At a glance. One lesson of about 60 minutes. Students explain why a trapezium's area is half of base-sum times height, choose the height from a diagram with extra lengths, and work backwards to a missing height or parallel side.
Calculator: no calculator. The numbers are small, and the structure of the formula is the learning. A calculator would only hide the bracket.
What students will learn
Know
- Area of a trapezium is ½ × (a + b) × h, where a and b are the parallel sides.
- The height is measured at a right angle to the parallel sides.
- The sloping sides are not used.
Understand
- A turned copy joins a trapezium into a parallelogram with base a + b, so the trapezium is half of it.
- ½(a + b) is the average of the parallel sides, so the area is the middle width times the height.
- Leaning the top edge changes the sloping sides but not the area.
Do
- Find the area of a trapezium from the parallel sides and the height, ignoring extra lengths.
- Find a missing height or parallel side by working backwards.
- Combine the formula with the area of a rectangle or triangle.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the formula, the height and the parallel sides (questions 1 to 9).
- Criterion B. Investigating patterns: predict what the second figure will do as the lean changes, test it, then explain in a sentence why the area does not change (the second figure and the what-if question).
- Criterion C. Communicating: show the height with a square corner mark, label the parallel sides, and write the calculation with the bracket (question 4 and the worked examples).
- Criterion D. Applying mathematics in real-life contexts: land, garden beds and cut tile, where the shape has to be seen as a trapezium first (question 3, question 7 and the rice field).
The one thing to do with the figure
Ask for the area before the copy is added. Cover the readouts, set 4, 10 and 5, and ask students to predict the parallelogram's base and area. Then press Add a turned copy and let them correct themselves.
In the second figure, set the lean to 4, then −3, with the sloping-side readout hidden by a sheet of paper. Ask which numbers can change. Most say the area will. The point is that the two sloping sides change and nothing in the formula notices.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | The rice field and the prediction. First figure: add the copy. |
| 10 min | Key idea. Students cut a trapezium from squared paper, copy it, turn the copy and tape it on. |
| 10 min | Second figure, the middle line, then worked examples and the faded one in pairs. |
| 5 min | Question 1, the check. If many choose B, go back to the second figure. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 35 cm²; 30 m²; 7. |
|---|---|
| Worked examples | 54 m²; 240 m²; height 6 cm. |
| Faded example | 72 cm². |
| 1. Which calculation gives the area | A. B uses the sloping side as the height, C forgets the half, D multiplies the parallel sides. If many choose B, go back to the second figure and set the lean to 6. |
| 2. Area of a trapezium | 32 cm². ½ × 16 × 4 = 32. |
| 3. Parallel sides in either order | 56 m². ½ × 14 × 8 = 56. |
| 4. Sloping sides are extra | 128 cm². ½ × 32 × 8 = 128. |
| 5. Missing height | 6 cm. ½ × 30 = 15 and 90 ÷ 15 = 6. |
| 6. Missing parallel side | 15 m. 2 × 84 ÷ 7 = 24, then 24 − 9 = 15. |
| 7. A cut-off corner | 18 cm². 60 − 42 = 18, or ½ × 6 × 6 = 18. |
| 8. Unknown in the parallel sides | 14. The parallel sides add to 34, so 2x + 6 = 34 and x = 14. |
| 9. Same sum, same height | A. B and C judge by the look of the sides, D thinks the sloping sides are needed. Both sums are 24 cm, so the middle widths are both 12 cm. The lengths are outside both figures' ranges. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B in question 1 | Takes a sloping side as the height. | Ask which length makes a square corner with the parallel sides. Point at the dashed line. |
| 64 or 43 in question 2 | Forgets the half, or halves only the first side. | Ask what the turned copy gives, and then what fraction of it one trapezium is. Write the bracket first. |
| 112 or 180 in question 3 | Forgets the half, or multiplies the parallel sides. | Say “the two parallel sides, end to end” and draw the copy joined on. |
| 160, 256 or 88 in question 4 | Uses a sloping side as the height, forgets the half, or uses only one parallel side. | Cross out the sloping sides on the diagram first. Then ask which two sides are parallel. |
| 3 or 2700 in question 5 | Divides without doubling, or multiplies. | Write ½ × 30 = 15 and ask what times 15 is 90. |
| 24, 3 or 33 in question 6 | Stops at the sum, skips the doubling, or adds the 9. | Ask what 24 stands for in the diagram. Then ask how much of it is the 9. |
| 42 or 36 in question 7 | Reports the trapezium, or forgets the half of the triangle. | Colour the piece the question asks about. |
| 17, 28 or 11 in question 8 | Treats the sum as x, or forgets to halve. | Write “the two sides add to 34” and then write the sides as x and x + 6. |
| B, C or D in question 9 | Judges by the longest side or the shape, or thinks the sloping sides are needed. | Ask for the middle width of each trapezium: (4 + 20) ÷ 2 and (11 + 13) ÷ 2. |
Questions worth asking
- Why are the two parallel sides added and not multiplied? The copy is turned and put alongside, so the two sides lie end to end along the base of the parallelogram. Listen for “end to end”.
