an arm outstretched (which rotates with the child) to
emphasise the direction the child is currently facing provides the child with a visual
reinforcement. P.E. activities can be devised that emphasise this idea of whole- and partturns. Half turns can also be associated with turning to face the opposite direction. In a
similar way, multiples of whole turns can be introduced.
Fractions of a turn can be related to movements of the hands on a clock e.g. discussing
the moving of the minute hand halfway round the clock and its relevance to the term
‘half-past’.
Additional vocabulary can be introduced, namely ‘clockwise’ and ‘anti-clockwise’, to
define the direction of a turn. It helps to ask children to imagine standing in the middle
of a clock placed on the floor and thus turn in the direction the hands of the clock would
turn. Children can then ‘discover’ the notion of equivalent rotations:
¼ turn clockwise is equivalent to ¾ turn anti-clockwise
½ turn clockwise is equivalent to ½ turn anti-clockwise
¾ turn clockwise is equivalent to ¼ turn anti-clockwise.
Typical activities:
•
Angles
Asking children to turn on the spot in multiples of 90° (both clockwise and
anti-clockwise) to face one of four objects set at 90° intervals around the
child.
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© 2000 Andrew Harris
•
•
•
Asking the child to predict, before turning, which object they will be facing
after the rotation is completed adds a further mental visualization
requirement to the task.
A still harder task is that of asking children to predict the direction a friend
will end up facing after a rotation (harder because the child making the
prediction is not at the centre of the rotation so has to mentally envisage the
rotation from another perspective.
In oral work, the children may use two strips of card or two geostrips
fastened with a paper fastener to show the teacher the result of a whole, halfor quarter-turn. A further step would be to show the result of a rotation which
is more or less than a whole, half- or quarter-turn or a rotation which is more
than a quarter-turn but less than a half-turn, for example.
The activities above emphasise the dynamic aspect of angle.
(b) Introducing right angles (NNS Years 2 & 3)
An intermediate stage between using whole and fractional turns and the use of standard
units for angle (degrees) is that of measuring with right angles. Initially these can be
introduced to children as ‘square corners’ i.e. like the corners of a square. Children can
make a right angle with which to measure as follows:
Fold any torn piece of paper to get a straight edge:
Figure 1
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© 2000 Andrew Harris
Then, fold the paper in two again to get a shape which has a right-angle in one corner:
Figure 2
90°
Children can use the ‘right angle measurer’ shown in Figure 2 to compare with ‘real’
angles found within the classroom (e.g. on the corners of books, tables, windows etc.)
and with angles encountered in mathematical apparatus (e.g. 2D shapes made on
geoboards or on the surfaces of 3D shapes).
The ‘angle measurer’ can be opened out as in Figure 1 to measure angles equivalent to
two right angles.
By unfolding the paper fully to show all folds, cutting up one fold to the crossing of the
folds and folding one part of the paper behind (as in Figure 3 below), children can use
their angle measurer to measure angles equal to three right angles.
Figure 3
Typical ‘static’ experiences of angles include:
• sorting objects into those with some right angles and those without;
• comparing angles found in the classroom with the angle measurer;
• identifying angles less than a right angle and those greater than a right angle
(and similarly with less than and greater than 2 right angles or 3 right angles).
Typical ‘dynamic’ experiences of angles include:
• using a geostrip angle indicator to show multiples of a right angle and extend
this to show angles more or less than a multiple of a right angle;
• setting, for the children, the unit angle size of a Roamer to 90° by entering
and then using the Roamer as a rotating device.
CM
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2
P
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GO
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© 2000 Andrew Harris
Thus, using a right angle as a unit of turn, a clockwise turn of 1 right angle is
produced using the instructions
, an anti-clockwise turn of 2 right
angles is produced by entering
and so on.
Children can be asked for the instructions to enter for Roamer to end up facing
a given direction (e.g. facing the doll) or asked to predict which way it will
end up facing for a given set of instruction. Similar activities can be done with
other programmable floor robots.
• Similar activities using compass points North, South, East and West (instead
of the toys depicted above) can be attempted with a Roamer or similar
programmable floor robot.
