In this activity, students become familiar with mathematical knots by
observing them. By putting these observations into words they gain
experience in using language to describe spatial properties. They will
undoubtedly discover many basic properties of knots.
A set of about five large knots that will be the object of the
students' study. These can be any knots that you find interesting and
that will stimulate a variety of observations on the part of the
students. Any of the following knots would be suitable:
Have students brainstorm about what they already know about
knots: What is a knot? Where and how they are used? How they are
made?
Tell the students that knots are important to mathematicians,
too. By studying knots, mathematicians have developed theories that
are useful in many areas of science.
Show the students the knots that they will initially study.
Point out that these are mathematical knots
. The knots
that mathematicians think about have their ends joined to form a closed
loop. One reason for doing this is to be able to move the knots all
around, pulling and twisting on the loops to see what happens, and be
sure that the knots don't come apart.
Explain that one of the things that mathematicians do when they
study something like knots is to look at a few of them and begin
making clear and precise observations about them. At first they
notice the simplest things. They try to describe what they notice as
clearly as they can so that when they talk to each other about them,
each one understands what the other one means and they don't get
confused. Often mathematicians who live far away from each other
study the same things, and they write each other letters and talk over
the phone. So they have to be sure when they describe something that
the other person gets the right picture.
Explain that another reason mathematicians make and collect
observations is that it often turns out that when you list the simple
and easy things, other things begin occurring to you, and you notice
things that are more complicated and interesting.
Have the students begin making observations about knots. This
can be done either with the large group, or you can ask the students
to work in smaller groups that rotate among the knots. In small
groups, students should collect and write down their observations and
then report back to the larger group. Make sure the students
understand that no observation is too silly or too small. If they
have trouble saying exactly what they mean, have them help one another
to figure out the best way to put it in words. If they need to, it is
okay to invent new words to make something clear, as long as everyone
understands what the new word means and they can explain it to anyone
else.
Suggest to the students that a clear and accurate drawing of the
object of their observation can be useful.
While the students are making their observations, it is
important that they be able to touch and manipulate the knots as well
as look at them.
How a knot can be tangled to look different, then returned to
its former shape.
Ways to distinguish differences between knots.
How the "knottiness" of a knot might be measured.
The above list is not exhaustive of the kinds of observations
that students can make. Save their list, refer to it often when
studying knots, and add new items to it as they are discovered.
If there is not agreement about some points, put question marks
next to them.
When you finish up the activity, help the students to see that
they learned very many things about knots by using their own minds and
senses. They haven't studied any information from the outside yet.
Help the students to begin generating and listing questions
about knots that they could investigate. Explain that when we take a
close look at something that we have never examined carefully before,
not only do we see new things, but we also start wondering about
things that we never thought of before. After looking at knots, what
are some questions that come into your mind?
Materials 
