How many pets?
For more information see http://nzcurriculum.tki.org.nz/Keycompetencies
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.
Looks at the "biggest chunk" of the distribution from the graph.
Only a few had more than 2.
Looks at the small "tail" of the distribution numerically.
Looks at the "tail" and the "biggest chunk" of the distribution numerically.
[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.
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.
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

Confuses the sample size (number of children in the class) with the number of pets owned

Assumes the number mentioned by the person is the number of pets

Confuses frequency of each "bar" with the "Number of pets"

Comments about the stack with the highest frequency (mode)
Comments that lots of children in the class have 0 pets (nearmode)

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 wholenumber 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.
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.