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 30 - Consolidating Pi

Written by

Liz Henning

Organisation

NCETM

Age/Ability Range

Year 8 – level 2/3 pupils

How was the session/task introduced?
In the previous lesson the idea of π ≈ 3 was introduced in relation to circumference of circles. The subsequent lesson, described here, dealt with the concept of area of a circle.

Pupils had 2 coloured equal squares and one circle. They folded the square into 4 equal parts and discussed the area of this. The circle had diameter equal to the length of the square and it was established that the circle’s area would be less that the square’s – as it was smaller.

The word radius was refreshed from pupils’ memory and the pupils were asked to use this knowledge to describe the area of each small square. We had done a lot of work on square numbers in various contexts so pupils were comfortable with
2 x 2 = 2²
3 x 3 = 3²
a x a = a²

so producing r x r = r², where r was the radius of the circle, for each square seemed quite a natural process.

the total area of the larger square being 4r².

Pupils worked in pairs and one cut around the outline of the circle on the square template. Using the pieces pupils were asked to compare the approximate area of the circle with the area of the square. Some pupils noticed that the four cut off pieces were approximately equal in area to one of the smaller squares. Hence they decided that the area of the circle might be about 3r². ( 4r² - 1r² = 3r² )

From knowledge of π ≈ 3 one pupil explained that the area of the circle might be πr²
Some may criticise the method as not being proof of A = πr² but I feel these pupils gained a “home grown” understanding of the area of a circle building upon prior knowledge of area of squares and square numbers.

How was the session/task sustained?
Working from revision of previous knowledge through practical work and discussion to tousling with new concepts

How was the session/task concluded?
One pupil said he thought the area of a circle was πr² this provided a good fulcrum to hang the lesson conclusion around. He described his thinking to the class and they agreed. This was a Level 3 pupil who had his special needs helper sitting in the lesson with him, He had tackled a Level 6 task with confidence

What were the critical moments?
Building slowly on the pupils’ knowledge so as to allow the concrete preparation to be built up steadily. The boy announcing he thought the Area was πr²

What mathematics was learnt? (on plan and off plan) and what is the evidence of learning?
Evidence of prior learning was apparent in early stages of the learning where pupils were using terms and concepts met previously.

The concept that the square had a larger area than the circle was discussed and agreed by each of the groups.

The idea of the area being πr² was mooted by the boy and the rest of the class discussed this and decided that he was correct

How was that mathematics learnt?
By discussion and practical work

Other memorable outcomes The look on the special needs assistant’s face when she heard the boy speak with confidence about the mathematical concept and using correct notation such as A= πr²

Resources
CAME lesson outline
2 coloured squares per group
1 coloured circle, scissors, pencil

Downloadable PDF

Click here to download this lesson account in PDF format.

Values & Principles

Fluency in recalling facts and performing skills

Conceptual understanding and interpretations for representations

Strategies for investigation and problem solving

Awareness of the nature and values of educational system

Builds on the knowledge learners already have

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