At a glance. One lesson of about 60 minutes. Students find the area of compound shapes made of rectangles and triangles by splitting or subtracting, find the perimeter from the outside edges only, and work back to a missing length.
Calculator: no calculator. Nothing here needs one. The products are small, and the work is in choosing the split and which edges count.
What students will learn
Know
- A compound shape can be split into rectangles and triangles, and its area is the sum of their areas.
- A shape with a piece missing has area equal to the whole minus the piece.
- The perimeter of a compound shape is the sum of its outside edges only.
Understand
- Every correct split gives the same area.
- A line drawn to split a shape is inside it and is not part of the perimeter.
- Cutting a rectangular piece from a corner leaves the perimeter unchanged; cutting it from the middle of an edge adds two depths.
Do
- Choose a split or a subtraction and find the area, including the area of a trapezium.
- Find a perimeter from the outside edges, including missing lengths found by subtraction.
- Find an unknown length from a total area.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: area and perimeter of compound shapes (questions 1 to 6).
- Criterion B. Investigating patterns: the second figure. Ask students to predict, then test, what happens to the perimeter when a notch moves from a corner to the middle, and to state a rule for each (question 8 and the what-if).
- Criterion C. Communicating: showing the split lines on the diagram and writing the sum for each part in order, so another student can follow it (questions 2, 3 and 9).
- Criterion D. Applying mathematics in real-life contexts: courtyards, roofs and paving slabs, where tiles, edging and cutting each need a different measurement (questions 2 to 6).
The one thing to do with the figure
Do the split three ways, one at a time, and make students say the totals before they appear. Cover the numbers on the right of the figure with a sheet of paper, ask for the total, then uncover it. The point is that they match. Then change the notch size and ask whether it still works: it does, for every size.
In the second figure, ask “is the perimeter bigger, smaller or the same?” before each change of notch. The corner case surprises most: the area drops and the perimeter does not move.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | The courtyard and the first figure. Students predict the totals for each split. |
| 10 min | Key idea. Students draw an L-shape on squared paper and split it two different ways, then compare areas. |
| 10 min | Second figure, then the worked example; students finish the faded one in pairs. |
| 5 min | Question 1, the check. If many choose A, cross out the shared edge on a drawing. |
| 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; 26 m; 58. |
|---|---|
| Worked example | Area 108 m², perimeter 42 m. |
| Faded example | Area 51 m², perimeter 32 m. |
| Trapezium worked example | 24 + 8 = 32 m². |
| 1. Perimeter of joined shapes | B. A adds the two perimeters, C gives one square, D misses a side. If many choose A, draw the joined squares and cross out the shared edge. |
| 2. Area of an L-shape | 69 m². 84 − 15 = 69. |
| 3. Perimeter of an L-shape | 38 m. 2 × (12 + 7) = 38. |
| 4. Area with a triangle | 52 m². 40 + 12 = 52. |
| 5. Perimeter with a triangle | 28 m. 8 + 5 + 5 + 5 + 5 = 28. |
| 6. Area of a trapezium | 44 m². ½ × 22 × 4 = 44. |
| 7. Working back to a length | 7 m. 57 − 36 = 21, 21 ÷ 3 = 7. |
| 8. Notches and the perimeter | 24 cm. The perimeter of the square does not change. |
| 9. A missing piece is not part of the shape | A. B adds, C is the rectangle-round-it idea, D mixes up the two measurements. If many choose C, return to the first figure and use the whole-minus-notch button. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in question 1 | Adds the perimeters of the parts. | Cross out the joining edge on a drawing and ask whether it is on the outside. |
| 84 or 15 in question 2 | Gives the whole rectangle, or only the missing piece. | Shade what is patio, then ask what shape it is. |
| 27 or 69 in question 3 | Adds the four given lengths, or gives the area. | Label every edge of the L-shape, including the two found by subtraction. |
| 40, 64 or 12 in question 4 | Misses the roof, the half, or the walls. | Draw the split line and name the two parts. |
| 24, 36 or 20 in question 5 | Uses the roof height for the slope, includes the inside line, or leaves out the base. | Walk a finger round the outside and count the edges you pass. |
| 88, 55 or 56 in question 6 | Forgets the half, uses the sloping side as the height, or gives the rectangle round the trapezium. | Ask which length is at a right angle to the parallel sides, and whether two copies make a parallelogram. |
| 21, 19 or 11 in question 7 | Forgets the first rectangle, does not take it away, or reads the whole left edge (7 + 4) as the unknown side. | Draw both rectangles and label each area; mark the 4 m strip on the left edge. |
| 32 or 16 in question 8 | Adds or subtracts the cut edges. | Draw a square, cut one corner, and mark the old and new edges. |
| B, C or D in question 9 | Adds the missing piece, trusts the rectangle round the shape, or confuses area and perimeter. | Shade the patio and compare it with the rectangle. |
Questions worth asking
- Does it matter how you split a shape? No: every split covers the shape exactly once. Listen for “the pieces just have to fill it”.
