There’s More Than Meets The Eye

This week’s Puzzle Corner activity is a modification of a puzzle that has been around for many years. All the versions of the puzzle I have seen use two identical arcs that are placed one above the other. While the two arcs have the same length, the one on top seems shorter. This illusion persists even when the arcs are switched.

The illusion in There’s More Than Meets the Eye is much more dramatic than the illusion in the puzzle described above. There’s More Than Meets the Eye consists of two arcs placed in such a way that they appear to have the same length. In reality, the top arc is about an inch longer than the bottom arc. This difference in length is not apparent until the arcs are cut out and reversed. This reversal produces the opposite effect and greatly exaggerates the differences in length. To “see” the true size relationship between the two arcs, they need to be placed directly on top of the other. The challenge in this puzzle is to come up with a reasonable theory to explain the illusion.

I have used this puzzle for a number of years and it is always one of the student’s favorites. I use two arcs (similar in size to the arcs on the student sheet) cut from the rim of a paper plate. I introduce the puzzle by holding the longer arc above the shorter arc and asking students to tell me how they compare in size. The students invariably tell me that the two arcs are the same length. I then tease the students and tell them that the arcs are made of a special material that can be stretched. I pretend to stretch the top arc and move it under the other arc. This causes great consternation since the this arc now appears to be much longer than the other arc.

Next, I “stretch” the arc now in the top position and place it below the other. The two arcs again appear to be the same size. After “stretching” and switching the arcs several more times, I leave them at a center so that students have ample time to “play” with them. I challenge students to discover as much as they can about the arcs and about what happens when they are placed in various positions. I also ask them to come up with some theories to explain the illusions. At the end of the week we have a class discussion on the arcs. Students share their discoveries and theories. After this discussion, I pass out paper plates and each student makes a set of arcs to take home and share with their families.

There’s More Than Meets the Eye can be done as describe above or it can be done as a whole-class activity by giving each student a copy of the sheet on the next page. In either approach, students should be given ample time to “play” with the arcs. If using the whole-class approach, pairs of students can share arcs so that one student has both short arcs and the other has both long arcs. In this way, students can see how the illusion works for two identical arcs.

Look at the two arcs. What is their apparent size relationship? 
Cut out the arcs and reverse their positions. 
What happens? 
Experiment with the arcs and see what else you discover. Write about your discoveries.
Arcs2

Dave Youngs

Dave Youngs is director of the Graduate Math/Science Education program at Fresno Pacific University. In this position he works with both pre- and in-service teachers in the areas of math and science education. He is most interested in the elementary and middle school levels.

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