As well as my story of how we “discovered” pi, the carnival features a bunch of great maths ideas from the early and primary years right up to advanced mathematical adventures.

Do you remember learning to calculate the circumference, diameter and area of a circle? If your experience was anything like mine, you were given this random 3.14 number (which for some reason the teacher kept referring to as pie), and told to plug some other numbers into a formula which you were then expected to memorise for the test.

Not exactly magical.

I wanted my daughter’s introduction to the mysterious properties of circles to be different. So last week I set up an activity to help her discover the magic for herself.

What we did

I gathered a variety of flat circular objects (mostly lids), a ruler and some string, and was beginning to take some measurements in preparation for introducing the activity to C(9), when she came over and asked what I was doing (I love it when an activity starts that way).

I told C(9) that I was wondering whether, if I knew how much a big wheel measured across the middle, I could (without measuring) figure out the distance around its edge. I said I was starting out by measuring some smaller circles to see if I could find any pattern.

“Can I help?” she asked, grabbing a ruler and a lid.

Making a chart

As she worked, C(9) recorded her measurements in a chart.

Once she had four pairs of measurements, we were ready for the next stage. I asked C(9) what our goal was, and wrote down her words: “To find out what the length across the middle is in relation to the length around the edge.”

Finding a relationship

C(9)’s first suggestion was that we calculate the difference between the two numbers in each pair of measurements.

But no pattern emerged, and she wasn’t sure where to go next.

Session 2 – Functions

I began our next session by reminding C(9) of our goal: “To find out what the length across the middle is in relation to the length around the edge,” and asked whether she’d had any thoughts about what to do next (no).

Then I asked her if she remembered learning about functions in Life of Fred. I wrote out some pairs of numbers and asked her to guess the function and apply it to a new case (I used “add 2”). Then she did the same for me (she chose doubling), and we did a few more.

Session 3 – Algebra

After our play with functions, C(9) returned to trying to find a relationship between the numbers in her chart. She suspected multiplication was the key: “15.5 times something is 50. But how do we know what the something is?”

By chance, the previous week C(9) had learned how to balance simple algebraic equations (we’d picked a random chapter from Primary Grade Challenge Math), so I suggested that we try using algebra to calculate the missing something.

This may seem an unnecessarily complicated step, but given what we’d been doing recently, balancing equations was the easiest way for C(9) to see that if we want to know what we multiply x by to get y, we have to divide y by x.

She then solved the equation, first by estimating and then with a calculator. The first missing number came out as 3.2. (At this point we had a little recap of decimal place value and rounding, and I reminded C(9) that our initial measurements had been rough and ready, using string.)

C(9) did the same for each pair of measurements, and got h=3.2 for all but one set, which came out as 3.4. (I’m not sure why the numbers were so consistent, when pi rounds to 3.1. But I guess it’s not too big a margin of error.)

I was pretty excited at this point, but managed to keep quiet because although C(9) could see the pattern, she wasn’t exactly jumping up and down yet.

We decided to test our newly-found relationship between the lid’s “middle” and “edge” on a new circle. C(9) drew a circle with compasses and wrote out the formula, “middle x 3.2 = edge”.

She measured the circle’s diameter, multiplied it by 3.2 and wrote down the answer, 23. Then we measured around the circle using string and a ruler – 23cm! C(9) was genuinely gobsmacked, like we’d just performed a magic trick – what a great learning state!

Session 4 – Introducing “pi”

In our next session, C(9) and I read Sir Cumference and the Dragon of Pi, in which a young knight has to solve a puzzle involving the relationship between a circle’s diameter and circumference, in order to save his father’s life. After the knight solves the puzzle, the book talks about how the Greek letter pi is used to represent the mathematical constant which is the key to that relationship.

Many paths to pi

I’m sharing this story not because I think what we did is the only way to teach a child about pi. I’m a near-beginner at this living maths business, and I’m sure a better mathematician could have guided C(9) through the process of discovering pi much more efficiently.

I’m sharing what we did because we both learned so much as we happily worked together, and I’d love for others to experience that joy.

If I’d thought in advance about teaching C(9) everything she learned as we did this puzzle, I’d probably never have got round to starting, and who knows how she would have reacted if I’d listed all the concepts she was going to use beforehand. But as we worked though our puzzle, C(9) had a reason to learn each new skill, and I had real examples to work with to teach her.

