022i - Problem Solving with Volume

Lesson 22 — Problem Solving and Consolidation: Volume

Strand: Measurement | Descriptor: AC9M7M02 | Duration: 45 minutes

Learning Intentions

• To solve practical problems involving the volume and capacity of prisms.
• To convert confidently between cubic units and units of capacity.

Success Criteria

I can:

1. Convert between , , millilitres and litres.
2. Solve multi-step volume problems involving filling, emptying and costing.
3. Decide sensibly whether to round up or down in a given context.
4. Check that an answer is reasonable against the situation described.

Warmup

(7 minutes — explicit instruction on capacity, then practice)

The conversions to know:

Why : a cubic metre is cm in each of three directions, so . The factor is cubed, just as area’s factor was squared.

Practice:

1. L
2. L
3. mL

(Answers: L; L; ; .)

Activities

Activity 1 — Capacity and Filling Problems (14 min)

Pairs. Every answer needs correct units, a rounding decision, and a sentence.

Problem 1 — Aquarium. A tank measures cm by cm by cm. (a) Find its capacity in litres. (b) It is filled to of its depth — how many litres of water does it hold? (c) At L per minute, how long does that take?

Problem 2 — Rainwater tank. A rectangular tank is m by m by m. (a) Find its capacity in litres. (b) A household uses L per day. For how many complete days will a full tank last?

Problem 3 — Concrete. A garden path is a rectangular prism m long, m wide and m deep. Concrete costs 210$ per cubic metre and is sold only in whole cubic metres. Find the cost.

Problem 4 — Packing. A carton measures cm by cm by cm. Boxes measuring cm by cm by cm are packed inside. How many boxes fit?

Socratic scaffolding for Problem 4:

Prompt	Purpose
Understand: what is the unknown?	A whole number of boxes.
First instinct — divide the volumes?	. But is that always achievable?
What could go wrong with the volume method?	The boxes might not fit neatly, even if the volumes divide exactly.
Devise a safer plan	Check how many fit along each edge.
Carry it out	; ; .
Multiply	boxes.
Looking back	Here both methods agree, because every edge divides exactly. When would they disagree?
Extend	If the carton were cm long, the volume method would still suggest , but only boxes fit along that edge — cm is wasted. Always check edge by edge.

Answers: 1. (a) L (b) L (c) minutes. 2. (a) L (b) days. 3. purchased, costing 42012$ boxes.

Activity 2 — Working backwards and Comparing (10 min)

I do. A tank holds litres. Its base measures cm by cm. How deep is it?

You do:

1. A prism holds L. Its cross-sectional area is . Find its length.
2. A cube-shaped container holds litres. Find its edge length in centimetres.
3. Two tanks: one is m by m by m; the other is m by m by m. Which holds more, and by how many litres?

(Answers: cm; , so edge cm; first L, second L, so the second holds L more.)

Activity 3 — Investigation: Same Volume, Different Surface Area (8 min)

Pairs, centimetre cubes.

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

1. Find every possible box.
2. Calculate the surface area of each. (Recall: .)
3. Which box uses the least packaging? Which uses the most?
4. What advice would you give the company?

Socratic scaffolding:

Prompt	Purpose
How do you find every box systematically?	List all ways to write as a product of three whole numbers, taking .
Carry it out	, , , , , , , — eight boxes.
Predict before calculating: which will use least card?	Most students correctly guess the most cube-like.
Test the extremes	: . : .
Looking back	The most cube-like box is by far the most efficient — less than half the material.
Connect	This is the same principle as Lesson 19’s box investigation, and the reason so much packaging is roughly cubic.

Answers: 3. Least: at . Most: at . 4. Choose dimensions as close to equal as possible — but note that real packaging must also consider shelf display, stacking and the product’s own shape.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Convert: (a) to litres (b) to litres (c) L to .
2. A tank is cm by cm by cm. Find its capacity in litres.
3. A prism holds L and has a cross-sectional area of . Find its length.
4. Concrete is needed for a slab m by m by m. It is sold in whole cubic metres at 195$ each. Find the cost.
5. Reasoning. A student converts to . Explain the error and give the correct value.

Answers: 1. (a) L (b) L (c) ; 2. L; 3. cm; 4. exactly, costing 5852\ \text{m}^3 = 2,000,000\ \text{cm}^3$. The student used the linear factor.

Common Misconceptions

Misconception	How to pre-empt it
or .	Show the reasoning explicitly. Contrast with area’s .
Confusing mL with L when converting from .	Anchor: mL exactly; a litre is of them.
Rounding down when purchasing materials.	Ask “can you buy of a cubic metre of concrete?” for every costing problem.
Rounding up when counting how many objects fit.	Contrast directly with the above: fitting rounds down, buying rounds up.
Dividing volumes to count packed boxes without checking the edges.	Activity 1 Problem 4 addresses this head-on with the cm extension.
Assuming a bigger surface area means a bigger volume.	Activity 3 disproves it — all eight boxes have identical volume.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A cube has volume . What is its surface area?

Answer

Edge cm, so .

E2 (AMC Junior style). A rectangular tank m by m by m is half full. How many litres of water does it contain?

Answer

E3 (Challenge). Water flows into an empty tank measuring cm by cm by cm at litres per minute. After how many minutes will the water be cm deep?

Answer

E4 (Challenge). A solid metal cube of side cm is melted down and recast as a rectangular prism with a cm by cm base. What is its height?

Answer

Volume is conserved when material is recast — only the shape changes.

E5 (Challenge). A swimming pool m by m has a floor sloping evenly from m to m deep. How many litres does it hold when full?

Answer

The cross-section is a trapezium: . Then litres.

Homework

1. Convert: (a) to litres (b) to litres (c) L to (d) to .
2. A fish tank is cm by cm by cm. Find its capacity in litres.
3. A rainwater tank is m by m by m. (a) Find its capacity in litres. (b) A garden uses L per week — for how many complete weeks will a full tank last?
4. A concrete driveway is m long, m wide and m deep. Concrete costs 185$ per cubic metre, sold in whole cubic metres. Find the cost.
5. A prism holds L and has a cross-sectional area of . Find its length.
6. A carton is cm by cm by cm. Boxes are cm by cm by cm. How many fit? Check edge by edge.
7. A tank cm by cm by cm is filled at L per minute. How long to fill it completely?
8. Reasoning. Two boxes have the same volume but different surface areas. Explain how this is possible, with an example.
9. Challenge. A cube of side cm is melted and recast as a triangular prism of length cm whose cross-section has base cm. Find the height of the triangular cross-section.

Answers: Q1 — (a) L (b) L (c) (d) . Q2 — L. Q3 — (a) L (b) weeks. Q4 — , cost 740304 \times 3 \times 2 = 2412010= 12V = 1000\ \text{cm}^3A = 1000 \div 25 = 40\ \text{cm}^2\tfrac12 \times 8 \times h = 40h = 10$ cm.