International Baccalaureate · PYP 2

Numbers to 1000

⌚ ~210 min ▢ 24 blocks ✓ 110 exercises ➣ Prerequisites : Numbers to 100, Ordinal Numbers and Patterns

I Place Value

Discover From Ones to Hundreds

To build bigger numbers, we group smaller ones together.

First, we group 10 ones to make 1 ten: = $10$ ones = $1$ ten =
Next, we group 10 tens to make 1 hundred: = $10$ tens = $1$ hundred =
A hundred is therefore also 100 ones: it holds $10$ tens, and each ten holds $10$ ones. = $1$ hundred = $10$ tens = $100$ ones

We can use a place value table to see how many hundreds, tens, and ones are in a number.

Hundreds	Tens	Ones
2	4	3

This table shows we have 2 hundreds, 4 tens, and 3 ones. This makes the number 243.

Definition — Base 10 system

In the base-10 system, the place of a digit tells us its value. We can show a number in many different ways:

With digits: $$243$$
In expanded form: $$\begin{aligned}[t] 2\text{ hundreds }&+& 4\text{ tens }&+& 3\text{ ones }\\ 200 &+&40&+&3 \end{aligned}$$
With words: two hundred forty-three
In a place value table:

Hundreds Tens Ones

2 4 3
With cubes:

Zero is a special digit. It acts as a placeholder to show that a place is empty. For example, in the number 20, the zero shows that no ones are left on their own: the ones place is empty.

Report an error

Method — Regrouping Tens into Hundreds

The same number can be written in more than one way. When a place holds ten or more, we regroup: every $10$ tens make $1$ hundred.

One place overflows. $14$ tens $= 10$ tens $+\ 4$ tens $= 1$ hundred $+\ 4$ tens.
A place overflows next to another. $2$ hundreds $+\ 15$ tens: the $15$ tens give $1$ hundred and $5$ tens, so $2$ hundreds $+\ 15$ tens $= 3$ hundreds $+\ 5$ tens.
Check. Regrouping never changes the number, only the way it is written: $14$ tens is $140$, and $1$ hundred $+\ 4$ tens is $140$ too.

Example — The Same Number, Other Units

The number $635$ is $6$ hundreds, $3$ tens and $5$ ones. The same quantity can be counted with other units.

How many tens does $635$ contain altogether? And how many ones altogether?
Which number do $5$ ones and $5$ hundreds and $13$ tens make?

Answer

Each hundred holds $10$ tens. Counting the tens of the $6$ hundreds gives $10$, $20$, $30$, $40$, $50$, $60$: the $6$ hundreds are $60$ tens. $$\begin{aligned}[t] 635 &= 6\text{ hundreds} + 3\text{ tens} + 5\text{ ones}\\ &= 60\text{ tens} + 3\text{ tens} + 5\text{ ones} && \text{(each hundred is 10 tens)}\\ &= 63\text{ tens} + 5\text{ ones} \end{aligned}$$ In the same way, each hundred is $100$ ones and each ten is $10$ ones: $600 + 30 + 5$, so $635$ is also $635$ ones. Do not confuse the digit with the number. The tens digit of $635$ is $3$: it says what sits in the tens place. The number of tens is $63$: it counts all the tens, those gathered in the hundreds too. To read it, hide the ones digit: what is left, $63$, counts the tens.
The order of the words does not change the number: each amount goes back to its own place. Then the $13$ tens overflow: $10$ of them make $1$ hundred. $$\begin{aligned}[t] 5\text{ ones} + 5\text{ hundreds} + 13\text{ tens} &= 5\text{ hundreds} + 13\text{ tens} + 5\text{ ones}\\ &\qquad\text{(in order: hundreds, tens, ones)}\\ &= 5\text{ hundreds} + 1\text{ hundred} + 3\text{ tens} + 5\text{ ones} && \text{(10 tens = 1 hundred)}\\ &= 6\text{ hundreds} + 3\text{ tens} + 5\text{ ones}\\ &= 635 \end{aligned}$$

The same number has now been written in four ways:

$6$ hundreds, $3$ tens and $5$ ones;
$63$ tens and $5$ ones;
$635$ ones;
$5$ ones and $5$ hundreds and $13$ tens.

