Angles and Related Concepts
Ch. 3, page 1
Angles and Related Concepts
Learning from Tian about Angles
It was one of those magical moments which teachers witness from time to time. Tian from
Year 2 and I were discussing which angles were smaller on the pieces of the tangram set. Tian
had clearly thought that being sharper meant that it must have been bigger but it did not seem to
be what I, the teacher, was saying. Being a second-language English speaker, young Tian had
learnt to revise her current meanings of words quite rapidly. Ah, she thought, larger angle
means the opposite to sharp. If they are blunt, then they are larger!
I was not the only one learning from Tian.
Victor was Tian’s friend and they were working cooperatively with their other friend,
James. They bent over the pieces of the tangram set comparing the angles on the different pieces
trying to order the angles as small, middle-sized, and large. I stand back, watch, and listen
joining in occasionally with their conversation.
2.01 Victor has gathered together some pieces, as if matching all the small angles.
2.02 Victor picks up the parallelogram for the middle-sized angle, discards it, and then
chooses the middle-sized triangle for the middle-sized angle. He puts his thumb and forefinger around the right angle and says "This is the middle angle." James watches.
2.03 Tian is quite clear about which angle is smallest and which is middle-sized and she has
been drawing both the small and middle-sized angles in the book.
2.04 James matches several angles against the drawing of the small angle....
2.05 Victor picks up the large triangle as if showing the small angle, "This is the biggest."
2.06 Tian takes it and says "No, it isn't," and places it on the drawing of the small angle. The
teacher confirms, "That's the small angle."
2.07 Victor has picked up the parallelogram, tests it against the drawings of the small and
middle-sized angles and draws the large angle into the book.
2.08 He goes back and draws along the whole length of the arms.
2.09 The teacher suggests they make points (angles) by joining smaller points together. Victor
puts points on top of each other.
2.10 James puts two small points together. The teacher says he has made the middle-sized one,
praises him, and suggests that he can draw it in their book. James nods his head but he
doesn't look convinced that he knows what he did. He puts two more together on top of
the middle-sized point of the large triangle.
2.11 Tian has now made the big point with the small angle of one small triangle and the
middle-sized angle of the other small triangle. Victor nods approval while he covers the
large triangle with the other shapes.
The discussion indicated how Victor, who seemed to know what was meant by the size of
points (the word generally used by these children to refer to angle), temporarily considered that
he should be comparing the overall size of the piece (paragraphs 2.02 & 2.05) or the size of the
sides of the shapes (paragraphs 2.08). The interaction between students helped Victor to clarify
what was meant by "the point of the same size" (paragraph 2.06). Through manipulations,
Victor (paragraph 2.11) spent some time relating the operational concept that making congruent
shapes means making equal angles.
During a later activity Victor explained that James had not made a right-angled triangle with
three matchsticks but that he had just made an equilateral triangle in another orientation. Victor
himself had made the right-angled isosceles triangle with the long side horizontal from two
matchsticks and a longer stick and he had checked it with the tangram piece which he put on top
("a lid," he called it). The teacher asked the children what was meant by a bigger angle. Tian
replied more spread out and picked up the tangram right-angled triangle, put an angle of the
pattern-block equilateral triangle on top of the right angle matching one arm, pointed to the
uncovered part of the right angle, and said, ”See it is bigger.”
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Angles and Related Concepts
Ch. 3, page 2
Later Victor was matching angles on different shapes. At one stage, he missed a right angle
on a triangle but when it was turned for him he explained that it was blunt and that the two
triangles were the same so the two right angles had to be the same.
What we Know of Students Learning about Angles
Learning about Specific Concepts
A number of Australian researchers have investigated the development of specific
spatial concepts: parallels, perpendiculars, angles in general, and symmetry. Early
spatial development has also been investigated in some of the more general studies that
have already been reviewed. Owens (1997) summarised results on spatial items of three
Basic Skills Tests used in NSW and noted the difficulties with items involving
symmetry and interpreting diagrams.
Parallels
Mitchelmore (1992a) sought an explanation for an earlier finding that when young
children copied parallel lines crossed by an oblique transversal, their copies were
usually not parallel. He found that Grade 2 children in fact recognised their copies were
not accurate, but were reluctant to alter their first attempts. When they were explicitly
given the chance to improve their drawings, they eventually drew accurate parallels.
Mansfield and Happs (1992) were interested in older children. By asking Grade 8
children to judge whether given pairs of curves were parallel, they showed that—
although most of them could give an acceptable definition of parallel lines—their
conceptions of parallelism were still basically global. Teaching activities designed to
make properties of parallels (and, in particular, the concept of the distance between two
lines) more explicit were found to be effective, at least in the short term.
Perpendiculars
Mitchelmore (1992a) found that children in Grades 2 and 4 have far greater difficulty
recognising oblique perpendiculars than parallels. He surmised that children normally
see right angles only as intersections of horizontal and vertical lines, and therefore do
not relate them to angles in general. Mitchelmore (1992b) suggested various teaching
activities designed to help children see right angles as special angles.
Angles
Mitchelmore and White have been carrying out an extensive investigation of the
development of general angle concepts among young children, based on a reformulation
of the theory of abstraction in mathematics learning (Mitchelmore, 1993; Mitchelmore
& White, 1998). This theory regards a concept as a product of the recognition of deep
similarities between superficially different objects, events, or ideas. In particular, the
angle concept is thought to develop gradually as children recognise more and deeper
similarities between physical angle experiences, going through three successive stages:
classification into physical angle situations such as walking up a slope and using
scissors; then into separate angle contexts such as sloping and crossing; and finally into
a general angle concept which includes all contexts (Mitchelmore & White, 1998).
The Need for Pre-Measurement Experiences
Most teachers have watched the frustrations of students trying to measure angles
with a protractor and wondered how we can improve students’ early learning
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Ch. 3, page 3
Angles and Related Concepts
experiences with angles. Close (1984) emphasised that it is most important that we
provide many pre-measurement experiences with angles before we ask them to measure.
Difficulties with the Concept of Angle
Many teachers have realised that students think equality of angles means equal
arms. Some students only recognise angles which are sharp or a right angle which is
drawn like an L. Indeed, when reflected some students have spontaneously called the
angle a left-angle. The term angle does not seem to be common in the vocabulary of
young children during play. Even adults find it hard to define the term angle because it
has many constructs and contexts from which we abstract the concept of angle. These
include angle as a measure of turning, angle between two rays (lines in two directions
from a point), direction, difference between the directions of the rays, and interior
angles of shapes. Most confusion reigns with our reference to a sharp turn around a road
corner when we are turning an obtuse angle. Colloquially we use angle in a metaphor
“what’s your angle?” meaning what point-of-view do you take.
Early Recognition of Angle
Young students intuitively recognise angles which seem to stand out for them and
help them to make distinctions between various shapes such as triangles and circles
(Piaget & Inhelder, 1956, p.30). Yet students have difficulties separating
(disembedding) an angle from the whole shape (Outhred, 1987; Owens, 1996). Outhred
showed that students found it hard to see some angles in pictures, to see straight-angles
in crossing lines, or to recognise right angles in closed figures when they were not in the
L position. Often they were unaware that an angle was bounded by two straight rays and
not with intersecting curves. Furthermore, it seems that young students recognise right
angles like corners or as a horizontal and vertical line without realising that they are part
of the family of angles (Mitchelmore, 1992b; Davey & Pegg, 1991).
Movement necessarily involves position concepts. Actions are described in
conjunction with directions such as left, right, or straight ahead. A particularly
important change of direction or turn is associated with the concept of angle. Early
learning is often stimulated by action and this turning of one arm of an angle away from
the other (opening) does seem to be one of the ways that children first begin to learn
about angles. However, they also begin to notice angles on shapes.

