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Introduction
Integers and Rational Numbers helps you connect a mathematical rule to a result you can explain and check. Work with signed fractions using common denominators. Explain what a negative quantity means rather than treating a minus sign as an instruction to discard it. This lesson belongs to algebra foundations and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in integers and rational numbers.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Find common multiples and interpret a fraction.
Previous lesson: Properties of Real Numbers
Key mathematical vocabulary
- factor
- An expression multiplied by another to form a product.
- ratio
- A comparison by division.
- expression
- A mathematical phrase representing a value.
- solution
- A value making the original conditions true.
- input
- A value supplied to a rule.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Signed rational arithmetic
Work with signed fractions using common denominators. Explain what a negative quantity means rather than treating a minus sign as an instruction to discard it.
A rational number can be written as a fraction of integers with a nonzero denominator. Integers, terminating decimals, and repeating decimals are rational. Negative numbers represent quantities on the opposite side of zero from positive numbers.
To add fractions, first express them using the same denominator, then add their signed numerators. To multiply, multiply numerators and denominators. To divide by a nonzero fraction, multiply by its reciprocal. Simplifying divides numerator and denominator by the same nonzero factor.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. Since 372=12·31, −12/372=−1/31.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. Both fractions represent signed amounts with the same unit size.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
Calculate −12/372 + 2/31.
- Since 372=12·31, −12/372=−1/31.
- The denominators now match: (−1+2)/31.
- Combine the numerators: 1/31. Reduce further if numerator and denominator share a factor.
Answer: 1/31
Verify: Both fractions represent signed amounts with the same unit size.
Avoid this mistake: Adding denominators when adding fractions: use a common denominator.
Example 2 · Basic
Calculate −13/403 + 2/31.
- Since 403=13·31, −13/403=−1/31.
- The denominators now match: (−1+2)/31.
- Combine the numerators: 1/31. Reduce further if numerator and denominator share a factor.
Answer: 1/31
Verify: Both fractions represent signed amounts with the same unit size.
Avoid this mistake: Canceling across addition: cancel factors, not individual terms.
Example 3 · Intermediate
Calculate −14/434 + 2/31.
- Since 434=14·31, −14/434=−1/31.
- The denominators now match: (−1+2)/31.
- Combine the numerators: 1/31. Reduce further if numerator and denominator share a factor.
Answer: 1/31
Verify: Both fractions represent signed amounts with the same unit size.
Avoid this mistake: Dividing by zero: zero can be a numerator, but never a denominator.
Example 4 · Advanced
Calculate −2/64 + 2/32.
- Since 64=2·32, −2/64=−1/32.
- The denominators now match: (−1+2)/32.
- Combine the numerators: 1/32. Reduce further if numerator and denominator share a factor.
Answer: 1/32
Verify: Both fractions represent signed amounts with the same unit size.
Avoid this mistake: Adding denominators when adding fractions: use a common denominator.
Example 5 · Challenge
Calculate −3/96 + 2/32.
- Since 96=3·32, −3/96=−1/32.
- The denominators now match: (−1+2)/32.
- Combine the numerators: 1/32. Reduce further if numerator and denominator share a factor.
Answer: 1/32
Verify: Both fractions represent signed amounts with the same unit size.
Avoid this mistake: Canceling across addition: cancel factors, not individual terms.
Common mistakes and corrections
- Adding denominators when adding fractions: use a common denominator.
- Canceling across addition: cancel factors, not individual terms.
- Dividing by zero: zero can be a numerator, but never a denominator.
Real-world applications
- Combine signed deposits and withdrawals. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Measure ingredients with positive and negative adjustments to a recipe. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Work with signed fractions using common denominators. Explain what a negative quantity means rather than treating a minus sign as an instruction to discard it.
A rational number can be written as a fraction of integers with a nonzero denominator. Integers, terminating decimals, and repeating decimals are rational. Negative numbers represent quantities on the opposite side of zero from positive numbers.
Frequently asked questions
What should I know before integers and rational numbers?
Find common multiples and interpret a fraction.
How can I check my work?
Both fractions represent signed amounts with the same unit size. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Adding denominators when adding fractions: use a common denominator. Canceling across addition: cancel factors, not individual terms. Dividing by zero: zero can be a numerator, but never a denominator.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Combine signed deposits and withdrawals.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Work with signed fractions using common denominators. Explain what a negative quantity means rather than treating a minus sign as an instruction to discard it.
- Adding denominators when adding fractions: use a common denominator.
- Canceling across addition: cancel factors, not individual terms.
- Dividing by zero: zero can be a numerator, but never a denominator.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
Continue learning
Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials