Temperature Change

At 6 AM, the temperature was $-8°C$. By noon, it had risen by $15$ degrees. What was the temperature at noon?

Identify the starting value: The temperature at 6 AM was $-8°C$ = Start: $-8$

Identify the change: "Risen by" means we add. The change is $+15$ = Change: $+15$

Write the expression: $-8 + 15$ = Expression set up

Calculate: Starting at $-8$, moving $15$ units right: $-8 + 15 = 7$ = $7$

Interpret the answer: Positive $7$ means above zero = $7°C$

Answer: The temperature at noon was $7°C$

Elevator Movement

An elevator starts at floor $3$. It goes down $7$ floors to the parking garage, then up $4$ floors. What floor is it on now?

Identify the starting position: The elevator starts at floor $3$ = Start: $3$

Translate the first movement: "Goes down" means subtract. Down $7$ floors: $-7$ = First move: $-7$

Translate the second movement: "Up" means add. Up $4$ floors: $+4$ = Second move: $+4$

Write and solve: $3 + (-7) + 4 = 3 - 7 + 4 = -4 + 4 = 0$ = $0$

Interpret: Floor $0$ is ground level = Ground floor

Answer: The elevator is now on the ground floor (floor $0$)

Bank Account Balance

Maria's bank account shows $-45$ euros (she owes the bank). She deposits $80$ euros and then pays a bill of $25$ euros. What is her final balance?

Identify the starting balance: Account shows $-45$ euros (debt) = Start: $-45$

Translate the deposit: Depositing money increases the balance: $+80$ = Deposit: $+80$

Translate the payment: Paying a bill decreases the balance: $-25$ = Payment: $-25$

Calculate step by step: $-45 + 80 = 35$, then $35 - 25 = 10$ = $10$

Interpret the result: Positive $10$ means she has money in her account = 10 euros in account

Answer: Maria's final balance is $10$ euros (she no longer has debt)

Submarine Depth Comparison

Submarine A is at $-120$ meters. Submarine B is at $-85$ meters. How much deeper is Submarine A than Submarine B?

Identify both positions: Sub A: $-120$ m, Sub B: $-85$ m = Positions identified

Determine which is deeper: $-120 < -85$, so Sub A is deeper (further below sea level) = Sub A is deeper

Find the difference: Difference = $(-85) - (-120) = -85 + 120 = 35$ = $35$ meters

Interpret: Submarine A is $35$ meters below Submarine B = $35$ m deeper

Answer: Submarine A is $35$ meters deeper than Submarine B

Mistake: Confusing 'how much deeper' with subtraction order

Why: Students often subtract in the wrong order when comparing values.

Correct: To find how much deeper A is than B, calculate the distance: $|(-120) - (-85)| = 35$ meters. Or think: from $-85$ to $-120$ is $35$ meters down.

Mistake: Ignoring negative signs in the answer

Why: Students calculate correctly but forget to consider what a negative result means.

Correct: Always interpret: negative balance = debt, negative temperature = below zero, negative floor = basement level.

Mistake: Adding when the problem says 'lost' or 'decreased'

Why: Key words like 'lost,' 'dropped,' and 'fell' indicate subtraction, not addition.

Correct: Lost 10 points means subtract 10: if you had $5$ points, now you have $5 - 10 = -5$ points.

Weather Forecasting

Meteorologists calculate temperature changes using integers every day.

If tonight's low is $-12°C$ and tomorrow's high is $8°C$, the temperature range is $8 - (-12) = 20$ degrees.

Football Yardage

Football teams track yards gained and lost throughout a game.

A team gains $12$ yards, loses $5$, then gains $8$. Net yardage: $12 + (-5) + 8 = 15$ yards.

Scuba Diving Depth

Divers track their depth below sea level using negative numbers.

A diver at $-15$ meters descends $10$ more meters to reach $-25$ meters.

Q: How do I know if an answer should be negative?

A: Think about the context: if someone owes money, is below sea level, or is at a temperature below freezing, the answer is negative. Your calculation will show this naturally if you set it up correctly.

Q: What's the difference between 'decreased by 5' and 'is 5 less than'?

A: 'Decreased by 5' means subtract 5 from a value: $x - 5$. 'Is 5 less than' compares two values: if A is 5 less than B, then $A = B - 5$.