options: Explain volume formulas and use them to solve problems.

Analyze, compare, create, and compose shapes.

Understand and apply theorems about circles.

Reason with shapes and their attributes.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Prove geometric theorems.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Solve real-world and mathematical problems involving area, surface area, and volume.

Use coordinates to prove simple geometric theorems algebraically.

Apply trigonometry to general triangles.

Prove theorems involving similarity.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Apply geometric concepts in modeling situations.

Define trigonometric ratios and solve problems involving right triangles.

Experiment with transformations in the plane.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Understand similarity in terms of similarity transformations.

Understand and apply the Pythagorean Theorem.

Make geometric constructions.

Classify two-dimensional figures into categories based on their properties.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Find arc lengths and areas of sectors of circles.

Translate between the geometric description and the equation for a conic section.

Understand congruence in terms of rigid motions.

Visualize relationships between two-dimensional and three-dimensional objects.
problem_activity: 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
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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.


correct_option_index: 7