1. Working With Ratios & Proportions: Facilitating Scientific Thinking

2. Why Ratios & Proportions? Ratios and proportions are extremely common in everyday values E.g. gas mileage, salaries, tire pressure, unit costs Ratios and proportions represent an extremely important and powerful way to describe patterns in the universe e.g acceleration, air pressure, solar flux, Hubble’s Law Ratio or proportional reasoning is a crucial, fundamental scientific thinking skill [Harold Arons quote]

3. Using Ratios & Proportions A Crucial Component of Scientific Reasoning Representing mathematical and physical relationships Communicating patterns Making predictions Analyzing data The most important thing in science is not so much to obtain new facts as to discover new ways of thinking about them. -William Lawrence Bragg The most important thing in science is not so much to obtain new facts as to discover new ways of thinking about them. -William Lawrence Bragg

4. Ratios, Proportions, & Fractions Basic Definitions Ratio: Compares part to part or one value to another that is related 1 Dodgers fan for every 2 Giants fans 30 miles for every gallon Proportion: Compares part to whole or one ratio to another 2 out of 3 fans are Giants fans 30 miles per gallon will allow you to drive 90 miles on 3 gallons of gas Fraction: A shorthand way of representing a proportion (part compared to whole) 2/3 of fans like the Giants

5. Representing Ratios & Proportions 10 Boys:30 Girls 25% Boys 3/4 Girls

6. Lab Activity #1 Ratios, Proportions & Fractions in a A Bag of Candy Overview Students predict and then determine the relative number of two colors of candies Students explore various ways to represent the ratio of colors, and practice explaining the ratio in everyday language Students distinguish between ratios, proportions and fractions using candy color data

7. Lab Activity #2 Comparing Mass and Volume For Various Materials Overview Students compare the mass and volume of various everyday materials Students represent the relationship between mass and volume using calculated ratios and a graph of mass vs volume Students practice explaining ratio relationships using everyday language

8. Lab Activity #3 Using Ratios to Measure Features in the Solar System Overview Students use basic image analysis tools and proportions to determine the size of lunar craters and solar flares. Using readily available images, students carry out their own investigation to determine the size of objects or features in the solar system.

9. Basic Image Analysis Salsa J: Analysis of astronomical (and other) images Standard image format from a camera attached to a telescope (a CCD camera) is called a “FITS” image (FITS = Flexible Image Transport System)

10. Astronomical Images All digital images are made up of pixels Each pixel represents a portion of an object or feature The color or brightness of each pixel is dependent on the amount and type of light that strikes the CCD chip

11. Salsa J: Basic Tools Locate & Open Program Open File: Full Moon Image Controls: Brightness (experiment!) Image Controls: Magnification (keep clicking image!) Image Data: Pixel Location Image Data: Counts (Corresponds to Photon Count) Image Analysis: “Slice” and Plot Profile (details to come) Image Info: Scale (details to come)

12. The Slice Tool A “slice” collects light data from each of the pixels through which it passes. A slice can be used to determine the dimension of features in pixels

13. A Slice Closeup Each pixel along the slice has collected a certain amount of light. The relative amount of light is reported as the “grey value”. The grey value is proportional to the number of photons striking that region of the camera detector (CCD chip)

14. Slice Plot Profile The Plot Profile records the counts (grey value) for each pixel along the slice on a graph.

15. Plot Profile

16. Measuring the diameter of the moon…in pixels. x-coordinates appear in results window after clicking on the plot profile where indicated

17. Measuring the diameter of the moon…in pixels. Diameter of Moon: 600-27=573 pixels

18. Scaling Pixels to Standard Units Simple Method: Provide students with scale that you calculate using Intermediate Method described below. Recommended for first-time run through. Intermediate Methods: Calculate ratio between known diameter of object and the number of pixels across the object in the image. Recommended for students after some practice is completed using Simple Method. See Handout: Salsa J Reference Sheet for details. Example: Involved Method: Use viewing angle for object of know distance. Use “small angle approximation” to convert viewing angle to calculate pixel scale in standard units (see handout for details). Recommended for more advanced students after adequate practice with above methods. Is sometimes necessary when only a portion of an object is imaged.

19. Determining the scale of each pixel. Pixel Scale:

20. Determining The Diameter of a Lunar Crater Step 1: Measure diameter of crater in pixels with using slice tool and plot profile. Diam of Crater = 58-4 = 54 pixels

21. Determining The Diameter of a Lunar Crater Step 2: Convert pixel diameter of crater to miles OR Ratio calculated previously Diameter = 204 miles

22. Determining the Size of a Solar Flare Image: Solar Flare.jpg Task: Given that the diameter of the Sun is approximately 864,300 miles, determine the dimensions of a solar flare in the image. How does this solar flare compare to the size of Earth? Keep an organized record of all data a calculations used in your solution.

23. Pixel Scale 2.90”/pixel This information is included with every image taken with an astronomical CCD camera. (only necessary for “Involved Method”) See “Small Angle Supplement” for details on using the above ratio to scale images into standard units.