How to explain it
The anchor students hold onto: The additive inverse of any number a is −a. A number and its additive inverse are always the same distance from 0 on opposite sides of the number line, and their sum is always 0.
You now know that a number and its opposite always sum to 0. The additive inverse is the key idea that makes integer addition with different signs work — next in the 7.NS.A.1 strand.
Worked examples
Example 1
Integers
Find: additive inverse of −7.
Step 1−7 is 7 units to the left of 0 on the number line.
Step 2Its opposite is +7 — same distance from 0, opposite side.
Step 3(−7) + 7 = 0. The additive inverse of −7 is 7.
AnswerThe additive inverse of −7 is 7.
Example 2
Special Cases
Find: additive inverse of 0.
Step 10 is at the origin of the number line.
Step 20 is equidistant from both sides — its own opposite.
Step 30 + 0 = 0. The additive inverse of 0 is 0.
AnswerThe additive inverse of 0 is 0.
Example 3
Real-World Context
Temperature: −12 and +12.
Step 1−12 and +12 are opposites: both 12 units from 0, on opposite sides.
Step 2−12 is the additive inverse of +12 (and vice versa).
Step 3(−12) + 12 = 0. Opposite temperatures combine to make 0.
Answer(−12) + 12 = 0; opposite quantities combine to make 0.
Common mistakes
What students write
Confuses the additive inverse with absolute value — writes the additive inverse of −8 as −8 (reasoning that the absolute value already has a sign), rather than +8.
The fix
The additive inverse changes the sign. The additive inverse of −8 is +8, not −8, because (−8) + 8 = 0. Verify: if the sum is not 0, the inverse is wrong.
Try this
Rivera says the additive inverse of −9 is also −9, because the absolute value of −9 is 9, and since there is already a negative sign, the inverse keeps the sign. Rivera's work: Additive inverse of −9 → |−9| = 9 → keep the negative sign → −9 Identify Rivera's error and find the correct additive inverse of −9.
What students write
Claims that two identical numbers like −3 and −3 are additive inverses because they have the same absolute value — confusing "same absolute value" with "additive inverse."
The fix
Additive inverses must sum to 0. (−3) + (−3) = −6, not 0. The additive inverse of −3 is +3, because (−3) + 3 = 0. Opposite signs, same distance from 0.
Try this
Tran says that −3 and −3 are additive inverses because they both have the same absolute value of 3 and the same sign. Tran's work: |−3| = 3 and |−3| = 3 → same absolute value and same sign → additive inverses Identify Tran's error and name the correct additive inverse of −3.
Teacher tip
Head off the two predictable errors before they happen. First: The additive inverse changes the sign. The additive inverse of −8 is +8, not −8, because (−8) + 8 = 0. Verify: if the sum is not 0, the inverse is wrong. Second: Additive inverses must sum to 0. (−3) + (−3) = −6, not 0. The additive inverse of −3 is +3, because (−3) + 3 = 0. Opposite signs, same distance from 0.