Lesson 1

Fermi Problems

1.1: Ant Trek (20 minutes)

Optional activity

In this Fermi problem, students estimate the time it would take an ant to run from Los Angeles to New York City. Monitor for different approaches to solving the problem, and select students to share during the discussion. In particular, look for:

  • A range of estimates
  • Different levels of precision in reporting the final estimate
  • Different representations (for example, double number lines and tables)

Launch

Arrange students in groups of 1–4. Provide access to four-function calculators. If students are doing their own research, provide access to internet-enabled devices. If conducting a gallery walk at the end, provide access to tools for making a visual display.

Before starting, ask students to come up with a guess about the answer and poll the class. Record all guesses.

Ask students to brainstorm the information they need to answer the question. Give the information provided when students ask for it, or provide access to internet-enabled devices so that students can find the information they need.

Vital information to have on hand includes:

  • The distance between Los Angeles and New York City is about 3,944 km.
  • An ant can run about 18 mm per second.

Also of interest is the fact that most ants do not live long enough to complete this trip. Many ants live for only a couple months. If students realize this, ask them how many ant lifetimes it would take for an ant to make this journey.

Conversing: MLR5 Co-Craft Questions. To use this routine to support student understanding of the context of the problem. Display a map that shows a route between Los Angeles and New York City. Invite students to work with a partner to create a list of possible mathematical questions that could be asked about the situation. Invite students to share their questions with the class and call students' attention to the the different ways students integrate measurements (distance traveled, speed) in their questions.
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

How long would it take an ant to run from Los Angeles to New York City?

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

Students may struggle to get started with no given information. Ask these students what they need to know in order to estimate how long it takes the ant to travel across the country. They may respond in a variety of ways, for example:

  • How far does the ant travel in a day (or a different period of time)? Give them the information that the ant travels 18 mm per second.
  • How far is it from Los Angeles to New York City? (They may ask for this distance in miles.) Give them the information that it is about 3944 km. 

Students may struggle to convert mm to km. Ask these students how many mm are in a cm? in a meter? How many meters are in a km?

Students may be distracted by other concerns, like what the ant will eat, whether or not the ant will travel throughout the winter, whether or not the ant can cross mountains. Help these students understand that it is okay to make simplifying assumptions so that the calculations are reasonable.

Activity Synthesis

Invite students or groups to share different solution approaches. Consider asking students to create a visual display and conducting a gallery walk. Highlight strategies that include keeping careful track of the information used including:

  • How far the ant has to travel
  • How fast the ant travels
  • A step-by-step analysis changing mm per second to km per day and eventually km per year

Make sure to discuss why student estimates are not all identical. For these open-ended problems that require many assumptions, there is not a single valid approach or answer. Some reasons for a variety of answers in this case include:

  • Students may take into account that the ant will need to rest or sleep.
  • Students may choose different speeds for the ant.
  • Students may round their answers differently.

If students report their answers in a variety of ways (to the nearest year, nearest month, nearest day, nearest hour, etc.), ask them what is most appropriate. Reporting to the nearest year is perhaps most appropriate, however, because there are so many hidden variables that influence the answer, including:

  • The speed is only approximate and would not remain constant for years.
  • The distance is also only approximate as the ant will need to circumnavigate many obstacles.
  • An answer to the nearest year communicates the general idea of how long it will take very effectively.

1.2: Stacks and Stacks of Cereal Boxes (20 minutes)

Optional activity

In this Fermi problem, students estimate the total volume occupied by all of the breakfast cereal purchased in a year in the United States.

Monitor for different approaches to solving the problem, and select students to share during the discussion. In particular, look for:

  • A range of estimates
  • Different levels of precision in reporting the final estimate
  • Different arrangements of boxes

Launch

Arrange students in groups of 1–4. Provide access to four-function calculators. If students are doing their own research, provide access to internet-enabled devices. If conducting a gallery walk at the end, provide access to tools for making a visual display.

Before starting, ask students to come up with a guess about the answer and poll the class. Record all guesses.

Ask students to brainstorm the information they need to answer the question. Give the information provided when students ask for it, or provide access to internet-enabled devices so that students can find the information they need.

Vital information to have on hand includes:

  • Every year, people in the U.S. buy 2.7 billion boxes of breakfast cereal.
  • A “typical” cereal box has dimensions of 2.5 inches by 7.75 inches by 11.75 inches.
Representation: Internalize Comprehension. Represent the same information through different modalities by using physical objects to represent abstract concepts. Some students may benefit by starting with an image, video, or a typical size cereal box to support visualization.
Supports accessibility for: Conceptual processing; Visual-spatial processing

Student Facing

Imagine a warehouse that has a rectangular floor and that contains all of the boxes of breakfast cereal bought in the United States in one year.

If the warehouse is 10 feet tall, what could the side lengths of the floor be?

