Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
Take any whole number between 1 and 999, add the squares of the
digits to get a new number. Make some conjectures about what
happens in general.
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
This problem encourages students to think about the properties of numbers. It could be used as an introduction to work on linear sequences and straight line graphs.
Now show the interactivity from the problem and alert the students that it does something slightly different (but don't tell them what!). Generate a set of numbers using Level 1 or 2, and give the class a short time to discuss with their partner what they think the computer has done. Do the same a couple more times, without any whole-class sharing, but giving pairs a little time to refine
their ideas. Then bring the class together and discuss what they think is going on. Link what they say to the terminology of "Table" and "Shift" that the computer uses.
Emphasise that the table should always be the largest possible, and the shift should always be less than the table. This example could be used to bring these ideas out:
Possible suggestions that might emerge:
But we are interested in
Ideally, each pair would now work at a computer to develop a method of finding the table and shift with ease. If that isn't possible, generate a dozen or so examples at appropriate levels, and write them on the board for the class to work on. Students could also work in pairs and create examples for their partners to work out.
Once students are finding the table and shift easily, bring the class together. Generate a new example and ask a pair to talk through their thinking as they work towards the solution, but ask them to stop short of actually giving the answer. The rest of the class could write the answer on mini whiteboards once they've heard enough to work it out. Repeat, giving other pairs the opportunity to
share their thinking.
Finally, allow the class some time to work in pairs on the questions at the bottom of the problem, and then discuss their ideas, emphasising the need to justify any conclusions they reach.
Here is an account of one teacher's approach to using this problem.