Use integers to describe location, direction, amount, and change

Ontario Grade 9 Mathematics

Number

Ontario MTH1W — Expectation B3.1

Every day, you encounter situations that involve opposites: owing money versus earning it, going up in an elevator versus going down, a temperature that rises versus one that falls. Integers give us a precise, shared language to describe all of these situations using a single number line. In this lesson you will learn how one type of number — the integer — can communicate location, direction, amount, and change all at once.

What you will learn

Use integers to describe a location on a number line or in a real-world context.
Use integers to describe a direction (positive or negative) and an amount.
Use integers to describe a change, including increases and decreases.
Choose the correct integer to represent a given situation and explain your reasoning.

What Is an Integer? A Quick Review

Before Grade 9, you worked with whole numbers like

0, 1, 2, 3, \ldots

and with negative numbers in contexts such as temperature. Now we bring those ideas together formally.

An integer is any number in the set

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

. Integers include all positive whole numbers, all negative whole numbers, and zero. Fractions and decimals such as

\frac{1}{2}

or

1.7

are not integers.

The number line is the clearest picture of the integers. Zero sits in the centre. Positive integers sit to the right of zero, and negative integers sit to the left. Every integer has an opposite: the opposite of

5

is

-5

, and the opposite of

-3

is

3

. The opposite of

0

is

0

itself. Knowing where a number sits on the number line — and what its opposite is — is the foundation for everything in this lesson.

Integers: $\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$ — no fractions, no decimals.
Positive integers are to the right of zero; negative integers are to the left.
Every integer has an opposite. The opposite of $n$ is $-n$ .
Zero is neither positive nor negative.

Using Integers to Describe Location and Direction

An integer can tell you where something is and which way it is pointing — at the same time. Consider a building with floors above and below ground. We can label ground level as

0

. Floor

3

is three storeys above ground, so we write

+3

(or simply

3

). The second level of an underground parking garage is two storeys below ground, so we write

-2

. The integer tells you both how far from ground level (the amount) and which way (above or below).

The same idea applies to a number line. If you are standing at position

0

and you move to position

-5

, you are now

5

units to the left of the start. The negative sign tells you the direction (left), and the

5

tells you the distance.

A signed integer therefore carries two pieces of information: its sign tells direction, and its absolute value (the size without the sign) tells the amount. For example,

|-7| = 7

means the location is

7

units away from zero, in the negative direction. You do not need to perform any calculation here — you simply read the sign as direction and the digits as amount.

Real-world locations that use integers include: altitude above or below sea level (sea level

= 0

), a bank account balance (money owed as negative, money held as positive), and temperature above or below zero degrees Celsius.

Location = sign (direction) + absolute value (amount from reference point)

A reference point of $0$ is chosen (ground floor, sea level, starting position, etc.).
A positive integer means above, right, or forward of zero.
A negative integer means below, left, or behind zero.
The sign gives direction; the digits give the amount (distance from zero).

Using Integers to Describe Amount

Sometimes an integer describes a pure amount — a count or a quantity — where the sign still carries meaning. Think about a bank account. A balance of

+250

means you have

250

dollars available. A balance of

-80

means you owe the bank

80

dollars. Both are amounts, but the sign tells you whether the amount is in your favour or against it.

Stock points, scores in some games, and points on a quiz can all be negative. A team that concedes

3

goals in a goal-difference table is recorded as

-3

. A student who loses

4

marks for an incorrect answer might have

-4

applied to their score. In every case, the integer precisely captures both the size of the amount and its nature (gain or loss, asset or debt).

It is important not to confuse the negative sign with subtraction. The integer

-4

is a number that represents a quantity. The operation of subtraction is a process you perform between two numbers. When you write

-4

to mean 'a debt of four dollars,' you are using

-4

as a description, not as an instruction to subtract.

Positive integers describe amounts that represent a gain, credit, or surplus.
Negative integers describe amounts that represent a loss, debt, or deficit.
The sign is part of the number itself, not an instruction to subtract.

Using Integers to Describe Change

A change tells you how a quantity moves from one value to another. Integers are ideal for this because change has both a size (how much did it move?) and a direction (did it go up or down?).

The word 'change' in mathematics means: final value minus starting value. If the temperature was

-3°C

in the morning and rose to CAD 4°C by noon, the change is

4 - (-3) = 4 + 3 = 7

. The change is

+7°C

, meaning the temperature increased by

7

degrees. If the temperature then dropped from CAD 4°C to

-1°C

in the evening, the change is

-1 - 4 = -5

. The change is

-5°C

, meaning it decreased by

5

degrees.

Notice the pattern: a positive change means the quantity went up or increased. A negative change means it went down or decreased. The size of the integer tells you by how much. This makes integers a compact, unambiguous way to record changes in any real-world measurement.

You can also think about a sequence of changes adding together. If your account changes by

-20

dollars on Monday (you spent

20

) and then by

+35

dollars on Wednesday (you deposited

35

), the total change is

-20 + 35 = +15

. Your account is

15

dollars higher than it was at the start.

