Class 5 Mathematics Chapter 1: We the Travellers — I

Mira's travel diary

Sanju and Mira are on a long train ride to meet their grandmother. Mira keeps a travel diary. At every station she copies the numbers she sees. At one platform a board shows 10,458. Sanju stares at it. He has never read a number this long. How do we even say it? Let us find out.

Counting in thousands

Begin at 1,000. Add one more thousand and you have 2,000. Keep going: 3,000, then 4,000, then 5,000. After 9,000 comes one more thousand. That is ten thousand. We write it as 10,000. It is the smallest number with five digits.

How ten thousand is built

Ten ones make one ten. Ten tens make one hundred. Ten hundreds make one thousand. Ten thousands make one ten thousand. Every step to the left is ten times bigger. So 10,000 is simply ten bundles of a thousand.

A new place called TTh

Our place value chart used to have four columns: Th, H, T and O. Now we add one more on the left. It is called TTh, for ten thousands. In 10,458 the digit 1 sits in TTh. So the number holds one ten thousand.

Reading the station board

Look at 10,458 place by place. One ten thousand. No thousands. Four hundreds. Five tens. Eight ones. Now say the whole thing. Ten thousand four hundred fifty-eight. Sanju reads it and grins.

Where the comma goes

A long row of digits tires the eye. So we put a comma after the thousands place. Then 13520 becomes 13,520. And 45867 becomes 45,867. The comma does not change the size at all. It only helps us read.

From words back to digits

Names work the other way too. Try twenty thousand eight hundred sixty-nine. Write 2 in TTh and 0 in Th. Write 8 in H. Write 6 in T and 9 in O. The number is 20,869.

Breaking a number open

Expanded form shows what each digit is worth. So 23,004 is 20,000 + 3,000 + 4. The 2 is worth twenty thousand, not two. The 3 is worth three thousand. Try 45,867 yourself. It is 40,000 + 5,000 + 800 + 60 + 7.

Zeros are not empty

Drop the zeros from 23,004 and you are left with 234. That is a much smaller number. A zero says that this place has none of its own. It still guards the place for every digit on its left.

Tens hiding inside a number

Papa has 7,934 rupees in his purse. How many ten-rupee notes is that? Count in tens. 7,000 is 700 tens. 900 is 90 tens. 30 is 3 tens. Together that makes 793 tens, with 4 rupees left over.

Remember

• 10,000 is the smallest five-digit number.
• The new place on the left is TTh.
• The comma sits after the thousands place.
• Expanded form gives every digit its true value.
• A zero holds a place open.

Scenario:

A class reads two town populations: 9,800 and 10,200. Ravi says the first is bigger. He counts four digits and five digits and still picks the four. His teacher asks the class to write both in a place-value chart. The second number needs a new column, the ten thousands place. The first stops at thousands. So the second is bigger by 400 people. Ravi sees where he went wrong. He had read the leading 9 as large and ignored how many places followed it. The class writes the rule down. Count the places first, then look at the digits.

Analysis:

A 9 looks bigger than a 1 and the number is smaller. The place-value chart settles it, because 10,200 needs a column that 9,800 does not.

Activity:

Find the population of two towns near you. Write both in a place-value chart and say which is bigger.

Title:

Build your own place value machine

Safety:

Nothing sharp is needed. Ask an adult before you cut the cards.

What You Notice:

The same five chits make very different numbers. The chit you drop on TTh changes the number the most.

A book of 200 pages, with about 50 words on each page, holds close to 10,000 words.

We build every number, small or huge, out of only ten digits, 0 to 9.

The order of the places — Ones, Tens, Hundreds, Thousands, Ten Thousands. — Read the chart from the right. Each step to the left is ten times bigger.

The showroom board

Sanju and Mira stop at an electric bike showroom. Six prices hang on the wall. The board reads 90,000, then 89,999, then 94,983. Below that come 49,900, 93,743 and 39,999. Mira wants the cheapest bike. Sanju cannot tell which one it is. So many nines look the same. Let us sort them out.

Count the digits first

Every one of those prices has five digits. So counting digits does not help here. But it is still the first check. A four-digit price like 9,990 would be smaller than all of them. More places always mean a bigger number.

Compare from the left

When two numbers are the same length, start at the left. Look at 90,000 and 89,999. The first has 9 ten thousands. The second has only 8. So 90,000 wins, even though it ends in zeros. One extra ten thousand beats nine hundreds and nine tens.

Putting them in order

Increasing order means smallest first. The prices line up as 39,999, then 49,900, then 89,999, then 90,000, then 93,743, then 94,983. Decreasing order is the same line read backwards. Mira takes the 39,999 bike.

Swapping two digits

Here is a puzzle. Take 10,593 and swap just two digits. Can you land between 11,000 and 15,000? Swap the 0 and the 3 and you get 13,590. From 48,247, swapping the 4 and the 2 gives 28,447, the smallest you can reach.

A hungry rabbit

Now a different question. A rabbit sits at 2,346 on a number line. Carrots are kept only at round numbers. The rabbit is tired, so it wants the fewest jumps. Which round number should it hop to?

Nearest ten

The tens on either side of 2,346 are 2,340 and 2,350. Going back to 2,340 costs 6 jumps. Going ahead to 2,350 costs only 4. So the nearest ten of 2,346 is 2,350.

