Curriculum Units – Level 5

If you ask many adults who claim to have difficulty with mathematics at what point they began to struggle, most will undoubtedly say “fractions” with an exclamation point in their voice. Most likely this is because their study of fractions centered around memorizing and using rules to add, subtract, multiply and divide fractions without ever understanding where these rules came from. Perhaps more importantly, they were never encouraged to develop a “fraction sense,” that is, an understanding of the underlying concepts such as fraction size, ordering, equivalence and combining. Thus the rules never made sense to them and, to this day, they probably cannot order fractions easily, readily estimate answers to computations with fractions or use benchmarks effectively. This unit has a very different approach to learning about fractions.

In this unit, students are introduced to two children, Tori and Jordan, who uncover hidden treasures in their grandparents’ attic from a general store that their great grandparents used to own. These treasures lead to an interesting exploration of fraction concepts. The focus of the entire unit is on making sense of fractions rather than on learning algorithms to perform computations. This is a significant difference from more traditional approaches. It is important for students to think about and picture the relative size of fractions and make estimates based on their mathematical thinking when ordering, comparing, adding and subtracting two fractions. Since fractions are such an important part of our everyday experiences, focusing on meaning rather than rules actually gives students a facility for understanding and working with fractions that will benefit them throughout their lives.

In the first chapter, students are exposed to a variety of models (specifically set, linear and area models) to name equivalent fractions. Using a variety of models helps students gain a firm grasp of equivalence and this, in turn, enables them to generalize and then apply their understanding to ordering and comparing fractions. When ordering and comparing fractions, students also learn multiple strategies, such as common numerators, common denominators, benchmarks and missing pieces of the whole, to analyze the size of fractions. Students also investigate the density of the real number system as they learn that between any two fractions, another one can always be found. This implies that there are no holes or gaps in the real number line. They also discover something quite exciting…that there are an infinite number of fractions equivalent to any given fraction!

In working with addition and subtraction of fractions in the second chapter, again the focus is on the meaning of the operations as they are used with fractions. Students first construct and discuss their own methods for adding and subtracting fractions of all sizes before they are introduced to the common algorithms. Because the emphasis is on making sense of problems, operations with like denominators are not separated from operations with unlike denominators. In real life, these situations arise simultaneously and students need to discover strategies that make sense working with both. Students initially use physical models, drawings and equivalent fraction charts that they used in the first chapter to develop written algorithms for the operations. After discussing and comparing their own methods, they compare them to the common algorithms and then decide which are most meaningful for them to use.