Addition as movement on a number line, the two rules of addition decided by whether the signs agree, the five properties of addition that those rules follow from, adding a long list of signed numbers efficiently, and modelling profits and losses as positive and negative quantities.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 2 — Properties of Real Numbers
Adding 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 2.3 Adding Real Numbers §2.3, pp. 78-84 — the lesson these objectives are drawn from
Warm-up
You have added and subtracted on a number line since primary school. All that changes is which numbers you are allowed to land on.
Discussion prompt
Start at 2 and move 6 units to the left. Where do you end up, and what addition did you just perform?
Hint: Count past zero and keep going.
Answer:
\[ 2 + (-6) = -4 \]
You land at negative four. Moving left is adding a negative, so what you performed was an addition and not a subtraction — even though the answer went down. That reading is what makes every rule in this lesson work: adding a negative and moving left are the same instruction.
Concept
Every addition of real numbers can be drawn as a movement on the number line. You add a positive number by moving to the right, and a negative number by moving to the left. Where you land is the sum.
closure property — The sum of any two real numbers is itself a unique real number. The real numbers are said to be closed under addition.
Because the real numbers are closed under addition, you never move off the line no matter what you add.
Figure (svg): A number line showing a jump starting at negative 2 and moving 5 units to the right, ending at 3
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.3 Adding Real Numbers §2.3, pp. 78-78
Section
Section 1
Concept
To add on a number line, start at the first number and then move by the second: right if it is positive, left if it is negative. The point you land on is the sum.
The direction comes from the sign and the distance from the digits — the same two-part reading as plotting a point in Lesson 2.1.
Figure (svg): A number line showing a jump starting at negative 2 and moving 5 units to the right, ending at 3
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.3 Adding Real Numbers §2.3, pp. 78-78 — the movement rules and Example 1
Picture it
Start on the negative side and move right past zero.
Figure (svg): A number line showing a jump starting at negative 2 and moving 5 units to the right, ending at 3
Five units to the right of negative two lands at three. The move crossed zero, which is exactly what makes the answer change sign — and drawing it removes any need to reason about that.
Worked example
This is Example 1 from the textbook. One moves right, one moves left.
\[ \text{Use a number line to find } \; -2 + 5 \; \text{ and } \; 2 + (-6). \]
Start at negative 2
Why: The first number sets the starting point.
\[ \text{start at } -2 \]
Move 5 units right, because 5 is positive
Why: Five units right from negative two crosses zero and lands at three.
\[ -2 + 5 = 3 \]
Start at 2 for the second sum
Why: New problem, new starting point.
\[ \text{start at } 2 \]
Move 6 units left, because negative 6 is negative
Why: Six units left from two crosses zero and lands at negative four.
\[ 2 + (-6) = -4 \]
Figure (svg): A number line showing a jump starting at 2 and moving 6 units to the left, ending at negative 4
\[ -2 + 5 = 3 \qquad 2 + (-6) = -4 \]
Verify: check which direction each answer moved
Why: The first sum ended up larger than where it started, which is what adding a positive must do. The second ended up smaller, which is what adding a negative must do. Both answers moved the direction their second number demanded.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.3 Adding Real Numbers §2.3, pp. 78-78
Sorting
For each addition, decide the direction of the move.
Sort into buckets
Sort each sum by the direction the second number moves you.
The direction depends only on the second number, never on the first. That is why the picture works even when both numbers are negative.
Worked example
Guided Practice 1 to 4. Draw each one before writing the answer.
\[ \text{Find } \; -4 + 5, \quad 1 + (-2), \quad -4 + (-5), \quad 0 + (-4). \]
Start at negative 4 and move 5 right
Why: Crosses zero and lands one unit past it.
\[ -4 + 5 = 1 \]
Start at 1 and move 2 left
Why: Crosses zero and lands one unit before it.
\[ 1 + (-2) = -1 \]
Start at negative 4 and move 5 further left
Why: Both moves go left, so they accumulate.
\[ -4 + (-5) = -9 \]
Start at 0 and move 4 left
Why: Starting at zero means the answer is simply the second number.
