Exploring
Two Dimensional
Shape
Doreen Drews
Naming 2 Dimensional (Plane) Shapes
Rectangles
Oblongs
squares
Triangles
equilateral
(3 sides equal)
isosceles
(2 sides equal)
right-angled
Hexagons
Pentagons
scalene
(no sides equal)
Circle
Octagons
2
Glossary for 2D shapes
Plane shape
A two-dimensional shape. Often referred to as ‘flat shapes’ by lower juniors.
Closed shapes
Open shapes
Polygons
Closed shapes with straight sides.
Regular Shapes
Shapes with equal length sides and all angles equal in size.
Quadrilaterals
Polygons which have four sides.
Parallelograms
A name given to quadrilaterals in which both pairs of opposite sides are parallel.
Vertex (plural vertices)
A point where lines or edges meet
3
Investigating Lines:
A starting point for work on 2D shape
For children geometry should be a powerful way of exploring and understanding the world
around them.
Straight lines and curved lines abound in the environment, both natural and constructed, and a
walk or ‘trail’ can act as a useful source for exploration.
Look for striped things.
Look at lines in the environment.
Curved, straight, wavy, horizontal, vertical, parallel.
Look at outlines.
(pictures by Rita Thompson)
4
Thick lines and thin
lines,
Look at the lines
in a thumbprint.
Look at the lines in leaf
rubbings
Draw some pictures of spirals,
diagonal, wavy and curly lines or stripes.
Find spirals in the
environment
A spiral
pattern
Use coloured strips to make
pictures with horizontal
and vertical lines. Look at
the intersections. What
shapes can you see? What
size are the angles?
Draw an outline of a shape with curved
lines and a shape with straight lines with
curved lines and a shape
Draw some intersecting lines.
Do a different pattern on
shapes with 3 corners,
4 corners, 5 corners.
(pictures by Rita Thompson)
5
Straight Lines
1.
Suggest that children try to draw a straight line without any aids - check efforts by using the fold
of a piece of paper, or tightly pulled string, or a ruler.
Search for straight lines in the classroom.
Horizontal and Vertical Lines
The horizontal can be demonstrated by showing how a ball will not roll if placed on a horizontal surface.
Part-filled transparent bottles are useful to show that water level is always horizontal no matter how the
bottle is tilted.
The vertical can be demonstrated by use of a plumb line (a lump of plasticine attached to the end of
string). Objects in the classroom or outdoors could be tested to see if they have vertical edges.
Discuss with children that all the lines they draw on their paper are horizontal (providing that their desk
is!), but that if their ‘line pictures’ are mounted on a vertical wall some lines would be vertical, some
horizontal.
The children could draw straight line pictures on squared paper and use two different colours to highlight
which lines would be vertical, which horizontal, when the picture is mounted on a wall.
horizontal. length
vertical. length
total length
8cm
12 cm
20cm
6
Perpendicular Lines
Two lines are perpendicular if the angle between them is 90º
horizontal
A vertical line is perpendicular to a horizontal line.
Children can use a piece of paper folded twice to check for right angles and perpendicular lines in the
classroom, environment, road signs etc.
Parallel Lines
Look for examples of parallel lines in the environment. Look at the edges of books, stripes on a jersey,
ladders, sides of a football pitch. Make straight parallel lines with the edges of a ruler or by folding a
rectangle of paper edge to edge to produce creases that are all parallel to these edges, or even with the
prongs of a fork! Ensure children realise that two parallel lines always remain the same distance apart,
that they have the same direction and will never intersect however far they go, unlike these:
Using isometric paper, ask children to identify, and colour in, the following shapes.
parallelogram
rhombus
parallelogram
equilateral
triangle
trapezium
regular hexagon
7
pentagon
Can they establish how many sets of parallel sides each shape has?
Can they find more shapes on the paper which have sets of parallel sides?
Do any shapes made have no sets of parallel sides?
This would could be continued by children being challenged to find different shapes made up from two
sets of parallel sides
e.g.
or
Straight Lines and Numerals
Write the numerals 0 9 on pieces of paper and ask the children to sort them for straight/curved lines.
