Filling bottles II
a) Shade this bottle to show it is \(1 \over 3\) full.  b) Shade this bottle to show it is \(1 \over 5\) full. 
c) Shade this bottle to show it is \(3 \over 4\) full.  d) Shade this bottle to show it is \(2 \over 3\) full. 
Y4 (11/2006)  
Student responses were coded into regions on the milk bottle as follows: • Region A – acceptable degree of accuracy • Region B  less accuracy [insufficient degree of accuracy] • Region C – correct side of a half • Region D – incorrect side of a half [incorrect]. The marking template on the next page can be photocopied onto acetate to help mark the accuracy of the students' work. 
a) difficult b) moderate c) difficult d) very difficult (for shading to region A) 
The most accurate answers were achieved by students who showed some evidence of how they partitioned the milk bottle into equal parts before marking their answer.
Common error  Likely misconception  
a) & b) 
Draws a mark in region D (^{1}/_{3} < ^{1}/_{2}) or (^{1}/_{5} > ^{1}/_{2}) 
Partwhole misconception of fractions This could indicate that students don't know how to show a fraction as a part of something. They may not be aware what the "whole" is, and are trying to find a fraction of some other "part", e.g., finding 1 2 of "half" the milk bottle. 
c) & d) 
Draws a mark in region D (^{3}/_{4} < ^{1}/_{2}) or (^{2}/_{3} < ^{1}/_{2}) 

a)
b) c) d) 
Draws a mark in regions B or C ^{1}/_{3} < ^{1}/_{2} ^{1}/_{5} < ^{1}/_{2} ^{3}/_{4} > ^{1}/_{2} ^{2}/_{3} > ^{1}/_{2} 
Lack of method to ensure accurate representation of fractions Students may know that ^{3}/_{4} > ^{1}/_{2} (or ^{2}/_{3} > ^{1}/_{2}), but they don't have a strategy to identify how far up the milk bottle the fraction is. 
 explain how they worked it out;
 explain how they know their method works (critique their answer and strategy);
 show this using diagrams, symbols or writing.
Students who indicated Region D may also need more exposure to understand that the top and bottom numbers in a fractional number show a partwhole relationship. Students could:
 investigate what the top and bottom numbers actually mean;
 explore fractions of 2dimensional shapes (and partitioning if required);
 try some simple fractions on a blank milk bottle, asking:
"How far is half way up the bottle?",
"How far up the bottle is this? [Indicating simple fractions]".
Students whose responses lay in regions B or C may need to clarify what "whole" they are finding the fraction of.
 Investigate how they know that is where the mark is supposed to be, and how they could check that this is correct.
 Encourage them to show how they constructed the answer, e.g., partitioning the milk bottle into equal parts and then adding each unit fraction to make the fraction shown.

If students are having difficulty, use questions like
"If the bottle was half full where would the mark be?", and "… a quarter full?",
"How far up the bottle is ^{1}/_{3} ?", and build up to ^{2}/_{3} .
Students whose responses lay in region B should be encouraged to show how they could check the accuracy of their answer. It may be an issue of needing more care when partitioning evenly.