Q1. If you are given any three points in the plane, will they form vertices of a triangle? Explain
your answer and feel free to use your DGE.
Q2. If you are given any three segments, can they be a
anged to form a triangle? Explain your
answer and feel free to use your DGE to explore.
Q3. Is the second statement of the theorem equivalent to the first? Provide a justification for your
esponse.
Q4. Which statement of the theorem do you prefer? Why?
Q5. If you were teaching, which statement would you want to provide to students? Explain.
Q6. Open up the sketch Triangle_Inequality.gsp and consider the questions that were included on
the worksheet, which is shown in Figure 2.1. Consider the six sets of side lengths that were
provided to students on the handout. Use the sketch to determine which ones will work and
which ones will not.
Q7. In what ways could this sketch and these questions be helpful to eighth-grade students who
are learning the triangle inequality theorem? How might this activity confuse students?
Q8. Develop a rationale for why the teacher selected the particular examples that he did. Are
there any that you would change? Explain.
Q9. Are there any sets of side lengths that you think students will have difficulty with? What
explanations and conjectures do you anticipate students will create?
Q10. The teacher made a decision to open and display the sketch to the class, explain some of its
components, and work through the first example on the worksheet. Do you agree with this
decision? In general, when do you think a teacher would want to work through an example with
a class and when do you think he or she would not want to do so?
Q11. During this episode, there is a 30-second pause (1:06–1:36). What do you think the teacher
and students are doing during this pause? Do you believe it was important for the teacher to
pause at this time?
Q12. When the teacher dragged the endpoints of the sides of the triangle and asked “Do you
think they’re going to meet?” many students said no. Was this surprising? Why do you think
they responded with this answer?
Q13. What else did you notice about what the teacher said or did during the beginning of this
lesson that you found interesting?
Q14. The students are working in pairs and sharing a laptop. What are the advantages and
disadvantages of providing these students with just one rather than two laptop computers?
Q15. What roles or jobs related to solving the problem has each student assumed? Are they
working effectively as a pair?
Q16. In response to the question, “Why was it impossible to construct a triangle with some of the
given lengths?” (5:45) David stated, “Because one side was too long or too short.” Is his answer
co
ect? If you were the teacher, how would you respond?
Q17. The conjecture they develop about a relationship among the lengths of the sides of the
triangle is “if there’s a long length, that’s your bottom, and so the two lengths on the sides, they
got to be very close to each other” (see time 6:29). What is your interpretation of what David
said? What question could you pose to David to assist him in making connections between this
statement and the triangle inequality theorem?
Q18. How did the students use the sketch? Was it helpful to them? Explain.
Q19. Measure the length of each side of quadrilateral ABCD. Is there a relationship among the
lengths of the sides of a quadrilateral? If yes, state the relationship. If no, explain.
Q20. Why or how are you convinced that the answer is yes or no?
Q21. Consider your investigation of the lengths of the sides of a quadrilateral. If you stated that
there was a relationship among the lengths of the sides, how did you convince yourself this was
true? How would you prove it is always true? If you stated that there was not a relationship, what
did you do to a
ive at this conclusion?
Q22. How can you assist students in understanding the difference between evidence and proof
when they are using DGEs? What tasks or questions could you pose?
Q23. Suppose a student used a DGE to create three non-collinear points, A, B, and C, and
connected the points with segments to create triangle ABC. Next, the student constructed the
isector of ∠ABC and stated, “For any triangle the angle bisector passes through the midpoint of
the opposite side.” Is this statement co
ect?
Q24. What do you think the student did with the DGE that led to this statement? How would you
espond to this student?
Q25. For each option, describe how you used your DGE to determine a location for the stadium.
Include a screenshot that shows the three cities and a proposed location for the stadium. Also, for
each option, compute the cost for road construction/ resurfacing.
Q26. Which option would you choose? Explain.
Q27. The planner has also suggested a third option that could take into consideration the
populations of the three cities. Describe a possible option that uses population information to
find a site for the stadium.
Q28. Create a different, fourth option. Describe and explain how it compares to the other three
options.
Q29. How does this task draw upon common topics that are studied in a high school geometry
course? Explain.
Q30. Would you prefer to introduce the geometric topics you identified in the previous question
first or have students consider solutions to the stadium problem first? Explain.
Q31. What do you notice about the intersection of the medians? Will this always be true?
Q32. Why is the intersection of the medians always in the interior of the triangle? Explain.
Q33. Explore your triangle centroid diagram. Create two or more conjectures. Be sure the
conjectures are stated in “if-then” form.
Q34. Select one of the conjectures and construct a formal proof.
Q35. A student claims, “I have proved that the medians of a triangle always intersect in the
interior of the triangle.” When you walk to his computer, he demonstrates his proof by showing
you that it is true “for all types of triangles.” He drags the triangle to make it obtuse, acute, right,
scalene, isosceles, and equilateral. He states, “Because I have shown it to be true for all cases, it
must always be true.” How do you respond?
Q36. Describe one method for partitioning the land between the two daughters. Will the method
work for any triangle? Explain.
Q37. Create a second method for partitioning the land. Explain the method and why it works.
Q38. Which of the two methods would you recommend? Why?
Q39. Suppose the man has a third child, a son. He now wants a method that he could use to
divide the triangular-shaped land among his three children. Describe a method and explain why
it works.
Q40. If students partitioned the land in more than one way, how would you sequence the two
methods in a class discussion? That is, which solution would you choose for students to present
first to the class?
Q41. How can solving a task such as this partitioning task help students develop a different
understanding of perimeter and area that goes beyond common formulas?
Q42. Create a new task with high cognitive demand that would use the concepts of perimeter
and/or area. How could this task be done in a DGE?