\[ 0 + (-4) = -4 \]
Figure (svg): The solution to Worked example four sums from guided practice shown as a ladder of expressions, one row per algebraic move
\[ 1, \quad -1, \quad -9, \quad -4 \]
Verify: notice which sums crossed zero and which did not
Why: The first two crossed zero and changed sign; the third did not cross and stayed negative; the fourth started at zero. Whether a sum crosses zero is exactly what decides whether the answer's sign matches the first number's, and the picture shows it directly.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.3 Adding Real Numbers §2.3, pp. 78-78
Trap
\[ 2 + (-6) \]
See the addition sign and move to the right
Why: The word add has meant get bigger for years, and the plus sign reinforces it.
\[ 2 + (-6) = 8 \quad \text{(wrong)} \]
The plus sign says that an addition is happening; the minus sign inside the brackets says which way. Both have to be read.
\[ 2 + (-6) = -4 \]
Read the sign of the number being added, not the operation sign
Why: Adding a negative moves left, which is why the answer can be smaller than where you started.
Adding no longer means getting bigger once negatives are on the table. It means moving, and the direction is carried by the number rather than by the operation.
Prediction
Predicting the sign before computing is a real check on the answer.
Predict first
In the sum negative 4 plus 5, does the answer end up positive or negative?
Correct: Positive, because the move right is longer than the distance to zero.
\[ -4 + 5 = 1 \qquad \text{since } 5 > |-4| \]
Why: Starting at negative four, zero is four units away to the right. The move is five units, which is one more than needed, so it crosses zero and lands one unit past it at positive one. Comparing the size of the move with the distance to zero predicts the sign every time, and that comparison is exactly what the opposite-signs rule formalises.
Matching
Four sums, four descriptions of the movement.
Match the pairs
Why: The first two cross zero and therefore change sign; the third never approaches zero and stays negative; the fourth starts at zero so the answer is just the number added. Whether a sum crosses zero is what decides the sign of the answer, and it depends on comparing the size of the move with the distance to zero.
Faded example
The start and direction are given. Supply the landing point.
Fill in the blanks
\text4 -3, \text-10 7 \text___ \;\rightarrow\; -3 + 7 = ___ \qquad \text___ -3, \text___ 7 \text___ \;\rightarrow\; -3 + (-7) = ___
Why: Moving right from negative three uses three of the seven units to reach zero and the remaining four carry past it to positive four. Moving left simply adds to the distance already travelled, landing at negative ten. Same numbers, opposite directions, and answers fourteen units apart.
Section
Section 2
Concept
Drawing a picture works but is slow. The two rules of addition do the same job arithmetically, and which one applies is decided before any calculation by looking at the two signs.
Notice that adding two numbers can require you to subtract, and that this is not a contradiction — it is what happens when the two moves point opposite ways.
Figure (svg): The two rules of addition, one for same signs and one for opposite signs, each in two steps
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.3 Adding Real Numbers §2.3, pp. 79-79 — the Rules of Addition box
Picture it
Check the signs first; the rule follows from that check.
Figure (svg): The two rules of addition, one for same signs and one for opposite signs, each in two steps
Every addition of two signed numbers falls into exactly one of these two columns, and deciding which takes about a second. Doing that first is what prevents the two rules from being mixed together.
Worked example
This is Example 2 from the textbook, both parts.
\[ \text{Find } \; -4 + (-5) \; \text{ and } \; 3 + (-9). \]
Check the signs of the first sum
Why: Both negative, so the same-sign rule applies.
Add the absolute values and attach the common sign
Why: Four plus five is nine, and the shared sign is negative.
\[ -4 + (-5) = -9 \]
Check the signs of the second sum
Why: One positive and one negative, so the opposite-signs rule applies.
Subtract the absolute values and attach the sign of the larger
Why: Nine minus three is six, and nine had the larger absolute value with a negative sign.
\[ 3 + (-9) = -6 \]
Figure (svg): Two arrows in the same direction combining, showing that same-sign addition adds the absolute values
\[ -4 + (-5) = -9 \qquad 3 + (-9) = -6 \]
Verify: check both against the number line picture
Why: For the first, two moves left from zero total nine units left. For the second, a move of nine left against a move of three right leaves six left. Both agree with the rules, which is the point: the rules are a shortcut for the picture rather than a separate system.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.3 Adding Real Numbers §2.3, pp. 79-79
Discrimination
Name the rule before doing any arithmetic.
Sort into buckets
Sort each sum by which rule of addition it needs.