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

  • Students may not think about how the boxes are packed. That is, they may make a volume calculation and then perform division. (Ask these students to think about how the boxes are placed in the warehouse.)
  • Students may struggle with unit conversions. (Encourage them to go step by step: How many inches are in a foot? How many square inches are in a square foot? How many cubic inches are in a cubic foot?)

Activity Synthesis

Invite students or groups to share different solution approaches. Consider asking students to create a visual display and conducting a gallery walk. Highlight strategies that include keeping careful track of the information used including:

  • What is the volume of each box?
  • How are the boxes arranged for storage?
  • How are the dimensions of the warehouse floor determined?

Make sure to discuss why different people have different estimates.

  • Some students may just calculate volume and not think of how the boxes are packed. (This is OK as the amount of empty space will not be large unless the boxes are placed haphazardly.)
  • Students may place the boxes in a different way.
  • Students may or may not take account of the small amount of space in the warehouse that is not filled by the boxes.

There are an infinite number of rectangles with suitable area to serve as the floor of the warehouse. Note, however, that having an appropriate area (e.g., 36,000,000 square feet) is not sufficient. For example, we could not have a warehouse that is 1 inch wide and 432,000,000 feet long. In general, if one of the dimensions is too small, it increases the amount of wasted space.

1.3: Covering the Washington Monument (20 minutes)

Optional activity

In this Fermi problem, students estimate the total number of tiles needed to cover the Washington Monument.

Monitor for different approaches to solving the problem, and select students to share during the discussion. In particular, look for:

  • A range of estimates
  • Different levels of precision in reporting the final estimate
  • Different choices made to simplify the situation

Launch

Arrange students in groups of 1–4. Provide access to four-function calculators. If students are doing their own research, provide access to internet-enabled devices. If conducting a gallery walk at the end, provide access to tools for making a visual display.

Show a picture of the Washington Monument.

A photograph of the Washington Monument.

Before starting, ask students to come up with a guess how many tiles it would take to cover the Washington Monument and poll the class. Record all guesses.

Ask students to brainstorm the information they need to answer the question. Give the information provided when students ask for it, or provide access to internet-enabled devices so that students can find the information they need.

Vital information to have on hand includes:

  • Dimensions of the monument:
A diagram of the Washington Monument.
  • Standard sizes for square tiles: side lengths of 1 inch, 6 inches, 8 inches, 1 foot, and \(1 \frac{1}{2}\) feet.

Note that 55 feet is the height of the pyramid at the top, not the height of each triangular face. Students would need the Pythagorean Theorem to find the height of these faces, so either tell them it is close to 57.5 feet or they can use the 55 feet for their estimate. This does not have a significant impact on the area calculation, as the pyramid at the top does not account for most of the surface area.

Action and Expression: Internalize Executive Functions. Provide students with a graphic organizer to capture information about the different assumptions they made when calculating the areas of the trapezoidal faces of the Washington Monument building.
Supports accessibility for: Language; Organization

Student Facing

How many tiles would it take to cover the Washington Monument?

An outline of the Washington Monument with one trapezoidal side and one triangular face.

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

Students may not account for the faces of the monument that you cannot see in the picture.  Remind these students that some faces are hidden. If three-dimensional building blocks are available, consider having students who have difficulty visualizing the monument build a small model. It is difficult to model the exact shape of the Washington Monument, but the physical model may help students make the needed calculations. 

Students may have difficulty converting units, especially if their tiles are measured in inches. Ask these students how many inches are in a foot and how many square inches are in a square foot. 

Activity Synthesis

Invite students or groups to share different solution approaches. Consider asking students to create a visual display and conducting a gallery walk. Highlight different methods of calculating the area of the trapezoidal faces of the Washington Monument including:

  • Putting two trapezoid faces (or all 4 faces) together to make a parallelogram
  • Decomposing each trapezoid face into a rectangle and two triangles and then rearranging

Discuss why different people have different estimates. Some reasons to highlight include:

  • Using different-sized tiles to cover the Washington Monument
  • Thinking about how some tile is wasted, namely the tiles that go on the edge of each face
  • Rounding

This problem has a different flavor than the two preceding ones. The only place estimation comes into play is with how to cover the top triangles and perhaps thinking about what happens to tiles at the edge of a face. What all the problems share, however, is that we cannot give an exact answer. To provide a reasoned estimate, keeping track of all assumptions being made is vital. This tells us not only how accurately we should report our answer but also helps us communicate clearly to others who may have made different assumptions.

Speaking: MLR8 Discussion Supports. Give students additional time to make sure that everyone in their group can explain their strategy for estimating the number of tiles using appropriate mathematical lanague (e.g., face, height, width, surface area, trapezoid, etc.). Be sure to vary who is called on to represent the ideas of each group. This routine will prepare students for the role of group representative and to support each other to take on that role.
Design Principle(s): Optimize output (for explanation)