Being stuck at home, I have really embraced crochet as a way to fill the hours. As a working engineer, I have fallen in love with the ease with which I can develop patterns, test ideas and visualise mathematical concepts using just a hook and some yarn. It occurred to me that it could be an excellent medium for children to explore mathematical concepts in an intuitive way, as well as keeping them occupied and learning a new skill. So I present: Teaching maths with crochet.

Needle and yarn craft is not often associated with the hard sciences but yarn work has a rich history and association with maths and science. Arguably the first examples of coding came from industrial weaving machines (Jaquard Loom) and fibre art has historically been a fabulous medium for representing academic concepts, starting with Daina Taimina famously using her crochet hook to create a realistic model of the hyperbolic plane which was difficult to represent with the paper models of the time (Daina Taimina) (Taimina's Hyperbolic plane pattern). The Crochet Coral reef project merges fibre art, mathematics and natural science to create a compelling and engaging record of the natural world. Fantastic work is going on around the globe to visualise with knitting and to understand its unique properties (Knitting is Coding)

Crochet is a marvelous medium for taking academic concepts and turning them into concrete geometry that you can touch and feel. It is easy to pick up, relaxing to do, uses very accessible materials and is so easy to undo and redo- promoting experimentation. So why not combine art and maths and use crochet to explore basic and more advanced mathematical concepts?

This Instructable will work you through a handful of basic crochet stitches, teach you how to make a simple granny square and explore the points at which these granny squares provide concrete examples of maths theories.

How it works:

I have tried to make this Instructable suitable for a range of abilities and experiences.

There are simple videos for each step which show the stitches and pattern sequence, a break down of the pattern visually, as well as freeze frames highlighting particular points of interest and simple written instructions. Pick and choose which level of instruction suits you.

At the end of the written instructions I discuss the maths concepts explored, breaking it down into basic concepts for young children, more complex activities for kids in primary school, and some advanced concepts that should relate to high school maths. Of course this is just the starting point, so the last step includes a series of ideas to for taking these concepts further and inspiring independent learning and discovery.

If this is a hobby you enjoy, I strongly recommend continuing the discovery on Youtube and maybe picking up a book on the subject. I have particularly enjoyed the DK Book Crochet: The complete step-by-step Guide

Supplies

- 2 colours of yarn of some kind with a little bit of thickness (wool, string, t-shirt yarn, plastic bags cut into strips)

(Its better for beginners to use a light coloured yarn, but whatever is to hand will do - granny squares are great for using up wool scraps)

- Crochet hook - try to find one that matches the wool you are using, in general an DK yarn uses a 4.00mm - 5.00mm hook. If that seems too much effort, a hook where the yarn fits easily into the crook is fine.

- Scissors - Just for trimming yarn

Step 1: First Steps to Crochet

So First thing first, how do you even crochet?

Crochet is, simply put, a loop of thread pulled through another loop, using a hook. Hold the hook in your dominant hand and thread the yarn around your non-dominant hand in a way that lets you pull the yarn from the ball tighter or looser. You can use my technique as an example, but you will find that you will have your own unique solution eventually.

To start with we make a small slip knot and thread it onto your hook. Then twist the hook so that you have the yarn (held taut by your other hand) caught around it, with the yarn nestled neatly into the crook of the hook. If this is tricky, try twisting the hook the opposite way around the yarn.

Pinch the knot part of the slip knot with your off hand (the one tensioning the yarn from the wool) then slide your hook towards your body, eventually pulling it and the yarn that was looped around it, through the loop of the slip knot.

*This can be a little tricky, so it might be worth doing this first stitch for little kids to start with*

You just Chained one! This is the foundational stitch of crochet. You can go ahead and chain 3 more times.

*Now would be a great time to kill 20 minutes just getting the hang of chaining. Make long chains, practice counting, get an understanding of how the tension of the wool from your off hand affects the size of your stitches. This is a particularly good idea for little kids because they haven't quite got the fine motor skills to do this super easily*

Now that you have 4 or 5 chains done, its time to join them into a ring, using a slip stitch. This is just like a chain stitch, only you push your hook through the first chain you made and then pull the yarn you gathered through both the first chain and the loop on your hook. Easy! So that's the first bit of crocheting, we will look at a new stitch or two and how to make more than string and loops in the next step.

Maths:

All Sized Kids

- Tension: Now is a great time to gain an inherent understanding of tension (pulling something from both ends). Experiment with making chains when holding the yarn really tightly and really loosely.

- Properties of the material: If you have different sized yarns, chain a string of 10 with each different sized yarn. See how they compare in length and width. Experiment with different sized crochet hooks, and how they interact with the different sizes of yarn. See how an inelastic material such as string becomes something bouncy and stretchy when crocheted.

