039i - Reasoning About Representations

Lesson 39 — Advantages and Disadvantages of Different Representations

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

Learning Intentions

• To compare different two-dimensional representations of the same object.
• To reason about which representation suits a given purpose.

Success Criteria

I can:

1. State what each representation shows accurately and what it distorts.
2. Choose the best representation for a stated purpose and justify the choice.
3. Explain why several representations are often needed together.
4. Identify information that is lost in a given representation.

Warmup

(6 minutes — same object, four pictures)

Display four representations of the same rectangular prism ( cm):

• A — its net
• B — plan, front and side elevations
• C — an isometric drawing
• D — a photograph taken at an angle

1. Which shows you the surface area most easily?
2. Which would you give a builder to construct it?
3. Which looks most like the real object?
4. Which would you use to work out the volume?

Answers to discuss, not settle immediately: A for surface area; B for construction; C or D for realism; B or C for volume (you need all three dimensions with true measurements).

Activities

Activity 1 — Explicit Instruction: the Comparison Table (14 min)

Build this table with the class rather than presenting it complete. Ask for each cell.

Representation	Shows well	Distorts or hides	Best used for
Net	Every face at true size and shape; surface area	Gives no sense of the 3D form; volume is not visible	Making the object; calculating surface area
Plan and elevations	True lengths and true angles in each view	No single picture of the whole; needs mental assembly	Building, engineering, architecture
Isometric drawing	All three dimensions at once; lengths to scale along each axis	Angles are distorted ( appears as or ); back faces hidden	Quick visualisation; instruction manuals
Perspective drawing / photo	Looks realistic; conveys depth	Lengths and angles both distorted; distant parts appear smaller	Art, advertising, showing appearance
Cross-section diagram	Internal structure	Only one slice; outside form not shown	Engineering, medical imaging, geology

The central principle to state:

Every two-dimensional representation of a three-dimensional object loses something. The question is never “which is best?” but “which loses what I can afford to lose?”

Worked demonstration — the distorted right angle. On the board, draw a cube isometrically. Measure the angle at the top corner with a protractor: it reads , not . Ask why the drawing still “looks right”. (Our visual system interprets the convention, but a measurement taken from the drawing would be wrong — which is exactly why builders do not work from isometric drawings.)

Activity 2 — Purpose-matching Task (10 min)

Pairs. For each purpose, choose a representation and justify in one sentence.

1. A carpenter must cut timber to build a storage box.
2. A furniture catalogue advertises the same box.
3. A student must calculate how much wrapping paper the box needs.
4. A plumber needs to see the pipework inside a wall.
5. An instruction leaflet shows how to assemble flat-pack shelving.
6. A geologist shows the rock layers beneath a hillside.

Answers: 1. Plans and elevations — true measurements to cut from. 2. Perspective drawing or photo — appearance sells. 3. Net — all faces at true size, so areas add directly. 4. Cross-section — reveals what is hidden. 5. Isometric — shows the whole object and how parts fit. 6. Cross-section — internal layering is the whole point.

Extension question: Which purposes could accept more than one answer, and why? Push students to defend, not just choose.

Activity 3 — Inquiry: what is Lost? (12 min)

Pairs.

Part A. A solid is described only by its plan: a rectangle of squares.

1. Sketch three genuinely different solids with this plan.
2. What information is missing?

Part B. A solid’s isometric drawing shows what appears to be a cube.

3. Could the solid be hollow? Could a cube be missing from the hidden back corner?
4. Which representation would settle the question?

Part C. A net is made of squares each of side cm.

5. What can you say for certain about the solid?
6. What can you not determine from the net alone?

Socratic scaffolding for Part B:

Prompt	Purpose
Understand: what does an isometric drawing actually show?	The faces visible from one particular direction.
How many of a cube’s faces can you see at once?	At most three.
So what is unaccounted for?	Everything behind — including the hidden back corner.
Could a cube be missing there?	Yes. The drawing would look identical.
Could the solid be hollow?	Yes — the drawing shows only the outer surface.
Which representation would settle it?	A cross-section, or a plan with the height of every column stated.
Looking back	Adding a second representation, viewed from a different direction, removes ambiguity that no single view can.

Answers: 1. Any solids with the same footprint but different heights — e.g. all columns high, all high, or a mixture. 2. The height at each position. 3. Yes to both — the isometric view cannot rule either out. 4. A cross-section, or a numbered plan. 5. It folds into a cube of side cm, with surface area and volume . 6. Nothing about the interior — whether it is solid or hollow, or what material it is made from.

