Directions for Using Worksheets on Thinking about 3D Shape (based on
Owens’ unpublished 3D test)
Equipment
Rectangular prisms (e.g., Cuisenaire rods, boxes or Lego pieces (3 shapes including
thin one), cone, cylinder, cube, open cubes which can be opened out, a square of
paper
Introduction
 Say these worksheets are about thinking in 3D space and about 3D shapes or
solids.
 Show at least 5 different-sized rectangular prisms such as cuisinaire rod, box for
disks, a thin and thicker Lego blocks. Whenever showing objects or 3D shapes turn
them slowly around and over. Ask What do you call these? (expect rectangular
prisms but if they say cuboid or square prism say that we will include them all as
rectangular prisms. All the sides are rectangles and they sit flat on the table with
the top parallel).
 Show a closed cube and an open cube. Ask for the names and why the difference
in name. Ask for the number of sides (6 and 5 respectively).
Show a triangular pyramid and a square pyramid. Ask What do you call these?
(expect pyramid and prompt for the two kinds). Show the difference in the bases
and that the pyramids come to a point with triangular side faces. Ask the class to
repeat the name and to say why they call it by that name.
 Show a cone, a cylinder. Ask for the names.
Worksheet A: Are the shapes the same? Recognising Rotated 3D Shapes
Say The diagrams or pictures on these pages are of 3D Shapes or solids. Look at the
front page. What is the first picture? (cone) Second? (cylinder). Read the instructions.
Student does the example first with help and then completes the questions
Worksheet B: Can you see the 3D shape?- Embedding and Disembedding 3D
Shapes
Join together 3 cuisinaire rods to form a rectangular prism. Ask what 3D shape has
been made? (rectangular prism). How many blocks of the same size were needed? (3)
Say This section is about putting blocks together and seeing which ones might fit
together to make rectangular prisms.
Worksheet C: Can You See it Another Way? - Folding and Unfolding 3D Shapes
Show the blocks together and sketch the front view on the board.
Take a piece of paper and fold it and tear out a shape in the middle of the open sides.
Ask a student to copy what it might look like when opened out.
Take the pyramids and cubes partially apart carefully and note the triangles and
squares.
Worksheet A: Are they the same?
Example
Look at the 3D shape under Smiley. If you turn it around
or over can it look like the shape shown below.
Circle Yes or No.
1. Yes
No
2. Yes
No
Look at the 3D shape under Smiley. If you turn it around
or over can it look like the shape shown below.
Circle Yes or No.
1. Yes
3. Yes
5. Yes
No
2. Yes
No
No
4. Yes
6. Yes
No
No
No
Three blocks have fallen on the floor. One block has a circle face, one has a triangle
face, and one has a rectangle face. We are looking down on them. Which one is
underneath on the bottom?
Circle the shape.
7.
Three sticks are marked with
Circle the shape of its mark.
8.
Which one is underneath on the bottom.
9.
Worksheet B: Can you See the 3D shape?
Example
Under Smiley is a finished picture. The picture below
is not complete.
Put in the lines to complete it like the finished one.
Under Smiley is a finished picture. The picture below
is
not complete.
Put in the lines to complete it like the finished one.
Example
Under Smiley is a building of blocks. Is the 3D shape
below one of the blocks?
Circle Yes or No.
Yes
No
Yes
No
Under Smiley is a building of 3 blocks. Is the 3D shape
one of the blocks?
Circle Yes or No.
1. Yes
4. Yes
No
No
2. Yes
No
5. Yes
3. Yes
No
No
Example
A 3D shape can be joined to the 3D shape under Smiley.
Can the shape below be joined to it to make a rectangular
prism?
Circle Yes or No.
Yes
No
Yes
No
Under Smiley are two blocks. Can they be joined to
make
the shape below.
Circle Yes or No.
Yes
Yes
No
No
Yes
No
Yes
No
Example
A 3D shape can be joined to the 3D shape under Smiley.
Can this shape be joined to make a rectangular prism?
Circle Yes or No.
Yes
No
Yes
No
A 3D shape can be joined to the 3D shape under Smiley.
Can the shapes make a rectangular prism?
Circle Yes or No.
1.
Yes
3. Yes
5.
Yes
No
No
No
2.
Yes
No
4. Yes
No
6. Yes
No
Example
Look at the block under Smiley. If you had some more
like this could you make the solid 3D shape below.
Circle Yes or No.
If Yes, write how many blocks you need.
Yes_____ No
Yes ____ No
1. Yes ____ No
2. Yes ____ No
3.
4. Yes ____ No
Yes ____ No
5. Yes _____ No
6.
Yes ____ No
Worksheet C: Can You See it Another Way?
Example
You can look from A straight down onto the object
under
Smiley. You can look at the sides from either B or C.
From which position do you see the view shown?
Circle A, B or C.
A,
looking down
B
C
A B
No
C
No
A
B C No
A B C
From which position are you looking at the object?
Circle A from the top,
B from in front,
C from the side,
or if you don’t know, circle No.
A
B
C
No
A B C No
A B C No
A B C
A B C No
A B C No
A B C
A
C
B
No
A
B
C
No
A B C No
A B C No
A B C
Example
A square piece of paper is folded in half taking the top to the bottom.
A hole is punched in and then it is opened up again. Will it look like the one shown?
Circle Yes or No.
Yes
No
Yes
No
A square piece of paper is folded in half taking the top to the bottom.
A hole is punched in it or a corner cut off and then it is opened up again.
Will it look like the one shown?
Circle Yes or No.
Yes
No
Yes
No
Yes
No
Yes
No
Example
The shape under Smiley is folded up. Can it form the 3D shape?
Circle Yes or No.
Yes
No
Yes
No
Yes
No
The shape is folded up. Can it form the 3D pyramid?
Circle Yes or No.
Yes
No
Yes
No
Can the rectangular paper be folded to make the hollow 3D shape shown?
Circle Yes or No.
If so, draw in the where you would fold the paper, if any hard fold is needed.
Yes
No
Yes
No
Yes
No
Can the triangular paper be folded to make the hollow 3D shape shown?
Circle Yes or No.
If so, draw in the where you would fold the paper, if any fold is needed.
Yes
No
Yes
No
Yes
No
S2 Lesson 1 - Pop-up Cards
Description
Students make simple pop-up cards in an investigative mode.
Purpose
Students experience how 3D can be generated and modified by folding backwards or
forwards. The cuts provide symmetrical changes.
Outcomes
SGS2.1 Makes, compares, describes and names three-dimensional objects
including pyramids, and represents them in drawings
Knowledge and skills
Working Mathematically
Students learn about
Students learn to
 identifying three-dimensional objects
 describe three-dimensional objects
in the environment and from drawings,
using everyday language and
photographs or descriptions
mathematical terminology
(Communicating)

