Integer Word Problems
Learn to translate real-world situations into integer operations and solve them step by step.
Definition
- *"below zero"* or *"under sea level"* → negative numbers
- *"increased by"* or *"gained"* → addition
- *"decreased by"* or *"lost"* → subtraction
- *"how much more/less"* or *"difference"* → subtraction
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Worked Examples
At 6 AM, the temperature was . By noon, it had risen by degrees. What was the temperature at noon?
Identify the starting value
The temperature at 6 AM was → Start:
Identify the change
"Risen by" means we add. The change is → Change:
Write the expression
→ Expression set up
Calculate
Starting at , moving units right: →
Interpret the answer
Positive means above zero →
Answer: The temperature at noon was
Common Mistakes
Confusing 'how much deeper' with subtraction order
Why it's wrong: Students often subtract in the wrong order when comparing values.
Correct: To find how much deeper A is than B, calculate the distance: meters. Or think: from to is meters down.
Ignoring negative signs in the answer
Why it's wrong: 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.
Adding when the problem says 'lost' or 'decreased'
Why it's wrong: Key words like 'lost,' 'dropped,' and 'fell' indicate subtraction, not addition.
Correct: Lost 10 points means subtract 10: if you had points, now you have points.
Interactive Visual
Click on numbers to select them. Adjust the range to explore different values.
Interactive Sandbox
Expression Calculator
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History
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Practice Problems
15 problemsThe temperature was in the morning. It rose by degrees. What is the temperature now?
Why It Matters
- Weather: A temperature of rises by degrees
- Finance: Your account has dollars (debt), then you deposit
- Geography: A submarine at meters descends more meters
- Sports: A football team gains yards then loses yards
Real World Applications
Weather Forecasting
Meteorologists calculate temperature changes using integers every day.
Example:
If tonight's low is and tomorrow's high is , the temperature range is degrees.
The temperature at midnight was . It dropped degrees by 3 AM, then rose degrees by 9 AM.
What was the temperature at 9 AM?
Step 1: Write the mathematical expression
Calculate:
Football Yardage
Football teams track yards gained and lost throughout a game.
Example:
A team gains yards, loses , then gains . Net yardage: yards.
A football team starts at the 30-yard line. They gain 15 yards, lose 8 yards, then lose 12 more yards.
What yard line are they on now?
Step 1: Write the mathematical expression
Calculate:
Scuba Diving Depth
Divers track their depth below sea level using negative numbers.
Example:
A diver at meters descends more meters to reach meters.
A diver is at meters. She ascends meters, rests, then descends meters.
What is her final depth?
Step 1: Write the mathematical expression
Calculate:
Key Takeaways
- 1Read carefully to identify starting values, changes, and what the question asks
- 2Translate key words: 'rise/gain/deposit' → add, 'fall/lose/withdraw' → subtract
- 3Use negative numbers for values below zero (debt, basement floors, underwater depths)
- 4After calculating, interpret what the answer means in the real-world context
Frequently Asked Questions
Glossary
- Integer
- A whole number that can be positive, negative, or zero
- Elevation
- Height above or below sea level (sea level = 0)
- Deposit
- Money added to an account (increases balance)
- Withdrawal
- Money taken from an account (decreases balance)
- Net change
- The overall result after all increases and decreases