Xmas Xmath
Written by Dave Didur
November 25, 2014 – It’s that time of the year to get out scissors, coloured paper, tape and glue
– and make mathematical Christmas decorations! Yahoo!! Actually, I intend to mix a little math
into traditional folding and cutting activities such as making chain garlands and snowflakes.
Parents can work with their youngsters for some of these projects and older children can work
on their own. Teachers could involve their students in making these objects – doing math while
making holiday decorations to spruce up the classroom.
Rational Number Paper Chains
Traditional Christmas chains are made
with alternating bands of red and green,
but a variety of colours makes for more
dynamic garlands.
Elementary school
students who have learned about rational
numbers or long division can create chains
as an arithmetic project (refer to my article
“Rational Numbers: Fractions, Decimals
and Calculators”). The teacher has to
provide ten different colours of construction paper. Each colour will represent
a different digit from 0 to 9. For example:
Pink = 0, Blue = 1, Orange = 2, Yellow = 3,
White = 4, Black = 5, Green = 6, Red = 7,
Purple = 8, Brown = 9.
The teacher could leave the task of making such associations to his/her students. Each 8 ½ X
11 sheet of paper should be folded in half lengthwise and then cut along the fold to produce two
4 ¼ X 11 sheets. Each of these would then be cut into one inch strips, resulting in a total of 22
strips that are 4 ¼” X 1” in size. If every student will be making a chain, I recommend cutting up
about 15 sheets of each colour, which would make 330 strips. These strips will be used to curl
into rings when building the paper chains. The teacher should come prepared with at least one
fraction for each student. Remember, a rational number in decimal form will either terminate or
form a repeating decimal that never ends. Dull examples should be avoided, such as fractions
like ¼ that terminate as 0.25, or fractions such as one-third that repeat in uninspiring patterns
like 0.333333333…(this last case would make a solid yellow chain, according to my colour
scheme given earlier). More interesting fractions are
2 3 5 8 4 13 17
, , , , , , ,etc. Dividing
7 13 17 41 19 29 31
such fractions on a calculator will only produce a limited number of decimal places (and the last
one may be rounded off), so it is better to have the students use long division to produce
something like 20 or 25 decimal places – or long enough that they can see the pattern emerge.
For example,
2
 0.2857142857142857142657142... (to 25 decimal places).
7
This rational
number has a period of 6 digits that repeat (285714), so if a paper chain was made to represent
this fraction it would be: Orange-Purple-Black-Red-Blue-White-Orange-Purple-Black-Red-etc.
Each student could take a small recipe-sized card, write his/her name on the card, the fraction
used and its decimal equivalent, and attach it to the beginning of his/her chain. The teacher
could mark this project, if desired, before hanging the chains around the classroom.
Irrational Number Paper Chains
Senior students who have studied pi (refer to my article “Calculating the Value of Pi”) and other
irrational numbers (such as 5 ) could make irrational number chains. As individual projects,
each student could work with a different radical ( 2 , 3 , 5 , 7 , 11,etc. ) and make a chain for
it. Of course, a calculator or an electronic spreadsheet will only produce a limited number of
decimals, as mentioned earlier for fractions. If the teacher is willing to spend time on an
enrichment topic, the students could be taught the algorithm for calculating square roots
manually – and they would then be able to calculate the decimal form for their radicals to 20 or
25 decimal places, and produce longer decorative chains.
A pi chain could be a class project. Students could access the internet and easily find pi
worked out to many hundreds of decimal places. A very long pi chain could be constructed that
would encircle the classroom – but a little statistical work could be incorporated into the fun as
well. The project could be: determine how the various digits are distributed in the decimal value
of pi. Suppose that the project began with 150 strips of each of the ten colours (each colour
represents one of the ten digits 0 – 9).
Pi to one hundred decimal places:
3.141592653589793238462643383279502884197169399375105
8209749445923078164062862089986280348253421170679…
As the chain is assembled, a chart like the following could be filled in.
No. of Decimal Digits
10
20
30
40
0s
0
0
0
1s
2
2
2
2s
1
2
4
3s
1
3
6
4s
1
2
3
5s
3
3
3
6s
1
2
3
7s
0
1
2
8s
0
2
3
9s
1
3
4
If the digits are normally distributed, we would expect to see about one of each digit in each
new string of ten digits. In the third row (when examining the first thirty digits of pi) we would
expect to see about three occurrences of each digit. In the chart, we see that the number of
zeroes is very low (none) and the number of threes is very high (six), while the rest occur 2, 3 or
4 times. What will we see in the first one hundred digits? The first 200? The first 300? Do the
digits appear to be normally distributed?
