Operations with Integers · Lesson 4 of 6
Integer Operations in Situations
“Translate changes in position, temperature, and profit into signed calculations.”
• Build an integer expression from an initial value and a repeated signed change. • Interpret multiplication in score and elevator stories. • Reframe division as a missing-factor question. • Apply division sign rules and avoid dividing by zero.
Repeated changes from a starting value
A signed rate tells us both the size and direction of a repeated change. A lift moving downward 3 metres each minute has change −3 metres per minute. After 45 minutes, its total change is 45 × (−3) = −135 metres. If it began at +15 metres, add that starting position: 15 + (−135) = −120 metres. Keeping the starting position visible prevents a common modelling mistake.
Problem
An elevator starts at ground level and descends 3 metres per minute for one hour. Find its signed position.
- 1.One hour is 60 minutes, and each minute contributes −3 metres.
- 2.Total change = 60 × (−3) = −180 metres.
- 3.Starting at 0, it ends at −180 metres, or 180 metres below ground.
Problem
The same elevator starts 15 metres above ground and descends for 45 minutes.
- 1.Its change is 45 × (−3) = −135 metres.
- 2.Add the initial position: 15 − 135 = −120.
- 3.It finishes 120 metres below ground.
A right answer contributes +5 points and a wrong answer −2 points in a 50-question challenge. Thirty right and twenty wrong responses give 30 × 5 + 20 × (−2) = 110 points. Similar modelling works for gains and losses or rising and falling temperatures. The full expression shows where each sign comes from.
Problem
A challenge has 30 right responses worth +5 each and 20 wrong responses worth −2 each. Find the total.
- 1.Right responses contribute 30 × 5 = 150.
- 2.Wrong responses contribute 20 × (−2) = −40.
- 3.Total = 150 + (−40) = 110 points.
Division asks for a missing factor
To find (−100) ÷ 25, ask: “25 multiplied by what gives −100?” Since 25 × (−4) = −100, the quotient is −4. Likewise, (−100) ÷ (−4) = 25 because (−4) × 25 = −100. This inverse relationship explains why equal nonzero signs give a positive quotient and unlike signs a negative quotient.
Problem
Find (−46) ÷ (−23).
- 1.Ask what number multiplied by −23 gives −46.
- 2.(−23) × 2 = −46.
- 3.The quotient is +2.
The magic grid activity invites you to choose one cell, cross out its row and column, and continue until each row and column has contributed once. Its rows share multipliers and its columns share multipliers, so every complete selection has the same product regardless of selection order. Try several selections and examine the row-and-column structure instead of assuming a lucky numerical coincidence.
| Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|
| 8 | −4 | 12 | −6 |
| −28 | 14 | −42 | 21 |
| 12 | −6 | 18 | −9 |
| 20 | −10 | 30 | −15 |
For example, choose 8, 14, 18 and −15: they occupy different rows and columns, and their product is −30240. Every row is a fixed row factor times the same four column factors. A complete selection uses every row factor and every column factor exactly once, so rearranging choices leaves the product −30240.
Division by zero is undefined. Check the divisor before using a sign rule. Also keep “starting value + change” separate from “change alone.”
Quiz
An elevator starts at +15 m and changes by −135 m. Its final position is
What is (−100) ÷ 25?
What is (−46) ÷ (−23)?
Which expression models a start of 32°C and a fall of 5°C per hour for 10 hours?
Which divisor is invalid?
Practice Problems
- A room begins at 32°C and cools 5°C each hour. Find the temperature after 10 hours.
- A business gains ₹8 for each of 3000 items and loses ₹5 for each of 5000 items. Find the signed net result.
- If 6400 loss-making items lose ₹5 each, how many gain-making items at ₹8 each balance them?
- Find 36 ÷ (−18) and (−46) ÷ (−23), checking each by multiplication.
- Fill the missing integer: (−3) × __ = 27; __ ÷ (−8) = 7.
- Try two different selections in the chapter’s 4 by 4 magic grid and explain what the row-and-column rule guarantees.
Key Takeaways
• A repeated signed change equals count × signed change per step. • Final position or amount equals starting value plus total change. • Positive and negative contributions can be combined in one expression. • Division finds the missing factor in a multiplication statement. • Equal nonzero signs give a positive quotient; unlike signs give a negative quotient. • Never divide by zero.