Unit Summary
In this unit, students gain familiarity with factors and multiples, then use that understanding to develop general methods and strategies to recognize and generate equivalent fractions as well as to compare and order fractions.
Students began their study of fractions in Grades 1 and 2, where students learned to partition rectangles and circles into halves, thirds, and fourths. In Grade 3, students developed an understanding of fractions as numbers rather than simply equal parts of shapes. Students worked with number lines, which help to “reinforce the analogy between fractions and whole numbers” (Progressions for the Common Core State Standards in Math, p. 8). Students also began their work with recognizing and generating equivalent fractions in simple cases, using a visual fraction model to support that reasoning. This also involved the special case of whole numbers and various fractions, e.g., $${1=\tfrac{2}{2}=\tfrac{3}{3}=\tfrac{4}{4}...}$$. Lastly, students began to compare fractions in cases where the two fractions have a common numerator or common denominator. Another key understanding from 3rd grade that students will rely on in this unit is their fluency with single-digit multiplication and division, aiding their understanding of factors and multiples and their relationship to fraction equivalence.
Students begin the unit by investigating factors and multiples within 100, as well as prime and composite numbers (4.OA.4). Then, in Topic B, students use their knowledge of factors and multiples as well as the fraction foundation built in Grade 3 to extend their understanding of and strategies to recognize and generate equivalent fractions. They use area models, tape diagrams, and number lines to understand and justify why two fractions $$\tfrac{a}{b}$$ and $$\tfrac{\left( n \ \times \ a \right)}{(n\ \times \ b)}$$ are equivalent, and they use those representations as well as multiplication and division to recognize and generate equivalent fractions. Lastly, they compare fractions with different numerators and different denominators in Topic C. They may do this by finding common numerators or common denominators. They may also compare fractions using benchmarks, such as “see[ing] that $$\tfrac{7}{8} < \tfrac{13}{12}$$ because $$\tfrac{7}{8}$$ is less than $$1$$ (and is therefore to the left of $$1$$) but $$\tfrac{13}{12}$$ is greater than $$1$$ (and is therefore to the right of $$1$$)” (Progressions for the Common Core State Standards in Math, p. 11). Throughout the discussion of fraction equivalence and ordering in Topics B and C, students’ work with factors and multiples, a supporting cluster content standard, engages them in the major work of fraction equivalence and ordering, e.g., by identifying a common factor of the numerator and denominator to generate an equivalent fraction in larger units.
Students engage with the practice standards in a variety of ways in this unit. For example, students construct viable arguments and critique the reasoning of others (MP.3) when they explain why a fraction $$\tfrac{a}{b}$$ is equivalent to a fraction $$\tfrac{\left( n \ \times \ a \right)}{(n\ \times \ b)}$$. Students use appropriate tools strategically (MP.5) when they choose from various models to solve problems. Lastly, students look for and make use of structure (MP.7) when considering how the number and sizes of parts of two equivalent fractions may differ even though the two fractions themselves are the same size.
Students will only work with fractions of the form $$\tfrac{a}{b}$$, including fractions greater than $$1$$. Students will develop an understanding of mixed numbers in Unit 5, where they will use fraction addition to see the equivalence of fractions greater than $$1$$ and mixed numbers. They also encounter all cases of addition and subtraction of fractions with like denominators, as well as multiplication of a whole number by a fraction in that unit. Then, in Unit 6, students will work with decimal fractions, developing an understanding of decimal notation for fractions, comparing decimal fractions, and adding decimal fractions with respective denominators $$10$$ and $$100$$. Students continue their work with fraction and decimal computation in Grades 5 and 6, including adding and subtracting fractions with unlike denominators by replacing the given fractions with equivalent ones. Thus, the property that "multiplying the numerator and denominator of a fraction by the same non-zero whole number results in a fraction that represents the same number as the original fraction" provides an important foundation for much of their upcoming work in Grade 4, as well as Grades 5 and 6 (Progressions for the Common Core State Standards in Math, p. 10).
Pacing: 17 instructional days (15 lessons, 1 flex day, 1 assessment day)