a door
simple angle tester
scissors
a slope
on a shape
Figure 1. Examples of turning of a line to form an angle in different everyday contexts.
Mitchelmore (1993) has pointed out that students find it easier to notice angles when
both rays are present (like scissors crossing) and almost as easily when one is present
and the other imagined as in a sloping ground. Students, however, find it more difficult
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Angles and Related Concepts
Ch. 3, page 4
to imagine an angle such as the angle at which a ball is hit in cricket. However, angles
embedded in scissors crossing is difficult for students to perceive or to decide on the
vertex (Mitchelmore, 2003; Chartres, 2005).
Young students need to notice, make, and compare angles. They need to mark
angles in their environment and this is best done with a simple folded strip of cardboard
(see Figure 1). This helps them to notice the angles. The tester strip has two arms which
turn apart and so students experience the idea of an angle as the amount of turning.
Young students can make angles by folding paper and making angles that are smaller or
bigger than right angles. They can make and compare angles by cutting out cardboard or
paper representations of different triangles or polygons. They can compare angles on
representations of shapes in structured materials such as tangram sets or pattern blocks.
The angles can be physically compared by overlaying them (see the story on Tian).
Turning and making angles can be fun when the thumb and forefinger or the arms
are used to make angles. Students can also be physically involved in making angles by
using a long stick or rope turning away from a fixed stick like radii on a large circle.
These activities help students to develop dynamic images of angles and to regard angles
as a family.
All this making of angles is a far cry from the student being given angles on a sheet
of paper and being expected to understand the differences in their size by measuring
with a protractor.
Developing Adequate Language
Language is a difficulty for students as shown in the story about Tian, Victor, and
James. Some researchers such as Fuys, Geddes, and Tischler (1988) suggest that
language such as non-standard vocabulary is closely linked with concept development.
However, our young friends have suggested that their conceptualisation has developed
faster than their language. Interestingly, language can be a way of drawing attention to a
particular aspect of a learning situation. Selective attention and visual analysis can be
important in students recognising what teachers are referring to when they talk about
angles.
Sometimes teachers concentrate on using the words acute, obtuse, reflex, right, and
straight. However, there is much more to language use in the development of the
concept as we talk about the angle contexts (sloping, crossing, turning, direction, arms)
than just these words.
Students often perceive right angles without realising the importance of the right
angle. Students talking with the class and teacher, for example, about the perpendicular
height in a triangle will assist students to note that this is a special line not just a line
“going down the middle” of the triangle. Recognition of perpendicularity and
parallelism and the importance of the properties associated with these kinds of lines will
be assisted by class discussion. It is too easy for the teacher to assume that the students
appreciate what is in some ways obvious.
Chartres (2005) found that students’ use of words was frequently at odds with what
teachers expected. Students used the word distance to discriminate the size of angles but
some referred to the distance between any two point on the two arms, the arms or the
other side of the triangle if the angle was on a triangle. He found students failed to use
instruments to help them decide size even when asked to be more accurate. The
instruments included simple jointed straws, string, rulers, protractors and rotators. On
concave quadrilaterals, students found it difficult to distinguish reflex angles as an
angle.
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Angles and Related Concepts
Ch. 3, page 5
Imagery of Angles
Along with experience and language development, students need to develop
imagery related to angles. Mansfield and Happs (1992) found some Year 6 students
imagined a protractor, others a right angle or half turn as a benchmark, and others an
angle of a polygon to assist them in estimating angle size. Owens (1993) has shown that
students need to develop their visual analysis and disembedding skills in order to
develop their concept of angles. As a result, students like Tian and her friends (even
though they were only in Year 2) can begin to learn about the family of angles.
Concept images are the summary images that students have for concepts but they
can also summarise misconcepts. For example, some students only have images of right
angles in the L position; others consider only right angles like the corner of a room as
angles; and others only consider acute angles as angles. Each of these limited
conceptions need to be extended by further learning experiences, activities that
encourage discussion, and a recognition by students of the imagery they are using.
Having dynamic images from the physical activities and from computer screens
generated by programs like Cabri Geometry, will help students establish adequate
concept images.
Measuring Angles
The measurement of angles can be developed in a similar way to measuring other
attributes like length. First the student must recognise an angle. Initially there can be
some direct and indirect comparisons, followed by the use of informal units for
measuring.
Students understand their measuring instruments and how to use them if they make
them (Owens, 1994). In this case, students can make a simple protractor with markings
every 30o (not that they at first know this is the size, it can just be an angle unit). By
having to mark off the angles from a reference line, they build up the understanding that
the instrument has markings giving the angle from the reference line just as the ruler
had markings giving the distance from the zero or reference point.
By marking their own protractors, they are much more able to think their way
through the myriad of numbers on a commercial protractor. By the way, it has been
found that the circular protractors with a moveable line to mark the turning and final
position of the ray of an angle is an effective tool for learning to read a protractor. (If
your protactor does not have this, make a hole at the centre and thread a piece of black
hat elastic through it, tying it off at the edge.)
Programming Lessons on Angles
There are some wonderful lessons on angles for early primary school. Below are a
number of ideas about angles that could be incorporated into a program for a particular
year. In general they are ordered so that the later activities could be done by older
primary school students.
Students need to recognise angles and then differences between angles. They need
to see angles in different contexts (on shapes, between arms, door openings, turnings,
directions, slopes) and later to abstract the idea of angle. Later students can articulate
differences and properties. The student needs to make and discuss angles.
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Angles and Related Concepts
Ch. 3, page 6
Learning Tasks for Readers
Angle Activities 1
experiencing
Noticing Angles
 Use a folded strip of card to notice angles around the classroom
(on objects, door ways, overhead projector) and in branches on
plants outside.
 Look for angles in crossings eg. scissors, embedded in pictures.
 Compare angles as bigger and smaller.
 Talk about turning the card arms apart as making the angle
bigger.
 Play semaphores to make right angles, straight angles, and half
of a right angle or third of a right angle with your arms.
Angles in Leisure
Ballet and other dancing. Discuss the angles of arms and feet in
different ballet positions or for different dance techniques.
 Roller blading. Discuss angles of feet when blading or skating,
and the effect.
 Collage making. Discuss the positioning of parts to make them
look attractive.
 Bicycles, skate boards, body boards. Discuss the angles used to
keep balance or jump.
 Netball. Discuss angle for throwing in different ways.
 Swimming. Best angle for arms and hands for efficiency with
different strokes.
 Football. Kicking angles.