When the time is right to teach J(8) about pi, I have no doubt our path will be different, with tangential learning that fits his needs – I’m looking forward to it.

Has your student “discovered” pi? I’d love to hear of the learning route you took.

The book tells the story of Radius, son of Sir Cumference and Lady Di of Ameter, who sets off on a quest to earn his knighthood. He takes with him a family heirloom – a circular medallion with mysterious numbers around its edge (the book comes with a cardboard copy of the medallion).

“What are these numbers around the edge of the medallion?” Radius asks. “No one knows,” Lady Di answered, “but may it bring you courage on your journey.”

During Radius’s quest we discover with him how to use the numbers on the medallion to measure right, acute and obtuse angles. With the medallion’s help, Radius succeeds in navigating a path through the perilous maze to complete his quest.

Maths Playtime

C(9) actually jumped ahead and read the book while I was working on something else with J(8)*. I came over to find her playing with the “medallion” (protractor). She’d drawn around it and marked 0, 90 and 180 degrees.

We talked about acute and obtuse angles, and I asked her what we might also call the 0 point (“360”.)

Then I pointed to the six o’clock position on the circle and asked how many degrees round that would be, counting clockwise from zero. I left the room to transfer some washing to the dryer – our “maths lesson” wasn’t meant to have started at this point. 😉

A few minutes later she came and found me with the answer – “270 degrees”. I asked how she’d worked it out. Finding out how a student’s minds works is such a valuable part of the mentoring process.

She told me she’d measured a degree with a ruler and found that “this amount” on the protractor [ten degrees] was the same as “that amount” on the ruler [a centimetre]. By working around the circle she’d found the answer. “Then,” she continued, “I realised there was a pattern – you add 90 each time you go round a quarter of the circle.”

I congratulated her on thinking like a mathematician!

If we used a formal curriculum, I’m sure my third grader would have “learned” about acute and obtuse angles by now and maybe even used a protractor to measure them. But what I love about this approach is seeing her sheer joy at figuring it all out for herself.

Hearing her animated explanation of how she’d solved the puzzle showed me without doubt that she really understood the concept. It also gave me valuable insight into her learning process, which is quite different from her brother’s.

If I’d asked J(8) the same question, I’m pretty sure he would have come straight out with the answer “270” – but he wouldn’t have been able to explain how he found it. Not having to “show your workings” when your brain doesn’t consciously do workings is one of the joys of homeschooling for the right-brained visual-spatial learner. Teaching J(8) to “backwards-engineer” and thus extend his thinking process (as well as pass exams, later on) is one of my long-term goals.

A final indication that C(9) took ownership of what she learned was that she decided to make a notebook page about what she’d learned for her maths journal.

*Incidentally, while C(9) was teaching herself about angles, I was helping J(8) understand the steps of long division using Life of Fred (Honey) – at his request. I love how a living maths mentoring approach means I can help each of my children learn in the way that’s right for them. (Which might be a different way next week – there’s never a dull moment!)

Tessellation is about regular patterns that split the plane up into lots of little tiles which fit together perfectly, without overlapping or leaving any gaps. Tessellation is fundamental to maths, because it’s all about symmetry.

We started with a cardboard square each (ours were about 5x5cm). We talked about how we could cover a page with squares without leaving any gaps.

First we cut a piece from the bottom of our square. We were careful not to cut the corners off, and we found it easiest to cut from corner to corner (to avoid having to measure where to reattach the cut piece on the other side). We slid the cut-off piece upwards, and attached it with tape to the top edge of the square.

Then we did the same on the left side of our square. We cut a piece out, slid it along to the right side, then reattached it.

I asked the children if we had added any cardboard to our shapes, or taken any away (no). We agreed, then, that our shapes should take up the same total amount of space as our original squares.

We traced around our shape on a blank piece of paper, then carefully moved it along and traced around it again. And again, and again until we’d covered the page.

Our tessellations looked so pretty, we decided to paint them.

J(8)’s didn’t cover his paper without gaps – he was adamant he wanted to create his art his way – but he understood the idea!

The artist M.C.Escher used tessellation to create amazing art. This BBC video clip is excellent!

Mathematicians know that their subject is beautiful. Escher shows us that it’s beautiful.

