Year 7 Maths Integers

Subtracting Integers

Q: Why is subtracting a negative the same as adding?

Follow the pattern 8 - 1 = 7, 8 - 0 = 8, 8 - (-1) = 9: taking away less makes the answer bigger.

Q: Is -3 - 2 equal to 5?

No. There is no double sign. Start at -3 and jump 2 left: -5.

Q: How do you find the difference between a positive and a negative?

Higher minus lower. 21 - (-6) = 27.

Q: How do you find a missing number taken away?

Start minus answer. In -2 - □ = 5, the missing number is -2 - 5 = -7.

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Theory

Subtracting a positive jumps left on a number line. Subtracting a negative is the same as adding its opposite, so it jumps right. This Year 7 maths page shows why \(7 - (-3) = 10\), how to change double signs, how to find a missing number and how to find differences such as temperature and time zone gaps.

Subtracting a positive moves left, even past zero: \(2 - 8 = 2 - 2 - 6 = -6\).

Subtracting a negative is the same as adding its opposite: \(-(-3) = +3\), so \(7 - (-3) = 7 + 3 = 10\).

Why? Follow the pattern. Each time the number taken away goes down by \(1\), the answer goes up by \(1\): \(8 - 1 = 7\), \(8 - 0 = 8\), \(8 - (-1) = 9\).

The difference between two values is higher minus lower, even across zero. A town at \(21^\circ\text{C}\) is \(21 - (-6) = 27^\circ\text{C}\) warmer than a town at \(-6^\circ\text{C}\).

\(2 - 6 = -4\): taking away \(6\) jumps \(6\) left.

\(-3 - (-5) = -3 + 5 = 2\): taking away \(-5\) jumps \(5\) right.

Change each double sign, then work from left to right:

Double sign	Becomes	Example
\(+\,(+)\)	\(+\)	\(2 + (+5) = 2 + 5\)
\(+\,(-)\)	\(-\)	\(2 + (-5) = 2 - 5\)
\(-\,(+)\)	\(-\)	\(2 - (+5) = 2 - 5\)
\(-\,(-)\)	\(+\)	\(2 - (-5) = 2 + 5\)

\[ a - (-b) = a + b \]

a - (- b) = a + b

\[ \text{difference} = \text{higher} - \text{lower} \]

difference = higher - lower

          Missing numbers. The number taken away \(=\) start \(-\) answer. For a missing start, add back: in \(\square - (-3) = 1\), \(\square = 1 + (-3) = -2\).
        

          Change only the double sign. \(-9 - (-4) = -9 + 4 = -5\). The first number keeps its sign.
        

How to subtract integers

Change every double sign: \(-(-)\) becomes \(+\) and \(-(+)\) becomes \(-\).
Jump on a number line, or use the adding rules.
Work left to right when there are three or more numbers.

How to find a difference

Write higher minus lower.
Change the double sign if the lower value is negative.
Add the sizes when the values are on opposite sides of zero.

How to find a missing number

Simplify the known part first.
Missing number taken away: start minus answer.
Check by putting the number back in.

Example 1 — Past zero

Use a number line to work out \(1 - 5\).

Solution

\(1 - 5\)	\(=\)	\(1 - 1 - 4\)
\(\)	\(=\)	\(-4\)

1 - 5 = 1 - 1 - 4 = - 4

\(-4\)

Example 2 — Double signs

Work out
(i) \(9 - (-7)\)
(ii) \(-12 - (-5)\)
(iii) \(-6 - 3\)

Solution

\(9 - (-7)\)	\(=\)	\(9 + 7 = 16\)
\(-12 - (-5)\)	\(=\)	\(-12 + 5 = -7\)
\(-6 - 3\)	\(=\)	\(-9\)

9 - (- 7) = 16, - 12 - (- 5) = - 7, - 6 - 3 = - 9

(i) \(16\)
(ii) \(-7\)
(iii) \(-9\) (no double sign: just jump left)

Example 3 — Continue the pattern

\(4 - 2 = 2\), \(4 - 1 = 3\), \(4 - 0 = 4\). Continue the pattern to find \(4 - (-1)\) and \(4 - (-3)\).

Solution

\(4 - (-1)\)	\(=\)	\(5\)
\(4 - (-3)\)	\(=\)	\(7\)

4 - (- 1) = 5, 4 - (- 3) = 7

Each step the answer goes up by \(1\), so \(4 - (-2) = 6\) comes between them.

Example 4 — Three numbers

Work out \(-7 - (-15) - 4\).

Solution

\(-7 - (-15) - 4\)	\(=\)	\(-7 + 15 - 4\)
\(\)	\(=\)	\(8 - 4\)
\(\)	\(=\)	\(4\)

- 7 + 15 - 4 = 4

\(4\)

Example 5 — Temperature difference

On a winter morning it is \(-6^\circ\text{C}\) on a mountain and \(21^\circ\text{C}\) on the coast. How much warmer is the coast?

Solution

\(21 - (-6)\)	\(=\)	\(21 + 6\)
\(\)	\(=\)	\(27\)

21 - (- 6) = 21 + 6 = 27

Higher minus lower: the coast is \(27^\circ\text{C}\) warmer.

Example 6 — The card taken away

Kim’s counter is at \(-2\). She takes away the number on a card and lands on \(5\). What number is on the card?

Solution

\(-2 - \square = 5\). She moved right, so the card is negative. The number taken away \(=\) start \(-\) answer.

\(-2 - 5\)

\(=\)

\(-7\)

- 2 - 5 = - 7

Check: \(-2 - (-7) = -2 + 7 = 5\). The card shows \(-7\).

Example 7 — Time zone difference

City A has a UTC offset of \(+9\) and city B has an offset of \(-3\). How many hours is city A ahead of city B?

Solution

\(9 - (-3)\)	\(=\)	\(9 + 3\)
\(\)	\(=\)	\(12\)

9 - (- 3) = 12

City A is \(12\) hours ahead.

Common pitfalls

Two negatives do not always make a positive. \(-3 - 2 = -5\). There is no double sign here, so just jump left.

Changing the wrong sign. \(-9 - (-4) = -9 + 4 = -5\). Only the double sign changes.

Swapping the order. \(4 - 9 = -5\) but \(9 - 4 = 5\). A subtraction cannot be turned around.

Subtracting sizes across zero. From \(-6\) to \(21\) is \(21 - (-6) = 27\), not \(15\).

Frequently asked questions

Why is subtracting a negative the same as adding?

Follow the pattern \(8 - 1 = 7\), \(8 - 0 = 8\), \(8 - (-1) = 9\): taking away less makes the answer bigger.

What does minus a negative become?

A plus. \(7 - (-3) = 7 + 3 = 10\).

Is -3 - 2 equal to 5?

No. There is no double sign. Start at \(-3\) and jump \(2\) left: \(-5\).

How do you find the difference between a positive and a negative?

Higher minus lower. \(21 - (-6) = 27\).

How do you find a missing number taken away?

Start minus answer. In \(-2 - \square = 5\), the missing number is \(-2 - 5 = -7\).

Subtracting Integers

📖 Theory

How to subtract integers

How to find a difference

How to find a missing number

Common pitfalls

Frequently asked questions

Theory