Algebra - Level 6

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:

• generating, in practical contexts, linear, quadratic, and other patterns, and finding and justifying the graphs and rules which describe them;
• using rules expressed in words or symbols to generate sequences;
• using geometrical patterns to investigate infinite numerical patterns, for example,

• using and interpreting formulae arising from practical situations, and sketching and interpreting graphs which illustrate everyday situations;
• investigating practical situations that are approximated by linear, quadratic, exponential, and trigonometric functions (such as the height above the ground of a person riding on a ferris wheel as a function of time);
• investigating families of functions (using a graphics calculator or computer) such as exponential functions and rectangular hyperbolae (including the nature of the inverse relationship);
• describing and interpreting significant features of graphs, for example, maxima, minima, rises, falls, plateaux, periodicity, symmetry, discontinuities, comparison of 2gradients.

Exploring equations and expressions
Students should be:

• using algebraic expressions to generalise from numerical instances in practical contexts;
• devising and using strategies for finding rules to represent practical situations, including finding rules for linear and quadratic number patterns by using first and second differences;
• interpreting inequalities and equations arising from practical contexts;
• forming equations (including linear, simultaneous, and quadratic) in practical contexts, and using a range of strategies to solve them, for example, numerical calculator, graphics calculator, computer software, trial and improvement, factorising;
• developing confidence in simplifying and rearranging algebraic expressions by:
  - substituting a range of numerical values into two forms of the same expression to confirm the equivalence;
  - simplifying expressions such as

Sample assessment activities

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

• Students find at least two rules for the number sequence, and show by re-arranging that the rules are equivalent.

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

• generate linear and quadratic patterns and find and justify the rule (A6);
• substitute values into formulae (A6);
• combine like terms in algebraic expressions (A5);
• factorise and expand algebraic expressions (A5);
• prove or refute mathematical conjectures (MP6);
• record in ways that are helpful for drawing conclusions and making generalisations (MP6).

Students substitute in formulae, rearranging if necessary, to find an unknown value. For example:

- In order to calculate the proper dosage of a drug, the following formula is used to calculate a person's body surface area, S m2, given their mass, W kg, and height, H cm.
S = 0.007184 x W^0.425 x H^0.725
Sally is 55 kg and is 1.75 m tall. Use the formula to calculate the surface area of her body.
- A school group rents a cabin for a cost, $C, given by C=3n+15 where n is the number of people in the group. If the cost was $45 how many people stayed in the cabin?

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

• form and solve linear equations, simultaneous equations, and simple quadratic equations (A6);
• substitute values into formulae (A6).

Students form simultaneous equations and use both (a) trial and improvement, or number patterns, and (b) elimination or substitution, to solve problems derived from practical contexts. For example: There are 11 competitors in a trike-and-bike race. Altogether there are 29 wheels. How many competitors are on trikes and how many on bikes?

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

• form and solve linear equations, simultaneous equations ... (A6);
• devise and use problem-solving strategies to explore situations mathematically (MP6).

Students sketch graphs of given functions or relations. For example:
y = (2x-3)(x+1)
y = 3x²

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

• graph linear, quadratic, and exponential functions, and relations of the form x²+y²=r² and xy = c (A6).

Students make conjectures about relationships. For example, in geometry, they explore the relationship between angles subtended by arcs at the centre and at the circumference of a circle, and propose a general rule.

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

• make conjectures in a mathematical context (MP6);
• prove or refute mathematical conjectures (MP6);
• generate patterns from a structured situation, find a rule for the general term, and express it in words and symbols (A5).

Working in groups, students devise systematic and efficient strategies to help to deduce results from unfamiliar problems. For example, if a bee in a honeycomb can travel from cell to cell only in the directions east or northeast, how many different paths to a particular cell are possible?

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

• devise and use problem-solving strategies to explore situations mathematically (MP6);
• record in ways that are helpful for drawing conclusions and making generalisations (MP6).

Sample development band activities

• Students investigate and report on mathematical structures, beginning with groups of order up to 5. The investigation should include the concept of isomorphism, and could include applications of group theory.