Mini Lesson 1: Surface to Volume Ratio
MATERIALS:


Handout ML 1.1 (Cardstock with printed patterns to make folded cubes)
Graphing Calculator or Excel
ESSENTIAL QUESTIONS:
 How do surface area and volume change with size?
 Why does decreasing size affect material properties?
KEY CONCEPTS:
 Surface area and volume change at different rates when size changes.
 The ratio of surface area to volume (SA/V) increases faster as size gets smaller.
 Surface area to volume ratio is important to understanding why properties change at the
nanoscale.
OBJECTIVES:
In this lesson, students will:


Explore changes in surface area and volume with size.
Use curve-fitting (regression) to find a mathematical model that explains the relationship
between surface area and volume.
Goal 1
Scale Concept
X
Goal 2
Curiosity &
Interest
Goal 3
Science
Process Skills
X
Goal 4
Nanoscience
Content
X
Vocabulary:


Surface area: the area of the outer surface of an object
Volume: the three dimensional interior space inside an object
PROCEDURES:
TEACHER NOTES:
Introduction/Hook:
Students should be able to see
that there are six sides that
each has an area of 1 m2 for a
total surface area of 6 m2. The
volume is 1 m3. Therefore the
surface to volume ratio is
For this activity, it would be helpful for students to visualize
the problem if you have a cube shaped box on hand.
Imagine that you have a box that is 1 meter wide, 1 meter long
and 1 meter tall. What is the surface area to volume ratio of
the box?
6 m2
 6 m 1
1m 3
What if we cut each size of the box in half, so that now it is ½
meter wide, ½ meter long, and ½ meter tall. What will happen
to the surface area to volume ratio?
Give students a few minutes
to reason through what they
think will happen. Students
may have trouble with
proportional reasoning.
Let students give answers and provide rationales for their
answers. Then work through the solution.
The new area of each side is ½ x ½ = ¼ m2. There are still six
sides, so the total surface area is 6 x ¼ m2 = 1.5 m2. So, when
the cube’s dimensions were cut in half, by what factor did the
surface area change? (The correct answer is ¼)
The new volume is ½ x ½ x ½ = 1/8 m3. By what factor did the
volume change? (The correct answer is 1/8)
1.5 m 2
 12 m 1
1 3
m
8
So, we see that when we cut the dimensions of the cube in half,
the new surface area was ¼ of the original surface area, and
the new volume was 1/8 of the original volume. The surface
area to volume ratio doubled when the side length was cut in
half.
Is there another way to solve this problem?
Will this pattern continue? What is the mathematical
relationship between surface area and volume?
In this lesson, you will figure out the relationship between
surface area and volume.
Create groups of 2 or 3 students for this activity. Students will
be building cubes, determining surface area and volume, and
use mathematical modeling to find the relationship between
surface area and volume.
Allow students a few minutes
to suggest other ways to solve
the problem.
Handout cardstock patterns for five cubes, with side lengths of
1 cm, 2 cm, 3 cm, 4 cm, and 5 cm. Cut out and assemble each
cube. For each cube, calculate surface area, volume, and
surface area to volume ratio.
After the table is filled out, have the group of students answer
the following questions:
1. Describe any patterns you observe in the data.
2. If you graph surface area to volume ratio vs. cube
length, what shape of graph would you expect to
see? Explain your choice.
3. Create a graph of surface area to volume ratio vs.
cube length using a graphing calculator or a
spreadsheet. What is the shape of the graph?
What does this tell you about the relationship
between surface area to volume ratio and cube
length?
4. Use the best-fit procedure to find the equation of
the line or curve of best-fit. Record this equation.
5. What type of function is your equation of best fit?
What does this tell you about the relationship?
6. What would the surface area to volume ratio be
for a cube that was 1/10 cm on each side? 1/100?
Explain your reasoning.
7. Write a paragraph that describes what you found
out about the relationship between surface area to
volume ratio and cube length.
8. What effect do you think surface area to volume
ratio could have on physical and chemical
properties? Why?
Debriefing:
Whole class debrief and discussion of findings.
Grouping:
Whole class for debrief and discussion. Individual for career
research and product creation.
Homework:
Differentiation for:
ELL
Twice-Exceptional;
Highly Gifted;
Differentiation:
Choice:
Resources:
Products:
Tiered questions/assignments:
Solution
Handout ML1.1 Blackline master for cube cutouts
Handout ML5.2 Activity Worksheet
Surface area to volume ratio vs. cube side length
Side
Length
(cm)
Surface
Area (cm2)
Volume
(cm3)
Surface Area
to Volume
Ratio (cm-1)
1
2
3
4
5
1. Describe any patterns you observe in the data.
2. If you graph surface area to volume ratio vs. cube length, what shape of graph
would you expect to see? Explain your choice.
3. Create a graph of surface area to volume ratio vs. cube length using a graphing
calculator or a spreadsheet. What is the shape of the graph? What does this tell
you about the relationship between surface area to volume ratio and cube length?
4. Use the best-fit procedure to find the equation of the line or curve of best-fit.
Record this equation.
5. What type of function is your equation of best fit? What does this tell you about
the relationship?
6. What would the surface area to volume ratio be for a cube that was 1/10 cm on
each side? 1/100? Explain your reasoning.
7. Write a paragraph that describes what you found out about the relationship
between surface area to volume ratio and cube length.
8. What effect do you think surface area to volume ratio could have on physical and
chemical properties? Why?