Levels 1 to 5

This set of exemplars is one of two that describe possible progressions in statistics. It shows the development in students' understanding of probability. (The other set shows the development in students' ability to display data.)

Probability is about calculating and expressing the likelihood of an event or series of events. Students experience "chance" from a very early age, but they need to be taught about its predictability, the application of its properties, and the language of probability. Many of the phenomena that students experience have random outcomes. It is important for students to realise that, although the outcomes of individual events are uncertain, the pattern of outcomes of random events is predictable. The applications of probability include insurance, weather forecasting, and all forms of card games and gambling.

Make probability comparisons using everyday language

At this stage in the probability progression, students understand that some events are more likely than others. They are able to use a range of everyday language to express the likelihood of an event occurring in a practical experiment, but are unable to assign a numerical value to this probability.

Assign numerical probability values to simple events

At this stage in the probability progression, students are able to determine the theoretical probabilities of the outcomes of simple events and to express them using numbers.

Identify all possible outcomes of more complex events

At this stage in the probability progression, students are able to identify all possible outcomes of a more complex event. Their strategies may include systematic listing, probability trees, or tables.

Use the possible outcomes to assign numerical probabilities

At this stage in the probability progression, students are able to assign probabilities to more complex events. (None of the level 4 achievement objectives relate directly to this learning step, although tree diagrams are mentioned as a way of finding all possible outcomes.)

Identify probabilities where listing outcomes is impractical

Students are now able to calculate probabilities for sequences of events for which listing would be impractical (for example, by using probability trees).

Background to the task

The task chosen as a context for this set of exemplars gives the teacher enough scope to observe the development in the students' understanding of probability through levels 1 to 5. The teacher asks the students to determine the chance of selecting red balls from each of a variety of buckets containing red and blue balls. The teacher tries to elicit each student's most sophisticated probability thinking about the task by presenting them with increasingly complex variations.

The task

The teachers asked the students whose work is shown in these exemplars to complete one of the versions of the task according to their level of development.

• Levels 1/2, 3i: The teachers presented the students with the buckets as shown in the diagram. They asked the students to give the probability, for each bucket, of selecting a red ball if they took one ball.
• Levels 3ii, 4: The teachers asked the students to determine the probability of getting two red balls if they randomly selected two balls from a bucket containing two red balls and two blue balls.
• Level 5: The teacher asked the students to determine the probability of getting one ball of each colour if they took two balls from a bucket containing two red and five blue balls.

Parallel tasks

The same task could be explored in many different contexts, including dice or card games, spinners, or lotteries.

References

Department of Education (1985-1989). Beginning School Mathematics: Cycles 1-8. Wellington: School Publications.

Ministry of Education (1999). Figure It Out, Levels 2-3. Wellington: Learning Media.

Ministry of Education (2000). Figure It Out, Level 3. Wellington: Learning Media.

Ministry of Education (2001). Figure It Out, Levels 3-4. Wellington: Learning Media.

Ministry of Education (1992). Mathematics in the New Zealand Curriculum. Wellington: Learning Media.