assignment
new doc 7 Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Chapter 3 questions Q1. Drag different vertices of the kite. Consider properties of sides, angles, and diagonals. Create a definition of a kite that encompasses all examples of kites. Q2. Reflect on your thinking processes. What properties did you consider important or not important? Did you create definitions that did not work? How did you know if a definition “worked” or not? Q3. Describe benefits and drawbacks of allowing students to interact with a constructed figure in a DGE to generate their own definitions versus a teacher providing a formal definition to students. Q4. When students are generating their own definitions, describe how a teacher can bring these different ideas together so the class is eventually working from a single definition. Q5. Determine if the five definitions of a square just listed are all equivalent. Explain. Q6. Based on the stated criteria for mathematical definitions, which of the five definitions do you believe is the best? Explain. Q7. Find or create a sixth definition of a square that is different from those that are listed. Explain why your definition is equivalent. Q8. Consider the five definitions of a square listed earlier. If you wanted to select one of the five to present to students, which would you select? Why? Q9. Of the five definitions of a square, which do you think would be most difficult for students to understand? Explain. Q10. Consider the following definition of a quadrilateral: “A quadrilateral is a four-sided polygon.” What do students need to understand in order to make sense of this definition? Create three examples and three nonexamples of quadrilaterals, based on this definition, with or without a DGE. Q11. Select one of the two definitions and describe how you would construct a square using this definition. Create your construction in a DGE. Q12. A definition does not explicitly state every property of the figure it is defining. Explore the construction of your square to describe additional properties (e.g., diagonals, symmetry, measures of angles). On a separate piece of paper, create a table like the one shown in Table 3.1 and record your properties. Q13. Analyze the two definitions just presented. Which properties of a square are highlighted? Is one of the definitions easier to use for constructing? Q15. Is a parallelogram a trapezoid using definition 1? What about definition 2? Explain. Q16. Which definition of a trapezoid was used in the Quads.gsp sketch? Explain how you determined your answer. Q17. Which definition of trapezoid do you prefer? Why? Q18. What are the benefits for teachers and/or students for using the second definition of a trapezoid rather than the first? What are the drawbacks? Q22. Is there another way you could create a hierarchical classification of quadrilaterals? Create a second method. Explain your decisions. Q23. Use the diagram you created in Q21 of a hierarchical classification system for quadrilaterals to determine whether the following statements are sometimes true, always true, or never true. A. A rectangle is a parallelogram. B. A square is a rectangle. C. A rectangle is a square. D. A cyclic quadrilateral is a parallelogram. E. A square is a trapezoid. F. A rhombus is a parallelogram. G. A square is a rhombus. H. A parallelogram is a rhombus. Q24. Which van Hiele level of thinking is needed for a student to make a list of properties of different quadrilaterals using a DGE sketch? Which van Hiele level of thinking is needed for a student to classify the quadrilaterals? Q25. How might students’ uses of a DGE to explore quadrilaterals and their properties influence their geometric thinking? Q26. Students often have difficulty determining whether statements, such as those in Q23, are always true, sometimes true, or never true. Use the van Hiele levels to provide an explanation for why this may be a difficult task for students. Describe how a teacher could assist students in answering questions such as these. Q27. Consider the topic “properties of quadrilaterals.” Create one task appropriate for a student who is at each of the five different van Hiele levels. Q33. Do you think the two brothers should use this method? Explain why. Q34. If the island was of a particular shape, would you suggest using this method? Explain. Q35. Use a DGE to determine another method for dividing the land fairly between the two brothers. Be prepared to explain why your method works. Q36. Suppose the brothers want to make sure they each have land that is on the water. Devise a method that will satisfy each brother. Be prepared to explain your solution. Q37. While considering various methods, the younger brother noticed that if you create the diagonals of the quadrilateral, measure the areas of each of the triangular regions created, and then multiply the area measures of the two nonadjacent triangles, the products are equal. Is this relationship always true? Explain. Q38. Describe the mathematical goals this task addresses. Q39. If a teacher posed this problem to students in a classroom, it is likely that students would develop more than one correct solution. Anticipate the different solutions students might develop. Describe how each of these solutions is related to the mathematical goals. Q40. For each of the solutions you anticipated, which would you have students share with the class? If you would share more than one solution, in what order would you have students present their solutions? Q41. What do you notice about triangle EFG? Create at least two different interesting conjectures based on the diagram. Q42. Select one of the two conjectures and determine whether it is never true, sometimes true, or always true. Q43. To determine whether a conjecture is always true, it is likely that you constructed a mathematical argument or proof. Describe how your work with the technology was related to the construction of your argument. Q44. Describe some ways that teachers can assist students in using the technology to gather data that can assist them in constructing formal mathematical arguments. Q45. Create a new problem that extends the given situation and provide a sample solution. Recall that problem-posing strategies were presented in Chapter 2. Q46. Consider the quadrilateral problem presented at the beginning of Section 5. Describe the dragging strategies that you used and the purpose for using that particular type of dragging. Q47. Were certain dragging strategies more or less useful in solving the problem? Explain. Q50. What type of quadrilateral is EFGH? How did you arrive at your conclusion? Q51. Determine the “given(s)” and the statement to “prove” and construct a formal argument to validate the claim you made in Q50. Q52. What prerequisite knowledge would students need to know in order to construct a proof such as the one that you created in Q51? Q53. Create four additional tasks that you could pose to students that are extensions or modifications of this problem.