Tools for Instruction
Explore Volume and Surface Area
Objective Calculate volume of spheres with numbers expressed
in scientific notation and write ratios to compare volumes.
Student develops deeper
understanding of skills
by making connections
across three domains, in
line with the Standards for
Mathematical Practice.
Materials Calculator (optional), Diameters of Planets (page 3)
This activity builds on students’ understanding of volume of various geometric figures and on computations with
large numbers. This activity requires students to calculate the radius, given a diameter, and then substitute the
radius into a formula to find the volume of a sphere. Students are likely to discover that it is more efficient to use
scientific notation to express the volumes and then compare them in ratios. These ideas provide a foundation for
work with scientific notation, using both positive and negative exponents.
This is intended to be a challenge activity, so refrain from providing answers. If the student is not successful, let
him or her know there will be an opportunity to return to try this challenge again at a later date.
Step by Step Instructional format follows
the gradual release of
responsibility model.
20–30 minutes
1 Discuss volume and surface area concepts.
Provides challenge activities for students
working on or above grade level. Student is
expected to work more independently with
teacher providing limited guidance as needed.
• Invite the student to share facts and understandings related to volume and surface area. Some examples
include:
Volume is measured in cubic units. Surface area is measured in square units.
The volume of a prism is the area of the base times the height. The surface area of a prism is the sum of the
areas of all of its faces.
The volume of a pyramid or cone is ​ 13 ​ the area of the base times the height.
··
• Review volume and surface area formulas, making sure the student understands that the volume of a sphere
is ​ 43 ​pr3 and the surface area of a sphere is 4pr2.
Graphic organizer helps
··
students record and track
their work and thinking.
2 Find the volume of the Earth.
• Give the student a copy of Diameters of
Planets (page 3) and have her use the
given diameters to find the volume of
Earth. (Radius of Earth: 6,378 km, Volume
of Earth: 1.09 3 1012 km3)
Planet
Diameter
Mercury
4,880 km
Venus
12,104 km
Earth
12,756 km
Mars
6,794 km
• Before starting, you may want to discuss
whether the student should use a
calculator and how the numbers should
be rounded.
Jupiter
142,984 km
Saturn
120,536 km
• Ask the student to express the volume
using scientific notation.
Uranus
51,118 km
Neptune
49,532 km
3 Predict the volume of Mars.
Radius
Volume
Surface Area
Reminders integrate important Common Core mathematical practices like
thinking about accuracy using rounding and modeling using scientific notation.
• Challenge the student to predict the approximate volume of Mars, based on the information about Earth,
and to explain his reasoning. Students should reason that because the diameter (and radius) of Mars is
roughly half of that of Earth, the volume of Mars will be about ​ 18 ​that of Earth.
··
• If desired, have the student verify his estimate by calculating the volume of Mars. (1.64 3 1011 km3)
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Number – Geometry I Level 8 I Explore Volume and Surface Area I Page 1 of 3
i-Ready Tools for Instruction
4 Repeat with surface area.
• Repeat Steps 2 and 3 for surface area. Have the student find the surface area of Earth and then use it to
predict the surface area of Mars. Be sure to have the student explain his reasoning. Point out any flaws in his
reasoning, encouraging him to reason carefully. (SA of Earth: 5.11 3 108 km2)
• Have the student calculate the surface area of Mars. (1.45 3 108 km2)
• If several students have completed the activity, then allow them to work in pairs to talk about and compare
their predictions.
• Otherwise, work directly with the student. Discuss questions such as the following:
Was your prediction correct, or close? If yes, what contributed to your success? If not, where was your error?
If you used scientific notation, was that helpful? If you didn’t use it, might it have been helpful?
Could you have had a more useful answer if you had rounded differently? Why or why not?
Check for Understanding
Helps teacher identify who has mastered
the concept and who hasn’t, and how to
address students’ misconceptions.
Scripting saves
time and takes the
guesswork out of how
to model converting
units using the table.
Ask the student to estimate the answer to the following questions, and explain his thinking in words or
equations. How many times the volume of the smallest planet is the volume of the largest planet? How many times the
surface area of the smallest planet is the surface area of the largest? (The ratio of the diameters of Jupiter to Mercury
is approximately 30 to 1 so the ratio of volumes will be the cube of that, or about 27,000 to 1. The ratio of surface
areas will be the square, or 900 to 1.)
Use the chart below to support the student as needed.
If you observe…
the student is not able to explain
the reasoning behind his answers
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The student may…
not understand what kind of
explanation is expected.
Then try…
providing a starting place. For
example, say: Start by telling me
the ratio of the diameters.
Number – Geometry I Level 8 I Explore Volume and Surface Area I Page 2 of 3
i-Ready Tools for Instruction
Includes printable table,
referenced on pg. 1, for student
use which saves teachers time.
Name
Diameters of Planets
Planet
Diameter
Radius
Mercury
4,880 km
Venus
12,104 km
Earth
12,756 km
Mars
6,794 km
Jupiter
142,984 km
Saturn
120,536 km
Uranus
51,118 km
Neptune
49,532 km
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©2013 Curriculum Associates, LLC
Volume
Surface Area
Number – Geometry I Level 8 I Explore Volume and Surface Area I Page 3 of 3