Class 5 Mathematics Chapter 5: Far and Near

A sprout too small to measure

Meera soaks some moong seeds overnight. In the morning she wraps them in a damp cloth. Next day a tiny white sprout has pushed out. She wants to measure it. It is much shorter than one centimetre. How can she write down its length? Her scale holds the answer.

Pick the unit first

Before measuring, think about the size of the thing. India Gate is about 42 metres tall, so metres suit it. A handkerchief is about 40 centimetres wide, so centimetres suit that. A deep well is measured in metres, and a mobile phone in centimetres.

Look between the numbers

Take out your scale. Between the mark for 1 and the mark for 2 there are small lines. Count the gaps between them. There are exactly ten. Every centimetre on every scale is split in the same way.

Ten little parts

Each of those ten small parts is called one millimetre. We write it as mm. So one centimetre is the same as ten millimetres. A millimetre is about as thick as two sheets of card.

Small things need small units

Now Meera can measure her sprout. A sprout, a small screw and a nail are all better measured in millimetres. Millimetres make a measurement more exact. That exactness is called precision.

Reading a length

Meera lays the sprout on her scale. Its end passes 4 and reaches the fifth small mark. So the sprout is 4 cm 5 mm long. She can also write the same length as 45 mm, counting only the small marks.

Changing centimetres into millimetres

To go from centimetres to millimetres, multiply by 10. So 6 cm becomes 60 mm. For 4 cm 5 mm, first make 40 mm, then add the 5 mm. That gives 45 mm. Bigger unit, smaller number.

Changing millimetres into centimetres

Go the other way by splitting the number into tens. Take 89 mm. It is 80 mm and 9 mm. The 80 mm make 8 cm. So 89 mm is 8 cm 9 mm. Try 514 mm. It is 51 cm 4 mm.

Drawing a line of 36 mm

Your ruler shows centimetres, not a line of 36 marks. So change the number first. 36 mm is 3 cm and 6 mm. Start at zero, go to 3, then count six small marks more.

Watching a sprout grow

Meera measures one seed every day for four days. She writes the length in millimetres each time. Day by day the number grows. A table of her lengths shows the growth much better than words do.

Same unit before you compare

Which is longer, 11 mm or 1 cm? They look different, so change one of them. 1 cm is 10 mm. Now compare 11 mm and 10 mm. The 11 mm line is longer, by just 1 mm.

Remember

• Choose metres for big things and centimetres for small ones.
• One centimetre holds ten millimetres.
• Multiply by 10 to go from cm to mm.
• Change both lengths to one unit before comparing.

Scenario:

A class measures a leaf with a scale. It ends between 6 and 7 centimetres. Nita writes 6 cm and Imran writes 7 cm. Their teacher asks them to look at the small lines between the numbers. There are ten small gaps in every centimetre. Each one is a millimetre. The leaf reaches the fourth small line past 6. So it is 6 cm and 4 mm. Nita sees why whole centimetres were not enough. The leaf does not stop at a big line, and the small lines are there for exactly that reason.

Analysis:

Two whole-centimetre answers and both short of the truth. The millimetre marks exist because things rarely end on a centimetre line.

Activity:

Measure four leaves or sticks in centimetres and millimetres. Write each length both ways.

Title:

Grow a sprout and measure it

Safety:

Ask an adult before using the kitchen, and do not eat seeds that smell bad.

What You Notice:

The sprout grows only a few millimetres a day, so centimetres would hide the change. The table shows the growth getting faster.

Every scale in every country splits one centimetre into the same ten parts.

Doctors measure the growth of a baby plant or a wound in millimetres, because the change is tiny.

Changing between cm and mm — Small unit, big number. — Millimetres are smaller than centimetres, so the number gets bigger. 6 cm becomes 60 mm.

