Class 5 Mathematics Chapter 10: Symmetrical Designs

The banner for sports day

Meera and her brother Arun are making a banner. It must say WELCOME in big paper letters. Arun starts on the letter M. He cuts one side, then the other side, and the two sides come out different. Meera laughs and says there is an easier way.

Meera's fold

Meera takes a sheet and folds it down the middle. She draws only half of the letter M against the fold. Then she cuts along her own line. She opens the sheet out. A neat M lies there, and both sides match.

What the fold line is

That fold line has a name. We call it a line of symmetry. It splits the shape into two halves. The halves cover each other when you fold. Nothing pokes out on either side.

The fold test

Do you want to test a line? Fold the paper along it. Look hard at the edges. If every bit of one half lands on the other half, the line passes. If a corner pokes out, it fails.

Letters that fold

Try the letter A. It folds down the middle, so it has one line. The letter B folds across the middle instead. The letter H folds both ways. So H has two lines of symmetry.

Letters that will not fold

Some letters are not so kind. Try to fold the letter F. No fold works at all. Try G, J, P and R as well. Ten capital letters have no line of symmetry.

One line or many

A shape can have more than one line. Fold a square down the middle and it matches. Fold it across and it matches. Fold it corner to corner and it still matches. A square has four lines.

The rectangle trap

Now take a rectangle that is not a square. It folds down the middle. It folds across the middle too. But fold it corner to corner and the halves slide apart. A rectangle has only two lines.

Lines in other shapes

A triangle with three equal sides has three lines. A kite has one line, down its long middle. A circle is the champion. Fold it through the centre any way you like and it matches.

Cutting in one go

This is why folded paper cutting is quick. Fold first. Draw half the shape against the fold. Cut, then open it out. A diya, a boat or a leaf comes out even on both sides.

Remember

• A line of symmetry gives two matching halves.
• Fold to test. Matching halves mean yes.
• Square four lines, rectangle two, equal-sided triangle three.
• The slanting corner line of a rectangle fails.

Scenario:

A class folds paper shapes to test for symmetry. Imran folds a rectangle down its middle and the halves match. He folds it corner to corner and one corner pokes out. He says the fold was careless. His teacher asks him to try the same fold on a square. There the corners match exactly. The class compares the two shapes. A square has four equal sides and the rectangle does not, which is why the diagonal fold works for one and not the other. Imran sees that the test was fine. The shape, not the folding, decided the result.

Analysis:

A poking corner is evidence about the shape, not the hand. The diagonal fold needs four equal sides, which a rectangle lacks.

Activity:

Fold cut-outs of a square, a rectangle and a triangle every way you can. Count the folds that match.

Title:

Cut a row of matching diyas

Safety:

Scissors are sharp. Ask an adult to help with the cutting.

What You Notice:

Every shape comes out even on both sides. The fold line is its line of symmetry, and the two halves cover each other when you fold again.

Rangoli artists often draw one half of a design and copy it across the middle.

The Taj Mahal looks the same on the left and the right of its middle line.

Testing a line — Fold, match, keep. — Fold the shape, see whether the halves match, and keep the line only when they do.

The firki at the fair

Ravi is holding a paper firki at the village fair. The wind pushes it and the four blades go round. Ravi stares at it. He cannot tell whether it has moved a little or a lot. Every blade looks like every other blade.

Nila's red dot

His friend Nila has an idea. She sticks a tiny red dot on one blade. Now they can follow that blade with their eyes. The dot travels round while the firki keeps looking the same.

A quarter turn

Nila turns the firki slowly. She stops when the red dot has gone a quarter of the way round. That is a quarter turn, which is 90 degrees. The firki looks exactly as it did before.

Half, three quarters, full

She keeps going. At half a turn the dot is opposite where it began. That is 180 degrees. Three quarters of a turn is 270 degrees. A whole turn is 360 degrees and brings the dot home.

What the two friends found

At every quarter turn the firki matched itself. The blades sat in the same places. Only the red dot showed that anything had happened. Four times in one full turn, the firki looked untouched.

Rotational symmetry

A shape has rotational symmetry when it matches itself before a full turn is over. The firki matches at a quarter turn, so it has rotational symmetry. Turning is a different idea from folding.

Counting the matches

We also count how many matches happen in one full turn. That count is called the order. The firki matches four times, so its order is four. A bigger order means a shape that turns more neatly.

Letters that turn

Stand the letter H on the page and turn it a half turn. It comes back looking the same. The letters I, N, S and Z do the same thing. Each of them has order two.

Digits that turn

The digit 8 matches itself after a half turn. So does the digit 0. Now try the digit 6. A half turn makes it look like a 9, so the digit 6 does not match itself.

