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 51 - Trigonometric Graph

Written by

Susan Wall

Organisation

NCETM

Age/Ability Range

16/17 year old AS students (mostly with grade C in GCSE – 2 hour lesson)

How was the session/task introduced? Transformation of trigonometrical graphs.

Lots of cards given to each pair of learners such as y = cosx; y = cos2x; y = 3cosx; y = -cosx; y = cos(x+1) etc plus similar sine ones.

They had no previous knowledge of sine and cosine graphs other than knowing y = cos x and y = sin x from previous work on solving trigonometrical equations. They had not met any of the others before.

They had to sort them into 2 groups using a criteria of their choice. Then their criteria were recorded on the board (initially, as expected, most separated the cards into sine and cosine sets). Then they put them back together again and sorted again into 2 groups but not using any of the criteria on the board. Then the criteria were written on board again. This was repeated once more. We got things like “all the ones with a number in front”, “all the ones with a number in front of the x”. No knowledge about transformations was expected. It was just to get them looking carefully at the structure of the functions e.g. sometimes the number is added on inside the bracket, sometimes outside…

How was the session/task sustained? Printed graphs of the functions were given out and each pair were given a criteria from the board to investigate. E.g. what difference does it make when there is a number in front of x? Then they had to explain their findings to the rest of the class. The pair who got the “what difference does it make if it is sine or cosine?” suddenly shouted out “they are the same – it makes no difference” during the investigation – they really had expected them not to be the same! Some rough notes were jotted down on the board by me as the findings were presented.

Using mini-whiteboards to check learning and understanding:

Started with some closed questions so that they could help each other, e.g. sketch y = 2sin x

Then giving a sample graph

on board and asking them to give a possible equation for it (but not the same as anyone also on their table).

Lots of discussion about possibilities.

Similarly for

Each pair was given some graphs such as

(given several versions of the same graphs) and asked to stick one graph on a large sheet of paper and find a card (from original ones) that was a possible equation. Mark on the graph the intercepts and describe the transformation from y = sin x or y = cos x. Then they either took another card that was possible or took a different graph. NB the printed sheets had been collected in by then!

Then if they wanted they could make up their own trigonometrical functions and not be constrained by the ones on the cards.

How was the session/task concluded? Final part of the lesson was the memory or pairs game played as whole class on the interactive whiteboard.

Underneath the numbers were all the transformations involving 4s, e.g. y = sin 4x under one, and one way stretch in the x direction scale factor ¼ under another. The class split into 4 – 5 teams and took it in turns to choose 2 numbers. When uncovered they could accept or reject. If they accepted the pair they could be challenged by others etc.

Using this approach the students worked each one out several times and had plenty of practice in identifying transformations and points mattered so there was an incentive to think it out correctly.

Finally when every number was uncovered they had to write how the functions paired up with the transformation involved in their notebooks.

What were the critical moments? Open questions on whiteboards and posters- really having to think about it.

What mathematics was learnt? (on plan and off plan) and what is the evidence of learning? How transformations are related to equations. They all could pair them up at the end and describe the transformation and evidence on open questions and associated discussions.

How was that mathematics learnt? Students discussing and thinking and explaining to each other.

Other memorable outcomes One groups of lads saying it was the best lesson they had ever had.

Resources Created by me.

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

Exposes and discusses common misconceptions and other surprising phenomena

Uses higher-order questions

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

very clear and inspiring account. I am new to teaching and looked at this as a follow up from my self evaluation. Thanks a lot.