Unit 9, Lesson 1
Fermi Problems (Optional)
Accelerated 6
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This activity is an opportunity for students to apply their understanding of rates to solve a Fermi problem estimating the time it would take an ant to run from Los Angeles to New York City. To do so, students need to consider various factors that could affect the time of travel, such as the distance between the two cities, the speed of travel, whether breaks are involved, and so on. As they consider relevant factors and make a plan to approach the problem, students make sense of problems and persevere in solving them (MP1). In making estimates and performing calculations involving large numbers, students practice attending to precision (MP6).
Monitor for different approaches to solving the problem, and select students to share during the discussion. In particular, look for:
Tell students that they will now answer the question about how much time it would take an ant to travel between the two cities. Remind students that a problem like this is called a Fermi problem—a problem that cannot be solved by measuring directly but can be answered by estimating and reasoning.
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. Provide access to four-function calculators.
Vital information to have on hand includes:
If conducting a Gallery Walk at the end, provide access to tools for making a visual display.
An ant is running from Los Angeles to New York City. How long will the journey take?
If students have trouble starting with no given information, consider asking:
Invite students or groups to share different solution approaches. Alternatively, consider asking students to create a visual display and conducting a Gallery Walk. Highlight explanations or visual displays that include keeping careful track of information such as:
The goal of this discussion is to make sure students understand that many different estimates are expected and correct. Here are some questions for discussion:
Also of interest is the fact that 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.
In this activity, students solve a Fermi problem involving geometry: estimating the total volume occupied by all of the breakfast cereal purchased in a year in the United States. To do so, students need to gather information about the number of households, estimate the size of a typical cereal box, and make assumptions about cereal consumption. This work engages them in aspects of mathematical modeling (MP4). As students make estimates and work with large numbers, they reason abstractly and quantitatively (MP2).
Monitor for different approaches to solving the problem, and select students to share during the discussion. In particular, look for:
If available, display a cereal box for all to see. Tell students that they are going to solve a Fermi problem about cereal boxes. Ask students to estimate how many boxes of that size could fill their locker, a storage cabinet in the classroom, or the classroom itself. Give students a minute of quiet think time and invite students to share their responses.
Next, display and read aloud the first paragraph of the task. If needed, clarify that a warehouse is a large building used to store goods or equipment. Give students a minute of quiet think time to make a guess. Poll the class on their responses and 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:
Arrange students in groups of 2–4. Provide access to four-function calculators. If conducting a Gallery Walk at the end, provide access to tools for making a visual display.
Imagine a warehouse that has a rectangular floor and that holds 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?
Invite students or groups to share different solution approaches. Consider asking students to create a visual display, and consider conducting a Gallery Walk. Highlight explanations or visual displays that include keeping careful track of information such as:
The goal of this discussion is to make sure students understand that there are many possible ways to approach the problem, and that different considerations will yield different estimates. Here are some questions for discussion:
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 (for example, 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.
In this activity, students estimate the total number of tiles needed to cover the Washington Monument. To do so, they apply what they know about units of measurement, tiling, and ways to find the area of two-dimensional figures. Students' answers will vary depending on, among others, the assumptions and estimates made about the situation, the size of tiles used, and the precision of rounding. Estimation may also come into play as students consider the height of each triangular face, how to tile the edges of each trapezoidal face, the thickness of any joints between the tiles, and so on.
Monitor for groups who use different approaches to determine the area of the faces of the monument, such as:
As students apply the mathematics they know and make assumptions, decisions, and approximations to solve a practical problem, they engage in aspects of mathematical modeling (MP4). In communicating their assumptions, selecting the units to use, and deciding on the level of precision in reporting results, students practice attending to precision (MP6).
Show a picture of the Washington Monument.
Ask students if they know the monument or have visited it. Invite students who are familiar with the monument to share what they know, such as its location and purpose, when and how it was constructed, and so on. Offer additional background information about the monument as needed to orient students to the context.
Next, invite students to guess the height of the monument. Poll the class on their responses, and record all guesses, including the units of measurement in each guess. Then read the question that students are to answer. Give students a minute 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.
Information students may need includes:
Length measurements of the monument.
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 that it is close to 57.5 feet, or they can use the 55 feet for their estimate. This does not have a significant effect on the area calculation, because the pyramid at the top does not account for most of the surface area.
Arrange students in groups of 2–4. Provide access to four-function calculators. If conducting a Gallery Walk at the end, provide access to tools for making a visual display.
Select work from students with different strategies, such as those described in the Activity Narrative, to share later.
The goal of this discussion is to share a variety of approaches to the same situation. Invite students or groups to share different solution approaches. Consider asking students to create a visual display, and consider conducting a Gallery Walk.
After all strategies have been presented, display 2–3 approaches from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:
How many tiles would it take to cover the Washington Monument?
If students do not account for the faces they do not see in the picture, consider asking: