038i - Drawing 2D Representations

Lesson 38 — Drawing and Interpreting Different 2D Representations

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

Learning Intentions

• To draw plans and elevations of a three-dimensional object.
• To construct isometric drawings of solids built from cubes.

Success Criteria

I can:

1. Draw the plan (top view), front elevation and side elevation of a solid.
2. Draw a solid on isometric dot paper.
3. Reconstruct a solid from its three views.
4. Count the cubes in a solid shown as a drawing.

Warmup

(6 minutes — guess the object, whole class)

Show three separate outlines on the board: a circle, a rectangle and a rectangle.

1. What everyday object could produce these three views from above, the front and the side? (A cylinder — a tin can.)
2. Now: square, square, square. (A cube.)
3. Now: circle, triangle, triangle. (A cone.)

Discussion: No single view identifies the object. Together, the three views usually do — and that is exactly why engineers and architects draw all three.

Activities

Activity 1 — Explicit Instruction: Plans and Elevations (14 min)

Definitions.

• The plan is the view looking straight down from above.
• The front elevation is the view looking horizontally at the front.
• The side elevation is the view looking horizontally from the side (state which side).

Key rule: each view is drawn flat, with true widths and heights. Nothing is drawn at an angle, and hidden depth is not shown.

I do — a solid made from cubes. Build, in front of the class, a solid: a row of cubes with one extra cube stacked on the leftmost.

Draw the three views on squared paper, narrating each:

• Plan (from above): a row of squares. (The stacked cube hides directly behind the one below it, so it adds nothing new from above.)
• Front elevation: a row of squares with square on top at the left — an L-shape.
• Side elevation (from the right): a column of squares.

Emphasise: what you cannot see matters. A cube hidden behind another does not appear in that view at all — which is precisely why one view is never enough.

We do: Build a base with one cube on top of one corner. Draw all three views together.

(Plan: a square of four squares. Front: two squares wide, with one extra on top at one end. Side: the same shape.)

You do: Four solids built from linking cubes, each requiring all three views drawn and labelled.

Activity 2 — Isometric Drawing (10 min)

Isometric dot paper required.

What isometric drawing does. It shows all three dimensions in one picture. Vertical edges stay vertical; the other two directions run along the dot lines.

The rules to state:

1. Vertical edges are drawn vertically.
2. Horizontal edges follow the diagonal dot lines — never horizontally across the page.
3. Parallel edges in the solid stay parallel in the drawing.
4. Lengths along each of the three directions are drawn to true scale.

I do: Draw a cube step by step on the board’s isometric grid, then a prism.

Important limitation to name now (it is developed fully in Lesson 39): an isometric drawing looks realistic, but angles are not shown truly. A cube’s corners appear as or on the page.

You do:

1. Draw a rectangular prism.
2. Draw a cube with one extra cube sitting on top of one corner.
3. Draw an L-shaped solid made from cubes.

Activity 3 — Inquiry: how Many Cubes? (10 min)

Pairs. Interpretation rather than drawing.

Here is the plan of a solid built from cubes. The number in each square shows how many cubes are stacked at that position.

1. How many cubes in total?
2. Draw the front elevation (looking from the bottom of the table as printed).
3. Draw the side elevation from the right.
4. Now the reverse: a solid has plan, front and side elevations all equal to a square. What is the smallest number of cubes it could contain? What is the largest?

Socratic scaffolding for Q4:

Prompt	Purpose
Understand: what do the three views tell you?	The solid occupies a region, and every row and column is “seen” as filled.
What is the maximum?	Fill the whole block: cubes.
Now the minimum. What does the plan force?	All four ground positions must be occupied — at least cubes.
What do the two elevations force?	Every row and every column must reach height somewhere.
How few columns of height will cover both rows and both columns?	Two, placed diagonally.
Count.	Two columns of height , plus two of height : cubes.
Verify	Heights — column maxima ✓ and row maxima ✓
Looking back	Minimum , maximum — so the same three views describe several different solids.

