Lesson Plan Template: Confirmatory or Structured Inquiry
Learning Objectives: What will students know and be able to do as a result of this lesson?
Students will use a real-world case study to measure the earth-sun distance using basic trigonometry. Students will get hands-on practice using basic concepts regarding angles, triangles, and triangulation. Students will acquire experience and knowledge in the following areas:
1. How to measure distances to faraway objects;
2. Vocabulary words: triangulation, parallax, transit, astronomical unit;
3. Relative distances in the solar system;
4. Following a defined procedure to achieve a numerical result;
5. Checking results with known values;
6. Identifying sources of error.
Prior Knowledge: What prior knowledge should students have for this lesson?
Students should have basic knowledge of geometry and trigonometry. The only trigonometry knowledge needed is how to use the Tangent function.
Guiding Questions: What are the guiding questions for this lesson?
What is a transit of Venus? - During a transit of Venus, the planet comes comes closer to earth than at any other time. A transit of Venus is the best opportunity to measure Venus' parallax and, therefore, its actual distance from earth. In addition to being extremely rare, transits of Venus help us to discover how big the solar system really is!
What is parallax? - Parallax is the apparent shift of an object's position when the viewing perspective changes. To illustrate parallax, have students extend an index finger and hold it at arm's length. Point the index finger so it points upwards. Now have students close one eye while keeping the other open. Ask students to notice what is in the background behind the finger, perhaps something on the classroom wall. Now have students switch eyes. By winking back and forth between left and right eyes, they should see a shift in the finger's position. That's parallax.
Introduction: How will the teacher introduce the lesson to the students?
One of the most important unsolved questions during the 18th and 19th centuries was: "How big is the solar system?" To answer this question, nine nations sent 120 observers to Asia and the Pacific to observe the rare 1769 transit of Venus. One of those explorers was Captain James Cook, who led his ship HMS Endeavor (NASA named a Space Shuttle after this famous ship) to observe the transit of Venus from the island of Tahiti. That spot today is known as "Venus Point."
Using satellite data, we can do in three hours what required James Cook three years to do during his epic voyage. All that's needed are the heart of an explorer and basic knowledge of geometry and trigonometry.
Two excellent NASA videos - each about 5 minutes long - give the historical context of Venus Transits.
Investigate: What question(s) will students be investigating? What process will students follow to collect information that can be used to answer the question(s)?
Students will use a defined procedure to analyze a set of satellite data from the 2004 transit of Venus. The data are from a NASA polar-orbiting satellite called "TRACE," which was launched specifically to observe the sun.
Data from the TRACE satellite - a satellite in polar orbit around earth - show a sinusoidal pattern in the path of Venus as it transits the Sun's disk. From its observed parallax, students use geometry and the tangent function to calculate the distance to Venus. After calculating the distance to Venus, students solve a simple proportion to obtain the earth-sun distance.
(1) The PDF file "Venus Transit Math" leads students through a five step procedure to calculate Venus' parallax and distance from earth. The final step requires solving a simple proportion to calculate the earth-sun distance.
(2) "Transit Geometry.png" illustrates the geometry of the satellite, earth, Venus, and the sun;
(3) "Venus Transit (NASA).gif" is an image taken in visible light by the TRACE satellite during the June 8, 2004 transit of Venus.
Analyze: How will students organize and interpret the data collected during the investigation?
Students will record their observations, calculations, and findings on a worksheet. Alternatively, students may use their class notebooks or a lab journal to record the results.
Closure: What will the teacher do to bring the lesson to a close? How will the students make sense of the investigation?
Review with the class a table of historic measurements of the earth-sun distance from 1672 to the current time. From this, students will see that knowledge is incremental and cumulative, building on the work of previous generations.
Planetary transits are important in modern astronomy because they are one of the main methods for discovering planets around other stars. The Kepler space mission, for example, uses the transit method to discover planets around other stars.
Teacher hands out a quiz with questions like the following:
S1. Given - The diameter of Venus is known to be 12,000 kilometers. At the time of the transit of Venus, its angular diameter is measured as 0.016 degree. Find - Using trigonometry, what is the distance from Earth to Venus in kilometers at the time of the transit?
S2. Given - Venus and Earth are 0.72 and 1.00 Astronomical Units, respectively, from the Sun. Find - Using a simple proportion, what is your estimate of the Astronomical Unit in kilometers? (Hint - use the result from S1.)
S3. What is a transit?
S4. What is parallax?
S5. In what year did Captain James Cook observe the transit of Venus from Tahiti? (circle one) 1639, 1769, 1882, 2004.
S6. In the exercise, what caused the sinusoidal (wavy) path of Venus as it transited the Sun's disk?
S7. In the exercise, why did you get the wrong answer the first time?
S8. In the exercise, how was that problem fixed?
S9. When will the next transit of Venus occur?
S10. What did you enjoy most (or least) about the exercise?
Teacher will use an initial assessment with questions (given orally or as a short quiz) such as the following:
F1. How often does a transit of Venus or a total solar eclipse occur?
F2. Why do people travel long distances to see these transits?
F3. What does a transit of Venus look like from space?
F4. What is the difference between a transit and an eclipse?
F5. When will a transit of Venus occur in my hometown?
F6. What is the earth-sun distance?
Teacher uses results to assess student knowledge in preparation for the lesson.
Feedback to Students
Teacher leads students through the process of using Method 1. Students record their answers on a worksheet. Teacher checks work during the lesson.
The following are opportunities in the lesson when students will provide evidence that teachers can use to give students feedback:
E1. Students record the largest and smallest image shifts.
E2. Students record the difference between the largest and smallest image shifts.
E3. Students calculate the radius, R, of the TRACE satellite orbit in kilometers.
E4. Students calculate 1st set of numerical results: (a) Apparent Parallax, P; (b) Apparent Venus Distance, D; (c) Apparent Sun Distance, L.
E5. Teacher pauses to explain why the 1st set of calculations leads to a result of Apparent Sun Distance, L, that is too large. Rationale - It's because the sun has parallax. Though much further away than Venus, the Sun's apparent position is shifting, too! The Sun's position is shifting up and down in the same direction as Venus, but not as much because it is further away than Venus. This leads to an "apparent" parallax of Venus which is too small. To obtain the true parallax of Venus, the sun's parallax must be added to the parallax of Venus.
E6. Students calculate 2nd set of numerical results: (a) True Parallax, P'; (b) True Venus Distance, D'; (c) True Sun Distance, L'.
E7. Students check their answers and report to the class.
E8. Students prepare a histogram that shows the distribution of results for the entire class.
E9. Teacher reviews final results with class.