Shapes that turn

A square matches four times in a full turn. A rectangle matches only twice. A triangle with three equal sides matches three times. A circle matches at every tiny turn you can make.

Full turns do not count

Every shape in the world matches itself after a full turn. That is why we never count the full turn on its own. A shape needs a match before the full turn to be called symmetric.

Remember

• A quarter turn is 90 degrees and a half turn is 180 degrees.
• Rotational symmetry means a match before the full turn.
• The order is the number of matches in one full turn.
• Firki four, square four, rectangle two, H two.

Scenario:

A class turns a paper pinwheel with four blades and counts how often it looks the same. The answer is four. Nita then turns a rectangle and gets two. She says every shape must match an even number of times. Their teacher hands her a triangle with three equal sides. It matches three times. Nita's rule breaks. The class works out what the number really counts. It is how many positions look identical before the turn is complete. A quarter turn is 90 degrees, a half turn 180, and a full turn 360, and the matches are spread evenly round that.

Analysis:

Even numbers held for two shapes and failed on the third. The count is of identical positions, which has nothing to do with being even.

Activity:

Turn four cut-out shapes through a full circle. Record how many times each one looks unchanged.

Title:

Spin your own firki

Safety:

A drawing pin is sharp. An adult must push the pin through and check the end.

What You Notice:

The blades look the same after every quarter turn. Only the red sticker tells you how far your firki has gone round.

The wheel on our national flag is built so that it matches itself many times in one turn.

A ceiling fan with three blades matches itself three times while it goes round once.

Telling the two symmetries apart — Fold folds, spin spins. — A fold tests reflection and gives two halves. A spin tests rotation and moves the whole shape.

The printing stall

Kabir is at a craft mela with his grandmother. At one stall an artisan from Rajasthan is printing cloth. She dips a carved wooden block in dye. She presses it down hard. She lifts it, and a petal is left behind on the white cloth.

One block, many prints

The artisan does not carve a new block for every petal. She uses the same block again and again. Each press leaves the same picture. Kabir watches her move the block a little each time.

Stamping round a dot

Now she marks a dot on the cloth. She presses the block above the dot. Then she turns the cloth a quarter turn and presses again. Four presses later, four petals sit round that one dot.

The design turns

Kabir turns his head and looks at the four petals. He gives the cloth a quarter turn. The design looks exactly the same. It matches four times in one full turn, so its order is four.

The design folds too

Then he tries the other test. He folds the printed square down the middle. The petals cover each other neatly. He folds it across the middle and it matches again. This design has both symmetries.

Three kinds of design

So a design can pass one test, the other test, or both. Some designs only turn. Some designs only fold. The prettiest ones, like this petal design, do both.

A design that only folds

Take one petal on its own, pointing straight up. Fold it down the middle and the halves match. Now turn it a quarter. It is lying on its side. One petal folds but does not turn.

A design that only turns

A firki is the other way round. Its blades all lean the same way. Turn it a quarter and it matches. Fold it any way you like and one half always leans wrong.

Why the block matters

If the carved block is even on both sides, the printed design can fold. If the block leans, the design can still turn but never folds. The block decides what the cloth can do.

Sorting your own designs

Cut out paper squares and triangles of the same size. Arrange them round a dot to make a design. Then sort your designs into three piles: only turning, only folding, and both.

Remember

• Stamping round a dot gives a turning design.
• Four presses at quarter turns give order four.
• Use the fold test and the turn test on every design.
• Some designs do one, some do both.

Scenario:

In Bagru, near Jaipur, a family has printed cloth for many years. Rekha carves the wooden blocks. Her father mixes the dye in a tank behind the house. Today they are printing a bedsheet for a hotel. Rekha marks a dot at the centre of each white square. She presses her petal block above the dot. Then she turns the cloth a quarter turn and presses again. Four presses fill the square. Her father checks the water tank, because printing uses a lot of water and every drop is carried from the well. The finished sheet carries the same petal design in eighty squares, and each one matches the rest.

Analysis:

The craft rests on exact quarter turns, careful counting and the care of a scarce resource.

Activity:

Count the printed squares on a dupatta or bedsheet at home and work out how many presses it took.

Title:

Print a turning design with a ladyfinger

Safety:

Only an adult should use the knife to cut the ladyfinger.

What You Notice:

The design matches itself at every quarter turn, so its order is four. If your prints stand upright, the folded halves will match as well.

Hand block printers in Rajasthan carve their wooden blocks with small chisels.

The same block can print hundreds of metres of cloth before it wears out.

Judging a design — Fold first, then spin. — Run the fold test, then the turn test, and write a yes or no for each before you decide.