Typical activities using dynamic aspects of angle combined with linear movement:
• make a child into a floor robot with other children giving instructions (made
up of rotations of multiples of a right angle and of forwards or backwards
linear movements) to move around a course;
• ‘random walk’ games:
Use a octagonal prism die marked in clockwise and anticlockwise rotations of
multiples of right angles (see Resources section) and either squared paper and
counters marked with a directional arrow or a square floor grid with children
as the counters. The counters or children start in the centre of the grid and the
die is rolled. Children/Counters turn the number of right angles specified by
the die and move along the grid lines to the next intersection or node in the
new direction they are now facing. Players take it in turns. The winner is the
first to reach the edge of the grid. For a more speedy version, players move
two intersections at a time instead of one.
• maze-type activities:
The child is asked to write the instructions to move through the maze. The
maze could be on squared paper or a floor grid or simply an arrangement of
suitable obstacles on the classroom floor. The child could be asked to walk
through the maze or to program a floor robot to traverse it, for example.
Angles
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© 2000 Andrew Harris
LT
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P
R
2
[]
3
W
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CE
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CM
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RT
GO
Similar activities can be done with Logo packages on a computer producing
trails like this:
FD 1
LT 90
FD 1
RT 90
FD 3
RT 90
FD 1
LT 90
FD 1
3. Standard Units (NNS Year 4)
The standard unit for angle is the degree and is denoted by °.
Introducing Degrees
Children can be introduced to degrees by a trial and improvement ‘discovery’ method
using a floor robot such as Roamer. The initial aim is to establish that a full turn is 360°
and then subdivde this.
Draw round the Roamer on a piece of paper to obtain a circle. Place
a sticky label on the Roamer's rim and make a mark on the circle
adjacent to the sticky label to show the starting point for turning.
LT
RT
Children take it in turns to program the Roamer to turn on the spot
entering different numbers for the size of the rotation in the
instruction sequence
CM
GO
1
2
3
4
P
R
[]
W
5
CE
6
7
8
9
0
S
T
C
until by 'trial and improvement' they arrive at 360° being equivalent to a whole turn.
In a similar way, half- and quarter-turns can be deduced to be 180° and 90° respectively
if the paper circle is folded in half or into quarters before placing the Roamer onto it.
Children then program Roamer to turn between the starting mark and the half- or
quarter-fold. Since children will know that quarter-turns and half-turns are equivalent to
1 right angle and 2 right angles respectively, they can thus recognise that any right angle
is the same as 90°.
Angles
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© 2000 Andrew Harris
In a similar way, half- and quarter-turns can be deduced to be 180° and 90° respectively
if the paper circle is folded in half or into quarters before placing the Roamer onto it.
Children then program Roamer to turn between the starting mark and the half- or
quarter-fold. Since children will know that quarter-turns and half-turns are equivalent to
1 right angle and 2 right angles respectively, they can thus recognise that any right angle
is the same as 90°.
[For more able children the activity can be extended into estimating less common angles
by programming Roamer to turn between the starting mark and other marks distributed
around the edge of the circle or by finding families of angles (some greater than 360°)
which result in Roamer facing the same direction e.g. 90°, 450°, 810° .]
Application in Work on Direction
Having established a right angle as being 90°, it is reasonably simple to introduce the
idea of half a right angle as being 45°. This can then be used in connection with the
standard eight compass points N, NW, W, SW, S, SE, E and NE where the angle
between each pair of compass points is a multiple of 45°. This has applications in
geography when learning about wind directions or in simple map work, for example.
Early Measurement of Angles
A set of angle cards (see Resources section) can be used as a precursor to formal use of a
protractor:
10°
etc.
20°
30°
Angles found in the classroom or encountered in work on shape or direction can be thus
measured to the nearest 10° by comparing the angles with the angle cards. In the same
way, it is also possible to begin to order angles in terms of their size.