Follow-Up Investigations: Cutting Paper Rings
When working with the paper chains previously, we made rings by taping the two ends of strips
of paper together. In this section we’ll make both normal rings and a new kind of ring called a
Mobius band (sometimes written as ‘Moebius’ instead of ‘Mobius’) – and we’ll see what
happens when we cut them in various ways. In a mathematics class, the investigations in this
section could follow up the chain-making exercises. For youngsters, it would seem to be magic
– wonderful, unexpected results. More advanced students could begin with these activities and
progress on to other topics related to this field of study. Resources for teachers of all grades
are included in the “Resources” section at the end of this article.
To begin, make four or five rings of each type (see below).
Activity #1: Take one regular ring and one
Mobius band. With a pencil, start drawing
a line in the centre of each ring (on the
inside) that is exactly in the middle – and
continue the line around the ring until you
meet the starting point. Examine the rings.
How many sides does each ring have?
Explain.
Activity #2: Use the rings from Activity #1.
With a pair of scissors, snip each ring in the
middle and cut along your line, going all the
way around. Are the results what you
expected?
Activity #3: Take one regular ring and one
Mobius band. Make a snip one-third of the
way from one edge and then continue to
cut – staying the same distance from the
edge of the ring – around the ring until you
meet the starting point. How many rings do
you get in each case? Are the results what
you expected?
These activities can be viewed as Experiments 1, 2 and 3 in the following video:
https://www.youtube.com/watch?v=BVsIAa2XNKc
Students can be encouraged to extend these
investigations by constructing other kinds of bands –
such as bands with two or three twists instead of just
one – and then cutting as before.
Activity #4: Take two regular rings. Turn one so that it
is 90 degrees to the other one – and tape them together
as shown at the right. Cut one ring in half all the way
around the middle. The pieces will tumble apart – but a
“bridge” connects the two pieces. Now cut the ‘bridge’
piece in half lengthwise, and carefully unfold the result.
Activity #5:
Do the same as above, but use two Mobius bands. First cut
one band in half by going around the middle. Again, pieces
will fall apart, but they will still be connected by a ‘bridge’
piece. As before, cut the ‘bridge’ piece in half and carefully
unfold the result. Perhaps this last activity is best saved for
Valentine’s Day!
Activities 4 and 5 are demonstrated in the following video:
https://www.youtube.com/watch?v=esxyahHBf6w
A very advanced investigation (best reserved for senior high school students) can be observed
in the video “Mobius Strip New Discoveries” at the following URL:
https://www.youtube.com/watch?v=I1Xs0G__KJM
The fascinating geometry associated
with the study of shapes like the
Mobius strip is called topology.
Earlier in this series of articles I
wrote about Euclidean Geometry,
Non-Euclidean Geometry, and
Fractals (the geometry of nature).
Topology developed out of geometry
and set theory. It concerns itself with
the study of shapes and spaces that
undergo continuous transformations
(such as stretching and bending)
without tearing or ripping.
Education.com gives a very good introduction to topology in its article
entitled “Unchanged: What is Topology?” It uses the example of a
ball of clay being transformed into a cube of clay, and then the
cube of clay being reshaped into a gingerbread man – using only
manipulations such as stretching (like forming a loaf of bread from
a hunk of dough). These three shapes are called topologically
equivalent because we are able to transform one into another
without making holes or cutting.
Another common example of topologically equivalent figures is a
donut and a coffee mug. Each has only one hole (for the donut, it
is in the centre; for the mug, it is the space between the handle
and the side of the cup). The interior of the cup is not a hole; a
piece of clay can be punched in the middle to make an
impression – and then the edges can be folded upwards to
make a bowl-like shape. A hole must pass right through an
object.
In this series of images, a donut is gradually transformed into a coffee mug by stretching and
reshaping. In the last step, a depression is made by pushing down into the figure to make the
bowl-like interior of the mug. As stated earlier, the interior is not a hole (a hole must go right
through a shape and come out the other side).
Consider a teapot without its lid. How many holes are in this object? The answer is two. One
hole goes through the handle (like in our coffee mug). The other hole goes in through the top
and comes out through the spout. Holes may curve and bend before exiting a shape. This
would make a teapot topologically equivalent to a 3D version of the numeral “8” (which also has
two holes).
The article “Topology and Children’s Intuition About Form” references statements by Piaget that
claim that young children are able to maintain topological distinctions before they can maintain
Euclidean distinctions, as demonstrated in how they draw various shapes. It’s amazing what we
know so early and forget so late!
Making Paper Snowflakes
Did we all do this when we were young? It was a
lot of fun. The folding techniques used before
cutting begins ensure that we get various
symmetries in the shapes. Each fold produces
a crease, which is an axis of symmetry for the
final figure. For younger children it is easier to
begin making 8-pointed snowflakes because the
folding is simple. All of the folds are straightforward. This video illustrates the process:
https://www.youtube.com/watch?v=zkEu2tTiXgY
6-pointed snowflakes require some trickier folds –
but the result is a more realistic snowflake (actual
snowflakes have six points and six axes of
symmetry). This video shows how it is done:
https://www.youtube.com/watch?v=rSZMM8_fFkc
For those who prefer to read written instructions and see photos, go to this web site:
http://www.instructables.com/id/How-to-Make-6-Pointed-Paper-Snowflakes/?ALLSTEPS
One of the properties of shapes studied in geometry
is symmetry. Consider the snowflake at the right.
