Year 7 Maths Integers

Integers Problems

Q: What is a directed number sentence?

A calculation where each change has a sign, such as -60 + 25 - 40.

Q: How do you find how far apart two positions are?

Higher minus lower. From -35 to 20 is 20 - (-35) = 55.

Q: When do you multiply in an integer problem?

When the same change repeats, such as a fee each week: 6 fees of $5 is 6 × (-5).

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Theory

To solve an integer problem, choose which direction is positive, write a directed number sentence, then calculate and answer in words. This Year 7 maths page works through temperatures, walks north and south, submarines, bank balances, quiz scores and time zones.

Step 1. Choose a direction. Decide what is positive, such as north, up, a deposit or a profit. The opposite direction is negative.

Step 2. Write each change as an integer. A drop of $6^\circ\text{C}$ is $-6$. A deposit of $\$20$ is $+20$.

Step 3. Write a directed number sentence. A change that repeats is a multiplication: $4$ withdrawals of $\$15$ is $4 \times (-15)$.

Step 4. Calculate, then answer in words. Say what the sign means: $-30$ m means $30$ m below the surface.

North is positive: $-3 + 4 = 1$, so the walk ends $1$ km north of home.

Up is positive: $-60 + 25 - 40 = -75$ m.

Common words and their signs:

Positive (+)	Negative (−)
above sea level, up, rise	below sea level, down, fall
north or east (if chosen)	south or west
deposit, profit, earn	withdrawal, loss, debt, fee
points won, above par	points lost, below par
ahead of UTC	behind UTC

\[ \text{distance between positions} = \text{higher} - \text{lower} \]

distance between positions = higher - lower

          A repeated change is a product. Five fees of \(\$12\) is \(5 \times (-12) = -60\), not \(5 - 12\).
        

          Give the direction in the answer. An answer of \(-120\) for a submarine means \(120\) m below the surface.
        

How to solve an integer problem

Choose which direction is positive.
Write each change as an integer, with its sign.
Write a number sentence; use $\times$ for a change that repeats.
Calculate using the order of operations, then answer in words.

Example 1 — Temperature drop

One afternoon the temperature reached $11^\circ\text{C}$. Overnight it dropped to $-2^\circ\text{C}$. How many degrees did it drop?

Solution

$11 - (-2)$	$=$	$11 + 2$
	$=$	$13$

11 - (- 2) = 11 + 2 = 13

It dropped $13^\circ\text{C}$.

Example 2 — North and south

Anya walks $3$ km south to a friend’s house. Together they walk $4$ km north. How far and in which direction are they from Anya’s home?

Solution

$-3 + 4$

$=$

$1$

- 3 + 4 = 1

North is positive. They are $1$ km north of Anya’s home, as in the first diagram.

Example 3 — Submarine moves

A submarine is at $-60$ m. It rises $25$ m, then dives $40$ m. Where is it now?

Solution

$-60 + 25 - 40$	$=$	$-35 - 40$
	$=$	$-75$

- 60 + 25 - 40 = - 75

It is at $-75$ m, which is $75$ m below the surface.

Example 4 — Profit and loss

A cafe’s profits for four weeks were $-\$130$, $\$90$, $-\$40$ and $\$160$.
(i) What was the total profit?
(ii) What was the mean weekly profit?

Solution

$-130 + 90 + (-40) + 160$	$=$	$80$
$80 \div 4$	$=$	$20$

- 130 + 90 + (- 40) + 160 = 80, 80 \div 4 = 20

(i) $\$80$
(ii) $\$20$ a week

Example 5 — Time zones

City P is UTC $+10$. City Q is UTC $-10$.
(i) How many hours is P ahead of Q?
(ii) It is hour $22$ (10 pm) in P. What time is it in Q?

Solution

$10 - (-10)$	$=$	$20$
$22 - 20$	$=$	$2$

10 - (- 10) = 20, 22 - 20 = 2

(i) $20$ hours
(ii) Hour $2$, which is 2 am

Example 6 — Quiz with marks off

A quiz has $10$ questions. A correct answer scores $+3$ and a wrong answer scores $-1$. Zoe answered every question and scored $14$. How many did she get correct?

Solution

Try a number correct; the rest are wrong. $5$ correct and $5$ wrong score $5 \times 3 = 15$ and $5 \times (-1) = -5$. $6$ correct and $4$ wrong score $18$ and $-4$.

$15 + (-5)$	$=$	$10$
$18 + (-4)$	$=$	$14$

15 + (- 5) = 10, 18 + (- 4) = 14

$5$ correct gives $10$, too low. $6$ correct gives $14$. Zoe got $6$ correct.

Example 7 — Repeated fees

Mia’s balance is $\$50$. A fee of $\$9$ is charged each month for $7$ months. What is her balance now?

Solution

$50 + 7 \times (-9)$	$=$	$50 - 63$
	$=$	$-13$

50 + 7 \times (- 9) = 50 - 63 = - 13

Her balance is $-\$13$: she owes $\$13$.

Common pitfalls

Leaving out the direction. $-75$ for a submarine means $75$ m below the surface. Say so in words.

Subtracting the wrong way. From $-35$ m to $20$ m is $20 - (-35) = 55$ m, not $20 - 35$.

Adding a repeated change once. Five fees of $\$12$ is $5 \times (-12) = -60$.

Mixing up profit and loss. A loss is negative and a profit is positive. Add them with their signs.

Frequently asked questions

How do you choose which direction is positive?

Pick the natural one: up, north, a deposit or a profit. Then the opposite is negative.

What is a directed number sentence?

A calculation where each change has a sign, such as $-60 + 25 - 40$.

How do you find how far apart two positions are?

Higher minus lower. From $-35$ to $20$ is $20 - (-35) = 55$.

When do you multiply in an integer problem?

When the same change repeats, such as a fee each week: $6$ fees of $\$5$ is $6 \times (-5)$.

What does a negative answer mean?

It means the opposite of the direction you chose as positive, such as below sea level or money owed.

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\(11 - (-2)\)	\(=\)	\(11 + 2\)
\(\)	\(=\)	\(13\)

\(-60 + 25 - 40\)	\(=\)	\(-35 - 40\)
\(\)	\(=\)	\(-75\)

\(-130 + 90 + (-40) + 160\)	\(=\)	\(80\)
\(80 \div 4\)	\(=\)	\(20\)

\(10 - (-10)\)	\(=\)	\(20\)
\(22 - 20\)	\(=\)	\(2\)

\(15 + (-5)\)	\(=\)	\(10\)
\(18 + (-4)\)	\(=\)	\(14\)

\(50 + 7 \times (-9)\)	\(=\)	\(50 - 63\)
\(\)	\(=\)	\(-13\)

Integers Problems

📖 Theory

How to solve an integer problem

Common pitfalls

Frequently asked questions

Theory