options: Find arc lengths and areas of sectors of circles.

Understand congruence in terms of rigid motions.

Reason with shapes and their attributes.

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

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

Apply geometric concepts in modeling situations.

Explain volume formulas and use them to solve problems.

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

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

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

Use coordinates to prove simple geometric theorems algebraically.

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

Experiment with transformations in the plane.

Understand similarity in terms of similarity transformations.

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

Define trigonometric ratios and solve problems involving right triangles.

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

Understand and apply theorems about circles.

Understand and apply the Pythagorean Theorem.

Make geometric constructions.

Analyze, compare, create, and compose shapes.

Prove theorems involving similarity.

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

Apply trigonometry to general triangles.

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

Prove geometric theorems.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
problem_activity: Activity
In this activity, students calculate the surface area to volume ratios of animals with different shapes and think about how this ratio might be important in biology. Note that this is the first time students will be using formulas for the surface area and volume of a sphere, and that these formulas are given without derivation. Students used the sphere volume formula in grade 8.
Launch
“Assume a spherical cow” is a common joke about physics. Sometimes in physics, everything about the world is made as simple as possible. There is no air resistance, and everything is modeled with ideal shapes like a sphere or a point. Tell students they will do the same today with spherical elephants and cylindrical snakes.
Student Facing
For a sphere with radius
\(r\)
, its volume is
\(\frac43 \pi r^3\)
and its surface area is
\(4\pi r^2\)
.
Let’s model an elephant with a sphere that has a radius of 4.5 feet.
What is the volume of the elephant?
What is the surface area of the elephant?
Let’s model a snake with a cylinder of length 3 feet and diameter 0.2 feet.
What is the volume of the snake?
What is the surface area of the snake?
Compute the surface area to volume ratio, or
\(\frac{SA}{V}\)
, for each animal.
Student Response
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Activity Synthesis
Ask students how the surface area to volume ratio could affect the biology of snakes and elephants. The goal is for students to notice that the snake has a much higher ratio and to think about the impacts this may have.
One aspect of the biology is that elephants are warm-blooded, so to maintain a consistent body temperature, their bodies need to minimize the surface area through which heat enters or escapes their bodies. Snakes are cold-blooded, so their bodies maximize surface area in order to absorb heat from the environment through their skin.


correct_option_index: 5

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