Download How Many Stars in the Sky?

Florida Math and Science Day 2009
Lesson Plan
Theme: Space Exploration
Lesson Title
How Many Stars in the Sky?
Grade Span
Content Emphasis
Targeted Benchmark(s)
3rd-5th
Math
Benchmark: MA.3.A.6.2
Solve non-routine problems by making a
table, chart, or list and searching for patterns.
Author(s)
School
District
Email address
Phone number
Jill Russ
James E. Plew Elementary School
Okaloosa School District
[email protected]
850-833-4100
Lesson Preparation
Learning goals: What will students be able to do as the result of this lesson?
Students will:
• solve non-routine problems by making a table, chart, or list and searching for
patterns.
• use random sampling to estimate the number of stars in the Milky Way.
• find the range, mean, and median of a set of data.
• solve problems that arise in mathematics and in other contexts.
Estimated time: Please indicate whether this is a stand-alone lesson or a
series of lessons.
The entire stand-alone lesson should take between 45 and 60 minutes.
Materials/Resources: Please list any materials or resources related to this
lesson.
•
•
•
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“The Sagittarius Star Cloud” handout, one copy for each group of 2 or 3 students.
“How Many Stars in the Sky” worksheet, one copy per student (pg. 7-8)
rulers
calculators
“Alternate Star Image” for lower ability or younger students
“Problem Solving Rubric”
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Teacher Preparation: What do you need to do to prepare for this lesson?
• Make one copy of the “The Sagittarius Star Cloud” handout (page 6) for each group
of 2 or 3 students.
• Make one copy of the “How Many Stars in the Sky” worksheet (page 7-8) for each
student.
• Make one copy of the “Problem Solving Rubric” (page 9) for each student for
assessment.
• Make copies of the “Alternate Star Image” (page 10) if needed for lower ability or
younger students.
Lesson Procedure and Evaluation
Introduction: Describe how you will make connections to prior knowledge
and experiences and how you will uncover misconceptions.
Ask students to share what they already know about stars. What do you know about
stars? How are the stars like our Sun? Have you noticed that you can see more stars in
the sky when you are in the country than when you are in the city? Why do you think
that is so? Have you ever looked at stars through a telescope? How many stars do you
think are in the Milky Way? As students are discussing, listen for any misconceptions
they have about the Sun and the stars.
Possible Math Misconceptions:
• Students may confuse the terms range, mean and median.
Students usually have only a vague understanding of what range, mean, and
median actually mean. They often learn a rote procedure for finding the answer
and don’t have any conceptual understanding of what mean and median
actually show about a set of data.
Possible Science Misconceptions:
• Students may believe that the Sun is not a star.
Students associate stars with the night sky because they can see stars clearly.
The Sun is a star. The Sun can be seen in the daytime because it is so close to
Earth. We can't see other stars during the day because the Sun's light
illuminates the Earth's atmosphere.
• Students may believe that all stars are exactly the same.
Twinkling and the small apparent sizes of stars make it difficult to perceive the
range of colors that different stars have. Unless a person is motivated to look
carefully or is naturally methodical, most people don't notice the colors. All
stars are not the same. Stars vary in brightness, color, mass, temperature, and
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age. Stars vary in brightness, color, mass, temperature, and age. Stars are
classified by colors as related to their surface temperature. The coolest stars are
orange, then red, yellow, green, blue and finally blue-white. The size of a star
on a photograph tells us about its brightness. Large star images mean a
brighter star. No camera can actually discern the actual size of a star because
they are so far away.
• Students may believe that stars are evenly distributed throughout the universe,
including within the solar system.
Because they may not be able to see many stars in the night sky, students may
think that the stars are spread evenly throughout the universe. The stars are
not evenly distributed, and areas of high concentrations and low concentrations
of stars can be seen in the star images. Students also need to understand that
stars are not part of the “solar system,” and understand the difference between
solar system, galaxy, and universe.
Exploration: Describe in detail the activity or investigation the students
will be engaged in and how you will facilitate the inquiry process to lead to
student-developed conclusions.
1.