Saris at the weaving centre

Shikha and Sonu help at a village weaving centre. They must sort saris and stoles by length. One label says 204 cm. Another says 2 metre 4 cm. A third says 240 cm. Are any of these the same cloth? To find out, the girls need one unit for all the labels.

A hundred in a metre

One metre is made of a hundred centimetres. So a metre stick holds a hundred small centimetre marks. To change metres into centimetres, multiply by 100. To go back the other way, look for whole hundreds.

Same length, two names

Take 204 cm. Two hundred of those centimetres make 2 metres. Four are left. So 204 cm is the same as 2 metre 4 cm. In the same way 540 cm is 5 metre 40 cm, and 240 cm is 2 metre 40 cm.

Odd names still work

A label may read 6 metre 150 cm. That looks strange, because 150 is more than a hundred. Change it all to centimetres. It becomes 600 plus 150, which is 750 cm. So that label matches a sari marked 750 cm.

Compare fairly

Which is longer, 456 cm or 5 m? Change the metres first. 5 m is 500 cm, and 500 is more than 456. Try another pair. 2 m 150 cm is 350 cm, and 3 m 50 cm is also 350 cm. Those two are equal.

Six tall statues

The Statue of Unity in India is 182 m tall. The Statue of Liberty is 93 m tall. Take one from the other and the gap is 89 m. The Motherland Calls stands 91 m high. Double it and you reach 182 m.

Tools for long distances

A scale is useless for a playground. A long tape can measure a hundred metres. A rodometer is a wheel you push along, counting turns. A rangefinder and a laser meter read a distance without walking it.

A thousand metres

Walk 1,000 metres and you have walked one kilometre. Kilo stands for thousand. We use kilometres for long distances, such as the road to the next town. Write it short as km.

Ropes into a kilometre

How many ropes of 100 m make a kilometre? Ten of them. Ropes of 200 m need only five. Ropes of 250 m need four. The shorter the rope, the more you need.

Planning a three kilometre race

Sheena is setting out a 3 km race. That is 3,000 metres. Water stations go every 500 m, so there are six, at 500 m, 1,000 m and so on. Children stand every 300 m to guide runners, so ten are needed.

The longest train ride

The Vivek Express runs from Dibrugarh to Kanniyakumari. That is 4,187 km of track. Jalpaiguri Road lies 983 km along, and Vijayawada lies 2,800 km along. Subtract to find the gap. It is 1,817 km.

Remember

• 100 cm make 1 m, and 1,000 m make 1 km.
• Change both lengths to one unit before comparing.
• Multiply to go to a smaller unit, and look for hundreds or thousands to go back.
• A gap between two points on a route is a subtraction.

Scenario:

The panchayat of Dhamora plans a race day on the village road. The course is 3 km long. Sheena marks water stations every 500 m, so runners never go thirsty in the heat. Jennifer asks children to stand every 300 m and point the way. The school van drives the course first to check it. Its meter reads 3,000 m, which matches the plan. A nurse waits at the halfway mark with a first aid box. The sarpanch says the same road carries farmers to the market, 12 km away. On race day the road is closed for two hours, and everyone walks instead.

Analysis:

A race is planned by dividing a long distance into equal steps. The same road serves the village every other day of the year.

Activity:

Walk from your gate to a shop nearby, counting your steps. Ask an adult how far it is, then work out roughly how many steps make one kilometre.

A rodometer is a wheel on a handle. Push it along and it counts the metres for you.

The Vivek Express takes about three days to cover its 4,187 km of track.

What kilo means — Kilo means a thousand. — A kilometre is a thousand metres, just as a kilogram is a thousand grams.

Comparing two lengths — One unit, then decide. — Change both lengths to centimetres, or both to metres, before you say which is longer.

Saji looks at his phone

Saji walks every day. By midday his phone showed 3 km 450 m. In the evening it says 4 km 650 m. He wants to know the whole day's walking. Adding these two is not quite like adding plain numbers. Let us see why.

