This document introduces continued fraction expansions by discussing rational approximations of real numbers using the mediant, or Farey sum, of two fractions. It describes an activity where students explore continued fractions by drawing and analyzing paths on a Farey diagram, which represents rational numbers as endpoints. The activity aims to reinforce fraction addition and introduce patterns in continued fractions that relate to operations on the path endpoints.

1. 3.3 Making an Unhappy Operation Sunny
This activity introduces continued fraction expansions. There are several ways
to do this. One way is via the question of finding the best rational approxima-
tion of an arbitrary real number. We choose to introduce continued fractions via
the mediant (Farey sum) because it allows us to reinforce the proper addition
of fractions, and it is directly related to the modular group.
Grade Level/Prerequisites: This activity is suitable for students in grades
4 – 6. It is certainly reasonable for students in more advanced grades as well.
Time: This will fit in a 90 minute session, if the participants are fairly quick
at picking up patterns. It may take two 90 minute sessions if the participants
take more time.
Materials: Chalk board with colored chalk, or white board with colored pens,
or similar to display computations to participants; Writing supplies (including
two colors) for participants; Participant hand-out; Sidewalk chalk.
Preparation: Read participant hand-out and try problems. Read leader guide.
Draw enough large (say six foot in diameter Farey Diagrams on the sidewalk
of parking lot for each fife participants to have one to explore. The Farey dia-
gram is an infinite collection of circular arcs with endpoints labeled by rational
numbers. A finite approximation is displayed in the figure below.
Figure 4: Farey Diagram
Objectives: Reinforce addition of fractions; Give a different way to understand
the number line; introduce continued fractions; discover patterns.
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2. Figure 5: Following the sun at the 2015 Summer Teacher Workshop
References/Authorship: Farey sequences have been studied since the early
1800s. Dave Auckly first saw them described by John Harer when Dave was still
a graduate student. He was reminded of them by Paul Melvin just one week
before the Summer 2015 Baa Hozho Math Camp, so when Bob Klein asked for
a session on continued fractions, Dave produced these notes. The start of this
activity was inspired by a math circle that Joshua Zucker ran on fractions. For
more information on this topic see [1].
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3. Sunny Paths and Fractions
There is a sunny way to look at fractions. Start with a warm-up question:
(1) Find a rational number between 5/17 and 6/23.
There is an unhappy way to combine two rational numbers. It is
a
b
¨
c
d
:=
a + c
b + d
,
With b > 0 and d > 0. This operation is not the way to add fractions.
(2) Why do we add fractions the way we do? Consider 2
3 + 3
2 .
(3) On a number line, or “number circle” connect 0 and ∞ = 1/0 with a
line. Given two numbers x and y connected with a line or circle, compute
z := x ¨ y, connect x and z with a circular arc, and connect y to z with
a circular arc. Repeat the process using −∞ = −1/0.
(4) Given a finite sequence of integers [a1, a2, · · · , aN ] make a path that starts
at 0 and walks to ∞, as you walk to ∞ find the path a1 steps over going
to the right if a1 is positive and to the left if a1 is negative. Then walk
along the new path looking for the path a2 steps over. Where will the
path [a1, a2, · · · , aN ] end? Try small examples and figure out the pattern.
Write a formula for the endpoint.
(5) Explain why we divide fractions the way we do.
There are a couple of operations on numbers that are relevant to this discussion.
The first is T(x) := x + 1. The second is S(x) = −1/x.
(6) Find how the T and S operations related to the end points of the paths
[a1, a2, · · · , aN ].
(7) Find 6 different continued fraction decompositions of 3/2.
(8) Quickly compute the endpoint of [−1, −2, 1, 1, −2].
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4. Teacher Guide/Solutions
Presentation It is often useful to have a story to go along with a session.
Perhaps the Farey diagram is a sun, a star or a dream catcher... You can make
up your own story. A very standard way to begin a math circle is with a warm-
up problem. That is how this session is written. Read the first problem to the
participants, but don’t give them the hand out yet.
We expect students at this level will be a bit stumped by this problem at first.
If a participant does come up with a solution, the solution is likely to be the
average of the two numbers and that is a bit messy to compute. If the students
are stuck, point out the following problem solving strategy:
Problem Solving Strategy: Change to easier – make up a new problem that
is similar to but easier than the original problem. This is one of the all-time
best problem solving strategies.
Have the students make up the new questions. You may get something with
less complicated fractions, or something with integers. Nudge them to make
even easier examples where easier can be taken to mean smallest numerators
and denominators in absolute value.
Create a table as in the following on the board as the participants create easier
examples.
between 10 and 15 find
between 1 and 2 find
between 0 and 1 find
between 0 and 1
2 find
Participants are likely to propose 1/4 as a number between 0 and 1
2 . It is in
between, but it is not the simplest number in between in the sense that there
is a fraction with smaller numerator and/or denominator, namely 1/3. After
explaining that you are looking for the simplest rational number between each
pair the students will be able to fill in the table.
