Problem 9.1: How you would teach this stuff
The Setup:
Reread Brown Section 2.1, especially Example 2 and figure 2.8. Then imagine you are teaching Theorem 6 to a high school student and that you with to make sure that your student understands the theorem in all its parts, not only the part of the theorem about the size of the angle in Theorem six, but also the importance of the order of the line.
You want to make sure that your student can answer every time, for special or general cases, either of these construction questions:
If you are given a rotation At, for some particular angle t and are given a figure with point A and a line m through A,
Your writing task to turn in:
a) Write down a clear and concise explanation of why Theorem 6 is true. This can be a complete proof or just a convincing argument that may leave out some details.
b) Make up two examples of the questions above and carry out the constructions in your examples.
c) Then write down 3 different ways that you can give your student so that, when confronted by the problem of deciding where to put the lines, s/he will be able to test the answer for reasonableness and will also have a memorable picture so that there is no chance the student will reverse the order of the lines. (Suggestions: Body movement, physical objects, special geometry figures, tracing the image of special points.)
d) Finally, show how the understanding of a-c be used to answer these related questions. Write down the reasoning as well as doing the construction.
Draw a figure with a point A and a line m through A.
Problem 9.2: A special example of composition if rotations
This problem has a construction in it, but the explanation is equally important. It has several parts. Please organize and label the steps in your work clearly using white space and figures of sufficient size so that it can be read without difficulty.
Let A and B be points in the plane and let A72 and B72 be rotations by 72 degrees centered at each of these points. Then the composition B72A72 is a rotation with center C by angle c. The composition A72B72 is a rotation with center D by angle d. (Theorem in Brown)
a) From the theorem in Brown, Tell what is angle c and what is the angle d. (Numerical answer)
b) Sketch (do not construct yet) a figure with A, B and C and label the sizes of all the angles in triangle ABC. While the sketch will be approximate the sketch should be clear and correct about the orientation of the angles of the triangle.) Be sure to check your answer for reasonableness and indicate what you did to see that the answer is reasonable.
c) Make a second sketch of the figure ABCD, writing in the angles in triangle ABC (again) and triangle ABD. What kind of shape is ABCD?
d) Tell how the vertices of the triangle ABC are 3 of the vertices of a particular regular polygon that we have studied. Then as a consequence of what you know about the polygon, tell what is AC and BC, if the distance AB = d.
e) Now for the construction part: On a new page, draw 2 points A and B, then construct the lengths AC and BC, writing a brief indication of the steps and your reasoning. Then use these lengths to construct the triangle ABC.
f) Further, using your method, continue to construct more vertices using the same ideas to construct the regular polygon with A, B, C as some of the vertices. Explain briefly your reasoning and the steps.
Problem 9.3
This is a graph paper version of one part of the lab given S = A90 and T = C180. One of the lab patterns has these rotations among the symmetries of the pattern.
On a sheet of graph paper, let point A have coordinates (0,0), point B (1,0), point C (1,1), point D (0,1). Suppose S and T are given as described. Find the centers of other rotations that are products ST, TS, SS, TT, STS, TST, STST, etc. (Clearly you will be looking for regularity in how these are arranged and at some point will conjecture the overall pattern.).
Problem 9.4
This is a graph paper version of the lab, given S = A90 and M = Rm, where m is the line BC.
Problem 9.5
Two lines m and n intersect at a point O. Denote by M and N the line reflections in m and n, respectively.
Consider these products:
MN, NMN, MNMN, NMNMN
a) Give a brief but convincing reason why none of these products can be a translation or a glide reflection.
b) Which products are rotations? Which products are line reflections? Tell your reasoning.
c) Draw two such lines m and n (at some general angle, not 90 degrees) and construct the mirror lines of the products that are line reflections. Explain your construction and your reasoning.
And LABEL the lines with the product so that this is clear.
Now look at a special case. Draw two lines m and n intersecting at a point O at an angle of 72 degrees (you can either copy this angle from a figure you already have, or this time you can just us a protractor. Label the lines.
Consider these products:
MN, NMN, MNMN, NMNMN, MNMNMN, NMNMNMN, MNMNMNMN, NMNMNMNMN, MNMNMNMNMN, NMNMNMNMNMN
d) Tell which of these are rotations and tell their rotation angle (you may wish to do some grouping to make this simpler to write and to read). Tell which rotations are inverses of each other. Which one is the identity?
e) Tell which of these are line reflections and draw carefully and label the mirror lines in the figure (construction is OK but not necessary).
Finally, consider these products for the same 72 degree figure:
NM, MNM, NMNM, MNMNM, NMNMNM, MNMNMNM, NMNMNMNM, MNMNMNMNM, NMNMNMNMNM, MNMNMNMNMNM,
f) Without going into detail for ALL of them, explain why these isometries are the same as the previous list.
g) Specifically, which isometry in the second list equals MNMNMNMN? Which one equals NMNMN. Justify your answer.