The Other Side of Zero · Lesson 2 of 10
Adding Signed Movements
“Follow lift journeys to see how signed movements combine and cancel.”
• Find a target floor from a starting floor and a signed movement. • Combine two or more signed movements. • Explain when opposite movements cancel. • Identify additive inverses, including the inverse of zero. • Write equations for different journeys to the same target.
A starting floor and a change
A lift journey contains two pieces of information: where you begin and how you move. If you begin at Floor +1 and go up two floors, you finish at +3. Addition records this change. The first number gives a position; the second gives the movement applied to it.
Going up is a positive movement and going down is a negative movement. This means that adding a negative movement can take you to a lower floor. Addition does not always mean that the final number becomes greater: it means that we combine the starting position with the specified change.
Problem
Start at the Food Court, Floor +1, and press +2. Where do you arrive?
- 1.The starting position is +1.
- 2.Move two floors up: +1 → +2 → +3.
- 3.The target is the Book Store, and (+1) + (+2) = +3.
Problem
Start at Floor +2 and press −3. Where do you arrive?
- 1.Move down three floors from +2.
- 2.The positions are +1, then 0, then −1.
- 3.The Toy Store is at −1, so (+2) + (−3) = −1. Count three movements, not four labelled positions.
Combining movements
You can also use addition when both numbers describe movements. Suppose you go up two floors and then down three. The second movement cancels the upward part and takes you one floor lower than where you began. The combined movement is −1.
Gurmit begins in the Toy Store at −1. He intends to go down two floors but accidentally presses + twice, then presses − three times. His combined movement is (+2) + (−3) = −1. His final floor is −1 + (−1) = −2. Notice the difference: −1 is the net movement, while −2 is the final floor.
Problem
Find the combined movement (+4) + (−3) + (−2).
- 1.Four up and three down leave one up: +4 + (−3) = +1.
- 2.Then move down two more: +1 + (−2) = −1.
- 3.The net result is one floor down. From Floor 0 the target would be −1; from Floor +5 it would be +4.
For two upward movements, add their sizes: +1 + (+4) = +5. Reversing their order, +4 + (+1), still gives +5. Two downward movements also combine their sizes, but the result remains downward. For example, −2 + (−3) = −5. With opposite directions, match the upward and downward parts; whichever has an unmatched part gives the result’s direction.
Undoing a movement
A journey can bring you back to where you started. If you go up three floors and then down three floors, the combined movement is zero. These movements are opposite in direction and equal in size. We call numbers such as +3 and −3 additive inverses.
The additive inverse of a number is the number that gives zero when added to it. The inverse of +3 is −3, and the inverse of −3 is +3.
Problem
Basant starts at 0, presses +3 by mistake, and wants to return to 0. What movement undoes it?
- 1.He is now at +3.
- 2.To undo three upward steps, he needs three downward steps, written −3.
- 3.Thus (+3) + (−3) = 0. The movement −3 is the additive inverse of +3.
Problem
Which number added to 0 gives 0?
- 1.A positive movement leaves zero upwards; a negative movement leaves it downwards.
- 2.Only no movement leaves the lift at the same level.
- 3.The additive inverse of 0 is 0, because 0 + 0 = 0.
| Number | Additive inverse | Check |
|---|---|---|
| +4 | −4 | 4 + (−4) = 0 |
| −4 | +4 | −4 + 4 = 0 |
| −3 | +3 | −3 + 3 = 0 |
| 0 | 0 | 0 + 0 = 0 |
| +2 | −2 | 2 + (−2) = 0 |
| −1 | +1 | −1 + 1 = 0 |
Many journeys, one destination
The same target floor can be reached from different starting positions. A lower starting position needs more upward movement, or less downward movement, than a higher starting position. Writing several correct equations helps us keep the start, movement, and target distinct.
Problem
Write three addition statements ending at −5.
- 1.From +2, go down seven: (+2) + (−7) = −5.
- 2.From −2, go down three: (−2) + (−3) = −5.
- 3.From 0, go down five: 0 + (−5) = −5. Each equation combines its own starting position with its own movement.
The + between the starting floor and movement tells you to apply the movement. It does not change a negative movement into an upward one. In +2 + (−3), move down three floors.
Quiz
Start at −1 and move +2. What is the target?
What is the net movement +2 + (−3)?
What is the additive inverse of −9?
Which pair cancels?
What is −1 + (+2) + (−3)?
Which equation reaches Floor −5?
Practice Problems
- Evaluate +1 + (+4), +4 + (+1), +4 + (−3), and −1 + (+2).
- Evaluate −1 + (+1), 0 + (+2), and 0 + (−2).
- Find the net movement −1 + (+2) + (−3).
- Write the inverses of +9, +7, −8, −5, and 0.
- From Floor −1, carry out Gurmit’s +2 then −3 journey. State both the net movement and final floor.
- Write journeys to −5 from starting floors +1, −4, and −5.
- Give three integers with sum −8 and explain your choice.
- Explain why adding zero leaves a floor unchanged.
+5, +5, +1, +1.
Key Takeaways
• Starting position plus movement gives the target position. • Signed movements can be combined by addition. • Equal opposite movements cancel. • Every integer has an additive inverse; zero is its own inverse. • The net movement and the final position need not be the same number.