Classroom Activity: How Far? How Small?

Main Idea: Students will use a simple set up to measure the change in the apparent size of an object as it moves away from them.

Learning Objectives:

• Observe how the height of an object appears to decrease as it moves away.
• Measure the apparent height of an object at several distances as it moves away.
• Graph the relationship observed.
• Describe and communicate a mathematical rule for the size-distance relationship.

Time: 30 to 45 minutes

Materials: Each 2-person team will need:

• scissors
• an 8 x 12 card
• a pencil
• a strip of stiff cardboard at least 30cm long
• a table at least 1m across
• a simple object such as a bottle or box 25-30cm tall
• a tape measure
• an easy to read ruler
• a copy of the How Far? How Small? Data Sheet

Skills Needed:

• Measurement accurate to the nearest 0.5cm
• Converting common fractions to decimals
• Graphing

Preparation: Each team should construct a viewer to improve the consistency of their measurements.

1. Measure and cut out a square with 1cm sides from near the middle of one end of the index card.



2. Fold the index card lengthwise along the bottom of the square hole, then open the fold to a right angle.



3. Tape the folded card to the end of the cardboard strip aligning the hole with the right edge of the strip. Make sure the tape does not block the hole.


Procedure:

1. Measure the actual height of the object to the nearest 0.5cm and record it in decimal form.




2. Stretch out a tape measure across a table so that 0 cm is even with the near edge of the table. Use tape to secure the tape measure in place.



3. Place the viewer at right angles to the table edge and aligned with the tape measure. Make the index card stick out exactly 10 cm. Secure the viewer with tape.



4. Sit or kneel on the floor and put one eye next to the hole in the viewer.

5. Hold or prop the ruler vertically, 0 end down, at the edge of the table so you can read it through the viewer.



6. Have your partner move the object 10 inches away along the tape measure. Look through the viewer and note where the top edge of the object now appears to be on the ruler. Using the Data Sheet, record this as the apparent height in decimal form.



7. Move the object away to 20, 30, 40, 50 and 60 cm. View and record the apparent height at each location.

8. Switch roles with your partner to collect a second set of data.


Analyzing the Results:

1. Make a graph of distance away versus apparent height.
   • Use your graph to make a prediction about the object's apparent height at a distance you did not test. Then check your prediction by actually making a measurement at that distance.
   • Make a prediction from your graph about how far away your object would need to be to look one inch tall. If your table is long enough, test this prediction as well.
2. Divide the actual height by the apparent height for each distance. The result is a ratio you can use for comparison with other teams who used different sized objects.

Drawing Conclusions:

• From your graph and/or ratios, describe the rule or relationship you found between the apparent height of the object and its distance away from you.