GRADE 6 RATIOS AND PROPORTIONAL RELATIONSHIPS
- Reasoning with ratios involves attending to and coordinating two quantities.
- A ratio is a multiplicative comparison of two quantities, or it is a joining of two quantities in a composed unit.
- Forming a ratio as a measure of a real-world attribute involves isolating that attribute from other attributes and understanding the effect of changing each quantity on the attribute of interest.
- A number of mathematical connections link ratios and fractions:
- Ratios are often expressed in fraction notation, although ratios and fractions do not have identical meaning.
- Ratios are often used to make "part-part" comparisons, but fractions are not.
- Ratios and fraction can be thought of as overlapping sets.
- Ratios can often be meaningfully reinterpreted as fractions.
- Ratios can be meaningfully reinterpreted as quotients.
- A proportion is a relationship of equality between two ratios. In a proportion, the ratios of two quantities remains constant as the corresponding values of the quantities change.
- Proportional reasoning is complex and involves understanding that -
- Equivalent ratios can be created by iterating and/or partitioning a composed unit:
- If one quantity in a ratio is multiplied or divided by a particular factor, then the other quantity must be multiplied or divided by the same factor to maintain the proportional relationship; and
- The two types of ratios - composed units and multiplicative comparisons - are related.
- A rate is set of infinitely many equivalent ratios.
- Several ways of reasoning, all grounded in sense making, can be generalized into algorithms for solving proportion problems.
- Superficial cues present in the context of a problem do not provide sufficient evidence of proportional relationships between quantities.
Lobato, J.E. (2010). Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning for Teaching Mathematics in Grades 6 - 8. Reston, VA: The National Council of Teachers of Mathematics, Inc.