what is this?
This is an example submission journey. It documents details of searching, questioning, and discovery of maths inside the things and spaces around us for an Third/Fourth Level (S1–S3) entry. Follow this story and learn how to catch the beauty of a discovery in a photo, title and commentary (credits)
This resource is part of a suite of materials and activities created to inspire entrants, and support teachers, and parents to enter maths inside: a photo competition open to everyone in Scotland.
maths inside: see different, make connections, celebrate!
complementary resources
→ Image Bank 1
→ displayed final submission
→ Shoelace Combinations example IDL activity with Es+Os
Shoelace Combinations
I recently bought a new pair of running shoes and I was messing around to try and find the best way to lace them up so that they were secure on my feet. It got me thinking – how many possible ways are there to lace up your shoes? I decided to use this for my maths inside entry and came up with the title
Shoelace Combinations
and commentary
I was tying my shoes and considered the different ways I could lace them up to stay on my feet
I also took the photo below
I wanted to improve this commentary by thinking more about the maths involved so I thought of some questions to answer. How many holes do the laces go through? Why do all the holes need to be used? What happens when a hole is used twice? I thought a little more about the number of holes the laces go through
Using only some of the holes will decrease the number of possible shoelace combinations. In some cases, it might be sensible to avoid using some of the holes but in other cases it isn’t. For example, I can choose to use the holes on only one side of the shoe but this would be daft as they wouldn’t stay on my feet. I can use a hole twice but doing this would increase the number of possible shoelace combinations. By thinking a bit more about this I now have the commentary:
I was working out the best way to lace up my running shoes to make sure they stayed on my feet. This got me thinking about the different ways I can lace them up and how many different combinations were possible. My trainers have 8 holes for the laces. I can choose to use only 6 of them, which would decrease the number of combinations I can choose from. I can choose to use a hole twice which would increase the number of combinations. Not all the combinations are sensible though because if I choose to feed the lace through only one side then my shoes won’t stay on my feet
What do you think of my commentary? Can it be further improved? I think so! I set myself some rules to make sure that the patterns I choose are sensible, and don’t leave me slipping out of my shoes. I wrote down what I wanted for the shoelace patterns
- I want the lace to pass through each hole only once
- I am only going to choose a pattern where my shoes will stay on my feet
- I want the pattern to start and finish at the top holes
With these rules I’ve created in mind I can add to my commentary:
I was working out the best way to lace up my running shoes to make sure they stayed on my feet. This got me thinking about the different ways I could lace them up and how many different combinations were possible. My trainers have 8 holes for the laces. I can choose to use only 6 of them, which will decrease the number of combinations I can choose from. I can choose to use a hole twice which will increase the number of combinations. Not all the combinations are sensible though. For example, choosing to feed the lace through only one side will cause my shoes to fall off my feet. To avoid these problems I set myself some rules. The lace can only pass through each hole only once, I can only choose a pattern where my shoes stay on my feet and my lace can only start and finish at the top holes. Setting these rules allowed me to reduce the number of possible shoelace patterns and also remove the patterns that weren’t a sensible choice
I also took another photo to show all of the combinations I had found with my rules
How can this photo be improved? I like the composition with the six shoes arranged in three pairs, but there is too much floor and not enough shoelaces framed in the for me to show off all these combinations! I decided to crop this photo for my final entry
further things to think about
There are even more questions I can think of to discover more about the lace combinations! How are the patterns the same when the shoes are laced up from the right or left first? Where are the differences? Can you have identical patterns? How can the number of patterns be reduced further? What new rules can be made? Why have patterns that are symmetrical? What patterns can be discovered using more than one lace? How many possible shoelace combinations are there? What rules do you want to have? Why?
Open to all ages with prizes in each level. You only need a mobile, the internet & curiosity! Enter on your own or as a team, mind to add the maths inside sticker, and submit in one, or in as many categories as you like. The photo should be your own, without changes, and for a chance to win, cannot be shared anywhere else. View the T&C for more information, and please do get in touch if you have any questions.
credits
This suite of resources are the fruit of a collaborative project between undergraduate and postgraduate students from the University of Glasgow — School of Mathematics & Statistics, Education Scotland, and Dr Andrew Wilson (maths inside Founder and Director)
The authors are Jordan Baillie, Nanette Brotherwood, Tanushree Bharat Shah, Lucas Farndale, Emma Hunter, Christopher Johnson, Harkamal Kaur, Christian Lao, Samuel Lewis, Kathleen McGill, Megan Ruffle, Yvonne Somerville, Andrew Wilson, and Yuanmin Zhu
The photos above are credited to Emma Hunter