"Chapter 3
Beginnings: Experimental Geometry
EXPERIMENTAL GEOMETRY
Students' early encounters with geometry will be linked with van Hiele's early stages of visualization and the beginnings of analysis and deduction through examining the properties of shapes
and beginning to classify them. Up to about the age of 11, shape recognition and acquiring
vocabulary, both the names for shapes and words used in describing their properties, position
and movement together with learning to measure length and angle will be dominant activities.
As they engage in exploring shapes in a variety of ways, besides learning appropriate language
and skills they will also be enhancing their spatial sense and laying the foundations for a more
analytical, deductive approach to geometry.
Measurement plays an important part in experimenting with geometry: length measurement
is developed first before moving on to the conceptually harder ideas of angle, area and volume.
It is a simple matter to compare two lengths by placing them alongside each other without
reference to a ruler: comparison comes before measurement. The same applies to angles: two
angles in a figure can be compared by superimposing them and that should precede making
measurements with a protractor. Comparisons of areas and volumes can be made at a simple
level by using plane figures made with squares and solids made with cubes. Making comparisons by fitting together or superimposing shapes is a step towards reasoning and deducing
general results, whereas measuring, while of obvious immense practical importance, focuses
attention on particular cases rather than encouraging a search for generality.
This preliminary work requires a wide range of experimental activities involving drawing,
looking at diagrams and pictures, cutting out, folding and fitting together. Card or plastic
polygons of all kinds are particularly valuable: the Mathematical Activity Tiles (MATs), produced by the Association of Teachers of Mathematics (see www.atm.org.uk) and described in
Pinel (1993), are a very useful and attractive tool for exploring the properties of shapes. Plastic
cubes that can be linked together are a useful resource for investigating three-dimensional
objects. Computer software such as SMILE, referred to in Chapter 2, which enables shapes to
be moved and compared on screen, provide another format for developing awareness of shapes.
Most mathematical activity brings together many different ideas simultaneously so that tasks
and problems will often not be purely geometrical, but involve other ideas such as counting,
calculation and, at a later stage, algebra. Absorbing this richness and making links between
different ideas is vital if students are to progress with mathematics in general and geometry
in particular. While it is important to focus narrowly on specific ideas from time to time,
ideas make much better sense when they are interrelated and used in a variety of interesting
contexts. In the sections that follow I look at various activities under the general headings of
shape recognition, symmetry, angles, accurate drawing, tessellations and transformations, and
illustrate how each provides opportunities to learn important words, to develop key ideas and to
take steps towards being able to reason and explain rather than guess and assert.

Beginnings: Experimental Geometry 31
RECOGNIZING SHAPES
Figure 3.1 shows a square in two different orientations. It is not uncommon to find that young
students recognize the left-hand diagram as a square, but not the second, which they will often
suggest is a diamond. This is shown in the following piece of dialogue which is not untypical of
the sort of discussion that can take place between teacher and student.
T: What is this shape?
[Shows a square with an edge placed horizontally]
A: A square.
T: If you turn it like this, what is it?
[Shows the same square turned through 45° so that it stands on one vertex.]
A: A diamond.
T: But you said it was a square before.
A: Now it is a diamond.

Figure 3.1 A square or a diamond?

Quite apart from the use of the word diamond instead of rhombus which is excusable at this
level, there is a very real difficulty in not accepting that a physical square is still a square
regardless of its orientation. The default image of a square has an edge placed horizontally and
that image is held so strongly that squares are not necessarily recognized when placed in a
different orientation. This is no doubt compounded by the fact that in this case the shape in the
second position is like a diamond in its default position and by the misconception that a shape
cannot be two things at once - both a diamond and a square. Similar difficulties arise when, for
instance, it is suggested that a square is special case of a rectangle or that a rectangle is a special
case of a parallelogram.
These difficulties can be overcome as students acquire greater familiarity with shapes through
having opportunities to talk about their properties. They are helped to use appropriate vocabulary and to see the links between shapes by classifying them according to their properties.
Merely naming examples and providing simple definitions is not sufficient. Handling actual
shapes and manipulating computer images accompanied by thoughtful discussion is needed to
build understanding and reduce confusion. As their thinking develops they can then begin
to appreciate that a definition is a statement of the minimum requirements for a shape, and that
other properties can be derived from it.
A rich task involving triangles, illustrated by Figure 3.2, is to ask students to create triangles
by joining points on a 3 by 3 square grid either on a pinboard (or geoboard) or directly on
paper with a square grid. The initial task, which I first encountered in SMP (1968), is to see how
many different triangles can be found. Figure 3.2 shows six different triangles: students need
to be challenged to see if they can find all the different triangles (there are more than the six
shown). That may result in some debate about what is meant by the word 'different'. Students

