Models – SS1 Further Mathematics Lesson Note (Third Term)

A model is a simplified representation of a real thing that helps us understand, study, explain or predict how that thing works.

Behavioural objectives

By the end of the lesson, students should be able to:

1. Explain what a model is and why models are important.
2. Identify the different types of models around them.
3. Describe how models help us learn and solve problems.
4. Recognise the benefits and limitations of using models.
5. Create simple models, including a mathematical model, to represent real situations.
6. Use models to make predictions and test ideas.

What is a model?

A model is a simplified copy or representation of something real that shows its most important parts and leaves out confusing details.

• Simplified: a toy car is much simpler than a real car, but it still shows the shape.
• Selective: a model keeps only the features that matter for its purpose.
• Useful: it makes a complicated thing easier to study.

What are simple examples of models?

• Globe: a model of the Earth.
• Toy aeroplane: a model of a real aeroplane.
• Map: a model of a real place.
• Weather map on TV: a model of weather patterns.

Why do we use models?

We use models to learn about things, test ideas, make predictions, and save time and money.

Learning and understanding

Models help us study things that are hard to study directly.

• Too big: the solar system.
• Too small: atoms.
• Too dangerous: volcanoes.
• Too expensive: building a real bridge.
• Too slow: how mountains form.
• Too fast: lightning.

Testing ideas

• Safe testing: models let us try ideas before real life.
• Example: engineers test a model bridge to check that the design is strong enough.

Making predictions

• Weather models: help predict whether it will rain tomorrow.
• Economic models: help predict whether prices will rise or fall.

Saving time and money

• Cheaper tests: it costs less to build and test a model aeroplane than a real one that might crash.
• Faster work: design problems are found early.

What are the types of models?

There are four types of models: physical, mathematical, computer and conceptual models.

What are physical models?

Physical models are three-dimensional copies of real things that you can touch and see.

Examples

• Scale models: model trains, dollhouses and architectural models.
• Anatomical models: plastic skeletons and heart models in science class.
• Prototype models: early versions of new products before mass production.
• Educational models: solar system models and volcano models for science projects.

Benefits

• Easy to understand: you can see and touch them.
• Show fit: good for showing how parts fit together.
• Show size: help you see size relationships.
• Hands-on: great for practical learning.

Limitations

• Incomplete: cannot show everything, such as how hot the sun really is.
• Different behaviour: may not work exactly like the real thing.
• Cost: detailed models can be expensive to build.

What are mathematical models?

Mathematical models use numbers, equations and formulas to represent real situations.

Examples

• Budget models: planning how to spend your allowance.
• Growth models: predicting how tall a plant will grow.
• Population models: estimating how many people will live in a city.
• Economic models: predicting how price changes affect buying.

Simple example: if you save ₦500 every week, the model is: Money saved = ₦500 × number of weeks.

Benefits

• Precise: gives exact answers.
• Fast: handles complex calculations quickly.
• Predictive: good for making predictions.
• Computer-friendly: can be used with computers.

Limitations

• Hard to follow: needs maths skills to understand.
• Missing factors: may not capture all real-world factors.
• Data needed: requires accurate data to work well.

What are computer models?

Computer models are programs that simulate (copy) real situations on a computer.

Examples

• Video game simulations: flight simulators and driving games.
• Weather simulations: programs that predict the weather.
• Business simulations: programs that test business strategies.
• Scientific simulations: programs that show how diseases spread.

Benefits

• Speed: can run many tests quickly.
• Long periods: can show things that happen over a long time.
• Safety: a safe way to test dangerous situations.
• Flexible: can be changed easily to test different cases.

Limitations

• Programming: only as good as the programming.
• Missing factors: may not include all real-world factors.
• Resources: requires computers and technical knowledge.

What are conceptual models?

Conceptual models are ideas or theories that help explain how things work.

Examples

• Food chain models: show how energy flows from plants to animals.
• Communication models: show how messages go from sender to receiver.
• Learning models: explain how people learn new information.
• Organisational models: show how companies are structured.

Benefits

• Organise ideas: help arrange complex ideas.
• Easy to share: simple to communicate to others.
• Show relationships: good for understanding how parts link.
• Low cost: need no special materials or technology.

