At the end of last year, the MTBoS (Math(s) Twitter Blog-o-Sphere) introduced me to this very interesting task: you have a cross made of four equal squares, and you are supposed to colour in exactly 1/4 of the cross and justify why you know it’s a quarter. I call it “Quarter the Cross”.
(Apparently, this problem originally came from the book “Great Assessment Problems”, by T. Dekker and N. Querelle, which you can find online at http://www.fi.uu.nl/catch .)
I was hooked from the moment I saw it. It had so many many possibilities for maths thinking and argumentation and creativity too! I couldn’t wait for the December session of One Hundred Factorial to try it out with others. And I was not disappointed: we all learned so much from each other and were inspired by each other to new heights of creativity and wonder.
Since then I have been trying to write a blog post about it, but every time I try, I get inspired one more time and end up working on new solutions for several days. Along the way I’ve learned a few brand new things; I’ve relearned a few things I’d forgotten; I’ve seen things I knew with new eyes and understood them better; and I’ve made brand new maths (for me anyway). And all of these intersect and combine to make even more new learning. How can I possibly organise all of this into a blog post?
So I’ve decided not to try to organise it in any particular order. Instead I am going to show you a collection of solutions, and a collection of ideas I used to make them. My hope is that you can enjoy going on a similar journey to mine.
If you do make any more solutions or have creative ways of explaining why mine work, you can tweet them with the Twitter hashtag #QuarterTheCross — I’d love to see anything you come up with!
UPDATE: I wrote a later post on how I implemented this in a classroom. Over there I have links to the Word documents with blank crosses to print for students, but you can get them direct here: here is the small version with five crosses, and here is the big version with one cross, both with a whole number of centimetres for the squares’ sides.
This is a picture of 100 of my solutions. In each, a quarter of the cross is coloured in red. You can explore a prezi of them here. A high-resolution picture can be downloaded here (PNG 4MB) and the original SVG file here.
Ideas used to construct my solutions
- Fraction addition and subtraction.
- Halving retangles and triangles using symmetry.
- Adding a bit to a shape and taking the same bit don’t change the area.
- Moving the apex of a triangle parallel to its base doesn’t change the area.
- Shearing a shape doesn’t change its area.
- Making a shape half as high and twice as wide doesn’t change the area.
- Shrinking all the lengths by half makes a shape with a quarter the area.
- The formula for the area of a trapezium.
- The formula for the area of a crossed trapezium.
- The Lune of Hippocrates.
- Pythagoras’ theorem.
- Rotational symmetry.
- The sum of a geometric series.