Sharing fractions
Show how you worked out each question below. 

a)  Tipene has 16 play station games. He gives half (\(1 \over 2\)) of them to his brother. How many games does he give his brother?  


b) 
Jodi buys 24 Kbars. She gives a quarter (\(1 \over 4\)) of them to her brother. How many Kbars does she give him? 



c) 
Ihipera has 15 picture books. She gives a third (\(1 \over 3\)) of them away to a friend. How many picture books does she give away? 


 using counting or visual grouping or mapping strategies to correctly solve the fraction problems.
 using additive strategies such as repeated addition (or repeated subtraction) to correctly solve the fraction problems.
 using a halving strategy to correctly solve question b).
 using simple reverse multiplication and division strategies to correctly solve the fraction problems.
Key competencies
Y4 (10/2010)  
a) 
8 Working involving any 1 of:
Many of these methods used could be different expressions of a halving strategy. 
very easy 
b) 
6 Working involving any 1 of: Working as above involving dividing by the denominator, short division, reverse multiplication, repeated halving, repeated subtraction, repeated addition, grouping objects, marks or tallies, or drawing a mapping diagram. 
moderate 
c) 
5 Working involving any 1 of: Working as above involving dividing by the denominator, short division, reverse multiplication, repeated subtraction, repeated addition, grouping objects, marks or tallies, or drawing a mapping diagram. [NOTE: Repeated halving would not be an appropriate strategy for this question] 
moderate 
NOTE: The main focus for the working part of these questions is on the use of an appropriate strategy i.e., the communication of the attempted calculation as well as a stepwise sequence that could lead to a correct answer.
This resource is about fractions as operators which can be seen as a different "personality" of fractions. This involves an understanding that a fraction can act as an operation upon a number (in this case a whole number). The number is treated as the whole of which to find the fraction. Ideally, this idea follows on from the understanding that fractions represent partwhole relationships (partwhole understanding of fractions), and these problems can be solved by identifying the part and the whole. Fractionoperator questions can be of two forms: find the whole given the part and find the part given the whole. These questions were all of the type, find the part given the whole.
Prior knowledge
Students should have a basic understanding of fractions as partwhole relationships for regions, sets and whole numbers.
Students used a range of simple multiplicative, halving, additive, counting, and visual grouping strategies to answer these questions. The most common sufficient (strategies that work) strategy that students used was halving for questions a) and b), and for question c) repeated addition, reverse multiplication and visual grouping were used in similar proportions.
A number of students also used "whole number" strategies to attempt to solve these maths problems. This misconception involves treating the numerator and the denominator as separate whole numbers and either subtracting or multiplying to get a wide range of answers. Some students also simply wrote the denominator as the answer (see the misconceptions below). As the questions got harder fewer students showed their strategies.
Common error  Likely misconception  
b) c) 
4 3 
Gives answer as denominator (Whole number misconception) Students wrote the denominator as their answer, e.g., ^{1}/_{4} of anything is 4. This could relate to them confusing ^{1}/_{4} means you get 4 (as an answer rather than 4 quarters  whatever they may be). Similarly for ^{1}/_{3} : ^{1}/_{3} means you get 3. 
a) b) c) 
14 [16  2] 20 [24  4] 12 [15  3] 
Subtracts the denominator (Whole number misconception) Students may know that finding a fraction of a number involves reducing the number, but they lack understanding of how to apply the fraction as an operator, so attempt to treat the fraction as a whole number and subtract – in this case the denominator of the unit fraction is subtracted. 
b)
b)

12 [^{1}/_{2} of 24]
8 [^{1}/_{3} of 24] 
Students confuse the unit fraction they are finding Students confuse a half and a quarter. Students confuse a quarter and a third – or confuse the factors when solving. 
b)
c)

18 10 
Close reading error Students work out how many remain rather than the portion given away. 
Cannot work out fractions of a whole number (Whole number misconception)
Students who worked out other incorrect solutions to the problems may need to look at more fraction as operator problems using materials (counters and sets) and smaller whole numbers that can be modelled by counters as required, e.g., ^{1}/_{4} of 8, ^{1}/_{5} of 10, etc. Students could also look at Fraction Soup animation or Finding the fraction of things.
See Fractional thinking concept map: Fractions as operators.
Students who used visual strategies
Students who used tallies and mapping diagrams should be encouraged to also write what is happening with the counts of the sets or tallies they may be using. This may lead them to recognise using numbers can be more efficient.
Close reading error
Students who found out how many remain would have answered a more difficult question.
This is likely to be a reading error. Get students to explain what they have found out. If they indicate they understand what they had done get them to reread the question to compare what they solved to what the question asked. These students could then explore fraction questions that involve larger numbers and simple nonunit fractions.
Insufficient or incomplete working
Students who showed insufficient or no working may not have experienced the need for working or not believe it is important to show working if they know the answer. Developing the ability to make a mathematical statement (such as how they worked it out) is a valid part of the mathematics learning. It can also provide significant formative information about their learning needs.
Students who showed their working as repeating or summarising information (e.g., ^{1}/_{4} of 24) from the question may need to discuss what constitutes working (or an argument), or even that working is their rationale for their answer. Students could share their working and identify if it is sufficiently explicit to be a justification for the answer they have given. As the use of multiple strategies is an important aspect of contemporary mathematics pedagogy (as well as a key indicator of an advanced stage in the Number Framework), value needs to be placed on encouraging students to communicate their strategies, both orally (see whole class discussion), and in writing.
Strategies
The most successful strategies that correlated to correct answers were repeated subtraction, halving [not a sufficient strategy for question c)], dividing by the denominator, reverse multiplication, and repeated addition. The success of these strategies ranged from 95% (halving) to 76% (repeated addition). Students who used marks, tallies, or mapping diagrams to solve these problems had a 69% success rate. However this visual strategy is less robust for larger numbers and developing an understanding of fractions as operators (upon numbers). For questions a) and b) students with the highest mean ability used halving strategy. For question c) students who used reverse multiplication followed by dividing by the denominator had the highest mean ability.
 Click on the link for further information about fractions as operators.
 To look at animations that explore finding fractions of a set see Students could also look at Fraction Soup or Toy holiday.
 See Student work samples (PDF) for a range of strategies that students' used to answer these questions.
Book 7: Teaching Fractions, Decimals and Percentages, 2008: Birthday Cakes and Fractional Blocks.