4.4. Calendar & Outlines

"The future belongs to those who prepare for it."--Ralph Waldo Emerson

April 24, 2002, Wednesday

Again, STAR testing will make this a 50-minute day instead of a 100-minute day. Students will practice "estimating" Cartesian coordinates written on the whiteboard and check their estimates with neighbors.

The purpose of the estimates is to explicitly develop a visual form of "number sense" in preparation for "exact" (or rather, "more exact") polar-to-Cartesian conversions. This way, when they begin doing "exact" conversions, they will be able to check their answers against the estimates to see if the answers are "reasonable" in addition to checking with peers.

NOTES: I'm not sure whether offering an explicit methodology for estimation would have been "better" than what students came to improvise on their own. While some were confused at first, judicious explanation as well as their peers' various methodologies (many drew auxiliary lines or used straightedges) gradually clarified the issues.

April 26, 2002, Friday

The last day of STAR testing; after a week of it I expect students to be burned-out, and hope to go easy on them toward the end of the period. Hardly anybody thinks well when s/he is stressed after a week of testing.

HOMEWORK #2 (GIF image) will be collected at the beginning of the period. Once again, students will estimate polar-to-Cartesian conversions. Then, some very light trigonometry review, primarily sine and cosine, will lead into "exact" conversions (or rather, "more exact" with round-off error.

NOTES: In practice, I ended up postponing the due date until Tuesday. I'm afraid I consistently underestimate the amount of time it will take to "cover" a topic so students have enough idea of what's going on to be able to make a meaningful attempt on the homework. I'm not averse to stretching their abilities once in a while, but on the other hand I don't want to overwhelm them before they're ready for a little overwhelming, either.

My mistake in writing the definitions of sine and cosine (I switched them accidentally) led to an unintended but near-perfect teachable moment as students protested that all their (more) exact conversions were coming out with the x- and y-values flipped. It only emphasized the importance of checking against estimates for "reasonableness.,"

April 30, 2002, Tuesday

Students will have whiteboard practice on plotting points in polar coordinates, then work on a textbook assignment from Integrated Mathematics 3 (Rubinstein et al., 1998) in randomized pairs: p. 437 #14-20 even, on converting polar coordinates based on a map of scientific research stations in Antarctica.

Students will receive HOMEWORK #3, due on Monday, May 6, which develops both polar-to-Cartesian conversions further through a textbook assignment in Integrated Math 3, p.436 #2-4, p. 438 #26, 32, 36, and 51 (Rubinstein et al., 1998). Some students will take QUIZ #2 (GIF image) while those who opted to do the homework instead can get an early start on HOMEWORK #3.

NOTES: The textbook assignment brought up the interesting question of scale, which was not explicitly shown on the textbook's map of Antarctica. A couple students who had become accustomed to counting off circles for distance were thus confused by coordinates given in miles. On the other hand, one student noted that "This is the first time all year I understood something." If only I could have done something rather earlier...

May 2, 2002, Thursday

After some brief review of polar-to-Cartesian conversions, I will begin to introduce first-quadrant Cartesian-to-polar conversions where students estimate the polar coordinates, then calculate the distance using the Pythagorean theorem and check against the estimates. Then I'll introduce the use of tangent (or, more accurately, arc-tangent) to calculate the angle. I'd like to start with first-quadrant conversions because students have a shaky grasp of domain and range, and are not deeply acquainted with trigonometric functions; I'd rather let it them become comfortable with the relatively "easy" first-quadrant conversions before they have to grapple with reflected, supplementary or complementary angles again.

During the second half of the period, students in groups of 4 will begin on TREASURE HUNT #1, PART 1, which was adapted from the Chapter 8 Unit Project in Integrated Mathematics 3 (Rubinstein et al., 1998), requiring them to draw or adapt a map and give directions/clues that .

NOTES: I keep forgetting how students can be such perfectionists when it comes to maps that I would just dash off. Also, the class talked me into letting them go out, which all but one group did. I'm still not sure it was wise, though my cooperating teacher thought it was okay and remained in the room while I made circuits to make sure all the students were within shouting distance/line-of-sight of the classroom. A couple groups had trouble staying on-task with all the distractions outside.

May 6, 2002, Monday

I will collect HOMEWORK #3 and hand out HOMEWORK #4 (GIF scan), which is more practice on conversions in both directions, due on May 10, Friday. Note that I misnumbered my copy and ended up renumbering the copies I made for the students, and problem #2 was taken from the Integrated Math 3 Practice Book, Practice 60 (McDougal Littell, 1995). The entire period will be devoted to a continuation: TREASURE HUNT #1, PART 2 (or view the GIF scan). All references to colored sheets come from the fact that I photocopy on different colors so students have an immediate way of distinguishing assignment X from assignment Y.

NOTES: I really hadn't expected this "treasure hunt" to take so long. In retrospect, I would be tempted to provide an array of maps that students could choose from. On the other hand, there's something to be said for their realization that it takes effort to make a map, especially a good one. On a more frivolous note, I wonder if the "icons" for the different roles are helpful or distracting. I do realize that the small, dense text is problematic and will have to be remedied in the future, preferably with a word processor and printer.

May 8, 2002, Wednesday

I will introduce Cartesian-to-polar conversions in quadrants other than the 1st quadrant after a brief warm-up on just that. The "trick" (if you can call it that) is to show how the arc-tangent gives a related angle and then use the graph and auxiliary lines to figure out how to derive the "correct" angle. Students will check, as always, against estimates and against their peers' calculations. Then students will wrap up TREASURE HUNT #1, PART 3 (or view the GIF scan) and prepare to give presentations.

