Primary & Middle School Mathematics · Grades 1–9

4Numbers to 100

Counting a big pile one by one takes forever. The great trick: bundle the objects in packs of ten, and count the packs. This is how the numbers up to $100$ are built — and why they are written with two digits.

4.1 Bundles of ten

Definition 4.1 (Tens and units)

A ten is a bundle of ten units. The number $34$ means $3$ tens and $4$ units:

34 = 10 + 10 + 10 + 4 .

The left digit counts the bundles, the right digit the loose units.

Thirty-four sticks: three bundles of ten (each tied with a band) and four loose sticks. No need to count to 34 one by one! — Thirty-four sticks: three bundles of ten (each tied with a band) and four loose sticks. No need to count to $34$ one by one!

Example 4.2 (Reading the numbers)

$20$ is two tens (twenty), $30$ three tens (thirty) … up to $90$ (ninety) and $100$ (one hundred: ten tens!). In between: $47$ reads “forty-seven” — forty for the $4$ tens, seven for the units.

The place table of 34: the 3 sits in the tens place, the 4 in the units place. This little table grows a new column every year! — The place table of $34$ : the $3$ sits in the tens place, the $4$ in the units place. This little table grows a new column every year!

Example 4.3 (Same digits, different numbers)

$27$ and $72$ use the same digits but are very different: $27$ is $2$ tens and $7$ units; $72$ is $7$ tens and $2$ units — much bigger! The place of a digit decides what it counts.

4.2 The hundred grid

The hundred grid: each row is one ten. One step right is +1; one step down is +10 (from the red 44 straight down to the blue 54). — The hundred grid: each row is one ten. One step right is $+1$ ; one step *down* is $+10$ (from the red $44$ straight down to the blue $54$ ).

Method 4.4 (Moving on the grid)

$+1$ / $-1$ : one step right / left;
$+10$ / $-10$ : one step down / up — only the tens digit changes: $37 + 10 = 47$ ;
counting by tens: $10, 20, 30, \dots$ — follow the last column of the grid straight down.

Example 4.5 (Counting by 2s and 5s)

By $2$ s: $2, 4, 6, 8, 10, 12, \dots$ (the even numbers). By $5$ s: $5, 10, 15, 20, 25, \dots$ — they end in $5$ or $0$ and make a nice pattern on the grid. Skip counting prepares multiplication (Chapter 10).

4.3 Comparing bigger numbers

Method 4.6 (Comparing with tens)

More tens wins: $61 > 58$ because $6$ tens beat $5$ tens — even though $8$ is more than $1$ ! Only when the tens are equal do the units decide: $43 < 47$ .

Example 4.7

Order $35$ , $53$ , $29$ : tens first ( $2$ , $3$ , $5$ ):

29 < 35 < 53 .

4.4 Exercises

Exercise 4.1 ★

How many tens and units in: $26$ ; $40$ ; $73$ ; $99$ ?

Solution

Solution of Exercise 4.1.

$26$ : $2$ tens, $6$ units. $40$ : $4$ tens, $0$ units. $73$ : $7$ tens, $3$ units. $99$ : $9$ tens, $9$ units.

Exercise 4.2 ★

Write the number: $5$ tens and $2$ units; $8$ tens; $1$ ten and $9$ units; $10$ tens.

Solution

Solution of Exercise 4.2.

$52$ ; $80$ ; $19$ ; $100$ .

Exercise 4.3 ★

Write in digits: sixty-three; forty; eighty-five; one hundred.

Solution

Solution of Exercise 4.3.

$63$ ; $40$ ; $85$ ; $100$ .

Exercise 4.4 ★

Compute by moving on the hundred grid: $23 + 10$ ; $67 + 10$ ; $45 - 10$ ; $80 - 10$ .

Solution

Solution of Exercise 4.4.

$23 + 10 = 33$ ; $67 + 10 = 77$ ; $45 - 10 = 35$ ; $80 - 10 = 70$ .

Exercise 4.5 ★

Continue: $10, 20, 30, ?, ?, ?$ . $5, 10, 15, ?, ?, ?$ . $12, 14, 16, ?, ?, ?$ .

Solution

Solution of Exercise 4.5.

$10, 20, 30, 40, 50, 60$ . $5, 10, 15, 20, 25, 30$ . $12, 14, 16, 18, 20, 22$ .

Exercise 4.6 ★

Copy and complete with $<$ or $>$ :

34 \;?\; 43, \qquad 58 \;?\; 61, \qquad 70 \;?\; 69, \qquad 99 \;?\; 100 .

Solution

Solution of Exercise 4.6.

$34 < 43$ ; $58 < 61$ ; $70 > 69$ ; $99 < 100$ .

Exercise 4.7 ★

Order from smallest to biggest: $46$ ; $19$ ; $64$ ; $91$ ; $41$ .

Solution

Solution of Exercise 4.7.

$19 < 41 < 46 < 64 < 91$ .

Exercise 4.8 ★

A gardener bundles $52$ flowers in bunches of ten. How many full bunches, and how many loose flowers?

Solution

Solution of Exercise 4.8.

$52$ is $5$ tens and $2$ units: $5$ full bunches and $2$ loose flowers.

Exercise 4.9 ★★

On the hundred grid, start at $36$ ; go two steps down and one step left. Where do you land? Which computation did you just do?

Solution

Solution of Exercise 4.9.

From $36$ , two steps down is $+20$ ( $\to 56$ ), one step left is $-1$ ( $\to 55$ ). You land on $55$ . The computation is $36 + 20 - 1$ , that is $36 + 19$ .