Comparing units (ratios, rates and proportion) description
This sub-element addresses comparing units in ratios, rates and proportions. A ratio describes a situation in comparative terms, and a proportion is taken to mean when this comparison is used to describe a related situation in the same comparative terms. For example, if the ratio of boys to girls in a class is 2 to 3, the comparison is the number of boys to the number of girls. Knowing that there are 30 children in the class, proportionally, the number of boys is 12 and the number of girls is 18. Applying the base comparison to the whole situation uses proportional reasoning. Proportional reasoning is knowing the multiplicative relationship between the base ratio and the proportional situation to which it is applied.
Learning to reason using proportion is a complex process that develops over an extended time. Proportional reasoning also includes numerical comparison tasks involving a comparison of different rates or ratios.[1] For example, if one dog grows from 5 kilograms to 8 kilograms and another dog grows from 3 kilograms to 6 kilograms, which dog grew more?
The sub-element of comparing units applies to Measurement, Interpreting fractions and Representing data.
Some students will communicate using augmentative and alternative communication strategies to demonstrate their numeracy skills. This may include digital technologies, sign language, braille, real objects, photographs and pictographs.
Each sub-element level has been identified by upper-case initials and in some cases lower-case letters of the sub-element name followed by ascending numbers. The abbreviation for this sub-element is CoU. The listing of indicators within each level is non-hierarchical. Subheadings have been included to group related indicators. Where appropriate, examples have been provided in brackets following an indicator.
[1] Cramer, K & Post, T 1993, ‘Connecting research to teaching proportional reasoning’, Mathematics Teacher, 86(5), May, pp. 404–407.
CoU1
Building ratios
- uses knowledge of fractions as part-whole relationships to divide and compare quantities
- represents and models ratios using diagrams or objects (in a ratio 1:4 of red to blue counters, for each red counter there are four blue counters)
CoU2
Ratios
- interprets ratios as a comparison between the same units of measure (students to teachers in a school is 20:1)
- expresses a ratio as equivalent fractions or percentages (ratio 1:1, each part represents ½ or 50% of the whole)
- uses a ratio to increase or decrease quantities to maintain a given consistency (doubling a recipe)
Rates
- interprets rates as a relationship between two different types of quantities (money per unit of fuel)
- uses rates to determine how quantities change
CoU3
Applying proportion
- interprets proportion as the equality of two ratios or rates
- uses common fractions and decimals for proportional division
- demonstrates how increasing one quantity in a ratio will affect the total proportion
- performs operations with negative integers involving rates (rates of descent or cooling)
- explains and applies the difference between direct and indirect proportion (direct – working more hours will result in earning more money; indirect – travelling at a greater speed will mean the journey takes less time)