Integers

Year 7 core

At the Year 7 level you should be able to:

place integers on a number line and order them from smallest to largest,
understand absolute value as distance from zero,
add and subtract integers, including when one or both are negative,
use integers to describe real-world situations like temperature, elevation, money, and floors.

Worked example 0 Real-world example: reading a weather map

Melbourne’s forecast shows overnight lows of $-2°$ C on Monday and $5°$ C on Tuesday. How much warmer is Tuesday night?

Identify the two temperatures: $-2$ and $5$ .
Find the difference: $5 - (-2) = 5 + 2 = 7$ .
Tuesday night is $7°$ C warmer.

Key idea: subtracting a negative is the same as adding — you move further right on the number line.

1. What integers are

The integers are the whole numbers, their opposites, and zero:

\ldots, -3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3, \ldots

Every integer has a place on the number line. Numbers get bigger as you move to the right, and smaller as you move to the left.

The integers from -10 to 10.

2. Comparing and ordering

$-5$ is smaller than $-2$ because $-5$ sits further to the left on the number line, even though the number $5$ is bigger than $2$ .

-5 sits to the left of -2, so -5 < -2.

3. Absolute value

The absolute value of an integer is its distance from zero on the number line. Distance is never negative.

|{-7}| = 7, \qquad |4| = 4, \qquad |0| = 0.

Absolute value

|a| = \begin{cases} \phantom{-}a, & \text{if } a \geq 0, \\ -a, & \text{if } a < 0. \end{cases}

4. Adding and subtracting - walking it out on the number line

Use the number line first, before any rules. Adding moves you right, subtracting moves you left.

Worked example 1 Start at -8, move 3 right

Evaluate $\;-8 + 3$ .

Start at $-8$ . Adding $3$ moves three steps to the right, so you land on $-5$ .

-8 + 3 = -5

-8 + 3 = -5.

Worked example 2 Start at 2, move 6 left

Evaluate $\;2 - 6$ .

Start at $2$ . Subtracting $6$ moves six steps to the left, so you land on $-4$ .

2 - 6 = -4

2 - 6 = -4.

5. The “minus a minus” trick

Two negative signs side by side flip into a plus. Think of it as “taking away a debt is the same as gaining money”.

Two signs meeting

a - (-b) = a + b.

Worked example 3 Subtracting a negative

Evaluate $\;4 - (-7)$ .

Rewrite $-(-7)$ as $+7$ :

4 - (-7) \;=\; 4 + 7 \;=\; 11.

6. Quick rules for add and subtract

Once the number line picture is clear, these shortcuts help for bigger numbers:

Adding and subtracting integers

Same signs: add the sizes, keep the sign

(+6) + (+3) = +9, \qquad (-6) + (-3) = -9.

Different signs: subtract sizes, keep the sign of whichever is bigger

(+8) + (-3) = +5, \qquad (-8) + (+3) = -5.

Turn subtraction into addition of the opposite

a - b \;=\; a + (-b).

7. A real-world example

Worked example 4 Temperature at 9 p.m.

At $6$ a.m. the temperature in Ballarat was $-4\,^\circ$ C. By noon it had risen by $9\,^\circ$ C. At $9$ p.m. it was $6\,^\circ$ C lower than at noon. What was the temperature at $9$ p.m.?

Temperature at noon: $-4 + 9 = 5\,^\circ$ C.
Temperature at 9 p.m.: $5 - 6 = -1\,^\circ$ C.

\boxed{-1\,^\circ\text{C}}

Practice: Year 7 core

Fluency

Ordering and absolute value

Order from smallest to largest: $\;-7,\ 3,\ 0,\ -2,\ -10,\ 5$ .
Which is smaller: $-4$ or $-9$ ?
Evaluate $|{-13}|$ .
Evaluate $|9| - |{-4}|$ .
True or false: $|{-6}| = 6$ .

Fluency

Adding and subtracting

Work out $\;-6 + 4$ .
Work out $\;-5 + (-8)$ .
Work out $\;12 + (-15)$ .
Work out $\;-3 + 10$ .
Work out $\;-9 - 3$ .
Work out $\;-7 - (-10)$ .
Work out $\;4 - (-11)$ .
Work out $\;-20 - (-20)$ .
Work out $\;0 - 7$ .
Work out $\;-2 + (-6) - (-4)$ .

Reasoning

Fill in the missing number

$\;-7 + \square = -2$ .
$\;\square + 4 = -1$ .
$\;-5 - \square = -12$ .
$\;\square - (-3) = 8$ .

