# Reaching for a Star... (and finding its diameter!)

#### Summary

In this activity, students review standard metric units of measurement and practice using basic science classroom instruments. To help students relate to these units, they create their own list of approximate representations for selected units of measurement using familiar classroom objects. Students learn that extreme distances can be measured by using the ratios of similar triangles. Triangles are generated by shadows cast by the tree and another, easily measured item. Finally, students are given a challenge to develop a method to approximate the measurement of the diameter of the sun using mathematical ratios.

[image right border]## Learning Goals

- To give students their own set of metric unit references helping them understand and remember quantities that those metric units represent.

- To challenge students with an 'extreme' measurement situation and allow them to develop a solution to that problem.

- To encourage the use of a simple mathematical ratios to solve a huge measurement problem.

## Context for Use

## Background

Familiarity with standard metric units of measurement is an important component of doing science and math. Effectively communicating one's finding with others is another important skill. Since English measurements are in common use in this country, students often struggle to relate to the quantities that metric units represent. This set of activities gives students practice with using the tools to obtain standard metric measurements of length, volume, mass, and temperature. Students will also develop a frame of reference when they create a chart of metric measurements of familiar objects. Students will also be challenged to create a measurement method for long distances: first they calculate the height of a tree using ratios and then they are challenged to do a similar calculation to measure the diameter of the sun. Research will show the students that the ancient Greeks were able to work out this problem by using mathematical ratios; no high-tech methods are required!## Description and Teaching Materials

### In-Class Activities

**Day 1**

1. Measuring activities are set up at various stations around the classroom: triple beam balances or digital scales for measuring mass, meter sticks and metric rulers for measuring various lengths, graduated cylinders and beakers for measuring volumes, thermometers for measuring temperatures. Each station should have an assortment of common objects or materials to be measured.

2. Students are introduced to the basic units of measuring length, volume, temperature, and mass in the metric system. The proper use of each of the measuring tools – and common sources of error - should be demonstrated by the instructor. Common mistakes include measuring from the edge of the ruler which may not be the beginning of the scale, not deducting the mass of the container when measuring a quantity of material on the triple beam balance, for example. Significant digits and sensitivity of the measuring devices should be addressed when using the various instruments.

3. Distribute a *Metric Measurement Reference Chart*to each student. Divide the students into small groups; one group for each of the stations. Working in groups, students should find an object that *approximately* measures each of the given amounts. Invite students to look beyond the provided objects if they want a challenge or wish to add more than one item to their chart.

To assist students who are struggling to match objects to measurements, a 'hint list' may be provided. Students work their way through each of the stations and complete the metric chart.

4. Review the completed charts with the students, checking any suspect measurements. Completed charts (and accurate) chart can be kept in the students' science binders for future reference.

5. Begin a group discussion on how to measure some 'extreme' distances or quantities. How could you measure the height of a tree, for example? How could you measure the distance to the sun? How about the diameter of the moon? Students can share their ideas with the class and discuss the challenges that making such measurements provides.

**Day 2**

6. Revisit the 'extreme' measurement problem and ask students to share their ideas. Explain to the students that the ancient Greeks were able to work out this problem by using their knowledge of angles and ratios. This part of the activity can be expanded if math is the focus of the lesson and students have more skills with geometry. If time allows, students can explore more by researching the work of Aristarchus and Eratosthenes. Explain to students that similar triangles (same angles but different side lengths) can be compared to obtain a missing side length.

a. Have students draw a right triangle (graph paper makes it easy) that has sides 3cm X 4cm X 5cm.

b. Now have students draw a similar right triangle with sides that are twice as long (6cm X 8cm X ?). Without measuring the 3^{rd} side, have students predict the size based on a ratio of the corresponding sides from both triangles. This method is outlined in detail on the Youtube videos listed below in Step 7.

c. Students should measure the sides to confirm that this ratio holds true.

7. Give students the challenge of measuring the height of a nearby tree (or building or flagpole). Allow students to discuss their ideas with each other and come up with possible solutions using the concept of similar triangles. If time permits, allow students to test out some their ideas if this can be safely done. You can show students any of the following YouTube videos: http://www.youtube.com/watch?v=31qq1zoQVHY&NR=1 or another good one is at http://www.youtube.com/watch?v=wHCFaGaKt-M&feature=related

Using any of the similar triangle methods shown, have students calculate the measurement of a nearby tall object. Students should compare their measurements for accuracy and discuss possible sources of error.

**Day 3**

8. Finally, ask students how they could use what they have learned about setting up ratios and similar lengths and apply it to really huge distances, such as those in outer space. How could they calculate the approximate diameter of the sun? Allow students to share their ideas with each other.

9. **Remind students that they should never look directly at the sun; this will result in damage to their eyes.*

A safe way to observe the sun is to create a pinhole viewer.

Create a pinhole viewer and a screen following the basic instructions at:

http://cse.ssl.berkeley.edu/AtHomeAstronomy/activity_03.html**.**

**Other sources of information can be found at:**

http://archive.planet-science.com/outthere/cap/pdf/cath_activity.pdf

http://www-outreach.phy.cam.ac.uk/resources/ESM/KS2/pinhole/index.php

http://www.youtube.com/watch?gl=CA&hl=en&v=zioSpV2yq24

A very simple set-up for the classroom is shown at:

http://hilaroad.com/camp/projects/pinhole_sun/sun_diameter.html

It will be helpful for the students to see how a pinhole viewer works by showing the projections of a flame through the pinhole (Experiment 1 on the Berkeley.edu site) before showing the projection of the sun's image (Experiment 2 ).

