Lesson Plan: Lesson plans were written by Maryland mathematics educators and could be used when teaching the concepts.
 Goal 2 Geometry, Measurement, And Reasoning Expectation 2.3 The student will apply concepts of measurement using tools and technology when appropriate. Indicator 2.3.2 The student will use techniques of measurement and will estimate, calculate, and/or compare perimeter, circumference, area, volume, and/or surface area of two-and three-dimensional figures and their parts.

### Lesson Content

Prisms: Areas and Volume

### Objective

Students will apply the formulas for surface area and volume of various prisms by first investigating the derivation of the formulas and then by applying the formulas in practice problems.

### Approximate Time

Two 45-minute lessons

### Prerequisite Concepts Needed

Students will need to be able to determine the area of triangles, quadrilaterals, and polygons and use the special right triangle formulas to solve special right triangles. You may need to review finding the area of a regular hexagon by first finding the apothem of the hexagon.

### Warm-Up/Opening Activity

Show students three dimensional models of many prisms. Discuss what part of each model represents the surface area and the volume.

Using the model of the rectangular prism, show where the length, width, and height of the model are in order to determine the volume of the prism. As you do this, have the students note the formula for finding the volume of a rectangular prism on the HSA formula sheet. Next look at the surface area formula for a rectangular prism. Trace your finger along each dimension of the object as it comes in the formula. Ask the students to explain why the formula for surface are of a rectangular prism is written as it is.

Practice calculating the surface area and volume for the rectangular prism shown below:

SA = 2lw + 2hw + 2hl = 2 · 3 · 6 + 2 · 8 · 6 + 2 · 3 · 8 = 180 ft²
V = l · w · h = 3 · 6 · 8 = 144 ft³

Have students explain which unit to use for volume and which unit to use for surface area (cubic units, and square units)

Repeat the process using other models: a triangular prism, a hexagonal prism, and a cylinder. Be sure to ask the students 'why' the formula is written as it is. Show relationships between the formulas for different figures. For example, length 'times' width is the area of the base of a rectangle, therefore using the area of the base 'times' height is the same formula for volume in each of the other two prisms.

Warm-up

A net is a two-dimensional pattern of a three-dimensional figure. Nets can be folded to create cubes, rectangular prisms, cylinders, etc. Use the Warm-up worksheet to get students to look at nets, and have them pick the ones that would fold into a cube, a rectangular prism and a cylinder. If you have time, ask students to cut out the nets and actually do the folding.

Ask students to draw a net of one of the models that you have presented in class.

### Development of Ideas

Initiate a discussion with students about the difference between surface area and volume (use the warm-up activity to help). Have a set of three dimensional figures to help guide your discussion. Show an example of a rectangular prism, a triangular prism and a cylinder (if you don't have good examples, just use a box and a can of soup). Remember, all figures we will discuss will be right prisms or right cylinders. They will only be regular when noted. Suggest that surface area will be the area that wrapping a gift would cover, and lateral area is the area that a label on a box or can may cover. Volume is the number of cubes that would completely fill the objects.

Activity One

Show students an empty cereal box. Ask them to estimate the volume, surface area and lateral area of the box.

Unfold the box, showing its net, and allow students to change their estimates if they want. (If you have enough boxes, give one to each group, otherwise, this can be a demonstration.)

Calculate the volume, surface area, and lateral area of the box. Get students to measure and to do the calculations. Suggestion: After measuring, let one-third of the class do each calculation (volume, surface area, and lateral area) - remember to use the HSA formula reference sheet.

Worksheet: Real World Applications

Activity Two – Investigation of volume and surface area of a rectangular prism

Worksheet: Volume and Surface Area of a Rectangular Prism

Each group of students needs an 8.5 in. x 11 in. piece of paper. Ask each group to create a box without a lid. Have one group cut a ½" square out of each corner, another a 1 in. square, and another with 1.5 in. squares. Continue until you get to 4in. if you have that many groups. Give each group tape so they can tape the sides together to make the box.

Initiate a discussion about why a group could not create a box using a 4.5 in. square.

Each group is to estimate then calculate the volume, and surface area of their 'box' (pretend the box has a top) completing the appropriate row in the table. Ask the students if the volume or surface areas of the boxes will be the same and justify their conclusion.

Ask students to show their boxes to each other. Have the students list the boxes in order from least to greatest by what they predict the volumes to be. Come to a consensus that the class can live with.

Fill the largest-predicted volume box with cereal. Poor that cereal into the next largest. Discuss what should happen if the volume is in fact larger (the cereal should flow over the smaller box). Continue on down the line and fix the order as necessary.

Have the students complete the table on the worksheet and mathematically verify their results.

Reflection: What dimension helped to determine the box with the greatest volume?

Activity Three – Investigation of volume and lateral area of a cylinder
This activity can be done as a group, or as a demonstration

Use a sheet of 8½" by 11" paper to create a cylinder by joining the top and bottom edges. The edges need to meet exactly, with no gaps or overlaps.

Use a second sheet of paper the same size to make a different cylinder, this time joining the left and right edges together. Mark the tall one Cylinder A and the other B.

Ask students if they think the surface areas are the same for each cylinder (pretend each has a top and bottom). Use mathematics to justify your answer. (Use this time to reinforce that the lateral sides of a cylinder are a rectangle.)

Ask students if they think the volumes are the same for each cylinder. Use mathematics to justify your answer.

Place cylinder B on a flat surface (you might want to use a box lid) and place A inside it. Fill cylinder A with cereal. Allow students to adjust their predictions as necessary. Slowly lift cylinder A so that the cereal falls into cylinder B. Make the conclusions: Since the cereal does not fill cylinder B, then the volume of B is larger than A.

Reflection: Justify why the two cylinders did not hold the same volume.

Since V = ; · r · r · h, the radius has a bigger effect on the volume than the height, so the cylinder with the greater radius will have the greatest volume.

Worksheet: Volume of Paper Cylinders

Show students a transparency of the cylinders on the worksheet, and ask them to predict the prism with the greatest volume. Have the students justify their answers by completing the table on the worksheet.

Worksheet: Determining Surface Area and Volume of Prisms Worksheets A and B

These two sheets can be used in many different ways. You can assign both worksheets to all of the students or allow students to complete different worksheets and share their work with students that did not complete that worksheet.

### Closure

Restate the objective. Ask students to explain in writing how the objective was met in their own words.