Problem solving

Real-world problems

The temperature on Mount Hotham was $-8\,^\circ$ C at midnight. It rose by $3\,^\circ$ C each hour until $6$ a.m. What was the temperature at $6$ a.m.?
A submarine is $120$ m below sea level. It ascends $35$ m, then descends $48$ m. What is its new depth, written as an integer?
Mira has a bank balance of -$45 (an overdraft). She deposits $120 and then pays a bill of $38. What is her balance now?
A lift is on floor $-3$ (basement level 3). It goes up $7$ floors, then down $5$ , then up $2$ . On which floor does it stop?
At dawn the temperature in Cooma was $-6\,^\circ$ C. By mid-morning it had risen to $4\,^\circ$ C. By how many degrees did the temperature rise?
A diver starts at sea level ( $0$ ) and descends to $-24$ m, then rises $9$ m to look at a reef. What is her depth now?

Extension

8. Multiplying and dividing integers

Think of $(-3) \times 2$ as “two lots of negative three”:

(-3) + (-3) = -6, \quad \text{so} \quad (-3) \times 2 = -6.

$(-3) \times (-2)$ means “take away two lots of negative three”. Taking away a loss is a gain, so the answer is $+6$ .

Sign rules for x and /

Same signs give a positive answer

(+)\times(+) = +, \qquad (-)\times(-) = +.

Different signs give a negative answer

(+)\times(-) = -, \qquad (-)\times(+) = -.

Division follows the same pattern

\frac{+}{+} = +, \quad \frac{-}{-} = +, \quad \frac{+}{-} = -, \quad \frac{-}{+} = -.

Count the negatives

An even number of negative factors gives a positive result; an odd number gives a negative result.

Worked example 5 Multiplying several integers

Evaluate $\;(-2) \times 3 \times (-4) \times (-1)$ .

Multiply the sizes: $2 \times 3 \times 4 \times 1 = 24$ .
Count negatives: three of them. Three is odd, so the answer is negative.

(-2) \times 3 \times (-4) \times (-1) = -24.

Practice: Extension

Fluency

Multiplying and dividing

Work out $\;(-6) \times 7$ .
Work out $\;(-8) \times (-5)$ .
Work out $\;9 \times (-4)$ .
Work out $\;(-12) \times (-3) \times 2$ .
Work out $\;(-2)^3$ .
Work out $\;(-5)^2$ .
Work out $\;-24 \div 6$ .
Work out $\;(-36) \div (-9)$ .
Work out $\;45 \div (-5)$ .
Work out $\;(-100) \div (-25)$ .

Reasoning

Sign reasoning and missing values

Without calculating, decide whether $(-17) \times (-23) \times (-4)$ is positive or negative. Explain how you know.
Fill in: $\;\square \times (-6) = 42$ .
Fill in: $\;(-48) \div \square = 8$ .
The product of three integers is $-60$ . Two of them are $-4$ and $5$ . What is the third?

Challenge

9. Order of operations with negatives

When an expression has several operations, use BIDMAS: brackets -> indices -> division & multiplication (left to right) -> addition & subtraction (left to right).

Worked example 6 Mixed operations with an index

Evaluate $\;-3 + 2 \times (-4)^2 - 10 \div (-5)$ .

Indices first: $(-4)^2 = 16$ , so the expression becomes $-3 + 2 \times 16 - 10 \div (-5)$ .
Multiplication and division, left to right: $2 \times 16 = 32$ ; $10 \div (-5) = -2$ . The expression becomes $-3 + 32 - (-2)$ .
Addition and subtraction, left to right: $-3 + 32 = 29$ ; $29 - (-2) = 29 + 2 = 31$ .

-3 + 2 \times (-4)^2 - 10 \div (-5) = 31.

10. $(-5)^2$ versus $-5^2$

Practice: Challenge

Reasoning

Harder reasoning

Evaluate $\;(-3)^3 + 5 \times (-2)^2$ .
Place $<$ , $>$ or $=$ between the pair: $\;(-5)^2 \;\square\; -5^2$ . Justify your answer.
Jamie writes " $-3^2 = 9$ ". Is Jamie correct? If not, what is the right answer and what mistake has Jamie made?
Tom thinks of an integer. He doubles it, subtracts $7$ , then multiplies by $-3$ . The result is $33$ . What integer was Tom thinking of?

Check your integers

Quick check: pick an answer and the page will tell you right away. Your score stays on this device only.