Step 2: Basic Granny Squares, Number Patterns and Algorithms

Granny squares are a relatively simple crocheted items (sometimes called motifs) that can be joined together to make big pieces of crocheted work. They are worked in the round, which means instead of in rows, you create the stitches to follow a circle, working counter clockwise if you are right handed.

We made the circle we need to start the granny square in the previous step.

Now, after the slip stitch, chain 3. This acts as the first treble crochet (even though it isn't, but it is the right height).

Now we learn a new stitch: The treble (UK crochet term).

This stitch is made by looping the yarn from your off hand around the crochet hook, like you would to make a chain. But now, instead of pulling the yarn looped around the hook through the loop on the hook, we stick the crochet hook through the centre of the chain circle that we made, and twist it round the yarn again and pull back out of the chain circle. We should have two twists of yarn on our hook now. Catch one more bit of yarn from your off hand with the hook and pull through two of the loops on your hook. There should be two more still on the hook. Grab some yarn and pull through the two remaining loops. Done!

Do this one more time, then chain 3. Make three more trebles, chain three, three more trebles, chain three, 3 more trebles.

Now chain three and use that handy slip stitch to connect the chain on your hook into the top of the first stitch you made this round.

At this point you can tie off your wool (make a chain, snip the yarn coming from your off hand, and pull the now loose yarn until your chain chain stitch becomes a knot.)

You can add a new colour by making a slip stitch through the gap made by your chain stitches. You don't need to worry about the loose end right now, just try to build your stitches over it, like in the video. If this doesn't work, you can weave it in with a needle at the end of your work.

Now chain three, add two trebles, chain three and then another 3 treble, all in the "Chain Space" also known as the gap between two of the bobbles you made before. This defines the corner. Keep going, adding your corner stitches in each chain space, and finish with a slip stitch when you reach the start.

Change colour again if you want. Do the same for round three as for round two, except now there will be a chain space between your corners. Add one set of three treble crochets in this space. Add the two sets of treble crochets separated by three chains at the corners you created in round two.

This is the end of your first basic granny square. Keep going and add rounds if you want to create a bigger shape.

Maths:

All sizes:

There are so many more stitches than just the chain, treble and slip stitch. Look up single crochet, double crochet, double treble etc and try making granny squares using these stitches in your bobbles. Try to guess what it will do to the size of the square. What happens if you use trebles in one round and doubles in another? What happens if you change sizes of yarn or hook between rounds?

Little sized kids:

- Look at the number patterns we achieve if we count the bobbles in each round. Can you guess how many will be in the next round? Crochet and find out!

Middle sized kids:

- This is a number pattern. See how each row goes up by the same number (find the difference between the numbers of bobbles per row by subtracting). Can you write a basic formula for this relationship? Number of bobbles in a row = Number of rows x 4. Predict the number of bobbles in a 5 round and a 6 round granny square.

- Once we move on to making new shapes (Step 4) what is the relationship between the number of rounds and bobbles in each round for these shapes?

- Area: How much height and width does an extra round add to the square? Can you guess how wide a 6 round granny square will be?

Big sized kids:

-These number patterns are called Arithmetic sequences. They are defined by the mathematical formula Tn = (a x n) + d. n means the number of rounds, Tn is the number of bobbles in a given round (if n = 2, Tn is the number of bobbles in round 2), a is the number of bobbles in the first round and d is the difference in the number of bobbles between the first and the second round. Use this formula to guess how many bobbles will be in the 8th round.

- Try to write a new arithmetic formula for the shapes you make in Step 4.

- Is there any pattern to the total number of bobbles? For round one we have 4 total, in a two round square we have 12, in a 3 round square 24. Is there a way to write a formula for this (hint, it won't be an arithmetic sequence, just write a standard formula).

Step 3: Using More Than One Colour, Understanding Fractions

Sometimes we want to make colour patterns that aren't based around rounds. We can do this by using two coloured yarns at the same time.

I will describe the more complex colour break up that we are creating: Quarters. To make halves, just change colours less often.

Begin by making your circle chain and the first bobble. Chain two. Drop your first coloured yarn (colour A) from your off hand, and pick up colour B. Use colour B to complete the third chain, just by pulling it through the loop on your hook as you normally would. Make the first two trebles of your second bobble. As you do this, capture the colour A yarn within your stitches (hold colour A against the chain that you are crocheting around, so that it is also crocheted around). This brings the other colour with you as you work around.

When you get to the third treble in colour B, only pull your yarn through the first two loops. Then switch to colour A to complete the last two loops. This stops weird bits of the wrong colour showing through where you don't want it to.

Continue around the round, swapping colours until you reach the start. You should have two bobbles of colour A and two of colour B. Each pair of colours should be opposite if you are making quarters.

When you make the slip stitch to finish round one, try to do it so that you pull the loose colour up with you (hook it up with the yarn you are making the slip stitch with). This means it will be in the right place when you start round two, but it doesn't super matter, so long as you keep crocheting over your unused colour as you start the round.