After I linked up our Pythagoras and the Knotted Rope activity at the Hammock Tracks Homeschool Review, Savannah offered to interview me about our full-time living maths approach for this week’s Homeschool Review. I jumped at the chance – I love talking about maths!

Tell me more about this full time living math approach

My children (aged eight and nine) don’t use any formal maths curriculum. Instead, we have a living maths routine. The move away from curriculum was gradual. I’d always liked living maths – the fun my children have with it, and how it gives them a sense of maths in the real world – but in my head “real maths” was the curriculum, and living maths was an extra. And we all know what tends to happen to “extras” in a busy homeschooling household!

Then I read Denise Gaskins’ book Let’s Play Math, which gave me the confidence to flip the balance. Let’s Play Math is one of those precious books which is both inspiring and practical – it makes you want to change, and tells you how to do it.

Here’s how our routine looks:

Monday – maths games like KenKen, Shut The Box or Yahtzee to practise arithmetic and maths facts.

Tuesday – oral story problems. We grab a whiteboard and take turns making up problems for each other. They learn from watching me solve their (usually very convoluted!) problems, and I learn how their minds work from seeing how they approach each problem. It’s a great opportunity for me to model, and the kids to practise, how to use notes and diagrams to solve real maths puzzles.

Wednesday – maths literature. We read aloud from a living maths book – maybe a mathematician’s biography or a maths picture book. I allow time afterwards to play with the ideas we’ve heard about. For example, when we read What’s Your Angle, Pythagoras we knotted ropes to make our own right-triangles and proved Pythagoras’ Theorem using Lego.

Thursday – manipulatives and hands-on geometry. Recently we’ve played with pattern blocks and tangrams, made geometric shapes with toothpicks and mini-marshmallows, and used isometric graph paper to make Maori taniko designs when we were studying the history of New Zealand.

Friday – children’s choice of any of the above.

As I was writing this, I put the question to my nine year old daughter:”Tell us more about this full-time living math approach.” Her reply: “We do more real life maths and story problems, which are really funny because you can make up extremely crazy things. And often we find maths in real life.”

What do you see as the benefits to this learning style?

Seeing my kids enjoy maths is very important to me, but in itself that wouldn’t be enough to satisfy me that a full-time living maths approach is right for our family. What does convince me is noticing my children beginning to think like mathematicians… Read the rest of the interview at Hammock Tracks

Now we’ve switched to a full-time living maths approach, we’re actually making time to play with some of the wonderful resources we’ve had on our shelves for years.

What’s Your Angle, Pythagoras?

On Friday we read What’s Your Angle, Pythagoras, a picture book which tells the story of how the young Pythagoras learns how to make a right-angled triangle using knotted rope, and discovers how to calculate the length of its hypotenuse using square tiles.

Obviously the book is mostly fictional, and it takes some historical liberties – the boy Pythagoras visits Alexandria, for instance, several hundred years before the city was built! – but these are discussed at the back of the book in a way that made my kids laugh and was a handy review of Ancient Greece and Alexander the Great.

How to make a right-angled triangle using rope

In the book, the young Pythagoras notices what happens when buildings are constructed with less-than-accurate right-angles. On a trip to Alexandria with his father, he learns how the Egyptians use knotted rope to overcome this problem.

We tried it out for ourselves. We tied eleven knots at equal distance along our rope before joining the ends in a final knot, so that we ended up with twelve short lengths of rope between each knot.

Then we used our rope to make different shaped triangles. We counted how many lengths of rope were on each side of each triangle.

To make a right-angled triangle, we found that we needed the sides to be 3 lengths, 4 lengths and 5 lengths of rope respectively.

(Top Tip: Take care to make the knots evenly spaced. C(9)’s rope worked perfectly for making right-angled triangles, whereas the one I helped J(8) make didn’t, oops!)

Using Lego to demonstrate the Pythagoras Theorem

While playing with floor tiles, the young Pythagoras in the story discovers that if he makes a square along each side of a right-angled triangle, the square on the longest side uses the same number of tiles as the other two sides’ squares put together.

We tried this for ourselves with 2×2 Lego bricks.

Pythagoras uses what he has learned to work out how long a ladder is needed to reach the top of a wall. He also helps his father calculate the sailing distance to Rhodes. Both excellent demonstrations of the usefulness of maths!

