030i - Consolidation Volume and Capacity

Lesson 30 — Consolidation and Check: Volume and Capacity of Right Prisms

Strand: Measurement | Descriptor: AC9M8M02 | Duration: 45 minutes

Learning Intentions

• To consolidate all strategies for finding the volume of right prisms, including composite prisms.
• To apply and convert confidently between units of volume and capacity in a substantial applied problem.

Success Criteria

I can:

1. Select and apply for any right prism, including composite solids found by addition or subtraction.
2. Convert confidently between units of volume and capacity within a multi-step problem.
3. Solve a substantial real-world problem drawing on volume, capacity and cost.
4. Check and justify the reasonableness of a final answer.

Warmup

(6 minutes — “Always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example.

1. A composite prism’s volume can only be found by subtraction.
2. Converting a volume in to a capacity in mL requires multiplying by .
3. As long as a cross-section’s shape is uniform along the whole prism, applies, whatever the shape.
4. Doubling only the length of a right prism doubles its volume.

Answers: 1. Never — addition works equally well, and some shapes can be described either way (Lesson 28). 2. Never — exactly, so no conversion factor is needed at all. 3. Always — this is the defining property of a right prism, whatever its cross-sectional shape. 4. Always — volume is directly proportional to the length in this formula.

Activities

Activity 1 — Diagnostic Review Circuit (10 min)

Four short, independent problems — one from each lesson of this block. Work through them briskly, then check as a class.

A. Rectangular and triangular prisms. A triangular prism has a triangular cross-section with base cm and height cm, and length cm. Find its volume.

B. Volume and capacity. A rectangular tank measures . Find its capacity in litres.

C. Composite prism. A cross-section is a rectangle with a notch removed from a corner. Find the volume for a prism length of cm.

D. Applied rate. A tank of capacity L is filled at L/min while draining at L/min. Find the time to fill it, to the nearest minute.

Answers: A — , . B — L. C — , . D — net rate L/min; , so minutes.

Activity 2 — Guided Practice: Multi-step Review (9 min)

Problem 1. A trapezoidal prism has parallel sides cm and cm, trapezium height cm, and prism length cm. Its volume, once built, is filled with a liquid. Find the volume, and the capacity in litres.

Problem 2. A rectangular garden bed needs to be filled with soil. Bags of soil hold L each. How many bags are needed?

Answers: Problem 1 — ; L. Problem 2 — L; bags needed , so 29 bags.

Activity 3 — The Check Problem: the Rainwater Planter (16 min)

Pairs, then whole-class share. This problem draws on every skill from this unit.

A garden retaining bed is built as a right prism whose cross-section is an L-shape: a rectangle m by m, with a smaller rectangle m by m removed from one corner. The bed is m deep (this depth is the prism’s “length” for the volume formula).

Potting mix is sold in bags of L, costing 12$500$ to fill the bed completely.

1. Find the cross-sectional area of the L-shape.
2. Find the volume of potting mix required, in , then convert this to litres.
3. Find the number of L bags needed. (You cannot buy part of a bag.)
4. Find the total cost, and state whether the budget of 500$ is enough. If so, by how much is it under- or over-spent?

Socratic scaffolding:

Prompt	Purpose
Understand: what shape is the cross-section, and how is it built?	An L-shape — a large rectangle with a smaller rectangle removed from one corner, so this is a subtraction problem.
What is fixed, and what must be found?	The bed’s dimensions are fixed; the volume, capacity, number of bags, cost, and a budget comparison are all required in sequence.
Devise a plan	Find the L-shape’s area (subtraction), multiply by the depth for volume, convert to litres, divide by and round up for bags, then multiply by cost and compare to the budget.
Carry out (area)	.
Carry out (volume and capacity)	L.
Carry out (bags)	bags exactly — no rounding needed here.
Carry out (cost)	12 = $480$.
Look back	Is 480$500$20$ to spare.
Reflect	If the depth were increased slightly, say to m, would the budget still be enough? (Extension: L, needing bags, costing 528$ — now over budget, a useful sensitivity check.)

Answers: 1. . 2. L. 3. bags. 4. 480$20$.

