Warm-up: Number Talk: Three Factors (10 minutes)
The purpose of this Number Talk is for students to mentally calculate a product which students will work with in context in this lesson. The first two products students may know from memory but if not, the idea of doubling \(8 \times 4\) to find \(8 \times 8\) can be helpful both for finding the value of \(8 \times 8\) and for starting a pattern that continues with the next product, \(8 \times 8 \times 2\), which is double \(8 \times 8\). The factors 8, 8, and 20 turn out to be the side lengths, in feet, of a standard container used on cargo ships. Students will examine these containers and the ships that carry them throughout the lesson.
- Display one expression.
- “Give me a signal when you have an answer and can explain how you got it.”
- 1 minute: quiet think time
- Record answers and strategy.
- Keep expressions and work displayed.
- Repeat with each expression.
Find the value of each expression mentally.
- \(8 \times 4\)
- \(8 \times 8\)
- \(8 \times 8 \times 2\)
- \(8 \times 8 \times 20\)
- “How did you find the value of \(8 \times 8 \times 20\)?” (I multiplied \(8 \times 8 \times 2\) by 10 because 20 is \(10 \times 2\).)
Activity 1: What a Waste (15 minutes)
The purpose of this activity is for students to make reasoned estimates about the volume of recyclable goods a school produces in a day, a week, a month, and a year. Students have seen cubic feet and cubic meters briefly in an earlier unit and it may be helpful to build a cubic foot to enable them to visualize this unit of measure and improve their estimates.
Answers to the questions here will vary widely based on the size of the school, the size of the recycling bins, and the exact estimates students make. Mathematically, the important point is that students use multiplication appropriately in their estimates and develop a sense for the staggering size of the waste that they will consider in the second activity.
Supports accessibility for: Social-Emotional Functioning, Language
- Gather a small recycle bin or trash can and a large recycle bin or trash can.
- “The United States ships recyclable goods like plastic to other countries for processing. We are going to estimate the volume of recyclable materials our school produces.”
- Show a recycling bin or trash can from class.
- Show a large recycling bin or large trash can from school (if possible).
- “How can we estimate the number of cubic feet the class recycling bin holds?” (We can measure the length, width, and height. We can build a cubic foot and put it inside the recycling bin to see how much space it takes up.)
- 1 minute: small-group discussion
- 5 minutes: individual think time
- 5 minutes: partner discussion time
- Monitor for students who
- make different estimates for the size of the recycling bins and how often they are filled
- make round estimates for the volume of recyclables the school produces and for the classroom
Estimate the value of each quantity.
- The number of cubic feet that the class recycling bin holds.
- The number of cubic feet that the school recycling bins hold.
About how many cubic feet of recyclable materials do you think your school produces in each amount of time? Explain or show your reasoning.
- a day
- a week
- a month
- a year
- Do you think all of the recyclable materials your school produces in a year could fit in your classroom? Show or explain your reasoning.
- Invite students to share their estimates for the volume of the different recycling bins.
- “How did you estimate the volume of the small bins?” (It looks like it’s about 2 feet tall and I think the base is maybe a square foot. So that’s 2 cubic feet.)
- “How did you estimate the volume of the large recycling bins?” (I think they’re about 3 feet tall with a base of about 4 square feet so that would be 12 cubic feet.)
Activity 2: Plastic Palooza (20 minutes)
The purpose of this activity is to use estimation to compare the amount of recyclable plastic produced by all elementary schools in the United States to the amount of recyclable plastics the United States exports every year for processing. In this task, an estimate for each school is provided. The new parts of this activity are considering all of the schools in the country, for which an estimate is provided, and the total amount of plastics exported by the United States each year, for which an estimate is also provided.
The numbers in this activity go beyond those expected for the grade 5 division standard but they are friendly and the quotient is small enough that students could find it by repeated subtraction or addition. It is also possible that they will use their place value understanding and their understanding of single-digit multiplication facts.
Advances: Writing, Speaking, Listening
- Groups of 2
- “You are going to compare an estimate for the amount of recyclable plastic produced by all elementary schools in the country with the amount of plastic the United States shipped in 2018 to other countries where it is recycled.”
- 5 minutes: individual work time
- 10 minutes: partner work time
- Monitor for students who
- recall that the they made the calculation for the first problem in the warm-up
- use the division algorithm to find the number of days it takes a school to fill one cargo container
- use the fact that \(3 \times 7 = 21\) and place value to find the value of \(210,\!000 \div 70,\!000\)
Your goal is to decide, by estimating, whether it is possible for all of the elementary schools in the country to produce enough recyclable plastic to fill the cargo containers that the United States ships each year.
- A standard cargo container for a ship measures 20 feet long, 8 feet wide, and 8 feet tall. What is the volume of the container?
- Each school makes about 40 cubic feet of recyclable plastic each day. How many days would it take for a school to fill one cargo container?
- In 2018 the United States exported about 210,000 cargo containers of plastic. There are about 70,000 elementary schools in the United States. How many cargo containers does each school need to fill in order to fill all of these containers?
- Do you think all the schools in the country produce enough plastic recyclables to fill all the cargo containers that the United States ships? Show or explain your reasoning.
- Invite students to share their solutions for finding the number of days it would take a school to fill one of the cargo containers with plastic.
- “How can you use the equation \(128 \div 4 = 32\) to find the value of \(1,\!280 \div 40\)?” (They have the same value because if 32 groups of 4 is 128, putting ten times as many in each group will give 10 times as much for the total.)
- Invite students to share their response for how many cargo containers each school will fill.
- “How did you find the value of \(210,\!000 \div 70,\!000\)?” (I know that \(3 \times 70 = 210\) and so this is also true for thousands.)
- Invite students to share their reasoning for the last question about all elementary school plastic recyclables.
- “Much of the debris that is in the Great Pacific Garbage Patch is plastic. Today, we got a sense of how much recyclable plastic all of the schools in our country produce.”
“Today we made estimates for the amount of recyclable plastic elementary schools might produce and compared this with the amount of plastic that the United States ships abroad.”
“What are some of the different estimates you made or worked with today?” (the volume of recycling bins, the amount of things we put in the bins each day, the number of schools and amount of recyclable plastics shipped)
“How is calculating with estimates the same as using exact values? How is it different?” (I still need to know what operation to use. But the round numbers are easier to calculate with.)
“If you knew that there were 68,372 schools rather than 70,000 and the U.S. shipped 207,364 cargo containers of plastic, would that change your answer to the question whether the schools could fill all of those containers? Why?” (No. I don't think I could find the value of the quotient but it should still be close to 3.)
“Estimation is important not only to check the reasonableness of answers but also we sometimes don't need an exact calculation to answer a question.”