Algebra - Level 7

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:

• investigating geometric and number patterns;
  - using designs from other cultures
  - using numerical approaches
  - modelling
  - using spreadsheets or computer software
• using graphs to illustrate and investigate sequences and series;
• investigating the underlying patterns in everyday situations;
• modelling real and simulated situations;
• exploring the use of algebra to express patterns in sequences and series, including arithmetic and geometric progressions;
• using calculators and computer graphing packages to plot accurately the graph of a general function from a given formula;
• investigating observed or given data to discover relationships;
• exploring the connections (including transformations) between types of graphs and their equations;
  - linear, quadratic, cubic polynomials
  - circles, rectangular hyperbolae
  - exponential functions of the form y = a^x - logarithmic functions of the form y =log a^x
  - trigonometric functions of the form y = a sin(bx+c), y = tan x
• investigating families of curves by considering such things as symmetry, periodic behaviour, maxima, minima, behaviour for large values of x and y, discontinuities, and asymptotes.

Exploring equations and expressions
Students should be:

investigating real and simulated situations using graphical, algebraic, geometric, and technological techniques, including:

• the manipulation and simplification of algebraic expressions, including rational expressions;
• the solution of linear and quadratic equations, and pairs of simultaneous equations, one of which may be non-linear;
• the solution of polynomial equations and the nature of their roots;
• the solution of trigonometric equations such as sin(x+a) = b, cos(ax) = b;
• the manipulation of exponents, including fractional and negative exponents;
• the concept, properties, and manipulation of logarithms.

Sample assessment activities

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

Students find unknown terms and sums in sequences and series derived from practical contexts. For example:

- The half life of caffeine in the body is three hours. Students model the amount of caffeine in a person's body with a sequence and explore the effects of having successive cups of coffee throughout the day.

- Students use a strip of paper to investigate the relationship between the number of times the strip is folded in half and the number of creases formed. They find at least two ways of describing the resulting sequence and orally present their investigation. (Does the direction of successive folds make a difference?)

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

• use sequences and series to model real problems and interpret their solutions (A7);
• effectively plan mathematical exploration (MP7);
• devise and use problem-solving strategies to explore situations mathematically (MP7);
• make conjectures in a mathematical context (MP7);
• use words and symbols to describe and generalise patterns (MP7).

Students form and apply an algebraic relation for data arising from an experiment. For example, given a table showing distance and time for a ball rolling down a smooth ramp, students write an equation for the distance-time relation and use it to calculate further distances or times.

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

• sketch graphs and investigate the graph of a function, using a calculator and plotting points if necessary (A7);
• use graphical methods to investigate a pattern in data and, where appropriate, identify its algebraic form (A7).

Students write and solve simultaneous equations involving at least one non-linear equation, arising from practical situations. For example, write a pair of equations for the amount each employee receives, and find the number of workers employed and the amount each gets, from a company which budgets to pay all employees equally from a total of $10 000. Each employee is paid a retainer of $300 plus an amount equal to four times the number of employees.

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

• write appropriate equation(s) or inequation(s) to describe a practical situation (A7);
• choose suitable strategies (graphic, numeric, algebraic, and trigonometric) for finding solutions to equations or inequations, and interpret the results (A7).

Students manipulate and simplify expressions such as:

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

• carry out appropriate manipulation and simplification of algebraic expressions (A7).

Students use their own methods to find the zeros of equations such as y=5x³ - 12x² - 16x + 8 and discuss the nature of the roots. (Satisfactory methods could include a table-building program or graphing software to isolate the roots between pairs of integers, or a successive approximation method, or a "zoom in" function on a calculator or computer.)

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

• choose suitable strategies (graphic, numeric, algebraic, and trigonometric) for finding solutions to equation(s) or inequation(s), and interpret the results (A7);
• effectively plan mathematical exploration (MP7);
• devise and use problem-solving strategies to explore situations mathematically (MP7).

Students develop mathematical models to describe familiar practical situations. For example, a ferris wheel has a radius of 8 m and makes a complete revolution in 12 seconds. Students develop a mathematical model that describes the relationship between the height, h, of a rider above the bottom of the ferris wheel (1 m above the ground) and time, t, seconds.

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

• model a variety of situations, using graphs (A7);
• write appropriate equation(s) or inequation(s) to describe a practical situation (A7);
• devise and use problem-solving strategies to explore situations mathematically (MP7);
• record in ways that are helpful for drawing conclusions and making generalisations (MP7).

Using a graphing package, or otherwise, students identify significant features of a graph and explain them.

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

• find by inspection, and interpret, maxima, minima, points of inflection, asymptotes, and discontinuities for given graphs (A7);
• sketch graphs and investigate the graph of a function ... (A7);
• use equipment appropriately when exploring mathematical ideas (MP7).

Students present proofs and/or analyse other students' early attempts at proving a proposition, and identify weaknesses in the arguments. For example:

• They prove that for any number x, 6x-10-x² is always negative.
• They prove or refute the statement, "There is only one value of n for which the positive integers n, n + 2, and n + 4 are all prime."
• They prove or refute that n 2 - n + 41 is prime for all n W.

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

• substitute values into formulae (A6);
• generalise mathematical ideas and conjectures (MP7);
• prove or refute mathematical conjectures (MP7).

Sample development band activities

• Students research and report on paradoxes in algebra, number, statistics, geometry, and logic.
  
• Students investigate practical situations where mathematical modelling is used and report on the model used, its usefulness, and limitations. The investigation could proceed from an examination of a simple example, such as the proposed use of a Fibonacci sequence to model the growth of a rabbit population.