Primary & Middle School Mathematics · Grades 1–9

3Subtraction: 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.

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, 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 $6$ .)

14 - 8 by counting up: from 8 to 14 there are 2 + 4 = 6 steps. — $14 - 8$ 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, $6 - 4 = 2$ . Draw the story when unsure!

3.4 Exercises

Exercise 3.1 ★

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

Solution

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

Solution of Exercise 3.2.

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

Exercise 3.3 ★

Count up to find the difference: $11 - 8$ ; $13 - 9$ ; $16 - 12$ .

Solution

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

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

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

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

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

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

Solution of Exercise 3.9.

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