BRIDGE CURRICULUM GUIDE________

MATH MEASURING ANGLES WITH A PROTRACTOR

MEASURING ANGLES WITH A PROTRACTOR Focus: You have used the Cartesian coordinate system to describe position and relative position of objects. Another method of determining relative position is angle measurement. When you hear expressions like “I did a 180," “make a right-angle turn,” or “perpendicular to,” this form of mathematical reasoning is being used. In this unit we will practice measuring angles with a protractor and learn how to convert from one measurement system for fractional degrees to another.

At outset of the module, review the general concept of relative position, referring to an example that makes use of the Cartesian system.

It may surprise you to know that our present system for measuring angles owes something to the ancient Babylonian Empire, situated along the Tigris and Euphrates rivers by what we now know as the Persian Gulf.

If you choose just to do the use of a protractor part the material is straightforward. If possible, attend corresponding technical sessions so you can make connections between the activity and degree-minute-second conversion.

For background on the Babylonian number system, please refer to materials your instructor provides. It will help you understand why we measure fractions of degrees the way we do.

Think About It: How do we still use base 60 in our system for telling time? How many degrees there are in a circle? How is this related to base 60?

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This module is divided into two parts. Both parts together could take more than 2 hours. If you do not have enough time, the parts can be done separately.

For a discussion of the Babylonian System, see Staszkow and Bradshaw, The Mathematical Palette, second ed, pp. 14-16. See also http://www-history.mcs.st-and.ac .uk/~history/HistTopics/Babyloni an_and_Egyptian.html. This site requires some knowledge of algebra. Additional references to Babylonian numbers are found at http://forum.swarthmore.edu/dr. math/problems/bragg1.10.97.ht ml

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BRIDGE CURRICULUM GUIDE________

MATH MEASURING ANGLES WITH A PROTRACTOR

Session Activities/ Outcomes: Upon completion of the session, you will: 1. Be able to measure an angle to the nearest degree with a protractor. 2. Recognize whether an angle is closest to 45 degrees, 90 degrees, or 180 degrees. 3. Be able to convert between degrees minutes and seconds, and decimal degrees. Activity 1 Your company time clock tells times in hours, minutes, and seconds. The office that handles your payroll, however, wants you to report times worked as decimal fractions of an hour. Create a conversion chart by following the directions below. Convert the times listed there.

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Form groups of three or four students. Many of them, especially those that have worked in shops, before will have well developed proportional reasoning. Help them explain alternate methods they use to solve these problems to the group if they come up.

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1. Label the lines in the first column so that the numbers start at 0, are equally spaced and end at 1. 2. Label the lines in the second column so that the numbers start at 0, are equally spaced and end at 60.

Students will need decimals in the left column labels. Some groups will immediately see this; others won’t. Provide adequate time for discovery.

3. 24 minutes is what decimal part of an hour? 4. .35 of an hour is how many minutes? 5. Estimate what decimal part of an hour 40 minutes is. 6. Estimate what decimal part of an hour 10 minutes is.

Create a spreadsheet that will convert the fractional part of a minute to a decimal. See the template provided. Can you make your spreadsheet convert back also?

Monitor the students to make sure they use formulas to make the spreadsheet conversion. This early in the semester they may need help with the syntax of the spreadsheet. The two most common ways to derive the formula will be multiply, divide by 60 and crossmultiply. After a group struggles for 10 minutes or so to get a formula you might want to provide the formula for converting from DMS to DD. Ask the students to confirm that it works using the conversion chart they made earlier.

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Activity 2 Using the illustration your instructor provides, develop an informal definition of an angle. Follow the instructions your instructor provides and, with a protractor, measure the angles on the attached piece of paper.

Show the students how to use the protractor. Measure a few angles for practice. On the handout, make sure the sides of the angles have varying lengths. This will help the students avoid the misconception that angle measure has to do with side length. Emphasize that it has to do with “wideness”. Two angles with the same measure and different side length can help make this clear. Tell students a 90-degree angle is a right angle. Draw one. Also show 45-degree angles and 180-degree (straight) angles. Use an overhead to show many different angles and to help students develop benchmarks. Ask students which angles are closest to 45, which to 90 and which to 180 degrees. Finally, introduce the notation and meaning of DMS notation for angles. Give a couple problems where you must convert DM to DD and vice versa. Ask students how to do this. Steer them to their spreadsheets if no one suggests this. A challenge question can involve converting 25.5075 to degrees

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MATH MEASURING ANGLES WITH A PROTRACTOR

minutes and seconds. (First convert .5075 to 30.45 minutes. Then convert the decimal minute to seconds .45 to 27 seconds, so 25 degrees 30' 45".) Think About It: It 1. Write a paragraph explaining to another student how to use a protractor to measure an angle. Remember that mathematical writing is detailed, benefits from examples and includes charts and graphs whenever possible. 2. How are minutes and seconds in angle measurements related to minutes and seconds in time measurements. Key Terms Protractor Angle Right angle Degree Minute (angle measure) Second (angle measure) Evaluation or Competency Testing Have students measure angles

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for you. Have students hand in their spreadsheets after showing they really work by converting some angles on it. (You can have them “drag” to convert multiple conversion slots and convert some angles you write on the board.) Have students write a paragraph explaining how to convert from degrees and minutes to decimal degrees using a calculator.

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