Fence or grass?

Two gardens use exactly the same length of fence. Do they hold the same amount of grass?

About 40 minutes
Class 6
Mathematics and money

What you will learn: Tell perimeter (the distance around) from area (the space inside), work out both for a rectangle, and use the right units.

What should I know first?

If this feels new, try these first. It is fine to go back.

Plan a reading corner Class 3
The 12-tile garden Class 5

Small test at the end: scroll down to the end of this page for five short questions that check what you have learnt. Tap an answer, then press Check answer. Go to the questions ↓

Jump to: Check yourself · For grown-ups · For teachers

Safety first

Measure on flat ground, away from roads, with a grown-up.

Keep it safe:

Measure on flat ground away from roads and drains, with a grown-up.
If you use pegs, counters or bottle caps as markers, keep them away from babies: nothing goes in the mouth.

You will need

Squared paperEach square stands for 1 square metre.
A long string and a metre ruler or tapeFor a real fence on the ground or floor.
Chalk or four markersTo mark the corners.

Time: About 40 minutes.

What do you think?

Dhruv says: "A garden with more fence around it always has more grass inside." Is he right?

Make a prediction and write one reason for it. Then tap the guess closest to yours to see how to test it. It is fine to change your mind after checking.

Yes, always

Let's test it. Draw a 6 by 4 garden and a 9 by 1 garden on squared paper. Count the fence and the squares for each.

No, not always

Can you find two gardens to prove it? Try a long thin one and a squarer one.

Sometimes

When does it work, and when not? Try a few rectangles on squared paper.

Not sure

That is fine. Draw two rectangles with the same fence on squared paper and count the grass in each.

Need a hint?

Fence is counted in metres around the edge. Grass is counted in squares inside.

The story

The school is making a small garden next to Dadi's chai stall, 6 metres long and 4 metres wide. Dadi needs to buy fencing to go around it and grass to cover the ground inside. "Both are 6 and 4," says Tanmay, "so it's the same number for both." Dhruv shakes his head. "One goes around the edge. The other covers the inside. I bet they are different."

The idea

Perimeter is the length all the way around a shape. You walk it, or you put a fence or a border along it.

For a rectangle: perimeter = length + width + length + width = 2 × (length + width).

Area is how much flat space a shape covers. You count it in squares: square centimetres (cm²) or square metres (m²).

For a rectangle: area = length × width, because there are "length" squares in each row and "width" rows.

A grid of 6 columns and 4 rows of squares, each 1 metre by 1 metre: 24 squares inside, and 6 + 4 + 6 + 4 = 20 metres around the edge.

Formula cards

From counting to a formula

Perimeter of a rectangle

What it finds: The length of fence, border or ribbon that goes all the way round.
The formula: Perimeter = 2 × (length + width)
Why it works: A rectangle has two lengths and two widths. Length + width goes halfway round; doubling it goes all the way.
When it works: For any rectangle. For an L-shape or a shape with a gap, walk the edge and add every side.
Units: A length: m or cm, never m² or cm².
Use it backwards: A pen with 18 m of fence and a length of 5 m: half the fence is 9 m, so the width is 9 − 5 = 4 m.

Area of a rectangle

What it finds: How many squares of grass, tiles or carpet cover the inside.
The formula: Area = length × width
Why it works: There are "length" squares in each row and "width" rows, so rows × squares in a row counts them all.
When it works: Only for a full rectangle. For other shapes, split them into rectangles or take away the missing part.
Units: Square units: m² or cm². Both sides in the same unit first.
Use it backwards: A lawn of 35 m² that is 7 m long is 35 ÷ 7 = 5 m wide.

Worked example

Fence and grass for the school garden

The garden is 6 m long and 4 m wide. How much fence and how much grass does Dadi need?

Fence (perimeter): 6 + 4 + 6 + 4 = 20 m. Or 2 × (6 + 4) = 20 m.
Grass (area): 4 rows of 6 squares = 6 × 4 = 24 m².
Check the units: fence is a length (m), grass covers a space (m²).

Dadi needs 20 m of fence and 24 m² of grass.

Try it

Draw the garden.
On squared paper, draw a rectangle 6 squares long and 4 squares wide. Each square is 1 m by 1 m.
Walk the edge.
Run your finger around the edge, counting the sides of squares. That is the perimeter in metres.
Count the inside.
Count the squares inside. Then count them faster: rows × squares in each row. That is the area in m².
Same fence, new shape.
Use the same 20 m of fence for a 9 by 1 garden, then a 7 by 3, then a 5 by 5. Find each area.
Make it real.
With a grown-up, mark a 2 m by 1 m rectangle on the floor or ground with string. Measure the string. Count 1 m squares inside with chalk lines.
Explain the difference.
Write one sentence on why the fence and the grass have different numbers and different units.

What you make (paper or offline)

Your goat pen plan

A grid of 5 columns and 3 rows of 1 metre squares: a goat pen 5 m by 3 m.

Draw a goat pen 5 m by 3 m on squared paper.

Write how much fence it needs and how much floor it has, with units.

Now design a pen with the same fence but more floor. What shape gives the most?

Make it on paper. Keep it with you. Nothing to upload.

What did we find out?

Read this after you have tried it. Look back at your guess and your reason first.

Perimeter is the distance around the edge. For a rectangle it is 2 × (length + width), in metres.

Area is how many squares cover the inside. For a rectangle it is length × width, in square metres (m²).

The same fence can hold very different amounts of grass: 6 by 4 holds 24 m², but 9 by 1 holds only 9 m².

Think back

Why does the long thin garden hold so little grass?

When would you need the perimeter, and when the area, in real life?

Investigate further

With 20 m of fence, which rectangle holds the most grass? Try every whole-number one.
Find the area of an L-shaped room by splitting it into two rectangles.

Can you use it here?

Find a rectangle at home: a table top, a mat or a window. Measure it and work out its perimeter and area.

Ask someone at home

Ask someone at home: if you were buying a frame for a photo, would you pay for the perimeter or the area? What about the glass?

No one free to join in? That's fine. You can do it by yourself with drawings, and ask a grown-up to help with the materials.

For grown-upsTap a heading to open it. The safety notes are at the top of the page.

Before you start

Squared paper, string and a metre ruler or tape. Keep pegs, counters and bottle caps away from babies; nothing in the mouth.
If you go outside, pick flat ground away from roads.

Questions that help

"Is this around the edge, or covering the inside?"
"What unit goes with that number?"
"Can you find a shape with the same fence but more grass?"

What's really going on

Many children mix up perimeter and area, especially for rectangles where both use the same two numbers.

Counting squares first, before the formula, shows why area is length × width.

Units matter: metres for lengths, square metres for areas. Writing them every time catches mistakes.

Other ways to do it

No squared paper? Draw a grid with a ruler, or use floor tiles as squares.
Use a book and sticky notes: border around, notes covering.

For teachersLearning aims, classroom ideas and what to look for.

Learning aims

Find the perimeter and area of rectangles and use the correct units.
Explain the difference between perimeter and area.
Show that shapes with the same perimeter can have different areas.

Subject links

Maths: multiplication, units of length and area.
Topic ladder 10.2 (lines, angles and symmetry) and 3.3 (area and volume).
Science and life skills: planning a garden, buying tiles or fencing.

These links are general. They have not been checked against a specific board syllabus.

In the classroom

Give each group 24 m of "fence" (string) and ask them to make the biggest pen on a square grid.
Measure the classroom floor and a desk; compare their perimeters and areas.

What to look for

Uses m for perimeter and m² for area without prompting.
Explains area as counting squares, not only "length times width".
Finds two rectangles with equal perimeter and different area.