019i - Volume of Rectangular Prisms

Lesson 19 — Volume of Rectangular Prisms

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

Learning Intentions

• To understand volume as the number of unit cubes that fill a solid.
• To calculate the volume of a rectangular prism using the established formula.

Success Criteria

I can:

1. Explain volume as “area of the cross-section × length”.
2. Calculate the volume of a rectangular prism using .
3. Use cubic units correctly (, ).
4. Find a missing dimension when given the volume.

Warmup

(6 minutes — building with cubes, pairs)

Give each pair a set of interlocking centimetre cubes.

1. Build a rectangle, one cube deep. How many cubes did you use?
2. Add a second identical layer. How many cubes now?
3. Add a third layer. How many now?
4. Without building it, predict how many cubes a tower would need.

Bridging question: What are you actually doing each time you add a layer? (Adding another — that is, multiplying the layer size by the number of layers.)

Activities

Activity 1 — Explicit Instruction: Layers and the Formula (12 min)

Definition. Volume is the amount of space a solid occupies, measured in cubic units. One is a cube measuring cm on every edge.

Building the formula from layers — do not just state it.

The general prism idea — introduce it now, as it carries into Lesson 20:

A rectangular prism is simply the case where the cross-section is a rectangle.

I do: A box cm long, cm wide, cm high.

State aloud: the base holds cubes, and there are layers.

Cubic units — emphasise. Length uses cm, area uses , volume uses . The exponent counts the dimensions.

We do: ; ; .

You do: Eight prisms from a worksheet, including two given as diagrams rather than dimension lists, and one requiring a unit conversion.

Activity 2 — Working backwards (8 min)

I do. A rectangular tank has volume , length cm and width cm. Find its height.

You do:

1. , cm, cm. Find .
2. , base area . Find the height.
3. A cube has volume . Find its edge length.
4. , cm, cm. Find .

(Answers: cm; m; cm (since ); cm.)

Note on Q3: this connects directly to Lesson 8 — a cube’s volume is an edge length cubed, just as a square’s area is a side length squared.

Activity 3 — Inquiry: the Box Problem (10 min)

Pairs, centimetre cubes available.

A manufacturer must design a rectangular box with a volume of exactly . All three dimensions must be whole numbers of centimetres.

1. Find as many different boxes as you can. Record them in a table.
2. Which design uses the least cardboard? (You will need surface area.)
3. What do you notice about the shape of the most efficient box?

Socratic scaffolding:

Prompt	Purpose
Understand: what makes two boxes “different”?	A different set of three dimensions. and are the same box.
How do you find them systematically?	Find all ways of writing as a product of three whole numbers.
Devise a plan	Start with , then , then , and list the remaining factor pairs.
Carry it out	, , , , , , , , . Nine boxes.
Now the cardboard. What is the surface area of a box?	.
Compare the extremes.	: . : .
Looking back — what do you notice?	The most cube-like box uses the least material. Long thin boxes waste cardboard.

Checks for Understanding

(5 minutes — exit ticket)

1. Find the volume of a prism cm by cm by cm.
2. A tank has volume and a base area of . Find its height.
3. A cube has edge length cm. Find its volume.
4. Reasoning. A student gives the volume of a prism as . What is wrong?
5. A box is cm by cm by cm. Another is cm by cm by cm. Which holds more?

Answers: 1. ; 2. m; 3. ; 4. The units — volume is measured in cubic units, so the answer is ; 5. Both are — equal.

Common Misconceptions

Misconception	How to pre-empt it
Using for volume.	Link the exponent to the number of dimensions every time: length , area , volume .
Adding the three dimensions instead of multiplying.	Return to the layers model — layers multiply, they do not add.
Confusing volume with surface area.	Ask “am I filling it or wrapping it?” before every calculation.
Believing a taller box always holds more.	The exit ticket Q5 confronts this. Compare several boxes with the same volume.
Mixing units, e.g. two dimensions in cm and one in m.	Require a conversion line first.
Thinking .	It is . Show the reasoning: .

Enrichment — Competition-Style Problems

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

Answer

Edge cm (since ). Surface area .

E2 (AMC Junior style). A rectangular prism has volume . If each dimension is doubled, what is the new volume?

Answer

Doubling every dimension multiplies the volume by .

E3 (Challenge). How many centimetre cubes fit inside a box measuring m by cm by cm?

Answer

Convert to centimetres: cubes.

E4 (Challenge). A rectangular water tank measures m by m by m. Given that litres, how many litres does it hold when full? If it is filled at litres per minute, how long does filling take?

Answer

Homework

1. Find the volume: (a) cm (b) cm (c) m (d) cm.
2. Find the missing dimension: (a) , cm, cm (b) , base area (c) for a cube — find the edge.
3. A shipping container is m long, m wide and m high. Find its volume, correct to one decimal place.
4. A fish tank measures cm by cm by cm. Find its volume in , then in litres ( litre).
5. A box holds exactly centimetre cubes. List four possible sets of whole-number dimensions.
6. Reasoning. Explain why equals , not .
7. Challenge. Two rectangular prisms have the same volume. The first is cm. The second has a square base of side cm. Find its height.

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) cm (b) m (c) cm. Q3 — . Q4 — litres. Q5 — e.g. , , , . Q7 — ; , so cm.