Fraction of a shape inquiry
Mathematical inquiry processes: Interpret; analyse structure; reason. Conceptual field of inquiry: Area; Pythagoras' Theorem and trigonometry; fractions and percentages.
The prompt has the potential to cover a wide range of topics in the school curriculum, including the areas of polygons and the circle, fractions, Pythagoras' Theorem, trigonometry, similar shapes and use of formulae. It has led to open inquiries that combine student questioning with teacher instruction in a mutually supporting process.
At the start of the inquiry, the teacher should establish that the three shapes are divided into five strips of equal width, perhaps by confirming a student's observation, and that the triangle is isosceles. Examples of students' questions and observations that have arisen in classroom inquiry are:
The fractions shaded of each of the shapes are not the same.
What fractions of the triangle and circle are green?
Which shape has a greater fraction shaded?
Is the triangle equilateral? Is it isosceles? Is the fraction shaded the same for an equilateral and isosceles triangle?
Do we need to know the height of the triangle to work out the area?
The base of the triangle and the diameter of the circle are divided into five equal parts.
To work out the area of the green part of the triangle, should you use the formula for a trapezium?
Over half the circle is shaded green.
Is the fraction of the circle shaded green smaller or bigger than three fifths?
Three-fifths of the square is shaded; for the triangle, the fraction shaded is greater than three-fifths. As finding the fraction of the circle shaded poses a much greater challenge (see mathematical notes), the teacher might guide students towards inquiry pathways involving other quadrilaterals, such as the trapezium, rhombus or parallelogram, before considering the circle.
Questions that extend the inquiry include:
What fraction is shaded if we use different types of triangles (including an obtuse-angled triangles)?
Why is the fraction shaded in the circle closer to the square than the fraction in the triangle?
What are the widths of the strips in a triangle and circle if the areas of the five regions are equal?
These are the initial responses of grade 5 pupils at the Vientiane International School (Vientiane, Lao PDR) to the prompt. The class has asked insightful questions, any one of which could lead to rich inquiry.
Two pupils' notes (below) show key vocabulary, calculations of the area of the circle, speculative propositions and provisional conclusions. It demonstrates a deeply mathematical process that combines procedural knowledge with plausible reasoning.
The reports shown in the picture below show the pupils' answers to their original questions, reflections on the inquiry and further questions. Overall, the class has produced a wonderful inquiry into the prompt.
The prompt was inspired by an idea on the MathArguments180 blog. During a series of 180 ideas for discussion in mathematics classrooms, number 89 (April 2014) featured different shaded shapes and asked which is the closest to being three-fifths shaded?