## A Model for Distance Between Two Points

### If "One picture is worth ten thousand words1," what's a model worth?

 Manipulative or Model       A model, which the instructor handles, or a manipulative, which the student handles, presents a method of instruction different from traditional lecture, or "chalk and talk." Distance between two points is an ideal topic for such a presentation. This article discusses two different models or manipulatives for use in a classroom or in a lecture hall setting, suggests a decrease in the use of "symbol-speak" and an increase in the use of our more abstract English language, and provides writing-to-learn or critical-thinking-type summary questions. About the Model       The goal is to concretely represent distance in one-, two-, and three-dimensions. To do this a rectangular box may be used. Corners A, (x1,y1,z1), and B, (x2,y2,z2), represent two points. The sides of the box model distance in one-space -- distance on a line. The diagonal of the base of the box models distance in two-space -- distance on a plane. Using an additional paper plane and the box, one may model distance in three-space.       Cut out the paper pieces. Crease then fold back and forth the dashed segments. Folding printed side in, construct the box. Use paper clips or fingers at the narrow ends to hold the box rigid. Examine the printing, the symbols and segments, as the box is constructed and make mathematical statements.       The dimensions of the box, 12 by 8 by 9, employ two sets of Heronian Pythagorean triples, 9-12-15 and 8-15-17, such that the hypotenuse of one triangle is the leg of another triangle. The length of segment (hypotenuse) BC is 15. The length of hypotenuse AB, the modeled distance between two points, is 17.       Coordinates of all vertices of the rectangular box are printed on the template. Reference points A, B, C, and D are included to facilitate discussion and to identify triangles ABC and BCD. The length, width, and height of the rectangular box are labeled as the differences in x- coordinates, written |x1-x2|, the difference in y-coordinates, written |y1-y2|, and the difference in z-coordinates, written |z1-z2| .

 The Classroom Model or Manipulative To assemble the model:   Click on the image for a large copy or here for a Word® document containing template, instructions, assignments.  1st: Make the desired number of copies of the template and distribute these with scissors to each student.  2nd: Cut out, leaving line segments and arcs intact, the three shapes on the template -- two rectangles and one net of rectangles and quarter circles.  3rd: The quality of the model improves if one scores all dashed lines with a pencil or ball-point pen. Be particularly accurate in scoring the radii of the quarter circles since these provide rigidity to the model and obviate the need for tape, glue, or paper clips to hold the rectangular box together.  4th: Fold the paper back and forth on each of the dashed segments.   5th: Fold the printed side into the interior of the box, and pieces of the quarter circles to the exterior of the box. Fold the quarter circles to the short ends of the box. Paper clips or fingers may be used to hold the box together for examination.   6th: Place one of the rectangular sheets containing segment AB in the container so segment AB lies between points A and B. If the rectangular sheet is too large to fit cleanly in the rectangular container, trim off the short segment BD and/or the short segment opposite BD. The model is now ready for use.       Copies of the rectangular box might be made of paper or on acetate. If made of paper, students could construct the model then hold it in their hands for examination. If made of acetate, each rectangle in the net should be cut out separately and taped with transparent tape for permanent assembly and prior to use on an overhead projector.

 The Lecture Hall Manipulative       The lecture hall manipulative requires additional pre-use work. It also requires yarn to be drawn through the paper and for the yarn to be taped. Paper clips are optional. Steps 1 through 4 below must be completed before use in the lecture hall. To assemble the container:   Click on the image for a large copy or here for a Word® document containing template, instructions, assignments.  1st: Make the desired number of copies of the template.  2nd: Cut out the net but keep it and the printing in the interior intact.  3rd: From the nonprinted side of the paper thread a length of yarn through the paper at point B using a thick needle. The yarn should be long enough to leave a tapeable length on the nonprinted side of the paper and to extend over the edge of the paper at the quartercircle centered at point C. Tape the yarn to the nonprinted side of the paper so that the yarn is held in place and the tape does not "cover" a dashed segment.  4th: [Optional] Clip two paper clips to each cut-threaded-taped sheet prior to distribution.  5th: Follow steps 3, 4, and 5 from the Classroom Model above.  6th: Once the box has been assembled, hold the yarn at point B so as to form the segment AB, the diagonal of the box, the distance between A and B.

