Class 5 Mathematics Chapter 11: Grandmother's Quilt

Nani's quilt

Nani has stitched a quilt from old kurtas and saris. It lies on the charpai, bright and warm. Now she wants a lace edging all the way round it. Tara and her brother Vikram want to help her buy the lace.

How much lace?

First they measure. The quilt is 9 units long and 6 units wide. Nani asks a simple question. How long must the lace be to go right round the edge?

Walking round the border

Tara walks a finger along the border. She goes 9 units, then 6 units, then 9 units, then 6 units. She adds as she goes. The four sides come to 30 units in all.

A shorter way

Vikram finds a quicker path. A quilt has two long sides and two short sides. So he works out 2 times 9, which is 18. Then 2 times 6, which is 12. Together that is 30 units.

The word for it

The length all the way round a shape has a name. We call it the perimeter. The perimeter of Nani's quilt is 30 units.

Choosing the lace

The shop has three reels. Red is 24 units, green is 30 units and blue is 18 units. Red and blue are both too short. Only the green reel reaches right round.

Two colours

Nani likes two colours together. So she buys 18 units of red for the two long sides. Then she buys 12 units of green for the two short sides. That is 30 units again.

All sides the same

Some shapes have sides that are all equal. A square napkin has sides of 7 cm, so its perimeter is 4 times 7, which is 28 cm. A five-sided mat with 4 cm sides gives 5 times 4, or 20 cm.

Shapes with odd sides

Not every shape is so tidy. A table mat may have sides of 6, 4, 5 and 3 cm. There is no short cut here. Add them all and you get 18 cm.

Perimeter is a length

Lace, rope, fencing and ribbon are all lengths. So a perimeter is measured in centimetres or metres, never in square units. Keep the unit in your answer.

Remember

• Perimeter is the distance all the way round the border.
• Add every side, or use 2 times length plus 2 times breadth.
• A square of side 7 cm has a perimeter of 28 cm.
• Perimeter is a length, so use cm or m.

Scenario:

A quilt is 150 cm long and 120 cm wide. Grandmother wants lace all round the edge. A class works out how much she needs. Ravi multiplies and gets 18,000. His teacher asks what that number would be measuring. The class sees that multiplying gives the cloth, not the border. They walk round instead: 150, 120, 150, 120. That comes to 540 cm. Ravi sees the difference. Lace goes along the edge, so the four sides are added, and the inside of the quilt does not come into it at all.

Analysis:

Both numbers are honest arithmetic and only one is lace. Adding the four sides is what measures a border.

Activity:

Measure a rectangular cloth or mat. Work out the lace for its edge and write how you did it.

Title:

Measure the borders at home

Safety:

Scissors are sharp. Ask an adult to help when you cut the string.

What You Notice:

The length of the string matches the total you added. Both numbers give the perimeter, so adding the sides really does walk the border for you.

Quilts made from old clothes are called kantha in Bengal and godhadi in Maharashtra.

A farmer buys fencing wire by the metre, which is the perimeter of the field.

Finding a perimeter — Walk the fence, add each plank. — Picture walking right round the shape, adding the length of every side you pass.

Covering the low table

Nani has three cloth patches left over. Tara wants to know which patch is the biggest. She lays each one on the low table. Some hang over the edge and some do not. Looking is not enough. They need a better way.

Shapes that fit together

Vikram fetches a box of cut-outs. He covers the table top with square cut-outs and no space shows through. He tries rectangles and then triangles, and both work too. Circles are hopeless. Gaps peep out between every pair.

The word for the covering

The part of the table that gets covered has a name. We call it the area. Area is the surface a shape takes up, not the distance round its edge.

Choosing one tile for everyone

If Tara counts triangles and Vikram counts squares, they cannot compare. So the world agrees on one tile. The tile is a square, because all four of its sides are equal.

The unit square

A square with sides of one centimetre is the unit for area. Its area is called one square centimetre. Count how many of them fit inside a shape and you have its area.

Squared paper helps

Nani hands them a sheet of squared paper. Each little square is one square centimetre. Tara traces a patch onto the paper. Then she counts the squares inside her outline.

Three patches counted

Patch P covers three squares across and four down, so its area is 12 square centimetres. Patch Q covers 8 and patch R covers 10. Now the order is clear. P is the biggest of the three.

Count inside, not round

Vikram nearly counts the squares that touch the outline only. That gives the border, not the covering. Every square inside the shape has to be counted, right up to the middle.

Halves join up

A traced leaf will not sit neatly on the lines. Count all the whole squares first. Then pair up the halves. Two halves make one more whole square.