Discussion to close: Why do architects produce plans, elevations, cross-sections and a realistic render for the same building? (Each audience and each purpose needs different information: builders need measurements, clients need appearance, engineers need internal structure.)

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Which representation would you use to calculate surface area, and why?
2. Name one thing an isometric drawing shows well and one thing it distorts.
3. Reasoning. Explain why a builder is given plans and elevations rather than a photograph.
4. A solid’s plan is a square. Give two different solids that match.
5. Reasoning. Why is more than one representation usually needed to describe an object fully?

Answers: 1. A net — every face appears at true size, so the areas simply add; 2. Shows all three dimensions in one picture; distorts angles (right angles appear as or ); 3. A photograph distorts both lengths and angles, so nothing can be measured from it reliably; plans and elevations preserve true measurements; 4. E.g. a flat slab, and a block — same footprint, different heights; 5. Every 2D representation loses some information about a 3D object. Combining representations lets each one supply what the others omit.

Common Misconceptions

Misconception	How to pre-empt it
Believing one representation is universally “best”.	Frame every judgement around purpose. The comparison table has no “winner” column.
Trusting angles measured from an isometric drawing.	Measure a drawn cube corner with a protractor and read aloud.
Assuming a drawing shows the whole solid.	Part B of the inquiry addresses hidden regions directly.
Thinking a net reveals volume.	The net gives surface area; volume needs the assembled dimensions.
Treating a photograph as a reliable source of measurements.	Perspective shrinks distant parts — demonstrate with a photo of a corridor.
Believing plans and elevations are unambiguous.	Lesson 38’s minimum/maximum cube task showed otherwise.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A cube’s net is made from six squares of side cm. State the cube’s surface area and volume, and say which of these the net shows directly.

Answer

Surface area — shown directly by the net. Volume — not shown directly; it must be deduced from the side length.

E2 (AMC Junior style). Three solids all have a plan that is a square, with heights as shown. Which has the greatest volume?

Answer

A cubes; B cubes; C cubes. B and C tie for the greatest volume, at cubes each — despite looking quite different.

E3 (Challenge). A rectangular prism has a net with total area . Its dimensions are whole numbers, and two of them are cm and cm. Find the third.

Answer

E4 (Challenge). An isometric drawing shows what appears to be a solid cube. What is the largest number of small cubes that could be missing from the interior without changing the drawing at all?

Answer

Only the single central cube is invisible from every direction — every other position touches at least one outer face. So at most 1 cube can be removed without altering any view.

E5 (Challenge). Explain why a cross-section of a prism looks the same wherever you slice it (perpendicular to its length), but a cross-section of a pyramid does not.

Answer

A prism is defined by having a constant cross-section along its length, so every perpendicular slice is congruent. A pyramid tapers towards its apex, so slices shrink as you move upwards — they are similar in shape but different in size. This is exactly why works for prisms and not for pyramids.

Homework

1. Copy and complete a comparison table for the net, plans and elevations, and isometric drawing, with columns “shows well”, “distorts or hides”, and “best used for”.
2. For each purpose, name the most suitable representation and justify your choice in one sentence: (a) Cutting card to make a box. (b) Showing a customer what a new kitchen will look like. (c) Giving a builder measurements for a wall. (d) Showing the layers inside a chocolate bar.
3. A solid’s plan is a rectangle of squares. Sketch three different solids with this plan and state how many cubes each uses.
4. A cube’s net is made from six squares of side cm. Find its surface area and volume, and state which the net shows directly.
5. Reasoning. Explain why angles measured from an isometric drawing cannot be trusted.
6. Reasoning. A friend says “a photograph is the most accurate way to show a 3D object.” Give two reasons to disagree.
7. A rectangular prism has a net of total area , with two dimensions cm and cm. Find the third dimension.
8. Challenge. Explain why two solids can share identical plans, front elevations and side elevations yet contain different numbers of cubes. Illustrate with an example.

Answers: Q2 — (a) net (b) perspective drawing or render (c) plans and elevations (d) cross-section. Q4 — surface area (shown directly); volume (deduced). Q5 — the isometric convention draws right angles as or , so a protractor reading from the page does not match the solid. Q7 — , so , giving cm. Q8 — the views show only silhouettes, so hidden interior positions can be filled or empty without changing any view; e.g. the case from Lesson 38 admits both and cubes.