Mathematics: Class 3 — Chapter 3: Double Century

3.1 The Story of Our Numbers

Tens of thousands of years ago, people started counting and keeping records of their belongings by making marks on cave walls and tree barks. Over time, they grouped items in sizes of 5, 10, 20, and 60.

Thousands of years ago, ancient Indians created a revolutionary method for writing any number, however large, using only ten symbols: $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$. This system is now used worldwide and made modern inventions like computers and mobile phones possible.

A critical part of this invention was the introduction of the symbol 0 to represent 'nothing' or the absence of something, which made the positional numeral system work.

3.2 Understanding 100 (A Century)

A Century represents the number $100$.

One more than 99: $99 + 1 = 100$
Bundles of Sticks: $10$ bundles of $10$ sticks each equals $100$ sticks.
Packets of Bindis: $10$ packets of $10$ bindis each equals $100$ bindis.
Sports Context: Scoring a century in cricket means scoring $100$ runs.

Making 100 in Different Ways

We can combine different pairs of numbers to make $100$:

$70$ and $30$ makes $100$
$60$ and $40$ makes $100$
$45$ and $55$ makes $100$
$75$ and $25$ makes $100$
$15$ and $85$ makes $100$

This can also be represented as jumps on a number line (e.g., jumping to $65$ and then jumping $35$ more to reach $100$).

3.3 Numbers Beyond 100

When we count beyond $100$, we use bundles of $100$ (which is $10$ bundles of $10$ sticks), bundles of $10$, and individual loose sticks ($1$s).

Counting from 101 to 110

$100$ and $1$ makes One Hundred One ($101$)
$100$ and $2$ makes One Hundred Two ($102$)
$100$ and $3$ makes One Hundred Three ($103$)
$100$ and $4$ makes One Hundred Four ($104$)
$100$ and $5$ makes One Hundred Five ($105$)
$100$ and $6$ makes One Hundred Six ($106$)
$100$ and $7$ makes One Hundred Seven ($107$)
$100$ and $8$ makes One Hundred Eight ($108$)
$100$ and $9$ makes One Hundred Nine ($109$)
$100$ and $10$ makes One Hundred Ten ($110$)

Place Value Representation (100s, 10s, 1s)

Numbers can be broken down into hundreds, tens, and ones:

123: $1$ hundred ($100$), $2$ tens ($10$s), and $3$ ones ($1$s).
104: $1$ hundred ($100$), $0$ tens ($10$s), and $4$ ones ($1$s).
120: $1$ hundred ($100$), $2$ tens ($10$s), and $0$ ones ($1$s).

3.4 Counting Beyond 150 up to 200

We can extend our counting up to $200$ (a Double Century).

Clap, Snap, and Pat Game

A fun way to represent numbers physically:

Clap: Represents $100$ (Two claps represent $200$)
Snap: Represents $10$ (Two snaps represent $20$)
Pat: Represents $1$ (Two pats represent $2$)
Example: Clap — Snap Snap — Pat Pat Pat represents $123$ ($100 + 20 + 3$).

Numbers from 150 to 159

150: $1$ hundred, $5$ tens, $0$ ones ($100$ and $50$)
151: $1$ hundred, $5$ tens, $1$ one ($100$ and $51$)
152: $1$ hundred, $5$ tens, $2$ ones ($100$ and $52$)
153: $1$ hundred, $5$ tens, $3$ ones ($100$ and $53$)
154: $1$ hundred, $5$ tens, $4$ ones ($100$ and $54$)
155: $1$ hundred, $5$ tens, $5$ ones ($100$ and $55$)
156: $1$ hundred, $5$ tens, $6$ ones ($100$ and $56$)
157: $1$ hundred, $5$ tens, $7$ ones ($100$ and $57$)
158: $1$ hundred, $5$ tens, $8$ ones ($100$ and $58$)
159: $1$ hundred, $5$ tens, $9$ ones ($100$ and $59$)

Representing 200 on a Number Line

We can show $200$ using different jumps on a number line:

$150 + 50 = 200$
$140 + 60 = 200$

3.5 Number Patterns and Games

Jumping Games on the Number Line

Jumps of 5: Counting by fives starting from $100$: $100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150$.
Jumps of 20: Counting by twenties starting from $100$: $100, 120, 140, 160, 180, 200$.

Predecessors and Successors (1 Less and 1 More)

Finding the number that is $1$ less or $1$ more than a given number:

For $160$: $1$ less is $159$, $1$ more is $161$.
For $129$: $1$ less is $128$, $1$ more is $130$.
For $187$: $1$ less is $186$, $1$ more is $188$.
For $134$: $1$ less is $133$, $1$ more is $135$.
For $158$: $1$ less is $157$, $1$ more is $159$.