Subtracting into the Negatives
Subtracting a larger number from a smaller number produces a negative result, because you move further left on the number line than your starting point. In Grade 6 Saxon Math Course 1 (Chapter 2: Problem Solving with Number and Operations), students find 2 - 5 by starting at 2 and counting back 5 steps, landing at -3. The shortcut: subtract smaller from larger (5 - 2 = 3), then apply a negative sign. Real-world contexts include temperature below freezing, bank overdrafts, and sea-level elevations below zero.
Key Concepts
Property If you subtract a larger number from a smaller number, the answer will be a negative number. To find $2 5$, you can start at $2$ and count back (to the left) five integers to get $ 3$.
Examples What is $8$ less than $3$? Start at $3$ and move left $8$ spaces: $3 8 = 5$. To solve $10 15$, we find the difference is $5$, but since we subtract more than we have, the answer is $ 5$. On a number line, calculating $4 9$ means starting at $4$ and moving $9$ units to the left, landing on $ 5$.
Explanation Ever tried to give away more cookies than you have? That's what subtracting a bigger number feels like! You start at your number on the number line and take so many steps backward (to the left) that you cross past zero into negative land. It's like owing a friend cookies you don't have yet!
Common Questions
How do you subtract when the result is negative?
Find the difference (larger minus smaller), then make the answer negative. For 3 - 8: 8 - 3 = 5, so 3 - 8 = -5.
What is 2 minus 5?
2 - 5 = -3. Start at 2 on the number line and count back 5 to land at -3.
Give a real-world example of subtracting into negatives.
Temperature of 3 degrees drops 7 degrees: 3 - 7 = -4 degrees. A bank account with $5 has a $9 charge: 5 - 9 = -$4 overdraft.
What is -4 minus 3?
-4 - 3 = -7. Starting at -4 and moving 3 more units left reaches -7.
How do you know before calculating that the result will be negative?
If the number being subtracted is larger than the starting number, the result is always negative.