- Is the area always between a × h and b × h? Yes. It is the average of them. Listen for “halfway” or “the middle width”.
- Does the sloping side ever matter? Not for the area. It matters for the perimeter. Listen for the distinction.
- What happens when the top side shrinks to nothing? A triangle: the formula still works. Listen for “the same as half base times height”.
Differentiation
A way in for anyone stuck
- Give cut-out paper trapeziums so students can turn a copy and see the parallelogram.
- Give a set square or the corner of a page, and have them mark the height before multiplying.
- Start with right-angled trapeziums, where the height is one of the sides.
Stretch
- A trapezium has parallel sides 6 cm and 10 cm and a height of 4 cm. What triangle with the same height has the same area? (Base 16 cm: ½ × 16 × 4 = 32.)
- Two trapeziums have the same area and the same height. The parallel sides of one are 4 and 12. What is the sum of the parallel sides of the other? (16 cm.)
Students learning in English as an additional language
“Parallel” and “perpendicular” are the two words that matter. Parallel lines stay the same distance apart and never meet. Say “at a right angle” and show the square corner. “Trapezium” is called “trapezoid” in some countries; the shape is the same.
| Word | Plain meaning | Example |
|---|---|---|
| trapezium | a four-sided shape with one pair of parallel sides | A garden bed with one straight side parallel to the opposite side |
| parallel sides | two sides that are always the same distance apart | The two rails of a railway track |
| height | the distance between the parallel sides, measured at a right angle | How far apart the two ends of a field are |
| sloping side | a side that leans and is not parallel to the other one | The edge of the field that goes at a slant |
| average | the number halfway between two numbers | The average of 2 and 8 is 5 |
| right angle | a square corner, 90° | The corner of a page |
Sentence starters for explaining aloud or in writing:
- The parallel sides are … and …, and the height is …
- I do not need the … cm side because …
- The area is half of the parallelogram because …
Two things not to say
“Add the top and the bottom.” Some trapeziums are drawn on their side. Say “add the two parallel sides”.
“Times the slanty one.” The sloping side is never part of the area. Saying it as a rule tells students that it is.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- FluencyA trapezium has parallel sides of 6 m and 8 m and a height of 5 m. Find its area.
35 m². - FluencyA trapezium has parallel sides of 3 cm and 9 cm and a height of 4 cm. Find its area.
24 cm².
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA flower bed in Nairobi is a trapezium with parallel sides of 3 m and 5 m and a depth of 2 m. Each square metre needs 2 bags of soil. How many bags?
16 bags: the area is ½ × 8 × 2 = 8 m², and 8 × 2 = 16. - SpacingFrom Year 7 Unit 7, lesson 3: a triangle has area 24 cm² and a height of 6 cm. What is its base?
8 cm.
3Challenge
Reasoning, generalizing and working backwards.
- MissingA trapezium has height 6 cm and area 54 cm². One parallel side is twice the other. Find both.
6 cm and 12 cm: the sum is 18, so a + 2a = 18 and a = 6. - ReasoningRectangle R is 12 cm by 5 cm. A trapezium has the same height and the same area, and its short side is 8 cm. How long is its long side?
16 cm: the sum is 24, so the other side is 24 − 8 = 16.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Every figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Squared paper, scissors, string and a few sheets of card do the same job: cut, turn, wrap and fold.