Further Mathematics - Senior Secondary 3 - Modelling

Modelling

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TERM: 2ND TERM

WEEK: 2
Class: Senior Secondary School 3
Age: 17 years
Duration: 40 minutes of 4 periods
Subject: Further Mathematics
Topic: Modelling
Focus: Introduction to Modelling, Dependent and Independent Variables, Construction of Models, Methodology of Modelling, Application to Various Sectors (Physical, Biological, Social, and Behavioral Sciences).
SPECIFIC OBJECTIVES:
By the end of the lesson, students should be able to:

  1. Define mathematical modeling and understand its importance in real-life applications.
  2. Identify dependent and independent variables in mathematical modeling.
  3. Construct simple mathematical models using given data.
  4. Understand the methodology of constructing mathematical models.
  5. Apply mathematical models to physical, biological, social, and behavioral scenarios.

INSTRUCTIONAL TECHNIQUES:
• Direct explanation
• Group work
• Guided practice
• Discussion
• Real-life examples
• Case study analysis

INSTRUCTIONAL MATERIALS:
• Chart showing various types of models (symbolic, conic, mental models, etc.)
• Whiteboard and markers
• Projector for displaying visual examples
• Worksheets for modeling exercises
• Access to real-world datasets

PERIOD 1 & 2: Introduction to Modeling and Dependent and Independent Variables

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction to Modeling

Explains the concept of mathematical modeling and its applications in solving real-world problems. Provides examples from physical sciences, engineering, and economics.

Students listen attentively and ask questions for clarity.

Step 2 - Dependent and Independent Variables

Introduces dependent and independent variables in the context of modeling. Demonstrates how independent variables influence dependent variables in mathematical equations. Example: In a simple physics model, distance is dependent on time.

Students take notes and ask questions to clarify their understanding.

Step 3 - Identifying Variables

Provides examples and asks students to identify the dependent and independent variables in each scenario. Example: In a model predicting population growth, time is the independent variable, and population size is the dependent variable.

Students participate in the identification activity and discuss in pairs.

Step 4 - Recap and Discussion

Reviews key points and opens the floor for questions. Encourages students to provide their examples.

Students discuss in pairs and ask questions about variable identification.

NOTE ON BOARD:

    
  • Modeling: A process of representing real-world phenomena using mathematical expressions or equations.
  • Independent Variable: The variable that is manipulated or changed.
  • Dependent Variable: The variable that responds or changes as a result of the independent variable.
    EVALUATION (5 exercises):
  1. Define mathematical modeling in your own words.
  2. What is the difference between a dependent and an independent variable?
  3. Give one example of a real-world situation where modeling is used.
  4. In the equation y=3x+5y = 3x + 5, identify the dependent and independent variables.
  5. Why are dependent and independent variables important in mathematical modeling?

CLASSWORK (5 questions):

  1. In a model predicting the cost of goods based on quantity purchased, identify the dependent and independent variables.
  2. Write a simple equation to model the relationship between the total cost and the number of items purchased.
  3. What is the dependent variable in a model that predicts the temperature based on time of day?
  4. Describe a real-life situation where you can apply mathematical modeling.
  5. Explain how the choice of independent and dependent variables affects the outcome of a model.

ASSIGNMENT (5 tasks):

  1. Research a real-world situation where mathematical modeling is used in biological sciences.
  2. Write a short essay explaining the significance of independent variables in mathematical models.
  3. In a social science model predicting voting patterns, identify the dependent and independent variables.
  4. Provide an example of a conic model in physics or engineering.
  5. Create a simple mathematical model to predict the cost of electricity consumption based on hours used.

PERIOD 3 & 4: Construction of Models and Methodology of Modeling

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Construction of Models

Demonstrates how to construct a simple mathematical model using real-world data. Walks through the process of formulating a mathematical equation. Example: Modeling the distance traveled by a car over time using a linear equation.

Students observe the teacher’s construction process and take notes.

Step 2 - Methodology of Modeling

Introduces the steps involved in building a model: identifying the problem, defining variables, collecting data, formulating an equation, and validating the model. Explains each step using examples.

Students follow along and ask questions as needed.

Step 3 - Guided Practice

Provides students with datasets (e.g., growth rates of plants over time) and guides them through the construction of a simple model. Encourages students to work in pairs.

Students work in pairs to construct their models, with teacher guidance.

Step 4 - Application to Real-Life Scenarios

Discusses various fields where mathematical modeling is applied (e.g., physical, biological, social, and behavioral sciences). Presents case studies and encourages students to analyze them.

Students analyze case studies and discuss potential applications of modeling in these fields.

NOTE ON BOARD:

    
  • Steps in Model Construction:
  1. Define the problem.
  2. Identify the independent and dependent variables.
  3. Collect relevant data.
  4. Formulate a mathematical equation.
  5. Validate the model by testing it against real-world scenarios.

 

EVALUATION (5 exercises):

  1. How do you construct a simple mathematical model for predicting rainfall based on temperature?
  2. Explain the methodology of mathematical modeling in your own words.
  3. What is the importance of validating a model in real-world applications?
  4. Construct a simple model for predicting the amount of money saved based on the number of months.
  5. What steps should you follow when creating a model for population growth?

CLASSWORK (5 questions):

  1. Use the following data to construct a simple model for predicting the speed of a moving object based on time:
    • Time: 1s, Speed: 5m/s
    • Time: 2s, Speed: 10m/s
    • Time: 3s, Speed: 15m/s
  2. In a model predicting the price of goods based on demand, what variables would you consider?
  3. Describe how you would validate a model for predicting energy consumption based on weather conditions.
  4. Construct a model to predict the number of customers visiting a store based on day of the week.
  5. In a biological model, what independent variables would you choose to study the growth rate of bacteria?

ASSIGNMENT (5 tasks):

  1. Research a mathematical model used in physical sciences and explain its application.
  2. Construct a model predicting the time taken for a vehicle to travel a given distance based on speed.
  3. In a social model predicting income based on education level, what variables should be considered?
  4. Explain the importance of dependent variables in economic models.
  5. Create a simple mathematical model to predict the number of hours of sleep a person needs based on age.