Chapter 4 questions
Q1. What is true about the size and shape of the two stick figures?
Congruent. Two figures are congruent if they have the same shape and size. Two angles are
congruent if they have the same measure. Two figures are similar if they have the same shape but
not necessarily the same size.
Q2. Use the A
ow tool to drag a point on the leg of the original stick figure (pre-image). Try
dragging a point on the arm of this stick figure. What else changes when you perform this
dragging? Why?
Q3. Measure the distance between the endpoint of the left leg of the original stick figure and the
co
esponding point on the left leg of the new stick figure. What other distances do you think
will be the same as this measure?
Q4. Measure several other distances. Select a point on the pre-image stick figure and measure the
distance from its co
esponding point on the image stick figure. Do this for several pre-
image/image pairs of points. Explain why all of these distances are the same.
Q5. Repeat the procedure to translate a stick figure to create a “chorus line” of stick figures that
are equidistance apart. Describe how you chose to do this. Discuss several alternative ways this
could be accomplished.
Q6. Drag a point on the pre-image stick figure. Which other points move? Explain. Drag a point
on one of the image sticks figures. Which other points move? Explain.
Q7. A student in a high school geometry class asks what the 5 and 0 represent (these are the
values that were typed into the polar coordinate dialog box to perform the translation). How do
you respond?
Q8. Translate the second stick figure to have three stick figures on the screen. What do you
notice about the size and shape of the three stick figures?
Q9. How is the location of the stick figures related to the translation vector?
Q10. Drag the line segment representing the translation vector so that the tail (B) co
esponds
with a point on the original stick figure. Where is the head of the vector located? Why?
Q11. Use the segment tool to create a segment that joins a point from the original stick figure to
its co
esponding image point. What do you notice about this segment and the vector?
Q12. Use the A
ow tool to drag the head of the vector, point A. What else moves? Why?
Q13. Predict what will happen if the tail of the vector is dragged. Drag point B. What else
moves? Why?
Q14. Mark vector AB by selecting point A and then point B and choosing “Mark Vector” from
the Transform menu. Select the original stick figure and translate it. Where does the image
appear relative to the original stick figure? Why?
Q15. What do you think will happen when you translate a line segment? What will be true of the
image of that line segment? What about a line?
Q16. What do you think will happen when you translate an angle? What will be true of the image
of that angle?
Q17. What do you think will happen when you translate two lines that are parallel to each other?
What will be true of the image of those parallel lines?
Q18. You have had the opportunity to perform several different translations. Based on your
experiences, describe a translation. Be sure to explain what stays the same and what changes.
Q19. Students often have difficulty in reasoning about vectors. Because representations of
vectors and rays are very similar, students confuse these two objects. Describe how you could
assist students in understanding differences between vectors and rays.
Q20. How does this introduction of translations in a dynamic environment, using dancing and
stick figures, compare with how you first learned translations? What are the benefits and
drawbacks of this approach?
Q21. Suppose a student has translated a line using a nonzero vector that is parallel to the line.
The student claims, “nothing happened!” How do you respond?
Q22. Explain how you could use this picture to explain to students that a translation would not be
the appropriate transformation to use to describe the positions of the dancers.
Q23. What do you notice about the size and shape of the two stick figures?
Q24. Drag a point on the leg of the original stick figure. What happens? Why? Drag other points
of the pre-image or the image stick figure.
Q25. Predict what will happen if you drag an endpoint of the mi
or line segment (point A). Drag
point A. What happens? Why?
Q26. Predict what will happen if you drag point B. Drag point B. What happens? Why?
Q27. Create a segment joining the hands of the two stick figures as shown below. What do you
notice about this segment and the line of reflection?
Q28. Have some fun; create other pairs of stick figures.
Q29. Based on your interactions with the sketch, provide a definition of a reflection.
Q30. Describe at least three different properties of reflections.
Q32. Select Let’s consider fixed points from an alge
aic context. Describe a linear function for
which there are infinite fixed points. Describe one or more linear functions for which there are
zero fixed points. Describe one or more linear functions that have exactly one fixed point.
Q33. Under a reflection, are there any fixed points? Explain.
Q34. Sometimes young students are taught to think about reflections as flips. What properties of
eflection are highlighted by thinking of reflections as flips? What properties of reflection are not
made as explicit when considering reflections as flips?
Q35. Consider the pre-image and image stick figures. Create a description of the relationship
etween the two stick figures using a synchronized swimming scenario.
Q36. What do you notice about the size and shape of the two stick figures?
Q37. Drag point C and describe what happens to the two stick figures. Explain.
Q38. Drag a point on one of the stick figures and describe what happens to the other stick figure.
Does it matter which stick figure you drag? Explain.
Q39. If you constructed a circle with center at point C that also passes through a point on the
original stick figure, what other point will the circle pass through? Use your DGE to test your
hypothesis.
Q40. Describe at least three different properties of rotations.
Q41. Are there any fixed points under a rotation?
Q42. To perform a rotation, most DGEs require that you input a particular angle measure. In the
earlier exercise an angle of 60 degrees was used. Describe how this design feature of the
technology may influence student thinking about rotations.
Q43. Are there particular angles of rotation that would be more or less helpful to use with
students? Explain.
Q44. Use the language of functions (e.g., domain, range, independent variable, dependent
variable, function, parameter) to describe the rotation activity you completed in Section 3.
Q45. Describe ways that you can assist students in thinking about geometric transformations as
functions.
Q46. What are the benefits of introducing students to geometric transformations as functions?
new doc 7
Scanned by CamScanne
Scanned by CamScanne
Scanned by CamScanne
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
ing 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
elieve 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
ather than the first? What are the drawbacks?
Q22. Is there another