Definition — A Thousand

We already know how to group tens into hundreds. We group one step further: $10$ hundreds make one thousand.

= $10$ hundreds = $1$ thousand =

To write this number, the three places we know are not enough: we need one more place, on the left, for the thousands.

Thousands	Hundreds	Tens	Ones
1	0	0	0

This number is written $1\,000$ and is read one thousand. In the table, the hundreds digit is $0$, the tens digit is $0$ and the ones digit is $0$: the three zeros say that nothing is left to place in those columns.
Careful — that does not mean one thousand holds no hundreds. It holds 10 of them, but they have all been grouped into $1$ thousand, so none is left on its own. In the same way, one thousand holds $100$ tens and $1\,000$ ones: the same quantity, counted in different units.

Method — The Number Just After 999

In $999$, every place is full: $9$ hundreds, $9$ tens and $9$ ones. Nothing can be added to a place without making it overflow. So we count one step forward and regroup each time a place reaches ten.

The ones overflow. $9$ ones and $1$ more make $10$ ones, that is $1$ ten and $0$ ones.
The tens overflow in turn. That new ten joins the $9$ tens: $10$ tens, that is $1$ hundred and $0$ tens.
The hundreds overflow in turn. That new hundred joins the $9$ hundreds: $10$ hundreds, that is $1$ thousand and $0$ hundreds.

	Thousands	Hundreds	Tens	Ones
$999$		9	9	9
$1\,000$	1	0	0	0

The number just after $999$ is therefore $1\,000$. Read the table column by column: the three full places have all become $0$, and the new place on the left has taken the $1$. One step forward has emptied three places at once and filled a new one.

Skills to practice

Counting Cubes in a Table
Writing Numbers From a Place Value Table
Finding the Digit
Counting Cubes
Writing Numbers From Hundreds, Tens and Ones
Writing Numbers From Expanded Form
Grouping and Regrouping
Breaking Down into Hundreds and Tens
Regrouping Tens into Hundreds
Writing a Number with Other Units
Reaching a Thousand
Buying Stamps

II Estimating a Quantity

Definition — Estimate

To estimate a quantity is to give quickly a number close to the real number, without counting the objects one by one. We say about: “about $40$ marbles”.
Counting gives the exact number: $43$. Estimating gives a close number: about $40$.

Method — Estimating with a Group of Ten

Circle a group of ten objects: it shows how much room ten objects take.
See how many groups like it would cover the whole collection, and count them by tens: about $3$ groups make about $30$.
Check by counting in tens: put the objects in groups of ten, count the tens, then the ones left over.

The objects are not always spread out in the same way, so an estimate does not need to be exact: it is good when it is close to the count.
A good estimate is often the ten closest to the count. For $48$ objects: $48$ is $2$ steps from $50$ and $8$ steps from $40$, so the ten closest to $48$ is $50$, and a good estimate is about $50$.

Example — Estimating with a Square of 100

Tom's card is covered with dots. The dashed square holds $100$ dots.

Without counting the dots one by one, estimate how many dots there are on the card.

Answer

The dashed square shows how much room $100$ dots take. About four squares like it cover the whole card. Counting them in hundreds: $100$, $200$, $300$, $400$. There are about $400$ dots.
To check, look at what is left. Each square holds $10$ columns of $10$ dots, so the $4$ squares cover $40$ columns, and $2$ columns of $10$ dots remain: $2$ tens. The card holds $400 + 20 = 420$ dots, and $420$ is close to $400$.