\text{Change} = \text{Final value} - \text{Starting value}

$\text{Change} = \text{Final value} - \text{Starting value}$
A positive change means an increase; a negative change means a decrease.
Multiple changes can be added together to find the overall change.
The size of the change tells you by how much; the sign tells you which direction.

Putting It All Together: Choosing the Right Integer

When a problem asks you to represent a situation with an integer, use this three-step approach. First, identify the reference point — what counts as zero in this situation? Second, decide the direction — is the value above or below, more or less than zero? Third, state the amount — how far is the value from the reference point?

For example: 'A submarine is

340

metres below sea level.' Sea level is

0

. Below means the negative direction. The amount is

340

metres. The integer is

-340

.

Another example: 'A player earns

50

bonus points.' The reference is

0

points. Earning means positive. The integer is

+50

, or simply

50

.

If you are describing a change, use the formula

\text{Change} = \text{Final} - \text{Start}

to determine the integer. Always check: does the sign of your answer match the direction of the change described in the problem?

\text{Change} = \text{Final} - \text{Start}

Step 1: Set the reference point (what is zero?).
Step 2: Decide direction (positive or negative).
Step 3: State the amount (how far from zero?).
For change problems: apply $\text{Change} = \text{Final} - \text{Start}$ and check the sign.

Four Uses of Integers: Real-World Examples

Use of Integer	Real-World Context	Reference Point (Zero)	Example Integer	What the Integer Means
Location	Building floors	Ground floor	$-2$	2 floors below ground
Location	Altitude	Sea level	$+850$	850 m above sea level
Direction & Amount	Bank account	Zero balance	$-75$	A debt of $75$
Direction & Amount	Game score	Starting score	$+12$	12 points gained
Change	Temperature	Starting temperature	$-9$	Temperature dropped 9°C
Change	Elevation on a hike	Starting elevation	$+230$	Elevation increased 230 m

Worked example

Describing Location, Amount, and Change with Integers

Mia is hiking. She starts at a trailhead that is

120

m above sea level. She climbs to a peak that is

475

m above sea level, then descends to a rest stop that is

60

m above sea level.
(a) Write an integer for each location using sea level as the reference point.
(b) Find the change in elevation from the trailhead to the peak.
(c) Find the change in elevation from the peak to the rest stop.
(d) Find the total change in elevation from the trailhead to the rest stop, and verify it using parts (b) and (c).

Step 1 — Set the reference point
Sea level is defined as $0$ m. Any location above sea level is positive. Any location below sea level would be negative. All three locations are above sea level, so all three integers will be positive.
$\text{Sea level} = 0 \text{ m}$
Step 2 — Write integers for each location (Part a)
Each altitude is already measured from sea level, so the integer is simply the altitude with a positive sign.
$\text{Trailhead} = +120 \text{ m}, \quad \text{Peak} = +475 \text{ m}, \quad \text{Rest stop} = +60 \text{ m}$
Step 3 — Change from trailhead to peak (Part b)
Use the formula: Change = Final − Start. The final location is the peak and the start is the trailhead. The result is positive, confirming Mia went up.
$\text{Change} = 475 - 120 = +355 \text{ m}$
Step 4 — Change from peak to rest stop (Part c)
Now the final location is the rest stop and the start is the peak. The result is negative, confirming Mia went down.
$\text{Change} = 60 - 475 = -415 \text{ m}$
Step 5 — Total change from trailhead to rest stop (Part d), Method 1
Use the formula directly: Final is the rest stop, Start is the trailhead. The negative result tells us the rest stop is lower than the trailhead.
$\text{Total change} = 60 - 120 = -60 \text{ m}$
Step 6 — Verify using parts (b) and (c)
The two individual changes should add up to the total change. Add the change going up and the change coming down.
$(+355) + (-415) = 355 - 415 = -60 \text{ m} \checkmark$

Answer: (a) Trailhead:

+120

m, Peak:

+475

m, Rest stop:

+60

m. (b)

+355

m. (c)

-415

m. (d)

-60

m, confirmed by

+355 + (-415) = -60

.

Check: The rest stop (

+60

m) is below the trailhead (

+120

m), so a negative total change of

-60

m makes sense. The individual changes also sum correctly:

+355 + (-415) = -60

. ✓

Common mistakes and how to avoid them

Writing a negative integer when the context is clearly a gain or increase. For example, writing

-50

for 'earned 50 dollars'.

Correction: Reread the situation and identify the reference point. Earned, gained, above, and increased all signal a positive integer. Write

+50

.

Confusing the negative sign with the subtraction operation. Treating

-4

as 'subtract 4' when it actually means 'a value of negative four'.

Correction: When

-4

stands alone as a number (a temperature, a balance, a score), it is a description of a quantity, not an instruction. Only treat a minus sign as subtraction when it appears between two numbers, as in

7 - 4

.

Calculating change as Start − Final instead of Final − Start. For example, finding the change from CAD 10°C to CAD 3°C

as

10 - 3 = +7 instead of

3 - 10 = -7

.

