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MAE 5868 E-FOLIO

MAE 5868 E-FOLIO. FLORIDA STATE UNIVERSITY SPRING 2006 MARSHA COPELAND. INTRODUCTION.

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MAE 5868 E-FOLIO

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  1. MAE 5868 E-FOLIO FLORIDA STATE UNIVERSITY SPRING 2006 MARSHA COPELAND

  2. INTRODUCTION This e-folio is a compilation of solutions to various problems. These solutions were discovered via three mediums: Geo-Sketchpad, Excel and Graphing Calculator. Discussions of the relevance of these problems and the tools used to solve them to educational standards in math and technology are also included.

  3. CONTENTS : Click on the link to access information: ISTE INFO STANDARD 1 STANDARD 2 REFERENCES STANDARD 3 STANDARD 4

  4. : ISTE INFORMATION • The International Society for Technology in Education (ISTE) is a nonprofit organization whose mission is to “provide leadership and service to improve teaching and learning by advancing the effective use of technology in education.” Great diversity is utilized to accomplish this mission. The organization is involved in professional development workshops, presenting innovative books and programs, research, and conducting forums and symposia to promote the use of effective technology in education.

  5. : ISTE INFORMATION • ISTE provides its members (>85,000 teachers, worldwide) with information, networking opportunities, and guidance in the use of technology in their programs. Forums, such as NECC, offers the opportunity to learn, exchange and survey the field of educational technology.

  6. : ISTE INFORMATION • ISTE supported Special Interest Groups (SIGs) allow members to “share best practices worldwide, sponsor workshops, review articles and programs and advance knowledge in their particular areas of interest.”

  7. : ISTE INFORMATION • ISTE is the home of NETS ( National Educational Technology Standards) which have been adopted by many districts, states and colleges. These can be accessed at http://cnets.iste.org/teachers/t_book.html BACK TO CONTENTS

  8. : STANDARD 1 National Educational Technology Standard 3: Students use technology tools to enhance learning, increase productivity and promote creativity. Students use productivity tools to collaborate in constructing technology-enhanced models, prepare publications and produce other creative works. LINK TO ACTIVITIES LINK TO REFLECTIONS BACK TO CONTENTS

  9. ACTIVITY 1 One of the Pythagorean Theorem typically uses squares constructed on each side of a right triangle. The area of the square constructed on the hypotenuse is equal to the sum of the squares constructed on each of the legs of the triangle. (50 Points). Instead of using squares, can we use equilateral triangles? How about regular hexagons? Would these or any other figures give the same results? Plan: I will construct a right triangle, then using the rotate and dilate tools, will attempt to construct a tessellation of equilateral triangles along each side to form a larger equilateral triangle on each side. Solution: See diagrams with measurements below. These are constructed in Geo Sketchpad and show that the areas of the shapes along each side of the right triangle’s legs sum to the area of the shape that shares a side with the hypotenuse.

  10. Analysis: Teaching view: Great activity to reinforce that there are many ways to solve a problem. Students can see that the Pythagorean theorem works for any shape Learning view: Students will have fun manipulating the shapes and sizes of the triangles, and verifying that the sum of the squares of the legs of a right triangle equal the square of the hypotenuse. As they manipulate the figures, they will learn the theorem. Reflection: Interesting problem…easy to construct, easy to see the outcome.

  11. ACTIVITY 2 A prime number is a positive integer, which is greater than one and evenly divisible only by itself and one.  For example, 2,3,5,7,11,13,17 are all prime numbers.  For centuries, prime numbers have fascinated both professional and amateur mathematicians. In 1742 C. Goldbach wrote a letter to the eminent Euler, stating therein the celebrated "Goldbach conjecture," essentially that every even integer exceeding 2 is a sum of two primes. For example: 2+2=4 3+5=8 3+7 =10 5+5 =10 Euler replied that Goldbach's conjecture did indeed seem plausible.  However, neither Goldbach nor Euler was able to prove it.  According to Hardy (1999, p. 19), "It is comparatively easy to make clever guesses; indeed there are theorems, like 'Goldbach's Theorem,' which have never been proved and which any fool could have guessed." Faber and Faber offered a $1,000,000 prize to anyone who proved Goldbach's conjecture between March 20, 2000 and March 20, 2002, but the prize went unclaimed and the conjecture remains open.

