Students quickly learn and understand the concept of area when it comes to rectilinear forms (forms whose perimeters are straight lines). The simplest example is a rectangle. Multiplying the base by the height gives the area in square units.
Dividing the rectangle into 1 x 1 squares allows us to visualize how many square units make up the area of the rectangle. If we multiply the base (6 cm) by the height (3 cm) of the above rectangle, we get an area of 18 square centimeters. By cutting up the rectangle, students can discover the formula (base x height or length x width) of a rectangle or parallelogram on their own.
Dividing the rectangle in half allows us the find the area of a triangle (half of a rectangle with the same base and height). It becomes more difficult, however, to count up the squares that make up one of the triangles above. From this, we discover that it is easy to find the area of rectilinear shapes by simply dividing them into rectangles and triangles. We can find the area of the trapezoid below, for example, without knowing the somewhat complicated trapezoid area formula. We simply divide the shape into a rectangle and two triangles.
Students ultimately ask the question, how do we find the area of curvy shapes? My initial answer is, it’s not easy! Finding the area of any shape bounded by curves is actually quite complex. Why? Because the concept of area comes from rectilinear shapes. Digital technology is based on mathematics (1s and 0s) and curves, in the digital world, are just a collecting of very small squares called pixels.
We begin to discuss the area of the simplest curved shape, the circle, in Grade 7 but we don’t actually discover the formula for the area of a circle until Grade 8. In the Grade 7 Geometry Main Lesson, the students discover irrational numbers like the golden ratio and pi. They discover pi by looking at the ratio of a circle’s circumference to its diameter. In Grade 8, students are asked to complete an exercise in order to discover how one might find the area of a circle. They have previously discovered that the circumference of a circle is pi x the diameter or pi x twice the radius which plays a crucial role in this exercise. They begin by cutting three equally sized circles out of paper into 4, 8 and 16 equal pieces respectively. The pieces are then reformed to create what ultimately begins to look like a rectangle.
The question is asked, what if the circle were cut into an infinite number of pieces? Because the rectangle would have a base of half of the circumference and a height of r, the formula for the area of a circle pi x r squared is discovered!
This is by no means a rigorous mathematical proof. It takes the subject of calculus to prove that this is actually the area of a circle. Grade 8 students are always amazed at how easy it is to find the area of rectilinear shapes compared with curved shapes. Even in calculus, areas of curved forms are found through cutting them into smaller and smaller rectangles, trapezoids and triangles. The famous astronomer and mathematician Johannes Kepler found the volume of wine barrels by chopping them into smaller and smaller circles. This would help Isaac Newton and G.W. Leibniz discover calculus several decades later.
By discovering this formula (the area of a circle) and not simply receiving it blindly, students will have a deeper understanding of what they are working with and where they are headed in mathematics.
Math Specialist, Andrew Starzynski
