Something that has captivated me from the Maths Twitter Blog-o-Sphere (#MTBoS) in the last few months is Quarter The Cross.
It is a classic low-threshold high-ceiling task. And, the more you experiment, the deeper and richer your mathematical understanding becomes. This can lead to some quite sophisticated and beautiful solutions.
The solutions above are six of mine. They are rather mundane when you look over the twitter hashtag #QuarterTheCross. In particular, my friend David Butler has produced more than 100 solutions. Check them out at his blog post. I love the connection that David draws between ways to algebraically and visually express fractions.
It delights me every time I see David’s work shared on Twitter, including images of use in classrooms.
— David Butler (@DavidKButlerUoA) April 12, 2016
Fraction Talks fosters this same creative thinking. Curated by Nat Banting, and with its own twitter account (@fractiontalks) and website (www.fractiontalks.com), it’s a marvellous collection of images for which the question ‘which fraction is shaded’ can be asked. Of course, the questions can go much deeper.
I included the following Fraction Talks image in the first assignment for the content course I teach for maths pre-service teachers (primary and middle). You can read my prompts here. The focus at university is often on ‘helping’ these students fill in the gaps in their procedural knowledge. A task like this is reasonably novel so I was curious as to how students would approach it without much direction. (Note that I’ve added the labels for convenience.)
‘Without measuring, explain what fraction of the large square is represented by the smaller shape labelled A.’
Of the 50 students who gave an explanation:
- 19 (38%) explained it was a quarter of a quarter.
- 18 (36%) divided the large square into 16 smaller squares.
- 4 (8%) noted that the side length of A fits four times along each side length of the larger square, and hence it’s 1/16 of the larger square.
- 6 (12%) found the fraction represented by B, and reasoned that A is half of B.
- 3 (6%) used other shapes in some way to find A.
‘If the area of C is 4 units2, what is the area of the large square?’
- 37 (74%) reasoned that 4 = (3/16)*total area, and solved for total area.
- 5 (10%) used proportional reasoning with C to find the area of 1/16, and then the total area.
- 5 (10%) used proportional reasoning with C to find the areas of all other shapes, then added them all together.
- 2 (4%) worked out how many times C fits into the large square.
- 1 (2%) used three Cs to make a square with area of 12 units2, found the length of the side of the square with sqrt(12), reasoned the length of the side of the large square, then found the area of the large square.
‘Shade exactly half of the large square. Explain how you know that this is exactly one half.’
I was expecting students to use the given shapes to shade exactly one half (eg D+B, A+2B+C) so I was surprised that many students ignored the shapes and shaded ‘across the middle’ of the large square.
However, many were able to reason that this is one-half:
- (Left image): 3B+A+(1/3)C = 1/2
- (Right image): 2B+(2/3)D = 1/2
I love the different reasoning that students deployed to tackle this question. Next offering I’ll choose a more varied fraction talk (including triangles) to draw out further reasoning.