Continue with round two. You will notice that the best place to swap colours is at the corner, so each corner should have three trebles in colour A, chain three in colour B and three trebles in colour B.

Work up as many rounds as you want.

Maths:

All Sized Kids:

- See how many different colours you can use to make a basic granny square. 1, 2 and 4 colours give you even splits, 3 colours won't create even colour blocks. The number of triangles that you can split a shape into tells you how many factors that number has. 4 has three factors, 1 (whole colour), 2 (even split with two colours) and 4(4 colours or two alternating).

- When making different shapes in the next step, see how many colour combinations you can make. How many even wedges can you split a six sided shape into? What about an eight sided shape?

Little Sized Kids:

- See that every shape can be made of one colour or as many colours as there are sides. That is because 1 and itself are factors of every number.

Middle Sized Kids:

- Some shapes will only split evenly with one colour or the same number of colours as its sides. These are Prime numbers. They are only divisible by themselves and one, unlike other numbers which have more factors. How many shapes can you make that have a prime number of sides?

- Make fractions. One half is easy, but how about showing one quarter, or three quarters. Can you use one colour to represent the missing portion of the fraction and another to show the portion you "have"?

Step 4: Beyond the Granny "Square", Understanding Shapes and Algorithms

We don't have to just stick to the basic square. Given our understanding of how to build corners using crochet, we can start altering the shapes we make by adding or subtracting corners. This is the really fun part of this exercise, when you get to have an idea, come up with a plan and execute. There will undoubtedly be a lot of unraveling and starting again, but that is part of the joy of discovery.

You will find that as the shapes get bigger you may need to add more chains to your initial ring to make room for the increasing number of bobbles. You may also need to reduce the number of trebles per bobble, and the number of chains at each corner, to keep the shape smooth and nice.

Maths:

Little Sized kids:

- We define shapes based on how many sides and how many corners they have. If a square has four corners, can you make a triangle by only crocheting three corners? What other shapes can you make?

Middle Sized Kids:

- Can you write a formula that links number of sides of a shape to the number of corners?

- Explore angles. Because each granny square is based around a central circle, the wedges that make up each shape each represent a portion of 360 degrees. As the number of triangles gets bigger, it makes sense that the angles need to get smaller and the wedges get thinner. Find out the angles by dividing 360 by the number of wedges.

Big Sized Kids:

- Crochet patterns are just like coding - they are instructions that you write down, give to someone else and which they use to create the thing you wanted to make. Can you code the pattern for a pentagon, a hexagon and a heptagon?

- When you build new shapes, do you have to adjust the number of stitches anywhere? How can you stop the finished shape getting "frilly". Altering the size and number of stitches, and the number of chains will give you different looking versions of the same shape. Try to make irregular squares, pentagons etc (they may end up looking rather strange).

Step 5: Joining Everything

Because eventually there will come a point where just making little scraps of crocheted whatevers gets boring, here is a quick way to join your motifs. It is a super duper basic joining technique based on the slip stitch.

Simply hold two pieces of crochet next to each other and push your needle through an edge stitch on both pieces. Grab some yarn, pull it through your crocheted pieces and the loop on your hook. That's a slip stitch through two pieces!

Keep going until you reach the end of your piece. You will notice that this creates a pretty serious seam on one side, and a less noticeable one on the other.

Maths:

Middle and Big sized Kids:

There are many more joining techniques. Look some up online and experiment. Try reading the crochet code (pattern) and turn it into a new type of seam (decoding it).

Step 6: Infinity and Beyond

Now that the basics of crotchet and granny squares has been learnt, here are a bunch of extension activities to try.

Little Sized Kids:

- Create a fraction scarf by joining together granny squares representing 1 whole, 1 half, 2 halves, 1 quarter, 2 quarters, 3 quarters and 4 quarters. Use the same colour for the missing pieces. Repeat to the length desired.

Do you notice how 2 halves, and 4 quarters look the same as a whole?

Middle Sized Kids:

- Make 3D shapes out of crocheted pieces. Make a pyramid, a cube etc.

- Make a tessellated quilt. Try to use several shapes to create a complex tessellation.

Big Sized kids:

- Can you make a crocheted D&D dice set? How will you mark the faces? What shapes do you need to make. Create a plan ahead of time using drawings.

- Make a prime number quilt (Prime number jumper)(Prime Number Blanket). What shapes will you use? How many colours do you need? How will you represent each factor?

- Use crochet to write in binary. Make a long chain first, then crochet on top. Use a treble crochet to represent one, a chain stitch to represent zero. Look up how to write the numbers 1 to 10 in binary. Make a bracelet with your phone number on and give it to a friend. How will you separate between numbers?

- Try to crochet the hyperbolic plane that Daina Taimina created, following the code published here: (Taimina's Hyperbolic plane pattern). Can you find other instructions to crochet mathematical shapes, or proteins or DNA?