explore what can be created from
folding and cutting to make threedimensional objects
Key ideas
Create nets from everyday packaging
Expected learning strategies
The student uses pattern and movement in her/his mental imagery and is developing
conceptual relationships. The student:


Predicts changes mentally modifying shapes and their attributes using motion or
pattern analysis
Discusses patterns and movements associated with combinations of shapes and
relationships between shapes.
Materials
Paper or cardboard per student and scissors. Coloured pencils. Examples of bought
and homemade pop-up cards.
S2 Lesson 1 - Pop-up Cards
Activities
Introduction





Show the student some purchased pop-up
cards and your own home-made ones.
Take a card and fold it. Ask students to
predict what might happen if you make a
cut at a slope on the fold.
Try it and open up the card. Ask student
how you might make the fold poke
inwards.
Try it and fold it hard. Re-close the card
and slowly reopen.
Ask students what it looks like and then
draw a face, say of a bird if they say it
looks like a beak.
Teaching Points and Questions
These activities can become art problem
solving activities.
Pop-ups can be made by sloping cuts on
fold, by two perpendicular cuts, by
diagonal folds at the top when sides are
cut away.
They can be decorated by gluing on
pieces or just colouring in to make a
picture.
What will happen if we make a cut at this
point?
What happens if we fold it inwards?
How can we make these cuts stick out?
Symmetry is a key feature of this work.
Activity (individual)
Students may find the beak easiest to do and
they can easily draw eyes around it.
Discuss the symmetry of the folds that go
inwards or outwards.
Glue a heart or similar cut-out on the front
of the rectangular folded cut.
If they are making cards, students can
decide what is an appropriate message to
write inside.
Push into card
Conclusion (whole class)
Which lines are symmetrical about the
Some students display their cards and show fold?
how they open up.
Could you predict what might happen if
you made another cut (or fold) parallel to
this one?
Students share ideas and help each other to
develop their cards.
S2 Lesson 2 – Shadow Shapes
Description
Students investigate the shadows made by a variety of 3 dimensional objects
Purpose
Students need to experience and imagine what a shape will look like from a different
perspective. They disembed the face-on view and realise some parts are then hidden
from view.
Outcomes
SGS2.1 Makes, compares, describes and names three-dimensional objects
including pyramids, and represents them in drawings.
Knowledge and skills
Students learn about
 comparing and describing features of
prism, pyramids, cylinders, cones and
spheres
 sketching three-dimensional objects
from different views including top,
front and side views
Working Mathematically
Students learn to
 compare features of three-dimensional
objects and two dimensional shapes
(Applying strategies, Reflecting)
 compare own drawings of threedimensional objects with other
drawings and photographs of threedimensional objects (Reflecting))
Key ideas
Name, describe, sort, make and sketch prisms, pyramids, cylinders, cones and spheres
Describe cross-sections of three-dimensional objects
Expected learning strategies
The student is developing mental images associated with concepts. She/he
demonstrates an increasing use of standard language. The student:
 generates images of shapes in a variety of orientations and with different features
 discusses shapes, their parts, and actions when the shape is not present.
The student uses pattern and movement in her/his mental imagery and is developing
conceptual relationships. The student:
 predicts changes mentally modifying shapes and their attributes using motion or pattern
analysis
 discusses patterns and movements associated with combinations of shapes and
relationships between shapes.
Materials
A variety of 3 dimensional objects, torch (or overhead projector), large sheets of
white paper Blu Tacked to the wall.
S2 Lesson 2 – Shadow Shapes
Activities
Teaching Points and Questions
Introduction (whole class)