Worked example
Guided Practice 5 to 8. Name the rule before computing each one.
\[ \text{Find } \; -3 + (-7), \quad -1 + 3, \quad 8 + (-3), \quad -2 + 3. \]
Negative 3 plus negative 7: same sign
Why: Three plus seven is ten, and both are negative.
\[ -10 \]
Negative 1 plus 3: opposite signs
Why: Three minus one is two, and three has the larger absolute value with a positive sign.
\[ 2 \]
8 plus negative 3: opposite signs
Why: Eight minus three is five, and eight has the larger absolute value with a positive sign.
\[ 5 \]
Negative 2 plus 3: opposite signs
Why: Three minus two is one, and three is positive.
\[ 1 \]
Figure (svg): The solution to Worked example four sums from guided practice shown as a ladder of expressions, one row per algebraic move
\[ -10, \quad 2, \quad 5, \quad 1 \]
Verify: check that every answer's sign matches the larger absolute value
Why: In the three opposite-sign sums the positive number had the larger absolute value each time, and all three answers came out positive. In the same-sign sum both were negative and so was the answer. The sign of every answer traces back to the rule that produced it.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.3 Adding Real Numbers §2.3, pp. 79-79
Error analysis
The student applied the rules of addition to four sums. Two answers are wrong.
Annotate
On: \( -4 + (-5) = -9 \qquad 3 + (-9) = 6 \qquad -3 + (-7) = -10 \qquad 8 + (-3) = 11 \)
Both errors would have been caught by a rough number-line check: negative nine is a longer move than positive three, so the answer must be negative, and eight plus a leftward move of three cannot exceed eight.
Elimination
The sum is 3 plus negative 9.
Eliminate the wrong options
What is the value?
Survives elimination: A
Why: The signs differ, so subtract: nine minus three is six. The sign comes from negative nine, whose absolute value is larger, so the answer is negative six. A number-line check confirms it — a move of nine left beats a move of three right, and the surplus is six units left.
Faded example
The rule has been named. Supply the two steps.
Fill in the blanks
-4 + (-5): \text9 |-4| + |-5| = -9, \text___ \;\rightarrow\; ___
Why: The absolute values four and five add to nine, and the sign shared by both original numbers is negative, so the answer is negative nine. Splitting the work into a size step and a sign step is what stops the two being confused, and it is exactly how the rule is written.
Socratic
The opposite-signs rule tells you to subtract inside an addition. That deserves an explanation.
Discussion prompt
Explain, using the number line, why adding two numbers with opposite signs requires a subtraction. Then say what the answer's sign is really recording.
Hint: Think about what two arrows pointing opposite ways do to each other.
Answer:
Two moves in opposite directions partly undo each other. If you move nine units left and three units right, the three cancel three of the nine and you are left with six units of leftward movement. Subtracting the absolute values is exactly the arithmetic of that cancellation.
The sign records which arrow was longer, since that is the direction the surplus points in. So the rule is really two separate questions: how much is left over, and which way does it go — and answering them separately is why the rule has two steps rather than one.
Section
Section 3
Concept
The rules of addition are a consequence of five properties. Naming them matters because later chapters will ask you to justify a step by property rather than by computing it.
The inverse property is the one that makes the next lesson possible: it is why subtraction can be rewritten as addition.
| Property | Statement | Example |
|---|---|---|
| Closure | a + b is a unique real number | 4 + 2 = 6 |
| Commutative | a + b = b + a | 3 + (-2) = -2 + 3 |
| Associative | (a + b) + c = a + (b + c) | (5 + 6) + 2 = 5 + (6 + 2) |
| Identity | a + 0 = a | -4 + 0 = -4 |
| Inverse | a + (-a) = 0 | 5 + (-5) = 0 |
Figure (svg): The five properties of addition, each stated in symbols with an example
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.3 Adding Real Numbers §2.3, pp. 79-79 — the Properties of Addition box
Picture it
Cover the middle column and state each property from its name.
Figure (svg): The five properties of addition, each stated in symbols with an example
Two of these — commutative and associative — are easy to confuse, and the next section is about telling them apart. The other three are each about a specific number: any number, zero, and the opposite.
