u1. We need two points to define a line. A different way to say this is that if you have two points, you will need at least a line to hold them. In this sense, what do you need at least three points to define? You don't need three points to make up a line because usually a line is made of two points and they continue going, they never end. 2. Two points define a line, but if there are two lines in a plane, and if they intersect somewhere, they \"define\" a single point. If there are two planes in space, and they intersect, what do they define? (Look at the walls of your room, where they meet. Imagine if there were no floor or ceiling, and the walls went up and down and side to side forever, what would the corner where they meet look like?) It would just be a straight edge. 3. In the figure below, compare the two rays ry_AB and ry_CD. Which one is longer? Why? Ry_AB, because A is the starting part and B is the ending point. 4. What is the difference between a plane and a piece of paper? Plane: Two-dimensional object of infinite proportions upon which you can have points, lines, angles, etc. Piece of paper: Flat object with straight angles. 5. I have ry_AY and ry_XZ. Given only what you've been told, is it possible that they could define an angle? No they don't have the same endpoint. 6. Consider two lines that intersect in a single point in a plane. How many rays would it take to draw the same picture? It would take four rays to draw the same picture. 7. Look at figure 1 (the last figure in the lesson above). Imagine that at the intersection of ry_EG and ln_DF we placed a point M. Which of the following most completely describes the new elements we have created? Explain your reasoning for choosing this answer. A. angle GMF and ls_EM B. angle GMF, ry_MD and ls_EM C.four rays D. one angle E. ry_MD and ry_MF F. ry_MG (X) 8. Imagine you printed out figure 1, enlarged it with a photocopier, and then compared the two figures. Everything would look larger, but many of these things would not actually be larger because they are really infinite. Knowing this, tell me how would ls_BC change? How would ln_DF change? How would ry_EG change? They're going to look bigger but not all. 9. Plane geometry as we study it now can be traced back to Euclid, and his book The Elements. Go HERE to find out more about that book, and find out what it means when they say "A point is that which has no part." Which of these best sums up that idea. Explain your reasoning for your choice: A. A point has no dimensions, it only defines a place in space B. Points are only on lines, never by themselves C.A point is only important if it defines a shape D. Points are very big E. Points are very small F. Points stand alone and have nothing more to them (I say that its F because it says it has no part so if it has no part the A stands alone with nothing more or nothing less.) - My answer is F. 10. Imagine you printed out figure 1 and enlarged it with a photocopier, then compared the two figures. Everything would appear larger. Would point B really be larger? No. 11. Does every ray contain a line segment? Yes. 12. How many points are on a line? A line has two points. 13. Does every line contain a ray? Yes. 14. In figure 1 above, could ls_BC become a line? Yes. 15. In figure 1 above, can ry_EH be used to create a line? Yes 16. If I have two rays, do I have a plane? No. 17. If I have an angle, do I have a plane? Yes. 18. If I have a line, do I have a plane? No. 19. If I have three points, do I always have a plane? No. 20. Does every ray eventually create an angle? Most likely. 21. Follow the steps below. Once completed answer the questions that follow. 1. Use a straight edge to draw the line segment AB. 2. Place the compass on one end of the line segment. 3. Set the compass length to any length longer than halfway down the line. 4. Without changing the compass width, draw an arc on each side of the line. 5. Again without changing the compass width, place the compass point on the other end of the line. Draw an arc on each side of the line so that the arcs cross the first two. 6. Using a straightedge, draw a line between the points where the arcs intersect. Where the line crosses segment AB is the midpoint of the line, and the line is perpendicular to the segment AB, so our new line is the perpendicular bisector of AB. Use your protractor and ruler to check your work. Did it work? If you follow these steps, will you ever get a line that is not the perpendicular bisector? Explain. No it will always be perpendicular it meets at a 90 degree angle. 22. Follow the steps below. Once completed answer the questions that follow. 1. Using a straight edge, draw acute angle ABC. 2. Place the point of the compass on the Angle's vertex. 3. Use the compass to draw an arc across each leg of the angle. 4. Keeping the compass at the same measurement, place the compass on the point where the arc crosses a leg and draw an arc inside the angle. 5. Without changing the compass setting repeat for the other leg so that the two interior arcs cross. 6. Using a straightedge, draw a line from the vertex to the point where the arcs cross The line bisects the angle. You can use your protractor to check your work. Try these steps again with an obtuse angle. Are there any differences? No diffrences. Consider (and answer) these questions about inductive and deductive reasoning. 