Discuss the different shapes and names
of the 3D solids
Discuss the different parts of the solidsfaces, edges, corners
Discuss the different views of the 3D
solids.
Select a geometric solid. Ask students
to predict what shape the sides will be.
What shape do you think the top of this
object will look like? What will the
base look like?
Demonstrate how to make a shadow
using a torch
What shapes will you see?
Do the shapes have different shadows?
How can we hold the 3D solid to make
a different shadow?
How can we make a shadow of the base
and top of the solid?
Encourage students to demonstrate how
they would hold the object to make a
different shadow.
Activity (pairs or small groups)





Students select a 3 dimensional solid to
investigate.
Predict the shape of the shadow. Draw
this on a small sheet of white paper.
One of the students holds the 3D solid,
another student uses the torch to project
the shadow onto a large sheet of white
paper
A member of the group traces around
the shadow and labels it.
Group members change roles and
repeat the procedure for the same 3D
solid but a different view. Eg top view,
side view.
Did the students predict correctly?
Do the shadows all look the same?
What is different? What is the same?
How can we make different shadows?
Teacher may need to demonstrate how
to hold the 3D object in different
positions.
Encourage students to make tracings of
all views of the 3D solids
Encourage student to use correct
terminology.
Conclusion (whole class)

Students compare and discuss their
shadows, predictions and tracings
Do the shadows always make regular
shapes?
Can we match a set of shadow drawings
to a specific 3D solid?
Can a set of shadow drawings match
more than one 3 dimensional solid?
Students ask each other questions.
Additional Activities
Students predict and draw the shadows that would be made for a variety of 3dimensional objects eg. A coffee cup, a pencil tin, a pencil case, a school bag, a
variety of toys, etc.
S2 Lesson 3 – Models from Drawings
Description
Students use blocks to make the models shown in diagrams. Students draw side, front
and plan views of the models.
Purpose
Interpreting diagrams is an important skill.
Outcomes
SGS2.1 Makes, compares, describes and names three-dimensional objects
including pyramids, and represents them in drawings.
Knowledge and skills
Students learn about
 comparing and describing features of
prism, pyramids, cylinders, cones and
spheres
 identifying three-dimensional objects in
the environment and from drawings,
photographs or descriptions
Working Mathematically
Students learn to
 describe three-dimensional objects
using everyday language and
mathematical terminology
(Communicating)
 compare features of three-dimensional
objects and two dimensional shapes
(Applying strategies, Reflecting)
Key ideas
Name, describe, sort, make and sketch prisms, pyramids, cylinders, cones and spheres
Expected learning strategies
The student is developing mental images associated with concepts. She/he
demonstrates an increasing use of standard language. The student:
 generates images of shapes in a variety of orientations and with different features
 discusses shapes, their parts, and actions when the shape is not present.
The student uses pattern and movement in her/his mental imagery and is developing
conceptual relationships. The student:
 Predicts changes mentally modifying shapes and their attributes using motion or pattern
analysis
 Discusses patterns and movements associated with combinations of shapes and
relationships between shapes.
Materials
Worksheet of diagrams per student. Cube blocks and rectangles (thick and thin) from
the attribute block sets per student.
S2 Lesson 3 – Models from Drawings
Activities
Teaching Points and Questions
A simple way of producing the dots on the
blackboard is to use an isometric dot
Show students how a 3D shape can be transparency on an overhead projector. The
drawn on isometric paper.
block should be held up so the edge is
facing the class when the drawing is
Explain that students will be using
commenced.
similar diagrams to work out how to
build the shapes with blocks.
Introduction (whole class)