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Teaching and Learning Geometry

Figure 3.2 Triangles on a 3 by 3 pinboard

may suggest that the same triangle in different orientations is different, the same issue as
discussed above with squares. Being sure that all the possibilities have been identified and that
the same triangle has not been repeated in a different orientation is a challenging task.
The set of triangles provides opportunities to identify isosceles triangles and right-angled
triangles and to realize that a triangle may be both isosceles and right-angled. Students will
often suggest that the triangle at bottom left is equilateral: that gives an opportunity to discuss why that is not so and to help them to see that they can be sure without making any
measurements.
The second task associated with triangles on a pinboard is to determine in how many
positions each triangle can be placed. This is a counting exercise which requires a systematic
approach, but it also involves geometrical skills related to transformations. Reflection, rotation
and translation can be invoked as a way of describing the relative positions taken up by a
particular triangle. For example, the small triangle at top left can be arranged in four different
ways in each square of the grid making 16 possible positions in total.
The pinboard task has all sorts of possible extensions: the same question can be asked
about quadrilaterals or other polygons and a square grid of larger size or an isometric grid
offer further possibilities.
Another task is to give students a pair of identical triangles and ask them to find and record
as many shapes as possible where the pair are placed edge to edge. They are then asked to
describe the quadrilaterals that they have created. Figure 3.3 illustrates the six possibilities with
a general triangle. For each common edge there are two possibilities which arise by reflecting
in the edge or by making a half turn about the midpoint of the edge. The first gives a kite
and the second a parallelogram. Since a triangle has three edges and there are two possible
quadrilaterals for each that gives a total of six configurations.
Other possibilities arise by using triangles with special features - isosceles, equilateral, rightangled and obtuse-angled. Each of these gives rise to further points for discussion. Allowing the
two triangles to overlap produces a very different set of examples to consider.
SYMMETRY
The diagrams of Figure 3.4 have each been created by placing four small right-angled isosceles
triangles on a 3 by 3 grid. There are many other possible arrangements which students can be

Beginnings: Experimental Geometry 33

Figure 3.3 Shapes with a pair of triangles

Figure 3.4 Observing line symmetry

asked to suggest. Diagrams like these can be used to improve students' ability to describe and to
form mental images of shapes and to decide how many lines of symmetry there are. The dialogue below, referring to the left-hand diagram, suggests some points for students to consider:
T: Tell me about this shape.
A: It's a small square in a bigger square.
B: There are four triangles at the corners.
T: What can you say about the triangles?
C: They are all the same size.
D: They have right angles.
T: Anything else?
E: They are isosceles triangles.
T: How much of the diagram is shaded?
F: Half of it.
T: How do you know?
G: Because four of the triangles will fit in the square in the middle.
T: Is there a line of symmetry?
H: Yes, down the middle - straight up and down.
I: And going across from side to side, so there are 2.
T: Any other ideas?
J: You could go from corner to corner.
K: And the other corners. So there are four lines of symmetry.

The third diagram does not have a line of symmetry, something which is not necessarily
obvious to students. They need to experiment with an arrangement like this to really convince
themselves before the idea of rotational symmetry is introduced in an informal way. Of course,
it is important to realize that some arrangements will display no symmetry at all and that
varying the colours of the components of designs like these may alter the symmetry.

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Teaching and Learning Geometry

Similar discussions can take place with other arrangements and students can be asked to find
their own, possibly designed to fulfil particular conditions. For example, they could be asked to
find a configuration which just has a horizontal line of symmetry or one that has no symmetry
at all, neither line symmetry nor rotational symmetry.
Folding and cutting paper are very useful ways of examining symmetry by providing
opportunities to ask students to predict what will happen by visualizing a mental image and
examining the properties it has. The left-hand diagram is a square of paper folded in half. The
right-hand edge in the diagram is the fold and a triangle has been cut out. Students are asked to
predict the form that the hole takes when the paper is opened out.

