Lesson 15

The purpose of this warm-up is to elicit students’ understandings of the relationship between the side lengths of a rectangle and its area. These understandings prepare students to reason about an unknown length or width of rectangles in the activities.

Students use their understanding about the relationship between multiplication and division, and their understanding of multiples of 10 to divide beyond 100.

• Groups of 2
• Display the image.
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “How did you make your estimate? How did you know it’s reasonable?” (Sample response: I know that the area is about 2,300 and one of the side lengths is 40. I know that \(40 \times 60 = 2,\!400\) and \(40 \times 50 = 2,\!000\), so the estimate is between 50 and 60, but closer to 60.)

• “Is anyone’s estimate less than 40 meters? Is anyone’s estimate greater than 60 meters?”
• “Based on this discussion does anyone want to revise their estimate?”

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 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).

• Groups of 2
• Explain what a mural is or show an image.

• 5 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who use multiples of 10 and 7 and then either add or subtract multiples of 7 to reach 189.

• 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. 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, \(140 \div 7 = 20\) and \(49 \div 7 = 7\), so \(189 \div 7 = 27\).)

This activity continues the work in the first activity. It uses a similar context and prompts students to reason about division, but the result of the division has a remainder, which students will need to interpret.

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

• If doing a gallery walk, create 3–4 posters to display during the activity that show or describe different strategies students are likely to use to solve the problem.  

• Groups of 2
• Give access to grid paper, in case students wish to use it to create an area diagram.

MLR7 Compare and Connect

• 2–5 minutes: independent or group work
• Give each student a sticky note.
• “Make one round to visit each poster. Place your sticky note on the poster with a strategy that matches your strategy or that makes the most sense to you.”
• “After your first round, make another round to visit 1–2 other posters that you didn’t select. Make sense of the strategy of the poster and be prepared to explain how it is different than yours.”
• 5–7 minutes: gallery walk

• Discuss the results of the gallery walk: “Which strategy seems to be most common? The least common? Why might they 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?” \((197 - 192 = 5\). Five tiles were not used.)
• Consider asking: “How are partial quotients a helpful strategy for finding the side length?” (Sample response: 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. \(197 - 180 = 17\). 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.)