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Filling bottles II

This task is about showing what fraction of a bottle is filled.

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.

Teaching and learning:

Using a milk bottle to explore fractions is similar to using a number line in that they both deal with length (1-dimension). Milk bottles were chosen because most students are familiar with fractional discussion about how full the bottle is, e.g., "The bottle is half empty", "There's about a quarter of a bottle left" etc. Also, students tend to find the apparent 2-dimensional nature of the milk bottle easier to visualise and break down into parts (partition).

Diagnostic and formative information:

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)	Part-whole 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.

Next steps:

Students may "guess" where the milk level is so it is important to encourage students to :

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 part-whole relationship. Students could:

investigate what the top and bottom numbers actually mean;
explore fractions of 2-dimensional 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.

	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)

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Filling bottles II

Filling bottles II