People and their pets

People and their pets

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This resource is about comparing two dot plots.

girls with a speech bubble statement

Lena wants to find out if homes in Ashvale usually have more pets than homes in Moananui.

Question

         

In which town do homes usually have more pets?

    • Ashdale

    • Moananui

    • You cannot tell which town usually has more pets in each home

    • Homes in Ashvale have about the same number of pets as homes in Moananui

Explain why you chose your answer. Hint: Look at the overall shape of each graph. How similar or different are they?
Task administration: 
This task can be completed with pencil and paper or online (with SOME auto marking).
Level:
4
Curriculum info: 
Maths, Statistics, Statistical literacy
Key Competencies: 
Using language, symbols, and texts
Keywords: 
dot plots, distribution, work samples
Description of task: 
Students compare the overall distribution of two graphs and describe their reasons.
Curriculum Links: 
This resource can be used as one source of evidence of students' understanding of comparing distributions visually to identify patterns, variations and relationships.
 
Key competencies
This resource involves interpreting a graph, and explaining why a person has a correct or incorrect interpretation of the graph.  This relates to the Key Competency: Using language, symbols and text.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Interpreting statistical and chance situations, sets 4-5
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y8 (10/2010)
a) D moderate
b) Any 1 of: 

  • Most of the data in both Ashvale and Moananui is between 0 and 4 pets. 
  • Both graphs have about the same shape and position along the scale, e.g., 
[Same] Because they have an even spread of pets throughout the numbers. 
[Same] Because … they are spread out fairly evenly. 
[Same] Because both of the graphs are nearly the same. 
[Same] Because you can see. 
[Cannot tell which town has more] because they look pretty even to me. 
  • The middle (mode, mean or median) of the data for Ashdale and Moananui is about 2. 
[No students in the sample did this.]
  • The number of pets is about the same (63 vs 64) 
[Same] Because there is only 1 pet difference between the two towns.
[Same] Because they are one only out from each other. [bold added for emphasis] 
[Same] Because one is 65 (sic) and the other is 69 (sic) so than mean they nearly have the same amount.
  • The number of pets per home is about the same. 
[Same] I added up the amount of pets in each town and divided it by the number of homes. The average for each was about 3. Although Moananui had 
a fraction more it was about the same.
  • Other correct statistical comparisons.
 very difficult
Teaching and learning: 

This resource is about comparing two statistical graphs which show the distribution of pets in two different towns. The key idea is to recognise that while the graphs differ, their overall shape (distribution) is approximately the same.

Part b) is "very difficult' partly because many students look at detailed features of each graph (such as the total number of pets in each town) and draw conclusions based upon them. This is a more numerical/algebraic approach, where students compare exact numbers or graphical features. Statistical graphics incorporate the idea of variation where small differences are of far less importance than the overall picture. Many students (and teachers too most probably) do not have this mindset that things do not have to be exactly the same to still be the same statistically.

Diagnostic and formative information: 
Types of student responses
  1. Makes invalid statements about the graph (e.g., there are 21 pets in each town).
  2. Gives a valid statement comparing individual features of the graph rather than a holistic statistical comparison (e.g., they have different numbers of pets (63 vs.64), different modes, different numbers at the maximum or minimums, the same range, etc.)
  3. Attempts an argument about the distribution of the data.
  4. Identifies that the distributions (graphs) are very similar and that the differences are of little importance.
 
Common incorrect responses

b)

Uses the number of homes (dots - 21) on each graph to make a comparison

  • [Same] I counted 21 pets in each home. 
  • [Same] I counted up the dots. 
  • [Same] I added the dots and they both equal the same answer. 

Uses the number of homes to make a comparison, ignoring homes with no pets (16 vs 18)

  • [Moananui] There are 18 pets in Moananui and only 16 pets in Ashvale. 
  • [Moananui] Ashvale = 16 homes and pets = 63; Moananui = 18 homes who have 64 pets. [Makes wrong conclusion from a rate argument]
Uses the number of categories with no dots to make a comparison 
[Same] They both have two homes with no pets.

