Writing percentages
| Y8 (11/2010) | ||
| a) | 10% | easy |
| b) | 7% | moderate |
| c) | 100% | moderate |
| d) | 175% | moderate |
| e) | 220% | moderate |
| f) | 45.3% | moderate |
| g) | 90.2% | difficult |
Based on a representative sample of 184 Y8 students.
This resource is about converting from decimals to percentages. It involves decimal fractions that represent a proportion, and relate to percentages by a factor 100.
Most students could write the more straightforward decimal fractions (less than 1 and no nested zeros) as percentages (e.g., 0.1 or 0.10 would be easier). A number of students added the percentage sign without changing the decimal fraction, e.g., 0.1 became 0.1%. Some students offset the decimal fractions by a factor of ten instead of 100 (e.g., 0.1 becomes 1%, 0.902 becomes 9.02%, etc). A small number of students could not convert the decimal fractions (even the simplest) to percentages. A notable number of students had difficulty with decimal fractions over 1 (including 1), and also what to do with nested zeros, e.g., they may have written 0.902 as 92% by dropping the nested zero.
What underlies this and many of the related misconceptions is the meaning of the percentage sign. It can often be helpful to look at the relationship between fractions and decimals, and then percentages, e.g., 0.4 as a decimal can be written as 40% or 100/3 as a fraction. Although there are many other equivalent fractions for the purposes of developing understanding around the meaning of percentage it may be best to focus on the hundredths, tenths or thousandths aspect of this relationship.
Students repeat the same decimal number as the percentage (e.g., 0.1 becomes 0.1%)
Students who wrote the same decimal number followed by a percentage sign for the simpler of the decimal fractions (e.g., 0.1 as 0.1%, and 0.07 as 0.07%) are likely to have no understanding of relationship between decimal fractions and percentages. Students may also be unaware that decimals can also be proportions: as well as being numbers, decimals can also represent decimal fractions, e.g., 0.5 of something, or 1.5 of something. These are directly comparable to the idea of fractions of something (fractions as operators). These students could firstly explore the proportional nature of percentages, and then relate these to decimal fractions (e.g., 75% = 0.75). The resource, Building percentages, looks at the use of simple percentages as proportions of a whole. These students should make connections between percentages and fractions "over 100", e.g., 75% is as 75/100 . Students could then start to recognise tenths, hundredths and the decimal counterparts, and recognise the 1:100 or factor of 100 relationship (i.e., per cent = per hundred).
Students adjust the decimal by a factor of ten
These students are aware that there is some factor of ten relationship between the decimals and percentages but confuse the relationship with the base 10 nature of the number system (e.g., 1.75 becomes 17.5%). Students could benefit by learning about the equivalence of fractions, percentages and decimals (decimal fractions) starting with half (1/2 , 0.5 and 50%), quarter (1/4 , 0.25 and 25%), and three-quarters (3/4 , 0.75 and 75%). This can be further developed by looking at 50/100 , 25/100 and 75/100). Students could be asked to explore, describe and make conjectures about equivalence between the examples of fractions and decimals–hopefully leading to an understanding that a percentage is a hundredth, i.e., 1% is 1/100 and so on. These students could also explore representing decimals and equivalent fractions on the abacus. Some students may find it easier to convert percentages to decimals, and then make conjectures. The resource, Health percentages, involves a range of examples of this.
Students who could not work with decimal fractions over one, decimals in the thousandths, or nested/place holder zeros (all decimal place value errors)
These students have indicated that they understand the relationship between decimal fractions of 2 decimal places (e.g., 0.23) and percentages of 2 significant figures (e.g., 23%). However, they have not fully understood that this relationship holds when decimal fractions are over 1 or involve parts of percentages such as 45.3% (thousandths, e.g., 0.453). Underlying this is that the decimal fraction multiplied by 100 will always yield the percentage as it ties to the definition of percentage. Students may feel they were meant to get rid of the decimal point to "make it a percentage" (e.g., for 45.3% remove the .3 because it must be a percentage not a decimal) or they may wonder how something could be more than 100% (e.g., 175%, so it must be 75%) or how a decimal fraction could be more than 1 (e.g., 1.75, so they only work with the "decimal part" – 0.75). It is likely that student needs to explore the relationship between decimal fractions and percentages and look at how they represent the same number (or proportion). Students could explore this equivalence through the use of 6-pronged abacus to represent whole number and decimal values and fractional values. Students may need to be reminded of the rule that governs using abacus (ten of one spike is equivalent to one of the next to the left).
Decimals and the fractional numbers can be placed on the abacus side by side so the “equivalence” can be seen. Attention can be drawn to the hundredths spike as the unit of a percentage (1% is 1/100). From here, fractional numbers and their decimal fraction counterparts can be created and checked, e.g., marking 3/100 on the abacus and then looking at what happens with the tenths. Ask how many? If there are zero tenths (in 0.03) then a zero place holder should be written. Similarly, for the hundreds, tens and ones on an abacus. The decimal, 0.3 can be marked on the abacus as above. As it is 3 tenths, 3/10 and 0.3 are the same. Students can then work out to show 0.35 and explore how to represent 35 hundredths.
Students could be asked to show and label a range of different numbers in different forms (decimal, tenths, hundredths, etc: 4/10 , 60/100 , 0.7, 0.25) involve the hundreds, tens and ones, so students do not develop the misconception that: all decimals are less than one, percentages are only whole numbers, or all percentages are less than 100%.