Multiplication of signed numbers: why a positive times a negative is negative and why two negatives give a positive, the counting rule for the sign of a product, the six properties of multiplication including the properties of zero and of negative one, simplifying products containing variables, and computing a change in position as velocity times time.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 2 — Properties of Real Numbers
Multiplying Real Numbers
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 93-98 — the lesson these objectives are drawn from
Warm-up
Multiplication by a whole number has always been repeated addition. That does not change when the number being repeated is negative.
Discussion prompt
Write out 3 times negative 2 as a repeated addition and work it out. Then say what you would expect negative 3 times negative 2 to be, and why.
Hint: The first part you can compute. The second you have to reason about.
Answer:
\[ 3(-2) = (-2) + (-2) + (-2) = -6 \]
Three copies of negative two total negative six, so a positive times a negative is negative. For the second, negative three is the opposite of three, so negative three times negative two should be the opposite of negative six — which is positive six. The answer is forced by what you already know rather than chosen.
Concept
In addition the sign of the answer depended on which number was larger. In multiplication it depends on nothing but how many of the factors are negative — an odd number of them makes the product negative, an even number makes it positive.
closure property — The product of any two real numbers is itself a unique real number. The real numbers are closed under multiplication as well as under addition.
That is a genuinely simpler rule than addition's, and it is worth noticing how different in kind it is.
Figure (svg): The sign rule for products shown as a count of negative factors, odd giving negative and even giving positive
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 93-93
Section
Section 1
Concept
Multiplication by a positive integer is repeated addition, and that settles the mixed-sign case. The case of two negatives then follows from the definition of opposites rather than being a separate convention.
\[ 3(-2) = (-2) + (-2) + (-2) = -6 \]
And since negative three is the opposite of three, negative three times negative two is the opposite of negative six, which is six.
Figure (svg): Three jumps of negative two shown on a number line, landing at negative six
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 93-93 — the derivation preceding the sign rules
Picture it
Each arrow is one copy of negative two.
Figure (svg): Three jumps of negative two shown on a number line, landing at negative six
Three leftward jumps of two units land six units left of zero. Nothing here has to be assumed — the answer is the same one Lesson 2.3 would have given for the sum.
Worked example
Repeated addition does the whole job when one factor is a positive integer.
\[ \text{Show that } 3(-2) = -6 \text{ using repeated addition.} \]
Write the product as repeated addition
Why: Three times something means three copies of it added together.
\[ (-2) + (-2) + (-2) \]
Apply the same-sign addition rule
Why: All three terms are negative, so the absolute values add and the negative sign is kept.
\[ -(2 + 2 + 2) \]
Compute
Why: Two three times is six, and the sign is negative.
\[ -6 \]
State the general conclusion
Why: A positive number times a negative number is negative.
Figure (svg): Three jumps of negative two shown on a number line, landing at negative six
\[ 3(-2) = (-2) + (-2) + (-2) = -6 \]
Verify: check with the commutative property
Why: Negative two times three should give the same answer, since multiplication is commutative — and it does, being two copies of negative three, which is also negative six. Two routes to the same answer confirm the reasoning rather than repeat it.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 93-93
Prediction
Reason from what you already know rather than from a memorised rule.
Predict first
Given that 5 times negative 3 is negative 15, what must negative 5 times negative 3 be?
Correct: 15.
\[ -5(-3) = -(5(-3)) = -(-15) = 15 \]
Why: Negative five is the opposite of five, so negative five times negative three is the opposite of five times negative three. That is the opposite of negative fifteen, which is fifteen. The answer is forced by the mixed-sign result together with the meaning of opposites, so it is not a separate rule to remember.
Worked example
This one cannot be done by repeated addition, because you cannot take negative three copies of anything.
\[ \text{Show that } -3(-2) = 6. \]
Recognise that negative 3 is the opposite of 3
Why: By the property of negative one, negative three is negative one times three.
\[ -3 = -(3) \]
Rewrite the product using that
Why: Negative three times negative two is the opposite of three times negative two.
\[ -3(-2) = -(3(-2)) \]
Substitute the result from the previous example
Why: Three times negative two is negative six.
\[ -(-6) \]
Take the opposite
Why: The opposite of negative six is six.
\[ 6 \]
Figure (svg): A derivation showing that 3 times negative 2 is negative 6, so negative 3 times negative 2 must be positive 6
\[ -3(-2) = -(3(-2)) = -(-6) = 6 \]
Verify: check that no other answer would be consistent
Why: If negative three times negative two were negative six, then it would equal three times negative two, which would make negative three equal to three. That is false, so positive six is the only value consistent with the properties already established.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 93-93
Trap
Two negatives make a positive because that is the rule we were told.
