---
title: "Counting to 20"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 1
exercises: 9
source: https://one-course.com/books/math/1/en/chapter/1-counting-to-20
---

# Chapter 1 — Counting to 20

How many marbles? Counting answers the very first question of mathematics. In this chapter we count small collections, write the numbers from $0$ to $20$, and compare: who has more?

## 1.1 Counting objects

**Method 1.1 (How to count).**

1. Touch each object once, saying the numbers in order: one, two, three, …;
2. do not skip any object, do not touch one twice (move the counted ones aside, or cross them out);
3. the *last* number said is how many there are.

The order of touching does not matter: counted left to right or right to left, $7$ marbles are $7$ marbles.

![Counting: one number for each marble, and the last number is the answer.](https://one-course.com/images/onecourse/chapters/math-1/g1-counting/fig-50975f87768f.svg)

*Counting: one number for each marble, and the last number is the answer.*

**Definition 1.2 (Zero).**

When there is nothing to count — an empty box — the number is *zero*, written $0$.

## 1.2 The numbers up to 20

**Example 1.3 (Reading and writing).**

The numbers from $0$ to $20$, in order:

$$
0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10,
$$

$$
11,\ 12,\ 13,\ 14,\ 15,\ 16,\ 17,\ 18,\ 19,\ 20 .
$$

After $9$, the numbers use *two* digits: $13$ is a ten and $3$ more; $20$ is two tens. (Tens take the stage in [Chapter 4](https://one-course.com/books/math/1/en/chapter/4-numbers-to-100#ch-g1-tens).)

![The number line from 0 to 20: every number has its place. Walking right, the numbers grow. The dots mark 7 and 13.](https://one-course.com/images/onecourse/chapters/math-1/g1-counting/fig-941744e1d1c3.svg)

*The number line from $0$ to $20$: every number has its place. Walking right, the numbers grow. The dots mark $7$ and $13$.*

**Example 1.4 (Just before, just after).**

On the line, each number has two neighbors: just before $14$ comes $13$, just after comes $15$. Counting backward — $10$, $9$, $8$, … — is walking left.

## 1.3 Comparing

**Method 1.5 (Who has more?).**

To compare two collections, make pairs: one from each collection, side by side.

1. If some objects of one collection are left without a partner, that collection has *more* ;
2. if nothing is left over on either side, there are *as many* on both.

With numbers, no pairing is needed: the one further right on the number line is the bigger — pairing and the line always agree. We write $12 > 9$ ($12$ is more) or $9 < 12$ ($9$ is less).

![Comparing by pairing: 7 blue marbles against 5 red ones — two blues (circled) are left over, so 7 is more than 5.](https://one-course.com/images/onecourse/chapters/math-1/g1-counting/fig-c79c4d02456c.svg)

*Comparing by pairing: $7$ blue marbles against $5$ red ones — two blues (circled) are left over, so $7$ is more than $5$.*

**Example 1.6 (First, second, third).**

In a race, the runners arrive in order: the *first*, the *second*, the *third*, the *fourth* … These words tell the *position*, not the amount: the third runner is one runner, at place number $3$.

## 1.4 Exercises

**Exercise 1.1 ★.**

Count the letters of your first name. Count the fingers of two hands. Count the legs of two cats.

**Solution of Exercise 1.1.**

Answers depend on the name; two hands have $10$ fingers; two cats have $8$ legs.

**Exercise 1.2 ★.**

Write in digits: seven; twelve; twenty; [zero](#def-g1-counting-zero). Write in words: $4$; $15$; $18$.

**Solution of Exercise 1.2.**

$7$; $12$; $20$; $0$. In words: four; fifteen; eighteen.

**Exercise 1.3 ★.**

Continue the counting: $8,\ 9,\ 10,\ ?,\ ?,\ ?$. Count backward: $15,\ 14,\ 13,\ ?,\ ?,\ ?$.

**Solution of Exercise 1.3.**

$8, 9, 10, 11, 12, 13$. Backward: $15, 14, 13, 12, 11, 10$.

**Exercise 1.4 ★.**

Which number comes just before $10$? Just after $16$? Between $12$ and $14$?

**Solution of Exercise 1.4.**

Just before $10$: $9$. Just after $16$: $17$. Between $12$ and $14$: $13$.

**Exercise 1.5 ★.**

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

$$
6 \;?\; 9, \qquad
14 \;?\; 8, \qquad
11 \;?\; 13, \qquad
20 \;?\; 19 .
$$

**Solution of Exercise 1.5.**

$6 < 9$; $14 > 8$; $11 < 13$; $20 > 19$.

**Exercise 1.6 ★.**

Draw $6$ circles and $8$ crosses. Pair them up: which are more, and by how many?

**Solution of Exercise 1.6.**

Pairing $6$ circles with $8$ crosses leaves $2$ crosses without a partner: the crosses are more, by $2$.

**Exercise 1.7 ★.**

Order from smallest to biggest: $12$; $7$; $17$; $2$; $10$.

**Solution of Exercise 1.7.**

$2 < 7 < 10 < 12 < 17$.

**Exercise 1.8 ★.**

Five children stand in line: Ana, Bob, Cleo, Dan, Emma. Who is second? Who is fourth? At which place is Dan?

**Solution of Exercise 1.8.**

Second: Bob. Fourth: Dan. Dan is at place $4$.

**Exercise 1.9 ★★.**

I am a number between $10$ and $20$. Counting by the line, I am three steps after $13$. Who am I? And who is three steps *before* $13$?

**Solution of Exercise 1.9.**

Three steps after $13$: $16$. Three steps before $13$: $10$.