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The prompt
The prompt
Mathematical inquiry processes: Explore, generate examples, reason and prove. Conceptual field of inquiry: Simplify fractions, general form of continued fractions, reciprocals, sequences.
Zack Miodownik, a teacher of mathematics in London (UK), designed an inquiry prompt with continued fractions. His design used a sequence that led, at the limit, to the Golden Ratio. Each simplified fraction is made up of two consecutive terms in the Fibonacci Sequence (see the illustration).
The prompt on this page is a modified version of Zack's. In taking out the integer at the start, the prompt focuses attention solely on the fractional part. The prompt leads to the reciprocal of the Golden Ratio, which represents the ratio of the width of the golden rectangle to its length. A proof is shown in the 'Lines of inquiry' section.
Question, notice, and wonder
In the question, notice, and wonder phase of the inquiry, students give their initial response to the prompt:
How do you simplify the fractions?
The sequence of the first three simplified fractions is one, a half, and two-thirds.
Does the sequence follow a rule?
Do the fractions go on forever?
How do you create the next fraction?
The teacher chooses between a structured, guided or open inquiry. A structured inquiry could start with one of the lines of inquiry below; an open inquiry would draw on students' experience of directing inquiries through the regulatory cards.
Finite continued fractions
The prompt features an infinite continued fraction, although presented as a sequence of finite fractions. If the initial focus of the inquiry is to be on finite continued fractions, the teacher should consider an alternative prompt such as the one below.
The lines of inquiry that develop from the prompt involve converting any rational number to a finite continued fraction using Euclid's algorithm or exploring the general form of continued fractions.
As an extension to the inquiry, students could research the application of infinite continued fractions to irrational numbers and square roots.
December 2024
Lines of inquiry
Lines of inquiry
1. Change the prompt
Students change the prompt in a systematic way by, for example,
replacing the numerator or denominator of the first fraction with another integer; or
using another integer instead of one.
(See other suggestions in the PowerPoint.)
2. Euclid's algorithm
Students can use Euclid's algorithm to convert any rational number to a continued fraction. There is a step-by-step guide in the slides.
You can find a more detailed explanation of the algorithm and of the connection between the diagram of squares within a rectangle and the continued fraction here (external link).
3. General form
Once students have been introduced to (or researched) the general form, they can explore patterns. How, for example, are the following finite continued fractions connected?
[2;1] [2;1,2] [2;1,2,1] [2;1,2,1,2]
or [1;2] [1;2,3] [1;2,3,4] [1;2,3,4,5]
4. Proof
The proof that the infinite continued fraction in the prompt leads to the reciprocal of the Golden Ratio will be accessible to many secondary school classes. Students can use the model to write their own proofs for other fractions.
An extension to this line of inquiry involves showing how the reciprocal of the reciprocal gives the Golden Ratio by rationalising a surd.
5. Irrational numbers
Students explore the general form for irrational numbers (π and e) and determine the accuracy of each estimation from successive finite continued fractions.
6. Square roots
Students can create a continued fraction for the square root of a number. For example, to find the square root of two, which is between one and two, students set up an equation and find x. They then substitute the expression for x into the equation until they have the required degree of accuracy.
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