I Place Value

Dr. Tariel's birthday cake holds many candles.

Counting them one by one would take a long time. We look for a faster way.
Since we have 10 fingers, we make groups of ten.

There are 4 groups of ten candles and 2 single candles left over.
4 tens and 2 ones make the number 42. Writing the number is quicker than drawing every candle.

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We can group 10 single cubes (ones) to make 1 ten-stick (ten). = $10$ ones $= 1$ ten =
To find out how many tens and ones are in a number, we can use a place value table. For the number 42:

Tens Ones

4 2

The table shows us that the number 42 is made of 4 tens and 2 ones.

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Definition — Digits

The symbols we use to write numbers are called digits. There are ten digits we use to write every number.

Word	Digit	Cubes
zero	0
one	1
two	2
three	3
four	4
five	5
six	6
seven	7
eight	8
nine	9

Definition — Representing 2-Digit Numbers

A 2-digit number is written with two digits: the tens digit followed by the ones digit. The 2-digit numbers go from 10 to 99. The place of a digit in a number tells us its value. We can show a number in many ways:

With digits: $$42$$
As tens and ones: $$ 4\text{ tens and } 2\text{ ones }$$
As a sum: $$40 + 2$$
With words: forty-two
In a place value chart:

Tens Ones

4 2
With cubes:

The Place Decides

Here are two numbers written with the same two digits, a $4$ and a $2$ — only their places have been swapped.

Tens	Ones
4	2

$\qquad$

Tens	Ones
2	4

On the left, $42$: four ten-sticks and two single cubes. On the right, $24$: two ten-sticks and four single cubes. Look at the cubes — there are many more on the left. The digits are the same. Only their place has changed, and the number is not the same at all. In a number, a digit does not carry its value on its own: it is its place that says whether it counts tens or ones.

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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.

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Example

A collection has $3$ tens and $17$ ones. Write it with a single digit for the tens and a single digit for the ones, then write the number.

$\quad+ \quad $

Answer

$17$ ones are more than $9$: ten of them make one more ten-stick. $$\begin{aligned}[t]3\text{ tens }+ \textcolor{colorprop}{17\text{ ones}} &= 3\text{ tens }+ \textcolor{colorprop}{1\text{ ten }+ 7\text{ ones}} &&\text{(ten ones make one ten)}\\ &= 4\text{ tens }+ 7\text{ ones}\\ &= 47\end{aligned}$$

$\quad+ \quad $

$\quad =\quad$

The collection is the number $47$: $4$ tens and $7$ ones.

Skills to practice

Counting Fruits in a Table
Counting Cubes in a Table
Counting Ten-Frames in a Table
Finding the Digit
Writing Numbers From Tens and Ones
Writing Numbers From Sums
Breaking Down Numbers into Tens and Ones
Drawing Tens and Ones
Grouping by Tens
Converting Ones into Tens and Ones
Regrouping Ones into Tens and Adding the Extra Tens

II Number Names

We now know how to write a number with digits: $42$ is $4$ tens and $2$ ones. But a number also has a name, the word we say out loud. Sometimes the name follows the digits, and sometimes it does not. Learning to go from the name to the digits — and back — is the second half of knowing a number.

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Definition — The Names of the Tens

Each whole ten has its own name.

Digits	Name	Tens
$10$	ten	$1$ ten
$20$	twenty	$2$ tens
$30$	thirty	$3$ tens
$40$	forty	$4$ tens
$50$	fifty	$5$ tens
$60$	sixty	$6$ tens
$70$	seventy	$7$ tens
$80$	eighty	$8$ tens
$90$	ninety	$9$ tens

From twenty on, each name ends in -ty, which means tens. From sixty on, the name even starts with the digit itself: sixty, seventy, eighty, ninety.

Definition — The Names from Eleven to Nineteen

Between ten and twenty, the names do not all work the same way.

$11$	$12$	$13$	$14$	$15$
eleven	twelve	thirteen	fourteen	fifteen

$16$	$17$	$18$	$19$
sixteen	seventeen	eighteen	nineteen

Eleven and twelve say nothing about their digits: they must be learnt by heart. From thirteen on, the name says the ones digit (thir- for three, fif- for five), then -teen for the ten: fourteen, sixteen, nineteen.

Method — From the Name to the Digits

Apart from the whole tens, which are one word (twenty, thirty…), the name of a number from $21$ to $99$ has two parts.

The first part gives the tens digit: forty-two $\rightarrow$ $4$ tens.
The second part gives the ones digit: forty-two $\rightarrow$ $2$ ones.

$$\text{forty-two} \;\rightarrow\; 4 \text{ tens and } 2 \text{ ones} \;\rightarrow\; 42$$ The other way round, we read the table of tens backwards: to say $58$, we look for $50$ — fifty — then we add the ones digit: fifty-eight.

Example

Write thirty-five with digits.

Answer

Thirty gives $3$ tens, five gives $5$ ones. $$\text{thirty-five} = 30 + 5 = 35$$

Tens	Ones
3	5

Method — The Names That Trick Us

Careful: some names do not follow the digits. The numbers from $13$ to $19$ say the ones first and the ten last: $$\underbrace{\text{four}}_{4} \; \underbrace{\text{teen}}_{\text{ten}} \qquad \text{but we write} \qquad 14 \;=\; 1 \text{ ten and } 4 \text{ ones}$$ We hear $4$ then ten, and we write $1$ then $4$. So fourteen is $14$, not $41$.

