Lesson 2 2025 Variables, Equations _ Graphs

Module Overview

  • Module: NCHE 171

  • Lesson: 2

  • Topic: Variables, Equations & Graphs

  • Lecturer: Dr. N.H. SEHERI

  • Institution: North-West University

Objectives

At the end of this lesson, students must be able to:

  • Describe what a variable is.

  • Explain why variables are required in a scientific investigation.

  • Provide correct relationships between variables.

  • Calculate the percentage composition using different variables.

  • Convert statements into equations.

  • Draw a graph from a given equation and determine whether X is directly or inversely proportional to Y.

Introduction

  • Definition of Variables:

    • Variables are quantities that can have different values under different conditions.

    • Important to identify and control variables that affect measurement in experiments.

Types of Variables

Independent Variables

  • Variables that the experimenter changes to observe effects on other variables.

  • Denoted by the letter x in experiments or graphs.

  • A change in the independent variable directly causes a change in the dependent variable.

Dependent Variables

  • The variable being tested and measured in an experiment.

  • Is 'dependent' on the independent variable.

  • Observed changes are recorded when the independent variable is altered.

    • Example: In an experiment measuring light’s effect on moth behavior, the amount of light is the independent variable, and the moth's reaction is the dependent variable.

Controlled Variables

  • Factors that are kept constant throughout an investigation.

  • Essential to avoid influencing the experiment's outcome.

    • Example: In testing plant growth with different water types, light amount will be a controlled variable.

Constant Variables

  • Variables held constant after identifying independent and dependent variables.

  • Examples include:

    • Concentration

    • Volume

    • Mass

    • Temperature

    • Pressure

    • Particle size

    • Time

Relationships Between Variables

  • Relationships can be expressed in three ways:

    • Statements

    • Equations

    • Graphs

Conversion of Statements into Equations

  1. Density: Density = Mass / Volume

  2. Ratio: Density defined as the ratio of mass to volume

  3. Fraction: Fraction = Part / Whole

  4. Percentage: Percentage = Fraction Ă— 100

Proportional Relationships

Directly Proportional

  • If Y is directly proportional to X, then increasing X will also increase Y in the same proportion.

    • Example: Doubling X will double Y.

Inversely Proportional

  • If Y is inversely proportional to X, then as X increases, Y decreases in the same proportion.

Activities

  • Convert Statements into Equations:

    1. Resistance (R) of a wire is directly proportional to its length (l) and inversely proportional to its area of cross-section (a).

    2. Concentration (cA) of solute A is defined as amount (nA) of A per unit volume (V) of solution.

Graphs

  • Graphs facilitate data display and analysis, making patterns and relationships visually identifiable.

  • Typically,

    • Independent variable is on the x-axis

    • Dependent variable is on the y-axis

  • A good graph structure shows significant features and findings clearly.

Types of Graphs

Line Graphs

  • Show continuous data over time.

  • Determine independent and dependent variables before plotting.

X-Y Scatter Plots

  • Individual data points can be connected to illustrate trends.

  • Useful in scenarios where incremental change from point to point is essential (e.g., titration curves).

Example of a Graph

  • A linear graph equation is generally expressed as y = mx + c.

  • Example: For the equation y = 3x + 2, build a table of values, e.g., (X: -2, -1, 0, 1, 2; Y: -4, -1, 2, 5, 8). This table represents the corresponding y-values for each x-value based on the equation y = 3x + 2.

Exponential/Scientific Notation

  • Definition: A method to present very large or very small numbers in a compact form that simplifies calculations.

  • Positive Exponent: Moving the decimal to the left.

  • Negative Exponent: Moving the decimal to the right.

Significant Figures

  • Definition: The exact digits known in a calculated number.

  • Precision Rule: A result from experimental data cannot be more precise than the least precise piece of information.

Determining Significant Figures

  • Non-zero digits are significant.

  • Zeros between non-zero digits are significant.

  • Zeros to the right of a non-zero number and a decimal point are significant.

Using Significant Figures in Calculations

  • Addition/Subtraction: The number of decimal places in the result should match the value with the least decimal places.

  • Multiplication/Division: The number of significant figures is determined by the value with fewer significant figures.

  • Rounding Rules: Specific rules apply for rounding off.

Dimensional Analysis

  • Definition: A problem-solving approach using dimensions or units as a guide for calculations. Also known as the factor-label method.

  • Conversion Factor: Used to express the equivalence of a measurement in two different units.

This lesson on emphasizes the role of variables in scientific investigations. It defines independent, dependent, controlled, and constant variables, explores their relationships, and outlines methods for converting statements into equations. The lesson also covers essential graphing techniques for accurately displaying data, along with key topics such as scientific notation, significant figures, and dimensional analysis for precise calculations.