• Participation and performance on the two cumulative projects in Part 2, Redecorating the Classroom and Drawing Plans, show how well students are using their knowledge of two-dimensional objects interacting with three-dimensional spaces. Look for their use of patterns, symmetry, and spacing.

• Observe students’ participation and performance related to the building of models and offering of predictions, conjectures, responses, and analyses during all activities in Part 1.

• Observing discourse offered throughout the lesson provides insight into the level of students’ understanding of the relationship between two-dimensional and three-dimensional representations.

Part 1 [IS.5 - Struggling Learners]

Have students visit NLVM’s Space Blocks applet available at http://nlvm.usu.edu/en/nav/frames_asid_195_g_4_t_3.html?open=activities&from=category_g_4_t_3.html.

Students should try to construct a given figure with X cubes and X surface area. This activity serves as a precursor to the real-world activities offered next.

Building Models

Break students into groups of three or four. Have students recreate two-dimensional drawings using building blocks. (Public domain drawings can be used.) The blocks should be in the appropriate scale of the drawings. Students actually create replicas of two-dimensional drawings and determine surface area and volume. This activity can be completed for a variety of real-world rectangular prisms/buildings.

Alternate activity idea: If you wish, give students a bucket of Legos and have them construct the buildings that way as well. This is a viable option if access to the Internet is not possible. Other materials may include plastic storage bins with snap lids, wooden blocks, and plastic cubes. Isometric paper (with regularly spaced dots) may also be useful in drawing two-dimensional representations of the models.

Swimming Pool Models

Break students into groups of three or four. You can also simulate the comparison of volumes of several different swimming pools via replications/models. Simply bring in various sized plastic prisms, with different heights, lengths, and widths. Label each prism as Pool A, Pool B, Pool C, etc. Have students predict which pool has the greatest volume/least volume. Then students should use measuring cups and water to fill each pool, while recording the number of cups added to each pool. Students can then make comparisons regarding the volume, using the total number of ounces added. Students could simply count the number of cups, but some containers may hold a fraction of a cup. Students should compare the results to their predictions and discuss why they made the original predictions and what they have learned.

Provide students with copies of the Swimming Pool Dimensions recording sheet (M-G-3-3_Swimming Pool Dimensions.docx).

Part 2: Redecorating the Classroom [IS.6 - All Students]

Give students the following scenario:

“Let’s say we plan to redecorate this classroom. We are going to retile the floor and paint the walls and ceiling. How do we determine the quantity of each material needed in order to accomplish our task?”

Break students into groups of three or four. Have students brainstorm the materials needed. Students should realize that they don’t actually need the paint and tiles, since the task is to determine how much of each is needed. Instead, students need measuring tools, such as a measuring tape. They also need a recording sheet to record the dimensions of each wall, the floor, and the ceiling (M-G-3-3_Classroom Dimensions.docx). This activity involves students in the everyday process of architects and carpenters, related to determination of surface area. Students have to visit a Web site for a home repair store, such as Home Depot, to determine how many square feet are covered per gallon of paint. They also have to choose a particular size tile and state the use of that size tile in their results. Conversions must be used. How many tiles fit into X square feet? How many square feet does one gallon of paint cover? Thus, how many gallons of paint do we need?

Also, invite students to use spatial estimation to determine how many of the various classroom objects could fill the space in the room. For example, approximately how many desks would fill the room, using all 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 like: Will this space be big enough?; What could this space hold?; How could we make it bigger?; Do the dimensions serve the intended purpose?; etc.

Pose this question: “If we were to design another classroom, what dimensions would give the largest volume?” [IS.7 - All Students]

“Consider ceiling height and length compared to width as constraints.”

Have students make a table for various dimensions, while finding surface area and volume of each. Students should examine patterns, formulate conjectures, and describe findings, offering supporting reasons and evidence.

Sample table:

Drawing Plans and Architecture

Students look at two-dimensional drawing plans for two different buildings to be built in New York City. Students look at building prices per square foot. Students must determine the best buy for the surface area and volume involved. Students visualize each solid and make estimations to determine the best choice in design according to price and value. Provide students with copies of the Drawing Plans activity sheet (M-G-3-3_Drawing Plans and KEY.docx).

Review Activity

Ask students to write a short mathematics article for the New York Times or the local newspaper. Online examples are available on the newspaper’s Web site. [IS.8 - All Students] Students should make the article innovative and creative, while focusing on one key topic from the lesson. For example, students might write an article on choice of architectural plans, providing the mathematical reasoning behind such choices.

Extension:

• Have students design their own restaurant. The restaurant should include various solids, such as cylinders, cones, rectangular prisms, cubes, triangular pyramids, square pyramids, spheres, hexagonal prisms, etc. The restaurant should be drawn in two-dimensional form on graph paper. Students can also replicate their restaurant using real-world manipulatives, such as candies, marshmallows, graham crackers, etc. They can use glue, toys that snap together, or modeling clay—anything that is safe and works for them.

• Students then estimate the surface area and volume of the whole restaurant and then estimate the surface area and volume of key components of the restaurant. For example, a student may estimate the volume of a cylindrical post in the front of the building. Since the replica is only a model, students need to find a way to use scale factors and proportions to relate the model to an actual real-sized restaurant building.