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.