February Math Workshop

Grade 5 Module 4: Multiplication and Division of Fractions and Decimal Fractions

Grade 5’s Module 4 extends student understanding of fraction operations to multiplication and division of both fractions and decimal fractions.  Work proceeds from interpretation of line plots which include fractional measurements to interpreting fractions as division and reasoning about finding fractions of sets through fraction by whole number multiplication.  The module proceeds to fraction by fraction multiplication in both fraction and decimal forms.  An understanding of multiplication as scaling and multiplication by n/n as multiplication by 1 allows students to reason about products and convert fractions to decimals and vice versa.  Students are introduced to the work of
division with fractions and decimal fractions.  Division cases are limited to division of whole numbers by unit fractions and unit fractions by whole numbers.  Decimal fraction divisors are introduced and equivalent fraction and place value thinking allow student to reason about the size of quotients, calculate quotients and sensibly place decimals in quotients.  Throughout the module students are asked to reason about these important concepts by interpreting numerical expressions which include fraction and decimal operations and by persevering in solving real-world, multistep problems which include all fraction operations supported by the use of tape diagrams.

Topic A begins the 38-day module with an exploration of fractional measurement.  Students construct line plots by measuring the same objects using three different rulers accurate to 1/2, 1/4, and 1/8 of an inch (5.MD.2).  Students compare the line plots and explain how changing the accuracy of the unit of measure affects the distribution of points (see line plots at the end of this page).  This is foundational to the understanding that measurement is inherently imprecise because it is limited by the accuracy of the tool at hand. 

Students use their knowledge of fraction operations to explore questions that arise from the plotted data such as, “What is the total length of the five longest pencils in our class?  Can the half inch line plot be reconstructed using only data from the quarter inch plot?  Why or why not?”  The interpretation of a fraction as division is inherent within this exploration.  To measure to the quarter inch, one inch must be divided into 4 equal parts, or 1/4.  This reminder of the meaning of a fraction as a point on a number line, coupled with the embedded, informal exploration of fractions as division, provides a bridge to Topic B’s more formal treatment of fractions as division.