Imagining Angles
 Use short boards to make ramps at different slopes and let toy
cars run down them without pushing.
 Discuss what happens at different slopes.
 Set the boards at different slopes in the air. Talk about the
slopes being the same as on the ground.
 Imagine angles between an edge of the classroom and a point in
the classroom - mark some with rope first. Do the same with
boxes.
 Have students draw plans of the buildings in the school and
discuss the angles people walk at when taking short cuts.
 Look at the angles between dots on board games such as
checkers.
 Play with the hose and talk about the angle of the water and
what happens.
 Fold papers and discuss the angles that will be made when
opened out.
 Use Cabri Geometry, Kidpixs or Word on the computer, draw
angles, estimate size, mark and measure the angle. Move the
end of the ray with the hand, discussing the variety of angles
being made. Use Drape to explore angles to make different
shapes.
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Angles and Related Concepts
Ch. 3, page 7
Informal Measuring Instrument
Use thin paper eg. circular or square tracing or baking paper,
fold in half then thirds then half again so that, on opening, there
are six diagonals through the centre. Discuss how big each
angle at the centre might be (30o). Draw one ray & label it 0o,
then mark other rays anticlockwise as 30o, 60o, 90o, 120o, 150o
....270o, 300o, 330 o, 360o on the first ray.
 Use this informal protractor to measure some angles.