Answers: 1. cubes; 2. Front elevation heights are the column maxima: , , — so a shape , , squares tall; 3. Side elevation heights are the row maxima: and — so and squares tall; 4. Minimum , maximum .

Checks for Understanding

(5 minutes — exit ticket)

1. A solid is a block of cubes, one cube high. Draw its plan, front elevation and side elevation.

2. From this plan with heights, how many cubes are there?

3. Reasoning. Explain why the plan alone is not enough to identify a solid.

4. Name the two things an isometric drawing shows well, and one thing it distorts.

5. A solid’s three views are all single squares. What is the solid?

Answers: 1. Plan — a rectangle of squares; front — a row of ; side — a row of (heights all ); 2. cubes; 3. The plan gives only the footprint — it says nothing about height, so many different solids share the same plan; 4. It shows all three dimensions at once and keeps lengths to scale along each direction, but it distorts angles ( corners appear as or ); 5. A cube.

Common Misconceptions

Misconception	How to pre-empt it
Drawing hidden cubes in a view.	Views show only what is visible in silhouette from that direction. Build the solid physically and look.
Drawing the plan as a 3D-looking sketch.	Plans and elevations are strictly flat. Use squared paper, not isometric.
Confusing the front and side elevations.	Label the viewing direction with an arrow on every diagram.
On isometric paper, drawing horizontal edges across the page.	Rule 2 — horizontals follow the dot diagonals. Check every drawing against it.
Assuming three views determine a unique solid.	Activity 3 Q4 disproves this directly.
Counting only visible cubes in a stack drawing.	Use the numbered-plan method, and build the solid to confirm.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A solid is built from cubes. Its plan shows heights as below. How many cubes are used, and what is the height of the tallest column?

Answer

Total cubes; the tallest column is cubes high.

E2 (AMC Junior style). A cube is built from small cubes. The whole outside is painted, then the cube is taken apart. How many small cubes have exactly three painted faces?

Answer

Only the corner cubes have three painted faces — there are corners, so 8 cubes. (Extension: edge cubes have two faces, face-centre cubes have one, and the single central cube has none. Check: ✓)

E3 (Challenge). A solid built from cubes has a plan, a front elevation and a side elevation that are all squares. What is the smallest number of cubes it could contain?

Answer

The plan forces all ground positions to be occupied — at least cubes. The elevations force every row and every column to reach height somewhere. Three columns of height , placed on a diagonal, cover all three rows and all three columns. Heights:

Total cubes. (The maximum is .)

E4 (Challenge). A rectangular prism has a plan of cm by cm and a front elevation cm by cm. What is its volume?

Answer

The plan gives length and width ; the front elevation gives length and height . So .

E5 (Challenge). Two different solids have identical plans, front elevations and side elevations. Give an example of each.

Answer

Use the case from Activity 3. Heights uses cubes; heights uses . Both give three full views, yet the solids differ.

Homework

1. A solid is a block of cubes, one cube high. Draw its plan, front elevation and right side elevation.

2. For each plan showing heights, find the total number of cubes:

   (a) (b)

3. Draw, on isometric paper, (a) a cube (b) a prism.

4. Draw the plan, front and side elevations of a solid made from a base of cubes with two cubes stacked on one corner.

5. A cylinder is viewed from above, the front and the side. Sketch all three views.

6. Reasoning. Explain why an architect draws plans and elevations rather than a single isometric picture.

7. Reasoning. A solid’s plan, front elevation and side elevation are all squares. Give two different solids matching this, and state how many cubes each uses.

8. Challenge. A cube built from small cubes is painted on all outside faces, then taken apart. How many small cubes have (a) three painted faces (b) no painted faces?

Answers: Q2 — (a) (b) . Q5 — plan: a circle; front and side: congruent rectangles. Q6 — plans and elevations show true lengths and angles, so they can be measured and built from; an isometric picture distorts angles and hides faces. Q7 — heights uses cubes; the full block uses . Q8 — (a) (the corners) (b) the inner block, so .