040i - Problem Solving with 2D Representations

Lesson 40 — Problem Solving and Consolidation: 2D Representations

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

Learning Intentions

• To solve problems using nets, plans, elevations and isometric drawings.
• To move fluently between different representations of the same solid.

Success Criteria

I can:

1. Convert between a net, a set of views and an isometric drawing.
2. Calculate surface area from a net and volume from dimensions.
3. Reconstruct a solid from partial information.
4. Justify which representation I would use for a given task.

Warmup

(6 minutes — rapid recall, mini whiteboards)

1. How many faces, edges and vertices does a triangular prism have?
2. What shapes make up the net of a rectangular prism?
3. What does the plan of a solid show?
4. Name one thing an isometric drawing distorts.
5. A cube’s net uses six squares of side cm. What is its surface area?

(Answers: ; three pairs of rectangles; the view from directly above; angles; .)

Activities

Activity 1 — Translating between Representations (14 min)

Stations or a graded worksheet. Each task requires moving from one representation to another.

Task A — net to solid. A net consists of two triangles (base cm, height cm) and three rectangles, each cm long, with widths cm, cm and cm.

1. Name the solid.
2. Find the total surface area.
3. Find the volume.

Task B — views to solid. A solid built from cubes has:

• plan: a rectangle,
• front elevation: a row of squares with one extra on top at the left,
• side elevation: a column of squares.

4. How many cubes could it contain? Give the minimum and maximum consistent with these views.

Task C — solid to views. A solid is a base of cubes with a single cube on the centre.

5. Draw its plan, front elevation and side elevation.
6. How many cubes in total?

Task D — dimensions to everything. A rectangular prism measures cm by cm by cm.

7. Find its surface area from a net.
8. Find its volume.
9. Sketch it isometrically.

Socratic scaffolding for Task A:

Prompt	Purpose
Understand: what does the mix of shapes tell you?	Two congruent triangles plus three rectangles means a triangular prism.
What do the three rectangle widths represent?	The three sides of the triangular cross-section: , and cm.
Which measurement is the prism’s length?	The cm shared by all three rectangles.
Surface area — where do you start?	Total the areas of all five pieces.
Carry it out (triangles)	.
Carry it out (rectangles)	.
Total	.
Now the volume — does the net give it directly?	No. You need .
Looking back	The net gave surface area immediately, but volume required reassembling the solid mentally.

Answers: 1. A triangular prism. 2. . 3. . 4. Minimum , maximum (see below). 5. Plan — a square; front and side elevations — a row of with one extra square on top of the middle. 6. cubes. 7. . 8. . 9. Student sketch.

Working for Task B Q4. The plan requires all ground positions filled. The front elevation shows a height of at the left column only, so exactly one of the two left-hand positions is high — but it could be either, and only one is needed. The side elevation confirms a maximum height of . So: base cubes plus at least extra gives a minimum of ; if both left-hand positions are high, the total is .

Activity 2 — Applied Problems (12 min)

Pairs. Every answer needs units and a justification of the representation used.

Problem 1 — Packaging. A gift box is a rectangular prism cm by cm by cm.

(a) How much card is needed to make it (ignoring flaps)?

(b) What volume does it hold?

(c) Which representation did you use for each, and why?

Problem 2 — Tent. A tent is a triangular prism. The triangular end has base m and height m, with sloping sides of m each. The tent is m long, and has a groundsheet.

(d) How much fabric is needed for the two triangular ends and two sloping sides?

(e) How much extra for the groundsheet?

(f) What volume of air does it enclose?

Problem 3 — Missing dimension. A cube has a net of total area .

(g) Find the edge length.

(h) Find the volume.

Answers:

1. (a) (b) (c) a net for the card (all faces at true size), and the dimensions for volume.
2. (d) two triangles ; two sloping rectangles ; total (e) groundsheet (f) .
3. (g) , so and cm (h) .

Activity 3 — Inquiry: Design a Package (8 min)

Pairs.

A company must package of product in a rectangular box with whole-centimetre dimensions.

1. Choose a box and draw its net accurately.
2. Calculate the card needed.
3. Compare with another pair’s design. Whose uses less card?
4. Which box shape uses the least card overall? Explain why.

