Mathematics · Book 1 · Grades 1–9

Primary & Middle School Mathematics

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=57 - 2 = 5

and read: “seven minus two equals five”.

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

Example 3.2 (Counting back)

For a small take-away, count backward on the line: 11−311 - 3: say “eleven”, then three steps back — “ten, nine, eight”. So 11−3=811 - 3 = 8.

3.2 How many more?

Example 3.3 (The difference)

“Nina has 99 marbles and Tom has 66. 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=3.9 - 6 = 3 .

Nina has 33 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 12−912 - 9, start at 99 and climb to 1212: “ten, eleven, twelve” — three steps. So 12−9=312 - 9 = 3. (Climb through 1010 for bigger gaps: 14−814 - 8: from 88 to 1010 is 22, from 1010 to 1414 is 44: difference 66.)

14 - 8 by counting up: from 8 to 14 there are 2 + 4 = 6 steps.
14−814 - 8 by counting up: from 88 to 1414 there are 2+4=62 + 4 = 6 steps.

3.3 Addition and subtraction are twins

Example 3.5 (Fact families)

The numbers 33, 44 and 77 form a little family:

3+4=7,4+3=7,7−4=3,7−3=4.3 + 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+4, doing −4-4 brings you back.

Example 3.6 (Checking)

Computed 13−5=813 - 5 = 8? Check with the twin addition: 8+5=138 + 5 = 13. ✓ A subtraction is right exactly when adding back works.

Example 3.7 (Which operation?)

“66 birds on the wire, 44 fly away”: take away, 6−4=26 - 4 = 2. “66 birds, then 44 more arrive”: put together, 6+4=106 + 4 = 10. “66 sparrows and 44 robins: how many more sparrows?”: difference, 6−4=26 - 4 = 2. Draw the story when unsure!

3.4 Exercises

Exercise 3.1 ★

Draw, cross out, and compute: 6−26 - 2; 8−58 - 5; 10−410 - 4.

Solution

Solution of Exercise 3.1.

6−2=46 - 2 = 4; 8−5=38 - 5 = 3; 10−4=610 - 4 = 6.

Exercise 3.2 ★

Count backward to compute: 9−39 - 3; 12−212 - 2; 15−415 - 4.

Solution

Solution of Exercise 3.2.

9−3=69 - 3 = 6; 12−2=1012 - 2 = 10; 15−4=1115 - 4 = 11.

Exercise 3.3 ★

Count up to find the difference: 11−811 - 8; 13−913 - 9; 16−1216 - 12.

Solution

Solution of Exercise 3.3.

11−8=311 - 8 = 3 (from 88 up to 1111); 13−9=413 - 9 = 4; 16−12=416 - 12 = 4.

Exercise 3.4 ★

Write the four facts of the family of 22, 66 and 88.

Solution

Solution of Exercise 3.4.

2+6=82 + 6 = 8, 6+2=86 + 2 = 8, 8−6=28 - 6 = 2, 8−2=68 - 2 = 6.

Exercise 3.5 ★

Compute, then check with the twin addition: 14−614 - 6; 17−917 - 9.

Solution

Solution of Exercise 3.5.

14−6=814 - 6 = 8 (check: 8+6=148 + 6 = 14); 17−9=817 - 9 = 8 (check: 8+9=178 + 9 = 17).

Exercise 3.6 ★

Complete the holes:

10−  ?  =7,  ?  −5=5,12−  ?  =12.10 - \;?\; = 7, \qquad \;?\; - 5 = 5, \qquad 12 - \;?\; = 12 .
Solution

Solution of Exercise 3.6.

10−3=710 - 3 = 7; 10−5=510 - 5 = 5; 12−0=1212 - 0 = 12.

Exercise 3.7 ★

“There are 1515 cherries in the bowl. The children eat 88. How many are left?” Write the subtraction and the answer sentence.

Solution

Solution of Exercise 3.7.

15−8=715 - 8 = 7. There are 77 cherries left.

Exercise 3.8 ★

For each story, choose ++ or −-, then compute: “99 ducks on the pond, 55 swim away”; “99 ducks, 55 geese arrive”; “Ali is 99, his sister is 55: how many years older is Ali?”.

Solution

Solution of Exercise 3.8.

“55 swim away”: 9−5=49 - 5 = 4 ducks. “55 geese arrive”: 9+5=149 + 5 = 14 birds. “How many years older”: 9−5=49 - 5 = 4 years.

Exercise 3.9 ★★

Mother duck had 1212 eggs. Some hatched; now 12−?12 - ? eggs remain and she counts 77 unhatched eggs. How many eggs hatched? (Which fact family is this?)

Solution

Solution of Exercise 3.9.

Family of 1212, 77 and 55: 12−7=512 - 7 = 5 eggs hatched (and 5+7=125 + 7 = 12 checks it).

Terms defined in this chapter

See all 395 terms in the glossary