Memorise the sign rules as four unrelated facts
Why: Four short rules are easy to memorise, so understanding them feels optional.
Memorised rules decay, and there is no way to check a half-remembered one. Two months later, does a negative times a negative give a positive or a negative?
\[ 3(-2) = -6 \;\Longrightarrow\; -3(-2) = -(-6) = 6 \]
Derive the rule from repeated addition and opposites whenever you doubt it
Why: The derivation takes about ten seconds and it cannot be misremembered, because each step follows from the one before.
The rule is not a convention. Given that three copies of negative two total negative six, positive six is the only value that keeps the arithmetic consistent.
Socratic
Repeated addition proves one of the two rules and not the other.
Discussion prompt
Explain why repeated addition works for 3 times negative 2 but cannot be used directly for negative 3 times negative 2. Then say what fills the gap.
Hint: Try to say out loud what negative three copies of something would mean.
Answer:
Three times negative two means three copies of negative two, which is a sensible instruction you can carry out. Negative three times negative two would mean negative three copies of negative two, and taking a negative number of copies of something is not an action anyone can perform — the idea of repeated addition simply runs out.
What fills the gap is the definition of opposites together with the property of negative one. Negative three is the opposite of three, so the product must be the opposite of what three would have given. That reasoning extends multiplication to negative factors in the only way consistent with everything already established.
Elimination
Four attempts to justify that two negatives give a positive.
Eliminate the wrong options
Which justification actually works?
Survives elimination: A
Why: The sound argument goes through opposites: negative three is the opposite of three, so its product with negative two must be the opposite of three's product with negative two. That is a chain of established facts rather than an analogy, and it explains why no other answer is possible.
Faded example
The chain is started. Supply the two missing values.
Fill in the blanks
4(-3) = (-3) + (-3) + (-3) + (-3) = -12 \qquad -4(-3) = -(4(-3)) = -(-12) = 12
Why: Four copies of negative three total negative twelve, and negative four times negative three is the opposite of that, which is twelve. The two blanks hold the same number deliberately: the second derivation reuses the first one's answer rather than computing anything new.
Section
Section 2
Concept
For a product of nonzero numbers, the sign depends only on how many negative factors there are. An odd number of negative factors makes the product negative; an even number makes it positive.
Zero negative factors counts as even, which is why a product of positives is positive.
Figure (svg): The sign rule for products shown as a count of negative factors, odd giving negative and even giving positive
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 93-93 — the Rules for the Sign of a Product box
Picture it
The middle column is the only thing that determines the sign.
Figure (svg): The sign rule for products shown as a count of negative factors, odd giving negative and even giving positive
Notice how little the numbers matter. A product of one negative and a hundred positives is negative; a product of two enormous negatives is positive. Only the count decides.
Worked example
This is Example 1 from the textbook. Count first, multiply second.
\[ \text{Find } \; -4(5), \quad -2(-5)(3), \quad -10(-0.2)(-4), \quad (-2)^4. \]
Count and multiply the first
Why: One negative factor, which is odd, so the product is negative. Four times five is twenty.
\[ -20 \]
Count and multiply the second
Why: Two negative factors, which is even, so positive. Two times five times three is thirty.
\[ 30 \]
Count and multiply the third
Why: Three negative factors, odd, so negative. Ten times 0.2 is two, times four is eight.
\[ -8 \]
Count and multiply the fourth
Why: Negative two to the fourth power is four negative factors, even, so positive. Two to the fourth is sixteen.
\[ 16 \]
Figure (svg): The solution to Worked example four products shown as a ladder of expressions, one row per algebraic move
\[ -20, \quad 30, \quad -8, \quad 16 \]
Verify: check each sign against its count
Why: One, three negatives gave negative answers; two, four gave positive ones. Odd counts produced negatives and even counts positives without exception, which is what the rule claims — and the sizes of the factors played no part in any of the four sign decisions.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 93-93
Sorting
Count the negative factors in each product. Do not compute the size.
