Multiplying and Dividing Negative Numbers
Questions designed to stretch thinking, reveal misconceptions, and spark mathematical reasoning.
Convince Me That…
Students must construct a mathematical argument for why each statement is true.
Look at the pattern: \(3 \times (-4) = -12\), \(2 \times (-4) = -8\), \(1 \times (-4) = -4\), \(0 \times (-4) = 0\). Each time the first number decreases by 1, the answer increases by 4. Continuing: \((-1) \times (-4) = 4\), \((-2) \times (-4) = 8\), \((-3) \times (-4) = 12\).
Alternatively, think of \((-3) \times (-4)\) as “the opposite of \(3 \times (-4)\).” Since \(3 \times (-4) = -12\), the opposite of \(-12\) is \(12\).
Both give \(-15\). We can calculate each side: \((-5) \times 3 = -15\) and \(3 \times (-5) = -15\). Multiplication is commutative — the order doesn’t matter, even when negative numbers are involved.
On a number line, \(3 \times (-5)\) means “three jumps of \(-5\),” landing at \(-15\). For \((-5) \times 3\), think of it as “the opposite of \(5 \times 3\)”: since \(5 \times 3 = 15\), the opposite is \(-15\). Both routes arrive at the same answer. Students who are unsure about commutativity with negatives can check several examples to build confidence.
Division asks: “what do I multiply \(-4\) by to get \(-20\)?” Since \((-4) \times 5 = -20\), the answer must be \(5\). The sign rule for division is the same as for multiplication: dividing two numbers with the same sign gives a positive result.
Alternatively, use the pattern: \((-20) \div 4 = -5\), \((-20) \div 2 = -10\), \((-20) \div 1 = -20\), \((-20) \div (-1) = 20\), \((-20) \div (-2) = 10\), \((-20) \div (-4) = 5\). As the divisor passes through zero and becomes negative, the answers switch from negative to positive.
\((-2)^2 = (-2) \times (-2) = 4\). Two negative signs multiply to give a positive. But \((-2)^3 = (-2) \times (-2) \times (-2) = 4 \times (-2) = -8\). The first pair of negatives cancel to give positive 4, but then multiplying by one more \(-2\) flips the sign back to negative.
The key rule: an even power of a negative number is always positive (the negative signs pair up and cancel), while an odd power of a negative number is always negative (there’s one unpaired negative sign left over).
The symbol \(\sqrt{\dots}\) specifically asks for the principal (positive) square root. So \(\sqrt{16}\) is just 4.
However, the equation \(x^2 = 16\) asks “what number, when multiplied by itself, equals 16?” We know \(4 \times 4 = 16\), but we also know that \((-4) \times (-4) = 16\). Therefore, \(x\) can be 4 or -4. This subtle distinction between the function (which gives one output) and solving the equation (which can have two solutions) is a key concept.
Give an Example Of…
For each prompt, provide: an example, another example, one no-one else will think of, and one someone might think works but doesn’t.
Example: \((-6) \times (-6) = 36\)
Another: \((-4) \times (-9) = 36\)
Creative: \((-0.5) \times (-72) = 36\) — using decimals or fractions opens up infinitely many pairs beyond whole-number factor pairs.
Trap: \((-6) \times 6 = -36\), not 36. A student might think “I’ve used a negative” but forget that both numbers must be negative for the product to be positive.
Example: \(3 \times (-5) = -15\)
Another: \((-7) \times 2 = -14\)
Creative: \((-0.1) \times 0.1 = -0.01\) — tiny decimals still follow the same sign rules. Or \((-1) \times (-1) \times (-1) = -1\) — three negatives give a negative product.
Trap: \((-4) \times (-3) = 12\), not \(-12\). A student might think “there’s a negative number, so the answer must be negative” — but two negatives multiply to give a positive.
Example: \(18 \div (-3) = -6\) — since \((-18) \div 3 = -6\)
Another: \((-36) \div 6 = -6\)
Creative: \((-1.8) \div 0.3 = -6\) — decimals work too. Or \((-60) \div 10 = -6\).
Trap: \((-18) \div (-3) = 6\), not \(-6\). A student might think “I’ve just changed which number is negative, so the answer stays the same.” But flipping the sign of the divisor flips the sign of the answer: \(-6\) becomes \(6\).
Example: \(n = -1\). Then \((-1)^2 = 1\) and \((-1)^3 = -1\). Since \(1 > -1\), we have \(n^2 > n^3\).
Another: \(n = -2\). Then \((-2)^2 = 4\) and \((-2)^3 = -8\). Since \(4 > -8\), it works.
Creative: \(n = 0.5\). Then \(0.5^2 = 0.25\) and \(0.5^3 = 0.125\). Since \(0.25 > 0.125\), it works. Also, negative fractions work: \(n = -0.5\). \((-0.5)^2 = 0.25\) and \((-0.5)^3 = -0.125\). Since \(0.25 > -0.125\), the rule holds for any number \(n < 1\) (except 0).
Trap: \(n = 2\). Then \(2^2 = 4\) and \(2^3 = 8\). Since \(4 < 8\), we get \(n^2 < n^3\), so this doesn’t satisfy the condition. A student might assume “squaring is always bigger than cubing” without checking.