Divide students into groups of 2 or 3 and distribute a black and white image of stars
to each group. Give each student a copy of the student worksheet. Tell students that,
since they cannot count stars during the day, they are going to count the number of
stars on an artificial sky-- a picture of the Sagittarius Star Cloud taken by the
Hubble Telescope. On the student worksheet, have the students record their
individual estimates of the number of stars in their picture, and then find the
average (mean) of their group’s estimates. Give each group an opportunity to share
their estimates. How many stars do they estimate are in their image?
2. Ask the students to count the number of stars in the picture. Hopefully, they will
soon realize that it is not practical or realistic to count every star.
3. Talk about the use of sampling to estimate quantities too large to count. There are
two principal ways of gathering quantitative data—by census or by sample. In a
census, every organism, object, or event is counted. Since it is usually impractical or
impossible to count every element, the preferred technique is sampling. For
example, rather than count all the grains of sand on the beach, you can count the
number of grains in a small area. Once that value is known, you can multiply to
estimate the number of grains of sand on the beach. Astronomers use sampling to
estimate the number of stars in our galaxy.
4. Have each group work together come up with a plan for estimating the number of
stars in the picture. They may come up with a plan such as dividing the picture into
sections, counting the number or stars in one section, and multiplying by the
number of sections. More advanced students may realize they need to adjust for
“fuller” and “emptier” sections of the picture.
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5.
Have students complete the student pages, using their calculators as needed to do
the calculations. How many stars were on your page? How does your count compare
to the counts of other groups? Did each group come up with the same number? Why
or why not?
6. Discuss what the mean, median, and range actually tell us about this set of data.
Discuss why mathematicians use different measures of central tendency. Outliers
are extremely high or extremely low data points that stick way out from the general
trend and may affect the mean and range. The median is not affected by extreme
values. The range is a measure of dispersion and it tells us the spread of the
numbers - that is, did some people get really high numbers and some get really low
numbers, or are all the numbers pretty close together?
7.
Discuss how students chose the sections of the picture to count. The larger a wellchosen sample is, the more accurately it is likely to represent the whole, but there
are many ways of choosing a sample that can make it unrepresentative of the whole.
Talk about how the stars are not evenly distributed throughout the picture, and that
choosing a “full” section or an “emptier” section would give very different results.
Have students share their ideas for ways to choose the “best” sections to count. One
of the most frequently used methods of sampling is random sampling. In random
sampling, each element has an equal chance of appearing in the sample.“Random”
means that you don’t actually get to pick which sections to count; you select sections
to count using something unpredictable. Discuss ways that students could use a
random method of selecting the squares to count. One possible method would be to
cut out squares of paper about 2 cm on each side, drop those squares onto the star
picture and trace around the squares.
8. The truthful answer to how many stars are in the sky is that no one really knows,
because there are just too many to count. Stars are not evenly spread out in our
universe but instead group together in galaxies. Our Milky Way galaxy alone is made
up of an estimated 100 thousand million (100,000,000,000 or 10^10 ) stars.
Counting the stars in our Milky Way individually, even counting non-stop at a rate
of 1 star every second it would take over 3,170 years! Rather than count individual
stars scientists look at how big and bright a galaxy is in order to estimate the
number of stars that make it up. When they know this they multiply this number by
the estimated number of galaxies in the universe. There are an estimated 10^10
galaxies in the universe. Stars are constantly being created and dying throughout
our universe but estimates as to the number of stars out there suggest a number
over 10^20 stars. That’s 10 with 20 zeros after it.
Application: Describe how students will be able to apply what they have
learned to other situations.
Use random sampling to estimate the number of grains of sand in a sandbox or the
number of blades of grass on the playground.
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Assessment: Describe how student knowledge is being assessed at the
appropriate cognitive level for the targeted benchmarks.
Ask students to share what they now know about stars. What do you know about stars?
How are the stars like our Sun? How many stars do you think are in the Milky Way? Are
all stars alike? As students are discussing, listen for any misconceptions they have
about the Sun and the stars.