Limitations

• Too simple: may leave out too much.
• Open to different readings: people may interpret them differently.
• Hard to test: difficult to test or prove.

How do the four types compare?

How do we build a mathematical model?

A mathematical model is built by turning a real situation into variables and an equation that can be used to calculate answers.

1. State the situation in words.
2. Choose letters for the quantities that change (variables).
3. Write an equation linking the variables.
4. Substitute values to calculate answers.
5. Check the answers against the real situation.

Worked example 1: savings

• Situation: Ade saves ₦500 every week.
• Model: S = 500w, where S is the money saved in naira and w is the number of weeks.
• Prediction: after 8 weeks, S = 500 × 8 = ₦4,000.

Worked example 2: delivery cost

• Situation: a bakery charges ₦1,500 for delivery plus ₦700 for each loaf.
• Model: C = 1500 + 700n, where C is the total cost in naira and n is the number of loaves.
• Prediction: for 6 loaves, C = 1500 + 700 × 6 = 1500 + 4200 = ₦5,700.
• Limit of the model: it assumes the price per loaf and the delivery charge never change.

How are models created?

A model is created in six steps: identify the purpose, choose the type, identify key features, build, test and validate, then use and refine.

Step 1: Identify the purpose

Decide what you want the model to do.

• What problem are you trying to solve?
• What do you want to learn?
• Who will use the model?
• How detailed does it need to be?

Step 2: Choose the type of model

Pick the type that best suits the purpose.

• Physical model: for showing shape and size.
• Mathematical model: for making calculations.
• Computer model: for complex simulations.
• Conceptual model: for explaining ideas.

Step 3: Identify key features

Decide which parts of the real thing are most important to include.

• What are the essential elements?
• What details can be left out?
• What relationships need to be shown?
• What level of accuracy is needed?

Step 4: Build the model

Create the model using suitable materials or methods.

• Gather: collect the materials or data.
• Follow: keep to your plan.
• Test: check the model as you build it.
• Adjust: make changes as needed.

Step 5: Test and validate

Check whether the model works as expected.

• Does it represent the real thing accurately?
• Does it serve its intended purpose?
• Are the results reasonable?
• What improvements could be made?

Step 6: Use and refine

Put the model to use and improve it over time.

• Use: apply the model for its intended purpose.
• Feedback: ask users what they think.
• Improve: make changes based on experience.
• Update: keep the model current.

What are examples of models in everyday life?

Models appear at home, at school, in business, in science and in technology.

At home

• Recipes: instructions that model how to make food.
• Furniture assembly instructions: diagrams that model how parts fit together.
• Budget spreadsheets: mathematical models of family finances.
• Exercise plans: models for improving fitness.

At school

• Classroom layouts: physical models showing where desks go.
• Lesson plans: conceptual models of how to teach topics.
• Science experiments: models of natural events.
• Study timetables: models for organising learning time.

In business

• Business plans: models describing how a company will operate.
• Organisational charts: models showing company structure.
• Financial projections: mathematical models predicting profits.
• Marketing strategies: models for reaching customers.

In science

• Atomic models: show how atoms are structured.
• Climate models: predict future weather patterns.
• DNA models: show the structure of genetic material.
• Ecosystem models: show how living things interact.

In technology

• App prototypes: early models of new software.
• Network diagrams: show how computers connect.
• User interface mock-ups: show how websites will look.
• System architecture: shows how technology systems work.

What are the benefits of using models?

The benefits of models fall into three groups: educational, practical and research benefits.

Educational benefits

• Simplify complex ideas: make difficult concepts easier to understand.
• Visual learning: help people who learn better by seeing.
• Hands-on experience: allow safe practice and experiments.
• Active engagement: keep learners interested and involved.

Practical benefits

• Cost savings: test ideas cheaply before spending heavily.
• Risk reduction: find problems before they cause real damage.
• Time efficiency: speed up design and testing.
• Communication: explain ideas to others clearly.

Research benefits

• Hypothesis testing: test theories in controlled conditions.
• Data collection: gather information that would be hard to get otherwise.
• Prediction: forecast future events or outcomes.
• Comparison: compare different options or scenarios easily.

What are the limitations of models?