Sort into buckets
Sort each product by the sign of its answer.
Not one of these decisions required knowing how big the factors were. The sign question and the size question are completely separate in multiplication, which was not true for addition.
Worked example
Guided Practice 1 to 4. Say the count out loud before writing anything.
\[ \text{Find } \; -3(5), \quad -2(-4)(5), \quad \tfrac{1}{3}(-3)(-2), \quad (-2)^3. \]
One negative, odd, so negative
Why: Three times five is fifteen.
\[ -15 \]
Two negatives, even, so positive
Why: Two times four times five is forty.
\[ 40 \]
Two negatives, even, so positive
Why: One third times three is one, times two is two.
\[ 2 \]
Three negatives, odd, so negative
Why: Two cubed is eight.
\[ -8 \]
Figure (svg): The solution to Worked example four from guided practice shown as a ladder of expressions, one row per algebraic move
\[ -15, \quad 40, \quad 2, \quad -8 \]
Verify: check the third one carefully
Why: One third times negative three is negative one, and negative one times negative two is two. Two negatives gave a positive, and the fraction did not affect the sign at all — it only affected the size, which is exactly the separation the rule promises.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 93-93
Error analysis
The student found four products by counting negative factors. Two are wrong.
Annotate
On: \( -4(5) = -20 \qquad -2(-5)(3) = -30 \qquad -10(-0.2)(-4) = -8 \qquad (-2)^4 = -16 \)
Counting is the whole method, and it has to be done to the end. Stopping at the first minus sign gives the right answer exactly when the count happens to be one.
Prediction
The count is all you need.
Predict first
A product has five negative factors and three positive ones. What is its sign?
Correct: Negative, since five is odd.
\[ \text{five negatives: } (-)(-)(-)(-)(-) = -, \text{ since two pairs cancel and one is left over} \]
Why: Only the count of negative factors matters, and five is odd, so the product is negative. The total number of factors is irrelevant, comparing counts of positives and negatives is irrelevant, and the sizes are irrelevant. That single-question test is what makes multiplication signs simpler than addition signs.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Addition | Multiplication | |
|---|---|---|
| What decides the sign | which number is larger | how many negative factors there are |
| Do sizes matter? | Yes | No |
| -4 and -5 combined | -9 | 20 |
The same two numbers give negative nine under addition and positive twenty under multiplication. The two operations use completely different sign logic, and mixing them up is the main hazard of this chapter.
Edge cases
The counting rule is stated for products of nonzero numbers. Find out why.
Discussion prompt
What happens to the sign rule if one of the factors is zero? Work out negative 4 times 0 times negative 5, count the negative factors, and say why the rule has to exclude zero.
Hint: Count the negatives, apply the rule, then actually compute the product.
Answer:
\[ -4 \cdot 0 \cdot (-5) = 0 \]
There are two negative factors, which is even, so the rule would predict a positive answer. But the product is zero, and zero is neither positive nor negative — so the prediction fails.
This is why the rule is stated for nonzero numbers. The property of zero overrides everything: any product containing a zero factor is zero, whatever else is present. Checking for a zero factor before counting signs is worth doing, since it can end the problem immediately.
Section
Section 3
Concept
Multiplication has closure, commutative, associative and identity properties that mirror addition's exactly. It adds two more: the property of zero and the property of negative one.
The identity for multiplication is one, not zero — a difference worth noticing, since zero is the identity for addition.
| Property | Statement | Example |
|---|---|---|
| Closure | ab is a unique real number | 4 times 2 = 8 |
| Commutative | ab = ba | 3(-2) = (-2)3 |
| Associative | (ab)c = a(bc) | (6 times 2)3 = 6(2 times 3) |
| Identity | 1 times a = a | 1 times (-4) = -4 |
| Property of zero | 0 times a = 0 | 0 times (-2) = 0 |
| Property of negative one | -1 times a = the opposite of a | -1 times (-3) = 3 |
Figure (svg): The six properties of multiplication, each stated in symbols with an example
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 94-94 — the Properties of Multiplication box
Picture it
The first four have addition counterparts; the last two do not.
