{"id":46,"date":"2024-05-24T10:41:56","date_gmt":"2024-05-24T10:41:56","guid":{"rendered":"http:\/\/localhost\/teachingsecondarymathematics\/?post_type=content&p=46"},"modified":"2024-08-07T10:34:13","modified_gmt":"2024-08-07T10:34:13","slug":"chapter-1-introduction","status":"publish","type":"content","link":"https:\/\/staging.routledgelearning.com\/teachingsecondarymathematics\/content\/chapter-1-introduction\/","title":{"rendered":"Chapter 1 – Introduction"},"content":{"rendered":"\n
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\n\tHome<\/a>\n<\/span>\/<\/span>\n\tResources<\/a>\n<\/span>\/<\/span>\n\tChapter 1 \u2013 Introduction\n<\/span><\/div><\/div>\n\n\n\n
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Chapter 1 – Introduction<\/h1>\n\n\n

The effective teacher of mathematics must be devoted to the profession. A teacher must create a stimulating atmosphere conducive to learning for all students. Whether in mathematics or in some other area, the teacher must possess the desire, passion, and patience to facilitate the learning of others. An effective teacher of mathematics wants to help erase the fear and anxiety that mathematics represents to so many students. A true teacher is always willing to learn new methods and strategies for introducing concepts to students.<\/p>\n<\/div>\n\n\n\n

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Exercises <\/h2>\n\n\n\n
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Exercise 1.1<\/h3>\n\n\n\n
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  1. If the first two terms of a Lucas sequence are 3 and 7, will the same sum be derived from a sequence starting with a 7, followed by a 3? Why or why not? Develop a series of questions you would use to assist a student in learning this result.<\/li>\n\n\n\n
  2. Determine how the answer to any Lucas sequence problem can be found quickly. Hint: Trachtenberg.<\/li>\n\n\n\n
  3. Locate some specific applications in nature of the Fibonacci sequence.<\/li>\n\n\n\n
  4. Is it true that the sum of the terms in a 20-term Lucas sequence can be found by finding the product of 11 times the 14th term? Would 22 times the 7th term work? Would 22 times the 14th term work? Is there any quick way to find the sum of the first 20 terms of a Lucas sequence?<\/li>\n\n\n\n
  5. Is there any set of numbers that could not be used as terms for a Lucas sequence? If so, provide an example.<\/li>\n<\/ol>\n<\/div>\n\n\n\n
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    Exercise 1.2<\/h3>\n\n\n\n
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    1. If the length of one side of a rectangle is doubled and the other tripled, what is the impact on the area of the rectangle? Does it matter if the roles are reversed as to which side is doubled and which is tripled? Why or why not?<\/li>\n\n\n\n
    2. If the base of a triangle is doubled and the height is quadrupled, what factor is the area multiplied by?<\/li>\n\n\n\n
    3. In a triangle, what factors can be used with the base and height as multiples and still keep the area the same? State a generalization for this situation.<\/li>\n\n\n\n
    4. Given a trapezoid with height h<\/em>, upper base b1<\/sub><\/em>, and lower base b2<\/sub><\/em>, what is the impact on the area if only the height is doubled? What is the impact on the area if only the length of b2<\/sub><\/em> is halved? Write a generalization about changing only one of the dimensions of a trapezoid. Don\u2019t forget the slant height.<\/li>\n<\/ol>\n<\/div>\n\n\n\n
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      Exercise 1.3<\/h3>\n\n\n\n
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      1. Some schools do not permit students to go outside the classroom to do an activity like the speed trap described earlier. How could the activity be completed inside the classroom?<\/li>\n\n\n\n
      2. Describe at least two other variations of the speed trap activity that students could perform.<\/li>\n\n\n\n
      3. Suppose the speed trap activity was being done at night. How could one person accomplish the task?<\/li>\n\n\n\n
      4. Describe at least two different applications you could use to entice students into establishing a laboratory or field experience that involves the mathematics they cover in the secondary school curriculum.<\/li>\n<\/ol>\n<\/div>\n\n\n\n
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        Exercise 1.4<\/h3>\n\n\n\n
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        1. Think of your secondary mathematics teachers. Hopefully at least one of them was a motivating factor in leading you to become a teacher of mathematics. List the characteristics of that teacher that helped you learn more about mathematics.<\/li>\n\n\n\n
        2. Do you think you could\/should become a clone (as far as teaching is concerned) of the teacher you described in part 1 of this exercise? Why or why not?<\/li>\n<\/ol>\n<\/div>\n\n\n\n
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          Exercise 1.5<\/h3>\n\n\n\n
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          1. Use a source like The World of Mathematics<\/em> (Newman, 1956), the VNR Concise Encyclopedia of Mathematics<\/em> (Gellert, Kustner, Hellwich, and Kastner, 1977), History of Mathematics<\/em> (Eve, 1967), Mathematics From the Birth of Numbers<\/em> (Gullberg, 1997) or the MacTutor History of Mathematics Index <\/em>(https:\/\/mathshistory.st-andrews.ac.uk\/<\/a>) to investigate a mathematical topic found in the secondary setting. Learn a new way of working with the topic. Present your conclusions in written form, giving appropriate bibliographic credit.<\/li>\n<\/ol>\n<\/div>\n\n\n\n
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            Exercise 1.6<\/h3>\n\n\n\n
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            1. Describe mathematical applications or extensions related to a hobby or personal interest you have.<\/li>\n\n\n\n
            2. This text briefly shows examples from the worlds of racing and space, depicting how mathematics and applications grew out of need. Select some area that interests you and develop a written outline showing at least four areas where mathematics was developed or applied that you think could be used to appeal to secondary students\u2019 curiosity.<\/li>\n\n\n\n
            3. Select something from the history of mathematics and describe how mathematics was created or refined to meet a need. Your description should be in a form that could be used to attract the attention of a secondary student.<\/li>\n\n\n\n
            4. The importance of mathematics in space exploration was even more evident in 1999. NASA sent a $125 million spacecraft to observe the climate of Mars. After a nine-month journey, the craft disappeared. NASA scientists were embarrassed to discover that the destruction of the craft was probably due to a failure to convert English units of measurement to metric ones. The Mars Climate Orbit flew too close to the planet and was destroyed. This was a costly mathematical miscalculation. Yet students can use this disaster to see that the mathematics that they learn and do in school can have serious ramifications in the real world. Explain how you could use this information in your mathematics classroom.<\/li>\n<\/ol>\n<\/div>\n\n\n\n
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              Exercise 1.7<\/h3>\n\n\n\n
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              1. Do the \u201cPoint D, which lies on one side of a triangle, is equidistant from all three vertices, A, B, and C\u201d problem a fourth way.<\/li>\n\n\n\n
              2. One president of the United States came up with a rather unique proof of the Pythagorean theorem. Who was the president? Duplicate the proof he did.<\/li>\n\n\n\n
              3. Show at least three different \u201cpicture proofs\u201d of the Pythagorean theorem.<\/li>\n<\/ol>\n<\/div>\n\n\n\n
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                Exercise 1.8<\/h3>\n\n\n\n
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                1. Make a set of cards like those described for the values less than 16, but make your set for all values less than 50. <\/li>\n\n\n\n
                2. Make a set of \u201chole\u201d cards for values less than 16 and demonstrate the trick to a middle-school class. Write a summary of the class reaction. <\/li>\n\n\n\n
                3. Select a topic or area like the one demonstrated with base 2, and then find at least three different places where it can be used in the secondary mathematics classroom.<\/li>\n<\/ol>\n<\/div>\n\n\n\n
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                  Exercises 1.9<\/h3>\n\n\n\n
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                  1. Use a dynamic geometry application or function plotting software to create a model of (x<\/em> + y<\/em>)2<\/sup> = x<\/em>2<\/sup> + 2xy<\/em> + y<\/em>2<\/sup>.<\/li>\n\n\n\n
                  2. Cut a right, circular-based styrofoam cone with four planes:
                    1. parallel to the axis<\/li><\/ol>
                      1. perpendicular to the axis<\/li><\/ol>
                        1. oblique to the axis<\/li><\/ol>\n
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                          1. parallel to the slant side.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n\n\n\n

                            Use this model to show the different conic sections.<\/p>\n\n\n\n