This section is a fast-paced, and we hope exciting, introduction to modeling and technology. The picture above shows the pattern created by shining the beam of a $19.95 laser pointer through a pattern of fine lines printed on ordinary transparency film by an ordinary Postscript laser printer.

Just a few years ago lasers were found only in laboratories. Now the same mysterious tools that are the stuff of science fiction and high tech eye surgery can be found in shirt pockets everywhere. To understand lasers and the seeming miracles they perform we need to understand light. This section is about light and waves, and also about other phenomena, like sound, that involve waves. It shows how modern technology -- not just graphing calculators and computers but everyday technology like portable casette tape recorders, photocopying machines, and laser pointers -- can link middle and high school teachers, students, and parents as they learn some fascinating science with incredible applications.

This course has an important subtext -- the reflective use of technology. A recently released report Does it Compute? The Relationship Between Educational Technology and Student Achievement in Mathematics from the Educational Testing Service confirms common sense --

This report presents new evidence on the effectiveness of educational technology. Analyzing data from the 1996 National Assessment of Educational Progress, it finds that the effectiveness of school computers depends upon how they are used; some uses are associated with improved student academic performance and school climate, while other uses are not.

As we use technology in this course, we discuss some of the principles behind its effective use. These principles can help you choose which technologies to use and how best to use them.

One of the most attractive uses of computers is for virtual or simulated experiments. They are cheap, safe, clean, and often very compelling. Although we like simulated experiments, we also believe that real "hands-on," "bench-lab," or "wet-lab" experiments are also important. This section has both kinds of experiments. You will need some equipment.

• A laser pointer that produces a bright and very tight beam. Compare laser pointers by projecting their beams side-by-side on a light wall. Choose the one that produces the brightest and smallest dot on the wall.

• A small bright flashlight.

• The transparencies described below..

• Graph paper.

• A Texas Instrument Calculator-Based Laboratory (CBL) with two microphones and one of the TI graphing calculators.

The best way to obtain the transparencies and graph paper is by printing them yourself using a Postscript laser printer. The Connected Curriculum Project, of which this course is part, maintains a library of files that can be printed on many Postscript laser printers using either transparency film or plain paper. For this section we use two of these files. If you haven't already installed the free Adobe Acrobat Reader, you should do so now by clicking the button below and following the instructions.

• Finely-ruled graph paper
  

  [Acrobat]

  [Postscript]

  You and your students can use this paper to draw graphs. Print as many copies as you need.

• Three patterns to be printed on transparency film
  

  [Acrobat]

  [Postscript]

  You and your students will use this file to do some hands-on experiments. Notice there are three copies of each of three patterns. Cut each sheet that you print into three pieces, each of which has one copy of each of the three patterns. Give each student two pieces, one printed on ordinary paper and one printed on transparency film.

You should see a pattern on your transparency film that is like the pattern shown below. Make a shadow by projecting this pattern on the wall using your small flashlight. You may need to dim the light in the room. You can do some simple experiments with this simple apparatus.

• While you hold the pattern steady, move the flashlight toward the pattern and then away from the pattern. As you move the flashlight toward the pattern, notice that the shadow grows larger. As you move the flashlight away from the pattern, notice that the shadow gets smaller.

• Now notice that the pairs of dots that are closer on the pattern produce shadows that are closer together on the wall.

The graph below represents this situation geometrically. Light radiates from the flashlight (indicated by the black dot at the right) in straight lines. The dots on the slide (indicated by the gray line in the middle of the figure) obstruct some of the light, causing shadows on the wall (indicated by the gray line at the left of the figure). Because light travels in straight lines, the shadow of each dot is on the line determined the light and the dot. The picture below is "live." You can move the light by dragging it with the mouse. Notice if you drag the light closer to the slide then the shadows on the wall spread out.

You can also move the dots on the slide by dragging them with the mouse. Notice as the dots get closer on the slide their shadows move closer on the wall.

Nothing we observed so far is surprizing, but stay tuned -- there are some surprizes in store. First we want to ask some questions that middle school geometry students can begin to answer. The underlying big question behind the small questions below is -- What is the relationship between the size of the shadow projected from a slide and the image on the original slide?

• In the figure below, measure the distance between the blue dot and the red dot on the slide (the gray line in the middle of the figure). Note that this figure is "dead." You cannot move the light or the dots on the slide around.

• In the figure below, measure the distance between the blue dot and the red dot on the wall (the gray line at the left of the figure).

• Find the ratio of the distance you measured on the slide to the distance you measured on the wall.

• In the figure above, measure the distance between the magenta dot and the red dot on the slide (the gray line in the middle of the figure).

• In the figure above, measure the distance between the magenta dot and the red dot on the wall (the gray line at the left of the figure).

• Find the ratio of the distance you measured on the slide to the distance you measured on the wall.

• Do you notice anything?

• Measure the distance from the light source (the black dot at the right of the figure) to the red dot on the slide.

• Measure the distance from the light source (the black dot at the right of the figure) to the red dot on the screen.

• Find the ratio of the distance from the light to the red dot on the slide to the distance from the light to the red dot on the screen.

• Do you notice anything?

With these problems as background, students would experiment using figures they draw themselves using graph paper. The idea is to let them discover the relationship between an image on a slide and its projection (or shadow) on the wall. This is a very open-ended question that students can answer with varying degrees of success.

• Some students may prove a theorem using similar triangles.

• Others may form conjectures based on many observations.

• Some students may experiment with a slide and a wall that are not parallel while others work within the confines suggested by the pictures above -- parallel slide and wall.

Students can check the theories they form using geometry against the reality of real slides and shadows. In a classroom students would be doing these experiments in small groups and talking about their theories while the teacher walked around the room interacting with different groups.

Notes on the reflective use of techology.

Even the simple observations they are making here have important practical consequences. For example, we have all seen presentations made with a projector that has been tilted by putting books under the front so that the projected image is high enough on the screen. The projected image is distorted -- wider at the top than at the bottom. The reason is that the slide and the screen are not parallel. Based on their work above, middle school students can suggest how projector manufacturers might try to solve this problem.

We have completed our first model for the behavior of light and shadows. We can't actually see what is happening between the light source and the slide and then between the slide and the wall but we have formed a mental picture or model of what is happening and that model enables us to make predictions that can then be verified or contradicted by experimental evidence. This is the modeling cycle --

• Begin with observations.

• Based on observation, construct a mental image or model.

• Use the model to make predictions and test those predictions by doing new experiments.

• If necessary, revise the model.

• Repeat the last two steps above, as necessary, to obtain better models.