WELCOME TO MY MARVELOUS MATH CLASS
Multiplication and Division of Fractions and Decimal Fractions
In Module 4, students learn to multiply fractions and decimal fractions and begin working with fraction division.
In Topic A, students explore 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. Students compare the line plots and explain how changing the accuracy of the unit of measure affects the distribution of points. Students use their knowledge of fraction operations to explore questions that arise from the plotted data. For measuring to the quarter inch, one inch must be divided into four equal parts, or 1 ÷ 4. This reminder of the meaning of a fraction as a point on a number line is very important.
Topic B focuses on interpreting fractions as division. Equal sharing with area models (both concrete and pictorial) provides students with an opportunity to understand division of whole numbers with answers in the form of fractions or mixed numbers (e.g., seven brownies shared by three girls, three pizzas shared by four people). Discussion also includes remainders as a fraction. Tape diagrams provide a linear model of these problems. Students also solve real-world problems and generate story contexts for visual models.
In Topic C, students find a fraction of a set ( 3/4 of 24) as multiplication of a whole number by a fraction ( 3/4 × 24) and use tape diagrams to support their understandings. Students apply their knowledge of a fraction of a set and previous conversion experiences (with scaffolding from a conversion chart, if necessary) to find a fraction of a measurement, thus converting a larger unit to an equivalent smaller unit (e.g., 1/3 minutes = 20 seconds and 2 1/4 feet = 27 inches).
In Topic D, students learn to evaluate expressions with parentheses, such as 3 × ( 2/3 − 1/5 ) or 2/3 × (7 + 9). They then learn to interpret numerical expressions, such as 3 times the difference between 2/3 and 15 or two-thirds the sum of 7 and 9. Students generate word problems that lead to the same calculation, such as “Kelly combined 7 ounces of carrot juice and 5 ounces of orange juice in a glass. Jack drank 2/3 of the mixture. How much did Jack drink?” Solving word problems allows students to apply new knowledge of fraction multiplication in context, and tape diagrams are used to model multi-step problems requiring the use of addition, subtraction, and multiplication of fractions.
Topic E introduces students to multiplication of fractions by fractions—both in fraction and decimal form. The topic starts with multiplying a unit fraction by a unit fraction and progresses to multiplying two non-unit fractions. Students use area models, rectangular arrays, and tape diagrams to model the multiplication. These familiar models help students draw parallels between whole number and fraction multiplication, as well as solve word problems. Just as students used unit form to multiply fractional units by wholes in Module 2 (e.g., 3.5 × 2 = 35 tenths × 2 ones = 70 tenths), they connect fraction-by-fraction multiplication to multiply fractional units‐by-fractional units (3.5 × 0.2 = 35 tenths × 2 tenths = 70 hundredths). Reasoning about decimal placement is an integral part of these lessons. Students will also find fractional parts of customary measurements and measurement conversion.
In Topic F, students extend their understanding of multiplication to include scaling. Students compare the product to the size of one factor, given the size of the other factor without calculation (e.g., 486 × 1,327.45 is twice as large as 243 × 1,327.45 because 486 = 2 × 243). Students build on their new understanding of fraction equivalence as multiplication by 𝑛/𝑛 to convert fractions to decimals and decimals to fractions. For example, 3/25 is easily renamed in hundredths as 12/100 using multiplication of 4/4 . The word form of twelve hundredths is then used to notate this quantity as a decimal.
Topic G begins the work of division with both fractions and decimal fractions. Students use tape diagrams and number lines to reason about the division of a whole number by a unit fraction and a unit fraction by a whole number. Students reason about how many fourths are in 5 when considering such cases as 5 ÷ 1/4 . Using this thinking as a backdrop, students are introduced to decimal fraction divisors and use equivalent fraction and place value thinking to reason about the size of quotients, calculate quotients, and sensibly place the decimal in quotients.
Topic H, begins with numerical expressions involving fraction-by-fraction multiplication. Students create and solve word problems involving both multiplication and division of fractions and decimal fractions.