Research in Benchmarks
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The concept of proportionality is essential to understanding much in science, mathematics, and technology. Many familiar variables, such as speed, density, and map scales are ratios (or rates) themselves. Students' understanding of proportionality progresses along three strands of benchmarks that address the relationship between parts and wholes, experience with and understanding of numerical descriptions and comparisons, and basic computation skills relevant to ratios and proportions. The next edition of Atlas will include several topics—number sense, measurement, shape, and scale—that relate to ratios and proportions.


The 6-8 benchmark "The expression a/b can mean different things..." provides a center point for this map, drawing from and contributing to every strand of development. Though it may seem no more than a technical detail about mathematical notation, this benchmark reflects the basic equivalence between fractions, division, and ratios—that they all can be expressed as quotients.

When something is bigger than something else, we can characterize the relationship by how much bigger it is or how many times bigger. (The trick, of course, is to decide when the additive or multiplicative relationship is appropriate.) The fact that a/b implies a special kind of comparison of a to b is critical to this map.

Two benchmarks on this map do not appear in Benchmarks. The K-2 benchmark "An important kind of relationship..." clarifies the importance of thinking about parts and wholes. The 6-8 benchmark "Some interesting relationships..." draws attention to a constant ratio between variables as an alternative to a constant difference (on which students often get stuck), and it also appears on the Describing Change map.

The benchmarks in the computation strand identify skills that contribute to working with ratios, rates, and proportions. Of course, general computation skills would include many more skills than are given here. Benchmarks for other, equally important skills (and the general topic of mathematical operations) can be found in Benchmarks Chapter 12: HABITS OF MIND and will be included in the next edition of Atlas.