The purpose of this activity is for students to find the product of a one-digit number and a two-digit number in ways that make sense to them. Students build on IM Grade 3 work with arrays to consider how to find the total number in an array without counting by 1. Students are not asked to find the answer, but instead share their strategies for doing so. This allows teachers to observe how students make sense of multiplying larger numbers.

Monitor for and select students with the following approaches to share in the Activity Synthesis:

• Decompose the two-digit factor by place value, such as partition the 13 columns in the array into 10 and 3 columns.
• Decompose one of the factors into multiple groups (for example, using doubling or tripling).

The approaches will later be displayed side by side to support students in finding the product of a two-digit number and one-digit number in ways that make sense to them. In addition to different ways to decompose the two-digit factor, focus the discussion on student drawings of arrays or diagrams that show the partitions they used and how the visual representation relates to expressions that involve the distributive or associative properties. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven’t shared recently. For an example for each approach, look at the Student Responses.

• Invite previously selected students to display their work side by side for all to see.
• Connect students’ approaches by asking:
  • “How are the ways to think about Elena's stickers the same and different?” (We decomposed one of the factors and showed it in a drawing, with an expression, or both. The difference is how we grouped or partitioned the stickers and we wrote it as an expression.) 
• Connect students’ approaches to the learning goal by asking:
  • “Which way of thinking about this product do you prefer? Why?” (Finding groups of 10 is easier because I can do the facts in my head. I like making 2 equal groups because I can use doubles.)

In this activity, students use strategies and representations that make sense to them to find products beyond 100. As before, the context of stickers lends itself to be represented with an array. The factors are large enough, however, it would be inconvenient, motivating other representations or strategies (MP2). Look for the ways that students extend or generalize previously learned ideas or representations to find multiples of larger two-digit numbers. While many of the student responses are written with expressions, students are not expected to represent their reasoning using equations and expressions at this time. Teachers may choose to represent student reasoning using equations and expressions so students can start connecting representations.

After students work on the first problem, pause to discuss some possible representations for finding the number of stickers. Each of the representations show different ways to represent the decomposition of a factor and students may decompose the factors in a variety of ways.

A. I created an array and decomposed it into smaller arrays.

B. I drew a diagram and decomposed it into smaller sections.

C. I decomposed the 21 and wrote one or more expressions or equations.

D. I decomposed the 9 and wrote one or more expressions or equations. 

Consider using the “four corners” structure to allow for movement and for interactions among students who might not typically interact. Post each of the four strategies in a different corner of the classroom. For the representations that use arrays or rectangular diagrams, it may help to give examples of decomposing using 1–2 samples of student work that you observe during the activity. Then ask students to move to a corner based on their reasoning strategy and representation.

• “In the Lesson Synthesis, let’s compare strategies to represent the problem and find the value of the products.”