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Working with frequencies

How can we use frequencies to promote insight into probability problems?
A set of experimental data entered into a frequency tree diagram
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?

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Teaching Probability

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