Skills to practice

Estimating with a Square of 100

III Number Names

Definition — The Names of the Hundreds

Each whole hundred has its own name: we say how many hundreds, then the word hundred. Ten hundreds make one thousand.

Digits	Name	Hundreds
$100$	one hundred	$1$ hundred
$200$	two hundred	$2$ hundreds
$300$	three hundred	$3$ hundreds
$400$	four hundred	$4$ hundreds
$500$	five hundred	$5$ hundreds
$600$	six hundred	$6$ hundreds
$700$	seven hundred	$7$ hundreds
$800$	eight hundred	$8$ hundreds
$900$	nine hundred	$9$ hundreds
$1\,000$	one thousand	$10$ hundreds

A number between two whole hundreds takes the name of its hundreds, then the name of the rest: $635$ is six hundred thirty-five, $208$ is two hundred eight. In the name of a number, the word hundred never takes an s: $200$ is two hundred. The last column of the table counts place-value units, like $2$ tens or $2$ ones: there, $2$ hundreds does take an s.

The name of a number is read in pieces. In English, a hyphen only joins the tens and the ones (thirty-five): it is the word hundred that shows where to cut.

The word hundred cuts the name in two: what comes before it counts the hundreds, what comes after it gives the end of the number. Six hundred thirty-five $\rightarrow$ six hundred $\mid$ thirty-five $\rightarrow$ $6$ hundreds, then thirty-five, that is $3$ tens and $5$ ones: $635$. If a single word from one to nine follows — seven hundred nine — it gives the ones, and the tens place stays empty: $709$; if that word is ten, twenty, thirty… — seven hundred ten — it gives the tens: $710$; if it is a word from eleven to nineteen — seven hundred twelve — it gives $1$ ten and the ones at once: $712$.
The word hundred stays singular, however many there are: five hundred sixty-two, never five hundreds. So it is the word in front that counts the hundreds — five hundred sixty-two is $562$.

Report an error

Saying “and”

Many English speakers, in Britain and in Australia for example, say and after the word hundred: $635$ is six hundred and thirty-five, $208$ is two hundred and eight. Both ways of saying it are correct, and they name the same number: the word and adds nothing to the value. A whole hundred has nothing after it, so no and: $700$ is seven hundred.

Report an error

Skills to practice

Writing Numbers from Their Names
Writing Numbers in Words

IV On the Number Line

Definition — Number Line

A number line starts at an origin, marked $0$, and shows the numbers in order, from smallest to largest; from one mark to the next, we always move on by the same step.

Method — Counting in Steps

On the number line, we can jump with equal steps.

By twos: $2$, $4$, $6$, $8$, $10$, $\dots$
By tens: $10$, $20$, $30$, $\dots$, $90$. We can start from any number: from $24$, we get $24$, $34$, $44$, $\dots$

To add $1$ is to say the number just after; to add $2$ is to add $1$ and $1$ again: $78$, then $79$, then $80$, so $78 + 2 = 80$.

Example — Counting in Hundreds

A number line does not have to count in ones. On this one, each mark is one hundred further than the one before.

Ten jumps of one hundred, starting from $0$, land on $1\,000$. These are the same $10$ hundreds that make one thousand — gathered into a cube in the definition, spread out along a line here.
A number that is not a whole hundred has no mark of its own: it goes between two marks. $350$ has $3$ hundreds and $5$ tens, so it stands between $300$ and $400$, halfway.

Example — A Number is a Distance from Zero

On a number line counting in ones, a point is marked. Starting from $0$, we need $7$ steps of one to reach it. Which number is written under it? And for a point $235$ steps away from $0$?

Answer

The number under a point is its distance from $0$ counted in steps of one: $7$ steps, the number $7$. For $235$ steps, the number is $235$: we can go there with $2$ steps of $100$, $3$ steps of $10$, then $5$ more — the same distance, counted faster.

