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.
WOW…I am not there yet with my little ones…but these are good exercises to remember! Enjoy your weekend!
Thanks, Melissa!
This is a great way to introduce pi. My son and I calculated the circumference of the mini-trampoline after measuring the diameter and a few other circular objects, but I left out the mystery part. We just did an example and since he didn’t have to think he forgot the next day. I think I will try your method next time we talk about pi. I’m not sure he has it down yet. Thanks so much.
Thanks, Julie. I anticipate reviewing this plenty of times in different ways, but when it’s this enjoyable, that’s ok. A mini-trampoline is a good example of a large round object. In another of our maths sessions this week, we played Speed on the iPad – so much fun!
You posted about the technique!! That is really very useful. Thank you Navigating By Joy!!
You do make me laugh, Loraine! Thanks for your lovely comment, I appreciate it 🙂
Once again I love the way you do Math with your kids. They are going to grow up just ‘getting it’. I wish I had done more of this with Keilee!! Just wonderful.
Ahh thank you, Karen, your words of encouragement mean a lot.
What a great way to introduce pi! You rock!
Thanks, Penny! 🙂
I love how you come up with these things. This is so much more work than assigning a few pages in a workbook (like I do!). I think we NEED to do this…because I don’t really even understand Pi!
Thank you, Theresa. I think most of the homeschool maths we do is me trying to understand what I never did at school – I just memorised the procedure and plugged in the numbers. If my children would do workbooks, we would almost certainly be doing a regular maths curriculum – but they just whined so much, I was forced to come up with an alternative 😀 And I must admit I wouldn’t change anything. Sometimes I appreciate having the more difficult (but interesting) path thrust upon me!
This is wonderful. I loved seeing the progress. Did you see my post where James did some coordinate graphing with Pi?
Thanks Phyllis. I couldn’t find the coordinate graphing post, but I am so glad I went looking because I found a bunch of your living maths posts I hadn’t seen before and will be using in future! I’d love to look at the post you mentioned, if you would be kind enough to give me a link.
That is a fun way to introduce pi 🙂 We’re not there yet in our math skills, but we’re learning more every day. I remember us reading about functions in Life of Fred and talking about some of the problems. 🙂
Thank you, Rebecca. 🙂 Life of Fred is great, isn’t it? I never would have thought to teach functions in that way if it weren’t for those books.
This is great Lucinda! Tell me something? I ask this as someone who would love to do maths this way but really don’t have the confidence to let go entirely of our maths curriculum. Each time your children discover something mathematical, do you see them remembering and applying it to new situations? I have my children do all the copious amounts of repetition so they don’t forget but living maths seems a much more sensible idea…..if it works. I’m not asking to scare you, I’m genuinely interested.
Oh and thank you, thank you thank you for supporting my children’s website. I so appreciate you doing so!
I’m loving the Little House website – so inspiring!
A few things come to mind about maths. My children really didn’t like any curriculum we tried. Even with Life of Fred they groaned at the exercises at the end of each chapter. I’ve always been interested in living maths, partly because I got through school maths by memorising formulae but never really understanding what I was doing – I wanted something different for my children (and for me, this time around!).
I’ve read that even if children learned nothing but the basics of how to add and subtract before they are 12, they would quickly catch up with their peers. So I guess I decided that, on balance, this living maths experiment was worth giving a go. We haven’t been doing it long enough for me to know definitively how it’s going, but my feeling is that the children are engaging with maths very differently now. They enjoy our sessions, and are becoming more flexible problem-solvers. Maybe it’s easy to remember things when they are not learnt as abstract concepts. There will be repetition, but I suppose in more of a spiral way than total mastery of each step before moving on.
I’m reading a lot of wonderful books about this approach, and everything I read makes me more committed to it.
I’m going to post an end-of-term review of how first term of living maths has gone, so hopefully I’ll pull together some more coherent thoughts then!
Thanks for a thought-provoking question 🙂
Wonderful!
Here is a guess at why your measurement of pi came out slightly greater than the actual value: string stretches. When you lay the string around the circumference, you lay it flat. But then when you spread it out straight to see how long the circumference is, I think it would be almost impossible to avoid stretching the string at least a bit. Especially if you tried to hold it straight while using the ruler.
Aha – thank you, Denise! I will share your stretchy-string idea with C(9) 🙂 And thanks for the vote of confidence!
What a MUCH better way to learn Pi! We celebrate Pi Day on March 14 each year with measuring lids from the house and doing the formula…and making a Pi(e). LOL
Fantastic – we will have to do that next year! Thanks, Jessy!
I LOVE this, Lucinda! What a wonderful way to learn about Pi. Thank you for sharing your process — it’s very inspiring and refreshing. The discovery process is so important as a step towards self-learning. I’m looking forward to reading more of your maths posts!
Thank you so much for your lovely encouraging comment, Hwee, it means a lot!
Thank you so much for sharing this! My girl can’t quote pi to quite a few decimal places, but we haven’t done much circle geometry. I like this approach and will definitely be giving it a go!
Hi Ingi, I’m so pleased to see you here – I’ve enjoyed your blog for ages! Thanks for taking the time to say hello 🙂 Lucinda