<p>In this unit, students will apply their visualization skills to real-world problems. Students will:&nbsp;<br>- use spatial visualization and estimate to tackle real-world problems.<br>- examine the relationships between solids based on their individual properties.</p>

<p>- What are the different solid properties and their relationships?</p>

<p>- Surface Area: The total area surrounding a three-dimensional figure.&nbsp;<br>- Volume: The capacity of a solid; the amount a solid can hold.</p>

<p>- building blocks (unit cubes, interlocking cubes, etc.)<br>- plastic rectangular prisms<br>- measuring cups<br>- Swimming Pool Dimensions recording sheet (M-G-3-3_Swimming Pool Dimensions)<br>- Classroom Dimensions recording sheet (M-G-3-3_Classroom Dimensions)<br>- Drawing Plans (M-G-3-3_Drawing Plans and KEY)</p>

<p>- Participation and performance on the two cumulative projects in Part 2, Redecorating the Classroom and Drawing Plans, demonstrate how well students apply their understanding of two-dimensional objects interacting with three-dimensional spaces. Consider their use of patterns, symmetry, and spacing.&nbsp;<br>- During all Part 1 activities, observe students' participation and performance in model construction and delivering predictions, conjectures, responses, and analyses.&nbsp;<br>- Observing student discussion throughout the lesson provides insight into their knowledge of the relationship between two-dimensional and three-dimensional representations.</p>

<p>Active Engagement<br>W: The lesson focuses on connecting visualization and comprehension of surface area and volume to real-world problems. Students must apply their knowledge and abilities to problems that professionals solve on a daily basis. Each activity requires group work, giving students the opportunity to discuss, debate, and present supporting arguments with other students. Students will be evaluated both during group activities and on individual activities.&nbsp;<br>H: The active engagement approach empowers students to develop their own understanding and and put students in charge of their learning. Students can assist and reinforce their grasp of dimensions and relationships in three-dimensional spaces by creating models with physical materials such as plastic blocks and tiles. The multiple discussions, debates, and problem-solving activities will keep students engaged throughout the lesson.&nbsp;<br>E: The lesson is separated into two parts. Part 1 focuses on imagery. Part 2 asks students to tackle two real-world situations. Students will see how mathematics may be applied outside of the classroom with topics such as architecture and design.&nbsp;<br>R: The open-ended review allows students to interpret the article as they see fit. Students can express their creativity by combining writing skills and mathematics knowledge, evaluating and using spaces.&nbsp;<br>E: Each activity prompts students to self-reflect and evaluate their understanding. Because each activity involves discovery learning, students do not observe the teacher model each situation. Instead, students must utilize measurements to define the problem, solve it, and assess the result.&nbsp;<br>T: Using multiple representations allows all students to learn. Students can feel comfortable and secure while studying and exploring when they have the support of their group members.&nbsp;<br>O: The lesson uses three-dimensional things such as buildings, rooms, pools, paint, and furniture. Students must use measures, areas, volumes, and viewpoints to make interpretations.</p>

