To support students' concepts of differentiating length from angle measure in the context of walking paths, developed in Lesson 2. The lesson introduces degree as measure of angle, a circular protractor as a measurement tool, and the use of the tool to calculate changes in the heading of a walk.

Tape a path on the floor with a 45 degree angle. Walk on the path and ask students to indicate what you are doing. Be sure to stop at the vertex of the angle and ask students how to indicate changing heading. Use pieces of tape to indicate what would have happened if heading had been unchanged. Be sure to build on the previous lesson's idea of a turn.

For example, WALK 10 steps (be sure to define) TURN RIGHT (1/8) of a whole turn. WALK 5 steps.

Introduce a circular protractor as a tool for measuring how much of a turn. Use the graphics transparency to label quarter turns and eighth turns. Then pick up the transparency and ask students how it could be used to measure the turn on the path taped on the floor.

Then pose the problem of measuring a turn that is about 30 degrees. How will we indicate that? This is intended to motivate the idea of a finer partition of the circle: the degree as 1/360 th of a whole turn. Revisit the number of degrees in a ¼, 2/4, ¾, 1 turn. Then ask students to stand up and to enact varying degree-turns: 90, 180, 270, 360, 45, 30.

The paths are:

PATH 1

1. Walk 6 steps.
2. Use your protractor to Change Direction: TR 90.
3. Walk 6 steps.
4. DRAW the path that you made on the sheet, starting at the X. (Use your protractor when you draw the path.)

PATH 2

1. Walk 6 steps
2. Use your protractor to Change Direction: TL 60.
3. Walk 6 steps.
4. DRAW the path that you made on the sheet, starting at the X. (Use your protractor when you draw the path.)

PATH 3

1. Walk 8 steps.
2. Use your protractor to Change Direction: TR 120
3. Walk 4 steps.
4. DRAW the path that you made on the sheet, starting at the X. (Use your protractor when you draw the path.)

PATH 4

1. WALK 3 steps.
2. Use your protractor to Change Direction: TR 30
3. Walk 6 steps.
4. Use your protractor to Change Direction: TL 130
5. Walk 4 steps.

PATH 5

1. Walk 5 steps.
2. Use your protractor to Change Direction: TR 90.
3. Walk 5 steps.
4. Use your protractor to Change Direction: TR 90
5. Walk 5 steps.
6. Use your protractor to Change Direction: TR 90
7. Walk 5 steps.
8. Use your protractor to Change Direction: TR 90
9. DRAW the path that you made on the sheet, starting at the X. (Use your protractor when you draw the path).

PATH 6

Write directions that will make a path that is a rectangle.

Then, students can invent their own path: GUESS MY DIRECTIONS

MAKE YOUR OWN PATH WITH AT LEAST 5 directions but no more than 8, beginning with FACE 0. Then, on the next page, draw your path. You will share the path with a partner, who will try to guess your directions.

So far, angles have been treated as motions-as changes in direction. Another perspective is to treat them as the union of line segments. We can reconcile these perspectives by treating one line segment as the original direction and the other line segment as representing the new heading-the new direction. The protractor is centered on the vertex with 0 on the line segment treated as the original direction. The change in direction is given by reading the protractor.

Try this first with a right angle. Then use it for acute and obtuse angles. Ask students to create angles and to measure them with their protractor.

Length as a Measure

It is likely that some students will want to use the protractor's arc as a measure of angle. This should be addressed by presenting a series of similar angles. 90 works. Then try to use length, either as the arc or as a line connecting the 2 opposite endpoints. Children should notice that the measure changes but does not do so when degrees are used. Of course, we later will want to use ratios of length to measure angles. But for now, we need to be certain that the measures are differentiated.