Their result could be represented on a diagram such as a Venn Diagram.
-
Where the numerals are placed depends on how they are made. Children could draw digital numerals and
discuss where they would be placed.
8
Exploring 2-D Shapes
Sorting
Using a mixed set of 2-D shapes open, closed, straight sides, curved sides, regular, irregular etc.
-
or
Using a set of cards with shapes drawn on card.
When describing and sorting shapes it is important to look at
their properties as well as their names children may readily identify a square as belonging to a set of
shapes with ‘four right angles’, but may not put it in a set of ‘rectangles’ or ‘parallelograms’.
-
Using Clixi/Polygon
CLIXI HOUSE GRID
Use Clixi to make some kind of house.
Record the number of shapes you used on the grid.
How many squares did you use? How many red shapes?
How many shapes altogether? is your friend’s house
different?
red
blue
9
green
yellow
Constructing 2-D Shapes
Construction may involve use of straws, lollysticks, geoboards, card, string, measuring instruments, paper
and pencil.
•
Using geostrips, or punched strips of card which can be fastened with split pins, polygons can be
constructed and manipulated to demonstrate differences i.e. between a square and a rhombus.
Geostrips are particularly useful to allow exploration of rigidity in shapes.
•
Using construction straws:- how many different triangles can you make from a set of straws of
varying lengths?
•
Using geoboards:- how many different polygons can you make on a 5 x 5 geoboard? How many
different regular polygons? Try making different shapes each with a perimeter of 12 units.
10
• Constructing Circles
a) Radius and Diameter
Drawing around the end faces of solids doesn’t help
children appreciate that circles have constant radius
and diameter. This can be done by putting one end of a
loop of string around a drawing pin and the other
around a pencil and pulling the loop taut.
If the length of loop is shortened, concentric
circles can be drawn, like those on a target.
The same method can be adapted to draw large
circles perhaps a circus ring, in the playground.
-
Children can also draw circles if different sizes using a
geostrip, by putting a drawing pin through one hole
and a pencil through another.
Both these methods provide a good introduction to
the use of compasses.
b) Circumference
Collect different-sized round containers with lids. Notice how each lid fits on in many
ways (unlike the lids of other shaped tins). Arrange the lids in order from the one with the smallest
diameter to the one with the largest diameter like this:
Ask the children to guess which they think has largest circumference, the next largest and so on.
Check with a thin strip of paper.
Choose one of the lids and ask the children to guess if the circumference is about twice, three or
four times the length of the diameter. Now wrap the strip of paper around the circumference,
cutting off its length. Stretch this across the lid and fold off the diameter length. The children will
see that the circumference is as long as 3 and a bit diameters, but should test all the other lids in
the same way. Ask them to test this relationship with a very large circle like the circus ring in the
playground.
• Using LOGO (Roamer, Floor or Screen Turtles)
• Using ATM Maths (see following pages from Mathematics Teaching June 1991)
11
Branches
From Camborne
Using black and white mats
in the classroom: some
ideas from an ATM
morning
Using the square
mats can you make
one continuous
line?
Can you make a
design with no
closed shapes?
How many square mats do you
need to make a small square?
How many more mats do you
need to make the next size
square?
Make a circle
using three
hexagonal mats
How many mats
do you need to
make
a triangle?
How many more for
the next ... ?
Create a pattern
. What happens
if you slide one
of the rows of
mats sideways?
How do you
place mats to
keep the pattern
open? What
happens if you
slide one row of
this pattern
sideways?
Arrange some
mats behind a
screen. Describe
the hidden
arrangement for
someone to copy.
12
.
Given 20 hexagons:
make a design with as
many triangles as
possible
How many mats do
you need to make a
circle?
make a design with no
closed shapes
How many mats do
you need to make a
triangle?
What can you do
with three mats?
How many lines of
symmetry are there
with three mats?
P
How do you
arrange
mats to
have the
maximum
number of
regions?
Take six mats and put them
together at random. What
patterns have you
made?
Use these ideas to
make a design
with ten tiles
twenty tiles...
make a design with as
many circles as
possible
.
