In this section, students investigate the quantities in a division situation, recall the relationship between multiplication and division, and revisit ways to represent this relationship.

First, students look at how the numbers in a division situation are related. Students observe—in concrete and abstract situations—how the size of the dividend and the divisor affect the size of the quotient. Next, students reason about division in terms of equal-size groups. They are reminded that:

• Multiplication is a way to find the total amount when given the number of groups and the size of each group.
• Division is a way to find an unknown factor, which can either be the number of groups or the size of one group, when given a total amount. This means division can be interpreted in two ways.

For instance, here are two ways to think about :

How many groups of 6 are in 12?
 

How much is in one group if there are 6 groups in 12?
 

The last lesson in the section is optional. It offers additional practice in interpreting and representing situations that involve equal-size groups, encouraging students to pay close attention to what the parts and numbers mean. 

A note about the term “group”:

Students may be most familiar with the idea of a group as a collection of people or objects. Clarify that the term is used more broadly here. A “group” can refer to a part, a batch, a bag, a section, or another quantity with a particular value. So “equal-size groups” can refer to collections with the same number of items or people in each, as well as parts with the same value, sections of equal length, bags of the same weight, and so on. As students reason about various multiplication and division situations, the meaning of “group” in each situation will become more intuitive.

A note about notation:

When writing multiplication equations to represent, for instance, “How many groups of 4 are in 12?” students may write either or as long as they understand what each factor represents. Because we tend to say “ groups of ” in these materials, we follow that order in writing the multiplication equation:

In this section, students reason about quotients of multi-digit numbers and work toward the standard algorithm for division.

The section begins with an optional lesson that revisits concepts from grade 5 and reinforces the role of place value in division. Students use base-ten diagrams to represent division and use strategies based on place value and properties of operations to find quotients.

Next, students calculate quotients by finding partial quotients, likewise building on a strategy developed in earlier courses. Students learn to organize their calculations of partial quotients vertically and to record the steps in a systematic way, which paves the way to long division.

Then students make sense of the standard division algorithm by analyzing worked examples, interpreting and explaining the numbers in each step. They learn that to use long division is essentially to find the partial quotients, but doing so one digit at a time and relying on the place of the digit to convey its value. Students practice using the algorithm—first to divide a whole number by a whole number, and then to divide a decimal by a whole number. Finally, students use equivalent division expressions and the algorithm to divide a decimal by a decimal.

This section deepens students’ understanding of the base-ten structure and extends their ability to add, subtract, and multiply decimals beyond hundredths.

Students begin by revisiting decimal operations in the context of money. They make sense of place-value relationships and use base-ten diagrams to represent decimals, their sums, and their differences.

Students recall that each place of the base-ten system has a value that is 10 times that of the place to its right and of the value of the place to its left. This means that 10 of a base-ten unit can be composed into 1 larger unit, and 1 base-ten unit can be decomposed into 10 of a smaller unit.

A base-ten diagram labeled “Diego’s Method.” There are 2 columns for the diagram. The first column header is labeled "tenths" and there are 4 rectangles. The second column header is labeled "hundredths" and there are 10 squares in that column. The last rectangle is circled with a dashed line and an arrow pointing from the rectangle to the column of squares is labeled “decompose.” The last three squares are crossed out.

Students reason about the composition and decomposition using both base-ten diagrams and vertical calculations (or standard algorithm). They apply their understanding to add and subtract decimals, regrouping the values as needed.

Next, students compare several methods for multiplying decimals. They express decimal factors as fractions, multiply the fractions, and rewrite the products as decimals. They also make use of the structure of base-ten numbers. They multiply decimal factors by powers of 10 to obtain whole-number factors, multiply them, and then divide the result by the same powers of 10.

Then students reason using area diagrams. They represent the factors as the side lengths of a rectangle, decompose them by place value (which decomposes the rectangle into sub-rectangles), find the partial products (the areas of the sub-rectangles), and add them to find the product of the original factors. Students see that these steps can be recorded concisely in a vertical calculation, and they make connections between the diagrams and the calculations.

In the last lesson of the section, students generalize the various ways of reasoning into an algorithm. They also summarize their observations about the placement of the decimal point in the product of decimals.

A note about language:

In these materials, the effect of multiplying a number by a power of 10 or 0.1 is described in terms of a shift in the places of the digits of the number, rather than a shift in the decimal point. The location of the decimal point is always between the ones and the tenths, rather than changing as a result of an operation. So instead of saying that multiplying 0.054 by 100 “moves the decimal point two places to right,” we say that it “moves the digits two places to the left,” as the 5 and 4 are now in the ones and tenths place, respectively. Because the decimal point does have a new position relative to the digits, it is fine if students describe their observation as the “decimal point moving” as long as they understand what is happening in terms of place value. This understanding will serve students well into their future studies.

In this section, students make sense of division situations involving fractions, gradually building an understanding that can be generalized into an algorithm.

The first two lessons explore “How many groups?” questions in various contexts. Students investigate situations in which the number of groups is a fraction greater than 1 or less than 1. For the latter, it makes sense to ask “What fraction of a group?” instead of “How many groups?” To represent and reason about different situations, students use tape diagrams and equations. For instance, “How many s are in 2?” can be represented in these ways:

Next, students explore cases in which the size of one group is unknown. They learn that questions such as “If there are 3 pounds in bags, how much is in 1 bag?” and “ of what number is 4?” can also be answered by dividing and can be represented with diagrams and equations. An optional lesson gives practice finding the amount in one group and solving various division problems in context.

As students use tape diagrams to reason repeatedly about division by a fraction, they look for structure and regularity in the process. Students generalize that to divide a number by a non-unit fraction is to figure how many unit fractions are in (find ) and then see how many groups of unit fractions can be formed (find ). They conclude that to find is to find , or . Finally, students practice applying this algorithm and using it strategically to solve problems involving constant rates between fractional quantities.

In this section, students apply their insights about multiplication and division, as well as prior knowledge about area and volume, to solve geometric problems.

Students first solve multiplicative comparison problems in contexts that involve fractional lengths in one dimension. They use their understanding of division to answer questions such as “How many times as tall or as far is this as that?”

Next, students reason about measurements of two-dimensional figures. They find the area of rectangles and triangles given base-height pairs and find an unknown base or height given the area and a known height or base.

Students then go on to reason about three-dimensional figures. They multiply and divide to find the volume of rectangular prisms with fractional edge lengths and to find an edge length when the volume of a prism and two other edge lengths are known. The work in this section builds on geometric concepts from grade 5, during which students found the area of rectangles with fractional side lengths and the volume of rectangular prisms with whole-number edge lengths.