Class 5 Mathematics Chapter 7: Shapes and Patterns

The baskets in the market

Ammu walks past a shop full of cane baskets. No two look the same. One has neat square holes. One looks like a chess board. She bends down and looks closely. Every basket is made from strips going over and under. Let us weave one ourselves.

Cutting the mat

Take a sheet 30 cm long and 20 cm wide. Fold it in half along the longer side. Cut straight slits from the folded edge. Stop 3 cm short of the top. Open it out. The sheet now has many slits, joined at the top and the bottom.

The first row

Cut eight strips, 3 cm wide, in two colours. Push the first strip across. Take it 1 under, then 1 over. Then 1 under, then 1 over again. Keep going to the end of the row.

The second row is the opposite

Now take a strip of the other colour. This time go 1 over, then 1 under. Wherever row 1 went under, row 2 goes over. That swap is what makes the chess-board look.

The part that repeats

Look at row 1 again. The words "1 under, 1 over" come back and back. That short piece is the repeating unit. Once you know the unit, you know the whole row. Every pattern has one.

A longer unit

Some mats use "2 over, 1 under". Now the unit is three slits long. The row goes over, over, under, then over, over, under. The second row of that mat is "2 under, 1 over".

Counting without counting

Which way does slit 12 go in a "2 over, 1 under" row? The unit is 3 long. Now 12 ÷ 3 = 4 with nothing left over. So slit 12 is the last step of a unit, which is under.

One more prediction

Try slit 10 in the same row. Here 10 = 3 + 3 + 3 + 1. So slit 10 is the first step of a unit. The first step is over. Multiplication tables help you jump ahead.

How many of each?

A row of 12 slits holds 4 whole units. Each unit has 2 overs and 1 under. So the row has 8 overs and 4 unders. Counting units is quicker than counting slits.

A row that starts oddly

Some rules begin with a step that is not repeated. One mat says "1 under, do not repeat, then 3 over, 3 under". So slit 1 goes under, and only then does the long unit begin.

Patterns on paper

You can draw the same idea on grid paper. Colour squares in a rule such as two blue, one white. Then repeat the rule across the row. Shift each row by one square and a new design appears.

Remember

• Find the unit, the short piece that repeats.
• Row 2 is often the opposite of row 1.
• Divide the slit number by the unit length to jump ahead.
• Shifting each row makes a brand new design.

Scenario:

A class weaves paper mats with the rule two over, one under. Nita follows it and her mat shows a clear slanting pattern. Imran forgets the rule halfway and his mat goes plain. Their teacher asks the class to find the repeating unit. It is three strips long: over, over, under. Once the children count in threes, they can check any row without starting again. Imran finds his mistake at strip eight. He sees that a rule has a length. Knowing it lets you check your work in the middle, not just at the end.

Analysis:

Imran did not need to unpick the whole mat. The repeating unit is three strips, so any row can be checked three at a time.

Activity:

Weave a paper mat with the rule two over, one under. Mark the repeating unit and check one row in the middle.

Title:

Weave a paper mat

Safety:

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

What You Notice:

A chess-board design appears. If you change the rule to 2 over, 1 under, the squares turn into slanting stripes.

Cane and bamboo baskets from Assam and Tripura use dozens of different weaves.

Cloth on a handloom is woven the same way, with threads going over and under.

Finding the repeating unit — Say it until it comes back. — Read the rule aloud. The moment the words start again, the unit has ended.

The floor at Bittu's house

Bittu is helping to lay tiles on the veranda. The tiles are square. They sit side by side with no space between them. Not one tile lies on top of another. The whole floor is covered. That is what tiling means.

Gaps and overlaps

A good tiling has two rules. There must be no gaps where the floor shows through. There must be no overlaps where one tile climbs on another. If a shape can do both, we say it tessellates.

Regular shapes

A shape with all its sides equal is called a regular shape. A square is a regular four-sided shape. A regular pentagon has five equal sides. A regular hexagon has six equal sides.

Four squares meet

Cut out some paper squares. Put one corner of each at the same point. Turn them round the point, edge to edge. Exactly four squares fill the whole turn. So squares tessellate.

Six triangles meet

Now cut out equilateral triangles. These have three equal sides. Put one corner of each at the same point. Six of them close the turn with no gap and no overlap. So equilateral triangles tessellate.

Three hexagons meet

Try regular hexagons next. Their corners are wider, so fewer are needed. Just three hexagons close the turn. That is why a honeycomb is made of hexagons.