Example

Write fourteen with digits, then say how many tens and ones it has.

Answer

The name says four, then teen. But a teen number always has one ten: $$\text{fourteen} = 10 + 4 = 14$$ So $14$ has $1$ ten and $4$ ones.

Skills to practice

Choosing the Right Name
Writing Numbers from Words
Writing Numbers in Words

III On the Number Line

Discover

A number line shows numbers in order, like a long ruler. The numbers get bigger as we move to the right.
To count big numbers faster, we can take big jumps of 10. This is called counting by tens. We say: “ten, twenty, thirty…”

One more jump of ten after $40$: which number do we reach?

Answer

Four jumps of ten reach $40$. One more jump of ten reaches $50$: $$40 + 10 = 50$$

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.

A Number Is a Distance from Zero

The number line starts at $0$: this point is its origin. From there, the line goes on to the right as far as we like — we only ever draw a part of it. A point placed on a mark of the line has a number. This number says how far the point is from $0$, counted in steps of one. The point below is $7$ steps of one away from $0$: it is the point $7$.

For a point far from $0$, we do not count every step of one: we count steps of ten, then steps of one. The point $43$ is $4$ steps of ten and $3$ steps of one away from $0$ — $4$ tens and $3$ ones, the same distance counted faster.

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Method — Placing a Number by Steps of One

On a number line marked in ones, we move on by $1$ from one mark to the next. To place a number:

Start from the mark of its tens: $52$ has $5$ tens, so we start from the mark $50$. A number less than $10$ has $0$ tens: we start from $0$, the origin.
Move on by as many steps of one as it has ones: $2$ steps for $52$.
Check by reading the point back: the labelled ten and the steps made give the number again — $50$ and $2$ more make $52$. When the number has ones, the steps left before the next ten complete $10$: $2$ steps after $50$, $8$ steps before $60$, and $2 + 8 = 10$.

Example

Place $37$ on this part of the number line.

Answer

$37 = 30 + 7$: $3$ tens and $7$ ones. We start from the mark $30$ and move on $7$ steps of one — $31$, $32$, $33$, $34$, $35$, $36$, $37$.

Check by reading the point back: $30$ and $7$ more make $37$; and $7$ steps after $30$, $3$ steps before $40$: $7 + 3 = 10$.

Skills to practice

Finding 2-Digit Numbers
Placing a Number on the Number Line

IV Reaching One Hundred

Discover

The class built $10$ ten-sticks.

Count by tens: $10$, $20$, $\dots$ How many cubes are there in all?

Answer

$10$, $20$, $30$, $40$, $50$, $60$, $70$, $80$, $90$ — and one more ten. We need a new number: one hundred.

= $10$ tens = $100$ =

Definition — One Hundred

Right after $99$ comes a new number: one hundred. We write it $100$: a $1$ followed by two $0$s — the first number with three digits. Counting by tens, ten ten-sticks make one hundred.

Example

Where does $100$ sit on this line?

Answer

On the mark just after $90$: ten steps of ten from $0$.

Skills to practice

Counting to One Hundred

V Estimating a Quantity

Discover

Léa's jar is full of marbles.

Tom says “about $10$”, Léa says “about $40$” and Sam says “about $90$”. Without counting the marbles one by one, say who is the closest.

Answer

The $10$ circled marbles fill a small part of the jar. About four groups like them fill the whole jar: there are about $40$ marbles. Léa is the closest.
To check, the marbles are put in groups of ten: $4$ tens and $3$ ones, that is $43$ marbles. $43$ is close to $40$.

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

About how many buttons are there? Choose $10$, $30$ or $70$.

Answer

One group of ten is circled. About $3$ groups like it cover all the buttons: about $30$.
Check by tens: $2$ tens and $8$ ones, $28$ buttons. $28$ is close to $30$, and far from $10$ and from $70$.

Example

Are there more or fewer than $20$ cubes? More or fewer than $50$ cubes?

Answer

$2$ groups of ten make $20$: the cubes cover more than $2$ groups like the circled one, so there are probably more than $20$.
$5$ groups of ten make $50$: the cubes cover fewer than $5$ groups, so there are probably fewer than $50$.
The count confirms the estimate: $3$ tens and $4$ ones, $34$ cubes, between $20$ and $50$.

Skills to practice

Choosing the Reasonable Estimate
More or Fewer?
Checking an Estimate

VI Going further

Skills to practice

Completing Equations with Pictures
Problems
Quiz

Digits	Name	Tens
\(10\)	ten	\(1\) ten
\(20\)	twenty	\(2\) tens
\(30\)	thirty	\(3\) tens
\(40\)	forty	\(4\) tens
\(50\)	fifty	\(5\) tens
\(60\)	sixty	\(6\) tens
\(70\)	seventy	\(7\) tens
\(80\)	eighty	\(8\) tens
\(90\)	ninety	\(9\) tens

\(11\)	\(12\)	\(13\)	\(14\)	\(15\)
eleven	twelve	thirteen	fourteen	fifteen

\(16\)	\(17\)	\(18\)	\(19\)
sixteen	seventeen	eighteen	nineteen