Geometry
Reformed pedagogy
Geometry
Geometry
Page 1
Some geometrical experiences: Overview
The following is the next phase in the Maths in
the Kimberley project and consists of some
lesson experiences. It follows on from
interviews with students at Kulkurriya and Yiyili
schools.
There are three aspects to the following. Note
that these do not address all aspects of
geometry required by the curriculum. Some key
aspects not addressed are transformations
(symmetry etc), and angles.
The focus on geometry is based on an often
stated assumption that the prevalence of
direction words in some Indigenous languages
implies that the learning of aspects of geometry
may be closer to Indigenous students’
experience than the learning of number. The
following is seeking to explore this.
The three lesson sequence, each of which
represents approximately one week of work, is
not intended to be sequential and so can be
taught in any order. They are written for upper
primary classes but are probably suitable for
secondary students as well.
If students do seem familiar with the concepts
then it would make sense to use this aspect of
the curriculum more actively to build
confidence, success, and connection to
mathematics and schooling. It may also be
possible to incorporate visual approaches to
other topics.
The three aspects are:



Perspective
Nets and properties of 3D geometrical
shapes
Maps (this one is not included yet)
I might add another on isometric drawing, if the
first ones go well.
It also seems that geometrical questions are
over represented in NAPLAN assessments so
some attention to this topic is likely be useful
for students and the schools.
For each aspect, the following presents:
An interesting aspect is that, in inspecting items
in NAPLAN assessments, it seems that the
majority of questions depend on knowledge of
conventions as much as knowledge of space
and location. One of the important conventions
is the way that 3D shapes are represented on a
2D page or screen. Metropolitan based children
are exposed to many such images and perhaps
the remote students do not have such
opportunities.

The approach proposed recognises that a
significant challenge for teachers is the process
of connecting English language, especially
specific mathematical terms, with student
experience. If concepts seem well developed in
relevant language then the goal of instruction is
to connect the English terms with local terms. If
some concepts do not seem well developed
then the goal would be to provide experiences
in the use of the various concepts.
Page 2

results from some data collection that has
informed the emphasis as well as a
sequence of the activities.
a sequence of experiences, not written as
scripted lessons, just as general ideas. It is
assumed that teachers will adapt these to
their own students.
The research questions at this stage are:
If lesson sequences are designed to address
specific geometrical concepts as identified by
student interviews:


Which aspects of those sequences are
engaging for students?
Which aspects of those sequences improve
responses of students to geometry items on
NAPLAN style assessments?
Geometry
Sequence 1: Perspective and location
Preliminary data collection
There are a number of items on NAPLAN that
require students to interpret diagrams and
drawings from different perspectives.
In an individual interview format, the students
were presented with 2009 NAPLAN items on
perspective and location that were relevant for
their level, with the items read to the students if
necessary.
In summary the results were:
 13/15 of the middle primary students could
choose the correct piece to go into a jigsaw
 Given a prism with an L shaped cross
section that required students to count
faces, 4/15 of the middle primary students
could choose the correct response (5/15
counted only the faces that could be seen)
 Given a horizontal view drawing of an ice
cream cone, 5/15 could choose a circle as
the top view.
 For the upper primary, given the net of a
cube, 5/13 could state which face is
opposite a nominated face
 Shown a cube with corners cut off and
asked to count the remaining faces, no
upper primary student could write the
answer correctly.
 Given a diagram of a rectangular prism, and
told that the sum of the faces, edges and
vertices is 26, 3/13 could choose the total
of the faces, edges and vertices of a square
pyramid.
There were also some comparative items that
were presented using objects that could be
touched or seen and not in the test item format.
The results from this were:


Given a small cube, 5/28 could name it,
4/28 could count edges, 18/28 could give
the number of faces, and 17/28 could
count corners.
Shown a photo of a structure made from
cubes, 4/28 could state how many cubes
were needed to make it, 1/28 could say
how many faces would be painted (one said
Geometry