The yellow line is called an “axis of symmetry”. The
part of the shape that is on one side of the yellow line
is a mirror-reflection of the part of the shape that is on
the other side of the line. An axis of symmetry is like
a mirror.
There are six places where we could place the yellow
line on the snowflake in order to get perfect
symmetries. We say that the snowflake has six axes
of symmetry.
The isosceles triangle pictured
at the left has one axis of
symmetry: the red line CD.
The equilateral triangle at the
right has three axes of
symmetry:
Line AD
Line BE
Line CF
Scalene triangles do not have
any axes of symmetry.
NOTE: A scalene triangle has three sides of different lengths, an isosceles triangle has two
equal sides, and an equilateral triangle has three equal sides.
If an entire class will be making paper snowflakes, it might be very decorative to
take all of the creations and put them on a wall or bulletin board in the shape
of a Christmas tree (as shown at the right). The same thing could be done at
home with the snowflakes made by your children.
Other Paper Ornaments and Decorations
Check these out:
1. Making a tree ornament
https://www.youtube.com/watch?v=jgfN5M8vwwM
2. Making a 3D snowflake (using shapes from #1)
http://www.wikihow.com/Make-a-3D-Paper-Snowflake
3. Making a paper lantern ornament
https://www.youtube.com/watch?v=u9CLt9VVUQY
4. Making a 3D Christmas tree decoration
https://www.youtube.com/watch?v=RzqFU-QfBG0
Have fun! Happy holidays!

This article is the ninth of a series of mathematics articles published by CHASA.
Marvellous Mathematics – Introduction
Euclidian Geometry – Article # 1
Non-Euclidean Geometry – Article #2
Rational Numbers – Fractions, Decimals and Calculators – Article #3
Continued Fractions – Article #4
Introduction to Fractals: The Geometry of Nature – Article #5
Solving Algebraic Equations (One Variable) – Article #6
Solving Systems of Equations in Two Variables – Article #7
Calculating the Value of Pi – Article #8
CHASA has received many communications from concerned parents about the difficulties their
children are having with the math curriculum in their schools as well as their own frustration in
trying to understand the concepts - so that they can help their children. The intent of these
articles is to not only help explain specific areas of history, concepts and topics in mathematics,
but to also show the beauty and majesty of the subject.
Resources
Mobius Strip Activities
Mobius Strip Cutting Magic -- https://www.youtube.com/watch?v=BVsIAa2XNKc
Paper Cutting of Connected Loops -- https://www.youtube.com/watch?v=esxyahHBf6w
Mobius Magic Ring – https://www.youtube.com/watch?v=f-19NLKxNUc
Making Paper Snowflakes
Cutting an 8-pointed Paper Snowflake -- https://www.youtube.com/watch?v=zkEu2tTiXgY
Cutting a 6-pointed Paper Snowflake – Video https://www.youtube.com/watch?v=rSZMM8_fFkc
Cutting a 6-pointed Paper Snowflake – Text
http://www.instructables.com/id/How-to-Make-6-Pointed-Paper-Snowflakes/?ALLSTEPS
Other Paper Ornaments & Decorations
Making a 3D Christmas Tree -https://www.youtube.com/watch?v=RzqFU-QfBG0
Tree Ornament -https://www.youtube.com/watch?v=jgfN5M8vwwM
3D Snowflake -- http://www.wikihow.com/Make-a3D-Paper-Snowflake
Paper Lantern -https://www.youtube.com/watch?v=u9CLt9VVUQY
Topology
Unchanged: What is Topology? -- http://www.education.com/sciencefair/article/mathematics_unchanged/
Topology -- http://en.wikipedia.org/wiki/Topology
Classroom Activities in Topology – Gr. 1 to 6
http://www.davidparker.com/janine/mathpage/topology.html
Topology and Elementary Students: Resources -http://mathforum.org/library/results.html?ed_topics=&levels=elem&resource_types=&topics=top
ology
Twists and Turns: Lesson Plans in Topology for Gifted Students
http://www.educationfund.org/uploads/docs/Publications/Curriculum_Ideas_Packets/Twists%20
and%20Turns.pdf
Mobius Strip New Discoveries -- https://www.youtube.com/watch?v=I1Xs0G__KJM
Topology and Children’s Intuition About Form -- http://newsavanna.blogspot.ca/2012/07/topology-and-childrens-intuition-about.html