Surface Area and Volume

Type of Unit: Conceptual

Prior Knowledge

Students should be able to:

Identify rectangles, parallelograms, trapezoids, and triangles and their bases and heights.
Identify cubes, rectangular prisms, and pyramids and their faces, edges, and vertices.
Understand that area of a 2-D figure is a measure of the figure's surface and that it is measured in square units.
Understand volume of a 3-D figure is a measure of the space the figure occupies and is measured in cubic units.

Lesson Flow

The unit begins with an exploratory lesson about the volumes of containers. Then in Lessons 2–5, students investigate areas of 2-D figures. To find the area of a parallelogram, students consider how it can be rearranged to form a rectangle. To find the area of a trapezoid, students think about how two copies of the trapezoid can be put together to form a parallelogram. To find the area of a triangle, students consider how two copies of the triangle can be put together to form a parallelogram. By sketching and analyzing several parallelograms, trapezoids, and triangles, students develop area formulas for these figures. Students then find areas of composite figures by decomposing them into familiar figures. In the last lesson on area, students estimate the area of an irregular figure by overlaying it with a grid. In Lesson 6, the focus shifts to 3-D figures. Students build rectangular prisms from unit cubes and develop a formula for finding the volume of any rectangular prism. In Lesson 7, students analyze and create nets for prisms. In Lesson 8, students compare a cube to a square pyramid with the same base and height as the cube. They consider the number of faces, edges, and vertices, as well as the surface area and volume. In Lesson 9, students use their knowledge of volume, area, and linear measurements to solve a packing problem.

Lesson OverviewStudents explore nets—2-D patterns that can be folded to form 3-D figures. They start by examining several patterns and determining which nets form a cube. Then, they sketch nets for rectangular prisms. They also find the surface area of the rectangular prisms.ELL: Remind students of the units used to measure area and volume. Use this opportunity to reinforce why square units are used for area (2-D) and cubed units are used for volume (3-D).MathematicsA net is a 2-D pattern that can be folded to form a 3-D figure. In this lesson, the focus is on nets for rectangular prisms. There are many possible nets for any given prism. For example, there are 11 different nets for a cube, as shown below.The surface area of a prism is the area of its net.Goals and Learning ObjectivesIdentify nets for cubes.Sketch the net of a rectangular prism.Find the surface area of a rectangular prism.

Lesson OverviewStudents build prisms with fractional side lengths by using unit-fraction cubes (i.e., cubes with side lengths that are unit fractions, such as 13 unit or 14 unit). Students verify that the volume formula for rectangular prisms, V = lwh or V = bh, applies to prisms with side lengths that are not whole numbers.Key ConceptsIn fifth grade, students found volumes of prisms with whole-number dimensions by finding the number of unit cubes that fit inside the prisms. They found that the total number of unit cubes required is the number of unit cubes in one layer (which is the same as the area of the base) times the number of layers (which is the same as the height). This idea was generalized as V = lwh, where l, w, and h are the length, width, and height of the prism, or as V = Bh, where B is the area of the base of the prism and h is the height.Unit cubes in each layer = 3 × 4Number of layers = 5Total number of unit cubes = 3 × 4 × 5 = 60Volume = 60 cubic unitsIn this lesson, students extend this idea to prisms with fractional side lengths. They build prisms using unit-fraction cubes. The volume is the number of unit-fraction cubes in the prism times the volume of each unit-fraction cube. Students show that this result is the same as the volume found by using the formula.For example, you can build a 45-unit by 35-unit by 25-unit prism using 15-unit cubes. This requires 4 × 3  × 2, or 24, 15-unit cubes. Each 15-unit cube has a volume of 1125 cubic unit, so the total volume is 24125 cubic units. This is the same volume obtained by using the formula V = lwh:V=lwh=45×35×25=24125.15-unit cubes in each layer = 3 × 4Number of layers = 2Total number of 15-unit cubes = 3 × 4 × 2 =  24Volume = 24 × 1125 = 24125 cubic units Goals and Learning ObjectivesVerify that the volume formula for rectangular prisms, V = lwh or V = Bh, applies to prisms with side lengths that are not whole numbers.

This lesson is from Tools 4 NC Teachers. In this lesson students will apply their previous knowledge of finding the volume of rectangular prisms to more complex figures which can be decomposed into rectangular prisms. Students will explore the additive nature of volume in these types of figures.

This lesson is from Tools 4 NC Teachers. In this lesson, students explore finding volume of two non-overlapping rectangular prisms using the context of a toy factory called Cubix. It is remixable.

This lesson is from Tools 4 NC Teachers. In this lesson students will explore volume as additive using concrete objects and pictorial representations. This is remixable.

This lesson is from Tools 4 NC Teachers. Students explore volume and determine the size (volume) of various boxes. This is remixable.

This task is from Tools 4 NC Teachers. Students determine the volume of joined, non-overlapping rectangular prisms. This is remixable.

This task is from Tools 4 NC Teachers. Students draw pictorial representations of rectangular prisms and examine the volume. This is remixable.

This task is from Tools 4 NC Teachers. Students consider ways to organize rectangular prisms into different sizes of moving trucks. This is remixable.