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   Introducing Distribution

    Measuring the Height of the School's Flagpole with an Inclinometer
    Task
 

How many feet high is the school's flagpole? Students measure the height of the school's flagpole using elementary trigonometry (properties of isosceles right triangles). The measurements are posted randomly on the board. Students work in pairs to create a display of the measurements that show "a pattern, if there is one."

 
   Rationale/ Purpose
 

Measuring the height of the flagpole grounds guided invention of distribution. The initial format of the distribution is a display. By contrasting displays, the intention is to ground the "shape" of the data. Various choices highlight different aspects of the data. It is anticipated that at least one invention will include intervals and frequencies. If no invention has these properties, then one is introduced, because interval is a pathway to the notion of density. The tool used and the method employed to estimate distance are later improved upon, so that subsequent measurements will be more precise. This has the effect of affecting the variation of measurements, but not the central tendency.

    Materials
 

Students are provided a cardboard inclinometer, or they construct one. You can find directions for constructing the inclinometer in the "Resources" section. The rationale for the inclinometer is based on triangle trigonometry. If the angle is 45 degrees, then the distance between the person and the object is the same as the object's height, plus the height of the person. Other angles could be employed, but this is a simple choice that relies on the trigonometry of the isosceles right triangle (See diagram). As we describe later, the unit of measure is a person's pace .

    Procedure
 

Students use the inclinometer to measure the height of the school's flagpole, four times. The inclinometer is used by holding it at arm's length and sighting along the top of the flagpole. Locate a position where the angle is 45 degrees and then walk from that spot to the flagpole, counting the number of paces. The height of the flagpole is then the sum of the number of paces (expressed in feet) and the person's height (also expressed in feet).

Pacing . The unit of length measure is a pace , introduced to create another visible source of prospective measurement error. A pace is a simple method of measuring linear distance by walking. A pace equals two normal steps, beginning and ending on the same foot. It is measured from the heel of the foot to the heel of the same foot in the next stride. It is easiest if the pacer says "and" during the first step, then the number at the end of the second step (and one, and two.). Pacing can be used anywhere: indoors, outdoors, in the woods, or in open fields. Pacing dates back to Roman times. The original Roman pace was slightly over 58 inches long. This has become known as the geometric pace, which measures about 5 feet for most people.

To obtain the measure of a person's pace, a few methods can be used. One is to set up a course of 50 feet with starting and ending lines. Start with the heel of the foot touching the line. Then have students pace the distance and record the number of paces.

Teacher's note. Ask students if there is reason to pace the same distance more than once. If students believe that multiple trials might be a sound idea, have each student walk the same 50 feet 5 times, recording the number of paces each time, and converting paces to feet/pace (divide 50 feet by the number of paces). Then ask students how they are going to think about a "best estimate" for the distance covered for each pace. This is an opportunity to think about how students think about measurement variation.

Students sight the flagpole with the inclinometer at a 45 degree angle and then paces from that point to the flagpole. Each student tries this four times, each time from a different perspective (so that the person is not starting from the same place each time).

Records . Students record each observation on an index card. On the front of the card is the measure in paces. The back is the name of student and number of observation. Students convert paces to a common unit of measure (e.g., feet) and add their height to each estimate to obtain an estimate of the height of the flagpole.

    Whole Group Conversation
 
The estimates are placed on post-it notes and tacked to the board (unstructured). Students talk about what they notice. Why are the measurements not all the same? ( Teacher note: Try to elicit potential sources of variation, including variation due to using the tool, variation in how a person paced [if the paces aren't all the same, what happens?], and other sources, such as calculation mistakes or rounding.)

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    Construct Displays

 

Estimates obtained by students are written as an unordered list and given to students. Students are provided paper (large chart paper is best) and asked to design a display that shows what they expect the height might really be ("best guess") and also differences in the measurements. The criterion is that someone who did not participate could get this information just by glancing at the display. It may be helpful to elaborate the goal as a display that shows a pattern in the data, if any patterns exist.

If no student invents an interval display, then one should be added by the teacher as representing the work of another class or simply to invite comment.

    Comparing Displays
 
When completed, students give the display to another pair, and then each pair describes the qualities of each display that they notice. The emphasis is on what each display allows us to "see" and what each display "hides." ( Teacher note: It is critical that the conversation be directed toward trade-offs rather than the "best" way to view the data.) The lesson concludes with attempts to explain any patterns that are visible in the displays. What about the process of measurement produces what is visible?
    Formative Assessment
 
Students' displays offer insights into how they are reasoning about the structure of the data. What aspects of the structure of the data did students appear to attend to? How did they explain the relation between measurement processes and the shape of the data?
    Students' Ways of Thinking  
Last Updated: April 13, 2006
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