Figure (svg): The six properties of multiplication, each stated in symbols with an example
The property of negative one is the bridge back to Lesson 2.2: multiplying by negative one is exactly taking the opposite, which is why the sign derivations in this lesson work.
Worked example
The skill is reading what changed, exactly as in Lesson 2.3.
\[ \text{Name the property: } \; 3(-2) = (-2)3; \quad (6 \cdot 2)3 = 6(2 \cdot 3); \quad 1 \cdot (-4) = -4; \quad -1 \cdot (-3) = 3. \]
In the first, the factors changed seats
Why: Three and negative two appear in the opposite order.
In the second, only the brackets moved
Why: Six, two and three are in the same order on both sides.
In the third, multiplying by one left the number alone
Why: One is the multiplicative identity.
In the fourth, multiplying by negative one gave the opposite
Why: Negative one times negative three is three, the opposite of negative three.
Figure (svg): The solution to Worked example name the property shown as a ladder of expressions, one row per algebraic move
\[ \text{commutative}, \; \text{associative}, \; \text{identity}, \; \text{property of } -1 \]
Verify: apply the two-question test to the first two
Why: Did the factors move, or only the brackets? In the first the factors moved, so commutative; in the second only the brackets, so associative. The same test that worked for addition works unchanged here, which is the point of the properties mirroring each other.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 94-94
Matching
Four statements, four properties.
Match the pairs
Why: The first two are the rearrangement pair, told apart by whether the factors or the brackets moved. The last two are each about one specific number: zero, which destroys any product, and negative one, which reflects a number to its opposite. Recognising which number a property is about is faster than recalling its name.
Worked example
The properties license rearranging a product to make it easy.
\[ \text{Evaluate } \; -4 \cdot 7 \cdot (-25) \; \text{ efficiently.} \]
Count the negative factors first
Why: Two of them, which is even, so the answer is positive.
Look for a convenient pair among the absolute values
Why: Four and twenty-five multiply to a hundred, which is far easier than any other pairing.
\[ 4 \cdot 25 = 100 \]
Reorder and regroup
Why: The commutative and associative properties allow the factors to be paired freely.
\[ (4 \cdot 25) \cdot 7 \]
Finish
Why: One hundred times seven is seven hundred, and the sign is positive.
\[ 700 \]
Figure (svg): The solution to Worked example use the properties to multiply efficiently shown as a ladder of expressions, one row per algebraic move
\[ -4 \cdot 7 \cdot (-25) = 700 \]
Verify: multiply left to right without rearranging
Why: Negative four times seven is negative twenty-eight; negative twenty-eight times negative twenty-five is seven hundred. The same answer by a harder route, which confirms the rearrangement was legal rather than lucky.
Trap
\[ \text{The identity property says } a \cdot 0 = a \]
Carry the identity from addition across to multiplication unchanged
Why: Zero was the identity for addition, and the two lists of properties otherwise look almost identical.
\[ (-4) \cdot 0 = 0, \text{ not } -4 \]
Zero destroys a product rather than leaving it alone. The number that leaves a product unchanged is one.
\[ 1 \cdot a = a \qquad 0 \cdot a = 0 \]
Keep the two properties separate: one is the identity, zero has its own property
Why: Each operation has its own identity — the number that changes nothing — and for multiplication that is one.
A useful check: an identity must leave every number unchanged. Try it on a specific number, and zero fails immediately.
Discrimination
Addition and multiplication have different identity elements.
Sort into buckets
Sort each statement by which operation it belongs to.
Elimination
The step is from negative 4 times 7 times negative 25 to the quantity 4 times 25, all times 7.
Eliminate the wrong options
Which properties license that move?
Survives elimination: A
Why: Reordering the factors requires the commutative property and regrouping them requires the associative property, and both are needed together. This is the same combination that made rearranging a long sum legal in Lesson 2.3, which is one reason the two property lists are worth learning side by side.
Socratic
Addition managed with five. Multiplication has six.
Discussion prompt
The property of zero and the property of negative one have no counterpart in the addition list. Explain what each one does that no addition property does, and say why addition does not need an equivalent of either.
Hint: Ask what adding zero does and what multiplying by zero does.
Answer:
The property of zero says a product with a zero factor is zero, which destroys all other information in the product. Addition has no number that does that — adding any number leaves a result that still depends on what you started with, so there is nothing to state.