Fun in the Playground
 Make a large circle with the class holding hands at arms length.
 Mark the centre where one student holds the end of a rope or
very long stick. Go around the circle marking points.
 Discuss how each point is exactly the same distance/length
from centre as all the others.
 Draw in the circle. Mark the circle on asphalt or mark as a
trench on a dirt playground.
 Mark in the North-South diameter using a compass to get it
correct. Use different colours for the two directions.
 Draw a short way along the rope in two directions and discuss
how this angle between the arms is still the same as the angle
if the rope goes to edge of the circle or edge of playground.
 Use the long stick or rope to make angles turning from the
north arm clockwise. Make it a game.
 Marching game. Students in two lines around semi-circle, go
up the south arm to the centre. At centre they can go straight,
or branch off at 90o, 45o, 30o etc (Partners should meet again at
the start of the south arm).
 Link the two arms of rope to hands of a clock, making one
shorter for the hour hand.
 Play cricket with batsman at the centre facing north. Class says
the angle that the ball is hit.
 Use an easterly arm and go anticlockwise to mark off angles
using the large board protractor and long rope or stick.
 Students do the same on small playground circles.
 Mark off angles around the circle from the north clockwise.
Line of sight (direction) and angles for turning
 Set up a teddy or little person in the centre of a board and place
a number of small toys around it. Let the teddy face one of the
toys and then turn it to face another toy. Find a turn from
facing one toy to another that is smaller than, the same as or
greater than the original toy. Think about the effect of how
close the toys are to the turning teddy.
 Use the cars on the street mat and discuss the angles that are
made – sharp, vee left, corner etc. How far does a car turn for a
sharp angle?
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Angles and Related Concepts
connecting ideas
Ch. 3, page 8
Teaching lessons
 Assess a Stage 2 student with the assessment tasks given
below. What can you say about the students’ imagery and
concept development for angles?

