Equations and Expressions: Level 6

The key idea that underpins this thread is that equations can be treated as objects, which can be acted on and transformed while preserving the same solutions.

At lower levels of the curriculum it is possible to view most equations as processes that give a result for any particular value of x, and therefore that the process can be reversed to solve for x.  At Level 6 this is no longer possible for many of the equations encountered.  Instead students need to act on equations by carrying out transformations.  They need to understand that when they do the same thing to each side of an equation, or re-arrange one side into an equivalent expression, they have created an equivalent equation with the same solution as the original.

The power of transforming equations can lead to an understanding of solution methods for: linear equations with unknowns on both sides, by doing the same thing to both sides; quadratic equations, by getting all terms to one side and factorising; simultaneous linear equations, by doing the same thing to both sides and combining to eliminate a variable; and inequations, by doing the same thing to both sides.

The purpose of solving equations should be to solve problems.  A powerful feature of algebra is that equations can be formed to describe practical problems.  These equations can then be solved without further reference to the original problem, until finally the answers of these abstract calculations are related back to the original problem context.  It is therefore important for students to appreciate that not all mathematical solutions (for example negative values) may be meaningful solutions to real problems. 

The material in this thread at Level 6 forms a progression between level 5 equations and expressions and level 7 equations and expressions.  Quadratic equations were introduced at Level 5 and in Level 7 students are expected to use the quadratic formula to solve them.  Level 6 is therefore a time for consolidation of solving by factorising.  Simple exponential equations of the form ax= b, which may be solved by inspection and basic knowledge, are encountered for the first time.  Familiarity with these is important as a foundation for an algebraic understanding of manipulating logarithmic expressions in Level 7.  Similarly familiarity with simultaneous linear equations in Level 6 provides a foundation for simultaneous non-linear equations in Level 7.  Solving equations is basic to a large proportion of mathematics.  At higher levels students need to know how to solve more and more equations and this knowledge needs to be virtually automatic, as a means to an end of solving the bigger problem.  Furthermore, the solution methods for solving equations become more difficult as the equations become more complicated.

The material in this thread also has strong connections with Level 6 patterns and relationships.  Graphs of equations are an important representation that provide further understanding of solutions.  This is particularly so for inequations in two unknowns, simultaneous equations and quadratic equations.  Situations where there are no solutions or an infinite number of solutions become much clearer when equations are represented graphically.  Graphics calculators therefore start to become particularly useful at this level. 

This key idea develops from the key idea of equations and expressions at level 5.

This key idea is extended to the key idea of equations and expressions at level 7.