Equal areas inquiry 1
Mathematical inquiry processes: Generate examples; analyse structure; reason. Conceptual field of inquiry: Area and perimeter; formulae; pi (π).
The prompt can be presented in words only or combined with one or both of the two sets of shapes above. If both sets are used, the inquiry can start with students' focus on the different triangles.
While this might seem a simple prompt, it can lead to a number of intriguing questions for students in lower secondary school.
How do you work out the area of an obtuse-angled triangle?
How do you work out the area of a circle?
Can the radius of the circle be a whole number if its area is a whole number?
Can the dimensions of the three shapes all be whole numbers or must there be decimals?
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 20cm2, for example, is the statement proved correct if the area of the circle is within one tenth of a square centimetre? ... or one half? Can the areas ever be exactly the same? Students might be able to use inverse operations to deduce that the radius of the circle is, in our example, √(20/π). Is it acceptable to leave the solution in terms of π?
These prompts can be used separately or in conjunction with the main prompt. On occasions, students' questions and comments lead the main inquiry into these pathways.
The areas of three different types of quadrilaterals are equal.
This prompt has led to students listing types of quadrilaterals and then using their knowledge of one type to attempt to deduce the formulas for the areas of others. Another (high-attaining) class decided to work algebraically by making formulae for area equal, such as lw = ½(a + b)h, in order to find values for the dimensions.
The perimeters of a rectangle, a triangle and a circle are equal.
This prompt has led to students constructing particular triangles that meet the condition of the prompt with a ruler and pair of compasses. It has also generated questions about shapes with the same areas (from the main prompt): Is it possible to have three shapes with the same area and the same perimeter? Is there a relationship between the perimeters of the three shapes with equal areas?
The volume of a cuboid, a triangular prism and a cylinder are equal.
The inquiry can be extended into three dimensions, giving rise to similar questions to those that arise with the main prompt.
Andrew Blair reports on a inquiry he developed with a year 8 mixed attainment class. The nature of the class led him to run a structured inquiry, in which he designed activities that addressed the students' questions and comments (above). Levels of motivation remained high during the inquiry because students could relate their learning to the starting points they themselves had created.
While the students had carried out inquiries before, the class 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 or responded to a peer's comment. Before the lesson, I had decided to restrict the regulatory cards I offered the class to these five.
However, when the time came, I judged the students required an immediate focus and handed out a sheet for them to discuss and work on (see two examples below). 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, the students 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.
I based the second lesson on the questions about whether it is possible to work out the area of a circle and, if it were, how to do so. We started with a discussion of the diagrams below 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 accurate to three decimal places. The students practised using the procedure of 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
The final lesson of the inquiry started by addressing the two remaining points from the initial questions and comments. The first related to the question about whether other shapes could have the same area. Students selected one of three tasks:
(1) Draw a rhombus, parallelogram and regular trapezium;
(2) Draw the three shapes with equal areas (by counting squares); or
(3) After completing task 2, make one cut to the three shapes and rearrange the pieces to make rectangles with equal areas.
The second point involved the perimeters of the shapes. After I explained why C = πd, students either practised finding the the lengths of circumferences or tried to establish if the shapes with the same areas (introduced at the end of lesson 2) had the same perimeters. The class decided that if the areas of a rectangle, a triangle and a circle are equal, it would be unlikely they would also have the same perimeter. Many in the class wanted to go further and say it was impossible, but no student could establish a solid reason why this might be so. The contention remained at the level of intuition.
Conceptual understanding through inquiry
The picture shows the initial responses to the prompt from Amanda Klahn's grade 4 (year 5) PYP class at the Western Academy of Beijing (China). The question about how to work out the area of the triangle and circle at the bottom demonstrates that pupils have identified the new concepts and procedures they need to make progress in the inquiry. The ability to recognise missing knowledge is a key part to developing as a self-regulated learner. 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!"
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.
Student questioning to launch inquiry
Amy Flood (Head of Mathematics at Mulberry Academy Shoreditch in London) posted the picture on twitter. It shows the questions and observations from 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:
Calculating the area of a circle;
Comparing the area and perimeter of shapes;
Finding a rectangle, triangle and circle with the same perimeter;
Deciding if it is possible for a rectangle, triangle and circle to have the same area (even with sides of different length); and
Developing a way of calculating area using formulae, rather than counting squares.
These are the questions of a year 8 class. The picture was sent to the website by Eduardo Abend who used Inquiry Maths prompts in the classroom and wrote a project about inquiry learning while training to be a secondary school teacher at London Metropolitan University. 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."
Claire Lee used the area prompt with her year 6 class of experienced inquirers. As her students are bi-lingual, Claire started with a pre-prompt – a preliminary question – about the mathematical vocabulary the class might need. Students arranged words related to the three shapes in a Venn diagram.
Once a consensus on the meaning of words was reached, the class generated a wealth of questions about the prompt that could have led into multiple pathways.
Claire also invited students to participate in directing the inquiry. They identified the need to learn how to work out the areas of the shapes and the circumference of the circle and how to use the first few digits of pi (see below). Claire used the students’ suggestions to plan the remainder of the inquiry.
Claire 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."