Achievement objective M87
In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:
 form and use trigonometric, polynomial, and other nonlinear equations.
Indicators
 Solves contextual problems by forming and solving equations.
 Solves quadratic and cubic equations with complex roots. These equations are likely to involve real coefficients so that solutions are complex conjugate pairs.
 Analyses the existence of solutions in the context of the situation.
 Nonlinear equations include:
 exponential
 logarithmic
 rational
 surd
 hyperbolic.
 Makes links to graphs of functions
M82, curve fitting
M84, manipulating expressions
M76 and
M86, and complex numbers
M89.
 See
key mathematical ideas on NZmaths.
Progression
M87 links from
M74,
M77 and to
M89.
Possible context elaborations
 Natural phenomena that result in periodic behaviours (for example, tidal patterns, sound waves, circular motion, harmonics …).
 Explore growth and decay situations.
 Completely factor a third degree polynomial, using long division and/or reminder and factor theorems.
 Given general solution for a trig equation, find the solutions that satisfy a particular context.
 Although numerical solutions (bisection and NewtonRaphson) do not need to be included, they could form part of an inquiry into dealing with nonlinear equations that can’t be solved with algebraic methods.

Activity: Big Bang Theory – Exact values

Graphing projectile paths

Investigating algebra: Exploring graphs

Practical situations involving trigonometric graphs: Includes pedal height of a bicycle, height of loaded oscillating spring.

Tidal estuaries: Investigating the times yachts can enter and leave tidal estuaries.
 Use of remainder and factor theorems.
Assessment for qualifications
NCEA achievement standards at levels 1, 2 and 3 have been aligned to the New Zealand Curriculum. Please ensure that you are using the correct version of the standards by going to the
NZQA website.
The NZQA subjectspecific resources pages are very helpful. From there, you can find all the achievement standards and links to assessment resources, both internal and external.
The following achievement standard(s) could assess learning outcomes from this AO:
Last updated September 17, 2018
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