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Statistical diagrams


This prompt is suitable for students between the ages of 10 and 14, although it could be made appropriate for younger children by presenting a pictogram and frequency polygon with the bar chart or for older children by adding a histogram. When the bar and pie charts have been presented as a pair, students have made comments related to the following:
    • Missing information - there should be labels on the axes of the bar chart (for example, 'frequency' on the vertical axis) and titles.
    • Reading the charts - the bigger the piece of the pie chart (or the higher the bar in the bar chart), the greater the amount represented; the charts tell you how many things there are in something; most of the time, pie charts are to do with percentages; the bar on the left is a quarter of the highest bar.
    • Construction of the charts - the pie chart has degrees; why is a pie chart harder to draw than a bar chart?
    • Interpretation - what do the charts represent?; the pieces of the pie chart are split into 5%, 20%, 25%, and 50%; the charts represent the same information; they show favourite school dinners.
An initial discussion might focus on the data set represented by the charts. How was the data collected? What sampling technique was involved? What was the size of the sample?
  
To continue the inquiry, students have decided to draw a scale on the bar chart, estimated the frequencies and drawn the pie chart accurately to test if the charts are drawn from the same data set. Whatever scale they use, students will end up drawing the same (or very similar) pie chart. An important lesson to draw out at this point is that pie charts show the proportion of quantities, not the quantities themselves. This can be emphasised by discussing a second prompt showing two pie charts.
   
The scatter graph introduces the concept of bivariate data and challenges students to compare its meaning with the two charts. Often, a class will argue that the graph is linked because of its similar shape to the bar chart. Students might interpret it as a time series with the line of best fit as a trend line (see box). The inquiry teacher could use the regulatory cards at this point to decide how to proceed. Students might ask the teacher to explain or provide resources so they themselves can inquire into the meaning and purpose of scatter diagrams.

Resources 
Prompt sheet (Bar and pie charts only)
Promethean flipchart    download

Making thinking visible through inquiry
The reports below come from grade 5 pupils at Castle Oaks Public School in Brampton (Ontario, Canada). As their teacher says, the pupils have made their thinking visible by asking and answering their own questions:
  • Do all the graphs represent the same thing but in different ways?
  • How can we figure out what they are trying to graph?
  • If the line is not connected to the X's is it still called a line graph?
  • What is the scale?
The spirit of inquiry in the classroom is summarised in the teacher's encouragement to others: "Keep engaging in open problems and critical thinking fellow math inquirer!"

You can follow the pupils' inquiries on twitter @Mr_AsAwesome5s.



  
Reasoning through inquiry
The picture shows the questions and observations from grade 5 students at the Western Academy of Beijing in China. Nathaniel Atherton, their teacher, reports on the inquiry that developed.

  
The students quickly began to identify the graphs, first by naming them and then by listing the missing information. They were able to name the first two easily but really struggled with the scatter graph, which they argued was like a line graph, but not a real line graph. The students then began analysing the values represented and drew connections between the growing patterns shown in each graph. The class debated whether all graphs were representing the same data.
  
Students then divided into sub groups to either prove or disprove the correlation between the graphs. Some groups focused on all three graphs while others looked at just the first two. The students quickly identified tools which would possibly aid in their inquiry including rulers and protractors. They specifically worked on the bar and pie chart, trying to develop a standard measurement unit in which they could compare the two sets of data.
  
The inquiry proved challenging yet engaging for the students and they enjoyed comparing their results.

Nathaniel Atherton is a grade 5 teacher and Grade Level Leader at
the Western Academy of Beijing. You can follow him on twitter @nat_atherton
  
Year 8 classroom inquiry
These are the comments and questions of
Ann Macdonald's year 8 class. The class used the regulatory cards to decide to inquire collaboratively. Ann describes how the inquiry progressed:
  
"All of the students were able to make up some data to fit the bar chart and the vast majority measured the vertical axis to create a scale. Very few had actually considered that the bar and the pie chart represented the same data, but a couple did. When this was brought up, the students wanted to draw their own pie charts, with those that knew how to helping the ones that didn't. Some figured it out once percentages had been mentioned. In the first lesson, no one thought to check whether their completed pie chart looked like the one on the prompt. The majority of the class thought the scatter graph was a 'line graph' so I decided to focus on the bar and pie charts. At the end of the first lesson, one girl volunteered to present how to draw a pie chart to the rest of the class as a re-cap. The rest of the class decided that this was a good idea."
During the first lesson, one student interpreted the scatter graph as a time series with the line of best fit viewed as a trend line. Ann distinguished the two by adding a time series to the prompt and, in the discussion that followed, drew out the knowledge that existed in the class about scatter graphs.
At the end of the second lesson, Ann used the learning journey to encourage students to reflect on how their inquiry skills had developed.

Ann Macdonald is a secondary school maths teacher in Brighton, UK. You can follow her on twitter @Mckyntyre.