Chapter 5 Test (Day 56)

Some people may question the wisdom of posting a test on Halloween. My thinking is that it's better for the students to take a test on Halloween and then have a homework-free night, when they will want to celebrate the holiday. Today I heard someone suggest that districts have a staff development day on Halloween -- students are simply in no mood to study at all on Halloween, in a way that can't be said of Columbus or Veterans Day -- the federal holidays for which schools actually close.



Meanwhile, I continued to look at the Glencoe Geometry text. I noticed that as bad as Section 1-6, on Two-Dimensional Figures is, Section 1-7, on Three-Dimensional Figures, is even worse. Students are now expected to cram in several surface area and volume formulas, all in the first chapter!


If I were teaching a class with the Glencoe text, I would cover Sections 1-1 and 1-2 (undefined terms), skip up to Sections 1-4 and 1-5 (on angles -- this is the one thing that makes Glencoe better than the U of Chicago, where angles don't appear until Chapter 3), and then just jump up and begin Chapter 2. Maybe I'd cover the names of the polygons in 1-6 and possibly even a polyhedron in 1-7, but certainly no formulas for area, volume, or distance. After all, the last four chapters (10 through 13) are already there to cover area, volume, and right triangles anyway.


So now I wonder, what is the fascination with the Distance Formula that both Prentice-Hall and Glencoe feel a need to place it in Chapter 1? Well, the first chapter of any geometry text should cover basic concepts, and while the Distance Formula isn't a basic concept, distance itself is. Perhaps the textbook writers feel that in order to cover distance fully, the Distance Formula is required.


But I don't want to torture my students with the Distance Formula right at the beginning of the year -- I agree with David Joyce that it should wait until after the Pythagorean Theorem. The U of Chicago's Section 1-8, on One-Dimensional Figures, contains all the information on length and distance that I'd want to cover at the beginning of the year. By the way, the student I tutored last night had no trouble with the area questions on his homework assignment, just those on perimeter of a triangle where the Distance Formula is required to find the length of one or two of the sides. (I wonder how he'll fare on the surface area and volume questions, though.)


As I mentioned earlier, today I subbed for a P.E. class. On the back of many of the students' P.E. uniforms were printed vocabulary words for English and formulas for math -- probably so that the students are still learning academics even during P.E. class. And as it turned out, one of the formulas was -- you guessed it -- the Distance Formula. Other formulas I saw include the Area of a Circle, Midpoint Formula, Pythagorean Theorem, and Quadratic Formula.

Now here are the answers to my scary Halloween test.

1-2. constructions (or drawings). Notice that a construction for #2 is halfway to constructing a square inscribed in a circle for Common Core.

3. 95 degrees.

4. 152.5 degrees.

5. 87 degrees.

6. 86 degrees.

7. x degrees. This is almost like part of Euclid's proof of the Isosceles Triangle Theorem (except I think that his proof focused on the linear pairs, not the vertical angles).

8. 27 degrees.

9a. x = 60

9b. 61, 62, 58 degrees.

10. 25, 69, 128, 138 degrees.

11. polygon, quadrilateral, parallelogram, rectangle, square.

12. kite.

13. rectangle.

14. false. A counterexample is found easily.

15. Yes, the perpendicular bisector of the bases.

16. 46. Yes, after seeing my student's assignment last night, I decided that even though I don't want to cover area so soon, perimeter is a concept that could be developed more in these early lessons. My question actually defines perimeter since my lessons haven't stressed the concept yet. This is the simplest possible perimeter problem that I could have covered, where only the definition of kite is needed to find the two missing lengths. I could have given an isosceles trapezoid instead, where the Isosceles Trapezoid Theorem is needed to find a missing side length. Or since I squeezed in the Properties of a Parallelogram Theorem in our Section 5-6, I could have even put a parallelogram here with only two consecutive side lengths given.

17. The conjecture is true, and is a key part of the proof of Centroid Concurrency Theorem.

18. Statements                     Reasons
      1. angle G = angle FHI 1. Given
      2. EG | | FH                   2. Corresponding Angles Test
      3. EFHG is a trapezoid 3. Definition of trapezoid (inclusive def. -- it could be a parallelogram)

19. Statements                   Reasons
      1. O and P are circles  1. Given
      2. OQ = OR, PQ = PR 2. Definition of circle
      3. OQPR is a kite         3. Definition of kite (inclusive def. -- it could be a rhombus)

20. Figure is at the top, then below it is quadrilateral. Branching out from it are kite, trapezoid. Then below trapezoid is parallelogram. Kite and parallelogram rejoin to have rhombus below. (Once again, these are inclusive definitions!)
I hope you have a wonderful Halloween.