Why do this problem?
Many standard questions give exactly the information required to solve them. In this problem, students are encouraged to be curious, go in search of the information they require, 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 and quadratic 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 students understand that more than one light can light up at once, and that each light is governed by its own rule.
Level 1 rules are linear sequences of the form $an+b$, with a and b between 2 and 12.
Level 2 rules are quadratic sequences of the form $an^2+bn+c$ with a=0 or 1
Level 3 rules are quadratic sequences of the form $an^2+bn+c$ with a=0, 0.5, 1, 2 or 3.
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.
Bring the class together to share insights and conclusions before moving on to A Little Light Thinking
, in which students are invited to find sequences that turn several Level 1 lights on simultaneously.
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?
Can you suggest a number bigger than 1000 that you think will turn on the light?
A Little Light Thinking invites students to explore turning on multiple Level 1 lights simultaneously.
Level 3 sequences can be used as a starting point for some detailed exploration into graphical representations of quadratic functions.
Shifting Times Tables offers students a way of thinking about linear sequences and opportunities to explore how they work.
Students could use a 100 square to record which lights turn on for each number they try.