  12. Finding prime numbers with EXCEL 1-Your assignment is to find ALL of the prime numbers between 1 & 50 using the Excel functions. You can use any function or create your own formula. We want to see your process. If students were to follow your completed steps then they should be able to generalize to find the prime numbers between 1 & n, where n is a positive Integer. (15 points) 2-As a follow up after you have found the primes, try to confirm or disconfirm the Goldbach conjecture for the numbers 1- 30. Basically find the primes (p1, q1) such that p1 + q1 = an even numbers between 1 & 30 (find for ALL even numbers). Sometimes you will find more than one pairs of primes whose sum makes up an even number (ex: 5+5=10; 3+7=10), find all possible alternatives. (30 points) PLAN: I will use a form of sieve to establish the primes, 2-30. then use excel to add each prime in turn to all of the primes established with the sieve. Once this is done, I will find sums of primes that result in all even numbers 4-30, making sure to notice all combinations and pairs. SOLUTION: First of all, list all integers from 2-50 in the first column. The number 1 is not listed because the smallest prime is 2. Then divide each integer in the list by 2. Those numbers not evenly divisible by 2 (except 2) are incorporated in a new list. Now each element in the new list is divided by 3, and all numbers except 3 not evenly divisible by 3 are used to make another new list. One need not divide by 4, since any number divisible by 4 is also divisible by 2 and would have been eliminated as a prime then. Now divide each element in the newest list by 5, follow this procedure for 7 also. Six, eight and nine need not be “tested” for the same reason there was no need to test 4. After checking for divisibility by 7, the final list includes all primes from 2-50.

  13. ANALYSIS: TEACHING VIEW: This is a good activity to allow students to get an understanding of prime numbers and the apparent “sporadity” of their occurrence. This activity uses technology to mimic the Sieve of Eratosthenes to find primes between 1 and n. Once the primes are found, then students would have to solve the problem of determining all the pairs of primes that add to even numbers…an activity that promotes problem solving. LEARNING VIEW: Students will find this activity engaging and interesting, providing them with a historical activity that can be related to technology and with a situation that makes it necessary for them to plan a strategy and problem solve to find a way to efficiently find all the pairs that add to an even number. REFLECTION: I enjoyed this activity. I am not particularly proficient with EXCEL, but was able to use this activity to become more adept at using it. If I used this with my students, I would be sure to help them with some of the function tools in the program and allow them to work in groups to brain-storm the problem and how best to set up the program in order to solve the problem. LINKACTIVITY 2 EFOLIO.doc TO DATA

  14. ACTIVITY 3 Using TI-83 Find the sum sequence from k=15 to k=19 (45 points)  (k2 +3k2/3 –5) PLAN: A sum sequence is as follows 1,2,3,6,12,24…so each element is the cumulative sum of those before it. Finding the sum sequence of (k^2 + 3k^(2/3) –5), for k= 15 to 19 is done in the graphing calculator by first forming a sequence of values generated by substituting the integers 15-19 into the expression in order. Once this is done, the cumulative sum function can be applied to the sequence to obtain the sum sequence

  15. SOLUTION: THE ENTRIES TO THE LEFT SHOW THE TERMS OF THE SEQUENCE FOTHE ENTRIES TO THE LEFT SHOW THE TERMS OF THE SEQUENCE FORMED BY SUBSTITUTING THE VALUES 15-19 IN FOR K IN THE EQUATION. RMED BY SUBSTITUTING THE VALUES 15-19 IN FOR K IN THE EQUATION.

  16. Now, having createdthe above sequence {238.246606, 270.0488126, 303.8344671, 339.6048564, 377.3611021}, this will be used to generate the sum sequence by adding the terms prior to get the next term in the new sequence: As can be seen from the screens above, the sum sequence generated is; {238.246606,508.2954186, 812.1298857, 1151.734742, 1529.095844}

  17. ANALYSIS: TEACHING VIEW: From a teaching standpoint, it is important to teach students to use the graphing calculator. Even more important is teaching them to use the manual that comes with it to accomplish the calculations they wish to do. I think that is the main focus of this activity and that is the way I would approach it with my students. I would give them the manual and the problem and ask them to attempt to learn to use the calculator by reading the manual. LEARNING VIEW: Students who become self-sufficient at learning how to use technology via the manuals and help guides in on the web will be better able to deal with anything they encounter in the careers they choose. This activity would be challenging, not so much in the calculations, but in finding the procedures and manipulating the calculator to accomplish what it is you wish to do. REFLECTION: It took me a while to figure out how to do this in the calculator, using the list menu and operations for lists. I could have simply used the home screen, accomplished the calculations myself and then summed to get the sum sequence. However, I felt it was important to learn to use the manual and determine the most effective way to accomplish the task. BACK TO STANDARD 1