Worked example
The skill being tested here is reading what changed between the two sides.
\[ \text{Name the property: } \; 3 + (-2) = -2 + 3; \quad (5 + 6) + 2 = 5 + (6 + 2); \quad -4 + 0 = -4; \quad 5 + (-5) = 0. \]
In the first, the numbers changed seats
Why: Three and negative two appear in the opposite order, and nothing else changed.
In the second, only the brackets moved
Why: Five, six and two appear in the same order on both sides; only the grouping differs.
In the third, adding zero left the number alone
Why: Zero is the additive identity, the number that changes nothing.
In the fourth, a number and its opposite gave zero
Why: Five and negative five are opposites, and their sum is zero.
Figure (svg): The solution to Worked example name the property shown as a ladder of expressions, one row per algebraic move
\[ \text{commutative}, \; \text{associative}, \; \text{identity}, \; \text{inverse} \]
Verify: apply the two-question test to the first two
Why: Did the numbers move, or only the brackets? In the first the numbers moved, so it is commutative; in the second only the brackets moved, so it is associative. Those two questions settle the pair that gets confused, and the other three properties each name a specific number instead.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.3 Adding Real Numbers §2.3, pp. 79-79
Matching
Five statements, five properties.
Match the pairs
Why: Closure says the answer stays inside the real numbers, which is a claim about where sums live rather than about rearranging. Commutative moves the numbers, associative moves the brackets, and identity is the specific claim about zero. Recognising which quantity a property is about is faster than recalling its name from memory.
Worked example
The properties are not just names — they license rearranging a sum to make it easier.
\[ \text{Evaluate } \; 17 + (-8) + 3 + 8 \; \text{ using the properties.} \]
Look for a pair of opposites
Why: Negative eight and eight are opposites, so their sum is zero by the inverse property.
\[ -8 + 8 = 0 \]
Move them together
Why: The commutative and associative properties allow the terms to be reordered and regrouped freely.
\[ 17 + 3 + (-8 + 8) \]
Use the inverse property
Why: The paired terms vanish.
\[ 17 + 3 + 0 \]
Use the identity property and finish
Why: Adding zero changes nothing, and seventeen plus three is twenty.
\[ 20 \]
Figure (svg): The solution to Worked example use the properties to add efficiently shown as a ladder of expressions, one row per algebraic move
\[ 17 + (-8) + 3 + 8 = 20 \]
Verify: add the four terms left to right without rearranging
Why: Seventeen plus negative eight is nine, plus three is twelve, plus eight is twenty. The same answer by a slower route, which confirms that the rearrangement was legal rather than lucky.
Trap
\[ (5 + 6) + 2 = 5 + (6 + 2) \]
Call it commutative because the two sides look different
Why: Both properties are about rearranging, and both statements have the same numbers on each side.
Nothing changed seats here. Five, six and two are in the same order on both sides, and only the brackets moved.
\[ (5 + 6) + 2 = 5 + (6 + 2) \quad \text{associative} \]
\[ 3 + (-2) = -2 + 3 \quad \text{commutative} \]
Ask one question: did the numbers move, or did only the brackets?
Why: Numbers moving is commutative; brackets moving is associative. Nothing else distinguishes them.
The two properties are genuinely different claims, and Chapter 7 will meet an operation that has one without the other.
Sorting
Each statement rearranges a sum. Decide how.
Sort into buckets
Sort each statement by what changed between its two sides.
The test never changes: read both sides left to right and compare the sequences. Different sequence means commutative; same sequence with different brackets means associative.
Elimination
The step is from 17 plus negative 8 plus 3 plus 8 to 17 plus 3 plus the quantity negative 8 plus 8.
Eliminate the wrong options
Which properties license that rearrangement?
Survives elimination: A
Why: Moving the terms into a new order requires the commutative property, and putting brackets round a new pair requires the associative property. Both are needed together, which is typical — most rearrangements of a long sum use the two in combination, and the inverse and identity properties then finish the job.
Socratic
You could add correctly without knowing a single property name.
Discussion prompt
Give one reason it is worth naming the properties rather than just computing. Then name an operation from earlier in the book that fails the commutative property, and say how you know.
Hint: Think about what happens when the numbers are replaced by letters.
Answer:
Once the numbers are replaced by letters you can no longer compute, so the only way to justify a step is to name the property that permits it. Every rearrangement in Chapter 3 and beyond is licensed by one of these five, which is why the names are worth having before the letters arrive.
