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Introduction
Independent and Dependent Variables helps you connect a mathematical rule to a result you can explain and check. Choose the input and output from context and label their units. A formula tells how the dependent variable changes with the independent variable. This lesson belongs to coordinate geometry and graphing 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 independent and dependent variables.
- 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
Read coordinate pairs and evaluate expressions.
Previous lesson: Plotting Points
Key mathematical vocabulary
- variable
- A letter representing an unknown or changing quantity.
- input
- A value supplied to a rule.
- output
- The result of a rule at an allowed input.
- expression
- A mathematical phrase representing a value.
- solution
- A value making the original conditions true.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Inputs and outputs
Choose the input and output from context and label their units. A formula tells how the dependent variable changes with the independent variable.
An independent variable is the input of a model; a dependent variable is the output calculated from it. Choosing an input causes the formula to specify an output, though this mathematical dependence does not automatically establish a scientific causal relationship.
Name each variable and its units. Put the input on the horizontal axis by convention and the output on the vertical axis. A table records matching input-output pairs. Context may restrict the inputs to nonnegative values, whole numbers, or a measured time interval.
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. The fare is calculated after distance is chosen or measured.
- 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. For d=5, C=78. The rule maps each distance to one cost.
- 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
A taxi fare is C = 12d + 18, where d is miles. Which variable is independent?
- The fare is calculated after distance is chosen or measured.
- Distance d is the input; fare C is the dependent output.
- The $18 initial charge does not depend on miles; each mile adds $12.
Answer: d (distance)
Verify: For d=5, C=78. The rule maps each distance to one cost.
Avoid this mistake: Swapping input and output in an ordered pair.
Example 2 · Basic
A taxi fare is C = 13d + 18, where d is miles. Which variable is independent?
- The fare is calculated after distance is chosen or measured.
- Distance d is the input; fare C is the dependent output.
- The $18 initial charge does not depend on miles; each mile adds $13.
Answer: d (distance)
Verify: For d=5, C=83. The rule maps each distance to one cost.
Avoid this mistake: Calling a constant rate the independent variable.
Example 3 · Intermediate
A taxi fare is C = 14d + 18, where d is miles. Which variable is independent?
- The fare is calculated after distance is chosen or measured.
- Distance d is the input; fare C is the dependent output.
- The $18 initial charge does not depend on miles; each mile adds $14.
Answer: d (distance)
Verify: For d=5, C=88. The rule maps each distance to one cost.
Avoid this mistake: Ignoring the meaningful input values in context.
Example 4 · Advanced
A taxi fare is C = 2d + 19, where d is miles. Which variable is independent?
- The fare is calculated after distance is chosen or measured.
- Distance d is the input; fare C is the dependent output.
- The $19 initial charge does not depend on miles; each mile adds $2.
Answer: d (distance)
Verify: For d=5, C=29. The rule maps each distance to one cost.
Avoid this mistake: Swapping input and output in an ordered pair.
Example 5 · Challenge
A taxi fare is C = 3d + 19, where d is miles. Which variable is independent?
- The fare is calculated after distance is chosen or measured.
- Distance d is the input; fare C is the dependent output.
- The $19 initial charge does not depend on miles; each mile adds $3.
Answer: d (distance)
Verify: For d=5, C=34. The rule maps each distance to one cost.
Avoid this mistake: Calling a constant rate the independent variable.
Common mistakes and corrections
- Swapping input and output in an ordered pair.
- Calling a constant rate the independent variable.
- Ignoring the meaningful input values in context.
Real-world applications
- Model taxi cost as a function of miles. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Relate a plant’s measured height to elapsed time. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Choose the input and output from context and label their units. A formula tells how the dependent variable changes with the independent variable.
An independent variable is the input of a model; a dependent variable is the output calculated from it. Choosing an input causes the formula to specify an output, though this mathematical dependence does not automatically establish a scientific causal relationship.
Frequently asked questions
What should I know before independent and dependent variables?
Read coordinate pairs and evaluate expressions.
How can I check my work?
For d=5, C=78. The rule maps each distance to one cost. 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?
Swapping input and output in an ordered pair. Calling a constant rate the independent variable. Ignoring the meaningful input values in context.
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: Model taxi cost as a function of miles.
- 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
- Choose the input and output from context and label their units. A formula tells how the dependent variable changes with the independent variable.
- Swapping input and output in an ordered pair.
- Calling a constant rate the independent variable.
- Ignoring the meaningful input values in context.
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