023i - Units for Area and Perimeter

Lesson 23 — Applying Appropriate Units: Area and Perimeter

Strand: Measurement | Descriptor: AC9M8M01 | Duration: 45 minutes

Learning Intentions

• To convert confidently between mm, cm and m, and between mm², cm² and m².
• To choose units appropriate to the size and context of a real measurement.

Success Criteria

I can:

1. Convert a length between mm, cm and m.
2. Convert an area between mm², cm² and m², using the correct squared conversion factor.
3. Solve composite-shape problems where dimensions are given in mixed units, converting before calculating.
4. Justify the choice of a sensible unit for a given real-world quantity.

Warmup

(6 minutes — rapid conversions, mini whiteboards)

1. cm mm
2. cm m
3. m cm

Answers: 1. mm; 2. m; 3. cm; 4. ; 5. .

Hook discussion: Would you measure the length of your pencil in mm, cm or m? Why? What about the distance to the office, or the thickness of a coin? What makes a unit “sensible” for a given job?

Activities

Activity 1 — Explicit Instruction: Extending Conversions and Choosing Sensible Units (10 min)

The mm conversion, derived explicitly:

I do:

Choosing a sensible unit — discuss as a class. For each of the following, decide mm, cm or m (for lengths) or mm², cm² or m² (for areas), and justify: the thickness of a phone screen; the area of a phone screen; the width of a classroom door; the area of a classroom floor; the length of a running track. (Note: for areas far larger than a classroom — a farm, a suburb — units such as hectares or are used, which you will meet in later years.)

We do: As a class, agree and justify a sensible unit for: a postage stamp’s area; a soccer pitch’s area; a fingernail’s width; a bridge’s length.

You do: Convert: (a) mm to cm (b) m to mm (c) to (d) to (e) to (f) to .

(Answers: cm; mm; ; ; ; .)

Activity 2 — Guided Practice: Composite Shapes with Mixed Units (12 min)

Real diagrams often mix units. Convert everything to one unit before doing anything else.

I do — a floor plan. A room is given as m long and cm wide, with a cm by cm rectangle removed from one corner.

We do — a T-shaped bracket. The bar is cm wide and mm tall; the stem is mm wide and mm tall, centred beneath the bar.

You do: Two further composite shapes with mixed mm/cm/m dimensions, requiring a full unit conversion before any area or perimeter calculation.

Activity 3 — Applied Task: Costing a Courtyard (11 min)

Pairs. Real quantities, real units, a real invoice.

A rectangular courtyard measures m by cm, with a m by cm garden bed left unpaved in one corner.

1. Convert all measurements to a single consistent unit.
2. Find the paved area. Pavers cost 65$ per square metre — find the cost.
3. Find the perimeter of the paved region. Edging strip costs 18$ per metre — find the cost.
4. Find the total cost of the project.

Socratic scaffolding:

Prompt	Purpose
Understand: what units are given, and are they consistent?	m, cm, m, cm — a mix.
Devise a plan	Convert everything to metres, since the costs are quoted per m² and per m.
Carry out (convert)	m; m.
Carry out (area)	; notch ; paved .
Carry out (perimeter)	Corner notch, so m.
Carry out (cost)	Pavers 371831.8 \times 18 = $572.40$4290.40$.
Look back	Would converting everything to centimetres instead give the same final dollar figure? Yes — provided the per-unit costs are also converted consistently.

Answers: 2. , 371831.8$572.40$4290.40$.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Convert: (a) cm to mm (b) m to cm (c) to (d) to .
2. A rectangular sign is given as m by cm. Convert to a single unit and find its area in .
3. A composite shape has overall dimensions cm by m, with an cm by m notch removed from a corner. Find its area in .
4. Reasoning. Explain why the conversion factor for area is the square of the conversion factor for length.
5. Would you measure the area of a book’s cover in , or ? Justify your choice.

Answers: 1. (a) mm (b) cm (c) (d) ; 2. ; 3. ; 4. Area is a product of two lengths, so converting both dimensions multiplies the conversion factor by itself; 5. — large enough to avoid an awkward mm² value, small enough that would give a tiny, hard-to-read decimal.

Common Misconceptions

Misconception	How to pre-empt it
Using the linear factor for an area conversion, e.g. .	Always show the squaring step explicitly: “the factor is , so the area factor is .”
Combining lengths in a calculation before converting them to the same unit.	Require a labelled conversion line as the first step of any mixed-unit problem.
Multiplying when converting to a smaller unit, or dividing when converting to a larger one — reversed.	Anchor with a concrete check: “more millimetres or more metres for the same length?”
Choosing an impractical unit for a real context, e.g. quoting a classroom floor in .	Practise the “sensible unit” discussion routinely, not just as a one-off activity.
Rounding an intermediate conversion before finishing a multi-step problem, causing the final answer to drift.	Insist on carrying full precision (or clearly noted fractions/decimals) until the final step.
Forgetting to convert a cost-per-unit rate to match the units used for the measurement.	Underline the rate’s units (e.g. "") and check they match the calculated area’s units before multiplying.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A postage stamp measures mm by mm. Express its area in .

Answer

E2 (AMC Junior style). A square tile has side length m. How many tiles, laid edge to edge with no gaps, are needed to cover a rectangular floor m by m?

Answer

E3 (Challenge). Two rectangles have equal area. Rectangle A is m by cm. Rectangle B is m by cm. Find .

Answer

E4 (Challenge). A composite shape’s overall dimensions are m by cm, with a cm by cm notch removed from the middle of one long side (not a corner). Using the Lesson 21 result about middle-of-side notches, find the shape’s perimeter without recomputing from scratch.

Answer

Convert to a single unit: cm.

Homework

1. Convert: (a) cm to mm (b) mm to cm (c) m to cm (d) cm to m.
2. Convert: (a) to (b) to (c) to (d) to .
3. A rectangular tabletop is given as m by cm. Find its area in .
4. A composite shape has overall dimensions m by cm, with a cm by m rectangle removed from a corner. Find (a) its area in (b) its perimeter in m.
5. A picture frame’s glass pane measures mm by mm. Express its area in .
6. Reasoning. A student converts to by multiplying by . Explain the error and give the correct value.
7. Challenge. Two rectangular plots have equal area. Plot A is m by cm. Plot B is m by cm. Find .

Answers: Q1 — (a) mm (b) cm (c) cm (d) m. Q2 — (a) (b) (c) (d) . Q3 — . Q4 — (a) (b) corner notch, m. Q5 — . Q6 — the linear factor () was used instead of the squared area factor (); correct value is . Q7 — ; gives cm.