Spinner probabilities

Spinner probabilities

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about identifying how likely spinners are to point to white.

Question

three spinners
 
Each spinner has an arrow.  All three arrows are spun. Which spinner is most likely to point to white?

    • Spinner 1 only

    • Spinner 2 only

    • Spinner 3 only

    • Both Spinner 1 and Spinner 2

    • Both Spinner 1 and Spinner 3

    • They are all just as likely

Explain why you decided on your answer.
Task administration: 
This task is completed with pencil and paper or online.
Level:
3
Curriculum info: 
Maths, Statistics, Probability
Key Competencies: 
Using language, symbols, and texts
Keywords: 
probability, spinners, independence
Description of task: 
Students decide which arrow on three spinners is most likely to point to white, and explain their reasons.
Curriculum Links: 
This resource can be used to help to identify students' understanding of comparing the likelihoods for simple situations involving chance.

At Years 5 and 6 students should be able to “order the likelihoods of outcomes for situations involving chance”. This task indicates that the following judgements of student achievement levels are reasonable:

  • Identify the correct multiple choice response and give a satisfactory explanation based upon the fraction of the circle that is white (1/4) indicates students are at curriculum Level 4.
  • Identify the multiple choice part of the question correctly, and give a reasonable explanation indicates students are at curriculum Level 3.
  • Identify the multiple choice part of the question correctly, and no explanation indicates students are early curriculum Level 3.
Key competencies

This resource involves justifing a conclusion using written communications, which relate to the Key Competencies: Using language, symbols and text, and Thinking.
For more information see http://nzcurriculum.tki.org.nz/Key-competencies

Learning Progression Frameworks
This resource can provide evidence of learning associated with
Interpreting statistical and chance situations, sets 3-4
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y4 (06/2009)
F (They are all just as likely) and
Explanation such as:

  • There is 1 chance in 4 of pointing to white for each spinner.
  • Each spinner is 1/4 white.
  • The proportion of white is the same for each spinner.

very difficult
(both correct)

 

Teaching and learning: 
  • Many students who answered (F) did not have a valid reason for their choice.
  • While this question is very difficult, it gives an excellent indication of students' understanding of probability.
  • It also would be ideal for a class or group discussion. Click on Mathematical classroom discourse for more information about this.
Diagnostic and formative information: 
Common error Explanation: likely misconception
B (Spinner 2)
or
E (Spinners 1 & 3)

Location of arrow (closest to white)
This assumes that the starting position of the arrow positively influences where it finishes up; e.g.,
[Spinner 2] because the arrow is closest to white.
Because spinner 2 was nearly pointing to white.
B (Spinner 2)
or
C (Spinner 3)		 Length of spinner
Various responses that the length of the spinner influences the outcome; e.g.,
Because spinner 2 is the smallest arrow.
Because spinner 3 has the longest arrow.
C (Spinner 3) Size of the spinner
The size of the spinner affects the probability of where it lands; e.g.,
[Spinner 3] because it is the biggest.
Size of the white region
The bigger the white area affects the probability of landing on it; e.g.,
Because 3 is biggest, and has a bigger white space.
F (They are all just as likely) Location of arrow
This assumes that the starting position of the arrow positively influences where it finishes up; e.g.,

  • the starting position of the arrow negatively influences where it finishes up: e.g.,
    [They are all just as likely] because none of the spinners are on white.
  • the starting position of the arrow positively influences where it finishes up; e.g.,
    [They are all just as likely]

because they are all pointing at the same part wicht (sic) is next to white.
because none of the spinners are on white.
They are all pointing on the same place.

Equi-probability
Student may see that any outcome is equally likely regardless of its actual probability; e.g.,
[They are all just as likely] because it can land anywhere.
Ask the student to elaborate on their answer. If it is still unclear, ask the student "In Spinner 1, what colour is it most likely to point to?" If they answer "grey", then they have this misconception.

NOTE: The errors in the right hand column may also be associated with other wrong responses to the first part of the question.

Examples of students' responses

Correct reasoning

  • [They are all just as likely] because they are all quarters.
  • [They are all just as likely] because they had the same fraction of white, (1/4)
  • [They are all just as likely] because if you look carefully 1/4 of each spinner is white so they are all as likely as each other.

Possibly correct (Question the student further)

  • [They are all just as likely] because they all look similar.

Other incorrect reasoning (see diagnostic box for others)

  • I chose it [Spinner 2] because it has different colours.
  • [Both spinner 1 and spinner 2 because] Because both spinner 1 and spinner 2 look the same.
  • [Both spinner 1 and spinner 3 because] They only have a quarter of white and spinner three had more colours.
  • [Spinner 1] because I tested it with my pencil.
Next steps: 
Location of arrow

1. Positive influence.

This is when the student thinks that the last spin will affect the outcome of the next spin by making it more likely to be the same or as close to it as possible. It is usually expressed by students as "The arrow in spinner 2 is closest to white."  If the spinners are fair then one spin does not affect the next spin. This is called "independence". If the spinner is not fair, then it may stop close to where it previously stopped because of the way it has been made. It is very had to make a fair spinner. Have a discussion about the spinners being fair.

2. Negative influence

This is when the student thinks that the last spin will affect the outcome of the next spin by making it more likely. It may be expressed by students as ""All the spinners are pointing away from white."  This assumes that the spinner was a "memory" and "thinks" it must be white's turn because it hasn't happened for a while. Even adults have this misconception.  Have a discussion, where students debate the point. Ask (after discussion) "How does the arrow 'know' how to point to white?"

Size of spinner / size of white area

Students need to realise that it is the relative size of the white area (i.e. ¼ of the spinner) rather than the absolute size of the spinner or the white area.