Mathematical inquiry processes: Identify properties; generate more examples; analyse structure. Conceptual field of inquiry: Area; fractions; ratio.
Area and fractions
What is the area of each section of the flag? What fraction of the flag does each area represent? What fraction of the flag is a particular colour?
What is the ratio of length:width for the designs? Are they always the same? Is there an international convention for flags?
What is the ratio of each part of the designs in the prompt to the other parts?
Is it possible to design flags with a given ratio? What would a flag look like, for example, if its three parts were in the ratio 1:2:4? And what if rectangles were not permitted?
Amanda Klahn used the prompt with her grade 4 PYP class. The pupils’ questions and comments cover a wide variety of topics:
Identifying and naming shapes;
Finding ratio and using ratios to create new shapes;
Area, including calculating the areas of a circle and a trapezium;
The meaning of π; and
At the time of the inquiry, Amanda was an IB teacher at the Western Academy of Beijing (China).
The genesis of the flags prompt
The flags prompt has undergone changes since it first appeared on the website. The suggested first version of the prompt was the US flag. However, Inquiry Maths prompts are internal to the subject, which means they take a mathematical equation, diagram or statement as the object of inquiry. Moreover, they are designed to have 'less to them, but more in them" in order for students to develop their own contexts if they so wish. (Read more about the issues created by a 'real life’ prompt here.)
The second version of the prompt featured two designs (see illustration) that, while being made-up, were readily identifiable as 'flags'. However, on occasions the exact same issues arose in the classroom as if the prompt had featured real flags. In fact, the issues were worse for the flags not being real. On one occasion, rather than focus on the mathematical potential of the diagrams, students began to argue about which countries the flags represented and, indeed, whether they did represent a country at all.
To attempt to rectify these issues, the colours were removed from the third version (the current prompt). In the one classroom trial to date, this prompt did result in students focusing on the measurements within the diagrams. Although one student remarked in the orientation phase that "they look like flags", the class decided to inquire into the ratios of different areas.
Inquiry teachers can learn a lot from the development of the flags prompt. If you are aiming for a wide-ranging, cross-curricula inquiry, then the first version of the prompt could link geography, politics, history and art with mathematics. However, if the focus has to be on mathematics - perhaps because of the demands made by the curriculum or structure of the school timetable - then the third version might be more appropriate.
The problems experienced with the flags prompt highlight the difficulties of using an artefact in order to generate a specifically mathematical inquiry. 'Real life' items can carry too much contextual information to make them successful starting points.
Inquiring into flags of nations
The flags prompt was inspired by David Aaron (a year 6 teacher in Blackpool, UK) who wrote to Inquiry Maths about an inquiry he had initiated with the Stars and Stripes. This was reminiscent of an inquiry that Alrø and Skovsmose include in their book on the Inquiry-Cooperation Model about constructing the Danish flag.
The Stars and Stripes
David described how he displayed the flag and went through the inquiry sequence, whereby the pupils made observations and asked questions. They came up with:
What is the area of the flag?
What is the area of the rectangle containing the stars?
What is the area of the striped compound shape?
What is the area of a small stripe? What is the area of the longer stripe?
What is the perimeter of a star ? What's is the total perimeter of all the stars?
How many right angles are there within the flag?
How many lines of symmetry are there?
An intriguing question about the Stars and Stripes concerns the rectangular array of the stars. What if there was one more state in the union? Two more? One less? What would the arrays look like in these circumstances? Is there an optimum array for each number of states? Can you produce an algorithm that gives that optimum array?
Alrø and Skovsmose report on an inquiry that starts with a teacher's question, "What does the Danish flag look like?" After the teacher and students get in contact by sharing their perspectives on the question, the class decides to zoom in on an inquiry to construct the Danish flag given its dimensions, including the width of the white stripe.
The authors discuss an episode when the teacher's deductive 'quizzing' of a group leads to a discrepancy between perspectives. The teacher and students have different procedures, both valid, for positioning the horizontal strip on the flag. The teacher's intervention, Alrø and Skovsmose argue, leads to an unnecessary interruption to the inquiry and brings a temporary end to the involvement of a girl who had suggested the students' method.
Alrø, H. and Skovsmose, O. (2002). Dialogue and Learning in Mathematics Education: Intention, Reflection, Critique. Dordrecht: Kluwer Academic Publishers.