Information for course "Unit 4 - Multiplying And Dividing Fractions"
Description: Work with fractions in grade 6 draws on earlier work in operations and algebraic thinking, particularly the knowledge of multiplicative situations developed in grades 3 to 5, and making use of the relationship between multiplication and division. Multiplicative situations include three types: equal groups; comparisons of two quantities; dimensions of arrays or rectangles. In the equal groups and comparison situations, there are two subtypes, sometimes called the partitive (or measurement) and the quotitive interpretations of division. Students are not expected to identify the three types of situations or use the terms “partitive” or “quotitive.” However, they should recognize the associated interpretations of division in specific contexts (MP7).
For example, in an equal groups situation when the group size is unknown, division can be used to answer the question, “How many in each group?” If the number of groups is unknown, division answers the question, “How many groups?” For example, if 12 pounds of almonds are equally shared among several bags:
There are 2 bags. How many pounds in each bag? (partitive)
There are 6 pounds in each bag. How many bags? (quotitive)
In a comparison situation that involves division, the size of one object may be unknown or the relative sizes of two objects may be unknown. For example, when comparing two ropes:
A rope is 12 feet long. It is twice as long as another rope. How long is the second rope? (partitive)
One rope is 12 feet long. One rope is 6 feet long. How many times longer than the second rope is the first rope? (quotitive)
In situations that involve arrays or rectangles, division can be used to find an unknown factor. In an array situation, the unknown is the number of entries in a row or a column; in a rectangle, the unknown is a length or a width measurement. For example, “The area of a rectangle is 12 square feet. One side is 6 feet long. How long is the other side?” If the rectangle is viewed as tiled by an array of 12 unit squares with 6 tiles in each row, this question can seen as asking for the number of entries in each column.
At beginning of the unit, students consider how the relative sizes of numerator and denominator affect the size of their quotient. Students first compute quotients of whole numbers, then—without computing—consider the relative magnitudes of quotients that include divisors which are whole numbers, fractions, or decimals, e.g., “Is 800÷110 larger than or smaller than 800÷2.5?”
The second section of the unit focuses on equal groups and comparison situations. It begins with partitive and quotitive situations that involve whole numbers, represented by tape diagrams and equations. Students interpret the numbers in the two situations (MP2) and consider analogous situations that involve one or more fractions, again accompanied by tape diagrams and equations. Students learn to interpret, represent, and describe these situations, using terminology such as “What fraction of 6 is 2?,” “How many 3s are in 12?,” “How many fourths are in 3?,” “is one-third as long as,” “is two-thirds as much as,” and “is one-and-one-half times the size of.”
The third section concerns computing quotients of fractions. Students build on their work from the previous section by considering quotients related to products of numbers and unit fractions, e.g., “How many 3s in 12?” and “What is 13 of 12?,” to establish that dividing by a unit fraction 1b is the same as multiplying by its reciprocal b. Building on this and their understanding that ab=a⋅1b (from grade 4), students understand that dividing by a fraction ab is the same as multiplying by its reciprocal ba.
The fourth section returns to interpretations of division in situations that involve fractions. This time, the focus is on using division to find an unknown area or volume measurement. In grade 3, students connected areas of rectangles with multiplication, viewing a rectangle as tiled by an array of unit squares and understanding that, for whole-number side lengths, multiplying the side lengths yields the number of unit squares that tile the rectangle. In grade 4, students extended the formula for the area of rectangles with whole-number side lengths to rectangles with fractional side lengths. For example, they viewed a 23-by-57 rectangle as tiled by 10 13-by-17 rectangles, reasoning that 21 such rectangles compose 1 square unit, so the area of one such rectangle is 121, thus the area of a shape composed of 10 such rectangles is 1021. In a previous grade 6 unit, students used their familiarity with this formula to develop formulas for areas of triangles and parallelograms. In this unit, they return to this formula, using their understanding of it to extend the formula for the volume of a right rectangular prism (developed in grade 5) to right rectangular prisms with fractional side lengths.
The unit ends with two lessons in which students use what they have learned about working with fractions (including the volume formula) to solve problems set in real-world contexts, including a multi-step problem about calculating shipping costs. These require students to formulate appropriate equations that use the four operations or draw diagrams, and to interpret results of calculations in the contexts from which they arose (MP2).