Example of Worksheet
Conclusion (whole class)
How did you decide where to attach the
Students discuss how they decide where block?
to place pieces and how many pieces to How did you decide how many blocks to
use.
use?
Teachers might find it easier to make up
new worksheets using the isometric paper.
S2 Lesson 4 - Faces and Nets
Description
Students use boxes to predict and investigate the nets of 3 dimensional objects.
Purpose
Students need to imagine and predict how the faces of a 3D shape will fold down to
make a net. By considering size and how the faces might be connected, students are
developing dynamic imagery.
Outcomes
SGS2.1 Makes, compares, describes and names three-dimensional objects
including pyramids, and represents them in drawings
Knowledge and skills
Students learn about
 creating nets from everyday packages
e.g. a cereal box
Working Mathematically
Students learn to
 compare own drawings of threedimensional objects with other
drawings and photographs of threedimensional objects (Reflecting))
 explore, make and describe the variety
of nets that can be used to create
particular three-dimensional objects
(Applying strategies, Reasoning,
communicating)
Key ideas
Create nets from everyday packages
Expected learning strategies
The student uses pattern and movement in her/his mental imagery and is developing
conceptual relationships. The student:
 Predicts changes mentally modifying shapes and their attributes using motion or
pattern analysis
 Discusses patterns and movements associated with combinations of shapes and
relationships between shapes.
Materials
A variety of different-shaped boxes, glue, coloured wrapping paper or magazine
paper, scissors, and paper and pencils.
S2 Lesson 4- Faces and Nets
Activities
Teaching Points and Questions
Introduction (whole class)


Discuss the different parts of the boxesfaces, edges, corners and use their threedimensional names.
Demonstrate how to cut around the box so
that it can be flattened out.
Talk about the difference between an
edge and a face
Do all the boxes have the same shaped
faces?
What makes the boxes different from each
other?
Are there different ways to cut the box?
Activity (individual)







Students select a box to investigate.
Before students cut around the box, the
teacher asks them to predict the net of the
box.
Students draw their prediction on a piece
of paper.
Trace around one of the faces of the box.
Cut out and paste onto the box.
Students cut along some of the edges of
the box so that it can be flattened out.
Students trace around the shape.
Fold the net to make the box.
Did the students predict correctly?
Did the students draw the shape of the
face of the box in similar proportions?
Ask students to predict what the box will
look like if it was flattened out. Allow the
students time to investigate their boxes
before trying to draw the shape.
Do all the drawings look the same?
How many faces can you see in your
drawing? Is this the same as the box?
Is your tracing the same as others’ in the
class? Why/ Why not?
Conclusion (whole class)


Students compare and discuss their
drawings.
Discuss the shapes that can be seen.
Encourage the students to look for
different ways to fold the nets.
Do they look the same? What is the same
about your drawings? What is different
about your drawings?
Encourage students to use correct
terminology.
Students can ask each other about
alternative nets.
Additional Activities