I would never have thought to teach my kids the Pythagoras Theorem at the ages they are (8 and 9) – all we did was read a picture book. But that living book inspired us to play, and before we knew it we were formulating mathematical proofs. Another living maths success!

We learned how Fibonacci brought Hindu-Arabic numerals to Europe, which had until then been using Roman numerals. Here we paused to talk about place value and how much harder it must have been for kids to do written arithmetic without a zero!

Next we puzzled over Fibonacci’s famous rabbit problem. (In short, if a pair of rabbits has two babies every month, how many rabbits do you have at the end of the year?)

J(8) got overwhelmed and ran off to the trampoline at this point. But I was delighted that C(9) – who also has “if I can’t do it perfectly straight away, I’m outta here” tendencies – stayed with the puzzle long enough to spot the pattern which gives us the Fibonacci Series. (J(8) will be ready for this level of engagement and reasoning in his own time!)

Day 2 – Fibonacci Numbers in Nature

We’d read in Blockhead how the Fibonacci Series is found throughout nature, so on our walks for the rest of the week we looked for examples.

Most daisies, for example, have thirty-four petals (a Fibonacci number).

(Top tip: don’t split the petals, thinking they’re two. The first time I counted fifty-nine. Next daisy, I carefully kept each petal intact and I got thirty-four exactly.)

Daisies have 13 (easier to count) sepals (another Fibonacci number).

Fibonacci numbers are found in so many places besides plants – they crop up everywhere, from fine art, to galaxies, to pineapples. What a lot we still have to explore!

Day 3 – KenKen Puzzles: Arithmetic and Logic Practice

KenKen – Japanese for “cleverness” – is an arithmetic logic puzzle invented by a Japanese maths teacher. It’s a similar to Sudoku but the digits in each mini-grid combine together to make a given number, using prescribed operation signs. Hard to explain but once you’ve done one or two you get it!

We downloaded the free KenKen iPad/iPhone app, which allows you to start with very easy puzzles using just addition and the numbers 1-3. I can see this providing hours of maths fact practice!

Day 4 – Pattern Blocks: Exploring Symmetry and Tessellations

Let’s Play Math suggests investing in manipulatives that are, among other things, strew-able. Pattern blocks have definitely passed that test this week.

Pattern blocks give kids the chance to explore pattern building, geometric shapes, tessellation, symmetry and all that other mathematical stuff in an open-ended way. I’m looking forward to looking at these concepts in greater depth over the course of our maths playtimes.

Day 5 – Story Problems

This was the simplest day in terms of set-up, and perhaps the most fun, which came as a welcome surprise to me. All we needed was a portable whiteboard and our imaginations (and a bit of patience waiting for J(8) to finish each of his long complicated stories!).

We took turns, and I think the children learned at least as much from setting me problems (and watching me work through them out loud and on the whiteboard) as they did solving them.

Here are some of the problems we came up with:

Story problem I started with

“If our puppy Harvey can skateboard at 5 metres per second, and the playground of our home ed centre is 20 metres long, how long would it take Harvey to skate from one end to the other?”

Story Problem by C(9)

(Who has recently been caring for her first flowering pot plant.) “If you water a plant every day, it grows one new flower every three hours. But you only water it every other day, so it grows half the number of flowers. How many flowers does it grow in a fortnight?”

Story Problem by J(8)

[Brace yourself.] “A man digs a hole 5 metres deep in 24 hours. If he sleeps 12 hours a night and has two 11 minute tea-breaks a day, how deep is the hole after 10 years?”

[I gave you the condensed version. The digging man (an escaping convict?!) ended up doing so many other things, we lost track. Once we’d negotiated relevant facts, I gamely worked out how far into the Earth’s core the man had burrowed.]

Verdict on Week One

We enjoyed each of our maths playtime sessions SO MUCH.

In addition to our living maths, J(8) also asked me to read Life of Fred: Goldfish to him every day. We worked buddy style through the questions at the end of each chapter.

There was also a lot of spontaneous maths play – and not just by the children!

So where are we going with this?

My goals for this term are for C(9) and J(8) to play with maths concepts, have fun with numbers and discover a bit of maths history.

My role will be to strew interesting materials, make suggestions, read aloud and – most importantly – observe. I love quietly playing detective, noticing what each child is drawn to, what comes naturally, and what might benefit from more practice playtime.

