027i - Volume and Capacity

Lesson 27 — Explicit Instruction: Relating Volume and Capacity, and Units of Measure

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

Learning Intentions

• To understand the relationship between volume and capacity, including and .
• To convert confidently between metric units of volume (, , ) and capacity (, , ).

Success Criteria

I can:

1. State the key volume–capacity equivalences: and .
2. Convert between metric units of volume by cubing the linear conversion factor.
3. Convert between units of capacity (, , ).
4. Solve a problem that requires converting between volume and capacity within a single solution.

Warmup

(5 minutes — recall and predict, mini whiteboards)

1. How many millimetres are in centimetre?
2. When converting an area from to , do you multiply by or ? Why?
3. Predict: when converting a volume from to , what power of do you think you will need, and why?

Answers: 1. ; 2. , because area has two dimensions, so the linear factor is applied twice; 3. Prediction — most students will correctly reason , since volume has three dimensions. Confirm this formally in Activity 1.

Activities

Activity 1 — Explicit Instruction: Converting between Units of Volume (12 min)

The rule. For a linear conversion factor of , the matching area factor is and the matching volume factor is .

I do — mm³ and cm³. Since , a cube of side cm can be packed with smaller cubes of side mm: along each edge, so small cubes fit inside.

I do — cm³ and m³. Since :

Worked conversion. Convert to , and to .

We do: Convert (a) to (b) to (c) to .

You do:

1. to
2. to
3. to
4. to

(Answers: 1. ; 2. ; 3. ; 4. .)

Activity 2 — Guided Practice: the volume–capacity Equivalence (10 min)

Key facts, modelled explicitly:

• of water occupies exactly of space — this is how the millilitre is defined.
• .
• (a kilolitre).

I do. A fish tank measures . Find its volume, then its capacity in litres.

I do — working in metres. A concrete water tank is a rectangular prism . Find its capacity in litres.

We do: A cooler box is . Find its capacity in litres.

You do:

1. A box measuring is filled with water. Find its capacity in litres.
2. A rainwater tank is a rectangular prism . Find its capacity in kilolitres.
3. A jug holds . What is this in ?

(Answers: 1. ; 2. ; 3. .)

Activity 3 — Problem-solving: Filling the Aquarium (13 min)

Pairs, then whole-class share.

An aquarium shop sells fish tanks by their advertised capacity in litres, but a customer needs to check the tank will fit a shelf measured in centimetres.

A tank measures .

1. Find the tank’s volume in , then its capacity in litres.
2. The shop recommends water conditioner tablet per L of water. How many tablets are needed to treat a full tank? (Round up — a partial tablet cannot be used.)
3. Conditioner tablets are sold in packs of for 6.50$. What is the minimum cost to treat the tank fully?

Socratic scaffolding:

Prompt	Purpose
Understand: what are you being asked to find across the three parts?	A capacity in litres, then a whole number of tablets, then a cost — three linked calculations.
Devise a plan for Part 1	Multiply the three centimetre dimensions to get , then convert directly to litres using .
Carry out Part 1	.
For Part 2, what operation finds “how many groups of L fit into L”?	Division: .
Carry out Part 2, then look back	. Since a partial tablet cannot be used, round up to tablets.
For Part 3, how many packs of are needed to get at least tablets?	, so round up to packs, giving tablets — more than enough.
Carry out Part 3	6.50 = $26.00$.
Looking back — why did both Part 2 and Part 3 round up, not to the nearest whole number?	Each answer represents a minimum requirement — under-treating the water or buying too few tablets is not an option, so rounding down would not be safe or possible.

Answers: 1. . 2. tablets. 3. 26.00$ (4 packs, 3 tablets left over).

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Convert to .
2. Convert to .
3. A box measures . Find its capacity in millilitres.
4. A tank has capacity . State this capacity in litres, and in .
5. Reasoning. Explain why , but .

Answers: 1. ; 2. ; 3. ; 4. ; ; 5. A millilitre is defined as the volume of exactly , but a litre is times larger than a millilitre, so it takes , not , to make litre.

Common Misconceptions

Misconception	How to pre-empt it
Using the linear or squared conversion factor for volume instead of the cubed factor.	Always ask “how many dimensions does this quantity have?” before choosing , or .
Believing .	Contrast a cm sugar cube with a litre carton side by side — the carton is cm per edge, i.e. times the volume.
Forgetting to convert all dimensions to the same unit before multiplying to find volume.	Insist on a labelled “convert first” line whenever mixed units appear in a question.
Rounding a “how many containers/tablets/bags are needed” answer to the nearest whole number instead of always rounding up.	Frame these as “minimum required” questions — model that rounding down would leave the job incomplete.
Confusing (kilolitres) with (millilitres) due to similar abbreviations.	Say the full word every time until fluent: “kilo” means , “milli” means .

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A cube has a volume of . What is its capacity in litres, and what is the length of one edge?

Answer

Capacity . Edge length: since , the edge is cm.

E2 (AMC Junior style). A rectangular tank long and wide is filled with of water. Find the depth of the water, in metres.

Answer

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

Answer

, so the tank is a cube — edge length cm.

E4 (Investigation). Explain why doubling every dimension of a rectangular tank multiplies its capacity by , not by , using both a numerical example and the formula .

Answer

Doubling all three dimensions doubles the cross-sectional area by a factor of (since area involves two dimensions), and then doubling the length multiplies volume by a further : overall . Example: a tank has volume ; doubled to , volume .

Homework

1. Convert: (a) to (b) to (c) to (d) to .
2. A box measures . Find its capacity in millilitres and in litres.
3. A tank is a rectangular prism . Find its capacity in kilolitres and in litres.
4. A carton holds of juice. How many glasses can be completely filled from one carton?
5. Reasoning. A student converts to by multiplying by , getting . Explain their error and give the correct answer.
6. Challenge. A rectangular tank has a square base of side cm and a height of cm. Its capacity is exactly L. Find .

Answers: Q1 — (a) (b) (c) (d) . Q2 — . Q3 — . Q4 — glasses. Q5 — the student used the linear factor () instead of the cubed factor (), since volume has three dimensions; correct answer is . Q6 — , so cm.