The property of negative one connects multiplication to opposites: multiplying by negative one reflects a number across zero. Addition already has opposites built into its inverse property, so it needs no separate statement. These two properties are exactly what make the sign derivations in this lesson possible.
Section
Section 4
Concept
Simplifying a product containing variables uses the same counting rule. Count the minus signs, decide the sign of the answer, then write the numbers and letters as a single product.
\[ -2(-x) = 2x \qquad 3(-n)(-n)(-n) = -3n^3 \]
The property of negative one is what makes this legal: each minus sign is a factor of negative one, and they may be gathered together.
Figure (svg): Multiplying by negative one shown as a reflection across zero on a number line
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 94-94 — Example 2, Products with Variable Factors
Picture it
The bridge between this lesson and the opposites of Lesson 2.2.
Figure (svg): Multiplying by negative one shown as a reflection across zero on a number line
Every minus sign in a product is a hidden factor of negative one. Gathering them is what turns a sign question into a counting question.
Worked example
This is Example 2 from the textbook. Count the minus signs in each.
\[ \text{Simplify } \; -2(-x), \quad 3(-n)(-n)(-n), \quad -1(-a)^2. \]
Count the minus signs in the first
Why: Two of them, which is even, so the product has no minus sign.
\[ -2(-x) = 2 x \]
Count the minus signs in the second
Why: The leading three is positive, and each of the three bracketed factors contributes one minus sign, giving three in total.
Apply the count to the second
Why: Three is odd, so the product carries a minus sign, and the three factors of n collect into n cubed.
\[ 3(-n) (-n) (-n) = -3 n ^{3} \]
Count the minus signs in the third
Why: The leading negative one contributes one, and the squared bracket contributes two, giving three, which is odd.
\[ -1(-a) ^{2} = -a ^{2} \]
Figure (svg): The solution to Worked example three products with variables shown as a ladder of expressions, one row per algebraic move
\[ -2(-x) = 2x, \quad 3(-n)(-n)(-n) = -3n^3, \quad -1(-a)^2 = -a^2 \]
Verify: substitute a value into each
Why: With x, n and a all equal to 2: the first gives 4, and 2 times 2 is 4. The second gives 3 times negative 8, which is negative 24, and three negatives make it negative. The third gives the opposite of 4, which is negative 4, and three negatives make it negative. All three signs match their counts.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 94-94
Sorting
Expand any powers before counting.
Sort into buckets
Sort each expression by whether its minus-sign count is odd or even.
The two items with squared and cubed brackets are the ones worth checking twice. An exponent multiplies how many minus signs the bracket contributes, and counting the visible signs on the page gets it wrong.
Worked example
Guided Practice 5 to 7. Count before you write.
\[ \text{Simplify } \; -8(-t), \quad -x(-x)(-x)(-x), \quad -7(-b)^3. \]
Two minus signs in the first, even
Why: So the product is positive: eight t.
\[ 8 t \]
Four minus signs in the second, even
Why: Four factors of the opposite of x, so the product is x to the fourth.
\[ x ^{4} \]
Count the third: one leading, three from the cube
Why: Four in total, which is even, so the product is positive.
\[ 7 b ^{3} \]
Write each answer as a single simplified product
Why: Numbers multiplied, letters collected as powers.
\[ 8 t, x ^{4}, 7 b ^{3} \]
Figure (svg): The solution to Worked example three from guided practice shown as a ladder of expressions, one row per algebraic move
\[ 8t, \quad x^4, \quad 7b^3 \]
Verify: substitute 2 into the last one
Why: Negative seven times the cube of negative two is negative seven times negative eight, which is fifty-six. And seven times two cubed is seven times eight, also fifty-six. The two agree, confirming the even count gave a positive answer.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 94-94
Trap
\[ -3(-n)^3 \]
Count two minus signs — one on the 3 and one on the n — and conclude the answer is positive
Why: The expression shows only two minus signs on the page, so two is the obvious count.
\[ 3n^3 \quad \text{(wrong)} \]
The cube means three factors of negative n, so the bracket contributes three minus signs rather than one. The count is four, which is even, so this one happens to come out positive anyway — but the reasoning was wrong and will fail next time.
\[ -3(-n)^3 = -3(-n)(-n)(-n) \]
Expand any power of a negative before counting
Why: An exponent multiplies the number of factors, and every one of those factors carries its own minus sign.
\[ \text{count} = 1 + 3 = 4, \text{ even, so } 3n^3 \]
Expanding takes a moment and removes the guesswork entirely. It is the same discipline as writing out the factors in Lesson 1.2.
