Virginia SOL 8 - 2020 Edition
Investigation: Maximizing volume and surface area
Lesson

### Objective

We will determine how changing one dimension of a rectangular prism effects the volume and surface area of the figure.

### Vocabulary

Begin this activity by defining the terms below.  Be sure to use these words correctly while you are discussing the investigation activity with your classmates.

1. Volume
2. Surface Area
3. Dimensions
4. Rectangular Prism
5. Scale Factor

### How do changes affect the volume?

#### Exploration

Copy the table below.  Then follow these instructions to use the first applet to explore how changing one of the dimensions of a rectangular prism will change the volume of the figure.  (Use the sliders to change the lengths of the sides of the figure and click the boxes to view the formula and the volume of the figure).

• In this activity, we will multiply only one dimension of the rectangular prism by a scale factor.
• It is helpful to choose the one dimension we plan to adjust so that its a multiple of the denominator of any fractional scale factors (see column $1$1 for the scale factors).
• Record the original length, width and height chosen in column $2$2 of the table (original dimensions).
• Find the volume of the rectangular prism and write the volume in column $3$3 of the table.
• Multiply one of the dimensions of the original rectangular prism by the scale factor given in column $1$1 of the table to form a new rectangular prism.  (Adjust the slider in the applet to the new value, so that we can see how the changed dimension changed the way the rectangular prism looks)
• Record the dimensions of the new rectangular prism in column $4$4 .
• Record the volume of the new rectangular prism in column $5$5.
• In column $6$6, use words to describe any observations that you can make about the changes in volume that are related to multiplying one dimension of the original rectangular prism by the scale factor.
 Created with Geogebra

Scale factor

(1)

Original  dimensions

(2)

Volume of original

(3)

New dimensions (after multiplying one side by scale factor)

(4)

Volume of new figure

(5)

Comment on observed differences

(6)

$\frac{1}{4}$14
$\frac{1}{3}$13
$\frac{1}{2}$12
$2$2
$3$3
$4$4

### How do changes affect the surface area?

#### Exploration

Copy the table below.  Then follow these instructions to use the applet below to explore how changing one of the dimensions of a rectangular prism will change the surface area of the figure.  (Use the sliders to change the lengths of the sides of the figure and click the boxes to view the formula and the surface area of the figure).

• In this activity, we will multiply only one dimension of the rectangular prism by a scale factor.

• It is helpful to choose the one dimension we plan to adjust so that its a multiple of the denominator of any fractional scale factors (see column $1$1 for the scale factors).

• Record the original length, width and height chosen in column $2$2 of the table.

• Find the surface area of the rectangular prism and write the surface area in column $3$3 of the table.

• Multiply one of the dimensions of the original rectangular prism by the scale factor given in column $1$1 of the table to form a new rectangular prism.  (Adjust the slider in the applet, so that we can see how the changed dimension changed the way the rectangular prism looks)

• Record the dimensions of the new rectangular prism in column $4$4 .

• Record the surface area of the new rectangular prism in column $5$5.

• In column $6$6, use words to describe any observations that you can make about the changes in surface area that are related to multiplying one dimension of the original rectangular prism by the scale factor.

 Created with Geogebra

Scale factor

(1)

Original dimensions

(2)

Surface area of original

(3)

New dimensions (after multiplying one side by scale factor)

(4)

Surface area of new figure

(5)

Comment on observed differences

(6)

$\frac{1}{2}$12
$2$2

#### Discussion questions

1)  Discuss the observations you made in each of the tables above with a partner.  Did you start off with rectangular prisms that had the same dimensions?  Were your observations about the result of changing the scale factor the same?

2) Can you write a general rule describing what happens to the volume of a rectangular prism if you multiply one dimension by a scale factor that is a whole number that is greater than one?  Share that rule with your partner.

3) Can you write a general rule describing what happens to the volume of a rectangular prism if you multiply one dimension by a unit fraction?  (Recall:  If you are multiplying a whole number $12$12 by the unit fraction $\frac{1}{3}$13, you get the same result as if you had divided the number $12$12 by the whole number $3$3. )

4) Can you write a general rule describing what happens to the surface area of a rectangular prism, if you multiply one dimension by a scale factor greater than one?

5) Can you write a general rule describing what happens to the surface area of a rectangular prism, if you multiply one dimension by a scale factor that is a unit fraction?

6)  Are the effects of multiplying one dimension of a rectangular prism by a scale factor the same on the volume of the new rectangular prism as they are on the surface area of the new rectangular prism?  Discuss your answer to this question with a partner.

### Outcomes

#### 8.6b

Describe how changing one measured attribute of a rectangular prism affects the volume and surface area