Figure 3.5 Symmetry by paper folding

T: I have folded this piece of paper and cut out a triangle on the fold. What shape will the hole be
when I open it out?
A: It will have four sides.
T: How do you know that?
B: Because the two sides you have cut are on both sides of the paper.
T: What do you call the shape you get?
C: A square.
D: No the sides are not the same. It's a rectangle.
T: Does anybody disagree with that?
E: The top two sides are short and the bottom two are longer, so it can't be a rectangle.
F: It's a kite.
T: Is that right?
[Opens out the folded paper to reveal the kite as in the right-hand diagram.]

Again this offers a rich variety of possibilities for experimentation and discussion. The shape
that is cut out can be varied and the number of folds can be increased. Various possibilities are
suggested below:
• Make two folds at right angles. What shapes do you get for the hole if you cut these shapes
from the corner:
square, rectangle, isosceles triangle, any other triangle, quadrilateral, kite
• Make three folds at 45°. What shapes do you get for the hole if you cut these shapes from the
corner:
isosceles triangle, right-angled triangle, any other triangle
Making folds at 60° or 30° and cutting out an assortment of holes creates attractive snowflake
patterns like that shown in Figure 3.6. These are much appreciated by young students and make

Beginnings: Experimental Geometry 35

Figure 3.6 A snowflake pattern

attractive classroom displays. They can lead to useful discussions about symmetry as well as
developing spatial sense when attempts are made to obtain particular effects.

ANGLES
The idea of angle is a much more difficult concept than that of length. It should be related to
turning from an early stage so that a right angle is related to a quarter turn and a straight angle
to a half turn. These can be linked to the commands 'right turn', 'left turn' and 'about turn',
but the language is a potential source of difficulty here. The link between right turns and
right angles might suggest that there is such a thing as a left angle! Right and left as turns relate
to turning clockwise and anti-clockwise, whereas right, as a description of an angle, is used
irrespective of the direction of turning, which is often not specified or particularly significant.
More facetiously perhaps there is also the contrast between a right angle and a wrong angle!
Angles based on quarter turns should be explored with young students by asking them to
make bodily movements and by looking at the many examples of turning that they can see all
around them - turning a door handle or a steering wheel, unscrewing the lid of a jam jar or
toothpaste tube, moving the hands on a clock face and so on.
LOGO is a particularly valuable piece of software for developing this sense of angle as
rotation and it can provide an introduction to degree measure well before students are ready
to learn how to measure accurately with a protractor. Figure 3.7 shows the first two sides of a
square produced by moving the cursor (often referred to as a 'turtle') forward 100 units, turning
right through 90 degrees and then moving forward another 100 units. The commands for this fd 100, rt 90, fd 100 - appear in the window at the bottom of the screen.
Some of the misconceptions that students have with angles have been considered in Chapter
2. One of the key ideas to emphasize is that an angle is linked to the idea of turning or rotation.
The way in which students learn to use a protractor should draw on and reinforce this link to
turning. Two difficulties that are encountered in using a protractor are significant here. Firstly it
is necessary to align the centre and the zero line of the protractor correctly. This is not a major
problem with a 360° protractor, but students often incorrectly place the bottom edge of a semicircular protractor along an arm of the angle they are measuring. Secondly, and much more
importantly, is the problem of deciding which scale of the protractor to read the number of
degrees from. Textbooks and teachers vary in the advice they offer about this. Experienced users

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Teaching and Learning Geometry

Figure 3.7 Creating a square or a staircase with LOGO

see which value is appropriate by unconsciously sensing whether the angle is acute or obtuse,
but this may not be the best approach with novices who may confuse the two words. It reinforces
the idea of angle as turning if they are encouraged to start from zero and count round to obtain
the angle they require, a procedure that becomes redundant as they become confident about
estimating the size of the angle as a guide.
It is not difficult to provide interesting tasks which reinforce measuring skills. After a few
initial practice examples on measuring and constructing angles, these skills can be related to
practical situations and used to develop further geometrical ideas. Scale drawings can be used
to solve simple problems. For example, measuring the angle of elevation to find the height of a
tree or using a pair of angles measured from two positions to determine the distance of a distant
object. Figure 3.8 shows how angles of 38° and 47° from two points A and B, 100 metres apart
on an east-west line, to a distant object, P, can be used to produce a scale drawing from which
the distance of the object can be determined. Ambiguity about what is meant by the distance of