Adding a millimeter ruler to the screen allows for easy measurement of the projected image. You can mount the pinhole viewer on one end of a meter stick and the screen on the other end. Clamp the meter stick to a ring stand or lean it on a chair and aim the pinhole at the sun. Adjust the angle of the meter stick so that the shadows cast by both the pinhole viewer and the screen are aligned.

Allow students time to explore how they might now calculate the diameter of the sun by using the pinhole viewer. They could start with using the pinhole viewer to project an image of known size such as a light bulb. Students can explore calculating the diameter of the light bulb using the pinhole device and then verify their measurements by actually measuring the diameter of the light bulb. Encourage students to think about the information they have and how they could use this information to calculate diameters. Do they need more information to calculate the diameter of the sun?

10. Discuss the students' ideas and lead them in the direction of setting up a ratio of measurements. Ratios are like analogies: this is to that as this other thing is to that other thing:

*the diameter of the sun is to the distance to the sun (from earth)*

as

*the* diameter of the pinhole image is to the distance from the pinhole to the viewer

To put the relationship in mathematical form:

diameter of the sun (km) = diameter of the pinhole image (mm)

____________________ ________________________________

distance to the sun (km) distance from the pinhole to the image(mm)

Then, solving for the unknown:

diameter of the sun (km) = distance to the sun (km) X diameter of the pinhole image (mm) distance from the pinhole to the image (mm)

Have students do the research or provide them with the approximate distance from the earth to the sun: 149,600,000 kilometers; and allow them to solve for the unknown to calculate the diameter of the sun. Have students defend their answers and explain why this method works as a method of calculation for extreme distances.

11. For a challenge, students can attempt to calculate the diameter of the moon using a similar technique.

12. Have students compare their answers to what modern scientists have calculated for the sun's diameter. Students should also be asked to analyze any possible sources for errors in their calculations.

Metric Measurement Reference Chart (Microsoft Word 2007 (.docx) 32kB Jul27 11)

### At Home Assignments

Any of the research components of this lesson can be completed for homework. These topics include the contributions of Aristarchus and Eratosthenes, the distance to the sun (along with seasonal variations), and the currently held calculation of the earth's distance to the sun.### Materials

- metric measuring tools: meter sticks and centimeter rulers, thermometers, graduated cylinders and beakers, triple beam balances- assorted items for measuring

- long tape measure for measuring the shadow of the tree and mirrors (or pans of water)

- Metric Measurement Reference Chart for recording data

- cardboard, foil, tape, meter sticks, ring stands and clamps to build pinhole viewers

- calculators

- research materials if investigating the work of Aristarchus and Eratosthenes.

## Standards

Massachusetts Learning Standards:Earth and Space Science, Grades 6-8(2006)

10. Compare and contrast properties and conditions of objects in the solar system (i.e., sun, planets, and moons) to those on Earth (i.e., gravitational force, distance from the sun, speed, movement, temperature, and atmospheric conditions).

Mathematics State Core Standards (2011)

7.RP Ratios and Proportional Relationships

Analyze proportional relationships and use them to solve real-world and mathematical problems.

7.EE Expressions and Equations

Use properties of operations to generate equivalent expressions.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

## Teaching Notes and Tips

- Students may struggle with the math portion of the activities, especially if they have not had much experience working with similar triangles and solving for unknowns using ratios. Check with the students' math teachers and either coordinate your teaching efforts, or time this lesson to follow the appropriate math units. Math teachers usually appreciate the reinforcement!

- Teacher Meeyoung Choi offers the idea of using a millimeter ruler attached to the viewer end of the pinhole device instead of using a grid or plain paper. This allows for easier measurement for any image that is projected onto the viewing screen.

- Building the components for the pinhole viewer would be a great project when you have some time to fill – the day before a holiday, for example. Then the pinhole lesson can go more quickly when the components are already built.

- Day 1 activities could easily be separated by quite a bit of time from Day 2 and 3 activities. Day 2 and 3 activities, however, should go together.

## Assessment

- Students should be able to explain the concept of similar triangles and how they can be utilized to measure very large distances

- Students should know the approximate measure of the distance from the earth to the sun and also the sun's approximate diameter in kilometers

## References and Resources

Information on building and using pinhole viewers can be found in many places.

Three good sources are:

http://cse.ssl.berkeley.edu/AtHomeAstronomy/activity_03.html.

http://archive.planet-science.com/outthere/cap/pdf/cath_activity.pdf

http://www-outreach.phy.cam.ac.uk/resources/ESM/KS2/pinhole/index.php

http://hilaroad.com/camp/projects/pinhole_sun/sun_diameter.html

Information on using similar triangles/distances to calculate unknown distances can also be found in many middle school math textbooks. On-line sources that can be shared with students include:

http://www.youtube.com/watch?v=31qq1zoQVHY&NR=1

http://www.youtube.com/watch?v=wHCFaGaKt-M&feature=related

The following website not only presents a good historical context for the activities, but also has detailed instructions on how to make a pinhole viewer and has a great review of using the math of similar triangles.

http://www.youtube.com/watch?gl=CA&hl=en&v=zioSpV2yq24

A good reference for distances and sizes of various solar system components: http://solarsystem.nasa.gov/planets/index.cfm

General references for information on the Earth, the Moon and Sun:

Elkins-Tanton, Linda T. The Earth and the Moon. New York: Facts on File, 2010. Print. ( a good discussion of the ancient Greek, Eratosthenes' efforts to measure the radius of the Earth – p. 6)

Elkins-Tanton, Linda T. The Sun, Mercury, and Venus. New York: Facts on File, 2010. Print. ( a good discussion of the ancient Greek, Eratosthenes' efforts to measure the radius of the Earth – p. 6)