Background A very traditional way to introduce continued fractions is via the
notion of best rational approximation to a given number. For example consider
finding rational approximations to
√
3. Since it is between 1 and 2 the best
integer lower bound is 1. The fractional part is
√
3 − 1 and the inverse of the
32

5. fractional part is (
√
3 − 1)−1 =
√
3+1
2 ≈ 1.37 The best integer lower bound for
this is 1. Thus we could take
√
3 ≈ 1 +
1
1
= 2 ,
as a better approximation. Repeat the process and notice that the inverse of
the fractional part of
√
3+1
2 is (
√
3+1
2 )−1 = 1+
√
3
2 ≈ 2.7. Thus the next rational
approximation would be
√
3 ≈ 1 +
1
1 +
1
2
=
5
3
.
Continuing the process one more time gives
√
3 ≈ [1; 1; 2; 1] := 1 +
1
1 +
1
2 +
1
1
=
7
4
.
As the dimensions of a standard business card are 3.5 × 2 inches and this is in
the ratio 7/4, we see that folding a standard business card so opposite corners
meet produces a very close approximation to a 60◦ angle. This is the basis of
many business card origami constructions.
Continuing the process leads to the infinite continued fraction expansion of
√
3:
√
3 = [1; 1; 2; 1; 2; 1; · · · ] := 1 +
1
1 +
1
2 +
1
1 +
1
2 +
1
1 +
1
· · ·
.
Presentation: With the understanding of the simplest rational number be-
tween a given two students should be able to fill in the table as follows:
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6. between 1 and 2 find 3
2
between 0 and 1 find 1
2
between 0 and 1
2 find 1
3
between 1 and 3
2 find 11
3 = 4
3
With enough data participants will guess that
a
b
¨
c
d
:=
a + c
b + d
,
is between a
b and c
d . Explain that the process of collecting data and looking
for patterns is a wonderful problem solving strategy.
Problem Solving Strategy: Collect data, look for patterns – Organize data
related to a problem. Perhaps solve the first few cases and look for patterns.
Warn the participants that it is only a guess that a
b ¨ c
d is between a
b and c
d until
someone can explain why this is the case. Make this explanation homework.
Background: Here is one explanation: Without loss of generality we may
assume that 0 ≤ a
b < c
d , b > 0 and d > 0. Set r = b
d and = cr − a > 0. Then
a
b
¨
c
d
=
a + c
b + d
=
cr − + c
dr + d
>
c(r + 1)
d(r + 1)
=
c
d
.
The proof of the other inequality is similar. The expression a
b ¨ c
d is known
as the mediant or Farey sum. However the notation for the operation is not
standard.
Solution (1): The number between 5/17 ¨ 6/23 = 11/40 is between 5/17 and
6/23.
Presentation: Some students try to add fractions via the mediant a
b ¨ c
d . Point
out that the sum of two positive fractions is larger that either summand. It is
worth discussing why we add fractions the way we do.
Solution (2): One can represent the fractions in question by cutting unit
blocks into 3 and 2 parts respectively and shading the correct number (2 or 3)
of parts for each fraction. One can not just add up the number of parts because
the parts have different sizes. One must subdivide the parts into smaller equal
pieces as demonstrated in the lower two pictures below.
Presentation: At this point you may give participants the hand-out. Draw
a number line and describe that some people will say the end will come back
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7. Figure 6: Adding fractions
to the beginning and draw the number line as a circle with zero marked on
the bottom, infinity (1/0) and negative infinity (−1/0) marked on the top.
This is the first stage in a recursive description of the Farey diagram. The
recursion now continues as follows: pick two extended rational numbers that
are connected by a line or circular arc. Connect each number to the mediant
of the two with a circular arc. Repeat. The result is the Farey diagram.
Solution (3): figure[!ht]
figure
Have several participants come to the board and add circular arcs to the di-
agram. After they understand how the diagram is made take them to the
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8. diagrams drawn with sidewalk chalk.
Now explain walks along the diagram. Any walk will be specified by a finite
sequence of integers, such as [3, −2]. Each walk will start at zero and move
along the path to infinity. Half way across the path identify the path meeting
the far end of the segment that is the first number of steps away with positive
counted to the right and negative to the left. Continue the walk to the end of the
first path, then start walking down the path that was identified earlier. Repeat
the process for each number in the list. For example, the path corresponding
to [3, −2] is displayed below.
