This unit *Pricing for Profit* is one of 5 units making up the Special Topic “Mathematical Modelling”.

Students approach the problem of how to price a product to maximise profit, in this case, items sold at a school fair to raise money. This unit involves students in iterative cycles of model improvement as they consider more sophisticated versions. As they work through the unit students see how varying the assumptions that they make affects the mathematical model they develop.

Additionally a lesson is devoted to developing knowledge and skills in model formulation.

Students produce a report to explain how they have developed the model, and how the model can be used to develop several viable alternative pricing strategies.

For classes who have not recently done Unit 1* Introduction to Modelling* a single ‘quick start’ lesson is provided.

Read the Teachers’ Guide (included in the download package) before using this resource.

### Lesson 1: A Simple Model

Working in the context of selling toys and biscuits at a school fair, students meet specialist terms, such as ‘profit’. They develop a spreadsheet and graph to model how profit depends on the number of items sold and the selling price. They consider underlying assumptions and how varying these will lead to a more sophisticated model, developed in later lessons.

### Lesson 2: Improving the Model

Students refine their model from Lesson 1 by addressing one major unrealistic assumption they made: that the number of sales is independent of selling price. They consider possible relationships between selling price and sales and derive (quadratic) relationships between price and profit.

### Lesson 3: Better Models with Spreadsheets

Students refine their models and implement them as spreadsheets. First, they make a spreadsheet where sales decrease as price increases for any linear decreasing pattern, and observe the effect of assumptions such as how many people buy at a given price. Then they include production costs in the model, and optionally make increasingly automated spreadsheets. There are good opportunities to use algebra.

### Lesson 4: Reflecting on Modelling

Students choose the model they think is best, and work in pairs to write report, critique the report of another pair and revise their reports in the light of their peers’ critique. They reflect on the process of modelling and consider the wider applicability of the model they have developed.

### Lesson 5: Formulating Models

Students focus on how to formulate models. First, students identify variables that are important in making a choice of what coffee to buy. Second, students select relationships between variables that might best model the likely spread of a bushfire. Third, they generate relationships to model various situations, graphically and algebraically.

Last updated November 9 2018.

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