 Discussing the Distances       The model "speaks" concretely.       To "speak" symbolically and to "speak" with words, one would need the following expressions, phrases, and sentences. The author encourages this "verbalization" as a necessary part of any manipulative activity. Avoid "symbol speak" by saying the words, not just reading the symbols.       The length and width, lengths BD and CD are sides of triangle BCD. The hypotenuse of BCD is segment BC. Its length is the positive square root of sum of the squares of the length and width.       Segment BC is a leg of triangele ABC. The other leg is segment AC, the height, or |z1 - z2| , or the positive difference between the coordinates. The length of the hypotenuse AB is now computable. It is the positive square root of sum of the squares of the height and length BC.       This may be simplified to its usual form. The distance between corners A and B is the positive square root of the sums of the squares of the length, width, and height of the rectangular prism. The distance between (x1, y1, z1) and (x2, y2, z2) is the positive square root of the sums of the squares of the differences in the corresponding coordinates of the points.

Summary Activities: From Hand to Head, Then, From Head to Hands

A necessary part of each concrete or manipulative activity should be an more abstract or a written/spoken/summary component. Some sample summary activities follow.

Activity 1. (Revision) Fill in the table.
Distance between 2 points
 Distance between 2 points in:     words, symbols, a picture One-space, a line Two-space, a plane Three-space, space

Activity 2. (Revision) Using the box as a prop, explain to a student absent from class, why the distance between two points may be computed using the formula

Activity 3. (Revision) Mentally rehearse a phone conversation with a friend in which you explain why distance in space may be computed using the above formula.

Activity 4. (Revision) Predict a formula for distance in n-space.
Discuss your formula with one other classmate and refine your formula.

Discuss the formula refined by you and your partner with another pair of students then with the class.

Present your formula to the professor with a written justification.

 After Word       The above article, exclusive of the words ", Then, From Head to Hands," was submitted for publication in June 1996, rejected in May 1997, and resubmitted in June 1997, with the following new passage on "chalk and talk vs. concrete thought," and, never published. The new passage introduces an new dimension to the use of manipulatives and concrete thought — reluctance on the part of sincere and knowledgeable professionals. This too is part of manipulative use in education.

 ‘Relinquishing Control of Center Stage'       The author passionately proposes the reader, and most certainly anyone reviewing the material presented, change the format of the class from teacher-center exposition to student-controlled experimentation and discovery before making a value judgement about concrete thought and the use of this or any other manipulative. Concrete thought does not happen abstractly unless one is experienced in concrete thought.       The professor or reviewer who chooses not to take model in hand HAS NOT EXPERIENCED CONCRETE THOUGHT. The professor or reviewer who has not taken model in hand has perhaps REPRESSED this language. See www.mathnstuff.com/papers/langu/page2.htm.

 Students Who Have No Difficulty With This Topic       One reviewer negated the value of the manipulative claiming "students have no difficulty with this topic." Who is to say that student would not gain more using the manipulative. Who is to say the student who appears to "get it" and memorizes a formula would not, with the manipulative, understand better the origin of the formula. Who is to say the student who doesn't make it to calc II where three-space distance is addressed, would not easily in algebra I or algebra II really "get it" through the concrete manipulative.

 Your Goals       If for you "chalk and talk" is not the only way to communicate mathematics and if the goals stated by the author and the AMATYC Standards match your own, experiment with the distance manipulatives and the debriefing contained herein.

 1 Fred R. Barnard, as recognized by William Safire in William, ‘Worth A Thousand Words," The New York Times Magazine, April 7, 1996, pg. 6 - 16.

 © August 1, 2003, A2 www.mathnstuff.com/papers/distman.htm