The stubborn pentagon

Place three regular pentagons round a point. A thin wedge of floor is left showing. Try to push a fourth one in. It will not fit, because it overlaps. So regular pentagons do not tessellate.

The octagon too

A regular octagon has eight equal sides. Two of them round a point leave a big gap. Three of them overlap badly. On their own, regular octagons do not tessellate either.

Two shapes together

A shape that fails alone can still help. Put regular hexagons and triangles side by side. Six-pointed stars appear, and the gaps close. Octagons work in the same way, with small squares filling the corners.

The same meeting every time

Look at the points where the tiles meet in such a pattern. At every marked point, the same set of shapes comes together. That sameness is what lets the pattern spread on and on.

Tiles all around us

Look at a bathroom wall or a footpath. Look at a honeycomb or a football. Tiling is everywhere. A tiler picks shapes that fit, because gaps would let in dirt and water.

Remember

• Tessellating means no gaps and no overlaps.
• Six triangles, four squares or three hexagons fill a turn.
• Regular pentagons and regular octagons do not tessellate alone.
• Two shapes together can tile a floor that neither can tile alone.

Scenario:

A class tries to tile a tray with paper shapes. Squares cover it with no gaps. Triangles do too. Then Meena tries circles and gaps appear between them however she pushes them together. She says she needs smaller circles. Their teacher asks her to try that. The gaps get smaller and do not go away. The class works out why. A circle has no straight edge to meet its neighbour, so there is always a space left at every join. Meena writes the test down. A shape tessellates only if copies of it leave no gaps and no overlaps.

Analysis:

Smaller circles make smaller gaps and never none. A curved edge cannot sit flush against another, so the gap is in the shape rather than the size.

Activity:

Cut many copies of three shapes and try to tile a tray. Record which ones leave gaps.

Title:

The round-a-point test

Safety:

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

What You Notice:

Four squares and six triangles close the turn exactly. Three pentagons leave a wedge, and a fourth one will not fit.

Bees build honeycombs out of hexagons, which tile with no wasted wax.

The floors of many old Indian palaces are tiled in star-shaped patterns.

Which regular shapes tessellate — Six, four, three: triangle, square, hexagon. — Six triangles, four squares or three hexagons close a turn. Pentagons and octagons cannot.

Farah's paper kite

Farah has a paper kite shaped like a rhombus. All four of its sides are the same length. Two sticks cross inside it, corner to corner. Those sticks cut the kite into four triangles. Farah cuts along them and lays the four pieces on the table.

Two pieces, one triangle

She slides two of the pieces together along their long edges. A new triangle appears. She measures its three sides. Two of them are the same length. The third is different.

Isosceles triangles

A triangle with two equal sides is called an isosceles triangle. Fold one in half along its tallest line. The two halves sit exactly on top of each other. So its two base angles are equal too.

Equilateral triangles

A triangle with all three sides equal is an equilateral triangle. Fold it any way you like. Every angle sits on every other angle. So all three of its angles are equal.

Scalene triangles

Now cut an equilateral triangle in half down the middle. Look at one half. Its three sides are all different lengths. A triangle with no equal sides is called a scalene triangle.

Four-sided shapes

Farah joins two pieces a different way and gets a four-sided shape. Any shape with four sides is a quadrilateral. Squares, rectangles and kites are all quadrilaterals. So is the rhombus she started with.

The kite shape

One of her shapes has two short sides side by side and two long sides side by side. Sides that sit next to each other are called adjacent. A quadrilateral with two pairs of equal adjacent sides is a kite.

The parallelogram

Another shape has its opposite sides equal instead. That is a parallelogram. Its corners lean over, and its opposite angles are equal to each other.

The rectangle

Now imagine a parallelogram whose corners are all right angles. That is a rectangle. So every rectangle is a parallelogram, but not every parallelogram is a rectangle.

Rhombus and square

A rhombus has four equal sides, but its corners lean. Push its corners upright and every angle becomes a right angle. Now it is a square. A square is a rhombus and a rectangle at the same time.

Shapes from a circle

Draw a circle and mark its centre. Draw two diameters and join their four ends. The shape is always a rectangle. When the two diameters cross at right angles, it is a square.

Remember

• Two equal sides means isosceles, three means equilateral, none means scalene.
• A kite has two pairs of equal adjacent sides.
• A parallelogram has equal opposite sides and equal opposite angles.
• A square is a rhombus and a rectangle together.

