The students used a variety of different methods and arguments to find and justify their answers. There were some differences of opinion and some heated debate. It took a lot of persuasive power to reach agreement.
In the end they decided that these were the fractions represented by each shape :
Shape no.1 : 1/8
Shape no.2 : 1/16
Shape no.3 : 3/16
Shape no.4 : 1/16
Shape no.5 : 1/8
Shape no.6 : 2/32 or 1/16
Shape no.7 : 3/32
Shape no.8 : 1/32
Shape no.9 : 1/32
Shape no.10 : 1/64
The students commented that it was easier to find the answers than to write up solutions. Several students were able to write partial solutions, transcribed below.
Shape no.1
M said : The no.1 shape is obviously 1/8 because if you look at half the shape you can see that splitting it into quarters leaving four 1's. Therefore, putting the halves together makes it 1/8.
KR said : Shape no.1 is 1/8 because if you look at the whole rectangle no.1 is in ¼ so you just half ¼ = 1/8
Shape no.2
M said : As we already know, the 1's are all 1/8. If you filled the whole box with 2's, it would fit 16 because they are half the size of a 1.
KR said : Shape no.2 is 1/16 because its half of shape no.1
Shape no.3
M said : As we can see, two 1's take up the same amount of space as a 2 and a 3. The 2, we know, equals half a 1, so then three 2's go into a 3, and because a 2 is 1/16 a 3 is 3/16.
KR said : No.3 is 3/16 because if you stack three of no.2 it will =3/16.
Shape no.4
M said : 4 = 1/16 because if you flip up the shapes on either side it fits perfectly, therefore it is basically the 1 cut in half and that means its half the fraction.
KR said : No.4 is also 1/16 because if you deform no.2 and squiez it in no.4 it will make that shape.
KH wrote up a complete solution for all shapes, and even checked at the end that the fractions added to 1. I have attached this as a pdf document.
Thanks for another argument-provoking maths problem!
Gina Meyers, coordinating for the 5/6 Maths Challenge Group