all of them), 25/28 could make it
(development was even across the grades),
23/28 could place a yellow cube to the left
of the structure (7/11 grade 4s), 14/28
could put a red cube north of the structure
(development was even)
Asked to draw the bird’s eye view of the
building they were in, 12/27 could do this.
Asked to draw the water tank from above
17/27 could do this
6/17 Kulkarriya were a able to draw the
shape of the homemaker building seen
from above, all of whom were from the
upper grades
In summary, nearly all of these items were
answered correctly by some students, and most
students were able to do some of the tasks,
including replicating a shape in a photo, and
some aspects of the properties of the cube. No
student was able to answer all of the items. In
general, the year 6/7 students did not seem
better able to do the tasks than the 4/5
students.
Overall goals of this sequence of experiences
on perspective and location
It is necessary to explain to the students why
they are doing this. The fundamental goal, that
you can say over and over, is “imagining”.
The capacity to interpret diagrams, drawing and
photos is important for some topics in
secondary level mathematics such as finding
the volume of objects, and practical problems. It
also helps in reading maps, interpreting drawing
of buildings, communicating directions,
describing objects and understanding
instructions. These are important life skills.
Hopefully some experience at these tasks will
also assist students in completing future
NAPLAN items as well.
The following some activities intended to
provide some relevant experiences.
Page 3
Step 1: Revising and
introducing the key terms
Explain the purpose to the students:
Explain that they are going to learn about
interpreting and making drawings of objects,
and how those objects look different from
different perspectives, and that this will help
them in their lives and also in their future
school learning. Explain that one of the key
issues for them is to learn the words that are
used to describe the ways things look.
is vertices), 2 dimensional, 3 dimensional, 2D,
3D, perspective, bird’s eye view.
Note that it is suggested that you do some
language activities everyday that you are doing
this work.
As a prompt to this discussion, you could find
some images that the students might recognise
(perhaps like the ones here). Ask them what the
logo or graphic might represent. How do they
know? How would they describe the image?
Teaching experiences:
It is assumed that you will use your usual
strategies for revising and introducing terms.
The strategies you use in AL may be useful. The
students need to hear you modelling the use of
the terms, and to say the words themselves
many times.
Other strategies could include
-
-
-
-
having labels on parts of the room to
show north etc, left right, above, below
etc.
emphasising direction words when
telling stories
asking the AEWs or other locals what
words would be used in Kriol or
language and have them talk to the
class about them. This could include
asking specific questions such as “how
do you show north? Left? near? above”
any other strategies that you find work
in literacy such as word searches
connect the terms to what they know
(“have you heard the word left before,
where? How do you remember left”)
asking what is similar and what is
different about, for example, a square
and a rectangle
Key terms for revision are left, right, above,
below, under, on top, side, north, south, east,
west …
Key new words for specific and formal
emphasis are face, edge, corner, vertex (plural
Page 4
Geometry
Step 2: Drawing and
making to instructions
interpretation. Repeat, adapting to the level
of success of the students.
As before you might ask one of the students
to give instructions. In fact they could give
the instructions in Kriol if they prefer.
Step 2: Drawing and making to instructions
Explain the purpose to the students:
Explain that it is often necessary to give
instructions and descriptions, and it is also
necessary to be able to interpret them. The
following activities give experience at using and
interpreting the necessary language.
Teaching experiences:
a. You will have to adapt this activity to the
equipment you have. The ideal is the sets of
Lego blocks with a base plate so that
students can hold up the shapes once they
have made them. However, you can do this
with any materials.
c. Have pile of cubes. Ask the students to:
i. Build something using 15 cubes.
Describe what they have built.
ii. Build something which is 3 cubes high
and 3 cubes wide made with 15
cubes.
iii. Something like “make a tower of 3
cubes, put a yellow cube to the left,
and blue cube to the north, …”
Repeat, adapting to the level of success of the
students.
Everyone has the same equipment (either
individually or in pairs), including you. You
make a shape but have it covered. Describe
what you have made, with the students
having to make what you say. For example,
you might say, there is a blue block on top
of the red block, there is a yellow triangle to
the left of that, there is a another blue block
to the north, and so on. Do this a number of
times creating different instructions each
time. If they are doing well, make the
instructions more complex. The review of
this is important. Have the students
compare what they build with the one that
you built. Are they the same?
You might ask one of the students to make
a shape and give instructions. In fact they
could give the instructions in Kriol if they
prefer.
b. Give the students instructions, and ask
them to draw them on a small white board.
For example, “draw a square, draw a
triangle on top of the square, draw a circle