Scenario:

A class cuts paper triangles and sorts them. Ravi measures the sides of one and finds two equal. He calls it equilateral. His teacher asks him to measure the third side. It is shorter. So only two sides are equal, which makes it isosceles. The class then measures the angles and finds the two base angles equal as well. Ravi sees the pattern. Where two sides match, two angles match. The children find an equilateral triangle too, with three equal sides, and all three of its angles turn out to be equal.

Analysis:

Two equal sides is not three. Measuring the third side is what separates an isosceles triangle from an equilateral one.

Activity:

Cut three triangles with different sides. Measure the sides and angles of each and name it.

Title:

Fold a paper kite shape

What You Notice:

The shape is a kite. Its two short sides sit next to each other, and so do its two long sides.

The paper kites flown at Makar Sankranti are usually rhombus shaped.

The word isosceles comes from old Greek words meaning equal legs.

Triangle names — Three equal, two equal, none equal. — Equilateral has three equal sides, isosceles has two and scalene has none.

Kite or parallelogram — Kite, next door. Parallelogram, across. — A kite has equal sides next to each other. A parallelogram has equal sides across from each other.

Nisha's big cube

Nisha glues 27 small wooden cubes together. They make one big cube, three along each edge. She dips a brush in red paint and covers the whole outside. When it dries, she pulls the small cubes apart. Some are very red. Some are not red at all.

What a cube has

First look at one small cube. It has 6 flat faces, and every face is a square. It has 12 straight edges where two faces meet. It has 8 corners where three faces meet.

Opening a cube out

Cut along some edges and lay a cube flat. You get a net of 6 squares joined together. One common net looks like a cross. Fold it back up and the cube returns.

Three views of one cube

A cube can be photographed from three sides. Each picture shows a different set of faces. Two faces that sit opposite each other never appear in the same picture.

The corner cubes

Back to the big cube. A small cube sitting at a corner has three of its faces outside. So it is painted on three faces. A cube has 8 corners, so 8 small cubes are like this.

The edge cubes

Next look along an edge, between two corners. Those small cubes have two faces outside. The big cube has 12 edges, with one such cube on each. So 12 small cubes are painted on two faces.

The middle of a face

Now look at the very middle of each flat face. That small cube shows only one face to the outside. The big cube has 6 faces, so 6 small cubes are painted on one face.

The hidden one

One small cube is buried right in the middle. No part of it touches the outside. So it gets no paint at all. There is exactly 1 such cube.

Checking the count

Add them up: 8 + 12 + 6 + 1 = 27. That matches the number Nisha started with. A count that adds back to the whole is a count you can trust.

Frames with cubes missing

The book shows big cube frames with small cubes taken out. Count the missing ones by layer. Work out how many the full cube would hold, then take away what you can see.

Two grand names

Some solids have many faces. An icosahedron has 20 faces, all equilateral triangles. A dodecahedron has 12 faces, all regular pentagons. Both of them have 30 edges.

Remember

• A cube has 6 faces, 12 edges and 8 corners.
• A net is a cube opened out flat.
• In a painted 27-cube: 8, then 12, then 6, then 1.
• An icosahedron has 20 faces and a dodecahedron has 12.

Scenario:

Meena's family makes wooden toys in Channapatna in Karnataka. They turn soft ivory-wood blocks on a lathe. The colours come from lac, a natural resin, and from vegetable dyes. Nothing harmful is used, because small children put the toys in their mouths. This week the workshop is making sets of small cubes. Twenty-seven cubes fit snugly into one square box. For a shop display, Meena stacks a whole set into one big cube and paints the outside red. Then she takes it apart again and sorts the small cubes by how much colour they show. Her little brother helps her count them.

Analysis:

A stacking puzzle becomes a counting rule, and a craft that uses safe natural dyes shows why materials matter.

Activity:

Stack 27 sugar cubes or dice into a big cube, then count the corner, edge and middle pieces.

A dice is a cube, and its opposite faces always add up to seven.

The name icosahedron comes from old Greek words meaning twenty faces.

The painted 27-cube — Eight, twelve, six, one. — Eight corners, twelve edges, six face middles and one hidden cube. They add back to 27.

Parts of a cube — Six faces, twelve edges, eight corners. — Say it while touching a dice. The numbers go six, twelve, eight.