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:

• The assumptions students make about the size of the boxes.
• The units of measurement students choose for the boxes and the warehouse.
• Whether or how measurements or estimates are rounded and converted in the process of problem solving.

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 side lengths and volume of each box.
• How the boxes are stacked or arranged.
• The side lengths and volume of the warehouse.
• Conversions from the smaller units used for the cereal box to larger units used for the building (or vice versa).

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:

• “Why did groups end up with different answers? Is that expected?” (It is expected because any approach will have different assumptions. A focus on volume will give a different answer than a focus on how many boxes fit in each direction.)
• “How did you begin to answer the question? What aspects of the situation did you think about first?" (We started by using one side of the box to decide how long another side of the warehouse could be. Since each box is 2.5 inches wide, we planned for 20,000 boxes to fit along one wall. That wall would need to be about 4,200 ft long. We then knew two side lengths of the warehouse, so we could solve for the third.)
• “Did you round any numbers while working on the problem? Which quantities did you end up rounding and why?” (We rounded when converting between units, for example when changing from inches to feet to miles. Rounding numbers to powers of 10 or to larger units made calculations and estimation easier. Larger units could also be more informative. For example, expressing the side length in miles might be more understandable than thousands of feet or inches.)

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:

• Composing two (or all four) trapezoidal faces to make a parallelogram and then finding its area.
• Decomposing each trapezoid face into a rectangle and two triangles and then finding the sum of their areas.

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).

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:

• “What do the approaches have in common? How are they different?”
• “Did anyone solve the problem the same way, but would explain it differently?”
• “How do these different representations show the same information?”