Subtraction fails the commutative property, and one example proves it: five minus three is two while three minus five is negative two. Division fails it too. Those failures are exactly why Lesson 2.4 rewrites every subtraction as an addition — the rewrite buys back the properties that subtraction does not have.
Section
Section 4
Concept
A long sum of signed numbers can be added left to right, but grouping is faster and safer. Total the positives, total the negatives, then perform a single opposite-signs addition.
The commutative and associative properties are what make this rearrangement legal.
Figure (svg): A bar model of four profits and losses combining into an overall profit
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.3 Adding Real Numbers §2.3, pp. 80-80 — Example 4, on adding several profits and losses
Picture it
Green bars run right for profit; red bars run left for loss.
Figure (svg): A bar model of four profits and losses combining into an overall profit
Six hundred and fifty of profit against three hundred of loss leaves three hundred and fifty. Grouping turned four mixed additions into two easy ones and one comparison.
Worked example
Example 4 in spirit. Profits are positive and losses are negative.
\[ \text{Find the total of } \; 250 + (-180) + 400 + (-120). \]
Separate the positives from the negatives
Why: Two profits and two losses.
\[ 250, 400 | - 180, -120 \]
Add the positives
Why: Two hundred and fifty plus four hundred is six hundred and fifty.
\[ 650 \]
Add the negatives
Why: One eighty plus one twenty is three hundred, and both are negative.
\[ -300 \]
Combine the two totals
Why: Opposite signs, so subtract: six fifty minus three hundred is three fifty, with the sign of the larger, which is positive.
\[ 350 \]
Figure (svg): A bar model of four profits and losses combining into an overall profit
\[ 650 + (-300) = 350 \text{ profit} \]
Verify: add the four terms left to right instead
Why: Two fifty minus one eighty is seventy; plus four hundred is four seventy; minus one twenty is three fifty. The same answer by the slower route, which confirms the grouping did not disturb anything.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.3 Adding Real Numbers §2.3, pp. 80-80
Ranking
Four moves, one correct sequence.
Put in order
Why: Separating comes first because everything else depends on it. The two group totals can be computed in either order, since each is a same-sign addition, and the final combination has to come last because it needs both totals. The whole method converts many mixed additions into exactly one.
Worked example
Same method, and this time the negatives win.
\[ \text{Find the total of } \; -45 + 30 + (-70) + 25. \]
Separate by sign
Why: Two positives and two negatives.
\[ 30, 25 | - 45, -70 \]
Add the positives
Why: Thirty plus twenty-five is fifty-five.
\[ 55 \]
Add the negatives
Why: Forty-five plus seventy is one hundred and fifteen, both negative.
\[ -115 \]
Combine
Why: Opposite signs, so subtract: one fifteen minus fifty-five is sixty, with the negative sign of the larger.
\[ -60 \]
Figure (svg): The solution to Worked example a sum that ends in a loss shown as a ladder of expressions, one row per algebraic move
\[ 55 + (-115) = -60 \text{ loss} \]
Verify: check the sign against the two group totals
Why: The negative group totalled 115 and the positive group only 55, so the negatives were larger in absolute value and the answer must be negative. Comparing the two group totals predicts the sign before the subtraction is done.
Trap
\[ 250 + (-180) + 400 + (-120) \]
Collect the numbers as 250, 180, 400, 120 and add them all
Why: Once the terms are being sorted, the minus signs are easy to leave behind with the brackets.
\[ 250 + 180 + 400 + 120 = 950 \quad \text{(wrong)} \]
Nine hundred and fifty would be the answer if the company had never lost anything. Two of the four months were losses, and the regrouping erased them.
\[ (250 + 400) + (-180 + (-120)) = 650 + (-300) = 350 \]
Carry each sign with its number into the group it belongs to
Why: A term is the number together with its sign, and moving one without the other changes the problem.
A quick check: the answer must be smaller than the sum of the profits alone, since the losses can only reduce it. Nine fifty exceeds the profit total of six fifty, which is impossible.
Estimation
A rough total first is a check on the exact one.
Predict first
Roughly what is the total of 250 plus negative 180 plus 400 plus negative 120?
Correct: About 350.
\[ 650 + (-300) = 350 \]
Why: The profits total about 650 and the losses about 300, so the result is a profit of roughly 350. Option B is the answer you get by ignoring the minus signs, and option C has the right size but the wrong sign — comparing the two group totals rules both out before any exact arithmetic.
