New offer! Get 30% off your first 2 months of Unlimited Monthly. Start your subscription for just £35.99 £24.99. New subscribers only T&Cs apply

Working with frequencies

How can we use frequencies to promote insight into probability problems?

Random sequences are interesting in their own right. However, data that is presented in the form of a list is not particularly easy to analyse.

You may have already arranged the results of your spinner experiment into a table of some kind – and our simulation spreadsheet included a simple table to summarise the outcomes.

We find that frequency trees are a useful form of representation to introduce at this stage, especially because they make it easier to develop a narrative – the ‘story’ of the data.

In the picture above, the teacher has recorded data from one group’s experiment as a frequency tree. Each of the four distinct paths through the tree represents a different ‘story’, that the teacher can unpick, as in the following dialogue based a real classroom discussion.

Teacher: Look at the top branches. What’s the story here?

Student: It’s sunny in the morning, and sunny in the afternoon.

T: And how about these two branches? [Indicating the outcome SR]

S: Sunny in the morning, rainy in the afternoon.

T: And these two branches? [RS]

S: Rainy in the morning, sunny in the afternoon.

T: And these? [RR]

S: Rainy in the morning, rainy in the afternoon. Better take an umbrella!

T: So we’ve got four possible outcomes. What are they?

S: Sunny all day, sunny then rainy, rainy then sunny, and rainy all day.

T: Could anything else happen if we stick with this model?

S: No.

We believe that:

• A dialogue like this helps students to understand both the sequence of events and the tree structure, so that they can be quite sure that all possible outcomes are included.
• Working through the tree structure in the way shown is an important preparatory step before the introduction of the more common (and abstract) probability tree diagrams.

Do you agree?