At a glance. One lesson of about 60 minutes. Students find the perimeter and area of rectangles, explain why area is length times width, give each answer in the right unit, find a missing side, and say what multiplying every side does to each.
Calculator: no calculator. Nothing here needs one. The products are small, and the point is the meaning of the two quantities.
What students will learn
Know
- Perimeter is the distance round a shape, in units of length.
- Area is the number of unit squares inside a shape, in square units.
- Area of a rectangle is length times width; perimeter is 2 times (length plus width).
Understand
- The same rectangle has two different measurements and they answer different questions.
- Area is rows times squares in a row, which is why it is a product.
- Multiplying every side by k multiplies the perimeter by k and the area by k times k.
Do
- Find the perimeter and area of a rectangle, with the right unit for each.
- Find a missing side from the area or from the perimeter.
- Predict and explain how both change when every side is scaled.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: perimeter and area of rectangles, units, and missing sides (questions 1 to 6).
- Criterion B. Investigating patterns: the scaling figure. Ask students to predict the perimeter and the area at times 3 before they slide, record the pairs for 1, 2, 3 and 4, and state the rule in their own words (question 6).
- Criterion C. Communicating: giving every answer with its unit, and writing in a sentence why the area alone cannot give the length of edging (question 9).
- Criterion D. Applying mathematics in real-life contexts: fencing and ground cover, where a student must decide which of the two measurements the problem needs (questions 7 and 8).
The one thing to do with the figure
Ask for a prediction, then change one thing. In the first figure make the length one metre longer and ask for both changes before anyone looks (perimeter 2 m more, area 4 m² more). Then make the width 1 m and ask “what has happened to the area, and to the perimeter?” The area is 6 m² and the perimeter is still 14 m. Take answers before you slide anything else.
In the second figure, cover the right-hand numbers with a sheet of paper and ask for the area at times 3 before uncovering them. Most students say “three times as big”: let the nine copies correct that without telling them.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | The plot and the two questions. Students decide which of fence and compost is the perimeter and which is the area, and what unit each is in. First figure with predictions. |
| 10 min | Key idea and the “add the sides” slip. Students draw a 5 by 3 rectangle on squared paper and count rows to build l × w themselves. |
| 8 min | Second figure, then the worked example; students finish the faded one in pairs. |
| 5 min | Question 1, the check. If many choose D, go back to the first figure and read both units aloud. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 144; 18; x = 13. |
|---|---|
| Worked example | Perimeter 34 m, area 66 m². |
| Faded example | Perimeter 44 m, area 112 m². |
| 1. Perimeter, area and units | A. B swaps the numbers, C stops after two sides, D swaps the units. If many choose D, go back to the first figure and read the two units aloud. |
| 2. Area of a rectangle | 120 m². 15 × 8 = 120. |
| 3. Perimeter of a rectangle | 105 cm. 2 × (32.5 + 20) = 105. |
| 4. Missing width from the area | 8 m. 72 ÷ 9 = 8. |
| 5. Missing width from the perimeter | 7 m. 38 ÷ 2 = 19, then 19 − 12 = 7. |
| 6. Scaling the area | 25 times. 5 × 5 = 25. |
| 7. From area to perimeter | 42 m. Width 98 ÷ 14 = 7, then 2 × (14 + 7) = 42. |
| 8. From perimeter to area | 81 m². Side 36 ÷ 4 = 9, area 9 × 9 = 81. |
| 9. Perimeter and area are different | A. B is a made-up doubling rule, C treats m² as m, D uses m² for a length. If many choose C, use 3 m by 4 m and 1 m by 12 m on the first figure. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| D in question 1 | Gets both numbers right and writes cm² for a distance. | Ask what a square centimetre looks like, and which number came from counting squares. |
| 46 or 23 in question 2 | Finds a perimeter, or adds two sides, when the question asked for area. | Ask them to draw the rows and count how many squares are in one row and how many rows. |
| 52.5 or 650 in question 3 | Adds only one length and one width, or finds the area. | Have them walk a finger round a drawn rectangle and count the sides touched. |
| 63 or 648 in question 4 | Subtracts or multiplies when the area must be divided by the length. | Say the sentence: 9 times what is 72? Then write the division. |
| 26 or 13 in question 5 | Takes away the length once, not twice. | Draw the rectangle, label two lengths and two widths, and cross the lengths off one at a time. |
| 5 or 10 in question 6 | Thinks the area changes by the same factor as the lengths, or adds the factor to itself. | Tile a 2 by 2 square with the first rectangle and count the blocks. |
| 21 or 28 in question 7 | Stops after finding the width, or adds only the long sides. | Ask for the number of sides the fence must cover, and write all four lengths. |
| 9 or 18 in question 8 | Stops at the side, or halves the perimeter. | A square has four equal sides: ask what 4 times the side must be. |
| B or D in question 9 | Invents a rule to get from m² to m. | Return to 3 by 4 and 1 by 12, which both have area 12 m² and different edges. |
Questions worth asking
- Can two rectangles have the same area and different perimeters? Yes. 3 by 4 and 1 by 12 both have area 12 m². Listen for “one is long and thin” and take it into lesson 2.
