---
title: "Numbers to 100"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 4
exercises: 9
source: https://one-course.com/books/math/1/en/chapter/4-numbers-to-100
---

# Chapter 4 — Numbers 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!](https://one-course.com/images/onecourse/chapters/math-1/g1-tens/fig-83c52fa834c5.svg)

*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!](https://one-course.com/images/onecourse/chapters/math-1/g1-tens/fig-77b3f12e78f5.svg)

*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).](https://one-course.com/images/onecourse/chapters/math-1/g1-tens/fig-828a889de4e0.svg)

*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. $+1$ / $-1$ : one step right / left;
2. $+10$ / $-10$ : one step down / up — only the tens digit changes: $37 + 10 = 47$ ;
3. 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](https://one-course.com/books/math/1/en/chapter/10-multiplication-first-steps#ch-g2-mult)).

## 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 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 of Exercise 4.2.**

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

**Exercise 4.3 ★.**

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

**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 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 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 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 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 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 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 $36 + 19$.