Primary & Middle School Mathematics · Grades 1–9

1Counting 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)

Touch each object once, saying the numbers in order: one, two, three, …;
do not skip any object, do not touch one twice (move the counted ones aside, or cross them out);
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.

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.)

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

If some objects of one collection are left without a partner, that collection has more;
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. — 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

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. Write in words: $4$ ; $15$ ; $18$ .

Solution

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

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

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

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

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

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

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

Solution of Exercise 1.9.

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