Put five mats to make a straight
line; and another four to make a
ut
wavy line. Using
3, 2 and then
make
1 another
mat, build
up
a
triangle
that
four to make
contains
at
least
1
closed
shape
a straight line; and
How many ways can you do
this?
Comments:
There is a need to play before
asking any questions
The quality of the questions
increases with the length of the
playing time.
Can we get children to ask
.
themselves
questions? (We
/
.
found others us questions
intrusive).
Working with others gives
richness and stimulates
discussion and questions
.
How many
Using three mats how many
different patterns can be
ways can you
made?
Which of these are
(a)symmetrical,
(b)asymmetrical?
Which have more than one
line of symmetry
how many patterns can you
make with just three lines of
symmetry?
Using hexagons:
play with making patterns.
Try to make a small triangle; make a larger
one around it; and a larger one around them;
and …..
by rotating one, how many different
patterns can you make?
lines of symmetry are there?
circles can you make?
13
Put five hexagons together to make
a continuous base line.
Add four hexagons above to make
a wavy line
Add three hexagons above to make
new shapes
Add two hexagons. Add one
hexagon to complete the triangle
Some part of the triangle must be
joined up. How many joined
shapes can you make?
Symmetry
Axes of symmetry
Horizontal
Infinite
Vertical
Axes of symmetry
Make some blots
Fold and rub.
Open out
The cut out shape is
symmetrical
Complete the pictures
.
.
Make each shape symmetrical
THE CHILDREN CAN USE
MIRRORS TO CHECK
REFLECTED IMAGES
Draw on axis of symmetry on each shape
Symmetrical shapes drawn on paper can be folded exactly into halves so that the two halves lie
exactly on top of each other. How many ways can we fold squares .. rectangles … circles
circles etc.
…
14
Back to front and upside down
Take a look at the face on the right. Not a very happy character
— but if you rotate the page through 180° you get a change of
mood and character.
Reflecting on letters
If you put a mirror alongside the letter A you will find that its
reflection looks the same.
But if you put the mirror underneath
A
…it doesn’t. The A is upside down.
If you put a mirror under the letter B
the reflection
is the same.
But if you put the mirror alongside the letter B its reflection is backto-front.
The letter A has vertical symmetry; the letter B has horizontal symmetry.
Which letters of the alphabet have horizontal or vertical symmetry? Which
have both? Which letters have no symmetry? Are there any letters whose
reflection stays the same no matter where you place the mirror?
15
ABCD
EFGH
I J KL
MNOP
Q R ST
UVWX
YZ
Line Symmetry (Bilateral)
The line of symmetry and half of some shapes are shown below. Using a mirror or tracing paper,
complete the shapes.
a)
b)
c)
d)
f)
e)
Rotational Symmetry
Mark the centre of rotation on the following shapes and find their order of rotation.
a)
b)
c)
d)
f)
e)
16
Rotation
Rotation through 180° and translation.
Reflection
Translation (slide)
Horizontally
vertically
17
diagonally
Tessellation
Tessellation literally means tiling, covering a flat surface with identical shapes which fit together
in a regular pattern in every direction.
A regular tessellation using only one
regular polygon.
A semi-regular tessellation using
two regular polygons.
It is not only regular shapes which tessellate.
You can design your own tessellating tile. Start with a shape which tessellates e.g. a rectangle.
Begin by writing some shapes from the rectangle and ‘sliding’ or rotating your cut out shapes into
a different position.
a) by rotation.
b) by translation (sliding)
Decorate your shape then draw round it to make a tessellating pattern.
(pictures by RitaThompson)
Tiling Patterns
Using these square mats, experiment to find how
many DIFFERENT tiling patterns can be generated.
Can you find a way of recording your pattern so that it could be duplicated
without looking at your square mats?
NOW try to create your own tiles! Using the sheets provided, choose two
colours and make IDENTICAL tiles. Cut them out and experiment to find a
variety of patterns. Stick your tiles down on paper to show your favourite
design. Can you find a way of recording your pattern?
