Lesson 19

The purpose of this Notice and Wonder is to introduce the idea of large amounts of trash floating on the water. In an earlier unit, students saw that large amounts of trash are shipped around the world. Before reconsidering trash in a volume context, this lesson focuses on the area that trash covers in some parts of the ocean. Students may notice and wonder many things about the image. The key discussion points are that the debris seen in the image is mostly trash and it covers a large area.

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• "The image shows what it looks like in some parts of the oceans. The small things floating on the water are trash."
• "How do you think trash gets into the ocean?" (People throw things away near the ocean. People throw things into the ocean when they are on boats.)

In previous grades, students worked with metric units of length. The purpose of this activity is to reason about the size of a kilometer. Students consider the distance they might walk or run, perhaps to and from school or during sports practice, and they also consider other lengths that they might be familiar with such as going across town or taking a bike ride or the distance to a friend's house. Building an intuition for large numbers or distances or other measurements takes a lot of time and practice. The intent of this activity is to begin this work informally. A wide variety of answers should be expected and the goal of the synthesis is to share this variety and come to a general agreement of approximately how far a kilometer, or some number of kilometers, is.

• Groups of 2
• Lay a meter stick on the ground.
• “Tyler walked from his classroom to the cafeteria. He said, ‘I think that’s about a kilometer.’ Do you agree with Tyler?” (No, a kilometer is a long distance, it's 1,000 meters, and it is not that far from a classroom to the cafeteria.)
• 1 minute: quiet think time
• 1–2 minutes: partner discussion
• Give students access to meter sticks.

• 2 minutes: independent work time
• 3–4 minutes: partner discussion
• Monitor for students who have different answers and explanations for the question about the distance between their home and school including:
  • less because I can walk there really quickly
  • less because it’s no more than 1 or 2 lengths of a soccer field
  • about a kilometer because I sometimes walk, so it’s not too far, but it’s also not real close
  • more than a kilometer as I take a bus and it's a long trip

If students do not recognize the distance of a kilometer, give them a meter stick and ask, “Estimate distances around the school that would be 1 meter, 10 meters, and 100 meters.”

• Invite students to share their responses for how far it is from school to where they live.
• “How many meters are there in a kilometer?” (1,000)
• “About how many steps does it take to go 1 meter?” (2)
• “About how many steps does it take to go 1000 meters or 1 kilometer?” (2,000)
• “Does 2,000 steps help communicate how far 1 km is?” (Yes. That's a lot but I can count to 1,000 and take that many steps so it’s not too far. No. 1,000 steps is a lot, too much for me to imagine.)

In the previous activity students developed an intuition for the size of a kilometer. The purpose of this activity is to calculate the area of different states in square kilometers and compare them to the area occupied by the Great Pacific Garbage Patch, a large collection of trash floating in the Pacific Ocean. This activity has several distinct goals:

• communicate the enormity of the Great Pacific Garbage Patch
• continue to build a sense of distances by looking at a map of the United States and thinking about the sizes of different states
• multiply large numbers, most likely using the standard algorithm
• calculate a product of two 3-digit numbers for the first time

Students calculate the areas of different states in order to grasp the enormity of a floating area of trash in the Pacific Ocean. It turns out that this area is substantially larger than most states! As they calculate areas of different states, students will use the standard multiplication algorithm and estimation. Estimation plays an important role because the area of New Mexico is a product of two 3-digit numbers. Students are prepared to make and understand this calculation but it also provides an opportunity to see the power of estimation because the area of New Mexico can be readily compared with the area of the Great Garbage Patch without calculating the exact area. For the last question, students use the area of the states they calculated and apply that to a map of the United States.

Students model with mathematics when they make assumptions about the states, for example that they are approximated by rectangles, in order to calculate or estimate their area and when they choose states whose area is approximately equal to the area of the Great Garbage Patch (MP4).

• Groups of 2
• “The Great Pacific Garbage Patch is a large area in the Pacific Ocean where trash has accumulated and floats on top of the water. We are going to compare the size of this part of the ocean to different states in the United States.”
• Display an image of the state you live in.
• “What do you know about the size of our state?”
• Highlight Rhode Island, Delaware, and New Mexico on the map in the student workbook so students can visualize the states and their relative sizes.

• 5–8 minutes: independent work time
• 5–8 minutes: partner discussion
• Monitor for students who:
  • use the standard algorithm to find the areas of Rhode Island and Delaware
  • use the standard algorithm to find the area of New Mexico
  • show the size of the garbage patch differently on the map of the United States

• Invite students to share their calculations for Delaware.
• “Is Delaware close in size to the garbage patch? How do you know?” (No, it is not close at all. 1,000,000 is a lot more than 7,392.)
• “Is the garbage patch more or less than 10 times as large as Delaware?” (Much more, because that’s only about 70,000 which is way less than a million.)
• “Is the garbage patch more or less than 100 times as large as Delaware?” (More, because that’s about 700,000 square kilometers.)
• “How could you estimate the area of New Mexico?”(596 is real close to 600 which is a nice number. 552 is harder, but if I say that’s about 500 then I get an estimate of \(600 \times 500\) or 300,000.)
• “How did you decide which area to circle on the map?”(New Mexico was the biggest and 3 New Mexicos would make about 1,000,000 square miles so I circled 3 states close in size to New Mexico.)