1. How many math teachers have you had in the last three years? Identify something that they all have in common. (Please don't say that they all teach math--we know that already). Based on your observations, what do you conclude about all math teachers? What kind of reasoning is that? I've had two math teachers in the last three years. The things they have in common are: knowing how to dress, liking the same music, being fun and always making me laugh, enjoying their job, always taking extra time out to help me with my work. Well I don't really know all I know is I liked my pervious math teachers. I'd say this is deductive reasoning. 2. Name a food that you have never tried, but think that you would not like. Why wouldn't you like it? What kind of reasoning are you using here? Think the reasoning through and explain it fully. Mussels, because they look slimy and it's in a shell. I'm using inductive reasoning here. 3. When you use inductive reasoning, you generalize from specific examples and discover some kind of rule. Give an example of a generalization. (You must give me both the specific example you are generalizing from and the generalization.) A baseball player every baseball player is going to talk about their team and which leagues they like. 4. When you use deductive reasoning, you may know something general, and use that information to predict something about something more concrete. Give an example of a prediction based on a generalization. You must state both the generalization and the prediction you draw from it. An artist, an artist will always gaze around and come up with different images they can paint. 5. There are two barbers in a town. Barber A has a clean shop and a nice hair-cut. Barber B has a dirty shop and a horrible hair-cut. If you want to get a good haircut, which barber should you go to? Explain why. Barber A, because why would I want a horrible hair cut? For questions 6-10, use the information below to make a chart and fill in what you can determine or guess about each person to find the answers. You may not be able to determine the answers for certain, but there is enough information to give a reasoned answer to every question. Two boys, Edward and Frank, and two girls, Rachel and Sally, all come from a different country: England, France, Russia, or Spain. They each have a favorite sport they play: tennis, golf, cricket, or baseball. (Click on each link to learn a little about each sport so you can make appropriate logical choices.) Edward's favorite sport requires a bat. Sally wrote a letter to Rachel, but it was written in Russian and Rachel could not read it. The person who plays golf played last week at a course in Costa Ballena. The person from France received a racquet for her birthday. Last week the person from Russia hit a home-run. Frank does not like to run. 6. What sport does Rachel play? Why would you guess this? Tennis, because she received a racquet for her birthday. 7. What country do you think Edward is from? Why do you think this? England, because Frank is from Spain 8. Sally probably plays baseball. Why would I say this? Because she's from Russia and it says she hit a homerun. 9. Which of the four people plays golf? Why do you guess this? Frank, because he does not like to run. 10. What country is Frank from? Why do you think this? Spain, because people from Spain like golf. For questions 11-15, use the information you learned in Lesson 1 to help answer the questions. 11. Will every ray contain a line segment? Why? What kind of reasoning is this? Yes every ray will contain a line segment, this is inductive reasoning. 12. If I have two lines, will they cross? Explain fully. What postulate helps me answer this question? No. 13. If I have a line segment, and another point not on the line segment, how many different geometric elements from Lesson 1 might I have? What kind of reasoning did you use? (This question is worth 2 points.) 3, deductive reasoning. 14. I have three parallel lines, so I know I have a plane. What definitions or postulates from Lesson 1 would I use to prove that? Use at least three definitions, and explain how they work together to prove that there is a plane. (This question is worth 3 points.) Could you explain this a little more to me, lol I'm having a hard time with both 14 & 15. 15. If I have an angle, I know I have a plane. What definitions or postulates from Lesson 1 would I use to prove that? Use at least three definitions, and explain how they work together to prove that there is a plane. (This question is worth 3 points.) 1. What is p? Something is a dog 2. What is q? Something is a mammal. 3. "If something is not a dog, then it is not a mammal" is the: If it's a mammal. Then it is a dog, a dog is a mammal. 4. ~q => ~p for this statement is: On 5 through 7 your complex statement is "If x2>10, then x>0." 5. "If x > 0, then x^2 > 10" is the: A. Converse B. Counterexample C. Contrapositive D. Counterpostive E. Counterintuitive F. Contrary to popular belief 6. "If x is not > 0, then x^2 is not > 10" is the: A. Converse B. Counterexample C. Contrapostive D. Counterpostive E. Counterintuitive F. Counter on a web page 7. "x = - 4" would be an example of a A. Converse B. Counterexample C. Contrapositive D. Counterintuition E. Counterpositive F. Counter On 8 though 10, the complex statement is "Cars can take you everywhere." 8. "If it is everywhere, then a car can take you" is the: A. Converse B. Counterexample C. Counterpositive D. Counterintuition E. Counterpositive F. Counter 9. "If it is not everywhere, then a car cannot take you" is the: A. Converse B. Counterrunner C. Counterexample D. Conterpositive E. Counterpositive F. Counter 10. "A car can't take you to the moon" would be the: A car can't take you to the moon. For problems 11 through 12, your complex statement is "Small pinpricks of light in the night sky are stars." 