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PART I - Ratio and Proportion

You may not always be able to measure an object, but you can estimate the size based on your knowledge and experience of the object or other objects around it. Look at the picture below. Most of us have seen a fire hydrant in real life. Based upon your experiences, what is your estimate for the height of the fire hydrant and the post?

1. Fire hydrant _________________________ Post________________________________

2. Explain how you determined the height of each object.______________________________ _________________________________________________________________________

3. How could you find out if your estimates are correct?_______________________________ _________________________________________________________________________
_________________________________________________________________________

Look at the picture of the light pole. Guess the height of the light pole.



4. Estimate of the light pole ___________________________

5. How can you find out if your estimate is correct?_______________________________ _______________________________________________________________________

Scientists and mathematicians may need to know the size of objects. It is not always possible or convenience for them to go to the object and measure it. An oceanographer may want to determine the size of a coral reef at the bottom of the ocean or NASA may want to determine the height of a mountain on the moon. One way to do this is by using ratio and proportion.

You may have studied ratios and proportions in your math classes before. Look at this example:



6. What number will replace the question mark in the denominator of the second fractionto make it a true statement? _________________________________

7. Explain how you decided what number to use? _________________________________________________________________________ _________________________________________________________________________

There are several ways to find the missing number. One way is to see what number is used to multiple with the 2 to get the 4. Since the "2" is multiplied by 2, you would multiply the "5" by 2 also to get 10. Another way is to take the product of the two diagonal numbers and divided it by the remaining number to get the answer for the "?". You may have other ways to get the answer.

Practice your method with the following problems.

a.          ? = ________      

b.          ? = ________      

c.          ? = ________      



d. (5,15) (1.1, 3.3) What is the constant of variation for this set of ordered pairs? ______

(45, ______) (12.6, ______) (______, 12) (_____, 1.5)

e. (3,4) (75,100) What is the constant of variation for this set of ordered pairs? ______

(80, ______) (1.6, ______) (_______, 8) (______, 20)

PART II - Using Scion Image

The method in PART I works if you already know the numbers, but how do scientists get the numbers. We will look at a software program that can be used to determine these numbers.

USING TECHNOLOGY-SCION IMAGE

Scion Image is a computer program that can be used to find measurements of objects in pictures. With these pictures, it is easy to determine measurements of objects that would otherwise be inconvenient or impossible.

In Part II, you will find the measurements of objects using the tools in the program Scion Image.

Open Scion Image. (Follow the instructions that your teacher has given to open the program.)

When the program opens, it will look like this.



Select the maximize box in the upper right corner of the menu in order to have Scion Image fill the screen.

Open the picture PostPlug. Scion Image should display the fireplug picture seen earlier in this lesson. It may be necessary to lighten or darken the picture. To do this, click and drag the line on the map bar.

To the left of the picture there is a vertical bar labeled TOOLS, which contains a collection of icons. These icons represent the various tools that can be used in the Scion Image program. Notice the icon composed of a dotted segment and a small arrowhead that is located in the fifth row of the TOOLS menu bar. This is the measuring tool icon.

Left click on the measuring tool and notice that the icon changes to a black background. This indicates that the measuring tool is active. Now move the cursor into the picture.

1. How does the cursor change? _______________________________________________________________

Move the measuring tool cursor to the top of the fireplug. Click the left button on the mouse and drag the mouse so that a vertical line runs along the picture of the fire hydrant. When the vertical line gets to the bottom of the fire hydrant, release the left button. (If you also hold down the shift key the line will remain straight.) Move the cursor to the Analyze menu option. Click, pull down the menu, and move the cursor to Set Scale. Release the left button. The screen should look like this.



From this display, very important information can be obtained to determine measurements of objects in a picture. The first important information obtained from this display is the number found in the box that is labeled:

Measured Distance: ______Pixels.

A pixel is the smallest piece of the display that the computer can change. These pixels are arranged in rows and columns like this: .......

The pixels are placed together so closely that our eyes cannot see the space between two pixels. When we use the Scion Image measuring tool, the program is actually counting the number of pixels in the line from our starting point to the finishing point.

2. What is the height of the fire hydrant expressed in pixels? _____________

Using the computer's units of measure, we can now say that "the fire hydrant is ________ pixels tall."