Figure 7: Farey [3, −2] path
Students should identify the end-points of the 1-turn paths fairly quickly: [2] =
2, [−4] = −4, [a] = a. Two turn paths may take a bit longer to identify. One
has [1, 1] = 0, [1, 2] = 1/2, and [1, 3] = 2/3. Some students may guess that
[a, b] = (b − a)/b. This is not the case as one can see by looking at [2, 2] = 3/2.
The examples [0, 1] = −1, [0, 2] = −1/2, [0, 3] = −1/3 combined with the
above and [2, 3] = 12
3 = 5/3 and [3, 3] = 22
3 = 8/3 should allow participants to
guess that [a, b] = a − 1
b . Students can then move on to three turn paths.
Solution (4): The pattern that emerges is that the endpoint of the path
[a1, a2, . . . , an] is the continued fraction:
[a1, a2, . . . , an] = a1 −
1
a2 −
1
· · · −
1
an
.
36

9. At this stage, this is a good conjecture. Introducing the S and T transforma-
tions will help participants prove it.
Solution (5): One interpretation of
a/b
c/d
is how many copies of c/d fit in a/b. When a/b = 2 and c/d = 2/3 one can
draw two congruent rectangles. Divide each rectangle into thirds. Color two
of the parts red, two green, and two blue. Thus there are 3 copies of 2/3 in
2. Increasing c will make all of the parts larger, so there will be fewer parts.
Increasing d will make the parts smaller so there will be more parts.
Background: The operations S and T generate the modular group. They
also figure prominently in the Rational Tangles activity. It is natural to
explain why every rational number can be generated by sequences of S and T
operations applied to zero during the Rational Tangle activity. It is natural to
conjecture that every rational number is the end point of an arc in the Farey
diagram. This is also true. The teacher guide includes the common proof to
these results. Essentially the same proof shows that these operations generate
the modular group PSL2Z. The modular group may be viewed as the group
of fractional linear transformations, i.e. f(x) = ax+b
cx+d , with integer coefficients
and inverse function having integer coefficients. This is the same thing as the
set of integer 2 × 2 matrices having determinant one, modulo the equivalence
that any matrix is equivalent to its negation.
In our recommended presentation of the Rational Tangles activity, we tell par-
ticipants about the S and T operations. Tom Davis uses and outlines a version
of the activity in which participants have to guess the S and T operations from
the tangle moves. We essentially move the guessing of the S and T operations
to this activity when participants find the pattern in the endpoints of the paths.
Solution (6): Rewriting the continued fraction expansion using the S and T
operations yields:
[a1, a2, . . . , an] = Ta1
◦ S ◦ Ta2
◦ S · · · ◦ Tan
(0) .
This observation makes it easier to prove the expression conjectured in the
answer to problem (4).
Indeed, we can use the S and T operations to give a precise description of the
path associated to [a1, a2, . . . , an]. Call the directed path across the diameter
from 0 to infinity the fundamental path in the Farey diagram and denote it by
FP . Set
(a1, a2, . . . , an) = Ta1
◦ S ◦ Ta2
◦ S · · · ◦ Tan
.
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10. For any element A of the modular group one has a directed path joining A(0)
to A(∞). For simplicity we denote this by AFP .
The path associated to [a1, a2, . . . , an] is simply FP , (a1)FP , (a1, a2)FP ,
. . . , (a1, a2, . . . , an)FP . To see that this is the case note that T just translates
everything 1 unit to the right on the number line and makes the corresponding
motion on the “number circle.” Similarly the S operation is just reflection
through the point at the center of the circle. Thus Ta2 FP translates the path
FP over a2 steps. The S operation moves the point corresponding to infinity
around to zero. The Ta1 operation then translates this to the path a2 steps
from the a1 segment of the path. (Follow these steps by making some drawings.)
This process continues. Once this description of the [a1, a2, . . . , an] path is in
hand, it becomes clear why the endpoint of the path is Ta1 ◦S◦Ta2 ◦S · · ·◦Tan (0).
Solution (7): In fact we see that there is a one-to-one and onto correspondence
between paths in the Farey diagram and continued fraction expansions of a
given number. Thus to solve this, one just needs to read off six paths ending
at 3/2. For example, [2, 2] = [1, −2] = [3, 1, 3] = [0, −1, −3] = [1, 1, 1, 1, −1] =
[1, 1, 1, 2, 2] = 3/2.
If you really understand something you can do it backwards and forwards.
This is the essence of this problem. We started in problem (4) by finding the
endpoints of given paths. In this problem we were working backwards and
finding paths given an end point. This is one of the major problem solving
strategies:
Problem Solving Strategy: Work backwards – It is often easier to start at
the end result and work to the starting point than it is to go forward. This is
the point of Jim Tanton’s “Math Salute” See http://www.mathcircles.org/
node/668.
Solution (8): Just follow the path. The result is
[−1, −2, 1, 1, −2] = 3/2 .
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