<p><strong>Part 1</strong></p><p>Students should visit NLVM's Space Blocks applet, which may be found at <a href="http://nlvm.usu.edu/en/nav/frames_asid_195_g_4_t_3.html?open=activities&amp;from=category_g_4_t_3.">http://nlvm.usu.edu/en/nav/frames_asid_195_g_4_t_3.html?open=activities&amp;from=category_g_4_t_3.</a></p><figure class="image"><img style="aspect-ratio:1284/805;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_168.png" width="1284" height="805"></figure><p>Students should attempt to design a particular figure using X cubes and X surface area. This activity serves as a preparation to the real-world activities that follow.</p><p><strong>Building Models</strong></p><p>Divide students into groups of three or four. Make students recreate two-dimensional drawings with building blocks. (Public domain drawings may be used.) The blocks should be at the same scale as the drawings. Students make reproductions of two-dimensional drawings and measure surface area and volume. This activity can be completed for a wide range of real-world rectangular prisms and buildings.</p><p>Alternate activity idea: If you like, give students a bucket of Legos and have them create the buildings that way as well. If you do not have access to the Internet, this is an option. Other materials could include plastic storage boxes with snap lids, wooden blocks, and plastic cubes. Isometric paper (with regularly spaced dots) may also be useful for creating two-dimensional representations of models.</p><p><strong>Swimming Pool Models</strong></p><p>Divide students into groups of three or four. You may also use replications/models to simulate the comparison of volumes of multiple swimming pools. Simply bring in plastic prisms of all sizes, including heights, lengths, and widths. Label each prism: Pool A, Pool B, Pool C, and so on. Have students predict which pool has the most/least volume. Then, using measuring cups and water, students should fill each pool and keep track of how many cups they added. Students can then compare volumes based on the total number of ounces added. Students might simply count the number of cups, however certain containers may only hold a fraction of a cup. Students should compare the results to their original predictions and discuss why they made them and what they have learnt.</p><p>Give students copies of the Swimming Pool Dimensions Recording Sheet (M-G-3-3_Swimming Pool Dimensions).</p><p><strong>Part 2: Redecorating the Classroom</strong></p><p>Provide students with the following scenario:</p><p><strong>"Let's imagine we want to redecorate this classroom. We plan to retile the floor and paint the walls and ceiling. How can we determine the quantity of each material required to complete our task?"</strong></p><p>Divide students into groups of three or four. Have students brainstorm the materials required. Students should understand that they do not actually need the paint or tiles because the aim is to assess how much of each is required. Instead, students require measuring tools, such as a measuring tape. They also require a recording sheet to record the dimensions of each wall, floor, and ceiling (M-G-3-3_Classroom Dimensions). This activity incorporates students in the day-to-day processes of architects and carpenters, such as surface area determination. Students must visit a home repair store's website, such as Home Depot, to discover how many square feet are covered by a gallon of paint. They must also select a specific tile size and indicate its use in their results. Conversions should be used. How many tiles can fit into X square feet? How many square feet will one gallon of paint cover? So, how many gallons of paint do we need?</p><p>In addition, ask students to use spatial estimation to determine how many different classroom objects could fit in the room. For example, how many desks would be required to fill the room, taking up every available space from corner to corner and floor to ceiling? Such estimation and reasoning connects students to another everyday task common to architects and carpenters: asking questions such as: Will this room be large enough?; What could this space hold?; How could we make it larger?; Do the dimensions fit the intended purpose?; and so on.</p><p>Ask the following question <strong>"If we were to design another classroom, what dimensions would give the largest volume?"</strong></p><p><strong>"Consider ceiling height and length compared to width as constraints."</strong></p><p>Have students create a table with various dimensions and calculate the surface area and volume of each. Students should examine patterns, formulate conjectures, and describe findings, offering supporting reasons and evidence.</p><p>Sample table:</p><figure class="image"><img style="aspect-ratio:596/739;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_169.png" width="596" height="739"></figure><p><strong>Drawing Plans and Architecture</strong></p><p>Students examine two-dimensional drawing plans for two different buildings to be built in New York City. Students compare building prices per square foot. Students must decide the best deal based on the surface area and volume involved. Students visualize each solid and estimate the best design choice based on price and value. Give students copies of the Drawing Plans activity sheet (M-G-3-3_Drawing Plans and KEY).</p><p><strong>Review Activity</strong></p><p>Ask students to write a brief mathematics article for the New York Times or a local paper. Online examples are available on the newspaper's website. Students should create an imaginative and creative piece that focuses on one key topic from the lesson. For example, students could write an article about architectural plan selection, explaining the mathematical reasoning behind such selections.</p><p><strong>Extension:</strong></p><p>Allow students to design their own restaurant. The restaurant should have a variety of solids, such as cylinders, cones, rectangular prisms, cubes, triangular pyramids, square pyramids, spheres, hexagonal prisms, etc. The restaurant should be drawn two-dimensionally on graph paper. Students can even recreate their restaurant with real-world manipulatives like candies, marshmallows, and graham crackers. They can use glue, snap-together toys, or modeling clay—anything that is safe and suitable for them.</p><p>Students then estimate the total surface area and volume of the restaurant, followed by the surface area and volume of essential restaurant components. For example, a student might calculate the volume of a cylindrical post in front of the structure. Since the replica is only a model, students must figure out how to use scale factors and proportions to compare the model to a real-sized restaurant building.</p>