## Summary of learning goals

The sequence begins by reviewing earlier ideas of probability and everyday situations where probability is relevant. Students calculate probabilities in a variety of situations where outcomes are equally likely, as fractions, decimals and percentages. They learn about the variability of the outcomes in an experiment with a small sample size, and the stability of the outcomes for a large number of trials; an instance of the law of large numbers which is the basis of statistics. They use this important knowledge in two situations where they sample to make predictions about the populations.

## Rationale for this sequence

This series of lessons is carefully sequenced to lead the students to a deeper understanding of the applications for probability. The sequence starts by looking at the expectations of certain outcomes of chance events occurring, expressing simple probabilities as fractions, decimals or percentages including in games. In the second lesson, students explore the idea of “short-run variability and long-run stability”. The students will quickly observe wide variation in their results when they conduct small-scale trials or sample a small set of data. They then see that the variation in percentage frequency reduces as the number of trials increases and their results conform to expected theoretical probabilities. This gives students practical experience of how variation in the results reduces as the number of trials in a chance experiment increases. In the final two lessons, the sequence concludes by turning this situation around. This time we have an unknown theoretical probability, and the students’ job is to find it. We gather information through a survey, calculate a probability from the sample, and use it to estimate the actual probability of these events.

“Chance and data” are often taught as two very separate topics. Much can be gained, however, by considering them together. This is especially the situation in surveys and opinion polls, which are used find out information about a whole population in the common situation when there is no capacity to ask everybody. If you use a random sample which is large enough, you can get a good estimate of the probabilities involved. An important thing is that to make a good prediction not everyone needs to be asked.

**reSolve Mathematics is Purposeful**

Problem solving encourages students to ask questions and then find solutions using their knowledge of probability. A wide range of real world situations are included in the sequence, from simple instances of interpreting common language about everyday events to estimating the distribution of letters in an English text and conducting a capture-mark-recapture experiment as is used by ecologists to estimate wild animal populations. This sequence of lessons also provides an opportunity for students to develop an understanding of the relationship between fractions, decimals and percentages, as well as understanding the nature of probability in making good predictions.

**reSolve Tasks are Challenging Yet Accessible**

The lessons presented here start with easily accessible games that provide a common experience and observations that students will be able to quickly understand in real world terms. Students with different home backgrounds will be able to contribute different understandings.

**reSolve Classrooms Have a Knowledge Building Culture**

Games and activities that involve chance are fun and enjoyable and provide students with opportunities to work together. Through interaction with others and by allowing students to discuss their experiences, a community of learners can be established and reinforced within the classroom.