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
Select at least three students who have used different methods, and invite them to draw the image on the board (perhaps using colours to emphasise the order in which it was drawn).
"Without counting individual matches can you say how many matchsticks there are in the drawing?"
Next, hand out this worksheet. There are six different patterns with the simpler ones at the start. Invite students to work in pairs:
"With your partner, choose two or three of the six patterns and have a go at the questions. Make sure you can explain clearly how you worked out your answers, focusing on the order in which you would draw the diagram, like we did for the Seven Squares problem."
While students are working, circulate and listen to the conversations, identifying students who have really elegant ways of seeing the general case in the initial picture.
"I'm going to give you ten minutes to produce an A3 display showing one of the problems you worked on and explaining how you arrived at your solution."
Students could choose which problem to work on, and you could guide particular students towards problems where you have noticed them reasoning clearly.
Once they have produced their sheets, there are a number of different ways that sharing and feedback could be organised:
Students could spend time exploring the first three patterns before moving on to the harder cases.
Encourage students to draw a few examples of each pattern and notice how their drawings develop.