Nearest hundred

Now the carrots sit at hundreds. The neighbours are 2,300 and 2,400. From 2,346 it is 46 steps back and 54 steps ahead. Back is shorter. So 2,300 is the nearest hundred.

Nearest thousand

The thousand neighbours are 2,000 and 3,000. The halfway mark is 2,500. Our rabbit has not reached it yet. So it hops back to 2,000, the nearest thousand.

Two roundings that agree

Vijay rounds a number to the nearest hundred. Suma rounds the same number to the nearest thousand. They both get 7,000. Which numbers could they have started with? Try 7,030 and 6,999. Both land on 7,000 either way.

Remember

• Count the digits, then compare from the left.
• Increasing order means smallest first.
• Look at both neighbours and pick the closer one.
• Halfway or past it, go ahead; before it, go back.

Scenario:

A shop lists four cycles: 49,900, 39,999, 90,000 and 89,999. A class is asked which is cheapest. Meena picks 39,999 and Ravi picks 49,900. Their teacher asks them to compare digit by digit from the left. All four are five-digit numbers. The first digits are 4, 3, 9 and 8. The smallest first digit is 3, so 39,999 is cheapest. Ravi sees why he was misled. The nines at the end of 39,999 made it look large, but the leftmost digit decides first. The class notes that length is checked first and then the left.

Analysis:

Four nines at the end of a number look big and change nothing. When the numbers are the same length, the leftmost digit decides.

Activity:

Write four prices with five digits each. Put them in order from cheapest and say which digit decided each step.

Title:

The price tag sorting game

Safety:

No tools are needed for this game.

What You Notice:

Two prices that look very different can round to the same thousand. Rounding hides small gaps.

Bus and train boards often round the distance, so 498 km is shown as 500 km.

A town of 45,858 people and a town of 45,900 people sound the same when rounded.

Rounding to a neighbour — Look both ways, then hop the shorter way. — Find the round number before and after. Count the jumps to each and pick the smaller count.

A trip for the whole school

The notice board has good news. The whole school is going on a trip. There are 461 girls and 439 boys. Together that is 900 travellers. How many vehicles will they need? And how long will the road take? Sanju and Mira open their notebooks. Numbers are about to get busy.

Travel, long ago

Long before engines, people walked. They also rode bullocks, horses, donkeys, mules and elephants. In snowy hills, yaks and dogs pulled the load. In deserts, camels carried people for days. Boats came early too, and they crossed rivers, lakes and seas.

Travel, today

Today the same trip looks different. We ride cycles, motorbikes, cars, buses and trains. Ships cross the sea and aeroplanes cross the sky. Submarines go deep under water. Some people even travel in a spacecraft, far above the clouds.

How far in one hour

On foot you cover about 3 to 5 km in an hour. A cycle takes you 12 to 20 km. A motorbike does 40 to 60 km. A train can run up to 160 km. An aircraft covers 750 to 920 km. A spacecraft beats them all.

More hours, more kilometres

Suppose a cyclist keeps a steady 15 km every hour. In 2 hours she covers 30 km. In 3 hours she covers 45 km. In 4 hours she covers 60 km. Same speed, so we just keep adding 15.

Sharing 900 into vehicles

Now the vehicles. A car holds 4 people, so 900 people fill 225 cars. A tempo traveller holds 10, which means 90 of them. A boat holds 20, so 45 boats. A minibus holds 25, so only 36 minibuses.

Bigger vehicle, shorter list

Look at the pattern. The more seats a vehicle has, the fewer of them you need. An aeroplane holds 180 people, so the whole school fits into just 5. A bicycle holds 2, and that would need 450 of them.

Big numbers hiding nearby

Large numbers are not only on the road. A book has about 200 pages. Each page carries about 50 words. That makes nearly 10,000 words in one book. Count the books in your class and you may find another four-digit number.

Count the corners only once

A caretaker put horses along the four sides of a square stable. Five stood on every side. He said that made 20 horses. But the corner horses belong to two sides each. With all four corners filled there are only 16 horses.

Remember

• Speed tells you how far you go in one hour.
• Same speed each hour, so keep adding.
• More seats in a vehicle means fewer vehicles.
• Count shared corners only once.

Scenario:

The Class 5 children of a school in Bharatpur are going to a bird sanctuary. There are 900 travellers in all. The teachers can hire cars with 4 seats or minibuses with 25 seats. The road to the sanctuary is narrow and passes a wetland where birds rest. The forest guard asks for as few vehicles as possible. He says every extra engine adds smoke and noise, and the birds fly away from noise. The parents also worry about the cost of fuel. The children sit with their notebooks and work out both answers before the teachers decide.

Analysis:

Sharing a group by the seats of a vehicle gives the number of vehicles. Fewer vehicles mean less fuel, less smoke and quieter roads.

Activity:

Count the vehicles that pass your gate in ten minutes. Note how many people ride in each one.

Title:

The puzzle that always ends in nine

Safety:

Nothing here can hurt you. It is only paper and pencil.

What You Notice:

Every answer is a step of 9, such as 9, 36, 54 or 72. The last digit you reach is always 9.

One kilometre is 1,000 metres, so a 60 km ride is 60,000 metres long.

Distances at sea and in the sky are often measured in nautical miles, not kilometres.

Seats and vehicles — More seats, shorter list. — As the seats in a vehicle go up, the number of vehicles needed comes down.

Steady speed — Same speed, same jump. — At a steady speed every extra hour adds the same distance to the total.