Modelling Variations

Students learn how to model variations of direct and indirect proportions by setting up an equation.  They learn how to solve this equation to find k, the constant of proportionality.  As learning progresses students illustrate these linear and non-linear using graphs.

This unit takes place in and follows on from Ratio and Proportion.


Modelling Variations Lessons


Prerequisite Knowledge
  • Use ratio notation, including reduction to simplest form
  • Express a multiplicative relationship between two quantities as a ratio
  • Understand and use proportion as equality of ratios
  • Relate ratios to fractions
  • Express the division of a quantity into two parts as a ratio
  • Apply ratio to real contexts and problems (such as those involving conversion, comparison, scaling, mixing, concentrations)
  • Understand and use proportion as equality of ratios

Success Criteria
  • Understand that X is inversely proportional to Y is equivalent to X is proportional to 1/Y;
  • Solve problems involving direct and inverse proportion, including graphical and algebraic representations
  • Construct and interpret equations that describe direct and inverse proportion.


Key Concepts
  • The symbol ∝ is used to represent ‘is proportional to’.
  • Direct proportion and varies directly both include y ∝  x, y ∝ x2 and y ∝ x3
  • Indirect proportion and varies inversely both include y ∝ 1/x
  • k is used as the constant of proportionality
  • Students need to be able to associate the graphical representations with the various proportions.

Common Misconceptions
  • Students often struggle with writing the correct proportional formula from the written description. Writing indirect proportions is particularly difficult for most students.
  • Modelling the variation as a formula with the correct value of k is key to accessing this topic.
  • When students do write the correct formula they are often unable to correctly manipulate it to calculate unknown values.

 

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