Study guide
9.2 Operations with Positive and Negative Numbers & Integer Rules
California Basic Educational Skills Test CBEST
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1 Introduction, Scoring Architecture & California BSR Pathways
1.1 CBEST Format, Appointment Scheduling & Section-by-Section Timing 1.2 The 123-Point Passing Standard, Section Scaled Scores & 37-Point Rule 1.3 CTC Basic Skills Requirement (CL-667), the SB 153 Degree Option & the Nine Statutory Pathways
2 Reading: Critical Analysis & Argument Evaluation
2.1 Author's Main Thesis, Supporting Evidence & Structural Hierarchy 2.2 Identifying Unstated Assumptions & Evaluating Premise-Conclusion Validity 2.3 Statements That Strengthen or Weaken Arguments & Logical Fallacies
3 Reading: Author Tone, Bias & Multi-Text Comparative Analysis
3.1 Detecting Author Tone, Attitude, Bias & Intended Audience 3.2 Distinguishing Objective Facts from Subjective Interpretations & Value Judgments 3.3 Synthesizing Paired Passages, Contradictions & Differing Points of View
4 Reading: Comprehension, Sequencing & Inference Skills
4.1 Literal Comprehension, Chronology & Multi-Step Process Tracking 4.2 Formulating Inferences, Predictions & Cause-and-Effect Relationships 4.3 Accurate Paraphrasing, Summarizing & Translating into Graphic Organizers
5 Reading: Contextual Vocabulary & Research Reference Skills
5.1 Contextual Meaning, Structural Word Analysis, Prefixes & Latin/Greek Roots 5.2 Figurative Devices, Idiomatic Expressions & Discourse Transition Markers 5.3 Using Reference Materials, Indexes, Tables of Contents & Reading Visual Data
6 Mathematics: Estimation Strategies & Mental Math
6.1 Pre-Computation Bounding & Reasonableness Checks Without a Calculator 6.2 Estimation Methods for Large Additions, Subtractions, Products & Quotients 6.3 Rounding Principles, Decimal Places & Contextual Precision Rules
7 Mathematics: Measurement Systems & Applied Geometry
7.1 U.S. Customary Units of Length, Weight, Capacity & Temperature Conversions 7.2 Perimeter, Circumference, Area of Polygons & Basic Geometric Shapes 7.3 Elapsed Time, Timetables, Scheduling & Multi-Rate Work Planning
8 Mathematics: Descriptive Statistics & Educational Test Scores
8.1 Mean, Median, Mode, Weighted Averages & Range Calculations 8.2 Elementary Probability, Simple and Compound Events & Likely Outcomes 8.3 Educational Statistics: Percentiles, Stanines (1–9) & Standardized Test Reports
9 Mathematics: Whole Number Arithmetic & Signed Integers
9.1 Multi-Digit Arithmetic, Factorization & Order of Operations (PEMDAS) 9.2 Operations with Positive and Negative Numbers & Integer Rules 9.3 Number Line Relationships, Absolute Value, Inequalities & Distance
10 Mathematics: Fractions, Decimals, Ratios & Percentages
10.1 Equivalent Fractions, Common Denominators, Mixed Numbers & Operations 10.2 Decimal Operations, Place Values & Fraction-Decimal-Percent Conversion 10.3 Ratios, Unit Rates, Proportions & Scale Factor Calculations 10.4 Percent Computation, Base Values, Percent Increase/Decrease & Discounts
11 Mathematics: Algebraic Reasoning & Practical Problem Solving
11.1 Translating Real-World Problem Statements into Algebraic Expressions 11.2 Solving One-Variable Linear Equations & Basic Linear Inequalities 11.3 Applied Consumer Math, Supplier Invoicing, Unit Pricing & Budgeting
12 Mathematics: Tables, Graphs & Applied Logical Reasoning
12.1 Extracting Table Data, Spreadsheet Formats & Solving for Missing Entries 12.2 Analyzing Bar Charts, Line Graphs, Circle Graphs & Comparative Trends 12.3 Formal Logic: Connectives (And/Or/If-Then) & Quantifiers (All/Some/None)
13 Writing: Rubric Criteria, Rhetorical Architecture & Mechanics
13.1 The CBEST 4-Point Holistic Rubric & Six Core Diagnostic Performance Dimensions 13.2 Rhetorical Force, Thesis Formulation, Paragraph Transitions & Essay Unity 13.3 Syntactic Variety, Active Voice, Standard Usage & Avoiding Common Grammar Traps
14 Writing: Essay Types, Scored Models & Exam-Day Strategy
14.1 Type I Essay Masterclass: Personal Experience Narrative & Reflective Insights 14.2 Type II Essay Masterclass: Expository Analysis of Statements & Issues 14.3 Benchmark Essay Analysis: Comparing Level 4, 3, 2, and 1 Student Models 14.4 45-Minute Timed Essay Strategy, Outlining Protocols & Proofreading Checklists
9.2 Operations with Positive and Negative Numbers & Integer Rules
Key Takeaways Explain This Topic AI
Integer addition depends on whether signs match: add absolute values when signs are identical and retain the common sign; subtract absolute values when signs differ and adopt the sign of the number with the greater magnitude.
