Models of Fractions
Researcher: Which fraction is smaller, 6/4 or 6/5?
Chris: 6/4. Because 4 takes longer to get to 6 than 5 does.1
Like Chris, many students reason about fractions in ways that work in certain situations but not in others. Chris may have an image of a fraction as a quantity less than one–and as a whole with some pieces missing. His reasoning would be helpful if he were comparing the size of 4/6 and 5/6, but leads him to the wrong conclusion in the example above.
Students’ images of fractions are shaped by mathematics lessons and by real life experiences. What images of fractions do your students typically draw when asked to make sense of a fraction problem? The illustration below shows the fraction 3/4 shaded in eight different models.
Models of the Fraction 3/4
Each of the models above has different strengths and shortcomings for building student understanding. Tad Watanabe’s article "Initial Treatment of Fractions in Japanese Textbooks" argues that there is not one single “best” model, since ultimately students need to grasp that 3/4 represents several very different situations, such as 3/4 of a meter or 3 apples in a basket of 4 fruits. US textbooks rarely use a linear measurement context to introduce fractions, but it is common in other countries and in research-based programs, and it may help students see fractions as numbers that can go on a number line. In the next activity, you can experience fractions in a linear measurement context.
1 Adapted from Cramer, K. & Wyberg, T. (2007). When getting the right answer is not always enough: Connecting how students order fractions and estimate sums and differences. pp. 205-220. W. G. Martin & M. E. Strutchens (Eds.) The learning of mathematics. 69th Yearbook Reston, VA:NCTM, p.215.