The purpose of this activity is for students to apply their understanding of area and surface area to solve a problem about tents. It also familiarizes students with some considerations that are important in tent design, preparing students to design their own tent in the next activity.

Students are presented with an image of two tents of the same design but of different sizes. While the problem situation is fairly well defined, students are not initially given a problem to solve or any measurements. Instead, they have an opportunity to think about mathematical questions that could be asked, consider the information necessary to answer one of the questions, and make assumptions or estimates if needed. As students make sense of a situation, consider relevant quantities in context, and think about how to solve a problem with the mathematics that they know, they practice aspects of mathematical modeling (MP4).

As students work to solve the chosen problem, monitor for different assumptions students make about the tents (such as whether there is a floor panel inside the tent or whether the tents have an open side). Also monitor for various ways in which students reason about the surface areas (including their choice of units of measurement) and communicate their reasoning. 

This is the first time Math Language Routine 5: Co-Craft Questions is suggested in this course. In this routine, students are given a context or situation, often in the form of a problem stem (for example, a story, image, video, or graph) with or without numerical values. Students develop mathematical questions that can be asked about the situation. A typical prompt is: “What mathematical questions could you ask about this situation?" The purpose of this routine is to allow students to make sense of a context before feeling pressure to produce answers, and to develop students’ awareness of the language used in mathematics problems.

Select students or groups who made different assumptions about the two tents to share their thinking. Then, discuss how the differences in assumptions affect their answers to the question. 

Next, select those who reasoned about surface area in different ways to briefly present their solutions. Consider asking students:

• “How did you account for the areas of all the faces of each tent? How did you organize your calculations?”
• “What units of measurement did you use? Why did you choose to use that unit?” 
• “How is working in inches different from working in feet?”

The goal of the discussion is for students to see that different assumptions about a situation may affect the solutions to problems about that situation, and that different reasoning strategies may affect the problem-solving process. For example, calculating surface area in inches involves working with large numbers and doing so in feet involves working with decimals.

In this activity, students apply the concepts and skills from this unit to design a camping tent. They learn about the design problem, consider the parameters for the design, and ask clarifying questions. Then they make decisions about their tent, create a design and necessary representations of it, and estimate the amount of material needed to construct it. 

This work prompts students to draw on the reasoning strategies developed in this unit, such as:

• Decomposing polygons and rearranging the parts to find area.
• Drawing and labeling a net to account for all the faces of a polyhedron.
• Using formulas to facilitate area calculation.
• Calculating areas with appropriate degree of precision.
• Including appropriate units of measurement.

In creating their tent design, students may need to estimate lengths that cannot be computed exactly given their current mathematical knowledge. For instance, the length of a slanted edge of a roof panel may be calculated using the Pythagorean Theorem but students are not expected to do so at this point. To support students in making reasonable estimates, encourage them to make comparisons. Consider asking questions such as:

• “If the horizontal side of this right triangle is 5 feet and the vertical side is 3 feet, would the slanted side be longer or shorter than 5 feet? About how much longer?”
• “About how many times as long as this 3-foot-long side is that side?” 

As students make assumptions and decisions for their design and apply the mathematics that they know to solve a problem, they practice aspects of mathematical modeling (MP4).

Students also have an opportunity to choose tools strategically (MP5) as they develop and represent a three-dimensional object. Some students may find it useful to think in two-dimensional terms and start by drawing a net. Others may wish to build a physical model of their design or to use a digital drawing tool. Students should be encouraged to consider the tools at their disposal and choose those that would enable them to complete the task.

Given the open-ended nature and high cognitive demand of a design activity, students may benefit from seeing a sample final product along with the intermediate steps. Consider sharing an example of completed work (such as the provided sample student response) to help students understand the expected outcome.

Students will share and reflect on their designs in the next activity. Before moving on, engage them in a brief whole-class discussion. Invite students to share some of the things that they considered and decisions that they made as they were designing their tent. Also invite them to share some challenges that they encountered when trying to estimate the necessary amount of material.

This activity gives students a chance to explain and reflect on their work. Students share drawings of their tent design, an estimate of the amount of material needed, and the justification. Then, students compare their creations with their peers' creations and discuss not only the amount of material required, but also the effects that different designs have on that amount.

As students discuss in groups, notice how they reason about and communicate their work. Monitor for whether they: 

• Provide justification for their measurements and choices.
• Include drawings that reflect their decisions and show relevant measurements and labels.
• Explain clearly their process of calculating surface area.

Also notice how students compare the amounts of material needed for the different designs in their group. Monitor for students who recognize how design decisions (such as tent types, tent measurements, and arrangements of sleeping bags) affect the amount of material needed.

Much of the discussion will take place within the groups. Once groups have had an opportunity to share their designs, reconvene as a class. Consider displaying tent designs that used the most and the least amount of material and discussing with students:

• “All the tents were designed to accommodate 4 people and a minimum height inside the tent. How did the tents end up using very different amounts of material?”
• “What design decisions impact the amount of material needed?”
• “Can you give examples of decisions that had a major impact? What about decisions that had a lesser impact?”
• “When calculating the surface area of your tent, what reasoning strategies from this unit did you find useful?”