B. Number · Number: Number Sets, Powers and Rational Numbers in Action

Integers in Context: Location, Direction, Amount and Change

B3.1 A1 A2About 65 minutes

Learning goals

I can use integers to describe a location relative to a zero point, a direction, and an amount.
I can calculate a change as final value minus initial value, and explain what its sign means.
I can combine several changes to find a final location or amount.
I can solve multi-step problems with integers in contexts like temperature, elevation, money and coordinates.

Words to know

integer
A number in the set {…, −3, −2, −1, 0, 1, 2, 3, …}.

reference point (zero)
The point chosen to be 0, such as sea level, the freezing point of water, the starting line or a $0 balance.

opposite
The number the same distance from zero on the other side: the opposite of −7 is 7.

absolute value
The distance of a number from zero, written |−7| = 7. It is never negative.

change (Δ)
Final value minus initial value. Positive means an increase (or movement in the positive direction); negative means a decrease.

displacement
A change in position, with a direction. Moving 5 km west is a displacement of −5 km if east is positive.

Remember from Grade 8

In Grade 8 you added, subtracted, multiplied and divided integers. In Grade 9 the focus is on meaning: what an integer tells you in a real situation, and how to use it to describe location, direction, amount and change.

One number, four jobs

Location: The parking level P2 is at −2; a diver is at −18 m (18 m below sea level).
Direction: If east is positive, then −5 km means 5 km west.
Amount: A bank balance of −$40 means you owe $40. A hockey team's goal differential of +6 means it scored 6 more goals than it allowed.
Change: The temperature rose by 15 °C (+15) or fell by 8 °C (−8).

Every situation needs a reference point (the zero) and a positive direction. Always say what they are.

The change formula

change = final value − initial value

A positive change means the value went up (or moved in the positive direction).
A negative change means it went down.
The size of the change is its absolute value.

Mathematicians often write change with the Greek letter Δ (delta): ΔT = T(final) − T(initial).

A cold winter day in Winnipeg (example values). The temperature rose from −24 °C to −9 °C, a change of +15 °C, even though both temperatures are below zero.

Worked example

Example 1: Temperature change

On a winter day in Winnipeg the temperature is −24 °C at 7 a.m. and −9 °C at 3 p.m. Then it falls 11 °C overnight. Find the daytime change and the overnight temperature.

Daytime change = final − initial = −9 − (−24) = −9 + 24 = +15. The temperature rose 15 °C.
A fall of 11 °C is a change of −11. Overnight temperature = −9 + (−11) = −20.

The change was +15 °C, and the overnight temperature was −20 °C.

Worked example

Example 2: Elevation

A scuba diver is at −18 m (sea level is 0). She rises 7 m, then descends 12 m. A gull flies at +15 m. Where is the diver now, and how far apart are they vertically?

Diver: −18 + 7 + (−12) = −23 m, which is 23 m below sea level.
Vertical distance = higher − lower = 15 − (−23) = 15 + 23 = 38 m.

The diver is at −23 m; they are 38 m apart.

Combining changes

When several changes happen one after another, add them: the total change is the sum, and the final value is the initial value plus the total change. The order of the changes does not change the final value (addition is commutative), but it does change the values in between.

A cross-country skier starts at the trail map (0) with east as positive. She skis 6 km east (+6), then 10 km west (−10). Her final position is −4 km: 4 km west of the map. Her total displacement is +6 + (−10) = −4 km, but she skied 16 km in total.

Worked example

Example 3: Money

Liam's account has $45. Over a week: a $60 purchase, a $25 deposit and a $12 purchase. What is his final balance? What was the total change?

Write each transaction as an integer: −60, +25, −12.
Total change: −60 + 25 + (−12) = −47.
Final balance: 45 + (−47) = −2.

The balance is −$2 (he is overdrawn by $2). The total change was −$47.

Worked example

Example 4: Change in position on a grid

A point moves from A(−3, 4) to B(5, −2). Describe the change in each direction.

