*Nature packages things in peels and shells and other sorts of outer wrappings that we just throw away or compost. People package things in bags and boxes and cans and so forth. Which kinds of packages are the most efficient? Learners have fun with food as they develop a clear conceptual basis of understanding of percentages. Read the story behind Natural Packages here. *

**Day 1**

15 minutes: **Introduce Investigation and Central Question**

Begin by asking students the following questions and recording their predictions.

- Did anyone bring a lunch to school today?
- Does anyone have a banana?
- How do you eat a banana?
- How much of the banana doesn't get eaten? How much is just “packaging”?

Steer the discussion toward the idea of a percentage that will be used to define “packaging efficiency.” Example: If a whole banana weighs 120 grams and the peel weighs 30 grams, then the edible part must weigh 120 - 30 = 90 grams. The ratio of edible part to the whole thing, the packaging efficiency, is 90 : 120, or 90/120. A percentage is just a way of comparing one part of a whole to a standard of 100. It can be set up just that way: 90/120 = x/100. This is the key ratio. It may be read as “90 is to 120 as *x* is to 100.” Changing 90/120 to a decimal, we get 0.75 = x/100 and so *x* = 0.75 x 100 or 75 percent. The packaging efficiency of that banana was 75%.

It is important that they see the way the percent is connected to the ratio, so it may help to practice this a few times and to repeat the verbal formulation as you write the mathematical expression. For example, a banana that is more efficient in its packaging might have a weight of 140 grams for the whole thing and 110 grams for the edible part. Then you would say, “110 is to 140 as *x* is to 100.” You want them to see that the statement always ends with “as *x* is to 100.” That is the nature of the process. You are comparing a part of a whole to the corresponding part as if the whole were composed of 100 pieces. In this case 110/140 = 0.79 to the nearest one hundredth, and so its efficiency may be expressed as 79 percent-literally, 79 per 100; 110 per 140 is equivalent to 79 per 100, or 79%.

We can compute this ratio for anything that has a package and a useful part inside the package. Ask students what they would do to find the packaging efficiency for an orange. How about an apple? A bunch of grapes? (The packaging is the stems!) A carrot? (To peel or not to peel, that is the question.) How about a hard-boiled egg? A peanut in its shell seems straightforward but will present challenges as you discover that a single peanut shell is likely to be too light to be weighed on a normal electronic scale. These are all natural packages, and each will have a different efficiency. All will rely on this simple proportional relationship:

The central question for this investigation is this:

- How efficiently is this item packaged?

20 minutes: **Conduct Investigation and Gather Data**

Split the class into teams of two to four students. Each team will have a scale and a different food item (or items if you have time for multiple trials on each food). Make a chart on the board or consider using a Google Sheet™ to record and share data if students have networked laptop computers. Make sure students predict the packaging efficiency for their food before they conduct the investigation. See figure 1 for a sample data sheet for the class results.

Note: If possible, conduct this investigation in concert with another class working on the same investigation. Your students can then trade information and results, thereby increasing their sample size. This connection increases motivation to get good results and to express these data in ways that will be understood by others.

15 minutes: **Consider Results**

Take a look at the chart with its different efficiency percentages. As a way to review and solidify understanding, ask for a team or two to explain what they did in order to get their results. Consider the implications of the differences you are finding for the packaging efficiencies of the different food items.

- Why, for example, does the banana have a larger percentage of its mass devoted to packaging than does a hard-boiled egg?
- What is the function of the banana peel?
- What is the function of the eggshell?
- How are the environments in which a banana and an egg develop the same? How are they different?

These questions may feel more like science than math, but this allows math to extend beyond the walls of the math classroom and into the realm of sense making.

Finish this session by calculating an average efficiency percentage for all of these natural packages.

5 minutes: **Preview of Day 2**

This first day has been directed at natural packages, packages constructed by nature-but humans package food items too. Day 2 of the investigation is directed at a similar investigation, but this time directed at human packaging and at consideration of the ways in which natural and human-made packaging are similar and different. Ask students to think of some examples of human-made packages (e.g., peanuts in plastic jars or eggs in cartons). When we buy packaged food items at a grocery store, how efficiently are they packaged? Record some of their estimates.

Invite students to bring an item they would like to investigate on day 2. Let them know that the food will probably not be edible when the investigation is done!

Note: You should be prepared on day 2 to supply human-packaged food items for the class because some students may not bring items or may bring items that do not lend themselves easily to the investigation.

**Day 2**

15 minutes: **Introduce Human-Made Packages**

Collect the volunteered human-packaged food items at the front of the room, and supplement them with the items you brought. If there is one that corresponds to a food you investigated yesterday, start with that. For example, a plastic bag with dried banana chips recalls the banana. A box of raisins is the human-packaged equivalent of a bunch of grapes. Drinking a can or carton of orange juice is similar to eating an orange except that humans have intervened and repackaged the orange. Sort of. Investigate the packaging efficiency of one such item. With a jar of shelled peanuts, for example, you can weigh the whole thing, dump the peanuts into a bowl and weigh the jar, which is the packaging. Then, just as in day 1, you can calculate the packaging efficiency and see how it compares to the packaging efficiency of nature with the peanut and the peanut shell. This allows you to review the concept of the percentage as a ratio and the process for calculating percentages.

Assign an item or items to each team and have them collect the materials they will need to investigate their item(s).

15 minutes: **Conduct Investigation and Gather Data**

As on day 1, you will make a chart on the board or in a shared spreadsheet so that students can record their results as they become available. You will have a chance to circulate around the classroom and assist teams as needed.

15 minutes: **Discuss and Generalize Results**

When all of the data have been collected, calculate a mean for the human-made packages just as you did for the natural packages. How do they compare? Is this a fair comparison if we are really interested in the efficiency of the packaging? What do we mean by “efficiency” in this case? Is there a difference between the “wasted” part of a natural product and that of a human-made product? What is the function of the packages people manufacture, and how do they differ from the function of packaging in nature? Which kinds of packages are more likely to be recycled? Why? These questions allow students to see mathematics as a sense-making endeavor rather than a set of exercises.