- Why is the inside line not in the perimeter? It is not an edge of the shape. Listen for “nobody walks along it”.
- When does cutting a notch not change the perimeter? When the notch is in a corner and rectangular. Listen for “the new edges replace the old ones”.
- Can you find the area without splitting at all? Yes, by subtracting. Ask which is quicker for this shape.
Differentiation
A way in for anyone stuck
- Give squared paper with the compound shapes pre-drawn so students can shade each part a different colour.
- Provide a labelled diagram with the missing lengths marked with question marks, and fill them in as a class.
- Use L-shapes on a grid first, where the parts can be counted.
Stretch
- A shape has a rectangular hole in the middle of it that is 2 m by 3 m, inside a 9 m by 6 m rectangle. Find the area, and the total length of both the outer and inner edges. (48 m²; 30 + 10 = 40 m.)
- Show that the perimeter of any shape made by cutting rectangles from corners of a rectangle is the same as the rectangle's. (Each cut removes two edges and adds two equal edges.)
Students learning in English as an additional language
“Compound” means made of more than one part. “Notch” is a small piece cut out of an edge. Use the shape’s picture each time and say “outside edge” with a finger tracing it.
| Word | Plain meaning | Example |
|---|---|---|
| compound shape | a shape made up of two or more simpler shapes joined together | An L-shape is two rectangles |
| notch | a piece cut out of the edge or corner of a shape | A corner taken out of a rectangle |
| outside edge | a side that is on the boundary of the shape | The fence round a field |
| split | to divide a shape into pieces with straight lines | Cutting an L-shape into two rectangles |
| subtract | to take one number away from another | The whole area minus the missing piece |
| overall | the total length or width of the whole shape | An overall length of 12 m |
| trapezium | a four-sided shape with one pair of parallel sides | A paving slab that is wider at one end |
Sentence starters for explaining aloud or in writing:
- I split the shape into … and …
- The perimeter is … because the outside edges are …
- The line I drew is inside the shape, so …
Two things not to say
“Add the perimeters of the parts.” The shared edges are inside and would be counted twice.
“Multiply the longest length by the longest width.” That is the rectangle round the shape, and a shape with a piece missing is smaller than that.
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.
- SplitAn L-shape is made of a 6 m by 2 m rectangle and a 3 m by 2 m rectangle joined at the end. Find its area.
18 m²: 12 + 6. - SubtractA 10 cm by 6 cm card has a 2 cm by 2 cm square cut from the middle. Find the area.
56 cm².
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA floor is a 10 m by 6 m rectangle with a 3 m by 2 m cupboard in one corner. Carpet covers the rest. How much carpet?
54 m². - SpacingFrom Unit 7, lesson 3: a triangle with area 24 cm² has base 8 cm. What is its height?
6 cm: 48 ÷ 8.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningA square of side 8 cm has a square of side 2 cm cut from each of its four corners. Find the area and the perimeter.
Area 64 − 16 = 48 cm². Perimeter 32 cm. - Working backA rectangle is 12 m by 9 m. A square notch is cut from a corner and 92 m² is left. What is the side of the notch?
4 m: 108 − 92 = 16, and 4 × 4 = 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 and a few sheets of card do the same job: cut, slide and fold.