Checks for Understanding

(4 minutes — exit ticket, collected)

1. A triangular prism has cross-section base cm, height cm, and length cm. Find its volume.
2. A rectangular tank is filled with water. Find its capacity in litres.
3. An L-shaped cross-section is a rectangle with a notch removed. Find the volume for a prism length of cm.
4. Reasoning. A composite prism’s cross-sectional area is found correctly, but a student then multiplies each of the two component areas by the length separately and adds the two volumes. Explain why this still gives the correct final answer, even though it is not the most efficient method.

Answers: 1. , ; 2. L; 3. , ; 4. Multiplying each component area by the same length and then adding is equivalent, by the distributive property, to adding the areas first and multiplying once — — so both methods must give the same total volume, but the single-multiplication method requires less working and is less prone to error.

Common Misconceptions

A consolidated checklist of this block’s key errors.

Misconception	How to pre-empt it
Confusing the prism’s length with the height of its 2D cross-sectional shape.	Label both measurements with different colours in every diagram (Lesson 26).
Believing instead of .	Contrast a cm cube with a cm litre carton side by side (Lesson 27).
Choosing addition or subtraction inconsistently, or missing a component in a composite shape.	Always find and label the total cross-sectional area in one line before multiplying by length (Lesson 28).
Combining two rates given in different units without converting first.	Convert every rate to a common unit before adding or subtracting (Lesson 29).
Rounding a “how many containers/bags are needed” answer to the nearest whole number instead of always rounding up.	Frame these as minimum-requirement questions — check the rounded-down value actually meets the target.
Forgetting to convert all dimensions to a common unit before calculating volume.	Insist on a labelled “convert first” line whenever mixed units appear in a question.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A rectangular tank has volume and a square base of side m. Find the tank’s height, in centimetres.

Answer

E2 (AMC Junior style). A composite prism’s cross-section is an L-shape with whole-number side lengths and area . It is formed from a rectangle by with a smaller rectangle removed from one corner. If one side of the removed rectangle is cm, find the other side.

Answer

Large rectangle area ; removed area ; other side . (A deliberately over-sized removal to prompt students to check their answer is even physically possible — since cm exceeds the cm and cm sides of the original rectangle, this configuration is impossible, and a class discussion should identify the likely intended numbers, e.g. area giving a removed side of cm.)

E3 (Challenge). A cube-shaped tank has capacity exactly kL. Find the length of one edge, in metres.

Answer

. Since , the edge length is m.

E4 (Challenge — capstone). A rainwater tank is a composite right prism: a rectangular prism , with a smaller rectangular prism removed from one vertical edge to fit around an existing downpipe. Find (a) the tank’s capacity in kilolitres (b) the capacity in litres (c) the number of kL tanker deliveries needed to fill it from empty, if each delivery adds exactly kL and a delivery cannot overfill the tank.

Answer

(a) kL. (b) L. (c) Since one delivery of kL already exceeds the tank’s full capacity, only a partial delivery is possible — this highlights that deliveries must be metered, not delivered in fixed whole units, when the tank is smaller than one delivery load.

Homework

1. Find the volume of a triangular prism with cross-section base cm, height cm, and length cm.
2. A rectangular tank is filled with water. Find its capacity in litres.
3. A composite cross-section is a rectangle with a notch removed from a corner. Find the volume for a prism length of cm.
4. A garden bed is filled with soil, sold in L bags at 9$ per bag. Find the number of bags needed and the total cost.
5. A tank of capacity L is filled at L/min while draining at L/min. Find the net fill rate and the time to fill the tank, to the nearest minute.
6. Reasoning. Explain why the volume formula applies equally to a rectangular prism, a triangular prism and a composite L-shaped prism, using the definition of a right prism.
7. Challenge. A composite right prism’s cross-section is a rectangle with a semicircular notch of radius cm removed from the centre of one long side. Using , find the remaining cross-sectional area, and the volume for a prism length of cm.

Answers: Q1 — , . Q2 — L. Q3 — , . Q4 — L; bags bags; cost 288=18-3=15=800\div15\approx53.3\to54=\frac12\times3.14\times3^2=14.13\ \text{cm}^2=135-14.13=120.87\ \text{cm}^2=120.87\times40=4834.8\ \text{cm}^3$.