By the end of term in July I’ll have a lot more information about how the maths playtime approach is going. Then we’ll talk over our experiences and take it from there.

Let’s Play Math by veteran homeschooler (and maths blogger) Denise Gaskins is the maths book I’ve been looking for ever since we began homeschooling.

Three things set it apart from any other maths book I’ve come across:

1. It’s incredibly readable. I found myself going to bed an hour early every night to enjoy it, and had read it cover-to-cover within a few days.

2. It’s chock full of suggested resources. These alone are more than worth the cover price. The Kindle version allows you to click straight through to the linked websites – brilliant.

3. It’s comprehensive. Combined with all the linked resources, this book is going to transform how I teach my kids maths. No more dabbling in “real maths” but then running back to the workbooks when anxiety strikes (me) – with this approach I can teach my kids to think like mathematicians without worrying about leaving gaps.

Why Learn Maths?

Why do we teach our children maths? So they can become mathematically literate adults, able to calculate their taxes and mortgages? To pass exams which will allow them to get into college or the job market? Both good reasons.

The problem is, many of us are so anxious about failing to do these things that we deprive our children of perhaps the most important reason to learn maths of all: because maths is beautiful, and fun.

The “Aha!” Factor

Humans are hard-wired to enjoy puzzles. When we learn something new, we receive a hit of the feel-good hormone dopamine. When the new information comes as a surprise, we get a double dopamine hit. That’s why “Aha!” moments, like when we get the answer to lateral thinking puzzles, feel so good.

Let’s Play Math is about cultivating the “Aha!” factor in our children.

Living Maths – Where to Start?

Anyone who surfs the educational ‘net knows that there are plenty of creative maths ideas out there. But this abundance of resources can be overwhelming. As Denise says, “It seems easier to shove a textbook across the table and say, ‘Work two pages'”, leaving someone else to make all the decisions.

Let’s Play Math cuts through the overwhelm.

Here are some of my favourite topics covered in the book:

Hands-On Maths

I use hands-on methods throughout our homeschool, but I’ve never felt very confident with maths manipulatives. (On the rare occasion I do manage to bring them in I get very excited and blog about it.)

Let’s Play Math has a section on buying manipulatives (ask questions like “is it strew-able?”, “is it worth the storage space?”) plus a section on homespun manipulatives, together with lots of ideas for using them.

Maths History

“I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history”.

James Glaisher, quoted in Let’s Play Math

What is it about maths that has parents who use living books throughout our homeschools reaching for the textbooks?

History is full of men and women so fascinated by this subject that some of them overcame extremely oppressive circumstances to find a way to pursue their passions. When we share their stories we give our children a taste of the excitement of maths.

“What a shame it is that our children see only the dry remains of these people’s passion. Worksheet exercises are the bare, abstract skeletons of what once were living puzzles.”

Denise Gaskins

Let’s Play Math suggests devoting one maths lesson a week to maths history, and offers plenty of ideas on how to choose good living maths books. There’s even a whole chapter entitled “4,000 Years of Stumpers” – puzzles that have challenged mathematicians throughout the ages.

Story Problems

Denise suggests that we might measure homeschool maths success by whether or not our children fear story problems, and the book is full of tips and resources for using story problems effectively. One of my favourite is to take turns, adults included – getting the chance to challenge Mum always goes down well in this house! Taking turns makes maths into a game.

Older Children

My two are only 8 and 9 at the moment but after reading the chapter on the “Transition to ‘Higher’ Math” I believe we can use this approach throughout all our homeschool years, including those when my children might be taking exams.

The book suggests different approaches for the teen years depending on whether a child has had a good taste of the “Aha!” factor during the elementary years. Once a teen is ready for textbooks:

“Don’t be fooled by your own experience of dry or tedious math classes: textbook mathematics is still math the mathematician’s way, as mental play. But it is no longer the play of a child dabbling in the shallows… No, this is the play of the athlete, who works hard at training and enjoys seeing his muscles grow firm, who can’t wait to test himself against a new and challenging opponent.”

Denise Gaskins

Let’s Play Math as a Supplement

I intend to use Let’s Play Math as our maths “spine” but, like Project-Based Homeschooling, I think you can do as much or as little of it as fits with your individual homeschool style. Read the book, feel inspired, and do whichever activities sound like fun to you.

