i
i
Activity 3
DIVIDING A LENGTH INTO EQUAL
N THS EXACTLY
(1)
(2)
(3)
1/3
P
For courses: geometry, precalculus
Summary
Students are asked to come up with ways to, say, fold the side of a square piece
of paper into perfect 3rds or 5ths or some other odd division. The aim here is to
develop exact methods, not approximations.
After students have tried this for a while, or perhaps in a later class, give them
the handout. This shows an origami routine that the students will discover produces a landmark for folding perfect 1/3rds. The students are then asked to generalize this method.
Content
This activity is mostly geometry, although it’s a problem that can be solved using
both synthetic and analytic methods. In fact, if the problem is solved analytically,
nothing more than finding equations of lines and their point of intersection is used,
making this a nice hands-on activity for a precalculus class.
Handouts
There are two handouts that take two different approaches to the same task: folding a square piece of paper into perfect thirds. The first one shows students the
folding method and challenges them to discover what it is doing. The second one
explains what the method is doing and challenges them to prove it.
Both of these handouts can be motivated by asking students beforehand to try
coming up with their own methods of folding thirds exactly.
Time commitment
Plan on reserving at least 30 minutes of class time for this activity, which includes
folding time, student work time, and discussion afterwards.
27
i
i
i
i
HANDOUT
What’s This Fold Doing?
Below are some origami instructions. Take a square and make creases by folding
it in half vertically and folding one diagonal, as shown. Then make a crease that
connects the midpoint of the top edge and the bottom right-hand corner.
P
Question 1: Find the coordinates of the point P, where the diagonal creases meet.
(Assume that the lower left corner is the origin and that the square has side length 1.)
Question 2: Why is this interesting? What could this be used for?
Question 3: How could you generalize this method, say, to make perfect 5ths or
nths (for n odd)?
i
i
i
i
HANDOUT
Folding Perfect Thirds
It is easy to fold the side of a square into halves, or fourths, or eighths, etc. But
folding odd divisions, like thirds, exactly is more difficult. The below procedure
is one was to fold thirds.
(1)
(2)
(3)
1/3
P
Question 1: Prove that this method actually works.
Question 2: How could you generalize this method, say, to make perfect 5ths or
nths (for n odd)?
i
i
i
i
30
Activity 3
SOLUTION AND PEDAGOGY
Since the two handouts are similar, we’ll focus on solutions for the first one.
Question 1: Synthetic approach
Assume that the square has side length one and consider the labeling in the figure
below. Denote the coordinates of P with ( x, x ). Then AE has length x, so EB has
length 1 − x. Also, EP has length x.
C
F
P
A
D
E
B
Then BDC and BEP are similar. Thus |CD |/| PE| = | BD |/| BE|, which
becomes
1/2
2
1
=
⇒ 2 − 2x = x ⇒ x = .
x
1−x
3
This could also be proven by using the similar triangles ABP and CPF.
Question 1: Analytic approach
Assume that the square sits in the xy-plane, with A at the origin and B at (1, 0).
Then P lies on the intersection of two lines: y = x and y − 1 = −2( x − 1/2).
Combining these to find their intersection gives x − 1 = −2x + 1, or 3x = 2, or
x = 2/3.
Obviously, the answer to Question 2 is that this can be used to fold the square
into thirds exactly. Try it!
Question 3
The picture below shows how to generalize this method to fold the side of a square
into n equal divisions, where n is odd. Instead of using the 1/2 vertical line, make
a vertical line at x = (n − 2)/(n − 1) (or 1/(n − 1) away from the right side).
1/(n−1)
E
i
B
i
i
i
Dividing a Length into Equal N ths Exactly
31
Finding this line should not be too hard, since n − 1 is an even number (since n
is odd). (If n − 1 equals something like 6, then you’d have to find a 1/3 point first
and then fold this in half to get a 1/6 mark. So in a sense this method is recursive.)
The same approaches to Question 1 will give that the point at which the two
diagonal creases in this general case meet is ((n − 1)/n, (n − 1)/n), which can
then be used to fold the paper into equal nths.
Pedagogy
As mentioned previously, students appreciate learning methods of folding perfect thirds a lot more when they’ve spent some time themselves trying to develop
them. There are many other methods for doing this kind of thing (some of which
will be described at the end of this section), and if students come up with methods
of their own then they should be studied and proven. In fact, if someone comes
up with the method provided in the handout, then that’s the best context in which
to investigate proofs and generalizations. Thus, if the students’ own explorations
go well, there may be no need for the handout.
The first handout may seem more advanced, but I’ve been surprised at how
able some students are at figuring out what the method is doing. Nonetheless, the
first handout does set students up for an analytic proof, since finding the coordinates of P is most easily done by finding the equations of the crease lines.
The second handout places more emphasis on developing proof-building skills.
Most students come up with the similar triangles proof, but the analytic approach
can be a very useful one in a variety of geometry problems and uses nothing more
than basic precalculus material. In a geometry course students are often delighted
to learn that they can solve some problems using such simple techniques. So if
all groups develop synthetic geometry proofs, make sure to drop some hints to
students who finish early about thinking of the paper as being in the xy-plane, so
that equations of the lines can be found. Usually this is all that needs to be told for
students to run with this and develop the analytical proof described above. (And
note that the second handout gives no hints about an analytic proof, unlike the
first handout.)
The general method is also easy for students to figure out, if they first try a simple case. Students who are stumped on how to generalize should be encouraged
to try an example, like folding 1/5ths. To make 1/5ths with this method requires
only using a vertical line at the x = 3/4 position instead of the 1/2 position. This
is pretty straightforward for students to figure out and can lead to the complete
generalization.
Other methods
As mentioned previously, there are many other methods for folding 1/3rds, 1/5ths,
or general 1/nths. A few will be shown here without proof.
Below is shown a way to achieve 1/3rds that follows naturally from one of the
methods for folding a 30◦ -60◦ -90◦ triangle (as seen in Activity 1). This does not
generalize to other 1/nths, however.
i
i
i
i
32
Activity 3
1/3
A different general method, shown below, was invented by Haobin Yu, a student at the 2000 Hampshire College Summer Studies in Mathematics. It uses the
premise, again, that divisions of 1/2n should be possible, and from this we get an
odd division 1/(2n + 1). It can be proven using similar triangles or the analytic
method used previously.
1/(2n)
1/(2n+1)
More methods for folding exact divisions can be found via web searches, in
particular see [Hat05] and [Lan04-1]. Any of these methods could be assigned for
homework exercises or extra projects.
i
i