  18. . REFLECTION ON STANDARD 1 Each of the activities chosen to model possible ways to approach this standard represent each of the three tools used in solving the problems in this class. The Pythagorean Theorem problem was solved via the geo sketchpad program and demonstrates creativity, and model construction that could be used by a student to solve the problem presented. Prime numbers and the Goldbach conjecture was solved using Excel. Students using this tool can, as modeled by the solution, increase productivity by organizing and manipulating the data and numbers via the EXCEL tools. The third problem, Sum Sequence, was accomplished with a graphing calculator. The use of the program in the computer to quickly solve the problem helps students realize the importance of technology to enhance their learning and increase their productivity. BACK TO CONTENTS

  19. . STANDARD 2 National Educational Technology Standard 6: Students use technology resources for solving problems and making informed decisions. Students employ technology in the development of strategies for solving problems in the real world. LINK TO ACTIVITIES LINK TO REFLECTIONS BACK TO CONTENTS

  20. ACTIVITY 1: 8. X-Intercept Transformation: Identify the change in the x-intercepts of a function when you reverse its leading coefficient and constant term.(25 pts) a.       What happens to the x-intercept of y = ax + b when you reverse the leading coefficient, a, and the constant, b? b.      What happens to the x-intercept(s) of y = ax2 + bx +c when you reverse the leading coefficient, a, and the constant, c? What happens to the x-intercepts of a polynomial in the form y = axn + bxn-1 +cxn-2+...+z when you reverse the leading coefficient, a, and the constant, z? PLAN: Working with the linear equation, I plan to graph pairs of equations with the leading coefficients switched, then trace the lines to determine the x-intercept of each, and compare the x-intercepts to determine their relationship to each other. I will also try this with the quadratic equations, then with higher order polynomials. If the trace will not “stop” at exactly y=0, I will use the programming function of the calculator to program in the quadratic formula and find the roots or x-intercepts that way. On higher level polynomials I can use the “solver” mode or the graphing mode and zoom in on the x-intercept until I can get close enough coordinates to make a determination about the x-intercepts.

  21. As can be seen from the graphs of the linear equations above, if the leading coefficients are exchanged, i.e. the values for the slope and the y-intercept are exchanged, the x-intercepts are reciprocals of each other. Remember, the x=intercept results when y = 0. Using the programming capabilities of the graphing calculator, I programmed in the quadratic formula to help find the x-intercepts or roots of various quadratic equations and proceeded to enter in values for a, b, and c using the form of quadratic ax^2 + bx + c = 0, exchanging the leading coefficients a and c and finding the different values for the x intercepts. FOR A=1, C=-2, THE ROOTS ARE {1,-2} FOR A=2, C=1, THE ROOTS ARE {1, -.5} 1 IS THE RECIPROCAL OF 1 AND –2 IS THE RECIPROCAL OF -.5

  22. FOR A=3, AND C=2, THE ROOTS ARE {2, -.33…} FOR A=2, AND A=3, THE ROOTS ARE {.5, -3} 2 IS THE RECIPROCAL OF .5 AND -.33 IS THE RECIPROCAL OF –3. FOR A=2 AND C=-4, THE ROOTS OR X-INTERCEPTS ARE {4, -.5} FOR A = -4, AND C = 2, THE ROOTS OR X-INTERCEPTS ARE {.25, -2} THE RECIPROCAL OF 4 IS .25 AND THE RECIPROCAL OF -.5 IS -2 Judging from the above examples, the resulting x-intercepts are again reciprocals of each other.

  23. c. By graphing and tracing higher degree polynomials, I found that the x-intercepts are always reciprocals of each other when the leading coefficients are exchanged.

  24. ANALYSIS: TEACHING VIEW: Teaching relationships and patterns in math is important. Using this activity helps students realize there are patterns that exist even if they are not at first evident or taught as part of the concept. LEARNING VIEW: Students will be challenged trying to figure out how they can find the x-intercepts for each type of equation. Suggestions of what they might use will be helpful, and allowing them to program the quadratic equation will also be interesting to them..(and they will have to completely understand the arithmetic in the formula). REFLECTION: This was an interesting activity. When I first thought about how to solve the problem, it seemed the graphing function of the calculator would be the most effective. However, the trace does not always give the coordinates you need to determine the exact x-intercept. It was necessary to know the exact values in order to determine exactly what the relationship between the x-intercepts are when you switch the leading coefficients. Therefore, I had to find another way to find the x-intercepts for quadratic and higher level polynomials. Programming in the quadratic formula gave me exact values for the quadratic equations, and this allowed generalization to higher- level equations.