Socratic scaffolding for Q4:

Prompt	Purpose
Understand: what is fixed and what varies?	The volume is fixed at ; the shape varies.
Have you met this before?	Yes — Lessons 19 and 22 investigated exactly this.
Recall the finding.	The most cube-like box uses the least material.
Which whole-number box for is closest to a cube?	.
Calculate its surface area.	.
Compare with an extreme.	: — well over twice as much.
Looking back	The net makes the comparison visible: a long thin box has a great deal of “side” area for very little content.

Answers: 4. at — the most cube-like arrangement, as established in Lesson 19.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. A cube’s net is made from six squares of side cm. Find its surface area and volume.
2. A rectangular prism is cm by cm by cm. Find its surface area.
3. A solid built from cubes has a plan showing heights . How many cubes, and what is its front elevation?
4. Reasoning. Which representation would you use to work out how much paint a box needs, and why?
5. A triangular prism’s net has two triangles (base cm, height cm) and three rectangles of length cm. Find the area of the two triangles.

Answers: 1. and ; 2. ; 3. cubes; the front elevation has column maxima and , so a square; 4. A net — it shows every face at true size, so the areas add directly to give the surface area; 5. .

Common Misconceptions

Misconception	How to pre-empt it
Reading volume off a net.	The net gives surface area; volume needs the assembled dimensions. Ask which question is being answered.
Assuming three views determine a unique solid.	Task B has a range of answers. Always ask for minimum and maximum.
Using for volume or for area.	The exponent counts dimensions. Check units on every line.
Forgetting a face when totalling surface area from a net.	Count the pieces on the net first and tick each off as it is calculated.
Using a slant side as a perpendicular height in a triangular end.	The tent problem gives both m (height) and m (slope) deliberately.
Choosing a representation without considering the purpose.	Require a justification sentence for every applied problem.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A cube has surface area . Find its volume.

Answer

, so and cm. Volume . (A pleasing coincidence: for a cube of side , the surface area and volume have the same numerical value.)

E2 (AMC Junior style). A rectangular prism has surface area and dimensions cm, cm and cm. Find and the volume.

Answer

Volume .

E3 (Challenge). A cube is built from small cubes and painted all over. It is then taken apart. How many faces of the small cubes are painted in total, and how many are unpainted?

Answer

The large cube’s surface consists of faces, each made of small faces: painted faces. Each small cube has faces, giving in total, so are unpainted. (Each of the small cubes has exactly painted and unpainted faces.)

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 its total surface area and volume.

Answer

E5 (Challenge). A solid built from cubes has a plan, front elevation and side elevation that are all squares. Its volume is cubes. Give one arrangement of column heights that works.

Answer

The plan requires all positions occupied, and each row and column must reach height . For example:

Total . Adjusting one to a gives . Students should check their own arrangement satisfies all three views.

Homework

1. A cube’s net is made from six squares of side cm. Find its surface area and volume.
2. A rectangular prism is cm by cm by cm. Find (a) its surface area (b) its volume.
3. A triangular prism has a net with two triangles (base cm, height cm) and three rectangles of length cm with widths cm, cm and cm. Find (a) the total surface area (b) the volume.
4. A solid built from cubes has a plan showing heights . Find (a) the number of cubes (b) the front elevation heights (c) the side elevation heights.
5. A cube has a net of total area . Find its edge length and volume.
6. A rectangular prism has surface area , with two dimensions cm and cm. Find the third.
7. For each task, state which representation you would use and why: (a) buying paint for a shed (b) showing a client the finished shed (c) giving a builder the measurements.
8. Reasoning. Explain why a net shows surface area directly but volume only indirectly.
9. Challenge. A cube is painted all over and taken apart into small cubes. How many have (a) exactly painted faces (b) exactly (c) none?

Answers: Q1 — , . Q2 — (a) (b) . Q3 — (a) (b) . Q4 — (a) (b) (c) . Q5 — , so cm and . Q6 — gives , so cm. Q9 — (a) edge cubes (b) face-centre cubes (c) central cube.