Many students, especially students with learning disabilities, have difficulty solving even
simple mathematical word problems. Non-routine problems and problems that do not
have one correct answer such as this activity are especially challenging. Most third,
fourth, and fifth grade students have not acquired the skills and strategies needed to
“decide what to do” to solve these types of problems in school and in their daily lives.
Therefore, after students have made an attempt to solve the problem on their own, it is
important to discuss appropriate methods of attacking the problem.
Students’ current level of problem solving abilities will be informally assessed using the
“Problem Solving Rubric.”
Teacher Self-Reflection: Record your thoughts on the lesson and describe
any modifications you would recommend based on the outcomes.
If students are having difficulty coming up on their own with a strategy for estimating
the number of stars, you should have a whole class discussion about possible ways to
solve the problem. One possible method of solving the problem is to use small squares of
paper to randomly choose which sections of the picture to count. Cut out about five or
six 2 cm x 2 cm squares of paper for each group. Drop the squares onto the star picture
and trace around them. Count the number of stars in each square. Find the average
(mean) of the number of stars in each square. Multiply the number of squares it would
take to cover the entire picture by the average number of stars. Alternatively, students
could divide the picture into squares and then choose a “full” section and an “empty”
section to count. Find the average of the number of stars in those two sections and use
that average to determine the total number of stars. Of course, their are many other
appropriate ways to approach this problem.
Due to the size of the numbers involved in calculating the mean, students will need to
use calculators. This computation may need to be done as a whole class. Also, you many
need to remind students of how to find the mean, median, and range.
This lesson can be modified for lower ability or younger students by using the “Alternate
Star Image” which shows a less dense concentration of stars.
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The Sagittarius Star Cloud
Photo No.: STScI-PRC98-30
Credit: Hubble Heritage Team (AURA/STScI/NASA)
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Name ____________________
Date ____________________
How Many Stars in the Sky?
Your task is to determine the number of stars on the page. Begin by making an estimate.
I estimate that there are _______ stars on the page.
The average estimate in our group is _____________ stars.
Method of Solving the Problem
Next, come up with a plan for finding the number of stars on the page. Describe how
you will go about finding the number of stars.
Collection of Data
Now, carry out your plan and calculate how many stars are on the page. Show your data
and your calculations here.
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Analysis of Data
Find out how many stars the other groups calculated and record their answers.
My Group _______________
Group 1 _________________
Group 2 ________________
Group 3 _________________
Group 4 ________________
Group 5 _________________
Group 6 ________________
Group 7 _________________
Group 8 ________________
Group 9 _________________
Group 10 ________________
Group 11 _________________
1. Find the mean number of stars.
2. Find the median number of stars.
3. How do the median and mean compare? _____________________________
___________________________________________________________
___________________________________________________________
4.Find the range of stars. _______________________
5. Why do you think there is such a variety of answers? ______________________
__________________________________________________________
__________________________________________________________
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Student’s Name __________________
Date_______________
Problem Solving Rubric
4
3
2
1
Interpretation
Finds all
important
parts of the
problem
Fully
understands
the problem
Shows good
understanding
of the problem
Strategy
Uses an
efficient and
effective
strategy to
solve the
problem.
Uses an
effective
strategy to
solve the
problem.
Uses a strategy Does not use a
to solve the
strategy to solve
problem but it the problem.
is not effective.
Accuracy
Calculations
are completely
correct and
answers
properly
labeled.
Calculations
Calculations
are mostly
contain major
correct, may
errors.
contain minor
errors.
A limited
amount of work
shown.
Calculations are
completely
incorrect
leading to an
incorrect
answer.
Mathematical
Concepts
Shows
complete
understanding
of the concepts
used to solve
the problem.
Shows
substantial
understanding
of the concepts
used to solve
the problem.
Shows some
understanding
of the concepts
needed to solve
the problem.
Shows very
limited
understanding
of the concepts
needed to solve
the problem.
Explanation
Explanation is Explanation is Explanation is
detailed and
clear.
a little difficult
clear.
to understand,
but includes
critical
components.
Explanation is
difficult to
understand and
is missing
several
components.
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How Many Stars In the Sky?
Shows some
understanding
of the problem
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Alternate Star Image
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