  25. ACTIVITY 2: • 1. Now use Excel to find the sum sequence from k=1 to k=10 (20 points) •  (k2 –7k+5) • 2. Now use Excel to find the sum sequence from k=15 to k=19 (20 points) •  (k2 +3k2/3 –5) • 3. Now use Excel to find the sum sequence from k=25 to k=33 (20 points) • ln x • Reflect on the differences, from a teaching window, between using TI and Excel when solving these. Which is better, IF ANY, to use? Why? LINK TO SPREADSHEET

  26. ACTIVITY 3 Suppose Sammy the swimmer at the tip of pier H wants to swim to the tip of pier I. Pier H is 2km long and pier 1 is 1km long. Since the swim is very long from H to I, Sammy thinks he will need to stip off at the beach to take a break (at point J). Sammy can stip at any point on the beach between the two piers. If Sammy takes the break, where should he sip on the beach if he wants to swim the least distance for the entire trip? What is the shortest distance Sammy can swim for the entire trip (30 points). Plan: I will construct the problem, then manipulate the distance between the piers and for each of these manipulations, drag the point on the beach to determine the shortest distance of his swim.

  27. Analysis: Teaching view: The depth of this problem is not evident at first. Students will have a problem because with the initial reading, there does not seem to be enough information. This is a good problem to help develop higher thinking skills. Learning view: Students will learn to delve deeper into seemingly simple problems. They will learn try the whole spectrum of a problem and avoid jumping to conclusions. Reflection: I enjoyed this problem. It was subtle and required higher level thinking skills. BACK TO STANDARD 2

  28. . REFLECTION ON STANDARD 2 The three activities chosen to model this standard are 1. X-Intercept Transformation, 2. Sum Sequence using EXCEL, and 3. the Swimmer problem. The first was done using a graphing calculator. Students will use the calculator to transform the coefficients in an equation and observe the changes in the x- intercept(s), and use these observations to make an informed decision of the affects of the transformations. The Sum Sequence carried out via EXCEL requires a student to develop an organized method for obtaining the values necessary using this program and is similar to situations in the real world. The swimmer problem solved using Geo-Sketchpad allows a student to manipulate the diagram and use the measurement tools to determine the solution(s) of the problem while realizing the subtlety of the problem. BACK TO CONTENTS

  29. . STANDARD 3 Curriculum and Content Area Standard (Mathematics Standards) 3: Mathematics instructional programs should include attention to geometry and spatial sense so that all students- • analyze characteristics and properties of two-and three-dimensional geometric objects • select and use different representational systems, including coordinate geometry and graph theory • recognize the usefulness of transformations and symmetry in analyzing mathematical situations • use visualization and spatial reasoning to solve problems both within and outside of mathematics LINK TO ACTIVITIES LINK TO REFLECTIONS BACK TO CONTENTS

  30. ACTIVITY 1 1.Tangential Circles: compare the area and circumference of a circle inscribed in a region between two intersecting circles: Two larger circles with equal radii pass through each other’s centers. A smaller circle can be created inside the overlapping region so that it is tangent to the other circles. Compare the area and circumference of the smaller circle to the area and circumference of the larger circle. Using geosketchpad, construct a circle with center A. Now shift-select a point on the circle and point A and construct a circle with the point on circle A as the center of this new circle. This will create a region or area of intersection. Choose a point onthe perimeter of this region on each circle. Construct a segment, and the midpoint of that segment. Use the midpoint as the center of another circle construction with the end of the radius on one of the large circles. Use the measure tool to have the program indicate the length of the radius, and the circumference and the area of one of the large circles and the small, inscribed circle. Shift-select the areas of each circle and use the plot as (x,y) to set up a graph which compares the areas of the circles. Trace this point on the graph as you move the endpoint of one of the radii to change the size of the two circles.

  31. IV. Closing Analysis: Learning window: Working with the geosketchpad to construct and manipulate the circles necessitates higher level thinking about the criteria of the problem and how the spaces relate to each other. It is interesting to “play” with the size of the circles and see how the data traces on the graph. It also becomes very clear not only how the areas relate to each other but how graphs can be used to make these relationships easier to see. Once I collected the data and looked at the graph, it was much clearer why these areas are related the way they are. Teaching window: As a teacher, I am always looking for ways to help students make connections between math concepts. The ability to tie the graph with the measures of the areas, then watch the graph form as the circles are manipulated in a strong way of accomplishing this. Not only will students be made to realize there is a connection, they will be able to visualize just why the relationship works as it does. V. Reflection: Given this type of technology, not only in a geometry classroom, but in an algebra classroom, I would definitely use it to promote the connections between different concepts and areas of mathematics. I enjoyed working with GSP to accomplish this and realized more about the relationship of the areas than I expected to. Once I was familiar with the program, it was relatively easy to construct and use the figures to determine the solution to the problem.

  32. ACTIVITY 2: Triangles in a Trapezoid: Determine the relationship of the areas of the triangles formed by the diagonals of a trapezoid. (15 points) How do the areas of the red triangles relate to each other ( triangles share a side with the legs of the trapezoid). How do the areas of the yellow triangles relate to each other? I. Plan: I will construct the trapezoid with diagonals and measure the areas of the triangles along with the lengths of the bases and the legs and also the area of the trapezoid itself. Comparisons will be done with these measurements in order to determine the relationships between the areas.

  33. Analysis: Teaching view: This would be difficult to use in a classroom unless you restrict it to dragging only the vertex that does not change the ratio of the unequal area triangles. Students will see that the relationships of the triangles remain constant. Learning View: Students will find the construction and manipulation of the shape fun and interesting. It would be difficult for them to see why the ratios of the areas change as they do when certain of the vertices are dragged and do not change when others are dragged.

  34. ACTIVITY 3 1. Varied Slopes and Intercepts: Discover what happens to the graph when you vary the values for m and b in the equation y = mx + b. (25 pts) . How does changing m affect the graph of the line? Now plug in different values for b, keeping m fixed. What does changing b do to the graph of the line? The equation of a line can be written in the form y = mx + b, where m and b are real numbers. Plug in different values for m, keeping b fixed. Make sure to use positive and negative values as well as fractional values, such as    PLAN: Using the graphing calculator I will enter various equations on the “y=” screen, first keeping the b fixed and varying the values for “m”, then keeping the m fixed and varying the values for “b”. Observing the changes in the lines when graphed and using the trace button to ascertain the values of “b” or the y-intercept will give me the information to answer the questions posed.

  35. Holding “b” the same, changing “m” or slope of the line…Positive slopes for this group of graphs: For negative “m” or slope of the line:

  36. Now, keeping “m” or the slope the same and varying “b” or the y-intercept: How does changing m affect the graph of the line? Since “m” represents the slope of the line, as can be seen in the first two sets of graphs, when “m” varies, the direction and “steepness” of the line varies. The larger the absolute value of “m”, the steeper the slope of the line. If the value of “m” is positive, the line slopes up from left to right, if it is negative, the line slopes down from left to right. What does changing b do to the graph of the line? The variable “b” in the equation of a line y= mx + b, represents the y-intercept of the line or where the graph of the line intersects the y-axis. Varying the values for “b” results in changing the point where the line crosses the y-axis, but does not change the slope of the line BACK TO STANDARD 3

  37. REFLECTION ON STANDARD 3 . Use of the Geo-Sketchpad lends itself well to incorporating standard 6 of the curriculum and content area standards for mathematics into the classroom. Two of the activities presented used this program. Students working to solve the Tangential Circles problem will be able to use visualization to solve the problem via the sketchpad. It is also possible to graph the data collected to a coordinate plane and represent the relationship between the areas of the circles using the graph that results. The Triangles in a Trapezoid problem allows students working with the sketchpad to visualize the shapes and how transforming the outer trapezoid affects the inner triangles. The third activity was the Varied Slopes and Intercept problem solved via the graphing calculator. Use of this tool for this problem enables students to represent the linear equations in a system other than pen and pencil and helps them recognize the usefulness of transformations in analyzing mathematical situations. BACK TO CONTENTS

  38. . STANDARD 4 Curriculum and Content Area Standard (Mathematics Standard) 5: Mathematics instructional programs should include attention to data analysis, statistics and probability so that all students- • pose questions and collect, organize and represent data to answer those questions • develop and evaluate inferences, predictions, and arguments that are based on data • interpret data using methods of exploratory data analysis LINK TO ACTIVITIES LINK TO REFLECTION BACK TO CONTENTS

  39. ACTIVITY 1 2. Transformations of Data: Explore what changes in the mean, median and mode as you add, subtract, multiply, and divide elements in a data set. (25 points) • Use spreadsheets to compute the mean, median, and mode for any set of data. • What happens to the mean, median and mode if you add 0.7 to each element of the data? (5 pts) • What happens to these values if you subtract 0.5 from each element of the data? (5 pts) • If the mean, median, and mode of the data set a, b, c, d are 16, 11 and 4, what are the mean, median, and mode of the data set a + x, b + x, c + x, d + x, where x is any real number? (5 pts) • Describe what happens to the mean, median and mode of a data set when you multiply each data point by a number k. (10 pts) LINK TO SPREADSHEET

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