This was the second Investigation of the LabLearner CELL Science and Art. In it, students continued the exploration of perspective and the illusion of depth in paintings.

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Begin this part of the Investigation by encouraging students to summarize their activities in Lab.  Prompt student discussion by posing the following questions:

1. Ask students: What were the main questions we wanted to investigate in this lab? Students should indicate that the questions were: Does the distance from which an object is viewed change its appearance? What is the relationship between the distance from an object and how its height appears?

2. Ask students: How would you summarize the types of experiments that you performed to investigate these questions? Students should indicate that in Trial 1, they placed a meter stick on the lab table.  The 0 cm end of the meter stick was at the end of the table and approximately 30 centimeters from their eyes.  They then measured the height and width of a centrifuge tube when it was placed at the 0, 20, 40, 60, 80, and 100-centimeter marks.  In Trial 2, they measured the visual angle or the angle at which light would enter their eyes when the centrifuge tube was at the 0, 20, 40, 60, 80, and 100-centimeter marks.

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Begin the analysis of the experiment by encouraging students to review and organize their data from Trial 1.

1. Tell students to locate Phase 2.8 Findings – Quantitative Data. This question asks students to graph their results from Trial 1.  

2. Encourage students to complete a line graph if they have not already done so.

3. As a class, review the two graphs. Notice that both the width and height decrease with distance from the eye.

4. Students may notice that, while both height and width decrease with distance from the eye, the two lines on the graph are not parallel to each other. This should NOT be interpreted to suggest that height and width change at different rates with distance from the eye. If this were to happen, objects would appear to change shape as they are moved further from the eye. We have seen that that doesn’t happen. To prove this (if the teacher feels their students are ready to think in terms of mathematical proportionality), if one looks at points on one curve, they will find that each point on one curve is proportional to the points on the other curve at the same distance from the eye. Consider the sample results plotted below:

Let’s take points from the best-fit curves for the 0 cm and 100 cm (~9 cm and ~2 cm) distances from the eye for the height measurement to the ~2 cm tube width at o cm and calculate the proportional expected 100 cm width at 100 cm from the eye. We can compare the ratios easily mathematically:

One will find that, while we chose the 0 cm and 100 cm distances from the eye as an example above, you can compare the ratios at any two distances from the eye and find that you can predict the width or height if you apply the proportionality method shown above. Proportionality is an excellent tool for many scientific and engineering findings. Note: Not all middle school classes/students may be ready for this analytic method. Therefore, it is omitted from the Student Guide webpage.

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When students have finished their graphs, ask a student volunteer to draw his or her graph on the board.  Discuss the completed line graph with the class.

a. Look at the x-axis on the graph. Begin on the left side:

Ask students: What happens to the distance from the eye as you move from left to right? The units increase in magnitude.

b. Look at the y-axis on the graph. Begin at the bottom:

Ask students: What happens to the units of height or width as you move from the bottom to the top of the graph? The units increase in magnitude.

c. Focus on the width of the centrifuge tube. For each measurement of distance along the x-axis, you plotted a point along the y-axis that corresponded to the width of the centrifuge tube. Look at the best-fit line that connects the points.

Ask students: Does the line go up or down? It goes down.

d. Ask students: What happened to the width when the distance from the eye increased? As the distance from the eye increased, the width of the centrifuge tube decreased.

e. Next, focus on the height of the centrifuge tube. For each measurement of distance along the x-axis, you plotted a point along the y-axis that corresponded to the height of the centrifuge tube.  Look at the best-fit line that connects the points.

Ask students: Does it go up or down? It goes down.

f. Ask students: Can you describe what happened to the height of the centrifuge tube when the distance from the eye increased? As the distance from the eye increased, the height of the centrifuge tube decreased.

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NOTE: There is a point at which some distortion will occur for some objects when viewed at very small visual angles.  For example, imagine looking up at a skyscraper and flagpole from directly below (right side of the slide). 

The overall shape of the building may appear slightly different than if the building were viewed from a farther away (left side of the slide).  This is because the distance between the top and bottom of the building is greater than the distance between the viewer and the building.  As a result, the top of the building has a smaller visual angle than the bottom because it is farther from the viewer than the bottom. However, this specific phenomenon involves small distances between the viewer and the object and very tall objects.

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This and the following slide illustrated the importance of maintaining proportion when depicting three dimensions and distance on a 2-dimensional surface.

Notice how odd the three pictures look when the height and width of objects change as they appear to decrease into the distance. In the bottom illustration, the basketball player’s width and height decrease proportionately, making the player seem further away but still clearly recognizable.

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This slide further illustrates the importance of maintaining proportion when depicting three dimensions and distance on a 2-dimensional surface.

This slide introduces the simple mathematics of proportions. Thus, if we measure the height and width of skyscraper A, which appears visually closest to us, we find a ratio of approximately 3 to 1 (height to width, 3:1). Notice that this ratio is maintained as the building is reduced in size (B and C), creating the illusion of distance.

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Continue the analysis by asking students why the height of the object appears to decrease as it is moved farther from the eye.

1. Remind students that in the previous Investigation, they concluded that the width of an object decreases as the object moves farther from the eye because the object takes up less of the field of view. Students also related the width of an object to the size of the visual angle that the object encompassed as it passed through the lens of the eye.  The smaller the visual angle, the more narrow the object appeared.