Figure 3.8 Determining the position of distant object

Beginnings: Experimental Geometry

37

the object can be discussed - do we mean the lengths AP and BP or the perpendicular distance
from P to AB1
Besides learning to measure lengths and angles, students should also learn to use a pair of
compasses to mark off lengths in producing accurate drawings. They can then learn to perform
constructions with straight-edge and compasses which are such an interesting feature of the
study of geometry.
Figure 3.9 shows the traditional 'flower' pattern that students encounter as an early exercise
in using a pair of compasses. The striking fact that it is so simple to construct six equally spaced
points round a circle makes it easy to draw a regular hexagon. The two other diagrams link this
to equilateral triangles. The right-hand diagram prompts interesting questions like determining
what fraction the area of the hexagon in the middle is of the area of the six-pointed star or of
the larger circumscribing hexagon. It also invites the idea of drawing another six-pointed star in
the smaller hexagon and then another yet smaller one and so on - an introduction to the
wonders of the infinitely small!

Figure 3.9 Constructions using six points on a circle

TESSELLATIONS AND TRANSFORMATIONS
Tessellations provide another attractive medium for developing spatial awareness and exploring
the properties of shapes. Figure 3.10 shows two tessellations of triangles. Creating such patterns
either with individual card or plastic triangles or by drawing is a valuable exercise in itself.
Observing and describing what is seen in the tessellation extends understanding of shapes and
the use of language. It is interesting to pick out other shapes in the tessellations: in both cases
it is easy to identify rows of parallelograms: in one case they are all facing the same way and
in the other facing in different directions on alternate rows. In the top example there are also
sets of parallelograms running diagonally whereas in the second case two rows of kites running
horizontally in opposite directions can be seen. Modifying the triangles so that they are
equilateral, isosceles or right-angled will offer other possibilities and, of course, there are a
whole range of possibilities using other polygons.
Figure 3.11 illustrates the three important transformations of reflection, rotation and translation which students should meet in an informal way at an early stage as a way of describing
properties of shapes and of exploring them. The tessellations considered above have involved
translations, and reflection and rotation lie at the heart of the two kinds of symmetry mentioned
previously. Exploring the effect of mirrors on both everyday objects and geometrical shapes
begins to develop the properties of reflection in an informal way.
Placing two mirrors at 60° and placing some objects between them emulates the effect of a
kaleidoscope by creating attractive patterns like that of Figure 3.12. The figure was created
with Cabri Geometry using the reflection command: it is a straightforward matter to produce

38

Teaching and Learning Geometry

Figure 3.10 Two tessellations of triangles

Figure 3.11 Transformations: reflection, rotation ana" translation

Beginnings: Experimental Geometry

39

Figure 3.12/4 kaleidoscope pattern

such patterns on a computer screen. Moreover being able to move any one of the initial motifs
of the pattern by dragging them has the interesting effect of moving all its reflections, thereby
drawing attention to key properties of the transformation.

CONCLUSION
The early stages of learning geometry should provide students with an ability to recognize
shapes and to begin to explore their simpler properties while learning the names and other
language associated with them. The visual attractiveness and the practical investigations
involved make the subject very motivating. It is important though that time spent making
diagrams attractive and in doing practical work does not hinder the development of language
and the power to observe, describe and to classify shapes and the ability to measure and produce
clear and accurate diagrams as well as sketches which identify key features of a shape.
The long-term aim is to develop students' ability to think deductively as well as to build their
powers of visualization and geometrical intuition. Deductive thinking has its beginnings in
expecting students to explain why something is true and to make sense of the explanations given
by their peers and by their teachers. This requires discussion, as well as practical and written
work, and a developing sense of the importance of explanation and justification alongside
practical skills and factual knowledge.
This chapter has highlighted some important initial ideas and skills to do with recognizing
shapes and some of their properties, the importance of symmetry and the related transformations of reflection and rotation. The idea of angle is a crucial one, but it is a frequent
source of difficulty. The use of simple tasks with LOGO allied to actual bodily movements and
learning to measure with a protractor in a way that reinforces the link with turning are both
valuable aids in developing understanding.
In subsequent chapters these ideas and others will be developed further and placed in a
context where reasoning becomes increasingly important. Nonetheless there is always a role for
experiment in learning geometry at any level. When a new idea or a particular difficulty is
encountered measurement, cutting and folding, making models and other experimentation all
have their place in aiding understanding and developing geometrical intuition.

" write a reflection about that text. do not miss any important point.