Application in Work on Shape
Children should be aware already of the existence of right angles in common shapes
such as rectangles and squares. This can be extended to other sizes of angles, e.g. three
60° angles in an equilareral triangle. In so doing, children will be expanding their
understanding of the properties of common shapes and the way in which angles of
shapes are used in classifying shapes. Some angle properties of shapes that children
should know at this point are:
• the angles in squares and rectangles are all 90°;
• all the angles in an equilateral triangle are 60°;
• that an isosceles triangle has two equal angles;
• that an n-sided polygon has n angles (as well as n sides);
• that regular shapes have angles which are of equal size;
• that the angles in irregular shapes are not equal.
Children should be expected to make use of such angle properties of shapes in sorting
and classifying activities. The way in which angles contribute to defining various classes
of shapes should also be explored.
Angles
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© 2000 Andrew Harris
5. Measuring using Standard Devices
NNS Year 5
At this point children can be introduced to the protractor as the standard measuring device
for angles.
It is worth remembering that children still need plenty of ‘dynamic’ experiences of angles; it
is very easy to treat a protractor as a device for measuring ‘static’ angles only. For this
reason, it is helpful to use a protractor or angle measurer which has some kind of pointer
which can be moved around the scale by the child. This means that the pointer can be
aligned with the starting direction (indicated by the first line bounding the angle) and then
the pointer is rotated around the protractor until it co-incides with the final direction
(indicated by the second line bounding the angle). This approach maintains the idea that
angle is about measuring rotation.
A protractor with a 360° scale is generally better than one which offers a 180° scale since
this avoids the necessity for children to move the protractor part way through a
measurement for angles larger than 180°. Initially, the reading of a protractor should be to
the nearest 10°. Once the skill of using the protractor has been established, this can be
refined to reading the scale to the nearest 5°. Children should, using their protractor/angle
measurer where appropriate, be asked to measure, estimate, compare and order angles.
They should also be taught to draw angles which are multiples of 5°.
Refer to the booklet ‘General Progress in Measures’ to find out about common errors in
using a protractor.
Classification of Angles
Mathematicians classify angles into several categories according to size: acute angles,
right angles, obtuse angles, straight angles and reflex angles. These can be defined as
follows:
•
•
•
•
•
acute
acute angles - angles greater than 0° but less than 90°;
right angles - angles of exactly 90°;
obtuse angles - angles greater than 90° but less than 180°;
straight angles - angles of exactly 180°;
reflex angles - angles greater than 180° (Reflex angles are taught in Year 6 of
the NNS but noted here for the sake of completeness of the classification of
angles).
right angle
obtuse
straight
reflex
Children should be taught to identify practical examples of each type of angle in real and
mathematical contexts.
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© 2000 Andrew Harris
Application to Work in 2D Shape
Children’s knowledge of angle-related shape properties should be extended further to
include knowing that:
•
•
•
•
scalene triangles have three angles which are all different;
that perpendicular lines are at right angles;
that parallel lines never intersect to form an angle;
that congruent shapes have identical corresponding angles (as well as
corresponding sides of the same length) but that the orientation of the shapes
may be different;
• that similar shapes have identical corresponding angles (but corresponding
sides are only proportional in length and the orientation of the shapes may be
different);
• that angles on a straight line add up to 180° (supplementary angles).
a
b
a + b = 180°
Children should be given opportunites to apply such knowledge e.g. in classifying
triangles or finding polygons with perpendicular or parallel sides.
Rotation of Shapes
While shapes themselves contain
angles, a whole shape can also be
rotated through a given angle.
Interesting shape patterns can be made
by rotating a shape repeatedly by a
particular angle.
A Logo-type software package is one
way to explore this idea. Normally, one
writes a Logo procedure for drawing
the chosen shape. This is then called as
part of a repeated wider procedure
which alternately uses the first
procedure to draw the shape and then
turns the ‘turtle’ by the chosen rotation
(in between each drawing of the chosen
shape). For further information, see the
Logo booklet.
Rotating a Regular Pentagon repeatedly by 30°
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© 2000 Andrew Harris
NNS Year 6
The National Numeracy Strategy recommends that children in Year 6 are taught to measure
and draw angles using a protractor to an accuracy of 1°. This will involve reading the
smallest unlabelled markings on the protractor scale. Reflex angles are also introduced
within the Year 6 NNS teaching programme.
