Fraction of a polygon inquiry
Mathematical inquiry processes: Interpret, test cases, reason and convince. Conceptual field of inquiry: Fractions, area of polygons, trigonometry and Pythagoras' theorem, and geometric sequences..
The prompt shows three polygons with an even number of sides - a hexagon, an octagon, and a decagon. In each polygon, a shaded rectangle is drawn from the base to the top.
The statement in the prompt is about the fraction of each polygon covered by the rectangle. It makes the contention that the fractions decrease in a geometric sequence - that is, there is a constant ratio between consecutive terms.
While there is connection between the fractions and a pattern for students to find, the fractions do not form a geometric sequence and, therefore, the statement is false.
The prompt has proved to be intriguing for students aged between the ages of 14 and 17. In the orientation phase when the class is invited to question, notice and wonder, responses have included:
Why are there only polygons with an even number of sides?
What is a geometric sequence?
To work out the fraction shaded you have to know the areas of the shapes.
You could split up the hexagon and octagon with horizontal and vertical lines to work out the fraction, but the decagon is harder.
The fraction shaded gets smaller. The width of each rectangle is about three-fifths or two-thirds of the one before so it could be a geometric sequence.
To decide if the sequence is geometric you have to take into account the area of the polygon as well, not just the rectangle. The hexagon has a greater area than the decagon.
Do polygons with an odd number of sides fit the sequence? Do they have their own sequence?
Finding the fractions shaded
At the start of the inquiry, students' attempt to find the fractions by decomposing the polygon into triangles and rectangles or by working out the areas of the shaded region and the polygon.
This latter approach might involve setting the length of each side of the polygon and then using Pythagoras' theorem and trigonometry to work out the other dimensions.
In the hexagon with side length one, for example, the height of the shaded rectangle is √3 units and its area is √3 square units. The area of the polygon equals 3/2(√3) square units and so two-thirds is shaded.
Students need only Pythagoras' Theorem for the hexagon because it decomposes into equilateral triangles. For the octagon, and thereafter, they will require the use of trigonometry and a knowledge of interior angles.
The prompt was inspired by Vincent Pantoloni's post on social media. His diagram consists of the first five cases, starting with a square, and also shows the fraction shaded (in its simplest form) under each polygon. Vincent asks: Is there a pattern? Is it always a rational number?
The inquiry prompt is different in four ways:
By starting with the hexagon, it ls more likely that students make the correct generalisation about the design of the diagrams. The three polygons in the prompt all show the rectangular shaded region produced from the base, whereas the square, which can be considered as part of the sequence later in the inquiry, is atypical in this respect.
By deleting the values of n, there is more 'space' for students to engage with and wonder about the prompt. If the prompt is aimed just above the level of the class, then all students should be able to comment on the properties of the polygons and speculate about others that are not included.
By removing the fractions, students have to come up with their own methods to calculate the fractions in order to make headway in the inquiry. This opens up the possibility of students applying mathematical tools they already know or learning about and practising the use of new ones.
By replacing the questions with a statement, the prompt promotes curiosity and sense-making in the classroom. It encourages students to come up with their own questions, rather than respond to pre-determined ones. They have to infer the design principles behind the diagrams, to determine the meaning of the prompt, and to decide if its contention is true or false.
Lines of inquiry
1. Geometrical reasoning
While the teacher might want students to practise using the approach with Pythagoras' theorem and trigonometry already outlined, it is possible to find the fraction of the shaded region of all polygons without calculations.
By splitting the polygons into congruent triangles that meet at the centre, the teacher can show that the area of the shaded region is always equivalent to four of the triangles.
The number of congruent triangles enclosed by the polygon is equal to the number of its sides. So the fraction shaded is four over the number of sides.
2. Is the prompt true or false?
To be a geometric sequence, each term has to be multiplied by a common ratio. That is not the case with the sequence, and, therefore, the prompt is false.
One way to help students find the term-to-term multiplier is to use common denominators.
By writing the first two fractions in the sequence as four-sixths and three-sixths, it is clear that the second term is three-quarters of the first one. Similarly, the second and third terms can be converted to tenths - five and four respectively. Thus, the third term is four-fifths of the second one.
Intriguingly, there is a pattern. If n is the nth term in the sequence, the multiplier to find the next term is (n + 2)/(n + 3), which can be written as the recurrence relation:
3. Polygons with an odd number of sides
The inquiry becomes more challenging when the class considers polygons with an odd number of sides.
If students use the same rule to shade the polygon (that is, they produce two lines at the ends of and perpendicular to the base of the polygon), then the shaded region is an irregular pentagon.
The area of the shaded region can be calculated by decomposing it into rectangles and triangles and using trigonometry to find the dimensions of the new shapes.
The calculation to find the percentage of the pentagon covered by the shaded region (below) uses a different approach. The area of the two triangles outside the pentagon are subtracted from the rectangle created by producing the lines until they are level with the highest vertex.