Students open the boxes out and trace around the net.
Refold the box
A student is then asked to match the net to the box
Explain the reasons for the match.
Extension activity
 Students
predict the net of a cube.
 They make a net to cover a cube.
 They cut it up and decide how else the net might be made.
Hint
There are many nets. For example, the base and top can be joined to any of the other six
sides.
S3 Lesson 5 – Models from Different Perspectives
Description
Students draw pictures of a construction from different perspectives- in front of, from
above, from the sides.
Purpose
Students need to recognise 3D shapes from different perspectives. They need to
identify the front view and to visualise the hidden parts of the three-dimensional
shapes.
Outcome
SGS3.1 Identifies three-dimensional objects, including particular prisms
and pyramids, on the basis of their properties, and visualises, sketches and
constructs them given drawings of different views
Knowledge and skills
Students learn about
 visualising and sketching threedimensional objects from different
views
 showing simple perspective in
drawings by showing depth
Working Mathematically
Students learn to
 describe to a peer how to construct or
draw a three-dimensional object
(Communicating)
 reflect on own drawing of a threedimensional object and consider
whether it can be improved
(Reflecting))
 ask questions about shape properties
when identifying them (Questioning)
Key ideas
Identify three-dimensional objects, including particular prisms and pyramids, on the
basis of their properties
Expected learning strategies
The student uses pattern and movement in her/his mental imagery and is developing
conceptual relationships. The student:
 develops and uses a pattern of shapes or relationship between parts of shapes
 discusses patterns and movements associated with combinations of shapes and
relationships between shapes.
The student selects from a range of spatial strategies that are appropriate for a
particular problem or concept. The student efficiently uses imagery, classification,
part-whole relationships, and orientation. The student:
 assesses images and plans the effective use of properties of shapes and composite
units to generate shapes
 describes effective use of properties of shapes to generate new shapes.
Materials
Set of approximately 10 different, large blocks for each group of 6 students, paper and pencil
for each student
S3 Lesson 5 – Models from Different Perspectives
Activities
Teaching Points and Questions
Introduction (whole class)



Students sit in circle while the teacher
builds a tower using the large blocks.
Discuss the different ways the blocks can
be stacked on top of each other and side
by side to build a large construction
Ask four students from opposite points in
the circle to describe what they can see.
Encourage students to use correct
terminology and positional language.
How can the rectangular prism be stacked?
Does the cube look different if we stack it
on a different side (face)?
Can the cylinder be stacked easily?
What will it look like if I put two cylinders
side by side? What will it look like from the
front? What will it look like from the side?
Use various 3D shapes for the building,
describing the placement eg. ‘I will put a
two cubes on top of the cylinder and a
rectangular prism across the top of the
cube.’
Activity (pairs)




Students form groups of four.
Each group builds a tower in the centre of
the table with the blocks provided.
The students sit around the table and draw
the construction from their own
perspective.
Each group shuffles their drawings and
moves to another table. Discuss how to
rearrange the shuffled drawings at the new
table.
This activity works best if the students are
seated at tables with the construction in the
centre.
What shapes have you used to build your
tower?
How many shapes high is the tower?
What shape can you see from where you
are sitting? Do you think all the students at
the table can see the same blocks as you
can?
What helped you to work out from what
side of the tower the drawing was made?
Discussion (whole class)


Discuss the different strategies that
students used to draw their pictures.
Discuss the strategies that students used to
rearrange the drawings
Was it easy to make the drawing look like
the tower? Why? Why not?
What did you like about the activity?
Were the drawings easy to match with the
tower?
What did you do to work out what part of
the tower the drawing was of?
Did you count the blocks? Did you look at
the shapes?
S3 Lesson 6 ─ Paper Box Folding
Description
Students fold up paper to make a box and lid.
Purpose
This provides a useful skill as well as encouraging an appreciation of 2D to 3D
transformations. A neat comparison in surface area size is also used.
Outcomes
SGS3.1 Identifies three-dimensional objects, including particular prisms
and pyramids, on the basis of their properties, and visualises, sketches and
constructs them given drawings of different views
Knowledge and skills
Students learn about
 identifying and listing the properties
of three-dimensional objects
 constructing three-dimensional
models given drawings of different
views
 visualising and sketching nets for
three-dimensional objects
Working Mathematically
Students learn to
 rxplain why particular threedimensional objects are used in the
built environment or appear in the
natural environment (communicating,
Reflecting)
 rescribe to a peer how to construct or
draw a three-dimensional object
(Communicating)
 ask questions about shape properties
when identifying them (Questioning)
Key ideas
Construct three-dimensional models given drawings of different views
Expected learning strategies
The student uses pattern and movement in her/his mental imagery and is developing
conceptual relationships. The student:
 develops and uses a pattern of shapes or relationship between parts of shapes
 discusses patterns and movements associated with combinations of shapes and
relationships between shapes.
The student selects from a range of spatial strategies that are appropriate for a
particular problem or concept. The student efficiently uses imagery, classification,
part-whole relationships, and orientation. The student:
 assesses images and plans the effective use of properties of shapes and composite
units to generate shapes
 describes effective use of properties of shapes to generate new shapes.
Materials
Two squares of coloured paper per student. Scissors.
S3 Lesson 6 ─ Paper Box Folding
Activities
Teaching Points and Questions
Introduction
If a square is used for the lid and for the
Display an example of the box and lid. Tell box, what might we do to the square to
make the box smaller than the lid?
the students that a square of paper will be
Where will we cut the square to make it smaller?
used to make the box and another square
If students suggest cutting around all four
will be needed to make the lid.
sides, ask
Since the box is slightly smaller than the
How might we make fewer cuts and still
lid, discuss how to get a slightly smaller
have a square?
square.
Activity (individuals working as a class)
Instructions:

Take one square and cut about 5mm off
on two adjacent sides so you obtain a
Base
slightly smaller square.
edge
 Half the square and open. Turn at right
angles and half the square and open.
 Fold over the edges to run exactly along
the centre of the square. Open.
 Fold each edge over to the quarter fold
closest to it. Open.
Top edge
Cut
 Cut down the second fold from each end
of the top and bottom as far as the second
Tuck in
Folded in
fold running the other way.
when folded
 Now fold over the long sides which will
tuck into the short sides when they are
folded.
 This should now be the box.
 Fold the larger square in the same way to
form the lid.
Discussion (whole class)
What is the shape of the box?
How might you make a box with a
 Discuss the shape of the box and how it
might be modified to get different shaped rectangular base?
boxes.
How might you make the sides deeper?
Does anyone suggest another method?
Additional Activities

Boxes can be used for volume work.
 Students can write procedures and drawings for making the box.
Chatter boxes can be made and the procedure described.
S3 Lesson 7 – Tree Planting
Description
Students are given a problem that requires visualising in three dimensions. They
consider the vertices of a triangular pyramid and its different orientations. It can be
linked into other lessons on tree planting.
Purpose
Students need to think three-dimensionally for this problem. The investigation makes
them think about the diagonals of squares and how three-dimensional shapes look in
different orientations.
Outcomes
SG3.1 Identifies three-dimensional objects, including particular prisms
and pyramids, on the basis of their properties, and visualises, sketches and
constructs them given drawings of different views
Knowledge and skills
Students learn about

visualising and constructing threedimensional models given a
description
Working Mathematically
Students learn to

SGS2.1 compare features of threedimensional objects and twodimensional shapes (Applying
strategies, Reflecting)

explain why the model is the same
despite its look in different
orientations (Communicating)
Expected learning strategies
The student selects from a range of spatial strategies that are appropriate for a
particular problem or concept. The student efficiently uses imagery, classification,
part-whole relationships, and orientation. The student:
 assesses images and plans the effective use of properties of shapes and composite
units to generate shapes
 describes effective use of properties of shapes to generate new shapes.
Materials
At least 6 straws or sticks and plasticine or joiners per group. Small boxes and string
and markers.
S3 Lesson 7 – Tree Planting
Activities
Introduction

Students form groups of 3 to 4
students.
 Set the problem for the students.
A farmer has a property with hills
and creek. He wants to plant four
special trees so that each tree is the
same distance from all the other
trees.
Use the straws to represent the distance
between the trees.
How can he plant them?
Teaching Points and Questions
The scenario is try to get the students to
think in three dimensions and not just
two-dimensions.
Initially many students will consider
planting on the vertices of twodimensional shapes.
Students are usually confronted with
the fact that the diagonal of the square
is longer than its sides. They will also
realise that a triangle only has three
vertices. Other shapes are good for
experimenting but will also help
students to realise the properties of
these shapes (e.g. a rhombus has
unequal diagonals)
Activity



Students work on the problem.
If there are groups with different
answers, they should look for reasons
why they might be permitted.
Students look at other threedimensional shapes in different
orientations and consider where the
vertices are. E.g in a rectangular
prism, they might be on walls and
floors and ceilings of a room.
One answer is the vertices of a triangular
pyramid with a horizontal base as if there
is a tree on the top of a hill and the others
planted at the base equally spaced. It is
also possible to have trees on two hills
with two in the valley or on three hills
and one in the valley. The pyramid is
turned partially over or inverted.
Some students start to see a picture and
want to make the sticks into trees. This is
fine so long as they still focus on the
distances between the trees.
Why do you think the problem talks about
hills and creeks?
What other kinds of shapes besides twodimensional shapes can we have?
Activity 2

