Learning Math: Measurement
Angle Measurement Part C: Geo-Logo (40 minutes)
Session 4, Part C
In Part C, we explore the interior and exterior angles of a figure as well as the relationship between the two. This activity will provide you with a different method of exploring angle measurement.
On a piece of paper draw the following shapes. Pay attention to the amount of turn you do with your pencil as you draw any two adjacent sides (the amount of turn will be equal to the exterior angle of the polygon at that vertex).
- Parallelogram that is not a rectangle
- Equilateral triangle
- Regular pentagon
- Regular hexagon
- Star polygon (five- or six-pointed star)
- n-gon (you decide the number for n)
- What is the relationship between the amount of a turn and the resulting interior angle?
- Why is this relationship important to understand?
Examine the polygons you drew where your turns were all in one direction (either all to the right or all to the left). What is the sum of the measures of the exterior angles of each of these polygons? Is this sum the same for all polygons? Why or why not?
Use your drawings to determine the relationship between the measure of central angles and the measure of exterior angles in regular polygons. Explain this relationship.
Try this with simpler shapes, such as an equilateral triangle.
|Video SegmentWatch this segment to see Mary and Susan experiment with Geo-Logo commands to create different regular polygons. They examine the relationship between the angle of turn and exterior and interior angles.
Do you think computer technology can enhance exploring mathematical tasks such as this one? Why or why not?
You can find this segment on the session video approximately 16 minutes and 47 seconds after the Annenberg Media logo.
Take It Further
In order to draw a non-intersecting star polygon, you must direct your pencil to turn both right and left. Print a copy of the star polygons and measure the exterior angles. What is the sum of the exterior angles in a star polygon? Explain this result using the commands.
Think about how turning in both directions affects the resulting exterior angles.
If you are working in groups, review and discuss Problem C3. For each of the polygons drawn in Problem C1, have group members add the amount of turns that were used to draw the shape and bring the turtle back to its starting position. Discuss the following questions: What is the total sum of the turtle turns for each polygon? (Or, in other words, what is the sum of the exterior angles of these polygons?) If, when drawing the polygons, the turtle had not gone forward but rather had stayed in one spot, what figure would the turtle have drawn?
- To form a square, try this: Repeat 4; Go Forward 4; Rotate Right 90; End Repeat.
- To form a non-square rectangle, try this: Repeat 2; Go Forward 5; Rotate Right 90; Go Forward 7; Rotate Right 90; End Repeat.
- To form a non-rectangular parallelogram, try this: Repeat 2; Go Forward 5; Rotate Right 60; Go Forward 7; Rotate Right 120; End Repeat.
- To form an equilateral triangle, try this: Repeat 3; Go Forward 5; Rotate Right 120; End Repeat.
- To form a regular pentagon, try this: Repeat 5; Go Forward 5; Rotate Right 72; End Repeat.
- To form a regular hexagon, try this: Repeat 6; Go Forward 5; Rotate Right 60; End Repeat.
- There are many different star polygons, but here’s one way to do it: Repeat 5; Go Forward 5; Rotate Right 144; End Repeat. The angle must be a multiple of the angle used to form a regular polygon of the same number of sides. If you wanted a star polygon without intersecting lines, try this: Repeat 5; Go Forward 5; Rotate Left 72; Go Forward 5; Rotate Right 144; End Repeat.
- To form a regular n-gon, theoretically, the commands would be as follows: Repeat n; Go Forward 5; Rotate Right (360/n); End Repeat.
- If the turn is x degrees, the resulting interior angle measure is (180° – x°).
- Answers will vary, but one important observation is that such turns measure exterior angles. In order to build a polygon with the correct interior angles, we must first subtract from 180 degrees to find the exterior angles.
The sum of the measures of the exterior angles is 360 degrees for all polygons drawn this way. One explanation for why this is true is that the cursor must make a complete circle and return to its original position, and there are 360 degrees in a circle.
In a regular polygon, the measure of each central angle is equal to the measure of each exterior angle:
Since the central angles total 360 degrees and the exterior angles total 360 degrees, each of these angles is also equal. (For non-regular polygons, these totals are still equal, but the individual angles are not.)
See commands in Problem C1 (h). The sum of the exterior angles is a multiple of 360 degrees, since the cursor will go around in a circle more than once as it moves through the points of the star. Answers will vary, depending on the size of the star polygon (specifically, the number of times the cursor must go around in a circle), but will always be a multiple of 360 degrees.
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.