A closer focus: Fractions, decimals and percentages
By Parveen Akhtar, Research Manager
Friday August 28 2020
In the question-level analysis of the key stage 2 2019 national curriculum tests, there were certain areas of the curriculum that pupils found significantly more difficult than others. Although pupils performed far better on the arithmetic paper than the reasoning papers in terms of percentage of items correct, a large proportion of the arithmetic questions that pupils found difficult were related to fractions and percentages. Of the ten most challenging questions, eight involved fractions or percentages. This area is one which has consistently posed difficulties for pupils – found in previous national curriculum test analyses as well as during the standardisation trial for the year 6 NFER Tests. The difficulties associated with this curriculum domain vary but when analysing the questions that pupils found particularly challenging, a possible number of reasons can be unearthed.
Fractions and percentages are areas of the curriculum which are further focused upon in key stage 3. It is therefore important that these misconceptions are addressed as early as possible when pupils enter year 7 in order to ensure that they have a secure understanding before moving onto more complex areas.
- Multiplying proper fractions and mixed numbers by whole numbers (5F5)
Pupils have consistently found multiplying proper fractions or mixed numbers by whole numbers difficult in the key stage 2 national curriculum tests. In the 2019 paper, pupils performed least well on a question assessing multiplying a proper fraction by a whole number, with only 43% of pupils achieving the mark. Pupils were required to multiply 5 ⁄ 6 by 540. It should be noted, however, that 16% of pupils omitted this question, of which 10% never even reached it. Nevertheless, 43% is still a low proportion for the arithmetic paper. When required to multiply a mixed number by a whole number (1 3 ⁄ 4 by 10) pupils also struggled, with less than 50% getting the correct answer. This has also been found in other papers including the 2018 national curriculum arithmetic paper, which was much simpler (1 1 ⁄ 2 by 40), and the NFER spring test where only a third of pupils were able to multiply a mixed number by a whole number. In the NFER year 6 autumn test this was even lower at 20% with higher, middle and lower achieving pupils struggling.
It is clear then that pupils find this area particularly challenging. When multiplying pairs of proper fractions, the process is relatively simple. Pupils must multiply the numerators together and then multiply the denominators together. It is interesting to note that pupils are confident at multiplying pairs of proper fractions, as evidenced in the NFER year 6 spring and autumn tests, even though this is a year 6 curriculum area. In contrast, when multiplying a proper fraction by a whole number the process is less straightforward. Pupils must recognise that they must multiply the numerator of the fraction by the whole number and then divide by the denominator. Recognising what multiplying fractions actually means would help pupils to overcome this difficulty. Therefore, it would be useful for pupils to have concrete and pictorial examples representing these types of calculation. Pupils can then relate the process of multiplying proper fractions with whole numbers to what it really means, thus creating a solid foundation.
One reason as to why pupils may struggle multiplying mixed numbers with whole numbers is that some may choose to convert these numbers into improper fractions; difficulty may arise here in the application of a skill from an earlier year’s programme of study. Interestingly, in the NFER autumn test, pupils found it difficult to convert numbers to improper fractions (in two questions, 51% and 48% gained the mark). Pupils would thus benefit from practice in converting mixed numbers. Of course, pupils do not have to convert mixed numbers into improper fractions to multiply by whole numbers; they could choose to multiply the whole number followed by the fraction. However, by providing practice and support across the range of methods, pupils can then confidently choose their preferred one or what is most appropriate for different questions.
- Adding and subtracting fractions with different denominators and mixed numbers using the concept of equivalent fractions (6F4)
Some pupils also found adding and subtracting fractions difficult when fractions had different denominators. For example, less than two thirds of pupils gave the correct response when required to subtract 1 ⁄ 4 from 8 ⁄ 9 i.e. fractions with denominators that weren’t related. When analysing common errors in the NFER autumn and spring tests it was found that middle and lower achieving pupils were more likely to find adding and subtracting fractions with different denominators most difficult.