Prediction
The sign of the answer can be decided from the group totals alone.
Predict first
A company records 120, negative 300, 90 and negative 40. Is the overall result a profit or a loss?
Correct: A loss, since the negatives total more.
\[ 210 + (-340) = -130 \text{ loss} \]
Why: The positives total 210 and the negatives total 340, so the negatives are larger in absolute value and the answer carries their sign. The exact figure is negative 130. Counting how many months were profitable tells you nothing — one large loss can outweigh several small profits, which is exactly what happens here.
Socratic
Adding left to right gives the same answer. Grouping is still usually better.
Discussion prompt
Explain two advantages of grouping by sign over adding a long list left to right. At least one should be about the kind of mistake it prevents.
Hint: Count how many opposite-signs additions each method requires.
Answer:
First, it reduces the number of hard steps. Adding left to right through six mixed terms means up to five opposite-signs additions, each one carrying a chance of a sign error. Grouping needs only one, with all the rest being same-sign additions that are simply sums of digits.
Second, it makes the answer's sign predictable in advance. Once you have the two group totals you can see which is larger and therefore what sign the answer must carry, so a sign error becomes visible immediately. Working left to right gives you no such checkpoint.
Section
Section 5
Concept
Signed numbers model any quantity that can go two ways: profit and loss, rise and fall, deposit and withdrawal. Deciding which direction counts as positive is a choice you make once and then keep.
The choice is arbitrary but it must be consistent — reversing it halfway through is what produces answers with the wrong sign.
Figure (svg): A bar model of four profits and losses combining into an overall profit
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.3 Adding Real Numbers §2.3, pp. 78-80 — the profit-and-loss context introduced with the lesson
Picture it
One convention, applied to all four months.
Figure (svg): A bar model of four profits and losses combining into an overall profit
Had losses been chosen as positive instead, every bar would flip and the answer would come out as negative 350 — still meaning a profit of 350, but only if you remembered the convention. Choosing once and stating it is what keeps the answer readable.
Worked example
Rises are positive and falls are negative, stated before anything is written.
\[ \text{A temperature starts at } -6 \text{ degrees, rises } 14, \text{ then falls } 5. \text{ Find the final temperature.} \]
State the convention
Why: A rise is positive and a fall is negative.
Write the whole change as a sum
Why: Start at negative six, add fourteen, add negative five.
\[ -6 + 14 + (-5) \]
Add the first two
Why: Opposite signs: fourteen minus six is eight, with the positive sign of the fourteen.
\[ 8 \]
Add the last term
Why: Opposite signs: eight minus five is three, positive.
\[ 3 \]
Figure (svg): The solution to Worked example temperature change over a day shown as a ladder of expressions, one row per algebraic move
\[ -6 + 14 + (-5) = 3 \text{ degrees} \]
Verify: check the story against the answer
Why: The temperature rose fourteen from six below zero, which certainly takes it above freezing, and then fell five, which leaves it a little above zero. Three degrees is consistent with that narrative, and an answer below zero would have contradicted it.
Translation
Four short situations. Rises, deposits and profits are positive.
Match the pairs
Why: Every one of these becomes an addition, including the ones the English describes as taking away. A loss, a withdrawal and a descent are all negative quantities being added rather than positive quantities being subtracted, and writing them that way is what lets the addition rules apply directly.
Worked example
Deposits positive, withdrawals negative. The final sign is the interesting part.
\[ \text{An account holds } 120. \text{ There are withdrawals of } 200 \text{ and } 40, \text{ and a deposit of } 90. \text{ Find the balance.} \]
State the convention
Why: A deposit is positive and a withdrawal is negative.
Write the sum
Why: One twenty, minus two hundred, minus forty, plus ninety, all as signed terms.
\[ 120 + (-200) + (-40) + 90 \]
Group by sign
Why: Positives total two hundred and ten; negatives total two hundred and forty.
\[ 210\text{ and } -240 \]
Combine
Why: Opposite signs: two forty minus two ten is thirty, with the negative sign of the larger.
\[ -30 \]
Figure (svg): The solution to Worked example an account balance shown as a ladder of expressions, one row per algebraic move
\[ 210 + (-240) = -30 \text{ dollars} \]
Verify: translate the sign back into words
Why: Negative thirty means the account is thirty dollars overdrawn rather than holding thirty dollars. Translating the sign back into the language of the situation is what turns a number into an answer, and it is the step that reveals whether the convention was applied consistently.