Simple rotations of shapes drawn on co-ordinate grids are expected at this stage. For
example, rotating the shape below by 90° or 180° around one of the vertices:
15
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4
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2
1
0
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
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© 2000 Andrew Harris
At this stage, children should also begin to learn about the sum of the interior angles of
common polygons.
The Sum of the Interior Angles of a Triangle
This is usually explored with children by asking them to draw any triangle, tearing off
the corners and putting the angles at the corners together, like this:
tear off corners
& arrange on
a straight line
B
C
B
A
A
C
Whatever triangle is chosen as the starting point (and it is a good idea to do this with
quite a few triangles ) the angles can be arranged to lie on a straight line (check with a
ruler). This indicates that
the sum of the interior angles of the triangle A + B + C = 180°.
If several children attempt this each with different triangles it can be shown to work for
several triangles. This is then usually accepted as adequate evidence that the sum of the
angles of any triangle = 180°.
Note that this procedure does not constitute a rigorous mathematical proof of this
mathematical statement since there are an infinite number of possible triangles and
therefore not all triangles have been tested by the procedure outlined above. However, it
is usually considered an adequate basis upon which to proceed for children at this level
of mathematics. The visual nature of the ‘proof’ helps to convince children of the truth
of the conjecture.
The Sum of the Angles in Any Quadrilateral
The fact that ‘the sum of the angles in any quadrilateral = 360°’ can be ‘proved’ in a
similar way to the procedure outlined above for triangles.
Draw any quadrilateral, tear off the corners and arrange around a common point:
B
C
tear off corners
& arrange around
a common point
A
D
A
C B
D
Then use the fact that a whole turn = 360° to deduce that A + B + C + D = 360° and so
‘prove’ that
the sum of the interior angles of any quadrilateral = 360°.
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© 2000 Andrew Harris
Note that the same reservations expressed above (regarding the procedure for ‘proving’
the angle-sum of triangles) about lack of mathematical rigor apply in this case also.
These new angle properties are used in explaining the tessellating abilities of triangles and
quadrilaterals (refer to the booklet ‘The Mathematics of Tessellation’ for more information
about this). A further result, namely that the sum of the angles around any point is 360°, is
also much used in explaining the mathematics of tessellation.
A Simple Clinometer
straw for
sighting
A clinometer is used for measuring angles
in the vertical plane. Key Stage 2 children
can use a simple clinometer constructed
using an isosceles right angled triangle as
shown opposite.
45°
cardboard
45°
This can then be used as a means of
finding the heights of very tall objects
plumb line such as trees or buildings.
Holding the piece of card so that the plumb line is parallel to the edge of the card, the
child moves forwards or backwards until the top of the tree or building is sighted in the
straw:
d
h
45°
Finding the height of a tree
45°
c
c
d
It can be seen that, at the point where the child sights the top of the tree through the
straw, a larger right-angled isosceles triangle exists which has part of the tree as its
vertical side. Using the properties of the isosceles triangle, this part of the height of the
tree is equal to the distance d from the child to the tree. The remaining part of the height
of the tree is equal to the height c of the child. So the height h of the tree can be
determined by measuring the distance from the child to the tree and the height of the
child and adding them together.
Angles
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© 2000 Andrew Harris
6. Extension Material and Additional Subject Knowledge
The following is supplied for working with very able pupils and for further developing
personal subject knowledge.
More Kinds of Angles
When two straight lines intersect, pairs of angles which are opposite each other (across
the intersection) are equal and are known as vertically opposite angles.
a
b
c
d
So a and b are vertically opposite angles and a = b.
Similarly, c and d are vertically opposite angles
and so c = d.
In the diagram above, it can be noted that angles a and d lie on a straight line and so
a + d = 180°. Such angles which lie adjacent on a straight line are known as
supplementary angles. Angles c and b, b and d, and a and c are also supplementary
angles. Each of these pairs of angles also totals 180°.
g e
p q
h f
r s
Where parallel lines are intersected by a straight
line we obtain two more kinds of angles.
