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Transcript
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. • • • • • • “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” Jill Russ How Many Stars In the Sky? Page 1 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 Jill Russ How Many Stars In the Sky? Page 2 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. Jill Russ How Many Stars In the Sky? Page 3 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. Jill Russ How Many Stars In the Sky? Page 4 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. Jill Russ How Many Stars In the Sky? Page 5 The Sagittarius Star Cloud Photo No.: STScI-PRC98-30 Credit: Hubble Heritage Team (AURA/STScI/NASA) Jill Russ How Many Stars In the Sky? Page 6 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. Jill Russ How Many Stars In the Sky? Page 7 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? ______________________ __________________________________________________________ __________________________________________________________ Jill Russ How Many Stars In the Sky? Page 8 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. Jill Russ How Many Stars In the Sky? Shows some understanding of the problem Page 9 Alternate Star Image Jill Russ How Many Stars In the Sky? Page 10