2. Tell students that part of their experiments in Investigation Two was to discover what happened to the appearance of the height of an object as it moved farther from the eye. To do this, students measured the height of the centrifuge near and far from the eye and the visual angle of the centrifuge tube as it entered the eye at different distances.

3. Review the definition and concept of visual angles. Use the questions below to help guide the discussion. In addition, refer to this slide which was seen in the PreLab session as well.

a. Ask students: When viewing an object, what is meant by the visual angle? The visual angle can either refer to the angle at which the light from an object enters the eye or the angle the light rays form as they pass through the inside of the eye to the retina.

b. Ask students: Look at Table A in your Student Guide.  Compare the outside visual angles for the centrifuge tube when placed at different distances from the eye. What happened to the visual angles as the centrifuge tube moved farther from the eye? The visual angles decreased as the centrifuge tube moved farther from the eye.

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4. Tell students that there is a special relationship between the visual angles outside and inside the eye.

a. Encourage students to look at the four diagrams on this slide.

b. These diagrams illustrate the movement of the centrifuge tube during Trials 1 and 2.

c. Ask students: Look at the visual angle outside the eye in the diagrams. Is it increasing or decreasing as the centrifuge tube moved farther from the eye? It decreased as the centrifuge tube moved farther from the eye.

d. Ask students: Look at the visual angle INSIDE the eye in the diagrams.  Is it increasing or decreasing as the centrifuge tube moved farther from the eye? It decreased as the centrifuge tube moved farther from the eye.

e. Ask students: What do you notice about the change in the visual angle INSIDE the eye and the visual angle OUTSIDE the eye as the centrifuge tube moved? Both visual angles decreased as the centrifuge tube moved farther from the eye.

5. Explain that the visual angles inside and outside the eye are similar to vertical ones.  Vertical angles are angles that are across from each other. Vertical angles have the same measurement.

6. Tell students that the visual angles inside and outside the eye are not exactly the same because of the substance inside the eye.  However, when scientists estimate, they often think of the two visual angles as vertical angles—having the same measurement.

7. Ask students: What happens to the height of the image as the centrifuge tube moves farther from the eye? The height of the image decreases as the centrifuge tube moves farther from the eye. This occurs because the height of the image is directly related to the visual angle of the object inside the eye.

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D. Explain the patterns that students just discovered and those they made in Investigation One and explain how we see depth.

1. Our eyes have a field of view. Objects in that field of view take up a certain portion or ratio. The closer the object, the more of the field of view is taken up. The farther the object, the less of the field is taken up. This applies to both the height and width of an object.

2. Another way to think about this is to describe what happens to the visual angles when an object is moved at different distances from the eye.  As an object moves farther from the eye, the visual angle decreases, and the image formed on the retina becomes shorter and narrower.  As an object moves closer to the eye, the visual angle increases, and the image of the object on the retina is taller and wider.

3. Discuss how the vanishing point is related to this concept: in terms of our eyes, a vanishing point is the distance that is so far away that an object takes up so little of the field of vision it appears to disappear.  At the vanishing point, the visual angles are so small that the image on the retina is almost non-existent.

4. Although not specifically addressed in this CELL, students may find it interesting that microscopes and telescopes use lenses to create a magnified image of objects so that a larger visual angle is shown to the eye.  If microscopes and telescopes are not used, the visual angle of the objects that a person observes is so small that the eye cannot resolve the image.

5. Conclude the PostLab by encouraging students to apply what they have learned by looking at the paintings in the following several slides.

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The following series of slides is designed to reinforce the concept of vanishing point and to illustrate how Renaissance and Post-Renaissance artists create a feeling of depth in their work by portraying the size of objects on a two-dimensional canvas the way in which the eye and brain perceive the objects in real life. In other words- the same object appears smaller when farther away than when close to the eye.  In the following paintings, the size of objects are sized to reflect this generality.

As students look at each painting, the red lines illustrate the vanishing point. As students view the lines that point to the vanishing point, they should realize that the objects along the lines become progressively smaller because the objects are becoming closer to the vanishing point.

Colored outlines are drawn around other objects to illustrate differences in size.

Ask students the questions on the slide.  Students should discuss that if a person were standing in the position of the artist they would “see” the tower that is in the distance as smaller—both more narrow and shorter – than the tower closer to the eye because of the difference in visual angles or the portion of the field of view taken up by the object.  The artist recognized this and used this relationship when painting.

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Ask students the question on the slide.

Students should discuss that the artist used the concept of vanishing point to achieve a feeling of depth or three dimensions in the painting. One of the depth cues for humans is that objects farther from the eye are shorter than objects closer to the eye.

Therefore, to create a feeling of depth or three dimensions in a two-dimensional painting, the artist painted the child as taller than the man to create the feeling that the child was in front or closer to the viewer than the man.

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Ask students the question on the slide.

Students should see this as another example of how the artist purposely paints the columns he “sees” in the back of the structure as shorter because this is how the eyes and brain perceive distant objects in real life. By painting the columns at different heights he creates a feeling of depth or three dimensions.

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Ask students the question on the slide.

Students should see this as another example of how the artist purposely paints the tree he “sees” in the back of the forest as shorter because this is how the eyes and brain perceive distant objects in real life. By painting the pine tree as very short, he creates a feeling of depth or three dimensions.  In this case, the pine tree is so small because it is at the vanishing point of the painting.