Research-based guidance and classroom activities for teachers of mathematics

Structures and relations

Structures and relations is a theme within quantities and algebraic expressions. Links to relevant activities and resources are on the right hand side of this page. 

Students know many numerical relations from their early work with arithmetic. School algebra is meaningful when it is interpreted to show the relations between numbers that students already know.

  • The additive relation, e.g. 2 + 3 = 5, is expressed as a + b = c, so we can find x if 2 + x = 5 and also graph the relation between x and y if
    x + y = 5
  • The multiplicative relation, e.g. 2 x 3 = 6, is expressed as c=ba, c/b = a and c/a = b
  • Addition and subtraction have an inverse relationship, one ‘undoes’ the other.
  • Addition and multiplication are both commutative
  • In 37 + 49 – 37 students can spot that they do not have to do calculations. a + b – a = b expresses the same relation.
  • This equation: 3(p+q) = 3p + 3q, expresses a fact that students often use in mental arithmetic

Students need to meet and understand a range of mathematical statements: formulae, equations, identities, properties, functions, and know what they mean and how to use them.

Students need to know how to ‘read’ algebra with meaning.

Students have to be fluent in transforming expressions into simpler or more usable forms, and how to use substitution of one expression for another to reduce complexity.

Research shows that holistic ways of relating algebra to situations are successful in helping students to learn the procedures, meanings and uses of algebra. Holistic teaching combines arithmetic, algebra, data, graphs and functions side by side.

Activities and resources