0 / 5

1. Which is the smaller number?
-5 sits further to the left on the number line, so it is the smaller number.
2. Work out 4 - (-7).
Two negatives turn into a plus: 4 - (-7) = 4 + 7 = 11.
3. Work out -9 + 4.
Start at -9, move 4 steps to the right: land on -5.
4. Which is correct?
The brackets say 'square the negative number'. (-5) x (-5) = 25.
5. At midnight the temperature was -8 deg C. It rose 3 deg every hour for 6 hours. What was the temperature then?
-8 + 6 x 3 = -8 + 18 = 10 deg C.

Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

Show the full answer key

This answer key follows the three sections of the practice: Year 7 core, Extension, and Challenge. Each answer gives the final result and a short method hint where the work is more than one step.

Year 7 core - answers

Fluency

Ordering and absolute value

$-10,\ -7,\ -2,\ 0,\ 3,\ 5$
$-9$ (further left on the number line)
$13$
$5$
True

Fluency

Adding and subtracting

$-2$ (start at $-6$ , jump $4$ right)
$-13$ (same signs, add the sizes)
$-3$ (different signs: $15 - 12 = 3$ , keep the sign of the larger size)
$7$
$-12$
$3$ (minus a minus: $-7 + 10$ )
$15$ (minus a minus: $4 + 11$ )
$0$
$-7$
$-4$ . Method: $-2 + (-6) = -8$ ; then $-8 - (-4) = -8 + 4 = -4$ .

Reasoning

Fill in the missing number

$5$ . Method: $\square = -2 - (-7) = 5$ .
$-5$ . Method: $\square = -1 - 4 = -5$ .
$7$ . Method: $-5 - \square = -12$ gives $\square = -5 - (-12) = 7$ .
$5$ . Method: $\square = 8 + (-3) = 5$ .

Problem solving

Real-world problems

$10\,^\circ$ C. Method: $-8 + 6 \times 3 = -8 + 18$ .
$-133$ m (or $133$ m below sea level). Method: $-120 + 35 - 48$ .
$37. Method: $-45 + 120 - 38$ .
Floor $1$ . Method: $-3 + 7 - 5 + 2$ .
$10\,^\circ$ C rise. Method: $4 - (-6) = 4 + 6$ .
$-15$ m. Method: $-24 + 9 = -15$ .

Extension - answers

Fluency

Multiplying and dividing

$-42$
$40$ (negative times negative is positive)
$-36$
$72$ (two negatives cancel, then multiply)
$-8$ (three negatives: odd)
$25$ (two negatives: even)
$-4$
$4$
$-9$
$4$

Reasoning

Sign reasoning and missing values

Negative. Reason: there are three negative factors, and three is an odd number, so the product must be negative.
$-7$ . Method: $\square = 42 \div (-6) = -7$ .
$-6$ . Method: $\square = -48 \div 8 = -6$ .
$3$ . Method: $(-4) \times 5 = -20$ ; the third number satisfies $-20 \times \square = -60$ , so $\square = 3$ .

Challenge - answers

Reasoning

Harder reasoning

$-7$ . Method: $(-3)^3 = -27$ ; $(-2)^2 = 4$ , so $5 \times 4 = 20$ ; then $-27 + 20 = -7$ .
$>$ . Reason: $(-5)^2 = 25$ and $-5^2 = -25$ . Since $25 > -25$ , the left side is greater.
Jamie is not correct. The correct answer is $-9$ . The notation $-3^2$ means $-(3 \times 3) = -9$ (the power applies only to the $3$ , with the minus sign in front). Jamie has read it as $(-3)^2 = 9$ , but without brackets the squaring does not include the negative sign.
$n = -2$ . Method: let the number be $n$ . Then $(2n - 7) \times (-3) = 33$ , so $2n - 7 = -11$ ; $2n = -4$ ; $n = -2$ . Check: $2(-2) = -4$ ; minus $7$ gives $-11$ ; times $-3$ gives $33$ .

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Year 7 core

1. What integers are

2. Comparing and ordering

3. Absolute value

4. Adding and subtracting - walking it out on the number line

5. The “minus a minus” trick

6. Quick rules for add and subtract

Adding and subtracting integers

7. A real-world example

Practice: Year 7 core

Extension

8. Multiplying and dividing integers

Sign rules for x and /

Practice: Extension

Challenge

9. Order of operations with negatives

10. (−5)2(-5)^2(−5)2 versus −52-5^2−52

Practice: Challenge

Answer key

Year 7 core - answers

Extension - answers

Challenge - answers

10. $(-5)^2$ versus $-5^2$