Add the like units

Keep kilometres with kilometres and metres with metres. So 3 km and 4 km make 7 km. Then 450 m and 650 m make 1,100 m. Now look at that metre answer. It is bigger than a thousand.

A spare thousand

A thousand metres make one kilometre. So 1,100 m is 1 km and 100 m. Hand that whole kilometre over to the kilometre side. 7 km plus 1 km gives 8 km. The answer is 8 km 100 m.

Or use one unit only

Saji's friend does it another way. She changes both lengths into metres. That gives 3,450 m and 4,650 m. Adding them gives 8,100 m. She then reads it back as 8 km 100 m. Both ways agree.

Rani's two ribbons

Rani has a ribbon of 3 m 75 cm and another of 2 m 25 cm. Add the metres to get 5 m. Add the centimetres to get 100 cm. A hundred centimetres is one whole metre. So Rani has exactly 6 m of ribbon.

The electricians' cable

Some electricians have a 63 m roll of cable. They use 16 m 75 cm in the first room. How much is left? Change both to centimetres. 63 m is 6,300 cm, and the used piece is 1,675 cm. Take one from the other.

Borrowing a metre

You can also keep the units and borrow. The centimetre column has 0 above 75, so borrow one metre. That metre becomes 100 cm, and 100 minus 75 leaves 25. Then 62 m minus 16 m gives 46 m. The answer is 46 m 25 cm.

Shirts for twenty children

A shirt for a ten year old takes 1 m 80 cm of cloth. How much do twenty shirts take? Multiply each part. Twenty metres, and twenty lots of 80 cm, which is 1,600 cm. That is 16 m. So the tailor needs 36 m.

Sharing a road

A road 12 km 600 m long is laid in six equal days. How much is done each day? Change it to metres first, which gives 12,600 m. Share that into six equal parts. Each day 2,100 m of road is laid, which is 2 km 100 m.

Cost by the metre

A shop sells bag cloth at 100 rupees for 5 metres. So one metre costs 100 divided by 5, which is 20 rupees. Ten metres cost 200 rupees. For 2,000 rupees you could buy 100 metres.

Feet and inches

Height is often given in feet and inches. Fetch a ruler and compare. One inch is about 2 cm 5 mm. Twelve inches make one foot, which is about 30 cm. So a person 4 feet tall is roughly 120 cm.

Remember

• Add and subtract like units only.
• A spare 1,000 m becomes a kilometre, and a spare 100 cm becomes a metre.
• Multiply or divide each part, then tidy the answer.
• One inch is about 2 cm 5 mm, and a foot is twelve inches.

Scenario:

A class adds two walks: 2 km 600 m and 1 km 700 m. Ravi adds the kilometres and the metres separately. He gets 3 km 1,300 m and says that is the answer. His teacher asks how many metres make a kilometre. One thousand. So 1,300 m is more than a kilometre. The class carries a whole kilometre across and writes 4 km 300 m. Ravi sees the step he missed. Adding the columns was right, and the answer was not finished until the metres were traded in.

Analysis:

Ravi's columns were correct and his answer was unfinished. Any count of metres past a thousand has to be traded for a kilometre.

Activity:

Add two distances in kilometres and metres. Check whether the metres pass 1,000 and carry if they do.

Title:

Measure the family in two ways

Safety:

Ask an adult before marking a wall, and do not stand on a stool to reach.

What You Notice:

The same person has two very different numbers, such as 120 cm and 4 feet. A smaller unit always gives a bigger number.

The Gulmarg Gondola in Kashmir is the second longest cable car in the world.

Height is still given in feet and inches in many Indian homes and on many forms.

Adding mixed units — Like with like, then tidy up. — Add metres to metres first. Afterwards, change any spare thousand into a kilometre.

Borrowing in a length sum — One metre is a hundred coins. — A borrowed metre becomes 100 cm, in the same way a borrowed ten becomes ten ones.