Fractions of sets

Fractions of sets

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about finding fractions of sets of objects.

Question 1Change answer

a) Write what fraction of objects has been circled.
 

Question 1Change answer

b) Write what fraction of objects has been circled.
 

Question 1Change answer

c) Write what fraction of objects has been circled.
 
 
 

Question 1Change answer

d) Write what fraction of objects has been circled.
 
 

Question 1Change answer

e) Write what fraction of objects has been circled.
 
Task administration: 
This task can be completed with pencil and paper or online.
Level:
2
Curriculum info: 
Maths, Number and Algebra, Number Knowledge
Keywords: 
fractions, part-whole fractions, sets
Description of task: 
Students identify fractions of sets of objects.
Answers/responses: 
  Y4 (11/2006)
a) 1/2 or 4/8 easy
b) 1/4 or 2/8 easy
c) 2/3 or 6/9 easy
d) 1/4 or 3/12 easy
e) 3/4 or 12/16 or 6/8 moderate
Based on a representative sample of 188 students.
Diagnostic and formative information: 
Averaged over the five questions, just under half the students wrote the count of circled items over the total number of items as their answer, and about a fifth of students gave the answer in its simplest form.

  Common error Likely misconception
a)
b)
c)
d)
e)

4
2
6
3
12

Whole number
The number of circled items is given as the answer indicating that they either do not know what a fraction is or do not know how to construct one.  About one-tenth of students answered with this misconception.
a)
b)
c)
d)
e)

1/4
1/2
1/6
1/12
1/3

"Unit fractions"
Student gives an incorrect unit fraction with the number of circled objects as the denominator.
a)
b)
c)
d)
e)

8/4
8/2
9/6 or 3/2
16/12 or 4/3
12/3

Inverts fraction
Gives the inverse of the correct answer, i.e. reverses the roles of the numerator and denominator.
a)
b)
c)
d)
e)		 4/4
2/6
6/3
3/9
12/4		 Ratio
The fraction is constructed by writing the number of circled items over the number of items not circled. This is a ratio.
Next steps: 
Whole number misconception
Students who write their answer as a whole number may need to develop an understanding about what a fraction represents, i.e., that if a set is partitioned equally into n parts then each part is called 1/n .  Students need to have more experience partitioning sets and naming the unit fraction/parts they have created.  Students beginning to understand fractions should be encouraged to use words to describe the parts, and delay the fractional notation until they have developed some understanding of what fractions represent.

Unit fraction misconception
Students who write their answer as a unit fraction of the items circled need to be aware of what the whole is that they are finding the part for.  Ask simple questions like "If this set is the whole set, then what part of the set is circled?"  This idea of the whole and the part is an important understanding of fractions for students to develop.

Inverted fraction misconception
Students who write their answer as the inverted fraction of the correct answer have developed some understanding about the part-whole relationship of fractions, but need to develop an understanding of the convention of how fractions are named.  Show some simple examples of sets with parts circled and ask what fraction of the set is circled.  Encourage students to recognise the relationship between the number of parts circled (top number) and the total number of parts (bottom number).
It may be helpful to remind students to ask themselves "What is the whole set? (referent whole) and "What is the part of the set?".

Ratio misconception
Students who write their answer as a ratio indicate an understanding that a fraction is a different kind of number to a whole number.  However, a fraction compares a part to the whole, whereas a ratio compares one part to another part.  Have students work with problems where they can identify what the whole is when they’re finding the part.
NOTE: Ratios are a concept that students will need to develop, so it is important to affirm students' ideas in this area.  The difference between a ratio and a fraction could be discussed and made explicit.  This may be a good opportunity to explore this difference.

 

Links to Numeracy
Equal sharing of sets or whole numbers simple problems and using a matching strategy is at Stage 4: Advanced counting to Stage 5: early additive part-whole.  Students who can construct and name non-unit fractions are at Stage 5: early additive part-whole (using repeating addition) to Stage 6: Advanced additive (partitioning using division) part-whole.