Primary & Middle School Mathematics · Grades 1–9

2Addition: First Steps

Three red marbles and two blue marbles: together, how many? Putting together and counting the whole is adding — the first operation, with its two famous signs $+$ and $=$ .

2.1 Putting together

Definition 2.1 (Addition)

Adding means putting together and counting everything. We write

3 + 2 = 5

and read: “three plus two equals five”. The answer of an addition is called the sum.

An addition in pictures: the two groups together make one group of five.

Example 2.2 (The order does not matter)

$2 + 3$ and $3 + 2$ put together the same marbles: both make $5$ . To compute $2 + 9$ , it is faster to start from the bigger number: say “nine”, then count on two — “ten, eleven”. So $2 + 9 = 11$ .

2.2 The ten friends

Example 2.3 (Complements to 10)

Two numbers that make $10$ together are “ten friends”:

1 + 9, \quad 2 + 8, \quad 3 + 7, \quad 4 + 6, \quad 5 + 5 .

Ten fingers show them all: raise $3$ fingers, and the $7$ folded ones are the friend. Knowing the ten friends by heart makes many additions fast.

The ten-frame shows every pair of ten friends at a glance: what is filled and what is missing always make 10 together. — The ten-frame shows every pair of ten friends at a glance: what is filled and what is missing always make $10$ together.

Method 2.4 (Crossing the ten)

To compute $8 + 5$ :

take from the $5$ what the $8$ needs to reach $10$ : that is $2$ (its ten friend);
$8 + 2 = 10$ , and $3$ remains from the $5$ ;
$10 + 3 = 13$ . So $8 + 5 = 13$ .

Passing through $10$ turns a hard sum into two easy ones.

8 + 5 on a ten-frame: the 8 blue counters take 2 of the 5 red ones to fill the frame — one full ten and 3 more, so 13. — $8 + 5$ on a ten-frame: the $8$ blue counters take $2$ of the $5$ red ones to fill the frame — one full ten and $3$ more, so $13$ .

8 + 5 on the number line: hop to 10 first, then the rest. — $8 + 5$ on the number line: hop to $10$ first, then the rest.

Example 2.5 (Doubles)

The doubles are worth learning by heart:

1+1=2, \quad 2+2=4, \quad 3+3=6, \quad 4+4=8, \quad 5+5=10,

6+6=12, \quad 7+7=14, \quad 8+8=16, \quad 9+9=18, \quad 10+10=20 .

Near-doubles come free: $6 + 7$ is one more than $6 + 6$ , so $13$ .

2.3 Little problems

Example 2.6

“Lea has $7$ stickers. She gets $6$ more. How many now?” Getting more means adding: $7 + 6 = 13$ (through ten: $7 + 3 = 10$ , then $+3$ ). Lea has $13$ stickers. Always finish with a little sentence!

2.4 Exercises

Exercise 2.1 ★

Draw the marbles and compute: $4 + 3$ ; $5 + 2$ ; $6 + 4$ .

Solution

Solution of Exercise 2.1.

$4 + 3 = 7$ ; $5 + 2 = 7$ ; $6 + 4 = 10$ .

Exercise 2.2 ★

Compute by starting from the bigger number: $3 + 8$ ; $2 + 12$ ; $4 + 9$ .

Solution

Solution of Exercise 2.2.

$3 + 8 = 11$ (start at $8$ , count on $3$ ); $2 + 12 = 14$ ; $4 + 9 = 13$ .

Exercise 2.3 ★

Give the ten friend of: $7$ ; $4$ ; $9$ ; $5$ ; $1$ .

Solution

Solution of Exercise 2.3.

Ten friends: $7 \to 3$ ; $4 \to 6$ ; $9 \to 1$ ; $5 \to 5$ ; $1 \to 9$ .

Exercise 2.4 ★

Compute through the ten (Method 2.4): $9 + 4$ ; $8 + 7$ ; $6 + 5$ .

Solution

Solution of Exercise 2.4.

$9 + 4 = 13$ ( $9 + 1 = 10$ , then $+3$ ); $8 + 7 = 15$ ( $8 + 2 = 10$ , then $+5$ ); $6 + 5 = 11$ ( $6 + 4 = 10$ , then $+1$ ).

Exercise 2.5 ★

Recite the doubles, then compute using them: $7 + 8$ ; $5 + 6$ ; $9 + 8$ .

Solution

Solution of Exercise 2.5.

$7 + 8 = 15$ (one more than $7 + 7 = 14$ ); $5 + 6 = 11$ (one more than $5 + 5$ ); $9 + 8 = 17$ (one less than $9 + 9 = 18$ ).

Exercise 2.6 ★

Complete the holes:

6 + \;?\; = 10, \qquad \;?\; + 5 = 12, \qquad 10 + \;?\; = 17 .

Solution

Solution of Exercise 2.6.

$6 + 4 = 10$ ; $7 + 5 = 12$ ; $10 + 7 = 17$ .

Exercise 2.7 ★

“In the bus there are $8$ children; $5$ grown-ups climb in. How many people are in the bus?” Write the addition and the answer sentence.

Solution

Solution of Exercise 2.7.

$8 + 5 = 13$ . There are $13$ people in the bus.

Exercise 2.8 ★

Tom has $6$ red cars and $7$ blue cars. Mia has $10$ cars. Who has more cars?

Solution

Solution of Exercise 2.8.

Tom has $6 + 7 = 13$ cars, Mia has $10$ : Tom has more.

Exercise 2.9 ★★

Three dice show $4$ , $6$ and $5$ . What is the total? (Add two dice first — choose the two that make a ten friend!)

Solution

Solution of Exercise 2.9.

Add the ten friends $4$ and $6$ first: $4 + 6 = 10$ , then $10 + 5 = 15$ . The total is $15$ .