Abstract
At the simplest level, proportional reasoning, whether involving ratio or proportion, relies on the ability to use multiplicative thinking rather than additive thinking. In other words, instead of describing a relationship between two quantities as being larger by three or smaller by five, the relationship would be described in terms such as triple the size, one fifth the size, four times greater, etc. The ability to use proportional reasoning is essential in developing a broad variety of mathematical concepts including similarity, relative growth and size, dilations, scaling, \(\pi\), constant rate of change, slope, speed, rates, percent, trigonometric ratios, probability, relative frequency, density, and direct and inverse variations.
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Notes
- 1.
A hit refers to batter safely reaches first base after hitting the ball into fair territory without the benefit of a fielder’s choice or an error.
- 2.
An at bat is different than a plate appearance. To qualify for an at bat in addition to batting against a pitcher, the batter must NOT have received a base on balls, have hit a sacrifice fly or bunt, been replaced by another batter before their at bat is completed, been hit by a pitch, been awarded first base due to interference, or had the inning end prior to the completion of the at bat.
- 3.
Thales (ca. 624–546 B.C.) was a Greek mathematician, astronomer and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. Aristotle considered him as the first philosopher in the Greek tradition.
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In this table, the ratio between two successive Fibonacci numbers (column 3) for the nth term (column 1) was calculated by the ratio \(\frac{{F_{n + 1} }}{{F_{n} }}\). The value of the Fibonacci number itself was calculated for \(n > 2\) as \(F_{n} = F_{n - 1} + F_{n - 2}\).
- 5.
Johannes Keppler (1571–1630), a German astronomer and mathematician.
- 6.
Jacques Binet (1786–1856), a French mathematician.
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Abramovich, S., Connell, M.L. (2021). Ratio and Proportion. In: Developing Deep Knowledge in Middle School Mathematics. Springer Texts in Education. Springer, Cham. https://doi.org/10.1007/978-3-030-68564-5_6
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DOI: https://doi.org/10.1007/978-3-030-68564-5_6
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