Notice the fourth line in the display labeled Units. Place the cursor on the arrowhead that is pointing downward. Click, hold, and pull the mouse downward until the preferred units are highlighted, then release the button. This specifies to the computer how you want the measurements to be expressed. We will select inches since this is the unit that was used to measure the fireplug.

Now notice the second line. It is labeled as Known Distance. In this space type in the number 30, since it is known that this is the actual height of the fireplug expressed in inches. As you type in the number 30, you will notice that numbers appear in the box labeled Scale change. Enter this number in the space below.

3. Scale: _____________ pixels per inch. This number expresses a ratio called the constant of variation.

4. Explain in your own words what this number means and how you might use it.
_______________________________________________________________________________
_______________________________________________________________________________

5. How does your original guess for the height of the fireplug compare to the actual height given for the fireplug? (If you guessed the measurement in feet, you may want to change the computer's calculated height for the fire hydrant into feet as well.)
_______________________________________________________________________________

Once all the pertinent information is obtained from the Set Scale Menu select OK to confirm the computer's calculations.

Use the measuring tool again, now measure the height of the pole next to the fireplug. Click on the Analyze menu and move to the option titled Measure. Even though it appears that nothing has happened, Scion Image has actually determined the measure of the pole in pixels. Using the Set Scale option in the Analyze menu, find the number of pixels it takes to measure the pole.

6. The height of the pole in pixels:_________________

Using this measurement, suggest the height of the pole in inches. _____________

7. Using the information you now have, calculate the height of the pole and record your answer on the lines below.

______________________________________________________________________________
______________________________________________________________________________

Select OK to confirm Scion's calculations.

We are now ready to use Scion Image tools to help us determine the actual height of the pole. Open the Analyze menu, and select Options. The Options screen allows you to specify the types of measurements that you want the program to determine. Make sure that only the Perimeter/length option is selected. (You do this by checking the white box in front of a particular measurement. If the box is not checked, clicking on the box will select the measurement. If the box is already checked, then clicking it will de-select that measurement.) Select OK to confirm your Measurement Options.

Looking again at the fireplug picture, make sure that your measurement line is drawn over the pole. If it is not, use the measuring tool to put a measuring line on the pole. Select the Analyze menu again. This time click on Show results. A table will be displayed with the title "Length". The length displayed will be the length that Scion Image suggests is the height of the pole.

Write this measurement down and compare it with your original guess as to the height of the pole and to your calculated height for the pole.

8. Original Guess Height ________ Calculated Height ________ Scion Image Height________

9. What do you observe about your original guess as to the height of the pole compared to your other two findings? ______________________________________________________________________________ ______________________________________________________________________________

10. Explain how Scion Image determined the height of the pole.

______________________________________________________________________________ ______________________________________________________________________________

In the File menu, opt to Close the PostPlug file. When asked to save, select NO. Let's continue determining measurements of objects in pictures using the Scion Image program. Open the file called LightPole. The picture should look like the picture below.



15. Notice the pickup truck with a camper in the lower left portion of the picture. You need to set the scale again. These trucks are known to be about 5.75 feet tall. Using this information, determine the following measurement:

a. The height of the truck in pixels._____________

b. The constant of variation. ________ pixels per foot.

c. The height of the light pole in pixels.________

d. Using the constant of variation, calculate the height of the light pole in feet.__________

e. Get the height that is calculated by SCION Image. ___________

f. Compare your estimate from Part 1 question 4 to d & e above.

___________________________________________________

When you have completed these exercises, close the picture and DO NOT SAVE.

PART III - Measuring the Curve

Now use the skills that you have learned in Part II to find the radius of the curve from our original problem. Open the file InsideCurve. Using the scale at the bottom of the image as your known distance, find the radius of the inside of the curve. A circle has been drawn on the curve for you to measure the radius. The circle was found by using a program called Geometer Sketchpad and geometric concepts that you learned in an earlier math class. Now, open the file OutsideCurve and repeat the process to find the radius of the outside of the curve. You will use these two measurements in the next part of this module. Record the results of the radius of the curve in the spaces below.

Radius of the Inside curve_____________________________________

Radius of the Outside curve ___________________________________