State College Area School District



The RulerPlease write your responses on separate paper(s)!1. Name some geometric objects that can be measured with a ruler.2. Name some geometric objects that cannot be measured with a ruler.3. Look at the centimeters side of your ruler. Each tick mark is 1 millimeter. How many millimeters are in one centimeter? (You can figure this out using your ruler!)4. If you measure a line segment as 3 cm and 4 additional tick marks, how long is your line segment (or how long should you report it)?5. If you do your best to measure a line segment with your ruler, how far off from the actual length do you think your measuring is? Explain why you think so.6. We write mAB=14.7cm to mean “The measure of line segment AB is 14.7 centimeters.” Sketch your own line segment, measure it, and then record its measure using the symbols above.7. Sketch a line segment with mCD=7.4 cm.8. Could there be more tick marks on your ruler than there are? Are there any lengths you cannot measure with your ruler?9. The Ruler Postulate says that: Each point on a line segment represents a different real number. Thus we can think of a ruler as a number line. What operation gives the distance between two numbers on a number line?10. Why does the Ruler Postulate imply that getting an integer length as a measurement on your ruler is unlikely?11. We write AB?CD to mean “Line segment AB is congruent to line segment CD.” What does it mean for two line segments to be congruent? 12. Sketch two congruent line segments and then write a mathematical statement about them using the congruence symbol above.13. Now turn to the inches side of your ruler. How many tick marks are between each inch?14. If you measure a line segment as 4 inches plus an additional 4 tick marks, is this 4.4 inches? Explain.15. Sketch and measure a line segment in inches. Write a mathematical statement using the symbols above about the length of your segment.16. What do you think is the biggest student misconception about measuring using inches?17. The English system (used only in the United States) has inches, feet, yards, and miles as units of measure. How many inches are in a foot? Feet in a yard? Feet in a mile?18. The Metric system (used everywhere else in the world and also in the US for science) has millimeters, centimeters, meters and kilometers. How many millimeters are in a centimeter? Centimeters in a meter? Meters in a kilometer?19. Use your ruler to estimate the number of centimeters in an inch. Then figure out how many centimeters are in a foot.20. What does it mean to measure with a ruler?The Protractor1. What is an angle?2. How many points are needed to determine an angle?3. Sketch an angle and label a point on each of its rays and on the vertex. In how many ways can you name this angle?4. Sketch two intersecting lines. Label the point of intersection and a point on each of the four rays. If you are trying to write to me about one of these angles, how many points do you need for me to be absolutely certain which angle you’re referring to?5. What does it mean to measure an angle?6. How does the protractor measure angles? What does the protractor really measure?7. The degrees we use are a unit of measure, but using 360 of them for a full rotation is a convention—a decision people made that we follow, but without a particularly meaningful basis. What’s another convention in math or in life that doesn’t really matter but following this convention is important? (My example is driving on the left or right side of the road!)8. Why do you think we use 360 degrees in the first place? There are some reasons historians think that this number was chosen.9. Using the convention that 360 degrees is a full turn, how many degrees is the measure of a straight angle, or around a line? 10. Axiom 3—that all right angles have the same measure—provides a basis for measuring angles. Prove that if there are 360 degrees in one rotation, then a right angle is 90 degrees.11. Explain why, if Axiom 3 were false, then we could not measure angles with a protractor.12. What really matters is the relationship between the measures of angles, not the degree values. If we defined a full turn to be 2π units (call them Wolf-Roots) instead of 360 degrees, what would be the measure of a straight angle? A right angle? The sum of the angles in a triangle?13. What does it mean for two angles to be congruent?14. What other kinds of angles are there? State, define, and sketch them.15. Does your protractor have two sets of numbers on it, one above the other? How do you know which set of numbers to read off?13. What do you think is the most common student misconception about measuring angles? What do you think is an easy way to check that your measurement makes sense?16. To bisect an angle means to cut it into two congruent angles. Sketch an example of an angle and then bisect it. 17. If you start with an obtuse angle and bisect it, what kind of angles are formed? Make a conjecture.18. Prove your conjecture! 19. Turn to page 38 of your textbook and define the four types or relationships of angles. 20. Prove that vertical angles are congruent without measuring them!21. What does it mean to measure an angle? ................
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