### Why do this problem?

This problem is a good way to engage students with many ideas in statistics. It can be accessed at a variety of levels.

### Possible approach

The activity works well as a group discussion. The most important features are the questions raised by the process. Students may come up with their own statistical questions in using this activity. It can be related to basic intuitive probability or more formal expectation analysis. Questions of convergence of the expression for the area can also be adressed informally or related to ideas surrounding the central limit theorem and the law of large numbers.

### Key questions

• What is the chance of a randomly thrown cell falling under a shape?
• Are there any cells which might be problematic? How might we deal with those?
• How reliable would you think that this method might be?

### Possible extension

Able students will want to focus on the question of creating an algorithm for deciding when to stop generating random squares. They can also relate this to work on the central limit theorem. They might like to consider a refinement of the algorithm which takes into account the boundary of the shape.

### Possible support

Just using the activity in a hands-on fashion to find areas by recording whether randomly generated squares fall under or to the side of a shape can really reinforce the understanding of basic ideas in statistics. Encourage students to note when questions or uncertainties in the process arise.