to the left, etc”. They can show you their
whiteboards to see whether they match
what you said. You can comment on their
Geometry
Page 5
Step 3: How many cubes?
They could also do this in pairs, with one
student making a shape and them asking the
questions of the other.
Teaching experiences:
Explain the purpose to the students:
Explain that these activities are for students to
experience the number of cubes used to
construct a shape even though those cubes
may not be immediately evident in the photo.
One of the tasks that the students found
difficult in the NAPLAN test was imagining
unseen but necessary aspects of objects.
Make a structure something like this photo
using cubes, ideally they should be coloured.
This will depend on the equipment you have.
You may have linking cubes so you will have to
adapt this.
Ask questions like:
“How many cubes did I use to make this?”
“If I painted the outside black including those
on the bottom, how many faces would be
painted black?”
Students can walk around your structure at
first, but for later questions have them just look
from one side.
Repeat this a number of times using different
structures.
Page 6
Geometry
Step 4: Taking a bird’s eye view
Explain the purpose to the students:
Explain that sometimes drawings and photos
show only one perspective of a shape, yet it is
necessary to imagine what the rest of the shape
might be. There is a particular view that is
called “bird’s eye”.
Teaching experiences:
a. Ask them to draw what the school bus
would look like to a bird flying directly above
it. What would be school bus look like to a
dog sitting on the road as the bus
approached.
be the object?“ (Note that there is a
range of possible anwers to this
question so consider all reasonable
suggestions. Repeat this for different
shapes such as rectangle, etc?
d. Have a photo of something around the
school. Ask students to say where the
photo was taken from. Ask them to draw
what the building would look like if the
photo was taken from the opposite
side? … if the photo was take from
above.
e. Have drawing of a building from above.
Ask them to suggest what building
might your drawing be?
b. Show a picture from google earth of the
community (on the next page – actually
you might be able to get a better one).
You might be able to put this on the
smartboard. Ask them to name the
various buildings or objects they can
see. Ask questions like “how do you
know that is the river?”
Explain that this image is old. What has
been build since this photo was taken?
c. Draw a circle on the board. Sat “This is
an object seen from above, What might
Geometry
Page 7
Page 8
Geometry
Step 5: Sorting the cards
Explain the purpose to the students:
Explain that this activity will give them the
opportunity to revise the work from the previous
lessons, and to match up some different views
of the same object made from cubes. Emphasis
that they will have to explain their reasoning.
Teaching experiences:
Step 6: Worksheet
Explain the purpose to the students:
This worksheet gives some revision and also
extends their knowledge of shapes.
Teaching experiences:
The students can work on the sheet individually
or in pairs. It is quite difficult, so you may need
to adapt it.
There are two sets of cards: an easy one to start
with, and a harder set if they can do this well.
The set of cards has 5 different representations
of objects made from what they mean. You
could ask them to read the individual cards
together. Ask them to interpret the information
on the cards. These are provided. Have some
large versions to read through, and discuss
The idea is that students, in pairs, match up the
different representations of the objects, and
then describe what they have done. Try to give
them as little help as possible.
You will need to decide whether this is too
difficult for the students depending on how they
have gone on the earlier steps. If you feel it is
too difficult, you might simplify the card set by,
for example, removing some of the cards.
After they have done that, there is a need to
review their answers. One possibility would be
to put the cards onto the smartboard. Another
might be to have a large version of the cards.
Another might be to have a set with Velcro on
the back.
Geometry
Page 9
I have 4
faces and
6 edges
I am a
triangular
prism
I have 6
vertices
and 9
edges
I have 12
edges
I am
square
pyramid
I have 4
vertices
I have 6
I am an
faces and
octahedron
8 vertices
I have 5
faces
My net is
My net is
My net is
My net is
I have 12
edges
I have 8
I am a
edges and
tetrahedron
5 vertices
My net is
I have 8
faces and
6 vertices
Page 10
I have 5
faces
I am a
rectangular
prism
Geometry
Revision sheet for perspective and location
Name ________________________
1. Draw the bird’s eye view of the building you are in.
2. Draw what your house looks like if you standing on the road.
3. Draw a square. Draw a triangle on top of the square. Draw a circle so that it is inside the
square. Draw another triangle on the right hand side of the square.
4. Look at this drawing.
a. How many cubes would be needed to
make this building?
b. Draw the bird’s eye view of the building
Geometry
c.
If you painted the outside of this building black, (the bottom is painted as well),
how many faces would be painted black?
d.
Which of these might be the view from the front?
Page 11
Preliminary data collection
There were 3 questions requiring analysis of nets
or related tasks in the Year 5 2009 NAPLAN. The
questions were asked individually to each
student, and read to them if necessary, to
minimise the chance that any difficulties were
with the language rather than the geometry. The
questions were:



