Streets or roads

Streets or roads

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about the best way to show data. 
Olivia drew a graph to show how many students in Room 12 lived on streets, roads, crescents, avenues, places, or ways.

Question 1Change answer

Hohepa said that this line graph is not a good way to show this data. Give a reason to support his opinion.

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: 
graph interpretation, line graphs, work samples
Description of task: 
Students decide if a graph is suitable to display category data and explain the reasons.
Curriculum Links: 
Key competencies
This resource involves explaining why a data display is inappropriate.  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 (11/2009)

Explanations for full credit fall into these main types:

  • The graph is for situations where the data is not category data. This is often accompanied by suggesting another graph, usually a bar graph. Fewer than 10% of students gave explanations such as these:
    Line graphs are usually used to show percentages and money usage. This type of graph is usually utilised to show growth or sales in business. Because their (sic) not trying to figure out the information over time.
  • There is nothing half way between the categories of the title of the house address (see detail of graph below – the line between Cresent & Avenue has no meaning).

Fewer than 10% of students gave explanations such as these:

The lines dont in this case represent anything. It is only the dots that do. A line graph shows the answers in between and that is rough [wrong]. Because the sections are joined. If the lines were not there it would be easier to understand.

Suggests another appropriate form of graphic such as a bar graph, dot plot, or pie graph or any other graphic suitable for discrete, nominal data. Do not accept a histogram (as this is for continuous data). Accept even if there are other errors or misconceptions in the explanation.

It is better to use a bar graph.
I think a pictograph would be a lot easier to do, easier to
understand the data and you could fit in more information.
A bar graph would show the information better.
A bar graph is more efficient and more accurate.
A pie chart would be better.

very difficult
Full credit

 

 

 

 

 

moderate
Partial credit
Based on a representative sample of 137 Y8 students.

NOTE: Three students gave these valid explanations that the category "Other" was needed.
  • Some students might live on farms.
  • There might be more st names like coldasack (sic).
  • There should be "other".

Ask these students "Is a line graph a good way to show this data?"
Students who got partial credit should be asked to justify why these other graphs are more suitable. This may indicate they have an understanding of discrete, nominal data, or may just indicate that they are more used to bar graphs, pies graphs etc.

Teaching and learning: 

The key concept that is being assessed is that the type of data influences the appropriate choice of graph. In this example the data is nominal, which is one kind of discrete data. For more on this, click on Types of data: Statistics.

Nominal data is non-numerical and there is no relationship between the categories. There is nothing half way between a Street and a Road (not a "Stroad" or a "Roeet"; a "Play" or a "Wace"). This means that there should be no lines joining the points.

Diagnostic and formative information: 
Common alternative explanations (do not target the type of data being plotted)
Sees the graph as hard to read (either for themselves or for others)

  • It is harder to follow than other graphs.
  • It may confuse some people in a way and it may be difficult for people to read.
  • Some people might not understand line graphs.
Thinks line graphs lack accuracy

  • Because it is not as accurate as other graphs.
  • Because the accuratcie (sic) isn't good enough.
  • Because it isn't very detailed or precise.
Says the graph does not have enough detail (about the total number of children or their names)

  • It does not show who they are & w[h]ere each person lives.
  • Because it do[e]sn't say how many children there are anywhere on the graph.
  • It does not have the number of the road.
  • Has no names of the people.
  • It doesn't tell you how much people there are and you have to look at the side [y-axis].
Graph will not be valid if more data is added or changed

  • Its possible that addresses will change ...
  • It could change having a new student.
Sees tables or tally charts as better options

  • Because it is easier to write the numbers down because some kids won't know how to use graphs.
  • Because a tally would be easier to read.
Sees the line graph as good

  • You know how many students are in street, Road, crescent, Avenue, Place and Way.
  • I think it is a good idea because it will show the information like other graphs.

Doesn't know that it is inappropriate to join the points on the graph. Does not appreciate the nominal, categorical nature of the data. It can only take on particular named values, such as Street, Road, etc. There is nothing that is halfway between "Street" and "Road". Joining the points implies there is.

Students who got partial credit had a similar mean ability than those who gave these misconceptions. An exception was that students who thought that the line graph was acceptable had a lower mean ability. Students who said that the graph should show the total number of children had higher mean ability than ones who gave other explanations.

Next steps: 

Sees the graph as hard to read
This may be the result of not being familiar with line graphs. Students may have only had exposure to graphs such as bar graphs, pictographs, pie charts, or dot plots. Students need to have experience drawing their own line graphs. To find some of these, use the keywords or click on the link line graphs AND graph construction.

Thinks line graphs lack accuracy
Ask students why they think the graph is hard to read or inaccurate. Some students may relate the lack of accuracy or the difficulty of reading the graph to the way Excel puts the category names (Streets etc) between the tick marks rather than aligned with the tick marks.

Sees tables or tally charts as better options
The student may need to have more experience constructing and reading various graphs, and looking at the trends, shapes, and patterns they show. Graphs show these better than tables do.

The graph does not have enough detail
Students need to have the idea that graphs and tables are there to get an overall picture of the trends or patterns in the data, rather than retaining information about individual people. The stem-and-leaf graph combines the individual data with the overall shape (distribution) of the data. Some computer programmes (for example Tinkerplots) allow individual names to be displayed.

Graph will not be valid if more data is added or changed
The student needs to realise that all graphs need to be updated when new data becomes available or data changes. This is easy with pictographs, dot plots, and to a lesser extent with bar graphs or histograms. Graphs like pie graphs or line graphs need to be redrawn. Computer packages can support students to focus on the meaning of the graph rather than spending time on construction.

Sees the line graph as good (potentially the other listed misconceptions have this issue as well)
For ALL these above alternative explanations, especially the last one:
Students need to be aware of different types of data and how to graph them.
Have a class or group discussion about what the lines on Olivia's graph mean

  • Ask "Is it OK to join the points like Olivia did?"
  • Get those who think it is OK, and those who think it isn't to defend their decision.
  • Ask "What kind of address has 7 houses?" — The answer seems to be "half way between Street and Road". You could ask students "What is halfway between a Street and a Road – Is it a Stroad or a Reet?"
  • Use examples to make students aware of the different types of data.
  • Get students to compare different types of graphs and what data types they should be used for. For more on the different kinds of data, click on Types of data: Statistics.

For more on conducting discussions click on Mathematical classroom discourse.

For teaching points or background knowledge about graphs click on Tables and graphs.
 
Figure It Out
  • Computer comparisons (Statistics, L3-4, pages 6-7).
  • Surf Stats (Statistics, L4, book 1, pages 8-9). 
  • Mad minute (Statistics, L4, book 1, pages 14-15).
For each type of graph in these examples, line graphs should only be used if the x-axis (horizontal) is measurement (continuous) data, even if this has been grouped into categories. The x-axis is often time data but can be other measurement data. For example the graphs in Fish figures or Stretching out (Figure It Out. Statistics, L3-4, pages 8 & 9) could have been drawn as line graphs by joining the middle of each vertical line of the histogram.