In this section, students use their knowledge of multiplication, place value, and area of rectangles to multiply one-digit numbers and numbers up to four digits, and to multiply pairs of two-digit numbers.

A key thread in this section is the idea of decomposing factors—particularly by place value—as a productive way of finding products. Students explore this idea with concrete and visual representations: arrays, base-ten diagrams, and rectangles with grids. As they decompose larger factors, students see the limits of these representations, motivating more efficient representations and strategies.

In IM Grade 3, students saw that rectangles can help us reason about multiplication—the side lengths of a rectangle can represent the two factors and its area can represent the product. As the factors become larger (for instance, ), it becomes necessary to draw rectangles whose side lengths are not to scale. When rectangles no longer accurately represent area, the term “area diagrams” is not used. Instead, “rectangular diagrams” is used in teacher materials and “diagrams” in student materials.

Later in this section, students encounter an algorithm that uses partial products, a different way to record the reasoning they used with diagrams. They learn that the partial products can be listed vertically, instead of inside the boxes of a rectangular diagram.

Students use this algorithm to multiply two-digit numbers, likewise connecting the partial products to the values in a corresponding diagram.

Algorithms that use partial products prepare students to make sense of the standard algorithm for multiplication, which students preview in this unit but will study closely in IM Grade 5.

In IM Grade 3, students made sense of division in relation to multiplication and equal-size groups. They reasoned about division problems in context and found whole-number quotients from two-digit dividends and one-digit divisors. In this section, students find quotients from larger dividends (up to four digits), investigate new division strategies and ways to represent them, and interpret division situations that involve remainders.

Students begin by solving problems in various situations, including those about equal-size groups and areas of rectangles. These experiences reinforce students’ understanding of the relationship between multiplication and division. They also build students’ intuition for the kinds of situations that involve division (including those where a remainder may be involved), before focusing on finding the value of quotients.

Students first reason about division problems in any way that makes sense to them, and later use base-ten representations. Students recall that to find the value of  , they could first put 4 tens and 4 ones into 4 groups (1 ten and 1 one in each group), and then decompose the remaining 2 tens into 20 ones and put 5 ones in each group.

Students see that as they distribute blocks of tens and ones into groups incrementally, they can decompose a dividend into parts and find partial quotients.

Later in the section, students take a closer look at division problems that do not have whole-number quotients and interpret the remainders in the context of the problem.

In this section, students generate geometric and numerical patterns that follow a given rule and analyze features of these patterns that are not explicit in their rule. They use these features to predict future terms in a pattern sequence. To make predictions, students use their understanding of operations and place value.

The section begins with patterns that are more concrete—such as shapes with features that change quantitatively and elicit addition or multiplication. It then moves toward patterns with repeating objects or numbers, which require students to reason more abstractly. Later in this section, students connect their experiences with the Choral Count instructional routine to make sense of numerical patterns. They generate and analyze numerical patterns that elicit what they know about factors, multiples, and place value in anticipation of the work of upcoming sections in the unit.

The section also progresses from a more informal expectation for the way students describe the features of patterns that are explicit in the rule to an increasingly more focused expectation. Features of patterns are any regularities that students notice in the pattern they generate. For the growing shape patterns describe the shapes in the pattern that stay the same and the shapes that change. It may also involve looking for ways to quantify different aspects of the pattern (such as the total number of shapes used) and looking for patterns in the numbers they use.

Rule: Start with 5, keep adding 5.

• The total number will have a pattern of odd, even, odd, even, odd, even because each time you add 5 you’ll either make a ten or add 5 to a ten.
• You’re adding a group of 5 so Step 1 will be , Step 2 will be , Step 3 will be , Step 4 will be , and so on.

When students analyze numerical patterns in later lessons, they may continue to describe features such as patterns in even and odd numbers or digits, but may also use what they know about multiplication to describe the numbers in a pattern as multiples or use expressions to describe each term.

In the final section of this unit, students engage with a variety of contextual problems that involve multi-digit numbers and all four operations. The problems can be approached in many ways, presenting students with opportunities to choose their strategies and representations strategically. Many of them also involve multiple steps and justifications, prompting students to practice constructing logical reasoning and critiquing the reasoning of others (MP3).