There are 4 pairs of corresponding angles in the
diagram opposite: e and f, g and h, p and r, and q
and s. Corresponding angles are equal so
e = f, g = h, p = r and q = s.
The other kind of angles shown are alternate angles (colloquially often known as ‘Zangles’ because of the shape made by the bounding lines around them).
In the diagram above, there are two pairs of alternate angles: p and f, q and h. Alternate
angles are equal so p =f and q = h.
Bearings
A bearing is a way of describing the direction from one location to another. Bearings can
be written in one of two ways:
• a two-figure bearing measured from North or South, indicating a turn towards
the west or the east (e.g. N25°E or S34°W);
• a three-figure bearing measured in a clockwise fashion from the direction of
North (which, by convention, is the bearing 000°) e.g. 098°.
The diagrams overleaf show examples of how three-figure bearings are used to describe
directions:
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© 2000 Andrew Harris
North
000°
North
000°
B
45°
A
260°
A
B
The bearing of B from A is 260°
The bearing of B from A is 045°
North
000°
North
000°
B
180°
45°
Reverse bearings can also be specified. In the examples
above the bearing of A from B would be found by
constructing another North line (see opposite). The two
North directional lines are in the same direction and are
therefore parallel lines. Consequently, it is possible to
identify some of the types of angles identified in the
previous section (‘More Kinds of Angles’) to work out
the reverse bearing.
The two angles marked 45° are alternate angles and are
therefore equal. The remaining part of the bearing angle
lies on a straight line and so is 180°. So the bearing of A
A
The bearing of A from B is 225° from B is 180° + 45° = 225°.
45°
The Sum of the Exterior Angles of a Polygon
The exterior angles of a polygon are those angles through which one would turn at each
vertex if one was to walk around the perimeter of the polygon.
Imagine that a pencil is placed at the vertex of a polygon A and rotated by an amount
equal to the exterior angle at that corner (Stage 1 below) and is then slid along the side
of the shape to the next corner B before being rotated through the exterior angle at that
corner (Stage 2 below) and so on until the pencil arives back at the original corner facing
in its original direction.
B
B
C
C
A
A
E
D
E
D
Stage 2 : Slide pencil along
to B and turn through the
exterior angle at B.
Stage 1 : Turn through
exterior angle at A.
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The total amount of rotation of the pencil on its travels around the exterior of the
polygon is found to be 360° and this is true for all polygons.
The sum of the exterior angles of a polygon = 360°.
[Note that this in itself is not a mathematical proof since we have not tested it in practice
for all polygons.]. For regular polygons only (where the exterior angles are equal), the
size of each exterior angle = number360°
of sides .
The Sum of the Interior Angles of a Polygon
A similar pencil-travelling exercise can be attempted for the interior angles of polygons.
B
B
A
C
1. Rotating around interior angle at A
A
C
2. Slide pencil to B and rotate around angle at B
B
B
A
C
A
3. Slide pencil to C and rotate around angle at C
C
4. Slide pencil back to A
On trying this for polygons with different numbers of sides, we find that the total
amount of rotation of the pencil (and, therefore, the sum of the interior angles) is
dependent on the number of sides the polygon has. After sufficient experimentation, the
following should be obtained:
Number of sides Number of complete rotations Number of right angles Rotation (in °)
Angles
3
1
2
2
180°
4
1
4
360°
5
1 12
6
540°
6
2
8
720°
7
2 12
10
900°
8
3
12
1080°
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© 2000 Andrew Harris
It thus becomes clear from the table above that adding an extra side (and therefore
angle) to a polygon adds an extra ½ turn or 180° or 2 right angles to the sum of the
interior angles of the polygon.
This pattern (in terms of the number of right angles) can be generalised as:
The sum of the interior angles of a n-sided polygon = (2n - 4) right angles.
