The Other Side of Zero · Lesson 5 of 10
Adding Integers with Zero Pairs
“Use positive and negative tokens to make integer addition visible.”
• Represent an integer with signed tokens. • Identify and cancel zero pairs. • Add integers of the same sign and of opposite signs. • Explain why different token collections can have the same value. • Connect tokens to movements and positions.
Numbers you can handle
A number line shows where numbers are and how movements act. Tokens provide another model that you can hold, draw, or rearrange. Let a green token labelled + have value +1 and an orange token labelled − have value −1. The labels matter; colours alone are not enough to explain the model.
Three + tokens represent +3. Four − tokens represent −4. To find the value of a mixed collection, combine the tokens’ values. A + token and a − token have total value zero, because +1 + (−1) = 0. Such a pair changes neither a starting floor nor a total value.
One +1 token and one −1 token together. Their combined value is zero, so adding or removing a complete zero pair leaves a collection’s value unchanged.
An empty collection represents zero. So does a collection of two positive and two negative tokens. These collections have different numbers of physical tokens but the same total value. Always distinguish the number of tokens from the number represented by the tokens.
Addition means combining collections
To add two integers, put down a token collection for each one and combine them. Then cancel as many zero pairs as possible. The remaining tokens show the result. You do not need to change the value or direction of either collection.
Problem
Use tokens for (+6) + (+4).
- 1.Lay out six + tokens for +6 and four + tokens for +4.
- 2.Combine them into a collection of ten + tokens. There are no − tokens, so there are no zero pairs.
- 3.The result is +10.
Problem
Use tokens for (−3) + (−2).
- 1.Lay out three − tokens and then two more − tokens.
- 2.Together they are five − tokens. No cancellation is possible.
- 3.The result is −5. Two negative addends combine into a larger downward amount, not a positive amount.
When both addends are positive, the result is positive. When both are negative, the result is negative. For two negatives, add the sizes and keep the negative direction. This is the same reasoning as combining two downward lift movements.
Addition with opposite signs
When the signs differ, each positive token can cancel one negative token. Keep removing complete pairs until one sign is exhausted. The sign of the unmatched tokens gives the sign of the result, and their number gives its size.
Problem
The attendant starts at Floor 0 and collects five + tokens and three − tokens. Which floor do they indicate?
- 1.The tokens record five upward button presses and three downward button presses.
- 2.Make three zero pairs, using three + tokens and all three − tokens.
- 3.Two + tokens remain, so (+5) + (−3) = +2. Starting at 0, the final floor is +2.
Problem
Find (+5) + (−8) with tokens.
- 1.Combine five + tokens and eight − tokens.
- 2.Remove five zero pairs. All five positives disappear, along with five of the eight negatives.
- 3.Three − tokens remain, giving −3. Thus 5 + (−8) = −3.
Problem
Use tokens for (−2) + (+6) and (+6) + (−2).
- 1.Both collections contain six + tokens and two − tokens.
- 2.Cancel two zero pairs in either case.
- 3.Four + tokens remain. Both sums are +4, showing that changing the order of two addends does not change their total.
Representations can look different
Suppose a collection has three + tokens and five − tokens. After three pairs cancel, its value is −2. Another collection with just two − tokens also has value −2. Adding a zero pair changes the picture, but not the number represented.
Problem
Find the values of a collection with 3 positives and 5 negatives, and one with 6 positives and 3 negatives.
- 1.The first loses three zero pairs and leaves two negatives: +3 + (−5) = −2.
- 2.The second loses three zero pairs and leaves three positives: +6 + (−3) = +3.
- 3.If each records movements from Floor 0, the attendant ends on −2 and +3 respectively.
Cancellation also explains additive inverses. If there are seven positives and seven negatives, every token pairs off and the result is zero. If you add no tokens at all, the value remains unchanged. These facts connect the token model to +7 + (−7) = 0 and −4 + 0 = −4.
Two negative tokens do not cancel: −1 + (−1) = −2. A zero pair requires one positive and one negative. Also, removing only one member of a pair changes the value.
Quiz
What makes a zero pair?
What is the value of 3 positives and 5 negatives?
What is −3 + (−2)?
Which collection represents zero?
What remains after cancelling zero pairs in +5 + (−7)?
What happens to a value when one zero pair is added?
Practice Problems
- Use tokens to find +6 + (+4), −3 + (−2), +5 + (−7), and −2 + (+6).
- Draw a collection with 6 positive and 3 negative tokens and state its value.
- Draw two different collections that both represent −3.
- How many zero pairs can be removed from 8 positives and 5 negatives? What remains?
- Represent +4 + (−4) and explain why its value is zero.
- A pocket has 2 positives, 7 negatives, and 4 more positives. Find the combined movement and the final floor if the attendant started at −2.
- Explain why 7 positives plus 2 negatives and 2 negatives plus 7 positives give the same answer.
- Why is “there are ten tokens, so the value is ten” an unreliable statement?
+10, −5, −2, +4.
Key Takeaways
• Each token represents +1 or −1. • A positive and negative token form a zero pair. • To add, combine collections and cancel complete zero pairs. • Unmatched tokens determine both the sign and size of the sum. • Adding or removing zero pairs preserves the represented value.