Today I subbed in a middle school self-contained special ed class. It's in my new district. Since this class is unrepresentative of the class I'd like to teach someday, I won't do "A Day in the Life" today.
As is typical for a class like this, the students go on a morning walk around the school. Of course, I can't resist singing "The Big March" during the walk. Some of the students enjoy it too much -- they start marching instead of walking!
Since I'm in a special ed class, I decide to start my Easter pencil and candy incentive today. Near the end of the day, the students have a writing assignment, so first I hand out pencils so that they have something to write on the worksheet with. Then I give out a tiny bag of M and M's to anyone who is able to complete the assignment. Fortunately, everyone earns the candy today.
Recall that middle schools in this district attend school for five periods under hybrid, since the last class is P.E. independent study. But students in this program attend five full days a week -- and so when the gen ed classes are dismissed, this class stays for the last period of the day. As is logical, this last class is P.E. -- they do P.E. by watching and dancing along to music videos in the classroom. The students definitely have fun during this P.E. class!
Today is Sunday, the third day of the week on the Eleven Calendar:
Resolution #3: We remember math like riding a bicycle.
There is a short math lesson on counting out coins, and I work with a group of three girls. One of them is able to count out 17 cents and 27 cents quickly, but another (her sister, in fact) struggles to count out the coins fully independently. Of course, coin values and totals are something that all students should be able to remember like riding a bike. We run out of time before I get any of the trio to count out 37 cents.
Interestingly enough, there is a coin problem on the Rapoport calendar today:
Allanna has nickels, dimes, and quarters. The average value of the nickels and quarters is 15 cents, as is the average of the dimes and quarters. What is twice the average of all the coins?
This problem involves coin ratios and averages, and so it's much more difficult than anything that we do in class today.
Since the nickels and quarters average 15 cents, there must be equal numbers of these two types -- but since the dimes and quarters also average 15 cents, there must be twice as many dimes as quarters. Thus the nickels, dimes, and quarters appear in a 1:2:1 ratio.
Each nickel, pair of dimes, and quarter make up a four-coin set worth 50 cents. So the average of each set of coins is 12.5 cents, so twice the average is 25 cents -- and of course, today's date is the 25th.
Lesson 13-6 of the U of Chicago text is called "Uniqueness." In the modern Third Edition of the text, uniqueness appears in Lesson 5-6. Recall that the lessons of the old Chapter 13 appear in various chapters of the new edition. Uniqueness of Parallels (Playfair's Parallel Postulate) now appears in Chapter 5 so that it can be used to prove Triangle Sum. In the old version, Triangle Sum is proved in Chapter 5 using a slightly different proof, since uniqueness doesn't appear until Chapter 13.
This is what I wrote two years ago about today's lesson:
The Glide Reflection Theorem only works when the preimage and the image have opposite orientation, not the same orientation. If a figure and its image have the same orientation, then we know that the isometry mapping one to the other is either a translation or a rotation. This case may be a bit tricky -- it could be that the easiest way is simply to translate A to A' and see whether this translation maps B to B' -- if not, then a rotation is necessary. But how do we find the center?
We know that the center of rotation is equidistant from A and A'. Thus it lies on the perpendicular bisector of AA'. For the same reason, the center lies on the perpendicular bisector of BB'. So where these two points intersect is the center of rotation. Notice that if these two perpendicular bisectors are parallel, then the above reasoning constitutes an indirect proof that there is no rotation mapping one to the other -- that is, there is a translation map instead.
Today's worksheet covers all of Lesson 13-6. This means that not only are there questions about the Glide Reflection Theorem, but also about uniqueness in general. A modern form of Euclid's original five postulates are given.
As usual, this worksheet contains the remnant of an old logic activity. The pandemic-friendly activity is planned for tomorrow's post.
No comments:
Post a Comment