Algebra - Level 5

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:

• developing an understanding of relations, and representing and interpreting them;
• interpreting relationships illustrated by points on a graph and representing such relationships in other ways;
• sketching and interpreting graphs which represent everyday situations;
• using rules given in words, symbols, flow charts, or graphs, for example, calculating the dose of medication needed for children of varying ages;
• generating, in practical contexts, linear, quadratic, and other patterns, and finding and justifying the graphs and rules which describe them;
• generating sequences from rules expressed in words and algebraically;
• investigating practical situations that are approximated by linear functions, and investigating the interpretation of the slope and intercept of lines drawn from practical contexts.

Exploring equations and expressions
Students should be:

• talking about the different ways rules can be expressed, for example, 6n + 24 = (n + 4) x 6 = 6 x (4 + n).
• using algebraic expressions to generalise from numerical instances arising in a practical context, for example, generalising the fact that the diagonal corners of any rectangular array on a calendar have the same sum, and expressing it in algebraic terms;
• developing an ability to solve equations in a problem context by:
  - exploring a variety of strategies for solving equations, for example, trial and improvement, forming patterns, balancing, and reversing flow charts;
  - exploring equations which have either no solution or multiple solutions;
  - representing problem constraints as algebraic inequalities;
• developing confidence in re-arranging and simplifying algebraic expressions by:
  - simplifying expressions by making tables, and by using the "rules of algebra", leading to simplifications such as x³ x x⁵ = x³⁺⁵ = x ⁸ and a¹¹ ÷ a⁴ = a^11-4 = a⁷ Examples might be: .
  
  - re-arranging linear equations and changing the subject of a formula in practical contexts;
  - using number experience, geometric models, and so on, to develop skills in factorising, and expanding them.

Sample assessment activities

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

• Students describe relations arising from practical contexts. For example:
  - A square garden is surrounded by square paving stones. What is the relationship between the length of a side of the garden and the number of paving stones needed?

Length of side of garden 1 2 3 4 ...
Number of paving stones 8 12 16 20 ...
- Students use patiki patterns from raranga (weaving) and
(a) determine the number of squares in each diamond of each colour;
(b) relate the number of squares to the length of the side;
(c) investigate and find the rule;
(d) explain and report the findings.

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

• generate patterns from a structured situation, find a rule for the general term, and express it in words and symbols (A5);
• use words and symbols to describe and generalise patterns (MP5);
• record information in ways that are helpful for drawing conclusions and making generalisations (MP5).

Students simplify expressions such as and

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

• simplify algebraic fractions (A5).

Students expand and factorise expressions such as 4y(2y + 5) and 3a + a².

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

• factorise and expand algebraic expressions (A5).

Students draw and interpret graphs of practical situations in context. For example:

- The graph illustrates a flag being raised on a flag pole. Write a description of the way the flag moves.

- Draw the height vs time graph for a sky diver from the time she leaves the ground in a plane until the time she reaches the ground again with a parachute.
- Students interpret a graph of New Zealand's sheep-meat export earnings over time. Do higher earnings necessarily mean higher prices are being paid?

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

• sketch and interpret graphs which represent everyday situations (A5);
• use their own language, and mathematical language and diagrams, to explain mathematical ideas (MP5).

Students write equations for practical situations and then solve them. For example, the band for the school dance charges $100 plus $80 per hour. $350 has been allocated to pay the band. Students find an equation and solve it to find how long the band would play for.

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

• solve linear equations (A5);
• use equations to represent practical situations (A5);
• devise and use problem-solving strategies to explore situations mathematically (MP5).

Students graph equations and interpret the gradient and "intercept" in relation to the equation and/or the practical situation represented. For example, the "Quick Fix" clothing company calculates the cost ($C) of repairs, using the formula C=10+15n where n is the number of hours needed for a job. Find the cost of a repair job taking (a) 3 hours (b) hours. Graph data for the "Quick Fix" clothing company and interpret the constants 10 and 15 in context. Explain the effect of changing the values of the constants.

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

• graph linear rules and interpret the slope and intercepts on an integer co-ordinate system (A5);
• interpret information and results in context (MP5);
• make conjectures in a mathematical context (MP5).

Sample development band activities

• Students investigate and report on the algebra of 2 x 2 matrices. They could extend the investigation to the applications of 2 x 2 matrices to transformations of plane figures in 2-dimensional co-ordinate geometry, or to the solution of pairs of simultaneous equations.