One of the most common errors was adding the two numerators together and adding the two denominators together (for example 2 ⁄ 3 + 1 ⁄ 6 became 3 ⁄ 9). In contrast, when adding and subtracting fractions with the same denominator, pupils performed much better with three-quarters giving the correct response. For example, in the 2019 arithmetic paper, 96% of pupils gave the correct response to 9 ⁄ 11 – 4 ⁄ 11. Although fewer pupils were able to combine fractions with the same denominator with mixed numbers, such as in 2019 when asked to subtract 4 ⁄ 7 from 1 3 ⁄ 7, it was still correctly answered by 75% of pupils - which is higher than where the denominators differ.
This pattern of performance has also been found with NFER’s Year 6 autumn and spring tests, suggesting that pupils did not recognise the need to find equivalent fractions, or struggled when finding equivalent fractions to find a common denominator. In the NFER Tests this was also demonstrated when pupils were required to order fractions with different denominators. This suggests that pupils require practice in recognising that a common denominator is required when adding and subtracting fractions and practice in finding equivalent fractions. Once they have done this, pupils seem to be able to confidently add and subtract fractions.
A further difficulty highlighted by varying performance across different types of questions is the inclusion of a mixed number. For example in one question, pupils are required to subtract 3 ⁄ 4 from 2 1 ⁄ 2. Here the denominators were related but the question involved subtracting a fraction from a mixed number, which adds to the complexity and less than two-thirds of pupils gave the correct response. It is also likely that because pupils were required to cross a whole number boundary, this created an additional challenge. This is further supported by pupils’ performance on Q35 in the 2018 national curriculum arithmetic paper, which involved subtracting 1 6 ⁄ 7 from 4 2 ⁄ 3. Pupils found this challenging with only 38% of pupils giving the correct response - a low proportion for the arithmetic paper. These performances suggest pupils need more practice with questions with mixed numbers in different forms, including crossing whole number boundaries.
- Solving problems involving ratio and proportion, e.g. percentages (6R2)
Overall pupils found percentage questions difficult. This was apparent in the 2019 and 2018 national curriculum arithmetic papers as well as NFER’s year 6 tests. However, closer inspection does show some strengths as well as common misconceptions. It is evident that when finding 10% or 20% of a number pupils are quite confident. For example, in the 2019 paper, pupils were able to find 20% of a four-digit number (3,000) relatively easily. This has also been found in the 2018 arithmetic paper with a large proportion of pupils finding 20% of 1,200 confidently. This may be due to pupils using the dividing by 10 approach to find 10%, a method they may have become confident using.
However, when required to find 36% of 450, pupils really struggled. There was also some difficulty when finding 35% of 320 and 51% of 900. For these questions, the most efficient method is to combine approaches: for example, to find 35% they can begin by finding 10% and multiplying by 3, and then finding 5% by halving 10%. For 36%, pupils would also be required to find 1%, which may be the root of the difficulty as pupils also really struggled to find 51%. It is possible that pupils are finding 1% and then multiplying this answer by the percentage required. This is much more difficult than finding 10% or 50% first as it also results in a decimal number, making the calculations particularly difficult. Therefore, it is possible that pupils have difficulty recognising the different ways to find the percentage of a number. Pupils would therefore benefit from identifying the most efficient ways of finding varied percentages of numbers after becoming secure in finding 10%. They should use their knowledge of 10% to find 5% and 1% and combine this knowledge. However, it is possible that they understand percentages but just struggle with the calculation aspect.
The NFER diagnostic guidance for the year 6 tests examines the percentage errors more closely and compares three different achievement groups. In developing the guidance, it was found that although higher achieving pupils were able to confidently solve percentage questions in the arithmetic paper, they struggled when percentages were tested within a context. This suggests, therefore, that lower and middle achieving pupils require practice with basic percentage questions, whereas higher achieving pupils require more practice with percentage questions presented in context.
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