Horizontal change: Δx = 5 − (−3) = +8. It moved 8 units right.
Vertical change: Δy = −2 − 4 = −6. It moved 6 units down.
You will use Δy and Δx again to find slopes in the Linear and Non-Linear Relations unit.

Δx = +8, Δy = −6

Hands-on activity

Weather log

You need: a weather app or the local news, paper, a pencil.

For 5 days, record the temperature at about 7 a.m. and at about 3 p.m. (If it is warm where you are, use the overnight low and the daytime high.)
For each day, find the change: afternoon − morning. Mark it + or −.
Also find the change from one morning to the next.
Which day had the largest change in size (absolute value)? Was any change negative? What weather caused it?
Add up the five morning-to-morning changes. Check that the total equals (last morning) − (first morning). Explain why that works.

Model answer: mornings −6, −2, 1, −8, −5 °C. Morning-to-morning changes +4, +3, −9, +3; total +1, and −5 − (−6) = +1 ✓.

Just for fun

In hockey, a player's plus-minus goes up by 1 when their team scores while they are on the ice and down by 1 when the other team scores (power-play goals do not count). So a defender can have a great game with zero points and still be a +3. Integers: they're not just for math class!

Your turn

A submarine at −120 m rises 45 m. Where is it now?

Find the change from −7 °C to −15 °C. Did it get warmer or colder?

A company had a profit of −$3500 in January and $1200 in February. What was the change?

An elevator goes from level P3 (−3) up 9 floors, then down 4. Where is it?

A point moves from (2, −5) to (−4, 1). Find Δx and Δy.

Which is farther from 0: −12 or 9? Use absolute value.

A team's goal differential went from +4 to −2 over three games. What was the change?

More practice: warm-up

What integer describes 8 m below sea level?

If north is positive, what does −3 km mean?

Find the change from 5 to −2.

What is the opposite of −14?

Find |−25|.

More practice: on your own

The temperature at Iqaluit was −31 °C at 6 a.m. and −18 °C at 2 p.m. (example values). Find the change and the average change per hour.

A hiker starts at an elevation of 240 m. She climbs 180 m, descends 95 m, then descends 210 m. What is her final elevation and her total change?

Your balance is −$18. How much must you deposit to have $50?

Write the changes that take a point from (0, 0) to (−7, 3) and then to (2, −5).

Over 4 days, a lake's water level changed by +3 cm, −5 cm, −2 cm, +1 cm. What is the total change? What is the mean daily change?

Challenge

In many buildings there is no floor 0: the floors go P2, P1, 1, 2, 3, … How many floors does an elevator travel going from P2 to floor 3? Why is 3 − (−2) = 5 wrong here?

The temperature at 6 p.m. was −4 °C. It changed by −2 °C each hour until midnight, then by +3 °C each hour until 6 a.m. What was the temperature at 6 a.m.?

Two divers start at sea level. One descends 3 m per minute; the other descends 5 m per minute. After how many minutes are they 14 m apart vertically? Explain.

Check your understanding

Quiz

The temperature changes from −3 °C to −11 °C. What is the change?

If east is positive, a displacement of −6 km means:

A bank balance of $20 has a $35 purchase. What is the new balance?

How far apart vertically are a bird at +12 m and a fish at −9 m?

A point moves from (4, −1) to (−2, −6). What is Δy?

A positive change always means:

For parents and teachers

Big idea: Expectation B3.1 asks students to apply their understanding of integers to describe location, direction, amount and change in many contexts. The key habit is to name the zero and the positive direction first, then use change = final − initial.

Common difficulties:

Subtracting in the wrong order (initial − final), which flips the sign of the change.
Thinking a positive change means a positive temperature. −24 °C to −9 °C is a rise.
Confusing distance and displacement. Distance is always positive; displacement has a sign.
Floor numbers. Real building numbering often skips 0.

How to help: Use real data: the weather, a bank or allowance record, sports standings. Ask, "What is zero here? Which way is positive? What was the change?"