Subtraction of signed integers is algebraically defined as addition of the additive inverse (a - b = a + (-b)), transforming double negatives into direct addition (a - (-b) = a + b).
In multiplication and division, matching signs yield positive results, while opposing signs yield negative results, regardless of which factor possesses the greater magnitude.
Exponents applied to signed numbers require careful parsing of parentheses: (-a)^n raises the negative value to the power n, whereas -a^n negates the positive power a^n.
Real-world signed scenarios—such as elevation changes, thermal fluctuations, and bank overdrafts—require translating descriptive language into formal signed expressions without dropping double negatives.
Last updated: September 2026
- 9.2 Operations with Positive and Negative Numbers & Integer Rules
- The Signed Number Domain (ℤ) on the CBEST
In the CBEST Mathematics subtest, operations on signed numbers assess whether examinees can manipulate negative quantities with absolute algebraic precision without a calculator. Negative values represent magnitudes below an established baseline zero: debts in accounting, depths below sea level, deficits in resource inventories, and temperatures below freezing. Mastery requires internalizing the fundamental algebraic symmetry of the integers (ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}), where every non-zero integer a possesses an additive inverse (opposite) -a such that a + (-a) = 0.
Rules for Integer Addition
Adding integers depends entirely on whether the addends share identical signs or possess opposing signs:
Case 1: Adding Numbers with Identical Signs
When adding two numbers with the same sign, sum their absolute values (magnitudes) and preserve the shared sign:
Positive + Positive = Positive: (+12) + (+15) = +27.
Negative + Negative = Negative: (-8) + (-14) = -(8 + 14) = -22.
Conceptual Model: If a school district incurs an operational deficit of $8,000 in October and a further deficit of $14,000 in November, the cumulative deficit is $22,000. On a number line, starting at -8 and moving 14 units further to the left arrives at -22.
Case 2: Adding Numbers with Opposing Signs
When adding two numbers with different signs, subtract the smaller absolute value from the larger absolute value, and assign the sign of the number that has the larger absolute value: Sign ( a + b ) = Sign of the term with max ( ∣ a ∣ , ∣ b ∣ ) \text{Sign}(a + b) = \text{Sign of the term with } \max(|a|, |b|) Sign ( a + b ) = Sign of the term with max ( ∣ a ∣ , ∣ b ∣ )
Example 1: (-17) + 11
Compare absolute values: |-17| = 17 and |11| = 11.
Calculate difference in magnitudes: 17 - 11 = 6.
Apply sign of larger magnitude (negative): (-17) + 11 = -6.
Example 2: 25 + (-9)
Compare absolute values: |25| = 25 and |-9| = 9.
Calculate difference: 25 - 9 = 16.
Sign of larger magnitude is positive: 25 + (-9) = +16.
Rules for Integer Subtraction: The Additive Inverse
Subtraction is formally defined in algebra as the addition of the additive inverse: a − b = a + ( − b ) a - b = a + (-b) a − b = a + ( − b )
Every subtraction problem can be converted into an addition problem by changing the subtraction operator to addition and reversing the sign of the subtrahend (the second number). This simple habit eliminates dropped signs.
- Subtraction Type Expression Transformation (a + (-b)) Simplified Result
- Pos. − - − Pos. (Smaller − - − Larger) 7 - 15 7 + (-15) -(15 - 7) = -8
- Pos. − - − Neg. (Double Negative) 12 - (-8) 12 + (+8) 12 + 8 = 20
- Neg. − - − Pos. -14 - 9 -14 + (-9) -(14 + 9) = -23
- Neg. − - − Neg. -20 - (-6) -20 + (+6) -(20 - 6) = -14
- Neg. − - − Neg. (Larger Magnitude) -5 - (-18) -5 + (+18) 18 - 5 = 13
- Deconstructing the Double Negative: a - (-b) = a + b
Examinees frequently stumble when subtracting negative quantities. Conceptually, subtracting represents removing a value. If you remove a debt of $50 from a balance sheet, your net worth increases by $50. Graphically on a number line, subtracting a positive steps toward the left, whereas subtracting a negative reverses direction and steps toward the right.
