Equal areas inquiry 2
Mathematical inquiry processes: Explore; test particular cases; change the condition; reason. Conceptual field of inquiry: Area; formulae; √2.
The prompt combines a diagram and statement. The contention about the area of the two regions in the diagram is accessible to all secondary school classes, but also arouses curiosity because the dimensions of the similar rectangles are not immediately obvious. A period of exploration leads to the realisation that the value of k is between 1 and 2. This inductive approach sees students narrow in on 1.41. A deductive approach (possibly introduced by the teacher) considers how the area of the white rectangle is wl and, therefore, the area of the larger rectangle is 2 wl. This leads to:
k2wl = 2wl ⇒ k2 = 2 ⇒ k = √2
As the inquiry opens up into more pathways (see below), the teacher should encourage students to adopt deductive reasoning as a more efficient approach than inductive exploration.
The concepts that could arise during the inquiry include, amongst others:
Area of shapes and volume of solids
Enlargement and scale factors (including fractions)
Lines of inquiry
The inquiry can go in different directions depending on which regulatory cards the students select or on the structure that the teacher gives the inquiry based on the students' questions and observations. The class might address the specific statement in the prompt before exploring different pathways:
(1) Exploration and verification
Substitute in values of l, w and k and attempt to determine a value of k that makes the statement in the prompt true.
Link the result to the structure of the diagram by reasoning deductively.
(3) Change the statement
Change the statement so that the area of the shaded region is twice, three times, etc the area of the small rectangle. In these cases, k = √3, √4, etc. Again, students should be encouraged to explain their findings.
(4) Change the diagram (reciprocal)
What happens if the constant k is switched to the smaller rectangle? In this case, the statement is true when k = 1 ⁄√2. As with the original prompt, students could change the statement to make the shaded area twice the size of the white rectangle.
(5) Inquire into other polygons
The inquiry could move into looking at other polygons (and two circles) under the same condition.
(6) Extend into 3 dimensions
Another extension is to consider three-dimensional shapes, changing the statement to include cuboids and volume. In the case of a cuboid, k = ∛2.
Students' responses to the prompt
Stem and leaves
A trainee teacher at London Metropolitan University used the equal areas prompt during her second school placement. She introduced the idea of a 'stem' and 'leaves' to think about the course of the inquiry. The stem is the concept that students should learn and apply and the leaves are concepts that they may apply or encounter as they develop their own lines of inquiry. Read the report here.
Questions and observations
Students’ questions and observations about the prompt that have arisen in class discussions include the following:
w stands for width and l stands for length.
What does k stand for?
We can make l, w and k any number we want.
You multiply the length and width of the small rectangle by the same number (k) to get the length and width of the big rectangle.
Is the shaded shape a rectangle? What if it’s a parallelogram
As the areas are equal, the area of the big rectangle is twice the area of the white one.
Is it possible? If you do l = 2 and w = 1 and k = 2, the area of the white rectangle is 2 and the area of the shaded region is 8 – 2 = 6.
We could change the shape to a triangle or other quadrilaterals.
Would it work in three dimensions?
On seeing the prompt on twitter, @ProfSmudge began to think about how students could visualise the solution to the statement in the prompt. He first talked of sliding the white rectangle down to the bottom-left corner (see second column in the diagram) to show the √2 scale factor, although concluded that it remains unclear. Rotating the inner rectangle, however, is effective when the rectangles are squares (see third column). If the length of the sides of the outside square is 2, the length of the sides of the inside square is √2, from which the scale factor can be deduced.
Genesis of the prompt
The prompt started out as a problem set by Richard Mills (a secondary school mathematics teacher in London) to his year 11 class. The problem, which was designed as a starter to a lesson, was as follows: Two rectangles are arranged with a vertex in common (see diagram below). The dimensions of the smaller rectangle are 40% of the dimensions of the larger rectangle. What percentage of the larger rectangle is shaded?
As the lesson developed, Richard started to pose other questions:
If the shaded area is 50% of the large rectangle, what percentage of the dimensions of the larger rectangle are the dimensions of the smaller rectangle?
What is the connection between the percentage of the dimensions and the percentage of the large rectangle shaded?
If the shapes were a different quadrilateral, would the answers be different?
By redesigning the problem as an inquiry prompt, the classroom is transformed. The learning activity becomes driven by students’ questions, rather than those set by the teacher. When students initiate the inquiry, the lesson becomes a collaborative venture to develop the inquiry pathways suggested by their questions and observations. This often leads to further questions and also to conjectures and generalisations.
One objection to this approach is that students' questions are not as sophisticated as those posed by a teacher. This may be so. However, the teacher as a participant in the inquiry can (and should) give more depth to her students’ questions. In so doing, she is teaching the class what makes a good question in a mathematical culture.