In this activity, students find the length of one side of a rectangle given the length of the other side and the area of the rectangle. This work builds on what students have done in IM Grade 3, where the area was within 100 square units. In this lesson, the area is a three-digit number beyond 100.

The use of tiles as a context and the presence of a grid allow students to see more concretely the relationship between a product and a factor, but the size of the product discourages students to count the tiles to find the unknown factor. Instead, students are encouraged to find multiples of the known factor or to decompose the product into parts (MP2).

• Select students to share their responses and reasoning. Display their responses for all to see.
• Highlight strategies that use multiples of 10 and 7 to find the side length of the rectangle by dividing or multiplying in parts. If no students use an area diagram in their reasoning, display an example for all to see (as shown in Student Responses).
• Invite students to write the equations they wrote. If no division equations are included, display division equations and ask students if they could be used to answer the question. (For instance, and , so .)

This activity continues the work in the first activity. This similar context prompts students to reason about division, but the result of the division has a remainder, which students will need to interpret. After this first informal encounter with leftovers, students will interpret quotients with remainders more formally in future lessons in this unit.

After students have had some independent work time, consider a Gallery Walk of strategies.

• Post 3–4 posters around the room, each showing a likely strategy for the last problem, such as using partial products, partial quotients, pictures, or words.
• Provide some blank posters for students to show additional unique strategies.

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

• Discuss the results of the Gallery Walk: “Which strategy seems to be most common? The least common? Why might the strategy be the most or least common?”
• “How many tiles did Tyler use for his mural? How do you know?” (Tyler uses 192 tiles because the mural is rectangular and there are 6 rows of tiles. 192 is the greatest multiple of 6 within 197.)
• “How many tiles were not used?” . Five tiles were not used.)
• Consider asking: “How was dividing (or multiplying) in parts a helpful strategy for finding the side length?” (I can use multiples of 10, which are easy to divide in my head. For example, the greatest multiple of 10 that is also a multiple of 6 within 197 is 180. I try to choose the greatest number so that I can keep track of the dividend. . 17 is left from the dividend to divide by 6, but 17 is not a multiple of 6, so I repeat the strategy. The greatest multiple of 6 within 17 is 12, and 5 is remaining.)