Introduction

Algebra is an essential tool in mathematics and physics. Students use algebraic skills to make generalisations and explore patterns in the world around them. Algebraic thinking begins at an early age, when young children represent mathematical operations with objects, their bodies (e.g. finger counting), images, drawings, and sounds. As children get older, they start to make deeper connections between algebra and formal arithmetic operations. Algebraic thinking involves noticing patterns and forming generalisations and is essential to support learners to make connections to real life. It is suggested that algebraic thinking involves three main strands (Blanton 2008):

– The study of structure in the number system

– The study of patterns relationships, and functions

– The process of mathematical modelling.

Algebra is often referred to as generalised arithmetic (van de Walle). To support students in generalising operations or patterns, they need opportunities to explore multiple examples and time to discuss what they observe, with particular emphasis on analysing the structure of what they are seeing. In this task, students are presented with a challenge: they must determine the missing number in the patterns presented. Depending on the learning context, additional structures and supports can be provided through teacher questioning.

Task Description

This rich mathematical task offers students the opportunity to reason about patterns and can serve as either an introduction to algebra through patterns or as a means to explore systems of equations. The task is designed to be accessible for all learners aged 10–16 and is applicable across various stages of mathematics learning journey. The task is presented as an open task, though more structure can be introduced through questioning. It can be used as a way for younger learners to explore patterns and for older, more experienced learners to reason about solving simultaneous equations. Students work collaboratively to explore visual and numerical patterns. They engage in problem-solving activities that develop their ability to generalise patterns and connect them to algebraic expressions.