Faded example
The expansion is done. Supply the count and the answer.
Fill in the blanks
3(-n)(-n)(-n): \text3 = -3n^3, \text___ ___
Why: The leading three is positive and contributes nothing, while each of the three negative n factors contributes one minus sign, giving three in total. Three is odd, so the answer is negative, and the three factors of n collect into n cubed. Counting and then writing is what keeps the sign decision separate from the algebra.
Elimination
The expression is negative 1 times the square of negative a.
Eliminate the wrong options
Which is correct?
Survives elimination: A
Why: The square of negative a is a squared, since two negatives pair off, and the leading negative one then makes the whole thing the opposite of a squared. The total minus-sign count is three, which is odd, so a negative answer is exactly what the counting rule predicts.
Socratic
Counting minus signs across a product is a shortcut for something the properties permit.
Discussion prompt
Explain, using the property of negative one and the commutative property, why it is legal to gather all the minus signs in a product and count them. Use negative 2 times negative x as your example.
Hint: Rewrite each minus sign as a factor of negative one first.
Answer:
\[ -2(-x) = (-1)(2)(-1)(x) \]
\[ = (-1)(-1)(2)(x) = (1)(2x) = 2x \]
Each minus sign is a factor of negative one by the property of negative one. The commutative and associative properties then allow those factors to be moved next to each other and multiplied first, and each pair of them gives one. So counting the minus signs is really counting how many factors of negative one there are, and pairing them off is multiplying them together.
An odd count leaves one unpaired factor of negative one, which is exactly what makes the answer negative. Nothing about the shortcut is a separate rule — it is the properties applied in a particular order.
Section
Section 5
Concept
The square of negative two and the opposite of two squared are different expressions with different values, and the difference is entirely in the brackets — exactly as in Lesson 1.2.
\[ (-2)^4 = 16 \qquad -2^4 = -16 \]
The first has four negative factors; the second has one, because the exponent takes only the two and the minus sign stands outside it.
Figure (svg): Two columns contrasting negative two to the fourth power with the opposite of two to the fourth power
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 93-93 — the Study Tip contrasting the two expressions
Picture it
Four negative factors on the left, one on the right.
Figure (svg): Two columns contrasting negative two to the fourth power with the opposite of two to the fourth power
Sixteen against negative sixteen, from the same digits. The bracket decides whether the minus sign is inside the repeated multiplication or outside it.
Worked example
The Study Tip beside Example 1 makes this point, and it is worth doing in full.
\[ \text{Evaluate } \; (-2)^4 \; \text{ and } \; -2^4. \]
Expand the first
Why: The bracket is raised to the fourth, so there are four factors of negative two.
\[ (-2) (-2) (-2) (-2) \]
Count and evaluate
Why: Four negative factors is even, so positive, and two to the fourth is sixteen.
\[ 16 \]
Expand the second
Why: No bracket, so the exponent takes only the two, and the minus sign stands outside.
\[ -(2 \cdot 2 \cdot 2 \cdot 2) \]
Count and evaluate
Why: One negative factor is odd, so negative, and two to the fourth is sixteen.
\[ -16 \]
Figure (svg): Two columns contrasting negative two to the fourth power with the opposite of two to the fourth power
\[ (-2)^4 = 16 \qquad -2^4 = -16 \]
Verify: check with an odd exponent to see the pattern break
Why: With a cube, negative two cubed is negative eight and the opposite of two cubed is also negative eight — the two agree. Powers of negatives differ from opposites of powers only when the exponent is even, which is a useful thing to know and a dangerous thing to rely on.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 93-93
Discrimination
Decide where each minus sign sits relative to the exponent.
Sort into buckets
Sort each expression by whether the minus sign is inside the power.
Worked example
Example 4 in spirit. A change in position is velocity times time.
\[ \text{A squirrel glides downward at } 4 \text{ feet per second for } 3 \text{ seconds. Find its change in position.} \]
Write the velocity with its sign
Why: Downward is negative, so the velocity is negative four feet per second.
\[ v = -4 \text{ft} / s \]
Write the formula and substitute
Why: Change in position is velocity times time.
\[ (-4) (3) \]
Count the negative factors
Why: One, which is odd, so the product is negative.
