Algebraic patterns and expressions

The patterns and expressions curriculum unit supports pupils to understand and express algebraic generalisations arising from how they 'see' the structure of a figural pattern. Pupils work in a dynamic environment to visualise and generate figural patterns and write algebraic expressions for those patterns. 

The activities encourage pupils to identify structures in the patterns to make sense of the letters and numbers in an algebraic expression.

One of the main goals of the unit is to help pupils develop a solid understanding of variables as representing quantities that vary instead of simply unknowns to be discovered.

It introduces the following 'hard to teach' mathematical ideas:

• the geometric structure of the pattern - seeing the general in the particular
• that 'variables' can be named and linked
• algebraic equivalence modelled through the different ways of seeing a pattern

The software has been designed to offer:

• dynamic graphical representation of figural patterns in  'Pattern player' and 'Pattern designer' windows
• dynamic linking between representations, particularly graphical and algebraic
• variables conceived as 'unlocked numbers' that can vary dynamically

Downloads

An overview of the 5 Investigations in the Patterns and expressions unit (PDF).

The pupil workbook and the teacher guide.

Resources for school based PD

The following resources are provided to support schools to involve more teachers in the department to teach the Cornerstone Maths algebraic patterns and expressions unit with confidence.

• A 3 minute introductory video about the algebraic patterns and expressions unit.
• A presentation to support a PD session to introduce teachers to the key mathematical ideas addressed by the unit and to gain hands-on experience with the software (PowerPoint and pdf document).
• PD tasks (for printing) - needed for the PD session

• A presentation to support a planning task to teach the landmark activity from the unit 'Investigation 2, Some lights are always on'

• The lesson planning proforma

• Examples of pupils' work

• and

Landmark activity – Investigation 2 Some lights are always on

Landmark activities are those in which the use of the technology prompts pupils (and teachers) to have an 'aha' moment about the mathematics.
In the patterns and expressions unit, the landmark activity relates to important considerations when defining algebraic variables. However, pupils will often need some careful support to use the software productively - and have a personal 'aha' moment.

Watch the following video clips to see how different teachers have used the software to support this dialogue:

Example 1 – A teacher who has stopped his year 9 class to give a whole-class input. The teacher has selected and displayed an appropriate example of one of the pupil's screens, which he then uses to prompt the pupils to think about how to make their two variables 'move together'.

Example 2 – A clip of a teacher having a discussion with a pair of boys about their work. The teacher prompts the boys to explore how to make their two variables 'move together'.

Some examples of pupils' written work:

and

Designing school-based assessments

The nature of the Cornerstone Maths activities results in some rich opportunities for accurate assessment of pupils' mathematical understanding.

The following Investigations (and particular questions) from the Algebraic patterns and expressions unit are especially effective:

• Investigation 1, Question 1a "Sketch any of the three figures in the Pattern Player"
• Investigation 1, Question 1c "Describe the pattern..."
• Investigation 1, Question 1 f-g-h "Copy your expression for total..." 
• Investigation 2, Question 1a-b "Describe the pattern's structure and predict the expression for the total number of lights."
• Investigation 2, Question 1 d-e "Explain how the numbers and variables in your pattern are related to the lights in the pattern" 

Some students' responses to these questions for a departmental discussion about assessment:

and