Fraction shapes II

Fraction shapes II

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about working out what fraction of a shape is shaded.
Write down what fraction of each shape is shaded.  
 
a)
fraction shape showing some shaded part
 
 
Fraction shaded _________
 
 
 
   b)
fraction shape showing some shaded part
 
 
Fraction shaded _________
 
 
 
 
  
c)
fraction shape showing some shaded part
 
 
Fraction shaded _________
 
 
 
   d)
fraction shape showing some shaded part
 
 
Fraction shaded _________
 
 
 
  
e) fraction shape showing some shaded part

 
 
Fraction shaded _________
 
 
   f)
fraction shape showing some shaded part
 
 
Fraction shaded _________
 
  
Task administration: 
This task is completed with pencil and paper only.
Level:
4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Keywords: 
fractions, part-whole fractions
Description of task: 
Students work out what fractions of shapes have been shaded.
Answers/responses: 
  Y6 (06/2006)
a) 1/3 or 2/6 very easy
b) 1/4 very easy
c) 3/4 or 6/8 easy
d) 1/4 or 2/8 easy
e) 1/5 moderate
f) 1/3 or 2/6 difficult
Based on a representative sample of 203 Year 6 students.
 
NOTE: For all questions accept as correct if students write the name instead of the notation for the fraction, but ask students to write answer as fractional notation.
Diagnostic and formative information: 

Correct responses
In questions a), d) and f), students who gave their answers in the simplest form did better overall than those who did not simplify.

  Common response Likely misconception
b)
e)		 1/3

1/3

 Visual misconception
The shape of the shaded part is presumed to be 1/3 – i.e., any shaded part that looks angled like 120° (or like a triangle for b) is 1/3 .
b)
d)
e)
f)		 1/2
2/4
1/2
2/3		 Uneven size of pieces
The shaded pieces of each shape are counted, and that number is put over the total number of pieces, irrespective of the size of the pieces.
f) 2/5
2/7
2/8		 The 2 shaded parts are counted and used for the numerator and  the size of the number of times they go into the whole shape (denominator) is estimated.
Next steps: 
Students who cannot identify the shaded fraction of the shapes may need to develop their understanding about partitioning and fractions as part-whole relationships:

Fractions as part-whole relationships
Students who constructed the name of the fraction by merely counting up the shaded pieces and putting that number over the total number of pieces, need to be aware when naming a fraction that each piece needs to be equal-sized.
Get students to partition a number of shapes into a range of parts.  Encourage students to explain how they know the partitions are even, and how they could justify this to somebody else (they may need to fold, or cut and overlay the pieces).  Then build up the partitioned pieces to make non-unit fractions and have students name them.  Encourage students to recognise the relationship between the bottom number (denominator) and the number of equal-sized pieces the shape is partitioned into.

NOTE: Drawing fractions is important and can help eliminate misunderstandings.

Students could explore the importance of the pieces being equal-sized in the resource Shaded fractions or Shaded Fractions II (both level 3)

Ratios
Students who constructed the name of the fraction by merely counting up the shaded parts and putting that number over the number of unshaded parts are setting up the fraction as a ratio.  A fraction compares a part to the whole, whereas a ratio compares a part to another part.  Ratio is an important understanding that students need to develop.  The occurrence of this misconception provides a real opportunity to explore the difference between ratios and fractions explicitly.