Multiply and attach the unit
Why: Four times three is twelve, so the change is negative twelve feet.
\[ -12\text{ feet} \]
Figure (svg): A flying squirrel's change in position computed as velocity times time, giving a negative displacement
\[ (-4)(3) = -12 \text{ feet} \]
Verify: check the sign against the story
Why: The squirrel was moving downward, so its position should end up lower than it started, which a negative change records. Had the answer come out positive, the model would be claiming the squirrel rose.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 95-95
Trap
\[ -2^4 \]
Read the whole thing as negative two raised to the fourth
Why: The minus sign is written immediately before the two, so it looks attached to it.
\[ -2^4 = 16 \quad \text{(wrong)} \]
Without brackets the exponent takes only the symbol directly in front of it, which is the two. The minus sign is applied afterwards, giving negative sixteen.
\[ -2^4 = -(2^4) = -16 \qquad (-2)^4 = 16 \]
Ask whether a bracket encloses the minus sign before applying the exponent
Why: A bracket puts the minus sign inside the repeated multiplication; without one it stays outside.
This is exactly the rule from Lesson 1.2 about what an exponent attaches to, applied to a minus sign rather than to a coefficient.
Prediction
Sometimes the bracketed and unbracketed versions give the same answer.
Predict first
Do the cube of negative 2 and the opposite of 2 cubed give the same value?
Correct: Yes, both are -8.
\[ (-2)^3 = -8 \qquad -2^3 = -8 \]
\[ \text{but } (-2)^4 = 16 \qquad -2^4 = -16 \]
Why: The cube of negative two has three negative factors, which is odd, so it is negative eight. The opposite of two cubed has one negative factor, also odd, so it is negative eight as well. The two agree whenever the exponent is odd and disagree whenever it is even, which makes an odd exponent a bad test case for whether someone understands the brackets.
Elimination
A squirrel glides downward at 4 feet per second for 3 seconds.
Eliminate the wrong options
What is its change in position?
Survives elimination: A
Why: Velocity times time gives negative four times three, which has one negative factor and is therefore negative twelve feet. The unit check confirms the operation: feet per second times seconds leaves feet, while adding or dividing them would not.
Socratic
The bracket question only changes the answer for some exponents.
Discussion prompt
Explain why the square of a negative number is always positive but the cube of a negative number is always negative. Then say what this implies about whether a squared quantity can ever be negative.
Hint: Count the negative factors in each case.
Answer:
Squaring means two factors, so two minus signs, which pair off and leave a positive. Cubing means three factors, so three minus signs, and one is left unpaired, making the result negative. Every even exponent pairs off completely and every odd exponent leaves one over.
This implies that no squared quantity is ever negative, whatever is squared. That fact will matter a great deal in Chapter 9, where an equation such as x squared equals a negative number turns out to have no real solution — for exactly the reason established here.
Comparison
Fill the blanks from memory before you scroll back. The first row is the whole method.
Comparison matrix
| Number of negative factors | Sign of the product | Example |
|---|---|---|
| zero (all positive) | positive | 4(5) = 20 |
| one | negative | -4(5) = -20 |
| two | positive | -2(-5) = 10 |
| three | negative | -10(-0.2)(-4) = -8 |
The pattern alternates with every extra negative factor, which is what odd and even are recording. Nothing in the table depends on how big any of the factors are.
Pattern
Whether the product has two factors or six, and whether it contains letters or not, the same five moves cover it.
Step two is the one people skip. An exponent multiplies how many negative factors a bracket contributes, and counting the minus signs visible on the page misses that.
OpenStax Elementary Algebra 2e, §1.4 Multiply and Divide Integers §1.4
Check
Counting negatives. Do the count before the arithmetic.
Check your understanding
What is negative 2 times negative 3 times negative 4?
Answer: A
Why: There are three negative factors, which is odd, so the product is negative. Two times three times four is twenty-four, so the answer is negative twenty-four. The sizes played no part in the sign decision.
Check
Brackets and exponents. Decide what the exponent takes.
Check your understanding
What is the value of the opposite of 3 squared — that is, negative 3 squared with no brackets?
