Surface Area & Volume

In this lesson, students are introduced to the concept of surface area. They identify the surface and attributes of figures that are measurable. They describe and distinguish the measures of surface area and volume of the figures. Given scenarios, students describe attributes which represent surface area and volume. Students apply the terms by identifying and distinguishing them in real-life scenarios.

Overall Expectations

• estimate, measure, and record perimeter, area, temperature change, and elapsed time, using a variety of strategies;
• determine the relationships among units and measurable attributes, including the area of a rectangle and the volume of a rectangular prism.

Specific Expectations

• estimate and measure the perimeter and area of regular and irregular polygons, using a variety of tools (e.g., grid paper, geoboard, dynamic geometry software) and strategies.
• determine, through investigation using a variety of tools (e.g., concrete materials, dynamic geometry software, grid paper) and strategies (e.g., building arrays), the relationships between the length and width of a rectangle and its area and perimeter, and generalize to develop the formulas [i.e., Area = length x width; Perimeter = (2 x length) + (2 x width)];
• determine, through investigation using stacked congruent rectangular layers of concrete materials, the relationship between the height, the area of the base, and the volume of a rectangular prism, and generalize to develop the formula (i.e., Volume = area of base x height) (Sample problem: Create a variety of rectangular prisms using connecting cubes. For each rectangular prism, record the area of the base, the height, and the volume on a chart. Identify relationships.);
• determine, through investigation, the relationship between capacity (i.e., the amount a container can hold) and volume (i.e., the amount of space taken up by an object), by comparing the volume of an object with the amount of liquid it can contain or displace (e.g., a bottle has a volume, the space it takes up, and a capacity, the amount of liquid it can hold) (Sample problem: Compare the volume and capacity of a thin-walled container in the shape of a rectangular prism to determine the relationship between units for measuring capacity [e.g., millilitres] and units for measuring volume [e.g., cubic centimetres].);

Introductory Activity

• Have students gather in small groups of three or four. Assign roles such as recorder, taskmaster, timekeeper and reporter.
• Post the following questions on the board, chart paper or overhead projector:
• What do you know about area? Give examples of where or how you observe area around you.
• What do you know about volume? Give examples of where or how you observe volume around you.
• Provide each group with chart paper. Instruct the students to make a two-column chart, one column for area and the other for volume.
• Monitor each group’s discussion, listening for misconceptions. Ask guiding questions to determine student understanding.
• Bring groups together for a class discussion of the questions. Record each group’s responses under the appropriate question. Discuss any questions and misconceptions.

Instructional Procedures:

Volume and capacity are used to describe the measure of space within a three-dimensional figure. Volume is expressed in terms using length units such as centimetres. Capacity is used to express liquid measures or the space of containers that hold liquid. For example, a can of paint has a surface area and a capacity. The surface area is represented by the measure of material used to make the can and the capacity represents the space in the can or the amount of paint (Van de Walle, 1998). At this point in the lesson it would be appropriate to begin to distinguish between the concepts of volume and capacity

Activities

Activity 1

• Collect different objects such as boxes and metal cans for students to identify and describe the surface of the object. Have enough objects so that each group has a box and a can.
• Discuss the concept of surface.
• Place students in groups as in the introductory activity. Give each group a box and a can. Ask,
• What is the surface of each object?
• How would you describe the shape of the surface?
• What shapes make up the surface?
• Instruct students to record their responses on the reverse side of the chart paper from the introductory activity. Observe the group discussions and make note of different ideas.
• Bring the class together to discuss the findings. Record the students’ responses on the board or overhead projector. Answer any questions that may arise from the discussions. Students should come to the conclusion that the surface is represented by the cardboard and metal used to make the objects.

Activity 2

• Facilitate a discussion to determine how to find the measure of the object’s surface. Ask questions such as:
• What attributes of the figures are measurable? (E.g., height, width, circumference, diameter, area of sides or material used, volume, perimeter of a face/side, weight)
• Which measure would be used to describe the surface?
• Is it important to know how much an object holds, to know the measure of the surface?
• How large was the sheet of cardboard used to make this box?
• If the can is cut from a sheet of metal, how much metal was used?
• Allow time for students to discuss this in their group before accepting answers from the class. Expect students to talk about measuring the area of the surface of each object.
• Ask questions to guide students to relate the unit of measure to the unit of measurement. Inform students that measuring the surface of an object is called surface area. Have students record this in their journals or notebooks.

Activity 3

• Present the following scenario to the class: Garry is wrapping a gift in a box for his friend.
• Have students select the object in the scenario that relates to the surface area and volume.
• Discuss any misconceptions. Allow students to refer to their notes from the introductory activity. (The box represents the surface area and the space inside of the box represents volume.)
• Present a different scenario and have students select the attributes of the objects that relate to surface area and volume. To keep cereal from becoming stale, Garry’s mother puts the cereal in a plastic container.
• Discuss any misconceptions. (The container represents surface area and the space inside or the cereal represents volume.)
• Present similar scenarios until students can distinguish the difference between the relationship of the object and surface area and volume, as needed.
• For example, making a homemade aquarium (surface area - glass used to make sides, volume or capacity-space for water)
• Football/basketball (surface area-measure of the material used, volume-the amount of air in the ball) Activity 4
• Have students write scenarios representing both surface area and volume. Select students to present a variety of scenarios to present to the class for discussion. Correct any misconceptions.
• Have students explain the difference between surface area and volume.

Scoring Guidelines:

• Assess through teacher observation. Make notes. Students should recognize the measurable attributes of shapes and figures and identify area as covering a two-dimensional shape and volume as filling a three dimensional object.
• Post-Activity Assessment:
• In this individual pencil-and-paper assessment, students explain the difference between surface area and volume and justify whether a scenario involves using surface area or volume.
• Distribute Surface Area or Volume? handout, See attachment A. Collect and evaluate understanding using the rubrics.

Materials & Resources

• Chart paper
• Overhead projector
• Boxes and cans
• Pencils
• Paper
• Transparencies
• Markers for overhead projector
• Geoboards

Rubric

Evaluate students’ understanding of distinguishing and describing surface area or volume. See rubric below:

Adequate Understanding

• Describes attributes of surface area and volume appropriately. Uses these attributes to describe how surface area and volume relates to each scenario. Appropriately selects objects from each scenario to correspond with surface area and volume.

Partial Understanding

• Describes some or a few attributes of surface area and volume appropriately. Describes how surface area and/or volume relate to each scenario with minor mistakes. Selects objects from each scenario to correspond with surface area and volume with minor mistakes.

Limited Understanding

• Shows little or no understanding of the difference between surface area and volume. Switches attributes in describing how surface area and volume relates to each scenario. Confuses objects from each scenario which corresponds to surface area and volume.