The limitations of models fall into three groups: simplification issues, accuracy concerns and interpretation challenges.

Simplification issues

• Missing details: important factors might be left out.
• Oversimplification: reality may be more complex than the model shows.
• Context differences: a model might not work in every situation.
• Scale problems: what works in a small model might not work at full size.

Accuracy concerns

• Data quality: a model is only as good as the information used to build it.
• Assumption errors: wrong assumptions lead to wrong conclusions.
• Measurement problems: some real-world factors are difficult to measure.
• Dynamic changes: the real world can change faster than the model is updated.

Interpretation challenges

• User understanding: people might misunderstand what the model shows.
• Overconfidence: users might trust the model too much.
• Misapplication: using a model for a purpose it was not designed for.
• Cultural differences: a model might not work the same way in different cultures.

How do we evaluate a model?

We evaluate a model by asking questions about its accuracy, usefulness and limits.

Accuracy questions

• How well does this model represent reality?
• What important things might be missing?
• How reliable are the data used?
• Has the model been tested and validated?

Usefulness questions

• Does this model serve its intended purpose?
• Is it helpful for making decisions?
• Can it make accurate predictions?
• Is it worth the time and cost to use?

Limitation questions

• What are the boundaries of this model?
• In what situations might it not work?
• What assumptions is it based on?
• How might it be improved?

How can students create their own simple models?

Students can build a physical, mathematical or conceptual model from a familiar situation.

Project idea 1: Classroom layout model

This is a physical model of your ideal classroom.

1. Measure your real classroom.
2. Choose a smaller scale, such as 1 cm on paper for 50 cm in the room.
3. Use cardboard or paper to make the furniture.
4. Test different arrangements.
5. Present your design to others.

Project idea 2: Daily schedule model

This is a mathematical model of your perfect day.

1. List all the activities you need to do.
2. Assign a time in hours to each activity.
3. Write an equation showing that the times add up to 24 hours.
4. Test whether everything fits in 24 hours.
5. Adjust priorities as needed.

Project idea 3: School lunch decision model

This is a conceptual model for choosing healthy lunches.

1. Identify the factors: nutrition, taste, cost and time.
2. Draw a decision tree or flowchart.
3. Test it with different lunch options.
4. Get feedback from classmates.
5. Refine the model using their input.

How do models help in problem solving?

Models help us identify a problem, develop a solution and plan how to carry it out.

Problem identification

• Current situation: shows what things look like now.
• Problem areas: highlights where problems occur.
• Impact: shows the effect of problems.
• Communication: helps explain issues to others.

Solution development

• Safe trials: try different approaches without real risk.
• Comparison: compare several options.
• Prediction: predict the outcome of each choice.
• Refinement: improve solutions before they are used.

Implementation planning

• Steps: show the process step by step.
• Resources: identify what is needed.
• Obstacles: predict possible difficulties.
• Timelines: create schedules.

What is the future of models?

Models are likely to become more realistic, more powerful and more widely used as technology develops.

Technological advances

• Virtual reality: more immersive and realistic models.
• Artificial intelligence: models that learn and improve themselves.
• 3D printing: easier creation of physical models.
• Cloud computing: more powerful computer models available to many people.

New applications

• Personalised education: models adapted to individual learning styles.
• Environmental protection: better models for understanding climate change.
• Healthcare: models for personalised medicine.
• Smart cities: models for improving urban life.

Evaluation

1. Define a model and give two examples.
2. State four reasons why we use models.
3. Name the four types of models. Give one benefit and one limitation of each.
4. List the six steps in creating a model.
5. A taxi charges ₦800 for booking plus ₦300 per kilometre. Write a mathematical model for the cost C of a journey of d kilometres, then find the cost of 5 km.

What practice activities can students try?

• Activity 1: find five different models in your daily life and classify each as physical, mathematical, computer or conceptual.
• Activity 2: choose something complex, such as your school cafeteria system, and design a simple model of how it works.
• Activity 3: compare two models of the same thing, such as two weather apps, and discuss their strengths and weaknesses.
• Activity 4: ask a family member which models they use in their work or hobbies.

Assignment

1. Develop a mathematical model for planning your weekly homework schedule. Define your variables, write the equation, use it to predict one result, and state one limitation of the model.