Lesson 2 Text ‘n’ talk

Episode 1

Reasoning Resources
Whole class preparation
Engagement with aspects of real life.

Recognising range of outcomes.
Engage the class in the context of buying a new mobile phone. Discuss the main features of mobile phones, perhaps listing pupils’ ideas on the board. Then focus on text messages and phone calls. If three people text each other how many text messages are sent altogether?Talk to your partner and think of a way to record this on paper. Remember that each person must text each of the others. Encourage pupils to show their simple result in more than one way.

Share the variety of methods on the board/OHP. Typical outcomes may be described in terms of full lists, abbreviations, half list/half diagram, full diagrams.

All wrong intuitive answers are due to inattention or logical errors, e.g. one more person adds one more message, or two more messages. Have a brief discussion of the advantages and disadvantages of each method, such as speed, clarity, efficiency. Then introduce the challenge. Ask for their intuitive prediction for the number of text messages if there are four people texting each other. There may be a variety from 7 to 24, which can quickly be listed and voted upon without discussion.
Small group work
There is also double counting for sending and receiving, which may be useful to discuss with lower attaining classes. Working in threes/fours the pupils should work out and show the messages on large sheets of paper with marker pen, or on OHT. Be careful that you have not missed out any of the text messages or counted any twice. Pupils who finish quickly should try for five people, first as a prediction then as showing all or as a calculation. Check using your systematic method. Write down any patterns that you notice and use this information to create a rule.
Whole class sharing and discussion
The structure of two variables is visible in all the recordings.

Mixture of listings and diagrams accepts variety in pupils’ preferences.

Easy large numbers in examples leads to general number and to symbolisation.

Number of people times number of people minus 1. (You don’t text yourself.)

n x (n-1) n(n-1)

n x n - n n^2-1
Collect and show different methods of recording. Sheets can be displayed on the wall. Typical outcomes will be variations on what was shown earlier, with some pupils switching to speedier methods and to calculations. Arrange a selection with an eye to showing the structure of the four people and the three messages they each send, eg:

Highlight the ‘order! and the ‘organised’ aspect to ensure nothing is missing or repeated, and the ‘grouping’ idea, to ease counting in ‘lots’ rather than one at a time. For the ’4 times 3’ idea circle the four and highlight the three for each of the ways of recording. Pupils should notice and explain in their own words the fact that the better systematic recordings show the structure of two variables: number of people, and the number of messages each sends (or receives) which is ‘1 less: The table is not easy, but the stars in each row can still be ‘lassoed’ to show that each person sends three messages. What happens with 5 people? With 10 people? What if all of the pupils in the class text each other? With 100 people? Can you find an expression to link the number of people with the total number of texts?

What if there were m people? Clarify how we use ‘n‘ The ‘n — 1’ phrase is a variable dependent on the value of n. The table allows the insight that the total number can also be found by squaring n then taking away 1. How that gives the same product as n x (n - 1) is a good challenge for some to tackle or just to keep in mind.


Thinking Mathematics Lessons Copyright © by Michael Shayer and Mundher Adhami. All Rights Reserved.

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