Angle in a Semicircle
By exploring different possible triangles ABC
where the diameter of the semicircle forms one
side and angle C is located at any point on the
curved side of the semicircle, it can be found
experimentally that the angle at C is always a
right angle [Note that this method does not
constitute a mathematical proof]. This result is
B usually described as
the angle in a semicircle is 90°.
C
C
C
C
A
Constructions of Angles with a Ruler and Pair of Compasses
Bisecting an Angle
To bisect (divide equally in two) any angle
with a pair of compasses, put the point of the
compasses at the centre of rotation R for the
angle and draw an arc which intersects the
directional lines bounding the angle at A and
B (as shown opposite).
A
R
B
A
R
C
Then put the point of the compasses at A and
draw an arc. Keeping the compasses at the
same setting, put the point of the compasses
at B and draw an arc which intersects the
previous arc at point C.
B
A
C
Finally, join points R and C. This line RC is
the bisector of the original angle.
R
Angles
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© 2000 Andrew Harris
Constructing a Right Angle
A right angle is constructed as follows. Draw a line
AB with a ruler. Set the compasses to be more than
half of the distance AB.
A
B
The point of the compasses is then placed at one end
A of the line and two arcs are drawn, one on each
side of the line (as shown opposite).
C
This process is repeated from the other end of the
A
B line B using the same setting for the compasses thus
producing another pair of arcs that intersect the first
two as shown at points C and D.
D
Finally, the intersections of the arcs at points C and
D are joined by a straight line drawn using the ruler.
This creates four right angles at point E. Any
unwanted lines can then be erased.
E
A
B
Note that this construction is also a method for
bisecting a straight line. It can also be used in
conjunction with the angle bisection method above
to produce subdivisions of 90° by repeatedly
bisecting the angle to get 45°, 22½° etc.
Constructing a 60° Angle
A 60° angle can be constructed as follows.
A straight line AB is drawn with a ruler.
The point of the compasses is placed at
one end of the line A and an arc is drawn
(as shown opposite) which intersects the
straight line at point C.
A
Angles
C
B
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© 2000 Andrew Harris
D
Keeping the compasses set as for the first
arc, the point of the compasses is placed at
point C and a second arc is drawn which
intersects the first arc at point D.
A
C
B
D
A straight line is then drawn which passes
through points A and D producing a 60°
angle at point A.
Note that if an additional straight line is
drawn joining points C and D an
equilateral triangle is thereby constructed.
A
C
B
Further sizes of angles (30°, 15°, 7½° etc.) can be constructed by producing a 60°
angle and then repeatedly bisecting it using the bisection method outlined above.
Common Misconceptions and Errors : Angles
Right/Left confusion
Children frequently confuse right and left. This can be for a number of reasons depending
on context. It could be owing to:
• failing to remember which direction is left and which is right;
• not understanding that left and right are relative terms (i.e. my left or right may
not be the same as yours - it depends on which direction each of us is facing);
• confusion between rotating (on the spot) towards the left or right and moving to
the left or right. This is a common problem which occurs if programmable floor
robot keys such as
and
are described as Left and Right. Most forms of
Logo also use Left and Right (or their abbreviations LT and RT) as the commands
for rotations. In both cases, these would be better described as clockwise and
anticlockwise since they are rotations not movements. In everyday speech,
however, when giving directions a movement towards the left or right is usually
implied.
East/West confusion
This is similar to the first of the above left/right confusions, namely, a failure to remember
on which side of the north-south axis each lies. This can be remedied by teaching children
simple mnemonics Never Eat Shredded Wheat (reading clockwise around the compass rose)
or the word ‘WE’ (read across the compass rose).
Angles
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Angle Size Dependent on Bounding Lines
Some children believe mistakenly that the size of an angle is determined by the length of the
directional lines at either limit of the rotation. A child entertaining this misconception
believes that the angle in Figure 1 below is larger than that in Figure 2.
Figure 1
Figure 2
Angle Size Dependent on Distance of Arc from Centre of Rotation
In Figure 1 above the curved line conventionally used to mark the angle is further from the
centre of the rotation than that in Figure 2. Some children believe mistakenly that the
distance from this curved line to the centre of rotation is the significant factor in determining
the size of the angle.
