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A “Problematic” Archer

Students often bring in questions that are taken or adapted from sources outside of the usual textbooks. This is especially true with regards to revision material when leading up to topic test or other major assessment. One question of this type that really caught my attention was from a "Smart Study" guide given to students preparing for a SAC (i.e. major test) on Functions and Graphs. The diagram is shown here:

(note this diagram remains the intellectual property of the copyright holder)

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A “Novel” approach to polynomial division…

During the last few weeks I have been working with some Year 11 students from a local private school that has a fairly good reputation. These students have been working, in particular, on cubic and quartic functions, including the use of polynomial division. The students in question are taught by two different teachers at the school, including their school’s Head of Mathematics, a highly experienced and long serving teacher by all accounts. All of these students are certainly above average students, but like most of their peers, have had little exposure to the long division method in primary school, mostly due to the prevalence of handheld calculators.

Whilst working with these students, each of them has independently demonstrated to me a short-cut method taught to them at school for polynomial division. This short-cut method is well known and quite useful when applied correctly. Unfortunately, when originally demonstrated to the students (judging by each of their theory books), their respective teachers each made the same monumental error and taught the students a technique that would work only in certain circumstances. The following are some of the examples given to these students in class:

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