Celtic Counting requires students to trace an under and over pattern – counting the Celtic loops. How many loops exist in this knot?

There are two.

How many loops does this knot have?

There are three. The following slides have larger knots, but students age 6+ should still be capable of tackling them – certainly collectively if not alone.

How many knots do square knot patterns have? These are actually easier to count because the loops are all like rectangles.

There are six.

This is a 6×7 knot. Students learning about relatively prime numbers, greatest common factors and lowest common multiples could try to come up with a general theory of how many loops are in these knots.

However, younger students can still tackle these puzzles to practice their tracing and counting skills.

This 6×7 knot has only one big loop.

This 6×8 knot has how many loops?

Just two.

This 6×9 knot has how many?

Three. Notice the symmetry.

This is just one of the loops in the previous pattern. Can the junior high student working on symmetry say anything about the symmetry of the loops in an nxm knot.

I’ll leave you with this bigger knot… Don’t go onto the next slide till you’ve solved it. Can you solve it without using paper?

There are more than two loops…

There are more than three loops…

There are four loops.

Three puzzle sheets are included in the pdf below. Here is the first.

Students should also be encouraged to create their own knots. The following sheet is a template that may help. All my knots have loops that alternate going over and under. That should be a rule for an introductory exploration.

Template to create knots…

Celtic Counting

(MathPickle, 2016)

In celebration of me going to Maths Week Ireland next week (October 15-23, 2016) here is a puzzle with a Celtic edge. Students can count the different loops (not an easy task for a six year old!) and older students can explore both symmetry of loops and can hypothesize a relationship to greatest common factors or lowest common denominations.

Here is a pdf. The first three pages are printable exercise sheets. The next page is a template for students to create their own knots. The rest of the presentation is for you to use in class as you see fit.

Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

MP1 Toughen up!

Students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP2 Think abstractly!

Students take problems and reformat them mathematically. This is helpful because mathematics lets them use powerful operations like addition.

MP3 Work together!

Students discuss their strategies to collaboratively solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

MP4 Model reality!

Students create a model that mimics the real world. Discoveries made by manipulating the model often hint at something in the real world.

MP5 Use the right tools!

Students should use the right tools: 0-99 wall charts, graph paper, mathigon.org. etc.

MP6 Be precise!

Students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

Students learn to identify patterns. This is one of the things that the human brain does very well. We sometimes even identify patterns that don't really exist 😉

MP8 Be lazy!?!

Students learn to seek for shortcuts. Why would you want to add the numbers one through a hundred if you can find an easier way to do it?


Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos 😉 For a free poster of MathPickle's ideas on elementary math education go here.

Gordon Hamilton

(MMath, PhD)