The shape of the lesson
By the end of the lesson, learners can
- Rectangles and triangles: In the lab, change the size of the rectangle and watch the area and perimeter change.
- Parallelograms and trapezia: Cut the corner off a parallelogram and slide it across: you get a rectangle with the same base and height, so the area is \(bh\).
- Working backwards: Many questions give the area and ask for a missing length.
- Rectangle: \(A = lw\).
- Triangle: \(A = \tfrac{1}{2}bh\) (the only one given in the exam).
- Parallelogram: \(A = bh\).
- Trapezium: \(A = \tfrac{1}{2}(a+b)h\), \(a\) and \(b\) parallel.
- Perimeter: add every side, including the sloping ones.
- Backwards: substitute what you know and solve for the missing length.
One picture every idea returns to
Area always uses a length and the PERPENDICULAR height: rectangle \(lw\), triangle \(\tfrac{1}{2}bh\), parallelogram \(bh\), trapezium \(\tfrac{1}{2}(a+b)h\). Only the triangle formula is in the List of formulas, so learn the other three. Perimeter is just the sum of the sides.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Inside and around.. Perimeter is the distance around a shape; area is the space inside it. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Rectangles and triangles | In the lab, change the size of the rectangle and watch the area and perimeter change. Project tab 2 of the lab (Area and perimeter lab, area and perimeter) and let a student drive it while the class predicts. Model the worked example Rectangle on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Parallelograms and trapezia | Cut the corner off a parallelogram and slide it across: you get a rectangle with the same base and height, so the area is \(bh\). Model the worked example Parallelogram on the board, one line at a time. Your turn: A trapezium has parallel sides 5.5 cm and 9 cm, and the distance between them is 6 cm. Find its area. | Pairs |
| 21 to 30 min | Teach and check: Working backwards | Many questions give the area and ask for a missing length. Model the worked example Missing side of a rectangle on the board, one line at a time. Your turn: A square has perimeter 36 cm. Find its area. | Jigsaw |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 16 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Working backwards, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Jigsaw
Split the content into expert pieces. Expert groups master one piece each, then regroup so every home team has one expert per piece, and each expert teaches their piece to the team.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Using the sloping side instead of the perpendicular height: \(\tfrac{1}{2} \times 10 \times 7 = 35\) is wrong. | The height is always the one that meets the base at a right angle. |
| In a trapezium the two sides you add must be the PARALLEL ones, and \(h\) is the distance between them. | Multiplying all three numbers together, or forgetting the half, are the usual slips. |
| Keep units straight: perimeter is a length (cm), area is in square units (cm 2). | Writing 40 cm 2 for a perimeter loses the unit mark. |
Area and perimeter of 2D shapes: worksheet
6 questions, 16 marks. Written in the style of the papers; no past paper question is reproduced.
A rectangle measures 9.5 cm by 4 cm. Find (a) its area (b) its perimeter.
Find the area of (a) a triangle with base 11 cm and perpendicular height 8 cm (b) a parallelogram with base 7.5 cm and perpendicular height 4 cm.
Without a calculator, find the area of a trapezium with parallel sides 4.6 cm and 7.4 cm, a distance of 3.5 cm apart.
A rectangle measures 16 cm by 9 cm. A square has the same area as the rectangle.
Find the difference between the perimeters of the rectangle and the square.
A trapezium has area \(72\text{ cm}^2\). The distance between its parallel sides is 8 cm, and one of the parallel sides is 7 cm long.
Find the length of the other parallel side.
A garden is a trapezium. Its parallel sides are 12 m and 18 m long and they are 9 m apart.
Turf is sold only in whole square metres at $6.40 per square metre. The gardener orders 10% more turf than the area of the garden, to allow for waste.
Calculate the cost of the turf.
Answer key and marking notes
- (a) \(9.5 \times 4 = 38\text{ cm}^2\)
- (b) \(2(9.5 + 4) = 27\) cm
- (a) \(\tfrac{1}{2} \times 11 \times 8 = 44\text{ cm}^2\)
- (b) \(7.5 \times 4 = 30\text{ cm}^2\)
- \(4.6 + 7.4 = 12\), and half of 12 is 6.
- \(6 \times 3.5 = 21\text{ cm}^2\)
- Area \(= 16 \times 9 = 144\text{ cm}^2\), so the square has side \(\sqrt{144} = 12\) cm.
