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Why do this problem?
Many standard questions give exactly the information required to solve them. In this problem, students need to go in search of the information and work in a systematic way in order to make sense of the results they gather.
The problem could be used to reinforce work on recording and describing linear sequences.
This task will require students to have access to computers. If this is not possible, Four Coloured Lights provides students the opportunity to make sense of numerical rules without the need for computers.
Shifting Times Tables
is a problem about linear sequences that could be used to prepare students for the thinking required in this problem.
Begin the lesson by dividing the board into two columns, one headed with a tick and the other headed with a cross.
Ask learners to suggest numbers, and write each suggestion in the appropriate column according to a rule of your own choice. Make it clear to the class that the activity is designed to model scientific enquiry, so they can come up with a hypothesis for your rule, but you will not confirm their hypothesis, you will only place numbers in the appropriate column.
Here are some suggestions for rules:
- Odd numbers
- Numbers which are 1 more than multiples of 4
- Numbers which are 2 less than multiples of 5
- Numbers which are 3 more than multiples of 7
Once the class have tried the activity with a couple of rules until all are reasonably convinced their hypothesis holds, move on to the main task.
To introduce the main task, show the interactivity and demonstrate entering a couple of numbers to see what lights up. Make sure learners understand that more than one light can light up at once, and that each light is governed by its own simple rule.
Students could then work in pairs at a computer, trying to light up each of the lights. Challenge them to develop an efficient strategy for working out the rules controlling each light.
While the class is working, note any particularly good ways of recording or working systematically, and highlight them to the rest of the class.
Ask students to think about what is special about a rule when all the 'light on' numbers
- are odd
- are even
- are a mixture of odd and even
Bring the class together to share insights and conclusions before moving on to A Little Light Thinking
Which numbers will you try first?
Which numbers will you try next?
How will you record your findings?
How many lightings are necessary to work out the rule for a light?
A Little Light Thinking invites students to explore turning on multiple lights simultaneously.
Learners could use a 100 square (Word, pdf) to record which lights turn on for each number they try.
In the Hint there is a version of the interactivity with just two lights which students might find more accessible.