Putting it into Practice

My favourite section of the book is “One Week of Real Mathematics”, which contains examples of what one week’s worth of math playtime might look like. I love having this starting point to show me what a balanced “maths diet” might look like.

My kids have a low tolerance for boredom and are very outspoken when something isn’t working for them, one consequence of which is that we threw out maths curriculum some time ago. Since then we’ve read maths biographies, played with platonic solids, and enjoyed solving story problems – with the odd workbook thrown in here and there when I get the “Argh! I’m leaving gaps!” anxiety.

I feel like I’ve been ambling in the woods – enjoying the journey but a bit anxious about where we’re going to end up and whether we’re going to reach our “destination” “on time” (whatever that means!).

I knew the well-travelled road (maths curricula) wasn’t for us, but I lacked confidence in my ability to guide my children through uncharted territory. Let’s Play Math is the map and the guidebook I’ve been looking for. With it in my hand I can’t wait to take my children by the hand and head off to explore the wonderful world of maths.

* I was not paid for this review. I bought my own copy of the book and I’m writing to share this great resource with other parents.

Our favourite resource this week was a charity shop find – the game Mastermind.

Mastermind is a logic game in which one player (the codemaker) places four differently coloured pegs (out of a possible eight colours) in a row, and the other person has up to twelve chances to crack the code.

The codemaker gives feedback by scoring each guess according to how many pegs of the correct colour it contains, and how many of those are also in the correct position.

I was amazed at how educational this game is – something I hadn’t noticed at all when I played it as a child. Playing recently my brain has at times felt quite scrambled figuring out the logic of my next guess!

One of the reasons Mastermind works so well in our house is because it’s not directly competitive. The codebreaker’s goal is to crack the code in as few guesses as possible and although theoretically the codemaker’s goal is for the code not to be guessed at all, in practice we were all rooting for the codebreaker!

We got out the popsicle sticks this week to help C(9) understand the meaning of fractions like ^{5}/_{8 }of 24.

Memorising Algorithms v. Understanding

C had no trouble calculating one eighth of twenty-four. There’s an easy-to-understand algorithm:

^{1}/_{8 }of 24 = 24 ÷ 8 = 3

But algorithms don’t facilitate real understanding.

This became apparent when we changed the numerator of the fraction. C is a bright girl and it wouldn’t have taken her long to memorise the extra step in the algorithm:

3/_{8 }of 24 = (24 ÷ 8) x 3 = 9

But I wanted her to understand what was actually happening here.

Bring on the popsicle sticks!

The equation C was working on at this point was 3/_{10} of 70 (because of the numbers involved, the answer “30” kept jumping in to her head). This was going to take a lot of popsicle sticks!

A Maths Story

We gathered 70 plain popsicle sticks. These, we decided, were 70 new children starting out in kindergarten. (Don’t ask me how this metaphor got into our homeschooling house.)

Then we found 10 coloured popsicle sticks. These were helpful grade 5 children who were looking after the kindergarteners for the day.

One by one, we distributed the seventy plain sticks equally between the ten coloured sticks, until each “older child” had seven “kindergarteners” to look after:

^{1}/_{10 }of 70 = 7

This step, naturally, involved many imaginary conversations between the children, some of whom were rather wilful!

Next step – different numbers of groups of “children” were taken “to see different parts of the school”. Of course it was important to count the number of children at each point, to make sure we didn’t lose any!

So:

Two of the groups went to see the science room. “Now then, how many children do we have here? Are you all going to fit in the science room? Let’s count. One, two, three … fourteen. Yes, you should be all right. Off you go.”

2/_{10 }of 70 = 14

Seven of the groups went to see the gym. “How many children is that? Goodness – forty-nine!”

7/_{10 }of 70 = 49

Three of the groups visited the art studio. “How many paintbrushes do we need for you all? One, two, three … twenty-one.”

3/_{10 }of 70 = 21

And so on. So much fun!

From Concrete to Abstract

C went on to complete a page of abstract problems with no trouble. As she practised, she came to recognise the algorithm as a convenient shortcut.

3/_{8 }of 24 = (24 ÷ 8) x 3 = 9

But thanks to ten minutes of popsicle stick fun, she now understood what the numbers represented.