Why does (-1) × (-1) = 1?

Pose the driving question and let learners physically experiment with the number line before any rule is stated.

• What does multiplying by a negative even mean?
• Predictions: does it flip direction, or shrink, or something else?
• Set the stage for a proof, not just a rule

One single-choice question asking learners to lock in their hypothesis before the proof unfolds.

• A single, explicit prediction
• Confronts the intuition that 'two negatives should be negative'

Anchor the investigation in the distributive property and the additive inverse, the only ingredients the proof will use.

• 0 = 1 + (-1)
• Multiplying anything by 0 gives 0
• a × (b + c) = a × b + a × c always holds

Step-by-step manipulation of the equation 0 = (1 + (-1)) × (-1), with each term highlighted as it expands.

• Replace 0 × (-1) with 0
• Expand (1 + (-1)) × (-1) using distributivity
• Recognize that -1 + ((-1) × (-1)) must equal 0

Walk through the logical chain that pins (-1) × (-1) to exactly one value: 1.

• (-1) × (-1) must be the additive inverse of -1
• The additive inverse of -1 is, by definition, 1
• Therefore (-1) × (-1) = 1 is not a choice — it is required

Clarify what the proof assumes and what it does not, so the rule's scope is honest.

• The proof depends on distributivity and the existence of additive inverses
• It does not explain why distributivity itself is true
• It applies to integers, rationals, reals — any field-like system

Learners use the same distributive argument to deduce another product, such as (-2) × (-3) = 6.

• Rewrite (-3) as -1 + -1 + -1
• Distribute (-2) across each -1
• Each (-2) × (-1) reduces by the proven rule

Tie the proof back to the original question and reflect on what changed in the learner's intuition.

• (-1) × (-1) = 1 because anything else would break distributivity
• The 'rule' is really a logical necessity
• The negative of a negative is not magic — it is arithmetic being honest with itself