Angle Size Dependent on Length of Arc
Some children believe (incorrectly) that the length of the arc which labels the angle is the
factor which determines the size of the angle. Such a child would believe that angle A below
is smaller than angle B.
B
A
Incorrect Perceptions of the Concept of Angle
Often children have little or no conception of angle as a measure of rotation. It seems likely
that this may be owing to a predominance of ‘static’ experiences of angle at the expense of
‘dynamic’ experiences. Common ideas expressed by children when trying to define angles
include:
•
•
•
•
‘The distance measured between two lines’;
‘The gap where two lines meet’
‘The area between two lines’
‘The spacing between two lines which meet at a point’ (from ‘Children Learning
Mathematics’, Dickson, Brown & Gibson (1984) ).
Relationships between Angles
The first APU (Primary) Survey (1980) indicates that children have difficulty recognising
relationships between angles (e.g. supplementary angles, corresponding angles etc.) and that
recognition of these diminishes when the angles concerned are part of larger, more complex
situations. For example, more children would be able to recognise the angles in Figure 1
below as supplementary angles (or ‘angles on a straight line’) and use this knowledge to
calculate the size of angle a than would be able to perform exactly the same task in the more
complex situation in Figure 2.
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150° a
a
Figure 1
150°
Figure 2
Resources
The resources supplied on the following pages are intended to support the teaching of angle at
Key Stage 1 and Key Stage 2.
The dice can be used in many games with groups (some of which are indicated on previous
pages) or, if enlarged, in whole class situations. They can also be used in association with
programmable floor robots such as Roamer and Logo-type packages.
The angle measurement cards can be used by children to measure and compare angles to an
acuuracy of 10°. Their use forms a helpful introduction to using the standard angle measuring
device (the protractor). Photocopy them onto card and then cut up each circleto obtain cards for
each of the different angles - they have been grouped in pairs whose sum is 360° to save both
space and the amount of cutting necessary.
The angle complement discs can be used to investigate pairs of angles which add up to 360°.
Angles
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© 2000 Andrew Harris
Clockwise and Anticlockwise Right Angular Rotations Prism
1 right angle
2 right angles
3 right angles
4 right angles
4 right angles
3 right angles
2 right angles
1 right angle
Instructions
Copy the prism onto thick card and cut out the net. Score all solid lines and fold flaps. Glue
flaps and thus construct the prism.
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© 2000 Andrew Harris
Multiples of 90° Dice
270°
270°
90° 180° 90° 180°
90° 180° 90° 180°
270°
270°
270°
270°
90° 180° 90° 180°
90° 180° 90° 180°
270°
270°
270°
270°
90° 180° 90° 180°
90° 180° 90° 180°
270°
270°
270°
270°
90° 180° 90° 180°
270°
90° 180° 90° 180°
270°
Angles
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© 2000 Andrew Harris
Left & Right Turns Dice
LT
LT
RT RT RT LT
RT RT RT LT
LT
LT
LT
LT
RT RT RT LT
RT RT RT LT
LT
LT
LT
LT
RT RT RT LT
RT RT RT LT
LT
LT
LT
LT
RT RT RT LT
RT RT RT LT
LT
LT
Angles
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© 2000 Andrew Harris
Clockwise/Anticlockwise Dice
Angles
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© 2000 Andrew Harris
°
°
70°
80°
280°
270°
12
0°
110
°
100°
290°
250°
260°
Angles
300°
90°
310°
320°
330°
60
50
40
30°
340°
°
350°
20°
10°
Angle Measuring Cards (Part 1)
28
240°
© 2000 Andrew Harris
210°
180°
180°
220°
170°
13
0°
14
°
160
Angles
200°
0°
230°
15
0°
Angle Measurement Cards (Part 2)
29
190°
© 2000 Andrew Harris
Angle Complement Discs
Cut out both circles
and then cut along
the marked radius
of each circle.
Slot the two circles
together along the
slit radii.
The two angles thus
created form
a complementary
pair of angles whose
sum is 360°.
Angles
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© 2000 Andrew Harris