Levels 1 to 5
Algebra is essentially a generalisation of number. Many of the laws that govern operating with number also apply to algebra. In the early levels, algebra is about patterns: making them, finding them, continuing them, describing them, and using them to solve problems. At the higher levels, these patterns are expressed in terms of "unknowns", which enables us to determine more about the patterns than is possible without algebra.
Copy a pattern and create the next element
 At this stage in the exploring patterns progression, students are able to copy given elements in a pattern, work out the next element in the pattern, and show it in some way.
Use a systematic approach to continue a pattern and find number values
 At this stage in the exploring patterns progression, students are able to use systematic counting to continue a pattern. This allows them to work out the next element in the pattern more efficiently and accurately.
Predict values using relationships between successive elements
 At this stage in the exploring patterns progression, students are able to recognise relationships between successive elements in a pattern and may be able to use a table to list values.
Predict values using rules
 At this stage in the exploring patterns progression, students are able to explain a rule to predict the value of any given element in a pattern. They no longer need to rely on knowing the previous element to work out any given element.
Find an algebraic expression for a relationship
 At this stage in the exploring patterns progression, students are able to state and use an algebraic expression for a relationship. They are able to use symbols and the variable "n" to express their rule.
Solve linear equations related to patterns
 Students are now able to use equations for a pattern to solve a range of problems related to that pattern. For example, they might use equations to solve the problem: "If 53 red tiles are used, how many blue tiles are used?"
Background to the task
The task chosen as a context for this set of exemplars allows enough scope to observe the development of an ability to understand and manipulate patterns through levels 1 to 5. Students are asked to copy and continue a pattern of coloured counters. If they are able to do this, they are asked to describe and use rules to predict subsequent elements of the pattern. The level of sophistication of the rules students are able to provide gives a good indication of the stage of the exploring patterns progression they have reached.
The task
The teachers asked the students whose work is shown in these exemplars to complete one or more of the following versions of the task, according to their level of development.
- Levels 1 to 2: The teachers asked the students to copy the pattern shown and display or draw the next element.
- Levels 3 to 5: The teachers asked the students to describe and use rules to predict subsequent elements of the pattern.
Note: Some students saw the pattern as a repeating one and repeated the first element as the third. Although this was not the intended pattern, it was a valid solution to the question. Their teachers told them that they had found one answer and asked them to try to find another. Together, they discussed what had changed between the first and second elements, looking firstly at the red counters and then at the blue ones.
Parallel tasks
The same progression could be explored in the context of any type of pattern.
References
Department of Education (1985–1989). Beginning School Mathematics: Cycles 1–8. Wellington: School Publications.
Ministry of Education (1992). Beginning School Mathematics: Cycles 9–11. Wellington: Learning Media, Ministry of Education.
Ministry of Education (1992). Beginning School Mathematics: Cycle 12. Wellington: Learning Media, Ministry of Education.
Ministry of Education (1999). Figure It Out, Levels 2–3. Wellington: Learning Media.
Ministry of Education (2000). Figure It Out, Level 3. Wellington: Learning Media.
Ministry of Education (2001). Figure It Out, Levels 3–4. Wellington: Learning Media.
Ministry of Education (1992). Mathematics in the New Zealand Curriculum. Wellington: Learning Media.
Ministry of Education (1996). Te Whāriki: He Whāriki Mātauranga mō ngā Mokopuna o Aotearoa/Early Childhood Curriculum. Wellington: Learning Media.
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