Toy car collections
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Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This resource is about comparing information on two graphs.
Ethan wanted to find an answer to his question.
He drew two graphs with the data he collected.
Task administration:
This task can be completed with pencil and paper or online (with SOME auto marking).
Level:
3
Curriculum info:
Key Competencies:
Keywords:
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 explaining a comparison of two graphs. This relates to the Key Competency: Using language, symbols and text.
For more information see https://nzcurriculum.tki.org.nz/Keycompetencies
Learning Progression Frameworks
This resource can provide evidence of learning associated with within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.Answers/responses:
Y8 (11/2015)  
a)  B  Boys in Naenae usually own fewer cars than boys in Taita  moderate 
b) 
Any 1 of:
Students who calculate the number of cars in each town, and comment that there are more in Taita may be partially correct. They would need to say that there were the same number of boys in each town, and indicate that they had done some calculation to receive full credit. This still focuses on the mean of the data, rather than the overall shape of the graphs. Typical partial responses were:
Because [I worked it out and] boys in Taita have more toy cars.
Because they have more cars  I counted.

very difficult 
Based on responses from 28 Year 6 students
Teaching and learning:
This resource is about comparing two statistical graphs which show the distribution of toy cars owned in two different towns. The key idea is to recognise that the overall shape of the graphs differ, with the data in the Taita graph being mainly to the right (i.e., bigger) whereas that from Naenae is mainly to the left (i.e., smaller).
Students should be encouraged to think statistically. Statistics is "the exploration of patterns and relationships in data" (NZC, p.26). Data always has variation in it, so students should be looking at the overall shape (distribution) of the graphs rather than precise details. Other mathematics tends to look at exact statements (e.g., 2 + 4 = 6), whereas in statistics things vary. For more on statistical thinking, click on either Developingstatisticalnumeracyprimaryschools , or the Probability concept map.
Part b) is "very difficult", partly because many students look at detailed features of each graph (such as the total number of cars in each town) and draw conclusions based upon them. This is a more numerical/algebraic approach, where students compare exact numbers.
Diagnostic and formative information:
Student uses the number of dots (21) on each graph to make a comparison
 They both have 21 toy cars.
 I counted up the dots.
 I added the dots and they both equal the same answer.
Student makes an incorrect visual comparison of the graphs
 Because the two graphs are almost the same.
 The graphs are similar.
Attempts a distributional or statistical statement
 Because, even though Car A has a longer stopping distance, it has less of it than Car B which has more at a longer distance.
 because 6/10 of Car A's tests were quite slow.
Next steps:
Uses the number of dots (21) on each graph to make a comparison
These students see each dot as representing 1 car. They may need to construct some dot plots of their own to develop the understanding that it is about the frequency of the dot plot not the number of dots. Use the ARB resources Marine fish or Waiting time to explore this.
Ask the student "How many boys are there in each place?" or "Are there more cars or more students in each town?"
Student uses the number of dots (21) on each graph to make a comparison
These students need to do more work with dot plots. They need to understand that each dot represents a frequency of occurences. They could do construct dot plots with either Waiting time or Zoo animals II or interpret dot plots with either Birth months or Time on homework I.
Makes an incorrect visual comparison of the graphs
Encourage students to put a circle around where most of the data is on each of their two graphs. They should then compare if the circles are in different places. Get them to compare where their peers' circles are located. See the resource Tennis ball throw
Attempts a distributional or statistical statement
These students have crossed or are crossing from exact (deterministic) thinking to statistical thinking. Get these students to talk upon their statements. This may indicate if statistical or deterministic thinking predominates for them. Also encourage them to think about the complete data set, rather than just specific features of it. Encourage visual responses, and discourage calculating just the average stopping distances of cars.
These students have crossed or are crossing from exact (deterministic) thinking to statistical thinking. Get these students to talk upon their statements. This may indicate if statistical or deterministic thinking predominates for them. Also encourage them to think about the complete data set, rather than just specific features of it. Encourage visual responses, and discourage calculating just the average stopping distances of cars.
People and their pets gives a scenario where the two graphs are similar rather than being different as they are for this resource.
Figure it out
For resources that look at the distribution (shape) of graphs visually:
For resources that look at the distribution (shape) of graphs visually:
 Population pyramid (Statistics, Book 2 L4, p. 10); and
 Uniform changes (Statistics L34, p. 2), particularly the first graph.
Click on this link to the Probability concept map for a discussion of variation and distribution.