Set another problem
Suppose you are an ant and you
want to walk from the corner near
the door to the opposite vertex on
the ceiling, what is the shortest root.
 Students take various rectangular
prisms and draw and discuss the
diagonals of the faces. They can also
If students draw the diagonals of the side,
they may still need string to compare
different lengths.
Light-coloured no-frills boxes are good to
draw on.
Students may need to make the skeletons
use the string or straws and discuss
the diagonals of the prism.
of the prisms to see the diagonals.
Is it shorter to walk along two edges or
across the diagonal?
Can an ant walk on the diagonal of the
prism?
Conclusion


Students draw conclusions about what
they have found in regard to the
vertices of turned three-dimensional
objects.
Students draw conclusions about the
different kinds of diagonals you can
have on a rectangular prism.
These problems really encourage students
to use their perceptions of objects and
visualising to develop conceptual
knowledge about the vertices and
diagonals of three-dimensional shapes.
S3 Lesson 8 – Nets and More Nets
Description
Students decide which of the pentomino shapes can be folded up to make an open
cube. They explore the patterns for systematically making all the nets of a cube. They
explore the similarities and differences of the nets of different rectangular prisms.
Purpose
Seeing limited examples of shapes or nets can restrict students’ development of
concepts. Students will use their dynamic imagery strategies and analytic ideas to
solve the investigations.
Outcomes
SGS3.1 Identifies three-dimensional objects, including particular prisms
and pyramids, on the basis of their properties, and visualises, sketches and
constructs them given drawings of different views
Knowledge and skills
Students learn about

visualising and sketching nets for
three-dimensional objects
Working Mathematically
Students learn to

describe to a peer how to construct or
draw a three-dimensional object
(Communicating)
Expected learning strategies
The student selects from a range of spatial strategies that are appropriate for a
particular problem or concept. The student efficiently uses imagery, classification,
part-whole relationships, and orientation. The student:
 assesses images and plans the effective use of properties of shapes and composite
units to generate shapes
 describes effective use of properties of shapes to generate new shapes.
Materials
For each group, stencils of the pentomino shapes (alternatively students can draw
these on grid paper), scissors, six squares, sticky tape (preferably removable), 2cm
square grid paper. The teacher may like a set of pentominoes cut out for discussion
purposes.
S3 Lesson 8 – Nets and More Nets
Activities
Introduction
 Ask
the students to visualise a cube
and to draw its net.
 Tell the students that they are going to
explore different nets for the same
three-dimensional shapes
 Students are given the pentomino
sheets so that each group of three
students has a set of 12, that is four
per student (or three per person for a
group of four).
 The problem is to imagine which of
the pentomino shapes fold to make
an open cube.
 Tell them that if they dispute and
cannot explain to each other they
should cut out the particular shape
and try it.
 Then they are to decide if there is
anything about the shapes that makes
them work or not.
Teaching Points and Questions
How many faces does it have?
Some schools have only introduced the
students to one net which is T shaped.
However it is possible for students to
have drawn different nets.
At this stage let students share a few
ideas but then move onto the
explorations.
Let students visualise folding up the
nets first and only let them cut out the
net if they are struggling.
Activity (groups of 3 or 4)
 Students
in groups decide which of
the pentomino shapes fold up or not.
 Then they have to decide what it is
about the net that means it will fold
up to make an open cube or not.
Why do you think it won’t work?
Why do you think it will?
This activity encourages dynamic
imagery but it also encourages students to
analyse two and three-dimensional
shapes.
Discussion
 Students
share the shapes they could
not visualise folding at first. They
explain how they overcame the
difficulty.
 Students share some of their reasons
for deciding whether a shape might
fold up.
 Have students draw yet another net
that will not work
Activity
Students should be referring to the angles
or corners and also the length of sides
The last part is to get them to realise that
the cube has all congruent (equal) square
faces.
 In
their groups of three, students begin
to draw all the nets of cubes.
 They should try to be systematic in
finding new nets
How are the nets joined onto the row of
squares?
Can you change the number in the row of
squares?
This is like three columns, could you have
more or less columns?
Do you have any repeats?
Discussion
 Students
can discuss the total number
Questions above
and how they decided systematically
on the number of nets.
 Students could also consider what
effect having a rectangular prism that
is not a cube would have on the
number of nets.
Activity
 Students
draw rectangular prisms.
Does having different sized faces make a
difference on the number of possible
nets? Why?
Discussion
 Students
share their ideas.
Question above