Correction: Always write Change = Final − Start. Ask yourself: where did I end up, and where did I begin? Subtract the starting value from the final value, in that order.

Forgetting that zero is a valid integer. Assuming every situation must use a positive or negative number.

Correction: If a quantity is exactly at the reference point — for example, a temperature of exactly CAD 0°C or a bank balance of exactly

0

dollars — the correct integer is

0

.

Using a decimal or fraction where an integer is required. For example, writing

-2.5

floors to describe a location in a building.

Correction: Check that the context calls for a whole-number value. Integers are whole numbers only. If the answer is not a whole number, re-examine the problem or the reference point chosen.

Lesson summary

An integer is any positive whole number, negative whole number, or zero. Fractions and decimals are not integers.
Integers describe location by comparing a position to a reference point of zero. The sign shows direction; the digits show the distance from zero.
Integers describe direction and amount together: a positive integer means above, right, gained, or ahead of zero; a negative integer means below, left, lost, or behind zero.
To find a change, use: $\text{Change} = \text{Final value} - \text{Starting value}$ . A positive result means an increase; a negative result means a decrease.
Multiple changes can be added as integers to find a total change.
Always set a clear reference point (zero), choose the correct sign (direction), and state the correct size (amount) when writing an integer for any real-world situation.

Check your understanding

Question 1

A diver is

18

metres below the surface of the ocean. Using sea level as the reference point, which integer best describes the diver's location?

$+18$
$-18$
$0$
$18.0$

Show answer and explanation

-18

Sea level is

0

. Below sea level is the negative direction. The diver is

18

metres in the negative direction, so the correct integer is

-18

. The option

+18

would mean above sea level. Zero would mean exactly at sea level.

18.0

is a decimal, not an integer.

Question 2

A city's temperature was

-6°C

at midnight. By noon it was

+2°C

. What integer describes the change in temperature from midnight to noon?

$-8$
$-4$
$+4$
$+8$

Show answer and explanation

+8

Change = Final − Start

= 2 - (-6) = 2 + 6 = +8

. The temperature increased by CAD 8°C, so the change is

+8

. A common error is to calculate

2 - 6 = -4

, forgetting that subtracting a negative number is the same as adding a positive.

Question 3

Javier has a bank balance of

-45

dollars. Which statement correctly interprets this integer?

Javier has $45$ dollars in his account.
Javier owes $45$ dollars to the bank.
Javier deposited $45$ dollars.
Javier withdrew $0$ dollars.

Show answer and explanation

Javier owes

45

dollars to the bank.

A negative bank balance means the account is overdrawn — the account holder owes money.

-45

means Javier owes

45

dollars. A positive balance would mean he has money available. Zero would mean the account is exactly empty.

Question 4

A hiker's elevation changed by

-320

m and then by

+150

m. What is the total change in elevation?

$+470$ m
$-470$ m
$+170$ m
$-170$ m

Show answer and explanation

-170

m

Total change

= (-320) + (+150) = -320 + 150 = -170

m. The hiker descended more than they climbed back up, so the total change is negative.

-170

m means the hiker's final elevation is

170

metres lower than the starting elevation.

Key terms

Integer: Any number in the set $\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$ . Integers include positive whole numbers, negative whole numbers, and zero, but not fractions or decimals.
Positive integer: A whole number greater than zero, such as $1, 2, 3, \ldots$ Positive integers are written with or without a $+$ sign and sit to the right of zero on a number line.
Negative integer: A whole number less than zero, such as $-1, -2, -3, \ldots$ Negative integers are always written with a $-$ sign and sit to the left of zero on a number line.
Opposite: Two integers are opposites if they are the same distance from zero but on different sides of the number line. For example, $+5$ and $-5$ are opposites.
Reference point: The value chosen to represent zero in a real-world context. For example, sea level, ground floor, or a starting balance. All other values are described as positive or negative relative to this point.
Change: The difference between a final value and a starting value. Calculated as: Change $=$ Final $-$ Start. A positive change means an increase; a negative change means a decrease.
Absolute value: The distance of a number from zero on the number line, always written as a non-negative value. For example, $|-7| = 7$ and $|7| = 7$ . The absolute value tells you the size (amount) without the direction.
Number line: A straight line where every point represents a number. Zero is at the centre, positive integers are to the right, and negative integers are to the left. Equal spaces between consecutive integers show that they are evenly spread.

Continue through MTH1W

View the complete MTH1W Ontario Grade 9 Mathematics curriculum and lessons

Use the free MTH1W test and exam review guide

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation B3.1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question

B3.1 · Use integers to describe location, direction, amount, and change

Ontario Grade 9 Mathematics

Ontario MTH1W — Expectation B3.1

What you will learn

What Is an Integer? A Quick Review

Using Integers to Describe Location and Direction

Using Integers to Describe Amount

Using Integers to Describe Change

Putting It All Together: Choosing the Right Integer

Four Uses of Integers: Real-World Examples

Describing Location, Amount, and Change with Integers

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MTH1W

About this lesson and its review