Common Core State Standards
Mathematics: Expressions and Equations
(8.EE.c.7) Analyze and solve linear equations and pairs of simultaneous linear equations
(HS.Modeling) Graphing utilities, spreadsheets, computer algebra systems, and dynamic geometry software are powerful
tools that can be used to model purely mathematical phenomena as well as physical phenomena
(NGSS.MP.2) Reason abstractly and quantitatively
(NGSS.MP.4) Model with mathematics
(NGSS.HSN.Q.A.1) Use units as a way to understand problems and to guide the solution of multistep problems;
choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays
ELA/Literacy
(RTS.1112.1) Cite specific textual evidence to support analysis of science and technical texts, attending to important
distinctions the author makes and to any gaps or inconsistencies in the account.
(RTS.1112.2) Determine the central ideas or conclusions of a text; summarize complex concepts, processes, or
information presented in a text by paraphrasing them in simpler but still accurate terms.

Next Generation Science Standards
Science and Engineering Practices
(HSPS21) Analyze data using tools, technology or models (computational, mathematical) in order
to make valid and reliable scientific claims
(HSESS11) Develop a model based on evidence to illustrate the relationships between systems or components of a system
(HSESS12) Construct an explanation based on valid and reliable evidence obtained from a variety
of sources including students own investigations, theories, simulations and peer review
(MSESS13) Analyze and interpret data to determine scale properties of objects in the solar system.
[Clarification Statement: Emphasis is on the analysis of data from Earthbased instruments, spacebased telescopes,
and spacecraft to determine similarities and differences among solar system objects. Examples of scale properties include
the sizes of an object’s layers (such as crust and atmosphere), surface features (such as volcanoes), and orbital radius.
Examples of data include statistical information, drawings and photographs, and models.]

Individual Math Challenges
Exploring the Atmosphere of Pluto
The atmosphere of Pluto comes and goes as its temperature changes during the long Pluto year.
This problem uses a simple equation to help you compare atmosphere thicknesses as temperature
and constituents change.
[Problem Page] 
[Answer Key]
Pluto: The Twilight World
The sun is so far away from Pluto that High Noon is only as bright as twilight here on Earth. That means
solar panels used for converting sunlight into electricity have to be much larger. This problem explores
solar brightness and its impact on generating electricity in the remote reaches of our solar system.
[Problem Page] 
[Answer Key]
Exploring the Dwarf Planets
Scientists can learn a lot about the inside of a dwarf planet by making very precise measurements of its
diameter and mass. From these measurements, average densities (mass divided by volume) can be figured out.
The density of an object gives us a clue as to whether it is mostly rocky or mostly icy.
[Problem Page] 
[Answer Key]

More Math Challenges
Exploring the Dwarf Planets
Scientists can learn a lot about the inside of a dwarf planet by making very precise measurements
of its diameter and mass. From these measurements, average densities (mass divided by volume) can be
figured out. The density of an object gives us a clue as to whether it is mostly rocky or mostly icy.
[Grade: 68  Topics: density; mass]
Exploring the Dwarf Planet Ceres
Counting and measuring craters on Ceres can provide insights into the cratering process that created
its surface. It appears to have fewer large craters than scientists had expected to see, a possible
indication that most of the material that came together to form Ceres was smaller asteroids.
[Grade: 68  Topics: scale; proportion]
Triton: The Twin of Pluto?
Triton is only slightly larger than Pluto. Both worlds have similar surface materials, such as nitrogen,
methane and carbon monoxide. Their diameters, masses and densities are amazingly similar.
[Grade: 68  Topics: volume=area x height; rates;volume of a sphere]
The Amazing Journey to Pluto!
Deep space missions can take up to 10 years from development to launch. For New Horizons, it took
close to 20 years from the time that scientists conceived of the mission to the time it reached its destination!
[Grade: 68  Topics: speed; time; unit conversion]
Exploring Interplanetary Communication
On July 14, 2015, NASA’s New Horizons spacecraft reaches dwarf planet Pluto and begins sending data back to
Earth. At that time, the distance from Earth to Pluto is 4.8 billion kilometers. At the speed of light,
oneway radio signal travel time is 16,000 seconds or 4 hours and 27 minutes.
[Grade: 68  Topics: distance; speed; time]
Modeling the Interior of Pluto
This interactive Excel spreadsheet lets you
create a model of the interior of Pluto based on its diameter, mass and the densities of ice and rock.
[Grade: 68  Topics: volume of a sphere; density]

Math Problem Resource Guides
Exploring the Dwarf Planets  Elementary (2015)
10 Problems
This book describes how missions like New Horizons and Dawn are created, built, launched and how they return data back to
Earth. Essays describe spacecraft design, STEM careers that form the basis for these missions, along with a series of 10
Space Math challenges to encourage a deeper understanding of the issues. A discussion of dwarf planets and why Pluto was
'demoted' is also provided.sic scales and proportions, fractions, scientific notion, algebra and geometry.
[PDF: 10 Mby ]

Math Problem Resource Guides
Exploring the Dwarf Planets  Advanced (2015)
11 Problems
This book provides 11 math problems for middle and high school students that explore various aspects of the dwarf planets
such as their atmospheres, sunlight changes, and the Kuiper Belt. Students also learn about the other known dwarf planets:
Eris, Makemake, Haumea and Ceres. The Answer Key is is available here, and also in the back of the book
[PDF: 3 Mby ]
