Unit 4: Dividing Fractions
Lesson 11: Using an Algorithm to Divide Fractions

In the previous lesson, students began to develop a general algorithm for dividing a fraction by a fraction. They complete that process in this lesson. Students calculate quotients using the steps they observed previously (i.e., to divide bya/b , we can multiply by a and divide by b), and compare them to quotients found by reasoning with a tape diagram. Through repeated reasoning, they notice that the two methods produce the same quotient and that the steps can be summed up as an algorithm: to divide by a/b, we multiply by b/a (MP8). As students use the algorithm to divide different numbers (whole numbers and fractions), they begin to see its flexibility and efficiency.

Unit 5: Arithmetic in Base Ten
Lesson 9: Using the Partial Quotients Method

Prior to grade 6, students reasoned about division of whole numbers and decimals to the hundredths in different ways. In this first lesson on division, they revisit two methods for finding quotients of whole numbers without remainder: using base-ten diagrams and using partial quotients. Reviewing these strategies reinforces students’ understanding of the underlying principles of base-ten division—which are based on the structure of place value, the properties of operations, and the relationship between multiplication and division—and paves the way for understanding the long division algorithm. Here, partial quotients are presented as vertical calculations, which also foreshadows long division.

In a previous unit, students revisited the two meanings of division—as finding the number of equal-size groups and finding the size of each group. Division is likewise interpreted in both ways here (MP2). When using base-ten diagrams or dividing by a small whole-number divisor, it is often natural to think about finding the size of each group. When using partial quotients, it may be more intuitive to think of division as finding the number of groups (e.g., 432 / 16 can be viewed as “how many 16s are in 432?”).

Unit 4: Dividing Fractions
Lesson 15: Volume of Prisms

In this lesson, students complete their understanding of why the method of multiplying the edge lengths works for finding the volume of a prism with fractional edge lengths, just as it did for prisms with whole-number edge lengths. They use this understanding to find the volume of rectangular prisms given the edge lengths, and to find unknown edge lengths given the volume and other edge lengths.

Problems about rectangles and triangles in the previous two lessons involved three quantities: length, width, and area; or base, height, and area. Problems in this lesson involve four quantities: length, width, height, and volume. So finding an unknown quantity might involve an extra step, for example, multiplying two known lengths first and then dividing the volume by this product, or dividing the volume twice, once by each known length.

In tackling problems with increasing complexity and less scaffolding, students must make sense of problems and persevere in solving them (MP1).

Unit 4: Dividing Fractions
Lesson 7: What Fraction of a Group?

In the previous three lessons, students explored the “how many groups?” interpretation of division. Their explorations included situations where the number of groups was a whole number or a mixed number. In this lesson, they extend the work to include cases where the number of groups is a fraction less than 1, that is, situations in which the total amount is smaller than the size of 1 group. In such situations, the question becomes “what fraction of a group?”.

Students notice that they can use the same reasoning strategies as in situations with a whole number of groups, because the structure
(number of groups) x (size of groups) = (total amount)
is the same as before (MP7). They write multiplication equations of this form and for the corresponding division equations.

Throughout the lesson, students practice attending to details (in diagrams, descriptions, or equations) about how the given quantities relate to the size of 1 group.