Mathematics

Intermediate

50 mins

Teacher/Student led

Learning Goals

Learning Outcomes

Lesson Files

+80 XP

What you need:
IWB/Projector/Large Screen
Square tiles

Comparing Area and Perimeter

Explore how two shapes can cover the same area but have different perimeters. You'll measure area by counting squares and perimeter by adding side lengths, discovering that the way a shape is arranged changes the distance around its edge.

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1 Getting Started
2 Watch and Notice
3 Draw the Two Rectangles in Your Copy
4 Try It Together
5 Class Challenge
6 What Did We Notice?
7 What's Next
Pupil practice

1 - Getting Started ~4 mins

Here are two rectangles drawn on squared paper.

Rectangle A: a long thin strip, 1 square tall and 6 squares long.
Rectangle B: a chunky block, 2 squares tall and 3 squares long.

Count the squares inside each one.

Now have a think before we work anything out: if an ant walked all the way around the edge of each one, would it walk the same distance both times? Hands up for your guess.

👇 Mark your steps as done as you complete them.

+5 XP

2 - Watch and Notice ~9 mins

Let's look at these rectangles together. For each one, count the squares that fit inside (that is the area) and then walk right around the edge, adding every side (that is the perimeter). The first two shapes cover the same area, so watch what happens to the distance around. The third one is a bigger shape where both the area and the perimeter are different, so keep an eye on both numbers.

👇 Mark your steps as done as you complete them.

+10 XP

3 - Draw the Two Rectangles in Your Copy ~3 mins

COPYBOOK MOMENT

Illustration for Draw the Two Rectangles in Your Copy In your squared copy, draw two different rectangles that each cover 8 squares. Write the perimeter beside each one, then ring the rectangle with the longer way around.

+10 XP

4 - Try It Together ~8 mins

Now it is your turn to work out the numbers. The shape-measurer will show a rectangle, but this time it hides the area and the perimeter until you enter your answer and check it. We will work three rectangles in turn. First a 2 by 4. Then a 1 by 8. Then a 3 by 3. For each one, count the squares inside to find the area. Then add all four sides to find the perimeter. Enter your answer, press Check, and say whether the distance around is getting longer or shorter as the shape changes.

Shape Measurer

+15 XP

5 - Class Challenge ~8 mins

Key point

Here is the investigation. Each time, use exactly 12 square tiles. Where you have your own tiles, work on your own; where a group of four shares one set of 12, take turns arranging them. Make as many different rectangles as you can and write down the area and the perimeter of each one.

First, make any rectangle from all 12 tiles and record its area and perimeter.
Then find a second, different rectangle from the same 12 tiles.
Which of your rectangles has the smallest distance around?
Explain why the area stayed 12 every single time while the perimeter kept changing.

Using exactly 12 square tiles each time, how many different rectangles can you make, and what is the area and perimeter of each?

Give each pupil 12 square tiles (or 12 shaded squares on squared paper) where possible; where tiles are short, seat pupils in groups of four sharing one set of 12 and take turns. They arrange all 12 into a rectangle, record the sides, the area and the perimeter, then rearrange into a different rectangle and record again. Repeat to find every rectangle possible from 12 tiles (1 by 12, 2 by 6, 3 by 4). Individual pupils bring one rectangle to the board to record its measurements for the class.

Make any rectangle from all 12 tiles and record its area and perimeter.
Find a second, different rectangle from the same 12 tiles.
Which of your rectangles has the smallest distance around?
Why did the area stay 12 every time while the perimeter changed?

Ways to start:

Lay all 12 tiles in a single long row — what are the sides of that rectangle?
Try two rows of 6 tiles — how does the shape look now?

Stretch:

Put your rectangles in order from the longest way around to the shortest — what do you notice about their shapes?
If you had 16 tiles instead, which rectangle would have the shortest perimeter?

Record: table with columns: sides (e.g. 2 by 6), area (squares), perimeter (units)

Answers & strategies (teacher)

Make any rectangle from all 12 tiles and record its area and perimeter. — For example 2 by 6: area 12 squares, perimeter 2 + 6 + 2 + 6 = 16.
Find a second, different rectangle from the same 12 tiles. — 3 by 4: area 12 squares, perimeter 3 + 4 + 3 + 4 = 14 (or 1 by 12: area 12, perimeter 26).
Which of your rectangles has the smallest distance around? — The 3 by 4 rectangle, with a perimeter of 14 — the chunkiest rectangle has the shortest way around.
Why did the area stay 12 every time while the perimeter changed? — The 12 tiles never went away, so the space covered (area) stayed 12; but spreading them into a long thin shape puts more tile edges on the outside, making the distance around longer.

+15 XP

Pupil practice

Module 5 · Area, Perimeter and Volume Measures

Lesson 61 · Comparing Area and Perimeter

Download Activity Book page (PDF)

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