---
title: "Addition: First Steps"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 2
exercises: 9
source: https://one-course.com/books/math/1/en/chapter/2-addition-first-steps
---

# Chapter 2 — Addition: 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.](https://one-course.com/images/onecourse/chapters/math-1/g1-addition/fig-59aa18476292.svg)

*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.](https://one-course.com/images/onecourse/chapters/math-1/g1-addition/fig-97538174130f.svg)

*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 $8 + 5$:

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

Passing through $10$ turns a hard [sum](#def-g1-addition-def) 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.](https://one-course.com/images/onecourse/chapters/math-1/g1-addition/fig-cbcfc1a4c27e.svg)

*$8 + 5$ 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.](https://one-course.com/images/onecourse/chapters/math-1/g1-addition/fig-0351b9e15f27.svg)

*$8 + 5$ 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 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 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 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](#met-g1-addition-crossten)): $9 + 4$; $8 + 7$; $6 + 5$.

**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 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 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 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 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 of Exercise 2.9.**

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