International Baccalaureate · PYP 3

Numbers to 10 000

⌚ ~154 min ▢ 17 blocks ✓ 73 exercises ➣ Prerequisites : Numbers to 1000, Odd and Even Numbers

I Place Value

We already know how to group ones into tens, and tens into hundreds. To build larger numbers, we follow the same pattern.

We group 10 hundreds together to make 1 thousand: = $10$ hundreds = $1$ thousand =

Now our place value table needs a new column for thousands. This helps us organise and read large numbers.

Thousands	Hundreds	Tens	Ones
1	2	4	3

This table shows we have 1 thousand, 2 hundreds, 4 tens, and 3 ones. This makes the number $\mathbf{1\,243}$.

Definition — 4-Digit Number

A 4-digit number is a whole number written with four digits; it ranges from $1\,000$ to $9\,999$. In the base-10 system, the place of a digit tells us its value. We can show a number like $1\,243$ in many different ways:

With digits: $$1\,243$$
In expanded form: $$\begin{aligned}[t] 1\text{ thousand }&+& 2\text{ hundreds }&+& 4\text{ tens }&+& 3\text{ ones }\\ 1\,000 &+& 200 &+& 40 &+& 3 \end{aligned}$$
With words: one thousand two hundred forty-three
In a place value table:

Thousands Hundreds Tens Ones

1 2 4 3
With cubes:

Definition — Ten Thousand

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

= $10$ thousands = $1$ ten thousand To write this number, the four places we know are not enough: we need one more place, on the left, for the ten thousands.

Ten thousands	Thousands	Hundreds	Tens	Ones
$1$	$0$	$0$	$0$	$0$

This number is written $10\,000$ and is read ten thousand. It is the first number with five digits. It holds $10$ thousands, $100$ hundreds, $1\,000$ tens and $10\,000$ ones — all grouped, which is why the four places on the right show $0$.

Method — The Number Just After 9 999

In $9\,999$, every place is full: $9$ thousands, $9$ hundreds, $9$ tens and $9$ ones. We count one step forward and regroup each time a place reaches ten.

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

	Ten thousands	Thousands	Hundreds	Tens	Ones
$9\,999$		$9$	$9$	$9$	$9$
$10\,000$	$1$	$0$	$0$	$0$	$0$

The number just after $9\,999$ is therefore $10\,000$. One step forward has emptied four places at once and opened a fifth.

Method — Writing a Number in Many Ways

The same number $4\,635$ can be written in many ways. Going from one to another is the point.

In digits: $4\,635$.
In thousands, hundreds, tens and ones: $4$ thousands, $6$ hundreds, $3$ tens and $5$ ones.
In tens and ones: hide the ones digit, what is left counts the tens: $463$ tens and $5$ ones. In the same way, hide the last two digits: $46$ hundreds and $35$ ones.
As a sum of products: $(4 \times 1\,000) + (6 \times 100) + (3 \times 10) + (5 \times 1)$.
As a sum: $4\,000 + 600 + 30 + 5$.
In words: four thousand six hundred thirty-five.

To go back to the digits from any of these writings, put each amount in its place, from the thousands to the ones.

Method — Counting When a Unit Overflows

A collection is not always sorted neatly: it may hold more than nine of one unit. Then we regroup, from the ones upwards, before writing the number.

Ones: $17$ ones are $1$ ten and $7$ ones. The ones digit is $7$; the ten is carried to the tens.
Tens: $8$ tens and $1$ carried make $9$ tens. The tens digit is $9$.
Hundreds: $32$ hundreds are $3$ thousands and $2$ hundreds. The hundreds digit is $2$; the $3$ thousands are carried.
Thousands: $2$ thousands and $3$ carried make $5$ thousands.

So $17$ ones, $8$ tens, $32$ hundreds and $2$ thousands make $5\,297$. Check with the sum: $17 + 80 + 3\,200 + 2\,000 = 5\,297$.

Example

A company needs $1\,235$ air filters. To get a good price, it only buys lots of one hundred filters. How many lots must it buy to have enough?

Answer

Hide the last two digits of $1\,235$: it holds $12$ complete hundreds, and $35$ filters more. With $12$ lots the company would have $1\,200$ filters — not enough. It needs one more lot: $13$ lots, that is $1\,300$ filters, and $65$ will be left over.

Skills to practice

Counting Cubes in a Table
Counting from a Table
Finding the Digit
Writing Numbers from Thousands, Hundreds, Tens and Ones
Regrouping Hundreds into Thousands
Reaching Ten Thousand
Converting Between Units
Counting When a Unit Overflows
Going from Products to Digits and Back
Counting Lots of a Hundred

II Number Names

Definition — The Names of the Thousands

Past nine hundred ninety-nine, the names of the numbers use the word thousand: we say how many thousands, then the rest of the number as before.

Number	Name
$1\,000$	one thousand
$2\,000$	two thousand
$3\,000$	three thousand
$4\,000$	four thousand
$5\,000$	five thousand
$6\,000$	six thousand
$7\,000$	seven thousand
$8\,000$	eight thousand
$9\,000$	nine thousand
$10\,000$	ten thousand

So $4\,635$ is read four thousand six hundred thirty-five, and $2\,071$ is read two thousand seventy-one: the pieces of the name follow the places of the number.

Writing Numbers in Words

To write a number in words, say it and write what you say, place by place: four thousand, six hundred, thirty-five. In English the tens and the ones are joined by a hyphen: thirty-five, forty-two.

Report an error

Skills to practice

Writing Numbers from Their Names
Writing Numbers in Words

III On the Number Line

Example

On this number line, the marks do not go up one by one. What number does $x$ stand for?

Answer

First, read the step. From one mark to the next, the number goes up by $1\,000$: $0$, then $1\,000$, then $2\,000$, then $3\,000$. The step is $1\,000$, not $1$.
Then, take one more step. After $3\,000$ comes $3\,000 + 1\,000$.
So $x = 4\,000$.

Method — Placing a 4-Digit Number on the Number Line

A number line starts at $0$ and goes to the right; its marks are evenly spaced, and from one mark to the next we always add the same number, the step. Often only a part of it is drawn, which starts at another number. To place a number on it:

Read the step. Between two written numbers, count the steps and share the difference: from $2\,000$ to $2\,200$ there are $2$ steps, so the step is $100$.
Find the two written numbers around the number. $2\,700$ is between $2\,600$ and $2\,800$.
Count the marks from the nearest written number. From $2\,600$, one step of $100$: $2\,700$ is the very next mark.
Mark the point on the line.

{\small In green: $2\,700$, one mark after $2\,600$.}

Skills to practice

Reading a Number on the Number Line
Completing the Number Strip
Placing a Number on the Number Line

IV Going further

Skills to practice

Finding the Step of a Number Line
Problems
Quiz

	Ten thousands	Thousands	Hundreds	Tens	Ones
\(9\,999\)		\(9\)	\(9\)	\(9\)	\(9\)
\(10\,000\)	\(1\)	\(0\)	\(0\)	\(0\)	\(0\)

Number	Name
\(1\,000\)	one thousand
\(2\,000\)	two thousand
\(3\,000\)	three thousand
\(4\,000\)	four thousand
\(5\,000\)	five thousand
\(6\,000\)	six thousand
\(7\,000\)	seven thousand
\(8\,000\)	eight thousand
\(9\,000\)	nine thousand
\(10\,000\)	ten thousand