NOTES: One of the students opined that this treasure hunt sequence took far too long, and I agree. It may have made more sense to have students work on it gradually throughout the unit. I wish I'd done it that way. Also, having students "check in" with each other to get clues in sequence (which I believe is actually what the original unit intended) would have made more sense in that the final destination wouldn't be immediately apparent. Let those be my lessons for the future.

May 10, 2002, Friday

I will give QUIZ #3, a polar-to-Cartesian conversion with a randomly selected point, and collect HOMEWORK #4 (GIF scan). Students will give their TREASURE HUNT #1 presentations and give FEEDBACK (or view the GIF scan) to one other group.

NOTES: I'm ashamed I didn't think of giving the students a feedback form much, much earlier, considering they've been doing this for the better part of the semester. I wonder if I get someday get them to devise their own, using this as a template.

May 14, 2002, Tuesday

Students will see a brainteaser on "minimal moves" in chess using the bishop, showing a physically-adjacent square that nonetheless has an "infinite" distance from the starting square from the bishop's point-of-view: a new metric. They will also review finding Cartesian distances using the Pythagorean theorem (or its yet-more formulaic incarnation, the distance formula) and have a brief introduction to

After that, students will spend the rest of the period writing LETTER TO ZAP #1. Each student will receive an index card with a point (Cartesian or polar) and find his/her partner who has the same point. The pair will work together on writing a letter that explains polar and Cartesian coordinates, then how to convert their point (from polar to Cartesian or vice versa), and gives a real-life example of the procedure's applicability. At the end of the day I will hand out HOMEWORK #5, which includes polar/Cartesian conversion review and an optional chess-based distance-metric problem.

NOTES: Students were extremely frustrated by the brainteaser at first, but after some heated discussion managed to hash it out among themselves with a minimum of teacher-intervention (mainly to clarify phrasing). The letters to Zap the Martian, in the meantime, went relatively well. I intended it as a an exercise to solidify their learning, provide handy references around the room, and give them more practice in communicating math.

May 16, 2002, Thursday

Class will open with a warm-up on distance and angular distance that stretches their abilities a bit: the first with a non-origin coordinate (which we reviewed last time using the Pythagorean theorem), the second with both a negative and positive angle.

For entertainment, they will also see a brief demonstration of 3-dimensional Cartesian coordinates using masking tape on the floor (x- and y-axes), a meterstick (z-axis), and two different-colored paper airplanes (incentive).

Students will then return to their partnerships from Tuesday and work on GRADING LETTER TO ZAP #1 according to guidelines (some of which I should follow more consistently myself). Afterward they will return grades to each other, and each pair will then give feedback on the grading job they received. Again, I hope this will lead students not only to look carefully at the accuracy and clarity of the letters, but to think about the issues involved in grading. After they have completed all this, they will be invited to write up a comment on "What did I learn about grading?" on a large sheet of butcher paper taped to a cabinet at the back of the room, with markers provided nearby.

If time permits they will see a quite brief intuitive introduction to vectors as "arrows."

NOTES: Once again, students found the warm-up frustrating but a half-hour of people running around asking each other about approaches (and my writing up "experts" on the board as they emerged) led to the knowledge percolating through the class. As far as 3-dimensional coordinates went, students caught on fairly quickly. When one of my paper-airplane-holder volunteers "crashed" the airplane into the floor it was a good excuse for them to see what z=0 meant.

Also, students graded each other and accepted those grades in a generally tough-but-fair manner, including some candid remarks about "we got the grade we deserved." My heart skipped a beat when one student wrote on the "poster" that "it's easier to grade than actually do the work," but the majority of those who chose to comment felt the opposite way. I need to take that poster with me, or photograph it; I plan to use it as my "artifact" at my portfolio hearing. Not all students will become teachers per se, but it demonstrates my operating principle that my ideal is to make myself obsolete.

Students gave me the strangest looks when I affirmed that vectors were numbers, and that they can think of vectors as "arrows" to be added (among other things, but we're not there yet). Once I demonstrated adding vectors by drawing on the whiteboard and walking around the room, someone asked about subtracting them. Life is good. I wonder who'll think of multiplying...?

May 20, 2002, Monday

Class will open with a not-very-brainteaser on negative z-coordinates, then warm-up problem on "better" coordinate systems for describing a given situation. I will collect HOMEWORK #5, then briefly review what we know about vectors intuitively in connection with the question "What is a number?".

Students will begin GIVING DIRECTIONS: VECTORS (or download the RTF file) in randomized groups of 4. This activity is a follow-up to the previous GIVING DIRECTIONS (or download the RTF file), except it involves motion without reference to a particular starting-point, which is more readily described by vectors. I hope to see students making use of diagrams, words, coordinates, and other means as well.

Students will receive HOMEWORK #6 [scan forthcoming], due on Friday, May 24, which introduces vector addition and subtraction intuitively through drawings, then through Cartesian coordinates, and invites students to ponder the similarities between vector and point notation.

Finally, as there will be a code red drill on Wednesday, the last 20 minutes of the class will be devoted to making plans for a barricade. My cooperating teacher suggested letting the class know in advance and taking advantage of the chaos. I like the idea a lot.

NOTES:

May 22, 2002, Wednesday

The code red drill will no doubt occupy a good half hour of the class. Students will give presentations on GIVING DIRECTIONS: VECTORS, including

NOTES:

May 24, 2002, Friday88

ADDING "ARROWS" worksheet. Collect homework.

NOTES: Movie Superstar. Should have given a quiz. Sigh.

May 29, 2002, Wednesday

Vector-related vocabulary: scalar, vector, resultant, magnitude, norm, direction.

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 1 June 2002