 

Common non-statistical responses

b)

Uses specific features of the graph 

Uses the most or least common category to make a comparison, e.g.,

  • [Moananui] Because number 2 of Moananui homes has 2 more numbers than in Ashvale 
  • [Ashdale] There are more houses that doesn't have pets in Ashvale. 
  • [Moananui] because in Ashvale most of them don't have pets [i.e. mode is 0] and in Moananui they have more of two pets than anything else. 

Uses just the minimum or maximum to make a comparison, e.g.,

  • Ashvale has more homes without pets. 
  • [Ashdale] In Ashdale there are two homes with 10 pets whereas in Moananui there is only 1 home with 10 pets. 
  • Ashvale has many people with lots of [10] pets. 

Sees a small difference in the number of pets in each sample as indicating that town has more pets (often 63 vs 64) – i.e., ignores small statistical variation, e.g.,

  • [Same] I worked out that Ashdale homes usually have 63 pets, Moananui has 64 pets. 
  • [Moananui] because they have more pets in their town and they have the same number of houses as Ashdale. 
  • [Moananui] They have the same number of houses [21] but Moananui has more pets [64 vs 63]. 
  • [Moananui] Ashvale = 16 homes and pets = 63; Moananui = 18 homes who have 63 pets. [Makes wrong conclusion from a rate argument]

Miscounts the number of pets as being equal (often 63 in each town) , e.g.,

  • [Same] I worked out how many pets in both places and the answers were the same [63].

 

 

b)

Attempts at a distributional or statistical statement, e.g.,

  • [Moananui] because their pet numbers seem to be evened out more than Ashdale. 
  • [Moananui] because the first 4 (0, 1, 2, 3) have a lot more than Ashdale but Ashdale is more. 
  • [Moananui] because you can see that Moananui homes have more pets by just looking at it, but I still counted just to check my answer. 
  • [Same] Because homes in Ashvale have 2 people who have 10 pets, but in Moananui they have 1 person who homes 1 pets, and another with 9, which is almost the same. [Accepts a slight difference is not significant, but only looks at part of the graph]
Next steps: 

Uses the number of  homes (dots) on each graph to make a comparison or
Uses the number of  homes (dots) to make a comparison, ignoring homes with no pets or
Uses the number of categories with no dots to make a comparison
Some of these students see each dot as representing 1 pet. They may need to construct some dot plots of their own. Use the ARB resources Marine fish or Waiting time.
Other students see that each dot represents the number of homes in each town who were asked how many pets they had. Ask them "What if the 21 homes in Moananui each have between 5 and 20 pets?"

Uses specific features of the graph (e.g., mode, maximum, minimum, total number of pets etc.)
These students need to make the paradigm shift between numerical, algebraic thinking to statistical thinking. The former, referred to as deterministic thinking, looks at exact outcomes, whereas statistical thinking involves variation. This is a big shift. Lots of exposure to statistical investigations is essential. They could also be exposed to numbers of probability investigations. Each of these will help them to view things more statistically.

Click on the link for the probability concept map for a discussion of variation and distribution.

Attempts a distributional or statistical statement
These students have crossed or are crossing from exact (deterministic) thinking to statistical thinking. Get these students to expound upon their statements. This may indicate if statistical or deterministic thinking predominates for them. Encourage visual responses, and discourage those which calculate the exact number of pets. Students often fall back on exact methods as the most reliable, for example the student who said "[Moananui] because you can see that Moananui homes have more pets by just looking at it, but I still counted just to check my answer."

Figure it out
For resources that look at the distribution (shape) of graphs visually:

  • Population pyramid (Statistics, Book 2 L4, p. 10); and 
  • Uniform changes (Statistics L3-4, p. 2), particularly the first graph.