---
title: "Subtraction: First Steps"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 3
exercises: 9
source: https://one-course.com/books/math/1/en/chapter/3-subtraction-first-steps
---

# Chapter 3 — Subtraction: First Steps

Seven balloons, two fly away: how many are left? Taking away is *subtracting*, written with the sign $-$. And subtraction has a second face: it also answers “how many more?”.

## 3.1 Taking away

**Definition 3.1 (Subtraction).**

*Subtracting* means taking away and counting what is left. We write

$$
7 - 2 = 5
$$

and read: “seven minus two equals five”.

![A subtraction in pictures: crossing out is taking away.](https://one-course.com/images/onecourse/chapters/math-1/g1-subtraction/fig-85e8009a3e0b.svg)

*A subtraction in pictures: crossing out is taking away.*

**Example 3.2 (Counting back).**

For a small take-away, count backward on the line: $11 - 3$: say “eleven”, then three steps back — “ten, nine, eight”. So $11 - 3 = 8$.

## 3.2 How many more?

**Example 3.3 (The difference).**

“Nina has $9$ marbles and Tom has $6$. How many more has Nina?” Nothing is taken away, yet it is a subtraction: pair up the marbles, and count Nina’s marbles without a partner:

$$
9 - 6 = 3 .
$$

Nina has $3$ more. The answer of a subtraction is called the *difference*.

**Method 3.4 (Counting up).**

To find a [difference](#ex-g1-subtraction-difference), counting *up* is often easiest: for $12 - 9$, start at $9$ and climb to $12$: “ten, eleven, twelve” — three steps. So $12 - 9 = 3$. (Climb through $10$ for bigger gaps: $14 - 8$: from $8$ to $10$ is $2$, from $10$ to $14$ is $4$: [difference](#ex-g1-subtraction-difference) $6$.)

![14 - 8 by counting up: from 8 to 14 there are 2 + 4 = 6 steps.](https://one-course.com/images/onecourse/chapters/math-1/g1-subtraction/fig-63d77103d255.svg)

*$14 - 8$ by counting up: from $8$ to $14$ there are $2 + 4 = 6$ steps.*

## 3.3 Addition and subtraction are twins

**Example 3.5 (Fact families).**

The numbers $3$, $4$ and $7$ form a little family:

$$
3 + 4 = 7, \qquad
4 + 3 = 7, \qquad
7 - 4 = 3, \qquad
7 - 3 = 4 .
$$

Knowing one fact of the family gives the three others for free. Subtraction undoes addition: after $+4$, doing $-4$ brings you back.

**Example 3.6 (Checking).**

Computed $13 - 5 = 8$? Check with the twin addition: $8 + 5 = 13$. ✓ A subtraction is right exactly when adding back works.

**Example 3.7 (Which operation?).**

“$6$ birds on the wire, $4$ fly away”: take away, $6 - 4 = 2$. “$6$ birds, then $4$ more arrive”: put together, $6 + 4 = 10$. “$6$ sparrows and $4$ robins: how many more sparrows?”: [difference](#ex-g1-subtraction-difference), $6 - 4 = 2$. Draw the story when unsure!

## 3.4 Exercises

**Exercise 3.1 ★.**

Draw, cross out, and compute: $6 - 2$; $8 - 5$; $10 - 4$.

**Solution of Exercise 3.1.**

$6 - 2 = 4$; $8 - 5 = 3$; $10 - 4 = 6$.

**Exercise 3.2 ★.**

Count backward to compute: $9 - 3$; $12 - 2$; $15 - 4$.

**Solution of Exercise 3.2.**

$9 - 3 = 6$; $12 - 2 = 10$; $15 - 4 = 11$.

**Exercise 3.3 ★.**

Count up to find the [difference](#ex-g1-subtraction-difference): $11 - 8$; $13 - 9$; $16 - 12$.

**Solution of Exercise 3.3.**

$11 - 8 = 3$ (from $8$ up to $11$); $13 - 9 = 4$; $16 - 12 = 4$.

**Exercise 3.4 ★.**

Write the four facts of the family of $2$, $6$ and $8$.

**Solution of Exercise 3.4.**

$2 + 6 = 8$, $6 + 2 = 8$, $8 - 6 = 2$, $8 - 2 = 6$.

**Exercise 3.5 ★.**

Compute, then check with the twin addition: $14 - 6$; $17 - 9$.

**Solution of Exercise 3.5.**

$14 - 6 = 8$ (check: $8 + 6 = 14$); $17 - 9 = 8$ (check: $8 + 9 = 17$).

**Exercise 3.6 ★.**

Complete the holes:

$$
10 - \;?\; = 7, \qquad
\;?\; - 5 = 5, \qquad
12 - \;?\; = 12 .
$$

**Solution of Exercise 3.6.**

$10 - 3 = 7$; $10 - 5 = 5$; $12 - 0 = 12$.

**Exercise 3.7 ★.**

“There are $15$ cherries in the bowl. The children eat $8$. How many are left?” Write the subtraction and the answer sentence.

**Solution of Exercise 3.7.**

$15 - 8 = 7$. There are $7$ cherries left.

**Exercise 3.8 ★.**

For each story, choose $+$ or $-$, then compute: “$9$ ducks on the pond, $5$ swim away”; “$9$ ducks, $5$ geese arrive”; “Ali is $9$, his sister is $5$: how many years older is Ali?”.

**Solution of Exercise 3.8.**

“$5$ swim away”: $9 - 5 = 4$ ducks. “$5$ geese arrive”: $9 + 5 = 14$ birds. “How many years older”: $9 - 5 = 4$ years.

**Exercise 3.9 ★★.**

Mother duck had $12$ eggs. Some hatched; now $12 - ?$ eggs remain and she counts $7$ unhatched eggs. How many eggs hatched? (Which fact family is this?)

**Solution of Exercise 3.9.**

Family of $12$, $7$ and $5$: $12 - 7 = 5$ eggs hatched (and $5 + 7 = 12$ checks it).