Answer: A
Why: Without brackets the exponent attaches only to the three, so the expression is the opposite of nine, which is negative nine. It carries exactly one negative factor, and one is odd.
Check
Naming a property. Look at which number is doing the work.
Check your understanding
Which property does this statement illustrate? negative 1 times negative 6 equals 6
Answer: A
Why: Multiplying by negative one produced the opposite of negative six, which is six. That is precisely what the property of negative one states, and it is the property that connects multiplication back to the opposites of Lesson 2.2.
Real world
A submarine descends at 8 metres per minute. You want to know its depth 5 minutes from now, and also its depth 5 minutes ago, relative to where it is at this moment.
Discussion prompt
Using a negative velocity and letting a time in the past be a negative time, compute both changes in position. Explain why the second one comes out positive, and say what real fact about the submarine that positive answer is recording.
Hint: Both the velocity and the past time are negative quantities.
Answer:
\[ \text{future: } (-8)(5) = -40 \text{ metres, so 40 m lower} \]
\[ \text{past: } (-8)(-5) = 40 \text{ metres, so 40 m higher} \]
The second product has two negative factors, which is even, so it is positive. That positive answer records a real fact: five minutes ago the submarine was forty metres higher than it is now, because it has been descending ever since.
This is the clearest everyday demonstration that two negatives give a positive. A downward motion viewed backwards in time is an upward displacement, and the arithmetic says so without being told.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Does the sign of a product depend on how large the negative factors are?
Correct: No, only the number of negative factors matters.
\[ (-100)(-2) = 200 \qquad (-100)(2) = -200 \]
\[ \text{but } -100 + (-2) = -102 \qquad -100 + 2 = -98 \]
Why: In addition the sign of the answer came from whichever number had the larger absolute value, so it is natural to expect the same here — but multiplication works completely differently. Negative one hundred times negative two is positive two hundred, and negative one hundred times two is negative two hundred, and in both cases the sizes are identical. Only the count changed, and only the count decided.
Explain it
They can multiply positive numbers fluently and have just met negative numbers.
Discussion prompt
In no more than four sentences, explain how to find the sign of a product with several negative factors, and why the rule is a count rather than a comparison. Then give them the one case where the rule does not apply.
Hint: The exception involves a number that is neither positive nor negative.
Answer:
A usable answer: count how many of the factors are negative and ignore how big they are. If the count is odd the answer is negative; if it is even the answer is positive. It is a count because each pair of negatives cancels, so all that matters is whether one is left over.
The exception is zero. If any factor is zero the whole product is zero regardless of the signs, so check for a zero factor first — it ends the problem before the counting starts.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: Counting with powers is fixed by expanding every power before counting anything. Brackets are fixed by asking whether the minus sign is enclosed before the exponent acts. Property names are fixed by asking which specific number the property is about — one, zero or negative one — and using the two-question test for the other two. Variable products are fixed by counting minus signs first and doing the algebra second. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
At the top of a page write the derivation that three times negative two is negative six as repeated addition, then underneath it the derivation that negative three times negative two must be six. In the middle, draw a four-row table of products with zero, one, two and three negative factors, giving an example and its sign for each row, and mark the row where the pattern first turns negative. Underneath that, write the six properties of multiplication and star the two that have no counterpart in the addition list. At the bottom, write the square of negative two and the opposite of two squared side by side with their values, and circle the difference between them. Finally, in the margin, write the one situation in which the counting rule does not apply.
The circled difference should be the brackets, and the marginal exception should be a factor of zero. If you wrote anything else in the margin, look again at what happens to the counting rule when a zero appears.
Recap
Five things, and the second one is a genuinely simpler rule than the one you learned for addition.
| If the question says | Your first move is |
|---|---|
| Find the product | Count the negative factors |
| Simplify with variables | Expand any powers, then count minus signs |
| Evaluate (-2) to the fourth | Count four negative factors, so positive |
| Evaluate the opposite of 2 to the fourth | Count one negative factor, so negative |
| Find the change in position | Multiply velocity by time, keeping the signs |
Lesson 2.6 puts multiplication and addition together in one property: the distributive property, which is what lets a factor outside a bracket be shared out over a sum inside it.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.5 Multiplying Real Numbers §2.5, pp. 93-98 — everything on these slides traces back here
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