Related Variables - Algebra-1

1. Fundamental Concepts

Related Variables: Variables that are connected in a specific context or study, where changes in one variable may affect changes in another.
Independent Variable:
A variable that is actively controlled or manipulated by the researcher, considered the "cause" of changes in other variables, and its changes do not depend on other variables.
Dependent Variable:
A variable that is observed or measured, considered the "effect" resulting from changes in the independent variable, and its changes depend on the independent variable.
Core Relationship: Independent variable (cause) → Dependent variable (effect), which forms the basis for analyzing related variables.

Correlation: There is a tendency for interaction between independent and dependent variables, but it is important to note that "correlation ≠ causation"; causal relationships need to be verified through logic or experiments.
Single vs. Multiple Variables:
- A single independent variable can correspond to a single dependent variable (e.g., "study time → exam score");
- Multiple independent variables may also affect one dependent variable (e.g., "study time + learning method → exam score").
Operability of Variables: Independent variables need to be controllable (e.g., "temperature," "time"), and dependent variables need to be measurable (e.g., "speed," "score").
Extraneous Variables: Variables that may interfere with the dependent variable but are not controlled, which should be excluded in analysis (e.g., "rainfall" may be an extraneous variable when studying "fertilizer amount → crop yield").

Scenario: "Studying the effect of daily sun exposure time on vitamin D levels"
- Independent variable: Daily sun exposure time
- Dependent variable: Vitamin D levels
Scenario: "Investigating the relationship between the number of practice problems students do daily and their math test scores"
- Independent variable: Number of daily practice problems
- Dependent variable: Math test scores

Scenario: "A brand studies how phone charging time affects battery life, while considering the interference of different charging temperatures"
- Independent variable: Phone charging time (core variable)
- Dependent variable: Battery life
- Extraneous variable: Charging temperature
Scenario: "A teacher tries different teaching methods (traditional lecture/group discussion) and observes changes in students' classroom participation"
- Independent variable: Teaching method (traditional lecture/group discussion)
- Dependent variable: Students' classroom participation

Scenario: "Scientists study the effects of 'fertilizer amount' and 'light intensity' on plant growth height, while recording whether plant species are the same"
- Independent variables: Fertilizer amount, light intensity (multiple independent variables)
- Dependent variable: Plant growth height
- Controlled variable: Plant species (needs to be consistent to eliminate interference)
Scenario: "A company tests the impact of different advertising budgets and advertising durations on product sales"
- Independent variables: Advertising budget, advertising duration
- Dependent variable: Product sales

Clarify the research purpose: Identify causal relationships through keywords like "influence" or "effect on" (e.g., in "the effect of X on Y," X is the independent variable and Y is the dependent variable).
Questioning method:
- Ask "What is actively changed?" → Independent variable;
- Ask "What is observed or measured?" → Dependent variable.
Eliminate interference: Distinguish extraneous variables and focus on the core "cause-effect" relationship (e.g., in "running distance → weight change," "diet" may be an extraneous variable, which should be assumed constant).
Multivariate analysis: For multiple independent variables, clarify how each variable affects the dependent variable (e.g., analyzing the relationship between "temperature, pressure, and gas volume" by examining each variable's impact on volume separately).
Reverse verification: Judge by asking "If A changes, will B change?" (If changes in A lead to changes in B, then A is the independent variable and B is the dependent variable).