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   Measurement
    Measure Walks
    Goal
 

Students invent units of length and direction measure in the context of walks—a measure-motion connection. To represent paths as sequences of these units, given an origin (tacit-meaning that the beginning is ). For example, starting at a landmark, Walk 10 paces, Turn 1/4 of a whole turn to your right, Walk 25 paces. Where are you in relation to where you started?

To compare the utility of different invented units for length (or appropriated units like paces), and to compare different systems for representing walks.

 
    Related Principles & Big Ideas
 
 
    Activity Structure
 
  • Whole class discussion & Pair work
  • Prepare a few locations on the school grounds to play the treasure-hunt game described next. Each student pair will need a clipboard, paper, and pencil.
  • Set-up discussion: Today we're going to play a treasure-hunt game. We will give each pair a starting place. Your job is to write directions from the starting place to anywhere else on the yard WITHOUT using any landmarks. For example, you can't say: "Look toward the back of the school building," or anything like that. Your ending place (the place where you will bury your treasure) must be at the end of a path that includes at least one bend. Someone else must be able to follow your directions and whichever team writes the best directions (others can follow them and find the treasure) for three different walks wins.

    What will you need to think about?

<Alternative treasure-hunt game in classroom>

  • Set-up discussion:

    Direction giving:
  1. With your partner, write a series of directions to get from point A to point B through C (for example, Start(A): Book case, Through(B): Door, End(C): Blower) so that someone else could move easily from one point to another without further assistance. The directions must specify an exact route to follow. You may only use words, no drawings. You may not use any landmarks.
  2. Write how you and your partner came up with your directions. What did you do?
  3. What is the total distance?

Direction following:

  1. With your partner, follow the directions given to you exactly as written.
  2. With your partner, create a map showing the route you followed using the directions given to you.
  3. What is the total distance?

Follow-up discussion:

  1. In direction giving, what was hard? Why?
  2. What would have made direction giving easier? List as many ideas as you can.
  3. In direction following, what was hard? Why?
  4. What would have made direction following easier? List as many ideas as you can.

  • Teacher note. You might want to tape a path with a bend in the room as an object of discussion. Ask students how it might be measured. One way to measure the bend is as part of a whole turn (one rotation). Be sure to ask students to represent the turn. What is important about the turn to represent? Let students invent multiple ways of representing turns. We think of turns as angles but the relation between the static description of angles as unions of line segments depicted in textbooks and angles as turns is NOT obvious.

    The pathway to degrees is straightforward, if students have part-whole conceptions of fractions. Each degree is 1/360 th of a turn (see degree graphic). (see degree graphic).

    Teacher Support of Student Thinking
 

Teachers should prompt students to consider what is mathematical about their activity. Each component of the lesson offers opportunities for elaboration.

(1) Paths without turns are estimated by units of student choice. These units can be compared for their motivations (Why that unit?) and utility (What is it good for?). New units can be introduced. For example, a pace is two strides, signified by (and 1), (and 2). Each pace is about as long as the strider is high).

(2) Paths with bends (measured by turns) are estimated. (How can we measure something like a bend in a path? How can we represent how much we turned, if turns are used?)

(3) Directions to mystery locations are created by each pair and tested by another pair. Do the students wind up at the location expected? What does it take to create “good” directions? The idea here is to compare the virtues of different units of measure (e.g., paces are good for long distances, feet for shorter ones) and different ways of representing directions, especially turns. What is the difference between measuring a distance and a bend in a path? What are properties of good choices for units of measure?

Extension :

  • What shape is made by a measure walk of:

Start, Pace 10, rt ¼ turn, Pace 10, Rt ¼ turn, Pace 10, rt ¼ turn, Pace 10, rt ¼ turn?

  • How could I construct a measure walk for a rectangle?
    Assessment of Thinking
 

Assessing student thinking during instruction forms a solid foundation for changing, elaborating or simply continuing a lesson, as needed. Here we note some of the tasks and situations developed by teachers to both assess student reasoning and to support its development.

Play robot . Some teachers have found that students use the idea of a bend to begin to talk about degrees. Often, they do not connect degrees to turns or to the notion of 360 degrees in one rotation. One way to see what students are thinking is to ask students to PLAY ROBOT. Students stand up and the teacher or another student gives directions. For example, to diagnose how students are thinking about turns, the teacher (or another student) give turning directions: TR 1 (turn right 1 whole turn), TR 2 (turn right 2 whole turns), TL ½, TL ½ of 1/2 (The last example helps us see how students think informally about multiplication of fractions).

Representing turns . Letting students invent representations of turns is a window to their thinking about them. It also sets the stage for relating conventional ideas of angle to turns. During Play Robot , ask students to invent a means of representing the turn. Compare and contrast student inventions. What is preserved? What is left out? Which representations would enable someone not present to reproduce the same turn?

One potential scaffold is to let a vertical line segment stand for the initial heading before turning and the other line segment, the final heading, after turning. An arrow can represent the direction of turn.

Home-school connections . Ask students to create a path for getting from one place to another in their neighborhood, once with and once without landmarks. Then the two descriptions of the path can be contrasted and compared. Teachers might ask students to imagine when a path perspective might be better than one employing landmarks (e.g., when features of the terrain are not obvious, when one can't see from one end of a woods to another etc.)

MEASURE PATHS is a series of supplementary lessons that build from the path perspective introduced in MEASURE WALKS. The supplementary lessons introduce a circular protractor and some elementary explorations of polygon paths.

    Students' Ways of Thinking
 

When teachers have tried this lesson, students often display patterns of thinking. We summarize some of the most salient patterns here:     

Directions students wrote at the beginning of the class:

Example 1, 2, and 3 were created by students at the beginning of the class. As you may notice, the three examples show different levels of thinking. In example 1, the student used "Go" for distance and directions. This means that the student did not differentiate distance from direction. Example 2 illustrates that the student made a distinction of distance and direction. However, the direction is not specific: Turn was not quantified or otherwise marked. Steps were taken-as-shared but the length of each was not clear. Example 3 shows that the student used "left" and "diagonal" to indicate directions. Even though it is more specific than example 1 or 2, it was still not clear enough for other students to follow. These problems can be revealed by asking "What does this direction make you to follow?" "Why is it hard to follow this direction?" Watch video clips showing the teacher problematize students' directions and dicussing problems with students.

Example 1:

1) Start at the habitat table.

2) Go straight

3) Go right

4) Go left.

5) Stop

Example 2:

1) 10 steps in front of desk

2) (Turn) 5 steps

3) 11 steps (Straight)

4) (Turn) 12 steps

5) You are there.

Example 3:

1) Start door

2) 1 step forward

3) 5 steps left

4) 2 steps right diagonal

5) 4 steps forward

6) 5 steps left diagonal

7) End, where did you end up

 

    Directions students revised:

After the whole class discussion about how to provide good directions, students revised their directions to make them clearer. Meanwhile, a student brought up the issue that how he could express turns in more mathematical way. The teacher did ROBOT PLAY with students to help students connect directions with turns. Example 4 is revised by the same students who wrote example 3. Now they use fractions to give directions. Also, they are more precise about steps, like medium or large. Notice that in example 5, the student specified that they defined steps as heal to toe walks.

Example 4:

1) Start at the door

2) Take 1 step forward (large)

3) Turn ¼ of a circle left

4) Take 8 steps forward (medium)

5) Take 5 steps back (medium)

6) Turn ¼ of a circle right

7) Turn 5 steps forward (medium)

8) Turn ¼ of a circle left

9) Take 6 steps forward (medium)

10) Take ¼ of a circle right

Example 5:

You will start at the book case. Then turn right ¼ & walk 5 steps & stop. Then turn ¼ again to your front & now walk 18 steps then you can stop. Now walk 12 steps, then turn ¼ to your right. And walk 3 steps & turn ¼ around to your front. Now walk 15 steps then stop & turn left ¼ again. Walk 10 more steps & turn right ¼ around walk 10 ½ steps and now you're done. Do you know where you are at?

But remember to put your heal in front of your toes each time you take a step. I will see your feet.

Last Updated: March 31, 2005
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