A collection of memorable mathematics lessons that conference and colloquia delegates had observed or taught which they felt were successful. Each account refers to one or more of the values and principles in the report.

Lesson Account 19 - Factorising Cubics

Written by

Jane Imrie

Organisation

NCETM

Age/Ability Range

Post 16 – A Level. (some A/A* GCSE and doing Further

Mathematics, some C GCSE) Task was to understand how to factorise cubics and how this related to the graph of the cubic.

How was the session/task introduced? Whiteboards – reviewing all knowledge on quadratics – making clear connections between graph, equation and intercepts. (Open questions like ‘Give me a possible equation for this quadratic’, when a parabola was drawn on the board with only some defining points. Questions that encouraged the recognition of different forms of the quadratic equation)

Introduced a curve with only x intercepts

What can you say about the equation of this curve?

How was the session/task sustained?
All the time building on what learners knew or what teacher found they knew through careful questioning. Range of activities (some whole group work as in the white board session or the sharing of outcomes at the end; some work in pairs, including sharing with other pairs; small group work)

The whiteboard whole class activity was followed by an activity with the whole group with sets of cards containing linear factors. The teacher put an unfactorised cubic on board and asked learners to hold up the cards with possible factors. The work developed into small group activity factorising some cubic equations and relating to graph of the cubic.

How was the session/task concluded?
Activity (with element of choice and level of challenge) that allowed learners to use what they had learned. The teacher had prepared large pieces of paper each with a cubic equation in the form y = ax³ + bx² + cx + d. The challenge was defined by the values of a, b, c and d. Pairs of learners chose which one to tackle (usually choosing a challenging one). The task was to factorise and graph the function.

What were the critical moments?
Moving from quadratic to cubic – recognising that intersection with x axis gave similar info about factors for any polynomial and thus making connections with what they already knew. This was almost seamless, (see below)

What mathematics was learnt? (on plan and off plan) and what is the evidence of learning?

Connections between areas of mathematics

Mathematical communication.

Problem solving strategies (building on and using skills already acquired)

The factor theorem.

How was that mathematics learnt?

Discussing / articulating.

Trying / making mistakes.

Using prior knowledge.

Other memorable outcomes
Discussion with learners after the lesson – they had clearly understood how to factorise and relationship with graphs. They actually had to be prompted to recognise that cubics were completely new in this lesson because they had so clearly made the connections with previous learning of quadratics. I presented them with a quartic and, following discussion amongst themselves, they realised that the same techniques apply and were able to factorise and sketch it.

Resources
All “homemade”, but many similar ideas appear in ‘Improving learning in mathematics’.

Downloadable PDF

Click here to download this lesson account in PDF format.

Values & Principles

Conceptual understanding and interpretations for representations

Strategies for investigation and problem solving

Builds on the knowledge learners already have

Uses higher-order questions

Makes appropriate use of whole class interactive teaching, individual work and cooperative small group work

Encourages reasoning rather than ‘answer getting’

Uses rich, collaborative tasks

Creates connections between topics both within and beyond mathematics and with the real world