Trap
Withdrawals are negative, so write 120, then negative 200, then 40 for the second withdrawal because it feels like another negative amount already accounted for.
\[ 120 + (-200) + 40 + 90 = 50 \quad \text{(wrong)} \]
Apply the sign convention to some terms and not others
Why: After a few terms the convention becomes automatic rather than deliberate, and one term slips through unsigned.
The second withdrawal was recorded as a deposit, which changes the answer by eighty dollars and turns an overdraft into a positive balance.
\[ 120 + (-200) + (-40) + 90 = -30 \]
Write every term with its sign in one pass, before adding anything
Why: Separating the signing from the adding means each is done with full attention.
A useful habit: count the terms with minus signs against the number of withdrawals in the story. Two withdrawals must produce two negative terms.
Elimination
A diver is at 30 metres below the surface, descends 15 more metres, then rises 20 metres.
Eliminate the wrong options
Which sum gives the diver's final depth?
Survives elimination: A
Why: Below the surface is negative, so the diver starts at negative thirty; descending fifteen more adds negative fifteen; rising twenty adds positive twenty. The total is negative twenty-five, meaning twenty-five metres below the surface. Each wrong option gets exactly one of the three signs wrong, which is the usual way a signed model fails.
Missing information
A question can be perfectly well written and still be unanswerable.
Discussion prompt
A company records profits and losses of 250, 180, 400 and 120 over four months. What was the overall result? Say exactly what is missing, and give two different answers depending on how the gap is filled.
Hint: Look at what the numbers do not tell you.
Answer:
Which months were profits and which were losses is missing. The four figures are given without signs, so the sum cannot be formed.
\[ \text{if } 250 \text{ and } 400 \text{ are profits: } 650 + (-300) = 350 \]
\[ \text{if } 250 \text{ and } 400 \text{ are losses: } 300 + (-650) = -350 \]
The same four magnitudes give a profit of 350 or a loss of 350 depending entirely on which are which. A signed quantity is not fully specified by its size, and that is the whole reason this chapter exists.
Socratic
The choice is arbitrary. That does not make it unimportant.
Discussion prompt
Suppose a company decided that losses would be positive and profits negative. Would its calculations be wrong? Explain what would and would not change, and say what the one real requirement on a sign convention is.
Hint: Think about whether the arithmetic or the interpretation changes.
Answer:
The arithmetic would be entirely correct. Every quantity would flip sign, the sum would flip sign, and the magnitude of the answer would be identical. Negative 350 under that convention would mean a profit of 350, exactly as 350 does under the usual one.
What would change is readability: everyone reading the report would have to remember the unusual convention, and any figure combined with an outside source using the normal convention would be wrong. The one real requirement is consistency — apply the same choice to every term and state it explicitly, and any choice works.
Comparison
Fill the blanks from memory before you scroll back. The decision in the first row is made before any arithmetic.
Comparison matrix
| Same signs | Opposite signs | |
|---|---|---|
| What you do to the absolute values | add them | subtract the smaller from the larger |
| Which sign the answer takes | the shared sign | the sign of the larger absolute value |
| On the number line | two moves the same way | two moves that partly cancel |
Notice that adding can require subtracting and that this is not a contradiction — it is what partial cancellation looks like in arithmetic.
Pattern
Whether the sum has two terms or ten, the same five moves cover it.
Step one is where signed word problems are won or lost. Once every term carries its correct sign, the rest is arithmetic.
OpenStax Elementary Algebra 2e, §1.3 Add and Subtract Integers §1.3
Check
Same signs. Name the rule before computing.
Check your understanding
What is negative 7 plus negative 6?
Answer: A
Why: The signs agree, so the absolute values add: seven plus six is thirteen, and the shared sign is negative. On a number line this is two moves left, one of seven units and one of six, landing thirteen units left of zero.
Check
Opposite signs. Decide the sign before the size.
Check your understanding
What is 5 plus negative 12?
Answer: A
Why: The signs differ, so subtract: twelve minus five is seven. Negative twelve has the larger absolute value, so the answer takes its negative sign. A move of twelve left beats a move of five right, leaving seven units left of zero.
