Levels 5–9: Proportional Reasoning
Materials
Instructions
Place Sense of Percent Worksheet in front of the student and say, "Each of these squares represents one unit. Point to the first square and ask: "Can you show me 25% on this one please? … If shown as one quadrant, ask, "Is there another way you could show that? … How would you write the amount you have shaded? … Can you write that in another way?" …Point to the second square and ask, "Can you show me what 87% would look like on this square? … How would you write that? … Is there another way to write the amount you have shaded?"
Point to first shaded bar and say, "This is like the bar that appears on a computer screen when a web-site is loading … What percentage do you think has been loaded? … Could you write that please?" … Point to the second bar and say, "What percentage remains to be loaded? … How did you work that out?" … Note student’s response.
Point to the number line and ask, "Can you show me where 145% is on this line please?" …. Note student’s response and explore his/her thinking as appropriate. Retain worksheet.
If student’s response indicates a reasonable understanding of simple percent, proceed to the quiz. Questions should be asked orally and followed by a request to explain the thinking involved. Note explanations and stop at the first sign of difficulty.
Quiz questions
- 50% of 4075% of 72
- 40% of 820
- 12 1 / 2 % of 24
- 200% of 68
Advice rubric
The everyday use of percent would seem to imply that this is a relatively well understood aspect of mathematics. While this is undoubtedly true for simple percents such as 50%, 25%, 33⅓% and so on, this is not necessarily the case for percentages more generally, percentages greater than 1, or percentages less than 0.01. The very fact that many textbooks teach the rule, “to change a fraction into a percent multiply by 100 over 1 and divide”, suggests that percent is understood more as an operation than as a decimal fraction. This leads to errors in computation and encourages inappropriate calculator use, eg, using the percent key on a calculator when multiplying by the relevant decimal fraction would be more meaningful and efficient.
This task examines students’ understanding of percent, in particular, their capacity to represent, name and record percent, recognise the relationship between percent and decimal fractions, and work with percents greater than 1.
| Observed response |
Interpretation/Suggested teaching response |
|---|
| Able to show 25%, may show as a quadrant initially, records at least one amount in another way (most likely as a common fraction) but experiences some difficulty with remaining tasks, eg, may guess amounts for computer strip, provides little/no response to number line task and quiz items |
Limited understanding of percent, may not understand relationship to decimal fractions - Review halving, thirding and fifthing partitioning strategies. Have students use these to construct their own fraction diagrams and line representations for a wide range of fractions. Review key generalisations (see Partitioning (pdf - 199.51kb) paper)
- Use halving and fifthing strategies to construct diagrams and line representations for tenths and hundredths (see Partitioning (pdf - 199.51kb) paper), Use to locate given decimal fractions on an open number line. Practice representing, naming, and recording tenths and hundredths.
- Use real-world examples to compare and order decimal fractions.
- Link hundredths to percent and practice renaming percentages as hundredths and vice versa
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| Able to show 25% and 87% and record in some other way, makes a reasonable estimate of computer strips but unable to locate 145% and experiences difficulty with some quiz items, eg, may state what needs to be done but does not attempt this and/or makes arithmetic errors |
limited strategies for working with a wider range of percents - Extend the use of halving and fifthing strategies to construct diagrams and line representations for hundredths and thousandths (see Partitioning (pdf - 199.51kb) paper), Use to locate decimal fractions to thousandths (eg, 4.537) and percents (eg, 73%, 156%, 12½%, 453% etc) on open number lines
- Build knowledge of fraction equivalents beyond halves and quarters to recognise that 1 eighth is 12½% as is it is half of 25%, 33⅓% is 1 third, 10% is 1 tenth and so on. Model and practice the use of these in combination to find percentages mentally (eg, for 45% of a quantity, find 50 % and subtract half of 10%)
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| Completes all tasks relatively easily, may record percents both as common fractions and decimals, able to justify location of 145%, uses a range of appropriate strategies to solve quiz items |
Suggests a sound understanding of percent and an ability to use a variety of strategies to calculate - Explore the use of percent in an extended range of problems, discuss how percentage calculations can be performed on a calculator and why, eg, to find 7½% of $148,530, multiply 148.530 by 0.075 because 7½% is equivalent to 7.5 hundredths or 75 thousandths
- Consider introducing ratio more formally and the use of a wider range of complex proportional reasoning problems which require the use of percents (eg, see the Mixing Juices (pdf - 24.96kb) task)
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