At a glance. One lesson of about 60 minutes. Students explain why a parallelogram has area base times height and a triangle has half of that, choose the perpendicular height from a diagram with extra lengths, and find a missing base or height.
Calculator: no calculator. Nothing here needs one. The products are small, and the two dissections are the learning.
What students will learn
Know
- Area of a parallelogram is base times perpendicular height.
- Area of a triangle is half of base times perpendicular height.
- The height is measured at a right angle to the base and is shorter than the slanted side.
Understand
- Cutting and sliding turns a parallelogram into a rectangle of equal area.
- Two copies of a triangle make a parallelogram, so a triangle is half.
- Sliding a triangle's top corner along a parallel line changes the shape but not the area.
Do
- Find the area of a parallelogram or a triangle from its base and perpendicular height.
- Pick the right length when extra lengths are given.
- Find a missing base or height from the area.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the two formulas and the perpendicular height (questions 1 to 8).
- Criterion B. Investigating patterns: the two figures. Ask students to predict what stays the same as the lean or the corner moves, test it, and then write a general statement about all parallelograms and triangles (questions 7 and 8).
- Criterion C. Communicating: explaining in words or a labelled sketch why the triangle is half, and showing the height with a square corner mark (questions 4 and 9).
- Criterion D. Applying mathematics in real-life contexts: sails and cloth, where extra measurements on the real object have to be sorted into the one that is needed (question 3 and the sail worked example).
The one thing to do with the figure
Make the lean slider the argument. In the first figure, ask: “I lean it 4 cm further. What changes?” Take answers: the slanted side changes, the area does not. Press Cut and slide and let students say why.
In the second figure, cover the area line at the top with a sheet of paper and ask for the area at corner 5 and at corner −2 before they slide. Most expect the obtuse one to be different. The height outside the shape is the new idea: draw the extended base line on the board.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | The sail and the three lengths. First figure with the prediction on the lean. |
| 10 min | Key idea. Students cut a parallelogram from squared paper, cut the triangle off one end and slide it, then cut a second paper triangle and turn the copy. |
| 10 min | Second figure, worked examples, then the faded one in pairs. |
| 5 min | Question 1, the check. If many choose B, go back to the first figure. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 52 cm²; 38; 70°. |
|---|---|
| Worked examples | Parallelogram 112 cm²; sail 44 m². |
| Faded example | Parallelogram 90 cm², triangle 45 cm². |
| 1. Which length gives the area | A. B uses the slanted side, C halves, D finds the perimeter. If many choose B, go back to the first figure and set the lean to 4. |
| 2. Area of a parallelogram | 54 cm². 9 × 6 = 54. |
| 3. Area of a triangle | 36 m². ½ × 8 × 9 = 36. |
| 4. Triangle with sloping sides given | 48 cm². ½ × 12 × 8 = 48. |
| 5. Missing height | 8 cm. 36 × 2 = 72, 72 ÷ 9 = 8. |
| 6. Missing base | 15 m. 90 ÷ 6 = 15. |
| 7. Sliding the corner | 30 cm². ½ × 10 × 6 = 30. |
| 8. Comparing two areas | 30 cm². 60 − 30 = 30. |
| 9. A right angle gives the height | A. B and C use the sloping side as a base with a height that is not at a right angle to it. D forgets the half. If many choose B or C, go back to the second figure and set the corner to 0. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B in question 1 | Takes the slanted side as the height. | Ask which length makes a square corner with the base. Point at the dashed line. |
| 15 or 27 in question 2 | Adds the lengths, or halves a parallelogram. | Cut and slide: what rectangle do we get, and what are its sides? |
| 72 or 17 in question 3 | Forgets the half, or adds. | Draw the parallelogram around the triangle and ask what fraction of it the triangle is. |
| 60, 96 or 120 in question 4 | Uses the sloping side as the height. | Ask them to rule a square corner with a set square or the corner of a page. |
| 4, 2 or 324 in question 5 | Divides by the base without doubling first, or doubles the wrong way. | Ask what base times height must be for two copies, then divide. |
| 540 or 7.5 in question 6 | Multiplies, or halves a parallelogram. | Say it aloud: 6 times what is 90? |
| 42 in question 7 | Adds the slide to the base. | Use the second figure with the corner at 4 and ask what the base is. |
| 60 or 15 in question 8 | Reports one area, or half the triangle. | Write both areas, then subtract. |
| B or C in question 9 | Uses the longest side as a base with the wrong height. | Ask which two sides make the square corner. |
Questions worth asking
- Why is the slanted side longer than the height? The slanted side is the long side of a right-angled triangle whose other side is the height. Listen for the longest side being opposite the right angle.
- Does a triangle always have half the area of the parallelogram with the same base and height? Yes. Listen for the copy and turn: two make one.
- Can the height of a triangle be outside the triangle? Yes, when one angle is obtuse. Listen for “extend the base line”.
- Which side of a triangle can be the base? Any, with the height measured to that side. Try the 9, 12, 15 triangle with each.
Differentiation
A way in for anyone stuck
- Give cut-out paper parallelograms and triangles so students can really slide and turn them.
- Give a set square, and have them mark the right angle on the diagram before multiplying.
- Start with right-angled triangles only, where the height is one of the sides.
Stretch
- Show that the right triangle with sides 9, 12 and 15 has height 7.2 cm to the longest side. (Area 54 = ½ × 15 × h, so h = 7.2.)
- Two triangles share a base of 8 cm and have the same area. One has height 5 cm. What can you say about the other? (Its height is also 5 cm: the area fixes the height.)
Students learning in English as an additional language
“Height” has an everyday meaning (how tall) and a maths meaning (the length at a right angle to the base). Say “perpendicular height” and show the square corner. “Slant” means leaning.
| Word | Plain meaning | Example |
|---|---|---|
| base | the side you measure the height from | The bottom edge of a triangle |
| perpendicular height | the distance from the base to the top, measured at a right angle to the base | The mast of a sail |
| parallelogram | a four-sided shape whose opposite sides are parallel | A leaning rectangle |
| slanted side | a side that leans and is not at a right angle to the base | The edge of a parallelogram that leans |
| right angle | a square corner, 90° | The corner of a page |
| triangle | a shape with three straight sides | A sail |
Sentence starters for explaining aloud or in writing:
- The base is … and the height at a right angle is …
- I do not need the … cm side because …
- The triangle is half of the parallelogram because …
Two things not to say
“Height is the tall side.” A tilted shape's tallest side is the slanted one, which is wrong for area.
“The half is only for the pointy ones.” The half belongs to every triangle, whatever its shape. A student who drops it for an obtuse triangle has not seen the copy and turn.
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.
- ParallelogramA parallelogram has base 14 cm and height 6 cm. Find its area.
84 cm². - TriangleA triangle has base 14 cm and height 6 cm. Find its area.
42 cm².
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA triangular flag has a base of 90 cm and a height of 60 cm. How much cloth is in one flag?
2700 cm²: ½ × 90 × 60. - SpacingFrom Unit 7, lesson 2: a rectangle has perimeter 36 m. What is the greatest area it can have, with whole-number sides?
81 m² (9 by 9).
3Challenge
Reasoning, generalising and working backwards.
- MissingA triangle has area 50 cm² and height 10 cm. Find its base. Then say what base a parallelogram of the same area and height would have.
Triangle base 10 cm; parallelogram base 5 cm. - ReasoningA right-angled triangle has legs 6 cm and 8 cm and longest side 10 cm. Find its area using each leg as the base in turn.
½ × 6 × 8 = 24 cm² both times.
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 and a few sheets of card do the same job: cut, slide and fold.