Algebra - Level 8

Achievement objectives

Exploring patterns and relationships
Within a range of meaningful contexts, students should be able to:

Exploring equations and expressions
Within a range of meaningful contexts, students should be able to:

Suggested learning experiences

Exploring patterns and relationships
Students should be:

• using graphing, modelling, and calculator techniques to explore a range of divergent, convergent, and oscillating sequences and series (including exponential and logarithmic series);
• investigating graphing, using sketching, plotting, computers, and calculators;
• modelling practical situations from science, commerce, and the social sciences;
• using piece-wise functions arising from practical contexts;
• using graphical and algebraic techniques to explore and illustrate reciprocal and inverse functions and the relationships between them, including exponential, logarithmic, and trigonometric functions.

Exploring equations and expressions
Students should be:

• making selections and arrangements leading to permutations and combinations a nd the binomial theorem for positive integral index;
• investigating and modelling real and simulated situations using graphical, algebraic, geometrical, and technological techniques, including:
  - the concept of 3-space related to 3 dimensions;
  - considering the consistency and uniqueness of solutions, including 2 x 2 and 3 x 3 simultaneous equations;
  - the use and proof of remainder and factor theorems;
• exploring the extension of the number system to include complex numbers;
• working with complex numbers in the form a + ib and in polar form to solve quadratic and other equations, including those in the form zⁿ = a (for n a positive integer) and z= r cisų, and using De Moivre's Theorem;
• exploring relationships between trigonometric expressions and situations that give rise to them.

Sample assessment activities

These assessment activities are examples of the kinds of tasks which teachers could devise for their own assessment programme.

Using a calculator (or a computer program), students investigate the convergence of the sequences such as f(n) = and g(n) = (1 + )ⁿ and estimate values of limits where they exist. Other suitable sequences could be:

Using this example, teachers could assess students' ability to:

• investigate and interpret convergence of sequences and series (A8).

Students calculate possible permutations and combinations in context. For example, typical car registration plate numbers are A 657 HDC (Britain), MPQ 049 (Victoria), UN5367 (New Zealand). How many unique licence plate numbers are possible in the various states?

Using this example, teachers could assess students' ability to:

• investigate and find numbers of arrangements and selections from a number of objects (A8).

Students solve simultaneous equations in 3 variables by elimination. For example, they:

• solve by elimination

• solve exactly by elimination
  x + 0.5y + 0.33z = 1
  0.5x + 0.33y + 0.25z = 0.75
  0.33x + 0.25y + 0.2z = 0.55
• explain the differences in the solutions to the two sets of equations.

Using this example, teachers could assess students' ability to:

• use simultaneous equations to model real and simulated situations, and interpret their solutions in a given context (A8);
• interpret information and results in context (MP8);
• make conjectures in a mathematical context (MP8);
• use their own language, and mathematical language and diagrams, to explain mathematical ideas (MP8).

Students write a report that shows the need for numerical techniques (both bi-section and Newton-Raphson methods) for solving equations, including a visual description of the methods, and the effect of different starting values. Students solve a given equation by numerical methods and explore a function of their own choosing.

Using this example, teachers could assess students' ability to:

• use appropriate numerical methods and technology to solve non-linear equations (A8);
• use their own language, and mathematical language and diagrams, to explain mathematical ideas (MP8);
• report the results of mathematical investigations concisely and coherently (MP8).

Students use a calculator or graphics package and the remainder theorem to identify factors or to locate intervals containing roots to an equation.

Using this example, teachers could assess students' ability to:

• use and prove the factor and remainder theorems (A8);
• use equipment appropiately when exploring mathematical ideas (MP8).

Students find the real and complex solutions of equations of the form(x - 1)(x² - 2x + 4) = 0.

Using this example, teachers could assess students' ability to:

• solve any quadratic equation and equations of the form zⁿ= a (for n a positive integer) and z = r cisų(A8).

Students sketch and interpret the graph of, for example, y = tan-1x.

Using this example, teachers could assess students' ability to:

• sketch the graphs of inverse and/or reciprocal functions and explain relationships between them and the original functions (A8).

Students make conjectures and prove or refute generalisations. For example:

• They investigate and make conjectures about numbers which are the sum of 3 consecutive whole numbers, 4 consecutive whole numbers, 5 consecutive whole numbers, and so on.
• They show that for all natural numbers n, 3 is a factor of n³+2n.
• They prove previously unseen results and some standard results (Remainder Theorem, De Moivre's Theorem, Binomial Theorem, alternative formulae for the standard deviation, compound angle formulae, the rule for differentiating composite functions, nCr = nCn-r and so on).

Using this example, teachers could assess students' ability to

• investigate and find numbers of arrangements and selections from a number of objects (A8);
• generalise mathematical ideas and conjectures (MP8).

Sample development band activities

• Students investigate the algebra involved in special relativity, including limits as vc, for example, in the Lorentz-Fitzgerald contraction.
• Students investigate systems of formal logic &; Aristotelian, Boolean, modern.