There are a number of lessons at the end of this chapter. Others
can be found in NSWDET book on Angles. Try out some
lessons.

Look at the Syllabus. When do students begin work on angles?

What are the initial contexts for angles that students can use?

Find papers by Mitchelmore or Owens on angles.
Mitchelmore discusses angle contexts that are more difficult.
What are they?


He then talks about abstracting the angle concept from the
different contexts. One skill that was used by Tian in the
scenario at the start of this chapter helps students to link
contexts. What is it? What else could be used besides fingers to
mark the arms of the angle?

Check you know how to measure angles with a protractor.

Make a concept map for the concept of angle.

Write definitions of the term angle.
summarise
and record
Learning Tasks for Readers
Angle Activities 2
experiencing
Angles of Shapes (Lesson from the scenario above)
 Take pieces of seven-piece tangram set, mark angles with thumb
& fore-finger
 Compare by overlaying & ordering from smaller to larger
 Draw angles in order and label: small, middle-sized, large
 Take blocks from a pattern block set and find equal angles and
order
 Draw in order from smallest to largest
 Use the smallest point (on the narrow rhombus) to find out how
many are needed for each of the other angles & record it. (If
your set does not have this rhombus, use a piece of card with a
30o angle)
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Angles and Related Concepts


Ch. 3, page 9
As the right angle is 90o, then calculate the size of all the other
angles (don’t forget the largest angle on the narrow rhombus)
Draw up all sorts of closed shapes with some right angles and
mark all the acute angles, right angles, obtuse angles.
Using Angles
 Investigate the angle sum of a triangle and other polygons
 Discuss the angles of a LogoDrape turtle track which complete a
closed polygon or completes a full circle
 Investigating angles of regular polygons, especially the rhombus
and the angles made by its diagonals
 Investigating angles of regular polygons, especially the rhombus
and the angles made by its diagonals
 Fold a paper and again to form a right angle, then fold a sloping
angle so that when it opens up it will be a rhombus

Look at the perspective lines in pictures and draw in vanishing
points
Using Angles in Design
 A 3D sculpture. Take a very long 3cm piece of thick coloured
paper & fold in several places. Use this to make a sculpture
thinking how it might turn around & slitting paper half way at
the intersecting parts to slot the paper together
 Use arms of angles to create a picture. Different thicknesses or
colours can be used (paint to the Socerer’s Apprentice)
 Discuss the angles in traditional Pacific/Melanesian weaving
such as starting the Maori tapire (60o) or a PNG mat (45o).
Discuss the effects on the designs and continue some weaving
(see the instructions later)

connecting ideas



The drawing of a circle with rope or stick to mark the radius is
common in societies that have round houses.
The playground protractor can be linked to the formal use of the
compass.
Different contexts reflect different ways of abstracting the
notion of an angle: e.g., cutting, slope, turning, part of shape,
two arms, picture, direction. Which were you using in the above
activities?
How do the activities encourage visual imagery of the angle,
e.g., movement?
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Angles and Related Concepts
Ch. 3, page 10

Consider a class and select a set of activities that can be
interrelated to develop the concept of the angle.

Explain why you selected the activities.

How will you assist students who have less developed concepts
than you expect? How will you extend other students?
summarise
and record
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens