Equal areas inquiry 1

The prompt

Mathematical inquiry processes: Generate examples; analyse structure; reason. Conceptual field of inquiry: Area; perimeter and circumference; formulae; pi (π).

The inquiry prompt is designed for classes that are familiar with the concept of area and can work out the areas of a rectangle and a triangle. The unfamiliar part of the prompt will be the student-led search for a rectangle and a triangle with equal areas, and then learning how to find the area of a circle.

The intrigue in the prompt relates to the comparison of shapes with straight sides and the circle bounded by a curved edge. Students might ask in the question, notice, and wonder phase of the inquiry whether it is even possible to work out the area of a circle.

When presenting the prompt the teacher might add one of the diagrams below. In their initial responses to the prompt, students have measured the dimensions of the rectangle and the triangle, concluding that their areas are not equal. This realisation can lead into a phase of exploration as the class attempts to create examples of the two shapes with equal areas.

Degree of accuracy

An issue that regularly arises is the degree of accuracy appropriate for the area of the circle in comparison to the areas of the rectangle and triangle. If the areas of the rectangle and triangle are 20 cm2, for example, is the statement proved correct if the area of the circle is within one tenth of a square centimetre? Or should the class aim for a greater degree of accuracy?

As the inquiry develops, students might begin to doubt whether the areas can ever be exactly the same. They might be able to use (or follow a teacher's explanation of) inverse operations to deduce that the radius of the circle with an area of 20 cm2 is √(20/π). What does it mean to leave the solution in terms of π? Is that acceptable?

Lines of inquiry

The slides contain lines of inquiry based on three regulatory cards.

In a structured inquiry, the teacher directs the class to follow the sequence of actions on the cards in order. In a guided inquiry, students suggest a course of action by selecting a card and justifying their selection to the class.

For the meaning of each card in the context of the inquiry, see the descriptions below.

In their initial responses to the prompt, students will invariably ask how to work out the area of a circle. The teacher should be prepared for the question.

The diagrams below often lead to an illuminating discussion.

In the first diagram a square is drawn on the radius (with an area of r2). The square is then extended to all four quadrants in the second diagram. The area of the big square (4r2) is larger than the area of the circle.

However, the area of the internal square (2r2) in the third diagram is less than the area of the circle. Thus, the area of the circle is approximately 3r2. The teacher can then introduce the constant pi (π).

Students set out to find examples of a rectangle, a triangle, and a circle with equal areas. The first two cases are relatively straightforward.

Students count the number of squares inside the shapes or use the formulae. They realise that if the rectangle and triangle have the same base, the height of the triangle must be twice the height of the rectangle.

Taking the counting squares approach for the circle can only lead to an approximate area (see the resource sheet). For a more precise calculation, students will require the formula for the area.

Students have often suggested adding another shape to the prompt. While a pentagon has been the most popular suggestion, the teacher might use the following prompt to develop the inquiry:

The areas of three different types of quadrilaterals are equal.

Students find the dimensions of, for example, a rectangle, a parallelogram and a trapezium that have equal areas. The inquiry might develop into equating two algebraic formulae - such as lw = ½(a + b)h - to assist the search.

Other extensions introduce a new concept to the inquiry. Exploring the perimeter is a common suggestion and the teacher might again prompt the class:

The perimeters of a rectangle, a triangle, and a circle are equal.

Another new concept is volume, which extends the inquiry into three dimensions:

The volume of a cuboid, a triangular prism, and a cylinder are equal.

Question-driven structured inquiry

Andrew Blair reports on a structured inquiry with a year 8 mixed attainment class. He designed activities that addressed the students' initial questions and comments (see picture below). Levels of motivation remained high during the inquiry because students could relate their learning to the starting points they themselves had created.

Inquiry lesson 1

Even though the class had carried out inquiries before, it had a reputation for being challenging with some students having a poor attitude to learning. Nevertheless, in the initial phase of the inquiry, they all listened attentively as each pair posed a question about the prompt.

Before the lesson, I had decided to restrict the number of regulatory cards to five:

Find some examples.
Decide if the prompt is true.
Ask the teacher or a student to explain.
Practise a procedure.
Reason about the shapes.

However, when the time came, I judged the students required an immediate focus and handed out a sheet to start the inquiry (see the two examples pictured).

Students commented on the connection between the areas using the large squares as a unit of measure and the areas using the small squares.

At the end of the lesson, they had found the areas of the rectangle and triangle and had started to make suggestions for changing their dimensions in order to make the areas equal.

We also had estimates for the area of the circle (using the large squares) of between 30.5 and 33.5. Students reasoned that the small squares would give a more accurate estimation because there were more whole squares to count.

Inquiry lesson 2

I based the second lesson on two of the students' questions: Is it possible to work out the area of a circle? and How do you work out the area? We started with a discussion of these diagrams that link the area of the square on the radius to the area of the circle.

Students realised that the area of a circle must fall in the range 2r2 < A < 4r2. The class seemed to have settled on 3r2 before one girl tried to justify “slightly more than 3” because “the circle bends towards the outside.” I then introduced the idea of π as a mathematical constant, which we went on to use with an accuracy of three decimal places.

The students practised drawing the square on the radius of a circle and multiplying its area by π.

Two students who had independently researched the formula for the area of a circle after the first lesson then presented the formula A = πr2. They modelled how to calculate the area of the circle on the worksheet from lesson one by substituting the length of the radius (3.25) into the formula.

The area (33.2 accurate to one decimal place) was towards the top end of the estimates from lesson 1, which led to a short discussion about why that might be.

As lesson 2 drew to a close, another student presented her dimensions for a rectangle, a triangle and a circle that have the same area (taking π accurate to three decimal places):

Rectangle length 15.71, width 20
Triangle base 31.42, height 20
Circle radius 10

Inquiry lesson 3

The final lesson addressed the two remaining questions from the students' initial responses to the prompt:

Do other shapes have the same area?
Is it possible for the perimeters of a rectangle, triangle and circle to be the same?

To answer the first question, some students drew a rhombus, a parallelogram, and a regular trapezium, trying to create the three shapes with equal areas. They checked the areas by using strategies to count the squares.

Other members of the class attempted to answer the second question. I explained the formula for the circumference of a circle and students either practised finding the circumferences of circles or tried to establish whether the shapes with equal areas (introduced at the end of lesson 2) had the same perimeters.

At the end of the lesson, one pair of students presented their claim to the class. If the areas of a rectangle, a triangle, and a circle are equal, they argued, it would be unlikely that they would also have the same perimeter. In fact, they thought it would be impossible but could not offer a convincing reason why this might be so.

Improving behaviour through inquiry

Eduardo Abend, a trainee teacher of mathematics at London Metropolitan University, used the prompt with his year 8 class.

Eduardo reports on the inquiry: "It was the class's first Inquiry Maths lesson, so I made it quite structured. I had the plan of covering the area of a circle, moving from what the class already knew about calculating a circle's circumference to the new knowledge.

"The class was really receptive to the new style of teaching and we managed to do a practical activity to discover the area of a circle. We cut circles into small sectors and arranged them into a form as close to a rectangle as possible. From there it was a matter of using previous knowledge to find out the area of the circle."

Eduardo used the prompt as part of a project on inquiry learning for his university course.

He explains that he was inspired to carry out research after his experience with a year 7 class: "What surprised me the most was how differently the students behaved. And that seemed a great area for my research."

May 2018

Student-regulated inquiry

Claire Lee, a PYP teacher at Ecolint (Geneva), used the prompt with her year 6 class. As her students are bi-lingual, Claire started with a pre-prompt about the mathematical vocabulary the class would need during the inquiry. Students arranged words related to the three shapes in a Venn diagram.

Once the class had reached a consensus on the meaning of the words, students generated a wealth of questions:

Is the rectangle smaller than the circle? Is the triangle smaller than the others? Is the prompt true?
Are the perimeters and circumference the same?
How do we find the area and circumference of a circle? How do we find the area of a triangle?
Is there a reason for the order? Are the shapes arranged in the order of smallest to largest area?
If we added a new shape, how would we know what size it should be?

As experienced inquirers, Claire invited the students to participate in regulating the direction of the inquiry. They identified what they needed to know and learn to make progress (see the picture of post-it notes below).

Claire used students’ responses to plan multiple lines of inquiry. She describes how the inquiry developed:

"Students calculated the area of all three shapes. One of the questions raised in a previous lesson led me to write, 'I can use my understanding of the areas of these shapes to create a new shape that would fit into the same category.'

"Some students suggested that the shape needed to be around 16cm2 to fit between the rectangle and the circle. They created hexagons and pentagons with these areas using their newly developed skills on finding the area of the triangle.

"Others remarked that the diameter of the circle was approximately the same as the base of the triangle and the rectangle and therefore the width of the new shape should be similar. We have had a lot of maths content from this prompt."

February 2017

Exploring new concepts

The picture shows the initial responses to the prompt from Amanda Klahn's grade 4 PYP class at the Western Academy of Beijing (China).

In posing the question about how to work out the areas of a triangle and a circle (at the bottom centre of the picture), pupils have identified new concepts and procedures they need to make progress in the inquiry. The ability to recognise missing knowledge is a key step in developing as a self-regulated learner.

The design of Inquiry Maths prompts at just above the current understanding of the class encourages students to 'reach up' for new knowledge. The inquiry process then involves using the new concepts in the service of the pupils' inquiry. This makes the concepts relevant at the point of meeting them and connected to other concepts that form part of the inquiry.

During the inquiry, Amanda described how the class explored π before returning to the prompt, adding "I love how students are led to explore new concepts!"

Question, notice, and wonder

Amy Flood (Head of Mathematics at Mulberry Academy Shoreditch in London) posted this picture on twitter. It shows the questions and observations of her year 8 mixed attainment class.

Amy reports that she used a Frayer model "to develop the definition of 'area' just before this, which helped with comments." The questions can lead into a number of different lines of inquiry:

Calculate the area of a circle;
Compare the areas and perimeters of the shapes;
Find a rectangle, triangle, and circle with equal perimeters; and
Decide if it is possible for a rectangle, triangle and circle to have the same area (even with sides of different length).

January 2020

The picture shows the questions and observations from Helen Hindle's year 8 mixed attainment class at Park View School (Haringey, UK).

Students show they can calculate the areas of the rectangle and the triangle, taking the dimensions from the prompt sheet. One pair explains that the height of the triangle would have to be double the height of the rectangle for the areas to be equal.

The class is now ready to learn how to work out the area of the circle. As the inquiry develops, students are also interested in adding another shape and suggest a pentagon.

January 2022

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