Operations with Integers · Lesson 1 of 6
Signed Movements and Additive Inverses
“See how direction, distance, and zero pairs explain integer addition and subtraction.”
• Use a sum and a signed difference to identify a pair of integers. • Represent left and right movements with signed numbers. • Distinguish a movement’s direction from its magnitude. • Explain integer subtraction using zero pairs and additive inverses.
A pair from its sum and difference
If two numbers add to 25 and the first minus the second is 11, try nearby pairs and check both conditions. The order matters: 18 + 7 = 25 and 18 − 7 = 11, but swapping the numbers changes the difference to −11 without changing their sum. This is a useful first reminder that a negative result can communicate direction or order.
| First | Second | Sum | First − second |
|---|---|---|---|
| 10 | 15 | 25 | −5 |
| 20 | 5 | 25 | 15 |
| 18 | 7 | 25 | 11 |
| 7 | 18 | 25 | −11 |
Problem
Two integers have sum 27 and first-minus-second difference 9. Find them.
- 1.Try a pair with sum 27: 18 and 9.
- 2.Check the ordered difference: 18 − 9 = 9.
- 3.The first number is 18 and the second is 9.
Move left and right on a number line
A carrom coin starts at 0. We call a rightward movement positive and a leftward movement negative. If one strike moves it 5 places right and the next 7 places left, its position is 5 + (−7) = −2. The sign gives direction, while the magnitude 2 gives its distance from 0. The same addition rule works for both directions.
Problem
The first movement is −4 and the final position is 5. What was the second movement?
- 1.Write −4 + b = 5.
- 2.The move from −4 to 5 is 9 places right, so b = +9.
- 3.Check: −4 + 9 = 5.
Several movements can also be added in order. In the pattern 1, −2, 3, −4, ..., 9, −10, pair successive moves: each pair totals −1. Five pairs leave the coin at −5. A sketch can help predict the sign before doing the arithmetic.
Zero pairs make subtraction visible
A positive token represents +1 and a negative token −1. One of each makes zero, so adding a matched pair changes nothing. To calculate 7 − 18, begin with seven positives, add 11 zero pairs, and remove 18 positives. Eleven negatives remain: −11. Removing positives has the same effect as adding negatives; removing negatives has the same effect as adding positives.
The number that sums with an integer to make zero. The inverse of a is −a; in particular, the inverse of −18 is −(−18) = 18.
Problem
Calculate 4 − (−12) using the token idea.
- 1.To remove 12 negative tokens from four positives, add 12 zero pairs.
- 2.Remove the 12 negatives; 16 positives remain.
- 3.Thus 4 − (−12) = 16 = 4 + 12.
In 7 − (−18), the middle sign means subtract, while the sign attached to 18 describes that number. Subtracting −18 adds its inverse, +18.
Quiz
Numbers with sum 25 and first-minus-second difference −11 are
A coin moves +5 then −7 from 0. Where is it?
What is the magnitude of a movement −4?
What is 4 − (−12)?
What is the additive inverse of −18?
Practice Problems
- Find the ordered pair of integers with sum −7 and difference −1.
- Find the pair with sum 0 and difference −10.
- A coin starts at 0 and moves −6, +9 and −4. Find its final position and describe it.
- If a first movement is −4 and the final position is 5, draw or explain the missing movement.
- Use zero pairs to explain 7 − 18 and 4 − (−12).
- Find the total of 1 − 2 + 3 − 4 + ... + 9 − 10 by pairing terms.
Key Takeaways
• The difference first minus second changes sign when the numbers are swapped. • A signed movement records direction and magnitude. • From 0, add signed movements to find the final position. • A +1 and −1 token pair contributes zero. • Subtracting a number means adding its additive inverse.
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Next · Lesson 2
Multiplication with Signed Tokens