037i - Nets and Views of 3D Objects

Lesson 37 — Representing 3D Objects in 2D: Nets and Views

Strand: Space | Descriptor: AC9M7SP01 | Duration: 45 minutes

Learning Intentions

• To understand that a three-dimensional object can be represented on a two-dimensional surface in several different ways.
• To recognise and construct nets of common prisms.

Success Criteria

I can:

1. Name the faces, edges and vertices of a prism.
2. Identify whether a given arrangement of shapes is a valid net.
3. Draw a net for a cube, rectangular prism and triangular prism.
4. Match a net to the solid it folds into.

Warmup

(6 minutes — unfolding a box, physical)

Give each pair an empty cardboard box (a tissue box or similar) and scissors.

1. Count the faces, edges and vertices before cutting.
2. Carefully cut along some edges and flatten the box completely.
3. Sketch the flattened shape.
4. Compare with another pair — did you get the same flat shape?

Discussion: Different cutting choices give different flat shapes, yet all fold back into the same box. That flat shape is called a net.

Record on the board for a rectangular prism: faces, edges, vertices.

Activities

Activity 1 — Explicit Instruction: Nets (14 min)

Definition. A net is a two-dimensional pattern that folds up to form a three-dimensional solid, with no gaps and no overlaps.

The three tests for a valid net:

1. Does it have the right number of faces? (A cube needs exactly squares.)
2. Are the faces the right shapes and sizes?
3. Will it fold without overlapping?

Test 3 is the one that requires visualisation — and it is the one worth spending time on.

I do — cube nets. There are eleven distinct nets of a cube. Show three valid ones and two invalid arrangements of six squares.

A classic valid net is the “cross”: a column of four squares with one square attached to each side of the second square.

Invalid examples to display and discuss:

• A rectangle of six squares — folding leaves two faces overlapping and two holes.
• A straight row of six squares — it rolls into a tube, leaving no top or bottom.

Investigation prompt: “There are eleven cube nets. How many can you find?” Give squared paper and let pairs hunt. Do not tell them the eleven — the search is the learning. Rotations and reflections count as the same net.

Nets of other prisms — build together:

Solid	Net consists of
Cube	congruent squares
Rectangular prism	pairs of congruent rectangles
Triangular prism	congruent triangles rectangles
Square pyramid	square triangles

Note: the square pyramid is not a prism — its cross-section changes along its height. A useful contrast with Lesson 20.

Activity 2 — Constructing Nets Accurately (10 min)

Ruler and pencil work. Accuracy matters.

Task 1. Draw an accurate net for a rectangular prism measuring cm by cm by cm. Label every dimension.

Task 2. Draw a net for a triangular prism whose cross-section is a triangle with base cm and height cm (the sloping sides are cm and cm), and whose length is cm.

Task 3. Cut out Task 1 and fold it to check.

Checking prompt for Task 1: the three pairs of faces should measure , and . Have students verify the total surface area two ways:

Connection to prior work: the net makes surface area obvious — it is simply the total area of all the flat pieces. Contrast this with volume, which the net does not show directly.

Activity 3 — Inquiry: Which Net Folds to Which Solid? (10 min)

Pairs, printed net cards.

You have eight net cards. Four fold into solids; four do not.

1. Sort them into “valid” and “invalid”.
2. For each valid net, name the solid.
3. For each invalid one, explain precisely what is wrong.
4. Can you fix each invalid net by moving just one face?

Socratic scaffolding:

Prompt	Purpose
Understand: what makes a net invalid?	Wrong count, wrong shapes, or faces that overlap when folded.
Start with counting. What is quickest to check?	The number of faces — a cube needs exactly six squares.
If the count is right, what next?	Mentally fold it. Pick one face as the base and track where the others go.
A strategy for mental folding	Choose a base, fold the neighbours up as walls, then see whether anything is left for the lid.
For the arrangement — what goes wrong?	Two squares land on the same face; two faces have nothing.
Can you fix it by moving one square?	Yes — move a corner square to extend the arrangement into a valid net.
Looking back	Validity depends on the arrangement, not just the collection of shapes.

Extension for fast pairs: How many nets does a triangular prism have? (Nine, though students need only discover that the answer is more than one.)

Checks for Understanding

(5 minutes — exit ticket)

1. How many faces, edges and vertices does a triangular prism have?
2. What shapes, and how many of each, make up the net of a triangular prism?
3. Sketch one valid net of a cube.
4. Reasoning. Explain why a row of six squares in a straight line is not a valid net of a cube.
5. A net is made of square and triangles. What solid does it fold into? Is it a prism?

Answers: 1. faces, edges, vertices; 2. triangles and rectangles; 3. Any of the eleven, e.g. the cross; 4. Folding a straight row wraps it into a tube — four faces form the sides, but there is nothing left for the top and bottom; 5. A square pyramid; no, it is not a prism, because its cross-section shrinks towards the apex.

Common Misconceptions

Misconception	How to pre-empt it
Believing any six squares form a cube net.	Display invalid arrangements and fold them physically.
Thinking there is only one net per solid.	The cube has eleven. Run the search in Activity 1.
Counting rotations or reflections as different nets.	State the convention explicitly before the hunt begins.
Confusing a net with a 3D drawing.	A net is flat and shows true sizes; a 3D sketch distorts them.
Assuming a pyramid is a prism because it has flat faces.	Return to the prism definition: a constant cross-section.
Miscounting edges (double-counting shared ones).	Count systematically — top, bottom, then verticals.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A cube has a net made of six squares each of side cm. What is the cube’s total surface area and volume?

Answer

Surface area ; volume .

E2 (AMC Junior style). A rectangular prism measures cm by cm by cm. What is the total area of its net?

Answer

E3 (Challenge). On a standard die, opposite faces sum to . A cube net is folded so that the face showing is on the base. Which number is on top?

Answer

. The is on top.

E4 (Challenge). A triangular prism has a cross-section that is a right-angled triangle with legs cm and cm (hypotenuse cm), and a length of cm. Find the total area of its net.

Answer

E5 (Challenge). How many edges does a prism have if its cross-section is a hexagon?

Answer

edges on the top face, on the bottom, and verticals joining them: edges. In general an -sided prism has edges, vertices and faces.

Homework

1. State the number of faces, edges and vertices for: (a) a cube (b) a rectangular prism (c) a triangular prism (d) a square pyramid.
2. Draw an accurate net for a cube of side cm. Label the dimensions.
3. Draw a net for a rectangular prism cm by cm by cm, and calculate its total surface area.
4. What shapes make up the net of: (a) a triangular prism (b) a square pyramid (c) a rectangular prism?
5. Sketch two different valid nets of a cube.
6. Sketch an arrangement of six squares that is not a valid cube net, and explain why it fails.
7. A cube net has faces labelled so that opposite faces sum to . If one face shows , what is on the opposite face?
8. Reasoning. Explain why a net shows an object’s surface area clearly but not its volume.
9. Challenge. A prism has a pentagonal cross-section. How many faces, edges and vertices does it have? Explain your reasoning.

Answers: Q1 — (a) (b) (c) (d) . Q3 — . Q4 — (a) triangles rectangles (b) square triangles (c) pairs of rectangles. Q7 — . Q8 — the net lays out every face at true size, so their areas add to the surface area; volume depends on how the faces enclose space, which the flat net does not show. Q9 — faces ( rectangles pentagons), edges, vertices, using the , , rules with .