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The prompt
The prompt
Mathematical inquiry processes: Identify properties; generate more examples; analyse structure. Conceptual field of inquiry: Measurement; area and perimeter of 2-dimensional shapes; fractions and ratio.
The flags prompt can lead to a wide-ranging inquiry that incorporates different mathematical concepts. If a class is not used to inquiry learning, then the teacher should direct students' attention to a particular property of the designs to start the inquiry.
To help maintain a mathematical focus, the prompt deliberately features made-up flags. When real flags were used in earlier versions of the prompt, students could become drawn into debates about their connotations (see 'The development of the prompt' below).
Lines of inquiry
Lines of inquiry
Area and fractions
What are the dimensions of each region in the flags? What is the area of each region? What fraction of the flag does each region represent?
On the slides (with the gridlines displayed) the area of each flag is 60 square units. Regions A and B in the top flag are both 9 square units and region C is 42 square units. Therefore, as fractions of the whole, regions A and B both represent three-twentieths and region C is seven-tenths. Considering the bottom flag (and using square units), regions A and C are 16 each, B is 12, D is (16 - π) and E is π.
Ratio
What is the ratio of length:width of a flag? What is the ratio of each region to the others? In the top flag, for example, the ratio of the areas of A, B, and C is 3:3:14.
Design a flag
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? Is it possible to draw the design without rectangles?
Research
Students might decide to extend the inquiry into real flags. One line of research involves the ratios of height to width of different flags (see pictures below). Another idea is to carry out a mathematical comparison of two flags involving area, fractions, and ratios.
Question-driven inquiry
Question-driven inquiry
Amanda Klahn used the prompt with her grade 4 PYP class at the Western Academy of Beijing (China). The pupils’ questions and comments cover a wide range of topics that could lead into separate lines of inquiry:
Identifying and naming shapes;
Angles;
Finding ratios and using ratios to create new shapes;
Dimensions, perimeter and area, including calculating the areas of a circle and a trapezium;
The meaning of π; and
Finding ratios and using ratios to create new shapes.
February 2015
Real flags as prompts to inquiry
Real flags as prompts to inquiry
The flags prompt was inspired by David Aaron (a year 6 teacher in Blackpool, UK) who contacted Inquiry Maths about an inquiry he had initiated with the flag of the USA - 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 compound shape with stripes?
What is the area of a small stripe? What is the area of the longer stripe?
What is the perimeter of a star ? What 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 inquiry 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? Is it possible to produce an algorithm that gives that optimum array?
May 2014
Alrø and Skovsmose describe an inquiry about the Danish flag - the Dannebrog - in their book Dialogue and Learning in Mathematics Education: Intention, Reflection, Critique (pages 54-61).
The inquiry in a 6th grade class starts with the teacher's question, "What does the Danish flag look like?" Students work in groups of between two and five to reconstruct the flag from memory. When finished, they discuss the proportions of their designs and decide which one is most similar to the real flag.
Alrø and Skovsmose discuss the inquiry in terms of the teacher getting in contact, locating the students' perspective, and identifying the procedures they are using. Students are involved in advocating, thinking aloud, reformulating, and evaluating.
The development of the prompt
The development of the prompt
The flags prompt has undergone changes since it first appeared on the website because of the difficulty of using an everyday artefact to generate inquiry that is specifically mathematical. Real-life flags can carry political or geographical connotations that impede their use as successful starting points. (Read more about the issues created by a real-life prompt here.)
The first version of the prompt was the Stars and Stripes (see above). However, Inquiry Maths prompts are internal to the subject, which means they take an equation, diagram, or statement as the object of inquiry. They deliberately omit a wider context so students' attention is drawn to mathematical properties.
The second version of the prompt featured two designs (see illustration). Even though they were made-up, the designs were readily identifiable as 'flags'. The same issues arose in the classroom. On one occasion, rather than focus on the mathematical potential of the diagrams, students began to argue about which countries the flags represented.
To rectify these issues, the colours were removed from the third version (the current prompt). Although the designs look like flags, the absence of colour means they cannot be real.
The development of the flags prompt highlights the importance of the stimulus to inquiry. If the teacher is aiming for a cross-curricula inquiry, then a real flag could link geography, politics, and history with mathematics. However, if the focus is to be on mathematics only - perhaps due to demands from the curriculum - then a decontextualised prompt might be more appropriate.
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