The 12-tile garden

Twelve tiles can make more than one rectangle. Does every one need the same length of fence?

About 40 minutes
Class 5
Mathematics and money

What you will learn: Find the area and the perimeter of rectangles made of square tiles, and show that the same area can need different lengths of fence.

What should I know first?

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

Plan a reading corner Class 3

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: Practise · Check yourself · For grown-ups · For teachers

Safety first

Paper tiles are safest. Real tiles are heavy: a grown-up lifts them.

Keep it safe:

Paper or card tiles are safest. If you use real tiles, a grown-up carries them and nobody stands where one could fall.
Keep scissors closed when you pass them, and keep small counters or bottle caps away from babies: nothing goes in the mouth.

You will need

12 paper squares, all the same sizeCut from squared paper or card. Each one stands for a 1 m tile.
Squared paper and a pencilFor drawing every garden you make.
A stringTo be the fence. Mark it in tile-lengths with a pen.

Time: About 40 minutes.

What do you think?

Tanmay says: "Any garden made of all 12 tiles needs the same length of fence." 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, the same

Let's test it. Lay the 12 tiles in one long row, then in 3 rows of 4. Count the tile sides on the outside of each.

No, it changes

Can you show it? Find two shapes of 12 tiles and count the fence around each one.

Only for some shapes

Which shapes? Make every rectangle you can with all 12 tiles and count the fence for each.

Not sure

That is fine. Lay the tiles in a long row and in a block, and count the outside edges of both.

Need a hint?

Count only the tile sides on the outside. Where two tiles touch, no fence is needed.

The story

Dadi has 12 square stone tiles, each 1 metre by 1 metre, for a herb garden behind her chai stall, and an old roll of fence 15 metres long. "Twelve tiles is twelve tiles," says Tanmay. "Whatever shape we make, it needs the same fence." Tara is not so sure. "A long thin garden goes round a lot more," she says. "Let's lay them out and count."

The idea

Area is how much ground the garden covers. Count the tiles: every garden made of all 12 tiles has an area of 12 square metres (12 m²).

Perimeter is the length all the way round the outside. Count only the tile sides on the outside. A side where two tiles touch is inside the garden, so it gets no fence.

Twelve tiles make three rectangles: 1 by 12, 2 by 6 and 3 by 4. They need 26 m, 16 m and 14 m of fence.

So the same area does not mean the same perimeter. The squarer the rectangle, the less fence goes round it.

A grid 4 squares long and 3 squares high, all 12 squares green: 12 tiles inside, and 4 + 3 + 4 + 3 = 14 m of edge around them.

Formula cards

From counting to a formula

Area of a rectangle

What it finds: How much flat space a shape covers, counted in squares.
The formula: Area = length × width
Why it works: A garden 4 tiles long and 3 tiles wide is 3 rows of 4 tiles. Rows × tiles in each row = 3 × 4 = 12, without counting one by one.
When it works: Only for a full rectangle: four square corners and no tiles missing. For any other shape, split it into rectangles, or work out the full rectangle and take the gaps away.
Units: Square units: cm² for a poster, m² for a garden. Both sides must be in the same unit first.
Use it backwards: If you know the area and one side, divide: a 24 m² stage that is 6 m long is 24 ÷ 6 = 4 m wide. Check: 6 × 4 = 24.

Perimeter of a rectangle

What it finds: The distance all the way round the edge.
The formula: Perimeter = 2 × (length + width)
Why it works: A rectangle has two lengths and two widths. Length + width takes you halfway round; doubling it takes you all the way.
When it works: For any rectangle. For other shapes, walk the edge and add every side, including the sides around a gap.
Units: Lengths: cm or m. Never cm² or m², because nothing is being covered.
Use it backwards: If a fence of 16 m goes round a garden 6 m long, half the fence (8 m) is one length and one width, so the width is 8 − 6 = 2 m.

Worked example

Which garden fits the fence?

Dadi has 15 m of fence. Which rectangle of 12 tiles can she fence all the way round?

1 by 12: fence = 2 × (12 + 1) = 26 m. Far too long.
2 by 6: fence = 2 × (6 + 2) = 16 m. Just 1 m too long.
3 by 4: fence = 2 × (4 + 3) = 14 m. It fits, with 1 m left over.
All three cover 12 m², so the choice is only about the fence.

Only the 3 by 4 garden fits Dadi's 15 m of fence.

A table of the three rectangles made from 12 tiles. 1 by 12: area 12 m², fence 26 m. 2 by 6: area 12 m², fence 16 m. 3 by 4: area 12 m², fence 14 m.

Practise

Practise, with help if you need it

Tara's path is 3 tiles long and 2 tiles wide, but the top-middle tile is missing. How much ground do the tiles cover, and how long is the edge all the way round?

A grid 3 squares long and 2 squares high. Five squares are green tiles. The middle square of the top row is empty, making a notch in the top edge.

Work it out on paper and write one reason. Then tap the answer closest to yours.

5 m² of tiles and 12 m of edge

Write your reason first. Then check the edge a second way: walk your finger all the way round, into the gap and out again.

6 m² of tiles and 10 m of edge

That is length × width for a full 3 by 2 rectangle. But one tile is missing. Open "Show me the edge" and count only the tiles that are there.

5 m² of tiles and 10 m of edge

The tiles are right: 6 − 1 = 5. But the gap has its own edges. Open "Show me the edge" and walk all the way round.

Not sure

That is fine. Open "Show me the edge" below, then come back to this question.

Show me the edge

Count the tiles that are really there: 3 on the bottom row and 2 on the top row. That is 5 tiles, so 5 m².