- Perimeters: rectangle \(2(16 + 9) = 50\) cm, square \(4 \times 12 = 48\) cm.
- Difference \(= 50 - 48 = 2\) cm.
- \(72 = \tfrac{1}{2}(7 + b) \times 8 = 4(7 + b)\).
- \(7 + b = 18\).
- \(b = 11\) cm.
- Area \(= \tfrac{1}{2}(12 + 18) \times 9 = 135\text{ m}^2\).
- Add 10%: \(135 \times 1.1 = 148.5\text{ m}^2\).
- Whole square metres, and at least 148.5 is needed, so buy \(149\text{ m}^2\).
- Cost \(= 149 \times 6.40 = \$953.60\).
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
A parallelogram has base 11 cm, sloping sides of 6 cm and perpendicular height 5 cm. Find its area.
- A\(55\text{ cm}^2\)
- B\(66\text{ cm}^2\)
- C\(27.5\text{ cm}^2\)
- D\(34\text{ cm}^2\)
The diagram shows a trapezium. Find its area.
- A\(57\text{ cm}^2\)
- B\(114\text{ cm}^2\)
- C\(61.75\text{ cm}^2\)
- D\(504\text{ cm}^2\)
A triangle has area \(54\text{ cm}^2\) and base 12 cm. Find its perpendicular height.
- A9 cm
- B4.5 cm
- C648 cm
- D42 cm
The length of a rectangle is 5 cm more than its width. Its perimeter is 34 cm. Find its area.
- A\(66\text{ cm}^2\)
- B\(282.75\text{ cm}^2\)
- C\(72.25\text{ cm}^2\)
- D\(34\text{ cm}^2\)
A trapezium has area \(91\text{ cm}^2\). The perpendicular distance between its parallel sides is 7 cm, and one parallel side is 9 cm. Find the length of the other parallel side.
- A17 cm
- B4 cm
- C26 cm
- D13 cm
A triangle has base 18 cm and perpendicular height 8 cm. A square has the same area as the triangle. Find the perimeter of the square, correct to 3 significant figures.
- A33.9 cm
- B48.0 cm
- C18.0 cm
- D8.49 cm
Quiz key
Area of a parallelogram \(= bh = 11 \times 5 = 55\text{ cm}^2\). The tempting \(66\) uses the sloping side, \(11 \times 6\), which is not the perpendicular height. 27.5 halves as if it were a triangle, and 34 is the perimeter \(2(11 + 6)\), a length, not an area.
\(A = \tfrac{1}{2}(7 + 12) \times 6 = \tfrac{1}{2} \times 19 \times 6 = 57\text{ cm}^2\). The tempting 61.75 uses the sloping side 6.5 cm as the height. 114 forgets the half, and 504 multiplies all three lengths \(7 \times 12 \times 6\).
\(54 = \tfrac{1}{2} \times 12 \times h = 6h\), so \(h = 54 \div 6 = 9\) cm. The tempting 4.5 cm forgets the half (\(54 \div 12\)). 648 multiplies the area by the base, and 42 subtracts the base from the area.
Perimeter \(= w + (w + 5) + w + (w + 5) = 4w + 10 = 34\), so \(w = 6\) cm and the length is 11 cm. Area \(= 6 \times 11 = 66\text{ cm}^2\). The tempting 282.75 adds only two sides, \(w + (w + 5) = 34\), giving \(14.5 \times 19.5\). 72.25 treats the shape as a square of side \(34 \div 4 = 8.5\), and 34 confuses the perimeter with the area.
\(91 = \tfrac{1}{2}(9 + b) \times 7 = 3.5(9 + b)\), so \(9 + b = 26\) and \(b = 17\) cm. The tempting 4 cm forgets the half: \(7(9 + b) = 91\) gives \(9 + b = 13\). 26 cm stops before subtracting the 9, and 13 cm is \(91 \div 7\) with no further steps.
Triangle area \(= \tfrac{1}{2} \times 18 \times 8 = 72\text{ cm}^2\). Square side \(= \sqrt{72} = 8.485...\) cm, so perimeter \(= 4 \times 8.485... = 33.9\) cm. The tempting 48.0 cm forgets the half: \(\sqrt{144} = 12\) and \(4 \times 12 = 48\). 18.0 divides the area by 4, and 8.49 cm is only one side.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.