The Other Side of Zero · Lesson 3 of 10
Subtraction as the Movement Needed
“Find the signed change that takes a starting position to a target position.”
• Interpret subtraction as a missing addition. • Identify the starting and target positions in a subtraction. • Find upward and downward changes between signed floors. • Explain journeys across zero. • Check a difference by adding it to the starting position.
Another meaning of subtraction
You may know subtraction as taking objects away: ten books minus four books leaves six books. There is another useful meaning. Subtraction can find the change that makes one quantity equal to another. This meaning works even when our numbers describe positions below zero.
Suppose you have ₹10 and your sister has ₹6. To find how much she needs to reach ₹10, ask 6 + ? = 10. The missing amount is 4, which is also the answer to 10 − 6. We are finding a change from a starting amount of 6 to a target amount of 10.
The difference target − start is the signed change that, when added to the starting value, gives the target value.
Problem
Interpret 15 − 5, 100 − 10, and 74 − 34 as missing additions.
- 1.For 15 − 5, ask 5 + ? = 15. The change is +10.
- 2.For 100 − 10, ask 10 + ? = 100. The change is +90.
- 3.For 74 − 34, ask 34 + ? = 74. The change is +40.
From one floor to another
In the building, the starting floor is where your journey begins, and the target floor is where you want to arrive. If the target is above the start, the required movement is positive. If it is below the start, the required movement is negative. A target equal to the start needs movement zero.
The addition equation is Starting floor + Movement = Target floor. When the movement is unknown, subtraction finds that missing number. We subtract the starting floor from the target floor in that order. This order matters because reversing a journey reverses its direction.
Problem
Find the movement from Floor +2 to Floor +5.
- 1.The target is +5 and the starting floor is +2.
- 2.Count upward: +2 → +3 → +4 → +5. That is three upward steps.
- 3.Therefore (+5) − (+2) = +3. Check: +2 + (+3) = +5.
Problem
Start at −2 and target −1. What movement is needed?
- 1.Floor −1 is above Floor −2, even though both are below ground.
- 2.Move up one floor, so the required movement is +1.
- 3.Thus (−1) − (−2) = +1. Check: −2 + (+1) = −1.
The sign of the target by itself does not determine the sign of the movement. A negative target can be above a more negative starting floor. Likewise, a positive target can be below a larger positive starting floor. Always compare the two positions before deciding which way to move.
Journeys across zero
When a journey crosses ground level, it can be split into two parts: reaching zero, then continuing to the target. The two movements have the same direction, so their sizes combine. Thinking in two parts prevents us from losing a step at the ground floor.
Problem
Find the movement from +3 to −1.
- 1.The target is −1, so the expression is (−1) − (+3).
- 2.From +3 to 0, move −3. From 0 to −1, move −1.
- 3.Combine the changes: −3 + (−1) = −4. Thus −1 − 3 = −4. Check: +3 + (−4) = −1.
Problem
Find the movement from −2 to +2.
- 1.The expression is (+2) − (−2).
- 2.From −2 to 0, move +2; from 0 to +2, move +2.
- 3.The total movement is +4. Check: −2 + (+4) = +2. Therefore (+2) − (−2) = +4.
A signed change is not an unsigned distance
The distance travelled tells us how many units a journey covers; it has no downward or upward sign. The signed change also tells us the direction. Going from +3 to −1 covers four floor intervals, but its change is −4. Returning covers the same four intervals with change +4.
In subtraction, we usually need the signed change, so retain its direction. Equal starting and target floors give zero even if those floors are negative. For instance, (−2) − (−2) = 0 because no movement is needed. To reach zero from −2, however, move +2; this gives 0 − (−2) = +2.
Problem
Someone says (−4) − (−3) = +1. Is this correct?
- 1.The starting floor is −3 and the target is −4. The target is lower, so the movement must be negative.
- 2.Move down one floor: the difference is −1.
- 3.Check −3 + (−1) = −4. The claimed +1 would take you to −2, so it is incorrect.
For target − start, begin at the second number. In +2 − (−2), begin at −2 and aim for +2. Starting at +2 instead would answer a different question.
Quiz
What missing-addition question matches 10 − 6?
From +3 to −1, what movement is needed?
Which expression gives the movement from −2 to +2?
What is (−4) − (−3)?
What is 0 − (−2)?
How can you check that target − start = movement?
Practice Problems
- Evaluate (+1) − (+4), (+4) − (+1), and (+4) − (−3) using floors.
- Find 0 − (+2), 0 − (−2), and (−2) − (−2).
- Find (−1) − (+2), (−1) − (+1), and (+3) − (−3).
- Find the movement from −4 to −3, then the movement back.
- A lift goes from +6 to −5. Split the journey at zero and find its signed movement.
- For −5 + ? = +4, find the missing movement and write a subtraction statement.
- A person says that every subtraction makes a number smaller. Give an example from the lift that shows why this is wrong.
- Explain the difference between the distance and the signed movement from +5 to +2.
−3, +3, +7. The second number is the starting floor.
Key Takeaways
• Subtraction can find the missing number in an addition. • Target minus start gives the signed movement needed. • A higher target needs positive movement; a lower target needs negative movement. • Split a journey at zero when useful. • Check a difference by adding it back to the starting position.