Finding the fraction of things
| a) |
Jess has 8 toy horses. This is \(1 \over 4\) of her collection of toy animals. Show how to work out how many toy animals she has altogether.
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b) |
In Te Kanohi School 30 students have cell phones. This is \(3 \over 5\) of the students at the school. Show how to work out how many students there are in the school altogether.
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c) |
At a game of rugby there are 84 people in the crowd. \(4 \over 7\) of them support the home team. Show how to work out how many home team supporters there are.
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| Y10 (09/2008) | ||
| a) |
32 Working involving:
i.e., 8 = 1/4 , so 4 lots of 8 is 32;
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very easy very easy |
| b) |
50 Working involving:
i.e., 3/5 = 30, so 1/5 = 10, therefore 5/5 is 50;
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easy easy |
| c) |
48 Working involving:
i.e., 1/7 = 84 ÷ 7 = 12, so 4/7 = 4 × 12 = 48;
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easy easy |
Based on a representative sample of 161 students.
This resource is about fractions as operators which can be seen as a different "personality" of fractions. This idea follows on from the understanding that fractions represent part-whole relationships (part-whole understanding of fractions), and these problems can be solved by identifying the part and the whole. Some students set up the problem as a ratio problem and used ratios, pronumerals and cross multiplication to solve. These fractions questions were of two forms: find the whole, given the part and find the part, given the whole. Only a small number (well under 10%) of students made the error of finding the whole or the part when it was already given. Most of these were for question b).
Prior knowledge
Students should have an understanding of fractions as part-whole relationships for regions, sets, and whole numbers.
Strategies
The most common (45%) strategy across all questions that students used was applying multiplication and division appropriately. Other students (21%) identified a basic fraction and combined them together to get the solution. In terms of accuracy of these two strategies, fewer errors were made by the students who used the basic fraction strategy (98% compared to 94%), but essentially they are similar.
| Common error | Likely misconception | |
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a) b) |
2 18 |
Finds 1/4 of 8 rather than 8 as 1/4 . Finds 3/5 of 30 rather than 30 as 3/5 . |
| c) | 147 | Treats 84 as 4/7 of the whole and solves to find the whole. or divides and multiplies by the wrong numbers ((84 ÷ 4) × 7) |
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a) b) |
24 150 |
Miscalculates 8 × 4 = 24 Doesn't take into account the numerator and then multiplies the part by the denominator, i.e., 30 × 5. |
For students who may have misinterpreted the question, ask them to describe what the maths problem is and what they are finding out. Students who have only seen questions involving find the fraction of this whole may need to compare the different forms of fraction operator questions to see what is actually being asked. These questions are written in different forms to explore the understanding behind the problem, and to avoid students decoding and relying on a remembered process to solve. This is to encourage their ability to transfer the learning to other similar (but not the same) problems.
Insufficient or incomplete working
Students who showed insufficient or no working may have not 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 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 multiple strategies is an important aspect of contemporary mathematics pedagogy, value needs to be placed on encouraging students to communicate their strategies, both orally (see whole class discussion), and in writing.
Cannot work out fractions of a whole number
Students who worked out other incorrect solutions to the problems may need to look at more basic fraction problems and smaller whole numbers that can be modelled by counters as required, e.g., 3/4 of 12, 1/5 of 15, and If 2/3 is 6 what is a whole? Students could also look at Partitioning sets.
See Fraction thinking concept map: Fractions as operators.
Modifying resources for further exploration into fractions as operators
Students, who correctly showed their working and solved the problems, could look at other students' strategies as well as look at more complex problems of this kind. Teachers can copy and paste this resource to change the fractions, quantities and contexts.
- Birthday Cakes
- Fractional Blocks