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.

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).

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.

The goal is to make sure students understand that to maximize surface area for a set volume, a shape should be made flat and long.

Sort the data that 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.

The purpose of this discussion is for students to notice that the snake has a much higher surface area to volume ratio and to think about the impacts this may have. 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 effects that 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.

The goal is to make sure students understand what it means for an ant to have 200 times the relative strength of a human. Tell students to imagine the heaviest weight that 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 the human lift?” (An ant has 200 times the relative strength of a human. So, if a human were as strong as an ant, the human could lift 20,000 pounds.That is the weight of around 5 average-sized cars.)