International Baccalaureate · PYP 4

Numbers to 999 999

⌚ ~226 min ▢ 26 blocks ✓ 121 exercises ➣ Prerequisites : Numbers to 10 000, Comparing, Ordering and Rounding

I Place Value

Discover From Thousands to Ten Thousands

Ones, tens and hundreds are grouped by ten: $10$ ones make $1$ ten, $10$ tens make $1$ hundred, $10$ hundreds make $1$ thousand. To build larger numbers, thousands are grouped in the same way.
We group 10 thousands together to make 1 ten thousand:

= $10$ thousands = $1$ ten thousand =

Our place value table now needs a new, larger column for Ten Thousands.

Ten Thousands	Thousands	Hundreds	Tens	Ones
1	2	4	3	5

This table shows we have 1 ten thousand, 2 thousands, 4 hundreds, 3 tens, and 5 ones. This makes the number $\mathbf{12\,435}$. In the same way, $10$ ten thousands are grouped to form 1 hundred thousand.

Definition — 5-Digit Number

A 5-digit number is a whole number written with five digits; it ranges from $10\,000$ to $99\,999$. The place of a digit gives its value. We can show a number like $12\,435$ in many different ways:

With digits: $$12\,435$$
In expanded form: $$\begin{aligned}[t] 1\text{ ten thousand }&+& 2\text{ thousands }&+& 4\text{ hundreds }&+& 3\text{ tens }&+& 5\text{ ones }\\ 10\,000 &+& 2\,000 &+& 400 &+& 30 &+& 5 \\ 1\times 10\,000 &+& 2\times 1\,000 &+& 4\times 100 &+& 3\times 10 &+& 5\times 1 \\ \end{aligned}$$
With words: twelve thousand, four hundred thirty-five
In a place value table:

Ten Thousands Thousands Hundreds Tens Ones

1 2 4 3 5
With cubes:

Extending to 6-Digit Numbers

The place value table can always be extended: adding a new column for the hundred thousands lets us write 6-digit numbers, from $100\,000$ to $999\,999$. For example, the number $214\,365$:

Hundred Thousands	Ten Thousands	Thousands	Hundreds	Tens	Ones
2	1	4	3	6	5

$$214\,365 = 2\times 100\,000 + 1\times 10\,000 + 4\times 1\,000 + 3\times 100 + 6\times 10 + 5\times 1$$ In words: two hundred fourteen thousand, three hundred sixty-five.

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Skills to practice

Counting Cubes in a Table
Counting Cubes
Counting Cubes from a Table
Finding the Digit
Writing Numbers from Ten Thousands, Thousands, Hundreds, Tens, and Ones
Writing Numbers from Hundred Thousands to Ones
Writing Numbers from Expanded Form
Writing Numbers from Expanded Form with Multiplication
Writing Numbers in Figures from Words
Converting Between Units
Writing Numbers in Words
Organising and Counting Collections

II On the Number Line

A number line helps us see numbers in order. We can use it to count by ones, or we can make bigger jumps to count faster: from one mark to the next, the jump is always the same. With large numbers, the number line can jump by thousands, by ten thousands or by hundred thousands at a time.

Counting by tens: $0$, $10$, $20$, $30$, …
Counting by hundreds: $0$, $100$, $200$, $300$, …
Counting by thousands: $0$, $1\,000$, $2\,000$, $3\,000$, …
Counting by ten thousands: $0$, $10\,000$, $20\,000$, $30\,000$, …
Counting by hundred thousands: $0$, $100\,000$, $200\,000$, $300\,000$, …

Report an error

Definition — Number Line

A number line starts at an origin, marked $0$, and goes to the right. Its marks are equally spaced: from one mark to the next, the same number is always added.

Skills to practice

Finding Numbers
Continuing Number Sequences
Placing Numbers on a Number Line

III Comparing and Ordering

Method — Comparing Two Whole Numbers

To compare two whole numbers:

Count the digits: the number with more digits is the greater one.
If both numbers have the same number of digits, compare the digits from left to right, place by place: the first different digit decides.

To order several numbers, compare them and write them from the smallest to the largest (increasing order) or from the largest to the smallest (decreasing order).

Example

Compare the numbers, using $<$ or $>$:

$98\,765$ and $123\,456$
$214\,365$ and $214\,635$

Answer

$98\,765$ has $5$ digits and $123\,456$ has $6$ digits. The number with more digits is the greater one: $$98\,765 < 123\,456.$$
Both numbers have $6$ digits. Comparing the digits from left to right: the hundred-thousands digits are the same ($2$), so are the ten-thousands digits ($1$) and the thousands digits ($4$), but the hundreds digits differ: $3 < 6$. So: $$214\,365 < 214\,635.$$

Skills to practice

Comparing Two Numbers
Finding a Number Between Two Numbers
Ordering Numbers

IV Placing a Number Between Two Consecutive Multiples

Method — Placing a Number Between Two Thousands, Hundreds or Tens

To place a number between two thousands next to each other, read how many whole thousands it contains: the number lies between that many thousands and one more. It is written with two symbols at once, and read with the words between … and …: $4\,000 < 4\,635 < 5\,000$, read: $4\,635$ is between $4\,000$ and $5\,000$. Between two hundreds or two tens, it works the same way, with whole hundreds or whole tens.

