The box problem

This is a simple modelling task which can be used early in the course to demonstrate how students’ existing mathematical knowledge can be used to solve an optimisation problem. It provides an introduction to mathematical modelling and seeks to demonstrate that much of the mathematics they have previously learnt can have applications. It also provides an opportunity to introduce spreadsheet techniques that will be new for some students.

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Overview of task

This is a simple modelling task which can be used early in the course to demonstrate how students’ existing mathematical knowledge can be used to solve an optimisation problem. It provides an introduction to mathematical modelling and seeks to demonstrate that much of the mathematics they have previously learnt can have applications. It also provides an opportunity to introduce spreadsheet techniques that will be new for some students.

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Strands

Number and Measures; Algebra and Graphs

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Prior knowledge

Trial and improvement, algebra, graphs.

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Relevance to Core Maths qualifications

  • AQA
  • C&G
  • Eduqas
  • Pearson / Edexcel
  • OCR

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Suggested approaches

The task is suitable for group work, with opportunities for collaboration, discussion, independent working or homework.

Students might begin with a hand calculation of the volumes associated with different sizes of cut-out and present the results in a table and a graph. This gives the option of trial and improvement or a graphical solution. An algebraic approach would allow students to use a graph-plotter.

It makes sense to create a spreadsheet to replicate the initial solution to the 20cm x 20cm square before trying to adapt to different squares and rectangles.

Moving from a square to a rectangle of card complicates the problem slightly. Students might try a hand calculation method initially or go straight to an adapted spreadsheet or formula.

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Resources/documentation

  • Worksheet for students
  • Teacher guidance notes (a calculus solution is included for the teacher but it is not expected to be used in any detail with the students)

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Relevant digital technologies

Spreadsheet for modelling, graphing software, calculator.