Check
Naming a property. Look at what changed between the two sides.
Check your understanding
Which property does this statement illustrate? (2 + 7) + 3 = 2 + (7 + 3)
Answer: A
Why: The numbers appear in the same order on both sides — two, then seven, then three — and only the brackets moved. Regrouping without reordering is exactly the associative property.
Real world
A hiker starts at an elevation of 200 metres, climbs 450 metres, descends 180 metres, descends another 320 metres, then climbs 90 metres.
Discussion prompt
Choose a sign convention and state it, write the whole journey as one sum, and find the final elevation using the grouping method. Then say how far the hiker is above or below the starting point, and explain why that second answer is not the same number as the elevation.
Hint: Elevation is measured from sea level; the change is measured from where the hiker started.
Answer:
\[ 200 + 450 + (-180) + (-320) + 90 \]
\[ \text{positives: } 200 + 450 + 90 = 740 \qquad \text{negatives: } -180 + (-320) = -500 \]
\[ 740 + (-500) = 240 \text{ metres elevation} \]
The final elevation is 240 metres. The change from the starting point is 240 minus 200, which is 40 metres of net climb — a different number because elevation is measured from sea level while the change is measured from where the hiker began.
Keeping those two apart is the same distinction as position versus displacement, and it is why every signed quantity needs a stated reference point as well as a sign convention.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Can adding two numbers ever give an answer smaller than both of them?
Correct: Yes, whenever both numbers are negative.
\[ -4 + (-5) = -9 < -5 < -4 \]
\[ \text{but } 4 + 5 = 9 > 5 > 4 \]
Why: Negative four plus negative five is negative nine, which is smaller than both of the numbers added. This happens because both moves go left, so the total is further left than either move alone. The belief that adding always increases a total is true for positive numbers and false as soon as negatives are allowed, which is one of the main adjustments this chapter demands.
Explain it
They can add positive numbers fluently and have just met negatives on a number line.
Discussion prompt
In no more than four sentences, explain how to add two numbers when one is positive and one is negative, without using the words absolute value. Then tell them how to know the sign of the answer before doing any arithmetic.
Hint: Describe it as two moves that fight each other.
Answer:
A usable answer: picture two moves along a line, one going right and one going left. They cancel each other as far as they can, and whatever is left over is the answer. So you take the difference between the two distances rather than their total.
The sign comes from whichever move was longer, because that is the direction the leftover part points in. If the leftward move was bigger the answer is negative; if the rightward move was bigger it is positive — and you can see that before doing the subtraction at all.
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: Choosing the rule is fixed by always writing down whether the signs agree before touching the numbers. Opposite-sign signs are fixed by asking which move was longer, since the answer points that way. Property names are fixed by the two-question test: did the numbers move, or only the brackets. Word problems are fixed by signing every term in one pass before adding anything. 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
Draw two number lines across the top of a page. On the first, draw a same-sign addition as two arrows pointing the same way, and write the sum beneath it. On the second, draw an opposite-signs addition as two arrows pointing opposite ways, shade the part that cancels, and write the sum beneath it. In the middle of the page, write the two rules of addition, each in its two steps, and beside each write which of your two pictures it describes. Underneath, list the five properties of addition with one example each, and circle the one that will make subtraction possible in the next lesson. Finally, at the bottom, invent a four-term profit-and-loss situation, write it as a signed sum, and find its total by grouping.
The circled property should be the inverse property. If you circled the identity property, look again at what a subtraction has to be rewritten as.
Recap
Five things, and the second one is where nearly every mark in this lesson is won or lost.
| If the question says | Your first move is |
|---|---|
| Use a number line to find the sum | Start at the first number, then read the second's sign |
| Find the sum | Check whether the two signs agree |
| Name the property | Ask whether numbers or brackets moved |
| Find the overall profit | Group the positives and the negatives |
| A loss of 180 | Write it as negative 180 before adding |
Lesson 2.4 uses the inverse property to make subtraction disappear entirely: every subtraction is rewritten as adding the opposite, so the two rules you have just learned become the only rules you need.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 2 Properties of Real Numbers — Lesson 2.3 Adding Real Numbers §2.3, pp. 78-84 — everything on these slides traces back here
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