RANGOLI PATTERNS
Background information
Rangoli patterns are used by Hindu and Sikh
families to decorate their homes during festivals
like Divali. Some are pictorial, while others are
geometrical and abstract. Some of the patterns
are made on the door—step, from food—stuffs,
such as rice, lentils, split peas, and seeds, so that
everyone coming into the house would pass over
the design and have good luck. Various
techniques can be used in creating Rangoli
patterns.
When a design has been drawn on dotted paper,
(See instructions on ‘Creating Rangoli
Patterns’), it can be used as a basis for collage
work: block printing : embroidery; batik work,
etc.
Another type of Rangoli pattern is that which is
used by women to decorate their hands and foreheads during weddings and festivals. These are
drawn with ‘mendhi’ — a mixture of powdered henna, lemon juice and water, - and applied with
a sharp stick.
MATHEMATICAL CONTENT
Concepts/Knowledge of:
Symmetry, reflection, recognition and properties of 2 - dimensional shapes, tessellation. etc.
Skills:
Marking out a square grid.
Outlining the top left—hand quarter of the square.
Drawing lines accurately using pencil and ruler inside the quarter.
Observing Images of the lines formed when a mirror is placed on the axes of symmetry.
Plotting the reflections of these lines in two, then four axes of symmetry.
Colouring or shading regions neatly. Repeating the complete pattern.
Rangoli patterns can be fed into the various computer programs available for work on
Tessellation.
CREATING RANGOLI PATTERNS
(1) Select a square grid and draw a few lines by joining pairs of dots.
(ii) and (iii) Reflect these lines in each of the four axes of symmetry of the square.
(iv) Shade in the regions to accentuate some of the shapes.
.
Adapted from ‘Mathematics for all’ Wiltshire Ed. Authority
CREATING ISLAMIC PATTERNS
Patterns by Matthew Fryer (aged 12) of Avon Middle School, Salisbury
The pattern is initiated within the square by drawing in lines such as diagonals, joining midpoints
of lines, end so on — making sure that symmetry is maintained in the four axes of the square.
Parts of these lines are then rubbed out — again keeping to the symmetry — until a satisfactory
pattern has been produced. The completed pattern is then tessellated to produce a large design.
ISLAMIC PATTERNS
Islamic patterns are more elaborate than Rangoli patterns. They are geometrical and abstract and
can be found decorating many Islamic buildings. Although the method we have employed in
creating the patterns is developed from a square base, it needs to be stated that a variety of shapes
which tessellate can also be used.
This pattern is built up by drawing a number of ‘obvious’ lines within the square — ensuring
symmetry is maintained in all four axes--.(see illustration and foot—notes on next page). Parts of
these lines (and their images) are then selected and removed. The completed pattern is then
repeated. This stage might appear, in theory, to be laborious and time—consuming — especially
if the pattern has to be drawn. Our experience, however, has shown that the pupils not only
become deeply engrossed in the activity, but are amazed and excited as new shapes emerge from
the ‘gaps' to produce a complex and elegant final design. Colouring the regions produces a
striking effect. The patterns can be used as a basis for art work and an introduction to calligraphy.
MATHEMATICAL CONTENT
Questions similar to the following might be asked about the properties of the shapes within a
completed pattern.
GEOMETRY
Observe and identify the various geometric shapes in the illustration: right-angled triangles.
isosceles triangles, quadrilaterals, squares, kites, parallelograms. trapezia, etc. Find the reflected
images’ of these. Which of these shapes are regular? What are the angles? How do you know?
Which of the triangles are right—angled isosceles? What are the angles on the points of the star?
Area:
How many times bigger is the large square than the small one in the centre? How do you know?
How many right-angled triangles are there altogether? Are they the same size? Which is the
largest triangle? How many times larger is this triangle than the smallest one? How would you
calculate the area of the four—pointed star? What other shape has the same area?
Extension
Introduce circles into the square and observe the effect on the shapes which emerge in the design.
Draw Islamic patterns using a variety of shapes for bases e.g. triangles, hexagons, or
combinations of other polygons. Find out more about Islamic patterns e.g. connections with
Magic Squares (see Critchlow: Islamic Patterns).