- Why are area units squared? Area counts squares of side 1 unit. Listen for “it is a square, not a line”.
- Why does doubling every side make four times the area? Two rows of two copies. Listen for the picture, not just the rule.
- If you know the perimeter of a rectangle, do you know its area? No: many rectangles share a perimeter. Listen for a counter-example, which leads into lesson 2.
Differentiation
A way in for anyone stuck
- Give squared paper and let students shade the rectangle, then count rows aloud before they use l × w.
- Provide a card with “perimeter: round the edge, m” and “area: squares inside, m²” and a small picture of each.
- Start with the 3 by 4 rectangle only, then change one side at a time.
Stretch
- Question 8, then: a square has perimeter p. Write its area using p. (Side p ÷ 4, area p ÷ 4 × p ÷ 4, which is p² ÷ 16.)
- A rectangle's length is three times its width and its perimeter is 48 m. Find its area. (Eight widths make 48, so the width is 6 m, the length 18 m and the area 108 m².)
Students learning in English as an additional language
“Area” and “perimeter” are often swapped. Link perimeter to “peri”, which means round, and say each word with a gesture: a hand tracing the edge, a hand sweeping the surface.
| Word | Plain meaning | Example |
|---|---|---|
| perimeter | the distance all the way round the edge of a shape | A fence round a field |
| area | the amount of flat surface inside a shape, counted in unit squares | The ground a lawn covers |
| square metre (m²) | the area of a square 1 m by 1 m | A floor of 12 m² holds 12 of them |
| length, width | the two different side lengths of a rectangle | A door 2 m long and 1 m wide in a drawing |
| rectangle | a shape with four straight sides and four right angles | A page, a door, a phone screen |
| scale factor | the number every length is multiplied by | All sides times 3 has scale factor 3 |
Sentence starters for explaining aloud or in writing:
- The perimeter is … because I added …
- The area is … because there are … rows of …
- I wrote … for the unit because it is a …
Two things not to say
“Perimeter is the small one and area is the big one.” Neither is bigger: they are different kinds of number with different units.
“Multiply the sides.” Without the picture of rows, students use the rule for the perimeter as well.
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.
- PerimeterA rectangle is 7 cm by 5 cm. Find its perimeter.
24 cm. - AreaA rectangle is 7 cm by 5 cm. Find its area.
35 cm².
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA noticeboard is 90 cm wide and 60 cm tall. Ribbon goes round the edge. How many centimetres of ribbon?
300 cm: 2 × (90 + 60). - SpacingFrom Unit 3, lesson 6 (solving simple equations): a rectangle has area 54 cm² and width 6 cm. Write and solve an equation for its length.
6x = 54, x = 9 cm.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningA rectangle has perimeter 30 cm and one side 4 cm longer than the other. Find its area.
Sides 5.5 and 9.5: the sides add to 15 and differ by 4, so 9.5 and 5.5. Area 52.25 cm². - ScalingEvery side of a rectangle is multiplied by 6. By what number is the perimeter multiplied, and by what number the area? Then: how many square centimetres are in 1 m²?
6 and 36. And 1 m² is 100 cm by 100 cm, so 100 × 100 = 10 000 cm².
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.