Given the net of a cube, when asked to find
the face opposite a given face, 5/13 of the
upper primary students could answer (this
was a write-in answer).
Given the net of a pyramid, and asked to
match a drawing with a net, 4/13 of the upper
primary students could do this (choosing from
4 options)
Asked to find the sum of number of faces,
edges, and vertices of a square pyramid, 3/13
of the upper primary students could do this
(choosing from 4 options).
We also asked some more direct questions to the
middle and upper primary students.

Shown the net of a cube, 1/13 could draw the
net of a square pyramid.
In summary, while there are issues with names of
shapes, it seems that some students are able to
answer most of these questions, suggesting that
such concepts are accessible at this level. That
the majority of students could not, even in the one
on one interview situation, indicates that it is
worth seeking to improve their experiences in this
aspect of geometry.
Page 12
Overall goals of this sequence of experiences on
nets
It is necessary to explain to the students why they
are doing this.
The fundamental purpose is for students to
experience the conventions of connecting 2D
representations of 3D shapes. It seems a useful
step in doing this is through some experiences
with nets, in that these show directly how a 2D
sheet is converted to a 3D shape.
There is also a pleasurable aspect to this,
somewhat like an Art lesson, in that the shapes
when made are nice to look at.
There are also some problem solving aspects of
what is proposed.
There are 9 steps in this sequence. I have no real
idea how long each step will take (and will want
you to tell me that) but I think that all of the steps
will be needed.
Note that there is some great software that goes
with these types of activities which would be ideal
on the smartboard. You probably have some of
these already, but there is also plenty available
free on the web.
Geometry
Explain the purpose to the students:

sorting sets of shapes into groups and
describing the groupings
asking what is similar and what is different
about, for example, a square and a rectangle
any other strategies that you find work in
literacy such as word searches
Explain that they are going to learn about shapes
and the words we use to describe them. Once they
do this, it will be easier to understand and
describe what they are doing in maths.

Teaching experiences:
Key terms for revision are square, rectangle,
triangle, …
It is assumed that you will use your usual
strategies for revising and introducing terms. The
strategies you use in AL may be useful. The
students need to hear you modelling the use of
the terms, and to say the words themselves many
times.
Other strategies could include







having posters with the shapes drawn and
labelled
having labels on everyday objects (such as
this is a vertex, this is an edge, ...)
matching labels with the objects
emphasising shape words when telling stories
connect the terms to what they know (“have
you heard the word cube before, where?”)
asking students to “tell me everything you can
about this shape”
asking the AEWs or other locals what words
would be used in Kriol or Language and have
them talk to the class about them. This could
include asking specific questions such as
“what is the difference between a square and
a cube?” “what would you call this shape?”
Geometry

Key new words for specific and formal emphasis
are cube, net, face, edge, vertex (this is a better
word than corner), 2 dimensional, 3 dimensional
Further words that need to be developed include
rectangular prism, square pyramid, tetrahedron
(triangular prism).
Note that it is suggested that you do some
language activities everyday that you are doing
this work.
Page 13
Explain the purpose to the students:
Explain that whenever they see pictures or
drawings of 3D objects they have to imagine what
the object is like, not just what they can see
directly in the picture.
Teaching experiences:
Put a big die on the table. Ask students to tell you
everything they can about the die. Extend them
with questions like “what is the mathematical
name for this shape?” “can you tell which number
is on the bottom without picking up the die?”
“what is the number of the north side of the die?”,
what is the number to the left of the die?’ “what
number would a bird flying overhead see?” etc.
Explain the purpose to the students:
Explain that one of the key ideas in mathematics is
seeing ways that 3D objects (like a box) are
sometimes made out of 2 D shapes (like a square or
a rectangle).
Teaching experiences:
Find some boxes, ideally one in the shape of a cube
and another that is a rectangular prism (a box shape
that is not a cube).
Focus on the cube first. Ask them to work out how
many faces there are, how many edges, how many
corners.
Cut it up along the edges. You might need to prepare
this beforehand.
Ask the students to try to reassemble the box. If the
class is big, you might need to have a few boxes.
Repeat this for the rectangular prism box.
Page 14
Geometry
Explain the purpose to the students:
Explain the purpose to the students:
These experiences are the reverse of the previous
one. It might seem like a waste of time, but
actually it is the most important in this sequence
because the students need to have the
experience. The goal is that they see how the 3D
geometrical shapes can be made. It also gives
them practice as being careful, and doing things
artistically.
This worksheet gives the students practice at
imagining which nets can be made into the cube.
There is some interactive software that does this.
It is also a problem solving type experience.
Teaching experiences:
Give them the net of a cube (nets are provided). I
have provided two sets: one set has the flaps and
the other does not. The nets with the flaps are
made by gluing the flaps to the relevant face. The
ones without flaps are made by using sticky tape.
Choose which ever one you think will work better.
Maybe you could try both to see which one works.
Emphasise to them that they need to be careful,
in doing this, and fold carefully along the lines.
Teaching experiences:
The students can work on the worksheet
(attached) to imagine which nets can be made
into cubes and which cannot. They need to explain
their reasoning. You perhaps turn this into a
poster and do it as a whole class.
Ask them to:





decorate the faces in some way, while it is flat,
including with their name
fold and make into the cube, - you will need to
explain the need to fold carefully, the ways to
tape the edges, etc
once they have done this, ask them to
describe their cube to the class;
choose a few of the cubes and ask formal
questions like: What is the shape of one face,
how many faces, point to an edge, how many
edges, point to a vertex, how many vertices
(note the plural form – they might see this in
the next NAPLAN),
ask problem solving questions like “What is on
the face opposite the (you choose something)”
Geometry
Page 15
Just imagining
1. Which ones of these can be folded to make a cube?
a)
b)
c)
d)
e)
f)
g)
2.
This is the net of a cube.
When it is folded which number will be opposite the 1?
When it is folded which number will be opposite the 2?
When it is folded which number will be opposite the 3?
When it is folded which number will be opposite the 4?
When it is folded which number will be opposite the 5?
When it is folded which number will be opposite the 6?
Page 16
Geometry
Explain the purpose to the students:
Explain the purpose to the students:
This is the same as the making cubes from nets
but it makes a different object.
This is taking the same idea one step further.
Teaching experiences:
Given the net of a rectangular prism (two sets of
nets provided). The same comments about nets
and flaps etc above apply.
o
o
o
o
Geometry
decorate, including writing in the
name
describe what you think this
might look like when it is folded
fold and make into the
rectangular prism taping the
edges,
ask them to tell you everything
they can about the rectangular
prism,
 What is the shape of
one face, How many
faces, how many edges,
how many corners,
 What is on the face
opposite the (yellow
flower?)
If you need to skip any activities, then this is the
one to skip. However if the students are doing
well to this stage, I suggest you do it. It is more
important for seeing the beauty of these
shapes, but less important in terms of their
future NAPLAN tests.
Teaching experiences:
Give different nets to different students, such
tetrahedron, square pyramid, octahedron (again
two sets are provided)




decorate, including writing in the name
describe what you think this might look
like when it is folded. Maybe have some
pictures or even models of these.
fold and make
ask them to tell you everything you can
about the shape,
 What is the shape of
one face, How many
faces, how many edges,
how many corners,
 What is on the face
opposite the (yellow
flower?)
Page 17
Explain the purpose to the students:
Explain the purpose to the students:
This is a problem solving activity that requires
them to connect the different words,
descriptions and nets, and pictures together.
Explain that this gives them a chance to use the
knowledge they have learned in the previous
experiences.
This worksheet gives some revision and also
extends their knowledge of shapes..
Teaching experiences:
Teaching experiences:
The students can work on the sheet individually
or in pairs. It is quite difficult, so you may need
to adapt it.
There is a set of cards, with 5 different
representations, of 3D objects. I have included
some sets of cards and also a set of large
version cards.
Read through the large versions with them, and
discuss what they mean. You could ask them to
read the individual cards together. Ask them to
interpret the information on the cards.
The idea is that students, in pairs, match up the
different representations of the objects, and
then describe what they have done. Try to give
them as little help as possible.
You will need to decide whether this is too
difficult for the students depending on how they
have gone on the earlier steps. If you feel it is
too difficult, you might simplify the card set by,
for example, removing some of the objects
represented, or removing some of the cards.
After they have done that, there is a need to
review their answers. One possibility would be
to put the cards onto the smartboard. Another
might be to have a large version of the cards.
Another might be to have a set with Velcro on
the back.
Page 18
Geometry
REVISION OF NETS WORKSHEET
Name ______________________________
1. Write down everything you can about this object
2. Complete the missing parts of this table
Name
No. of faces
No. of edges
No. of vertices
3.
On the back of this page, draw the net of this object
4.
I am imagining an object that has at least one face that is a triangle. Draw what this
object might be, and draw its net.
Geometry
Page 19