11. The converse of the statement is: \"Small pinpricks of light in the night sky are stars\" 12. "Small pinpricks of light in the night sky might be satellites" is a(n) For problems 13 through 14 your complex statement is "Baseball players are athletes." 13. Which of the following is accurate? Explain your reasoning for choosing your response. A. The inverse of the statement is "If someone is a baseball player then someone is an athlete." -( A is right because if some plays baseball they're an athlete. ) B. The statement is "If someone is an athlete, then they are a baseball player." C.The statement can never be true. D. Baseball players all have great teeth and gums. E. The inverse of the statement is not true. F. The converse is: "Joey is a baseball player, and he is not an athlete." 14. What is q? A. Someone is an athlete B. Someone is a baseball player C. All baseball players are athletes. D. All athletes are baseball players. E. Baseball payer F. Athlete For problems 15 through 20, create Venn Diagrams to help you solve the problems. These are not easy diagrams, take your time and think through this carefully. 15. 500 students are enrolled in at least two of these three classes: Math, English, and History. 170 are enrolled in both Math and English, 150 are enrolled in both History and English, and 300 are enrolled in Math and History. How many of the 500 students are enrolled in all three? 1. 170 - 60 = 110 students taking Math and English only. 2. 150 - 60 = 90 students taking History & English only. 3. 300 - 60 = 240 students taking Math & History only. All three subjects equal 60 students and the total number of students is 500. Hints on 15 (highlight the following paragraph with your mouse to see them, they are in the form of questions you'll need to answer): You aren't meant to find out how many students are in the individual courses. How many students are you supposed to have counted? How many wound up being counted? What does the overage mean? How many times too many was a student counted if he was in all three classes? 16. 30 people are having lunch at my house. 16 of them want salads, 16 of them prefer pasta, and 11 of them want steak. 5 say they want to have both salad and steak, and of these, 3 want pasta as well. 5 want only steak, and 8 want only pasta. How many people want salad only? 18 people want salad only. Make a Venn Diagram from the following information to answer questions 17 through 20: 25 students played soccer 4 boys played soccer and baseball 3 girls played soccer and baseball 10 boys played baseball 4 girls played baseball 9 students played tennis 3 boys played soccer and tennis 3 girls played soccer and tennis 3 boys played baseball and tennis 1 girl played baseball and tennis 1 boy played all three sports 1 girl played all three sports Hints on the diagram (highlight the following paragraph with your mouse to see them): 17. How many students played soccer, but not baseball or tennis? Notice that the counts don't make sense as they are, because they're all inclusive. The soccer count includes every who plays soccer, even the students in the soccer and baseball, soccer and tennis, and the all three sport counts. The count for soccer and baseball includes the students who play all three sports. So you'll need to correct from the inside outward...first subtract the boy and girl who play all three sports from all the other counts, then subtract the dual-sport counts from the single sport counts. Put another way, this is like the gecko problem--the entire soccer circle including the soccer and baseball students and the soccer and tennis students and the students who play soccer and baseball and tennis, will add up to 25. A. 4 B. 25 C. 12 D. 6 E. 14 F. 9 18. How many students played soccer and baseball, but not tennis? 25. 19. How many students played just one of the three sports? 13. 20. How many girls played only baseball? 4. Problem A. Robin and John enter an archery contest. Robin is the better archer and wants to win, but not completely destroy John. The target is below. The diameter of the black circle is 4 inches. The radius of the white middle circle is 6 inches and the Radius of the red circle is 9 inches. 1. Robin knows that if he hits the white part of the target, he just slightly win, therefore not embarrassing John. What is the probability that Robin will hit the white part of the target (not the red and not the bull's-eye)? Show your work. 3.14(2)2= 12.56 3.14(6)2= 113.04 3.14(9)2= 254.34 2. After Robin shoots into the white area, John knows he cannot win, so he decides to set his sites on second place. To get second place, he needs to hit either the white circle or the bull's-eye. What is the probability that he can do that? Show your work. 3.14(2)= 12.56 , 3.14(6)2= 113.04 , 3.14(9)2= 254.34 Problem B. The following is a diagram of a shuffleboard table The table is 30 inches wide and 180 inches long. The width of the 2 and 3 rectangles is 12 inches. The sides of the 4 rectangle are 15 inches and the top and bottom are 12 inches. The 1 rectangle is twice as wide as the 2 3. What are the dimensions of the 1 rectangle? 4. What are the measurements of one of the 5 rectangles? 5. You are playing shuffleboard in P.E. class. To score points, your disc must land in a box on the other side of the table and you are awarded the number of points that is in that box. What is the probability of scoring 1 point? 6. What is the probability of scoring 2 points? 7. What is the probability of scoring 3 points? 8. What is the probability of scoring 4 point? 9. What is the probability of scoring 5 point? 10. What is the probability of scoring at all (any number of points)