Lesson 16

The purpose of this warm-up is for students to reason about how to manipulate a shape to enclose more space using a limited amount of material. In subsequent activities, students will apply this same kind of thinking to three-dimensional shapes.

Arrange students in groups of 2. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.

Some students may not be certain if a square is a rectangle. Remind them that a rectangle is defined as a quadrilateral with 4 right angles, and ask them if a square meets that definition.

The purpose for discussion is for students to understand that the square has the most area for a fixed perimeter. Select students to share their responses and explain why they think their enclosure has the most area. Consider these questions for discussion:

• “What are two opposite extreme cases that have very small area? What would be the medium between these two extremes?” (Two extreme cases would be if the fence was 1 foot wide and 89 feet across or 1 foot across and 89 feet wide. The medium between these two extremes would be a square enclosure, which would be 45 feet by 45 feet.)
• “What if we wanted to maximize the volume of a prism with fixed surface area instead of maximizing area of a rectangle with fixed perimeter?” (It seems likely that a cube would give us the maximum volume of a rectangular prism with fixed surface area.)

Students analyze the relationship between the dimensions and the surface area of a solid with fixed volume. As students identify patterns in the results of their classmates’ calculations, they are making sense of the problem and persevering to solve it (MP1).

Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

This activity involves thinking about the workings of a lithium ion battery, such as those in laptops or cell phones. Emphasize to students that under no circumstances should they attempt to open one of these batteries, as this is extremely dangerous.

Arrange students in groups of 4.

Prepare a 5-column table in which students can record the results of their calculations for all to see. It will be easiest to organize the data by putting the numbers in order (small, medium, large), rather than arbitrarily assigning direction (length, width, height). To illustrate this idea, invite students to calculate the volume of these rectangular prisms:

1. length = 2, width = 6, height = 7
2. length = 6, width = 7, height = 2

Point out that the assignment of the dimensions didn’t matter; the two prisms had the same volume. Tell students to record their data in the class chart by numerical order rather than in order of length, width, and height.

Some students may assume that because a square maximizes area for a set perimeter, a cube must maximize surface area for a set volume. It’s true that a cube maximizes volume for a set surface area, but not the other way around. Prompt these students to consider the results they are seeing from their classmates’ calculations to verify if this assumption holds true.

If students struggle to calculate the surface area of their prism, remind them that they can draw the faces of the prism to help them organize their thinking.

Sort the data students have collected in descending numerical order of the surface areas. Consider using a spreadsheet to organize the data and displaying it for all to see. Ask students, “What if we considered values that aren’t integers?” The important takeaway is that to maximize surface area for a set volume, a shape should be made flat and long. For a battery, this is accomplished by making the lithium into a thin foil that can be rolled up to fit inside the battery.

Consider discussing the general idea that packing as much surface area as possible into a small volume is often accomplished by folding. For example, the chemical reactions that occur in mitochondria to provide energy to cells in our bodies take place on the surface, which is why the inside of a mitochondria has many folds. Likewise, the chemical reactions that occur in the brain happen on its surface, which is why the brain has many folds. Intestines absorb nutrients through their surfaces, so they are folded to pack as much surface area into the digestive tract as possible.

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.

“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.

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.

Students investigate another application of surface area and volume: measuring strength based on cross-sectional muscle area.

Tell students to imagine the heaviest weight an average person can lift. For ease of calculations, consider using a number like 100 pounds. Now ask, “If a human had the relative strength of an ant, how much weight could they lift?” (An ant has 200 times the relative strength of a human. So, if a human was as strong as an ant, they could lift 20,000 pounds. That is the weight of around 5 average-sized cars.)