{\small Not to scale: the jumps of $100$, of $10$ and of $5$ are drawn the same length.}

Method — Placing a Number on the Number Line

Often only a part of a number line is drawn, which does not start at $0$. To place a number on it:

Read the step: the number added from one mark to the next.
Find the written numbers nearest to the number, before it or after it.
Count the steps from the nearest one, forward or back.
Mark the point on the line.

For example, place $370$ on this part of a number line, counting in tens from $300$, with one mark at each grid line.

The step is given: $10$. When it is not given, it is found on the line: from the written $300$ to the written $350$ there are $5$ steps, and counting in tens — $310$, $320$, $330$, $340$, $350$ — five steps land exactly on $350$.
The written numbers nearest to $370$ are $350$, before it, and $400$, after it.
$370$ is nearer to $350$: $2$ steps of $10$ from $350$, that is $360$, then $370$. From $400$, it would take $3$ steps back.
The point goes on the second mark after $350$.

{\small In green: $370$, two marks after $350$.}

Skills to practice

Counting in Steps of 2, 3 and 4
Counting by Fives, Tens and Hundreds
Placing and Reading Numbers on the Number Line
Placing a Number on the Number Line

V Positions up to the Hundredth

Discover

In the school race, $100$ children cross the finish line one after the other. Léa is the thirty-second. How many children finished before her? And how many after her?

Answer

Léa is in position $32$: all the positions from $1$ to $31$ are before her: $31$ children. After her come the positions $33$ to $100$: $100 - 32 = 68$ children. Check: $31 + 1 + 68 = 100$.

Definition — Ordinal Numbers up to Hundredth

The ordinal numbers say a position. Beyond twentieth, only the last word of the name changes: it becomes the ordinal already known up to twentieth (one $\rightarrow$ first, two $\rightarrow$ second, three $\rightarrow$ third, five $\rightarrow$ fifth …); a whole ten changes -ty into -tieth (forty $\rightarrow$ fortieth), and hundred becomes hundredth:

$21$ twenty-first	$40$ fortieth	$71$ seventy-first
$22$ twenty-second	$50$ fiftieth	$80$ eightieth
$23$ twenty-third	$60$ sixtieth	$81$ eighty-first
$30$ thirtieth	$70$ seventieth	$100$ hundredth
$31$ thirty-first	$90$ ninetieth	$91$ ninety-first

So we say twenty-first, forty-second, fifty-third, and never twenty-oneth: only the last word takes the ordinal ending. A position never changes the number: the eighty-first is the child number $81$.

Method — Two-Digit Positions and Those Before

From the position to those before: those before are one less than the position. The sixty-fourth has $63$ before it.
From those before to the position: if $47$ people are in front of me, I am the forty-eighth ($47 + 1$).
Those after: the total minus the position. The sixty-fourth of $100$ has $100 - 64 = 36$ behind.

Skills to practice

Naming the Position
Counting Those Before and After

VI Going further

Skills to practice

Completing Equalities with Pictures
Drawing Numbers
Problems
Quiz

Digits	Name	Hundreds
\(100\)	one hundred	\(1\) hundred
\(200\)	two hundred	\(2\) hundreds
\(300\)	three hundred	\(3\) hundreds
\(400\)	four hundred	\(4\) hundreds
\(500\)	five hundred	\(5\) hundreds
\(600\)	six hundred	\(6\) hundreds
\(700\)	seven hundred	\(7\) hundreds
\(800\)	eight hundred	\(8\) hundreds
\(900\)	nine hundred	\(9\) hundreds
\(1\,000\)	one thousand	\(10\) hundreds

\(21\) twenty-first	\(40\) fortieth	\(71\) seventy-first
\(22\) twenty-second	\(50\) fiftieth	\(80\) eightieth
\(23\) twenty-third	\(60\) sixtieth	\(81\) eighty-first
\(30\) thirtieth	\(70\) seventieth	\(100\) hundredth
\(31\) thirty-first	\(90\) ninetieth	\(91\) ninety-first