Always, Sometimes, Never
Is the statement always true, sometimes true, or never true? Students should justify their decision with examples.
For example: \((-2) \times (-3) \times (-4)\). The first pair gives \((-2) \times (-3) = 6\) (positive), but then \(6 \times (-4) = -24\) (negative). Likewise \((-1) \times (-1) \times (-1) = -1\). The result is always negative.
The key is to count the negative signs: an even number of negatives gives a positive product (the signs pair up and cancel); an odd number gives a negative product (one sign is left over). Three is odd, so three negatives always give a negative. Students who know “neg × neg = pos” sometimes assume adding more negatives keeps the answer positive — but the third negative flips the sign back.
When you multiply numbers with different signs, the result is always negative. For example: \((-3) \times 5 = -15\), \((-1) \times 100 = -100\), \((-0.2) \times 4 = -0.8\). There is no exception.
This catches students who muddle the sign rules. The rule is simple: same signs → positive, different signs → negative. This rule applies identically to both multiplication and division.
True for positive numbers: \(5 \times (-3) = -15\), and \(-15 < 5\). But it fails for zero: \(0 \times (-3) = 0\). Since 0 is equal to 0, it is not smaller (strictly less) than the original number.
False for negative numbers: \((-5) \times (-3) = 15\), and \(15\) is larger than \(-5\). The misconception here is “multiplying makes bigger” or its reverse, “multiplying by a negative always makes smaller.” The effect depends on whether the original number is positive, negative, or zero.
True when both are positive: \(3 \times 5 = 15\), which is positive, and both 3 and 5 are positive. But also false: \((-3) \times (-5) = 15\), which is positive, yet neither number is positive.
A positive product tells you the signs are the same — but they could both be positive or both be negative. Students who assume “positive answer = positive inputs” are forgetting the negative × negative = positive rule.
Odd One Out
Which is the odd one out? The challenge: make a valid mathematical case for each one being the odd one out.
Note: All three expressions are equal to -3. This question asks you to look at the structure and placement of the negative sign.
Explain the Mistake
Each example contains a deliberate error targeting a common misconception. Can you find where and why the reasoning goes wrong?
Answer: \(-20\)
Reasoning: “4 times 5 is 20, and there’s a negative sign so the answer is −20.”
The student noticed there are negatives and assumed the answer must be negative. This is the “any negative means a negative answer” misconception. They treated the two negative signs as if there were just one.
The correct answer is \(20\). When multiplying two negative numbers, the signs cancel: negative × negative = positive. The student correctly computed \(4 \times 5 = 20\) but then applied the wrong sign rule. A useful check: \((-4) \times (-5)\) must be the opposite of \(4 \times (-5) = -20\), so it’s \(+20\).
Answer: \((-3)^2\) is larger ✓
Reasoning: “Squaring and cubing both make negatives positive. \((-3)^2 = 9\) and \((-2)^3 = 8\). Since 9 is bigger than 8, \((-3)^2\) is larger.”
The answer happens to be correct — \((-3)^2\) is larger — but the reasoning is dangerously wrong. The student claims “cubing makes negatives positive,” which is false. Only even powers make negatives positive; odd powers keep them negative.
The correct calculation is \((-3)^2 = 9\) and \((-2)^3 = (-2) \times (-2) \times (-2) = -8\). So \((-3)^2\) is larger because \(9 > -8\), not because \(9 > 8\). The student got lucky here. If asked to compare \((-2)^3\) with \(-9\), their faulty logic would give \((-2)^3 = 8\) and then conclude \(8 > -9\) — when in reality \((-2)^3 = -8\) and \(-8 > -9\) only barely, for a completely different reason.
Answer: \(24\)
Reasoning: “There are three negatives. Two negatives make a positive, so I can cancel them: \(2 \times 3 \times 4 = 24\).”
The student knows that “two negatives make a positive” but applies it incorrectly to all three signs at once, as if the negatives simply disappear. This is the “cancel all the negatives” misconception — treating the sign rule as a blanket removal rather than working through the multiplication step by step.
The correct approach is sequential: \((-2) \times (-3) = 6\) (two negatives cancel → positive), then \(6 \times (-4) = -24\) (positive × negative → negative). The answer is \(-24\), not \(24\). The shortcut is to count the negative signs: an even count gives positive, an odd count gives negative. Three is odd, so the result must be negative.
Answer: \(9\)
Reasoning: “A negative squared is always positive. Negative 3 squared is 9.”
The student has confused \(-3^2\) with \((-3)^2\). This is the “bracket blindness” misconception — ignoring the crucial difference that brackets make. Without brackets, \(-3^2\) means \(-(3^2) = -(9) = -9\). The squaring applies only to the 3, not to −3.
With brackets, \((-3)^2 = (-3) \times (-3) = 9\). The notation matters enormously: \(-3^2 = -9\) but \((-3)^2 = 9\).
A specific trap: Typing \(-3^2\) into a calculator usually gives \(-9\) because the calculator follows Order of Operations strictly (squaring before negation). To square the negative number, you must type brackets: \((-3)^2\).