A full 3 by 2 rectangle has 2 × (3 + 2) = 10 m of edge. Where the tile is missing, 1 m of the top edge goes.

But the edge now dips into the gap: down 1 m, along 1 m, up 1 m. That is 3 m of new edge.

So the edge is 10 − 1 + 3. Work it out, then look at your answer again.

The same path: five green tiles with the top-middle square empty.

Close this box, look at the question again, and tap your answer. It is fine to change your mind.

Now a new problem, on your own

A patio is 4 tiles long and 3 tiles wide, but one corner tile is missing. How much ground is tiled, and how long is the edge?

Draw it on squared paper first. Count the tiles, then walk the edge.

See a worked answer

Tiles: the full patio is 4 × 3 = 12 tiles. Take away 1: 11 tiles, so 11 m².
Edge: at a corner, you lose 1 m on each of the two outside sides of the missing tile: 2 m less.
But the edge now runs along the two inside sides of the gap: 2 m more.
So the edge stays 2 × (4 + 3) = 14 m. Check by walking round on squared paper.

11 m² of tiles and 14 m of edge: taking away a corner makes the area smaller but leaves the perimeter the same.

Can you use length × width for the notched path straight away?

See what we think

Not straight away. Length × width counts every square of a full rectangle, and this one has a gap. Work out the full rectangle, then take the missing tile away: 3 × 2 − 1 = 5.

I can find the area and the perimeter of a shape that is not a full rectangle. Show a grown-up: draw a shape of 6 tiles with a longer edge than the 2 by 3 rectangle. Explain why the edge grew.

Build it on screen

Build Dadi's garden

Tap + and − to change the length and the width. Find every garden that uses exactly 12 tiles, and record each one.

Length

Width

Garden	Area	Fence	Fits 15 m?

Try it

Cut out 12 tiles.
Cut 12 squares of the same size from paper or card. Each one stands for a tile 1 m by 1 m.
Make every rectangle.
Use all 12 tiles to make a rectangle with no gaps. Then find every other rectangle you can. Write each one as "rows by tiles in a row".
Need a hint?
Start with 1 row of 12. Then try 2 rows: how many tiles in each? Then 3 rows. Does 5 rows work? Why not?
Count the fence.
For each rectangle, run your finger along the outside edge and count the tile sides you pass. Do not count sides where two tiles touch.
Fill in a table.
Make three columns: the garden, its area (tiles) and its perimeter (fence). What stays the same? What changes?
Test the 15 m fence.
Mark a string in 15 tile-lengths. Which of your rectangles can it go all the way round?
Two tiles, then three.
Put 2 tiles side by side and count the fence: 6, not 8. Add a third tile to the row. Does the fence grow by 4 or by 2? Explain why.

What you make (paper or offline)

Your 20-tile garden plan

A grid 5 squares long and 4 squares high, all 20 squares green.

Tanmay has 20 tiles. Draw every rectangle he can make with all 20.

Write the area and the fence for each, with units.

Which one needs exactly 18 m of fence? Which needs the most fence?

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.

All three rectangles of 12 tiles have the same area: 12 m².

Their fences are different: 1 by 12 needs 26 m, 2 by 6 needs 16 m, and 3 by 4 needs only 14 m.

Only the 3 by 4 garden fits Dadi's 15 m of fence, with 1 m to spare. The squarer the rectangle, the less fence it needs.

Think back

Why does the long thin garden need so much more fence than the 3 by 4 one?

When you add one tile to a garden, can the fence stay the same? Can it get shorter?

Investigate further

With 16 m of fence and whole metres, which rectangles can you make? Find the area of each. Which holds the most?
Draw a shape of 12 tiles that is not a rectangle and needs more than 26 m of fence. How long can you make it?

Can you use it here?

Find something at home made of squares in a rectangle: floor tiles, a chocolate bar, a calendar page. Count its area and its edge.

Ask someone at home

Ask someone at home: if you had to pay for a fence by the metre, would you want a long thin garden or a squarer one? Why?

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

12 paper squares (or any 12 same-size squares), squared paper, a pencil and a string.
Keep scissors closed when passed, and keep counters or bottle caps away from babies.

Questions that help

"Is this number covering the inside, or going round the edge?"
"Does that side need a fence, or do two tiles touch there?"
"Can you find a garden with the same tiles but less fence?"

What's really going on

Many children think the same area always has the same perimeter. Laying out real tiles shows it is not true.

Children often count the sides where tiles touch. Running a finger round the outside edge fixes this.

The formula comes after the counting, so the child knows why length × width counts the squares.

Other ways to do it

No paper squares? Use 12 biscuits, coins or bottle caps on a chalk grid.
Draw the gardens on squared paper only, and colour the edge.

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

Learning aims

Find the area and the perimeter of rectangles made of unit squares, with the right units.
Show that rectangles with the same area can have different perimeters.
Use both conditions together (tiles and fence) and work backwards from a perimeter to a missing side.

Subject links

Maths: multiplication facts and factor pairs of 12 and 20; units of length and area.
Leads to "Fence or grass?" (Class 6), where the same fence holds different areas.
Topic ladder 10.2 (lines, angles and symmetry) and 3.3 (area and volume).

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

In the classroom

Give each group 12 paper squares and a 15-unit string. Ask for every rectangle and which one the string fits.
Repeat with 20 tiles and an 18-unit string, then ask groups to set a puzzle like it for another group.

What to look for

Counts only outside edges, without counting the sides where tiles touch.
Writes m for the fence and m² for the area without being asked.
Checks an answer against both conditions before choosing it.