The right unit

A border is written in centimetres. A covering is written in square centimetres. Writing 12 cm for an area loses half the meaning, so always say square centimetres.

Remember

• Area is the covering, and perimeter is the border.
• Squares, rectangles and triangles tile, but circles leave gaps.
• One square of side 1 cm has an area of 1 square centimetre.
• Count every square inside, and pair up the halves.

Scenario:

A class covers a table top with square tiles of paper. It takes 8 rows of 5 tiles, so 40 tiles cover it. Meena then tries round coasters instead. However she arranges them, small gaps stay between them. She says she needs more coasters. Their teacher asks her to count the gaps. They do not close however many she adds. The class works out why. Squares meet edge to edge with nothing left over, and circles cannot. Meena writes the finding down. Area is the surface covered, and only shapes that tile a surface can measure it without gaps.

Analysis:

More coasters make more gaps, not fewer. A square's straight edges are what let it measure a surface exactly.

Activity:

Cover a book with square paper tiles and count them. Then try circles and write what happens.

Title:

How big is your palm?

What You Notice:

Your hand covers a number of square centimetres. The part squares along the outline pair up neatly, and a bigger hand always covers more squares.

Bees build honeycombs from six-sided cells, which tile with no wax wasted.

A tile shop asks for the area of your room before it counts out the boxes.

Telling area from perimeter — Fence round, field inside. — A fence is the perimeter and the field it holds is the area.

A patchwork on the floor

Nani starts a second patchwork. She lays her square patches on the floor in neat rows. Tara looks at the growing rectangle and starts counting the patches one by one. She loses her place twice.

Vikram sees rows

Vikram laughs and tells her to stop. He counts down one side and finds 6 rows. He counts across the top and finds 4 patches in each row. So the whole piece holds 6 times 4, which is 24 patches.

The rule

That idea works for every rectangle. Area of a rectangle is length times breadth. Here the length is 6 cm and the breadth is 4 cm, so the area is 24 square centimetres.

Checking against counting

Tara is not sure, so she tests the rule on a small patch. It is 3 squares across and 4 squares down. Counting gives 12 squares. The rule gives 3 times 4, which is 12 too.

When all sides are equal

A square is a rectangle whose sides are all the same. Take a square of side 5 cm. Its area is 5 times 5, which is 25 square centimetres.

The border of the same shapes

The border needs adding, not multiplying. For the 6 cm by 4 cm patchwork it is 12 plus 8, or 20 cm. The square of side 5 cm gives 4 times 5, which is 20 cm as well.

Big shapes, same rule

The rule does not mind big numbers. A field is 42 m long and 34 m wide. Multiply and you get 1428 square metres of ground.

Working backwards

Sometimes the area is given instead. A mango orchard covers 96 square metres. One of its sides runs 8 m. Divide 96 by 8 and the other side comes out as 12 m.

Same area, different border

Now a surprise. Three rectangles all cover 12 square centimetres. They are 1 cm by 12 cm, 2 cm by 6 cm and 3 cm by 4 cm. Their borders are 26 cm, 16 cm and 14 cm.

Bigger area, smaller border

It can even go the other way. A 3 cm square covers 9 square centimetres with a border of 12 cm. A 1 cm by 8 cm strip covers only 8, yet its border is 18 cm.

What that teaches us

So a bigger area does not promise a bigger border. Long thin shapes have huge borders. Fat, square shapes hold more inside for the same border.

Remember

• Area of a rectangle is length times breadth.
• Area of a square is side times side.
• The border is found by adding, never by multiplying.
• Two shapes with the same area may have different borders.

Scenario:

The children of a school in Wardha are making a kitchen garden. Their teacher marks a bed that is 12 m long and 5 m wide. The bed must be fenced with net, or the goats from the road will walk in. It must also be spread with compost before the seeds go in. The compost comes in sacks, and one sack covers 10 square metres of ground. Sheela is in charge of the net and Imran is in charge of the compost. Sheela walks right round the bed with a ball of string. Imran stands at one corner and looks across the whole bed. Each of them needs a different number.

Analysis:

The same plot gives two answers: a border length for the fence and a covering for the compost.

Activity:

Measure a vegetable patch or a flower bed near you and work out its fence length and its area.

Title:

Plan a small vegetable bed

Safety:

Ask an adult before you dig or move anything in the garden.

What You Notice:

The string matches the total you added, while the multiplying tells you how much soil the bed will swallow. The two numbers are never the same.

Farmers in India often measure land in bighas, but a bigha is still an area.

A painter works out the area of a wall before deciding how much paint to buy.

Choosing the method — Add round, multiply inside. — The border is found by adding the sides. The covering inside is found by multiplying them.