Algebra - Level 4

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:

• finding, and continuing, linear number patterns from practical contexts and finding and justifying the rules which describe them;
• developing an understanding of relations and representing and interpreting them;
• sketching graphs which represent familiar situations;
• interpreting a relationship illustrated by points on a graph, for example,

• representing a relationship by a point on a graph;
• writing stories and talking about graphs representing familiar situations;
• generating and graphing sequences from rules expressed in a variety of ways;
• developing strategies for finding rules for linear patterns arising in practical contexts, and using symbols to express these rules.

Exploring equations and expressions
Students should be:

• using, creating, and describing formulae derived from practical contexts, using words and symbols;
• solving number puzzles with whole number solutions which can be represented by simple linear equations such as 2 + 4 = 16.

Sample assessment activities

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

Students describe in words suitable rules to represent sequential patterns. For example, suitable descriptions for the following pattern would be "To get the number of lines in any drawing, multiply the number of squares by 3 and add 1." "To get from one value to the next just add 3."

Number of squares 1 2 3 4
Number of lines 4 7 10 13 ...
Predict the number of lines needed for 20 squares, and then graph the sequence.

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

• find a rule to describe any member of a number sequence and express it in words (A4);
• use a rule to make predictions (A4);
• devise and use problem-solving strategies to explore situations mathematically (MP4);
• use words and symbols to describe and generalise patterns (MP4).

Students

- solve informally problems based on relations and explain their reasoning. For example, one band for the school dance charges $150 plus $90 per hour and another charges $60 plus $105 per hour. If $375 has been budgeted for the band, how long would each band play for and which will play longer?
- find a word formula for problems such as the following: The class is going to the pictures in town. It costs $8.50 per person and $50 for the bus. What is the total cost for the class? What if 5 people are absent?

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

• find and justify a word formula which represents a given practical situation (A4);
• devise and use problem-solving strategies to explore situations mathematically (MP4);
• report the results of mathematical explorations concisely and coherently (MP4).

Students interpret graphical information derived from familiar contexts. For example, the following graph shows the depth of water in a bath. Write a story about the water level as a person takes a bath.

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

• sketch and interpret graphs on whole number grids which represent simple everyday situations (A4);
• interpret information and results in context (MP4);
• record information in ways that are helpful for drawing conclusions and making generalisations (MP4).

Students:

• explore the number patterns found in an array of ordered numbers, such as in a calendar, and investigate the effects of changing the dimensions of the array, for example, a calendar with only 6 days a week;
• investigate the sum of numbers in opposite corners of any rectangular array of numbers within a calendar.

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

• find a rule to describe any member of a number sequence and express it in words (A4);
• use a rule to make predictions (A4);
• devise and use problem-solving strategies to explore situations mathematically (MP4);
• report the results of mathematical explorations concisely and coherently (MP4).

Students investigate the conjecture that "The sum of 2 triangle numbers is always a square number". Students might test the conjecture by trying a range of values, and attempt to prove the conjecture by making use of a geometrical model such as:

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

• find and justify a word formula which represents a given practical situation (A4);
• interpret information and results in context (MP4);
• report the results of mathematical explorations concisely and coherently (MP4).

Students investigate simple strategy games. For example, they find and explain a winning strategy in the following game. Given 2 piles of counters, players take turns to remove at least one counter from any one of the piles. The winner is the person who removes the last counter or group of counters.

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

• use words and symbols to describe and generalise patterns (MP4);
• make conjectures in a mathematical context (MP4);
• record information in ways that are helpful for drawing conclusions and making generalisations (MP4).

Sample development band activities

Students investigate and report on a number of different games of strategy. For example:

• How many different games of noughts and crosses are possible? They could discuss the concept of isomorphism in this context, and perhaps extend the idea to other contexts.
  
• They find a winning strategy for "Nim" or "21 or Bust" on a calculator.