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- Title
A Fraction Problem.
- Authors
Holshouser, Arthur; Reiter, Harold
- Abstract
In recent years, mathematics educators have begun to realize that understanding fractions and fractional arithmetic is the gateway to advanced high school mathematics. Published in the current issue of the peer-reviewed journal Psychological Science, the study [7] found that understanding fractions and division at age 10 predicted success in algebra and overall math achievement in high school, even after statistically controlling for a wide range of factors including parents' education and income, and children's age, gender, I.Q., reading comprehension, working memory, and knowledge of whole number addition, subtraction and multiplication. Yet, U.S. students continue to do poorly when ranked internationally on fractional arithmetic. This essay is intended to help mathematicians interested in working with ambitious students who want to develop a deeper understanding of some fundamental ideas. We do this by posing and solving the following problem: If e/f is a positive rational number reduced to lowest terms, we call e+f the size of e/f. If 0 ⩽ a/b < c/d are given rational numbers (reduced to lowest terms), what is the fraction e/f of smallest size between a/b and c/d. We give two methods including a continued fraction method for finding e/f. At the end of the paper, we use the ideas developed here to generalize this problem in two different ways. Also, at the end, we pose several more problems. These problems are not intended to be used in a classroom with young students. On the other hand, they can all be used to deepen teachers' understanding of fractions and fractional arithmetic.
- Subjects
STUDY &; teaching of fractions; STUDY &; teaching of arithmetic; MATHEMATICIANS
- Publication
Mathematics Competitions, 2017, Vol 30, Issue 1, p25
- ISSN
1031-7503
- Publication type
Article