Integer Multiplication and Division Sign Conventions
Multiplication and division of signed integers follow strict structural rules based on the parity (even or odd count) of negative signs:
- The Two-Factor Sign Matrix
- (+) × (+) = (+) and (+) ÷ (+) = (+)
- (-) × (-) = (+) and (-) ÷ (-) = (+)
- (+) × (-) = (-) and (+) ÷ (-) = (-)
- (-) × (+) = (-) and (-) ÷ (+) = (-)
Unlike addition, the relative magnitudes of the factors have no influence on the sign of the product or quotient:
- (-2) × (-8) = +16
- (-48) ÷ (-6) = +8
- (-7) × (+9) = -63
- (+56) ÷ (-8) = -7
- Multi-Factor Products and the Parity Principle
When multiplying three or more signed factors, count the total number of negative factors:
An even number of negative factors produces a positive product .
An odd number of negative factors produces a negative product .
Example with 3 negatives (odd): (-2) × (-3) × (-5) = (6) × (-5) = -30.
Example with 4 negatives (even): (-1) × (-3) × (-2) × (-4) = (3) × (8) = +24.
Division by Zero: The Undefined Trap
A frequent conceptual trap on teacher credentialing exams involves division by zero:
0/a = 0 (for any non-zero integer a). Dividing zero items among 5 classrooms gives 0 per classroom.
a/0 is undefined . Division asks: "What number multiplied by the divisor yields the dividend?" There is no real number x such that 0 × x = 7. Never select zero when zero is in the denominator!
The Crucial Exponent Distinction: (-x)ⁿ vs. -xⁿ
One of the most heavily tested signed number traps on the CBEST is the distinction between a base enclosed in parentheses versus a base preceded by a negative sign:
Base Enclosed in Parentheses: (-a)ⁿ
The negative sign is bound to the base and multiplied n times:
- (-4)² = (-4) × (-4) = +16
- (-4)³ = (-4) × (-4) × (-4) = -64
- No Parentheses: -aⁿ
By the order of operations (PEMDAS), exponents precede negation (which is multiplication by -1). The exponent applies only to a, and the negative sign is affixed afterward:
- 4² = -(4²) = -(4 × 4) = -16
- 4³ = -(4³) = -(4 × 4 × 4) = -64
Notice that while odd powers produce identical values by coincidence, even powers produce opposing values: (-5)² = +25, whereas -5² = -25.
Applied Word Problem Contexts for Signed Integers
On the CBEST, signed number calculations are routinely contextualized in applied word problems:
- 1. Temperature Fluctuations
- Net Change Formula: Δ T = T_final - T_initial
If the temperature at Mammoth Lakes is -8°F at 5:00 AM and rises by 23°F by 1:00 PM, the new temperature is -8 + 23 = +15°F.
If the high temperature in Truckee is 14°F and drops to -9°F overnight, the temperature drop is 14 - (-9) = 14 + 9 = 23°F.
2. Vertical Elevation Above and Below Sea Level
Mean sea level is designated as elevation 0. Elevations above sea level are positive; depressions or underwater depths are negative.
Total vertical displacement between two locations: Displacement = Elevation_top - Elevation_bottom.
California features both the highest and lowest geographic points in the contiguous United States: Mount Whitney at +14,505 ft and Badwater Basin at -282 ft. Displacement = 14 , 505 − ( − 282 ) = 14 , 505 + 282 = 14 , 787 ft \text{Displacement} = 14,505 - (-282) = 14,505 + 282 = 14,787\text{ ft} Displacement = 14 , 505 − ( − 282 ) = 14 , 505 + 282 = 14 , 787 ft
3. Banking Ledgers and Overdraft Fees
Deposits and credits are positive (+x). Withdrawals, purchases, and debit fees are negative (-x).
If an account has a balance of $85, and transactions of -$120 and a -$35 overdraft fee occur, the updated balance is: Balance = 85 + ( − 120 ) + ( − 35 ) = 85 − 120 − 35 = − 35 − 35 = − $ 70 \text{Balance} = 85 + (-120) + (-35) = 85 - 120 - 35 = -35 - 35 = -\$70 Balance = 85 + ( − 120 ) + ( − 35 ) = 85 − 120 − 35 = − 35 − 35 = − $70