How many pets?

How many pets?

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources

Question

 
The graph above shows how many pets the children in Room 8 have. Which person has the best statement about this graph?

    • Nisha

    • Jason

    • Tim

    • Abby

    • They all made correct statements

Explain why you decided on your answer.
Task administration: 
This task can be completed with pencil and paper or online (with SOME auto marking).
Level:
3
Curriculum info: 
Maths, Statistics, Statistical literacy
Key Competencies: 
Using language, symbols, and texts
Keywords: 
graph interpretation, dot plots, distribution, work samples
Description of task: 
Students identify a correct statement about the comparing the overall distribution of two dot plots.
Curriculum Links: 
This resource can be used to help to identify students' understanding of identifying a correct statement about a data display.
 
Key competencies
This resource involves interpreting two graphs, and explaining which person has a correct interpretation of the graph.  This relates to the Key Competency: Using language, symbols and text.

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: 
    Y5 (05/2010)
a)
i) Abby
ii) Statements that identify that:
 
Most students had fewer than 3 pets; 
Looks at the "biggest chunk" of the distribution numerically. 
  • Because 8 have 0, and 1 has 10 and altogether it is 29. 
Looks at the "biggest chunk" of the distribution from the graph. 
  • Because most people are more on the other [left hand] side. 
  • Because most of the people by this graph have fewer than three pets. or
Only a few had more than 2. 
Looks at the small "tail" of the distribution numerically. 
  • Because only three people have more than three pets. 
  • Because on the graph only 3 people have 4 or 5 pets. 
Looks at the "tail" and the "biggest chunk" of the distribution numerically. 
  • Because … most children have fewer than 3 pets because only 5 have 3 pets or more and 18 have less than 3 pets. 
  • Because 24 children had fewer than 3 pets, and 3 people had more than 3 pets. 
  • I counted the numbers on each side. 
  • Because 0 1 2 have 6 or up pets 3 4 5 have 2 or less pets.

[Accept if students confuse "<" (less than) with ≤ (less than or equals), e.g., Only a few have more than 3 (Instead of "3 or more")]

 difficult
very difficult

NOTE: Some students Eliminate the three wrong options, for example: Abby – because Jason is wrong because there are 51 pets not 29, and Tim is wrong because most children have 1 pet, [and] Nisha was close but wrong. These students should be asked "Is Abby correct?" and then to explain why she is, ignoring the other three student's explanations.

Teaching and learning: 

The big idea behind this question is where most of the data in a graph lies (i.e., the shape or distribution of the graph). Interpreting statistical graphics is far more than identifying single features of a graph. Students should concentrate on what the graph is showing overall. In this case, students are asked to identify where most of the data lies, i.e. where "the biggest chunk" of the data is.

Students need to distinguish between most and most common. Most refers to where "the biggest chunk" of the data is, i.e., the overall pattern or distribution of the data. Most common refers to the most common category for the data (called the mode), which is a single feature of the graph.

Diagnostic and formative information: 

The misconceptions below suggest that the student is not yet performing at Year 7.

Likely misconception
Ignores the graph and uses their own ideas or experiences of what people have

  • Abby – Because I think people have 2 or 1 pets especially in the city.
  • Abby – Lots of people have 2 or 1 cats, even my cousin has 1 cat.
  • Tim – Because some kids might have more than 2 pets.
  • Tim – Because lots of people have more than 2 pets like my friends.
  • Nisha – Because Nisha might not have a pet or her sister or big brother might not have a pet.
  • Nisha – Because most people are poor so they can’t afford it.
  • Abby – Most people have one pet because it will be easier.
  • Jason – Because his class likes pets better than the others.
Confuses the sample size (number of children in the class) with the number of pets owned

  • Jason – Because I counted the dots and there are 29 dots.
  • Jason – because I counted them.
  • Jason – Because he counted all the children's pets
Assumes the number mentioned by the person is the number of pets

  • Jason – Jason’s class has the most pets because he has 29 pets and Abby’s class has fewer than 3 pets.
  • Jason – He has the most pets.
  • Jason – He has 29 pets.
  • Jason – Because [there are] 29 pets, and Abby has less than 3 pets.
  • Nisha – Because she doesn’t have to do [a] lot of things for the pet’s like getting them lots of food, washing them and cleaning up there mess every time. (Assumes Nisha has 0 pets)
Confuses frequency of each "bar" with the "Number of pets"

  • Tim – Most kids do have more than 2 except one [ i.e., 4 pets has just 1 dot, and confuses "<" (fewer) with "≤"]
  • Tim – Most have 2 or more pets except one.
Comments about the stack with the highest frequency (mode)

  • Tim – Because number 1 on the graph has most out of 1 to 4.
  • Nisha – Because 0 is the second heist and the[re] is no one who said  "1".
  • Abby – Most people had 1 [pet].

Comments that lots of children in the class have 0 pets (near-mode)

  • Nisha – Because lots have 0 and 0 is close to 1.
  • Nisha – On the graph most people have 1 pet and 0 is the closest to 1.
Next steps: 

Ignores the graph and uses their own ideas or experiences of the number of pets people have
Ensure these students realise that the statements in the speech bubbles relate to the graph that is shown. If they don't, stress the point and ask them to repeat the question. If they still give general stories, they need experience of graphing simple whole-number data.

Confuses the sample size (number of children in the class) with the number of pets owned
It may be helpful to ask these students "How many children are there in Room 8?" Many will answer "29". Get them to read Jason's statement again. Help them see that the question is about the number of pets, not the number of children in the class. If students still cannot interpret the statements or the graph, they may need to construct some dot plots of their own.

Assumes the number mentioned by the person is the number of pets
The students need to realise that each child in the question (Nisha, Jason etc.) is talking about the graph of the data and not about the number of pets they have themselves. This means that we don't know how many pets Jason (for example) has. He is unlikely to have 29. Once this is clarified, students should then attempt the question again.

Confuses the frequency of each "bar" with the "Number of pets"
Students need experience of constructing dot plots. This will help them realise that the number of dots represents the sample size. These students could also be asked "Point to the part of the graph that shows people who have two pets". They may well point to the two dots above "3" and "5 or more". They could then be asked "What does "3" or "5 or more" stand for – is it a person?"

Comments about the stack with the highest frequency (mode)
These students need to see that "most" refers to all the children represented in the graph, not just the single category where most data is (the mode). Students need to see where the biggest chunk of the data is. Get them to put a circle around the part of the graph which has most of the data (i.e., 0–2).
The student needs to see that while 1 is the most common number of pets, it contains well under half of the dots.
There may be a need to have a discussion about the difference between "most" and "most common". "Most" talks about the overall shape or distribution, while "most common" is the idea of mode.
Click on Mathematical classroom discourse for more on class discussions.

Unclear what a dot plot is
Students who make some of these errors may not be familiar with dot plots. Get these students to construct some bar graphs or pictographs (use the keywords: graphs, pictographs and graph construction). The students can then replace each unit of the bar with a dot to create a dot plot.

For students who got the question correct (or others too)
Have a discussion about whether this graph has enough information to answer the question "How many pets does a typical New Zealand child own?", and "If not, why not?" This could be followed up by exploring how to find out the typical number of pets children have. It could either be done by asking all children in New Zealand (the population of children), or by taking a group (sample) of students who represent or typify the population of New Zealand children. The former is very expensive. The latter, making deductions (inferences) about the whole population of children by using a good sample is the method that is used by many statistical studies.
Click on Mathematical classroom discourse for more on class discussions.