Between two	It contains	Writing
thousands	$4$ whole thousands	$4\,000 < 4\,635 < 5\,000$
hundreds	$46$ whole hundreds	$4\,600 < 4\,635 < 4\,700$
tens	$463$ whole tens	$4\,630 < 4\,635 < 4\,640$
neighbours	—	$4\,634 < 4\,635 < 4\,636$

The smaller the unit, the closer together the two numbers, and the more precisely they say where the number is; the closest of all are the two neighbours, the number just before and the number just after.
This only works if the number is not exactly a number of thousands (or of hundreds, or of tens, for those marks). $4\,000$ is exactly $4$ thousands: it sits on the mark, not between two marks.
Do not confuse the digit with the number of hundreds. The hundreds digit of $4\,635$ is $6$, but $4\,635$ contains $46$ hundreds: the digit says what sits in that place, the number of hundreds says how many there are altogether.

Example

Write $214\,365$ between two consecutive hundred thousands.

Answer

$214\,365$ contains $2$ whole hundred thousands, so it lies between $2$ and $3$ hundred thousands: $$200\,000 < 214\,365 < 300\,000.$$

Definition — Placing a Number Between Two Consecutive Multiples

Placing a number between two consecutive multiples of $10$ (of $100$, of $1\,000$…) means giving the two consecutive multiples between which it lies: the smaller one is the lower bound, the larger one the upper bound. The lower bound can be the number itself: to the hundred, $365$ lies between $300$ and $400$, and $300$ between $300$ and $400$.

Method — Finding the Lower and Upper Bounds

To place a number between two consecutive multiples of a place (ten, hundred, thousand…):

Determine the lower bound. Keep the target digit and all digits to its left, and replace all digits to its right with zeros.
Determine the upper bound. Keep the digits to the left of the target place value, add one to the target digit, and replace all digits to its right with zeros. (If the target digit is $9$, adding one carries over to the digit on its left.)

Example

Write the number $365$ between two consecutive hundreds.

Answer

The target place value is the hundreds. The digit is 3.
Lower Bound: Keep the 3, replace subsequent digits with zeros. The lower bound is 300.
Upper Bound: Add 1 to the hundreds digit ($3+1=4$), replace subsequent digits with zeros. The upper bound is 400.
Therefore, 365 lies between 300 and 400. This can be written as $300 \leq 365 < 400$. The sign $\leq$ reads “is less than or equal to”: the lower bound can be the number itself, as in $300 \leq 300 < 400$.

Skills to practice

Placing a Number Between Two Hundred Thousands
Placing a Number Between Two Consecutive Multiples

V Rounding Numbers

Proposition — Rounding Means Taking the Nearest Bound

To round a number to the ten (to the hundred, to the thousand…), place it between two consecutive multiples of $10$ (of $100$, of $1\,000$…): its rounding is the bound nearer to it. A number exactly in the middle of the two bounds is rounded to the upper bound, by convention.

To the hundred, $365$ lies between $300$ and $400$. It is $65$ away from $300$ and $35$ away from $400$: it is nearer to $400$, so $365$ rounded to the nearest hundred is $400$. The digit to the right of the target digit tells in which half, between the two bounds, the number lies: from $5$ to $9$, the upper half (the middle, $350$, included); from $0$ to $4$, the lower half. This is why the rounding procedure only looks at that digit.

Method — Standard Procedure for Rounding

To round a number to a given place (the ten, the hundred, the thousand…):

Find the target digit: the digit in the place the number is rounded to.
Look at the digit just to its right.
Apply the rule:
- If that digit is 5 or more (5, 6, 7, 8, 9), round up: add 1 to the target digit.
- If that digit is 4 or less (0, 1, 2, 3, 4), round down: the target digit stays the same.
Replace the digits after it by zeros: every digit to the right of the target digit becomes $0$.

The digits to the left of the target digit stay the same, except when a 9 becomes 10: if the target digit is $9$, adding 1 gives $10$, so write $0$ and add 1 to the next digit on the left (for example, $9\,999$ rounded to the nearest thousand is $10\,000$). A number exactly halfway between the two bounds, such as $350$ between $300$ and $400$, is rounded up by convention.

Example

Round the number $365$ to the nearest hundred.

Answer

The procedure is executed as follows:$$\begin{aligned}[t]&\textcolor{colordef}{\underline{3}}65 && \text{1. The target digit in the hundreds place is } \textcolor{colordef}{3}.\\ &\textcolor{colordef}{3}\textcolor{colorprop}{\underline{6}}5 && \text{2. The digit to its right is } \textcolor{colorprop}{6}.\\ &&&\text{3. As } \textcolor{colorprop}{6} \ge 5, \text{ the target digit is rounded up: } \textcolor{colordef}{3}+1 = \textcolor{olive}{4}.\\ &\textcolor{olive}{4}\textcolor{purple}{00} && \text{4. Subsequent digits are replaced with zeros.}\end{aligned}$$ Therefore, $365$ rounded to the nearest hundred is $400$.

Skills to practice

Rounding to the Nearest Ten
Rounding to the Nearest Hundred
Rounding to the Nearest Thousand

VI Going further

Skills to practice

Solving Problems with 6-Digit Numbers
Problems
Rounding and Bounding in Problems
Quiz